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GENERAL I ARTICLE
Planets Move in Circles ! A Different View of Orbits T Padmanabhan
T Padmanabhan works at the Inter University Centre for Astronomy and Astrophysics (IUCAA) at Pune. His research interests are in the area of cosmology, in particular the formation of large scale structures in the universe, a subject on which he has written two books. The other area in which he works is the interface between gravity and quantum mechanics. He writes extensively for general readers, on topics ranging over physics, mathematics, and just plain brain teasers.
The orbits of planets, or any other bodies moving under an inverse square law force, can be understood with fresh insight using the idea of velocity space. Surprisingly, a particle moving on an ellipse or even a hyperbola still moves on a circle in this space. Other aspects of orbits such as conservation laws are discussed. Yes, it is true. And no, it is not the cheap trick of tilting the paper to see an ellipse as a circle. The trick, as you will see, is a bit more sophisticated. It turns out that the trajectory of a particle, moving under the attractive inverse square law force, is a circle (or part of a circle) in the velocity space (The high-tech name for the path in velocity space is hodograph). The proof is quite straightforward. Start with the text book result that, for particles moving under any central force f (r) ~, the angular momentum J=r x p is conserved. Here r is the position vector, p is the linear momentum and r is the unit vector in the direction of r. This implies, among other things, that the motion is confined to the plane perpendicular to J. Let us introduce in this plane the polar coordinates (r, 8) and the cartesian coordinates {X, y). The conservation law for J implies
de
-= constantlr 2 d,t
== hr
2,
(1)
which is equivalent to Kepler's second law, since (r 28/2) = h/2 is the area swept by the radius vector in unit time. Newton's laws of motion give
m
dv _x
dt
dv
= fer) cos 8; m 2
dt
=f(r) sin 8.
(2)
-34--------------------------------~------------R-E-S-O-N-A-N-C-E--1-s-ep-t-e-m-b-e-rl-9-9-6
GENERAL I ARTICLE
Di viding (2) by (1) we get
The high-tech name for the path
m dv x = fer) r2 cos e; m dvy = fer) r2 sin e . de h de h
(3)
in velocity space is
hodograph.
The miracle is now in sight for the inverse square law force, for whichf(r) r2 is a constant. For planetary motion we can set it to fer) r2 = - GMm and write the resulting equations as dv x de
-GM
= -
h
cose;
dv
--L
-GM.
= __ Sine.
de
(4)
h
Integrating these equations, with the initial conditions Vx (e=O) =0; Vy (8 = 0) = u, squaring and adding, we get the equation to the hodograph: v 2x +(vy - u
+ GM/h)2 =
(GM/h)2
(5)
which is a circle with center at (0, u - GM/h) and radius GM/h. So you see, planets do move in circles! Some thought shows that the structure depends vitally on the ratio between u and GM/h, motivating one to introduce a quantity e by defining (u - GM/h) =e (GM/h). The geometrical meaning of e is clear from Figure 1. If e = 0, i.e, if we had chosen the initial conditions such that u = GM/h, then the center of the hodograph is at the origin of the velocity space and the magnitude of the velocity remains constant. Writing h=ur, we get u 2 = GM/r2 leading to a circular orbit in the real space as well. When 0 0 . T h e v ecto r x is in tw o d im en sio n a l sp a ce a n d w e lo o k fo r a sta tio n a ry b o u n d sta te
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w av efu n ctio n à (x ) w h ich sa tis¯ es th e eq u a tio n µ ¶ ~2 2 ¡ r ¡ V 0 ±(x ) à (x ) = ¡ jE jà (x ); 2m
(6 )
w h ere ¡ jE j is th e n eg a tiv e b o u n d sta te en erg y. R esca lin g th e va ria b les b y in tro d u cin g ¸ = 2 m V 0 = ~2 a n d E = 2 m jE j= ~2 , th is eq u a tio n red u ces to ¡ 2 ¢ r + ¸ ±(x ) Ã (x ) = E Ã (x ): (7 )
W e co u ld h av e d o n e ev ery th in g u p to th is p o in t in a n y sp a tia l d im en sio n . In D d im en sio n , th e D ira c d elta fu n ctio n ±(x ) h a s th e d im en sio n L ¡ D . T h e k in etic en erg y o p era to r r 2 , o n th e o th er h a n d , a lw ay s h a s th e d im en sio n L ¡ 2 . T h is lea d s to a p ecu lia r b eh av io u r w h en D = 2 . W e ¯ n d th a t, in th is ca se, ¸ is d im en sio n less w h ile E h a s th e d im en sio n o f L ¡ 2 . S in ce th e sca led b in d in g en erg y E h a s to b e d eterm in ed en tirely in term s o f th e p a ra m eter ¸ , w e h av e a p ro b lem in o u r h a n d s. T h ere is n o w ay w e ca n d eterm in e th e fo rm o f E w ith o u t a d im en sio n a l co n sta n t { w h ich w e d o n o t h av e. T o see th e m a n ifesta tio n o f th is p ro b lem m o re clea rly, let u s so lv e (7 ). T h is is fa irly ea sy to d o b y F o u rier tra n sfo rm in g b o th sid es a n d in tro d u cin g th e m o m en tu m sp a ce w av efu n ctio n Á (k ) b y Z Á (k ) = d 2 x à (x ) ex p (¡ ik ¢ x ): (8 )
T h e left-h a n d sid e o f lea d s to th e term [¡ k 2 Á (k )+ ¸ à (0 )], w h ile th e rig h t-h a n d sid e g iv es E Á (k ). E q u a tin g th e tw o w e g et ¸ à (0 ) Á (k ) = 2 : (9 ) k + E W e n ow in teg ra te th is eq u a tio n ov er a ll k . T h e left-h a n d sid e w ill th en g iv e (2 ¼ )2 à (0 ) w h ich ca n b e ca n celled o u t o n b o th sid es b y a ssu m in g à (0) 6= 0 . (T h is is, o f co u rse,
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n eed ed fo r Á (k ) in (9 ) to b e n o n zero a n d h en ce is n o t a n a d d itio n a l a ssu m p tio n .) W e th en g et th e resu lt 1 1 = ¸ 4¼ 2
Z
d2k 1 = 2 k + E 4¼ 2
Z
d2s : s2 + 1
The Dirac delta function, in spite of the nomenclature,
(1 0 )
is strictly not a function but what
T h e seco n d eq u a lity is o bp ta in ed b y ch a n g in g th e in teg ra tio n va ria b le to s = k = E . T h is eq u a tio n is su p p o sed to d eterm in e th e b in d in g en erg y E in term s o f th e p a ra m eter in th e p ro b lem ¸ b u t th e la st ex p ressio n sh ow s th a t th e rig h t h a n d sid e is in d ep en d en t o f E ! T h is is sim ila r to th e situ a tio n in th e electro sta tic p ro b lem in w h ich w e g o t th e in teg ra l w h ich w a s in d ep en d en t o f x . In fa ct, ju st a s in th e electro sta tic ca se, th e in teg ra l o n th e rig h t h a n d sid e d iv erg es, co n ¯ rm in g o u r su sp icio n . O f co u rse, w e a lrea d y k n ow th a t d eterm in in g E in term s o f ¸ is im p o ssib le d u e to d im en sio n a l m ism a tch .
mathematicians call a distribution.
O n e ca n , a t th is sta g e, ta k e th e p o in t o f v iew th a t th e p ro b lem is sim p ly ill-d e¯ n ed a n d o n e w o u ld b e q u ite co rrect. T h e D ira c d elta fu n ctio n , in sp ite o f th e n o m en cla tu re, is strictly n o t a fu n ctio n b u t w h a t m a th em a ticia n s ca ll a d istrib u tio n . It is d e¯ n ed a s a lim it o f a seq u en ce o f fu n ctio n s. F o r ex a m p le, su p p o se w e co n sid er a seq u en ce o f p o ten tia ls · ¸ V0 jx j2 V (x ) = ¡ ; ex p ¡ 2¼ ¾ 2 2¾ 2
(1 1 )
w h ere x is a 2 -D v ecto r a n d ¾ is a p a ra m eter w ith th e d im en sio n o f len g th . In th is ca se, w e w ill a g a in g et (7 ) b u t w ith th e D ira c d elta fu n ctio n rep la ced b y th e G a u ssia n in (1 1 ). B u t n ow w e h av e a p a ra m eter ¾ w ith th e d im en sio n o f len g th a n d o n e ca n im a g in e th e b in d in g en erg y b ein g co n stru cted o u t o f th is. W h en w e ta k e th e lim it ¾ ! 0 , th e p o ten tia l in (1 1 ) red u ces to o n e p ro p o rtio n a l to th e D ira c d elta fu n ctio n . (T h is is w h a t w e m ea n t b y say in g th e d elta fu n ctio n is d e¯ n ed a s a lim itin g ca se o f seq u en ce o f fu n ctio n s. H ere th e fu n ctio n s a re G a u ssia n s
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The essential idea is to accept that the theory requires an extra scale with proper dimensions for its interpretation and treat the coupling constant a function of the scale at which we probe the system.
in (1 1 ) p a ra m etrized b y ¾ . W h en w e ta k e th e lim it o f ¾ ! 0 th e fu n ctio n red u ces to d elta fu n ctio n .) T h e tro u b le is th a t, w h en w e let ¾ g o to zero , w e lo se th e len g th sca le in th e p ro b lem a n d w e d o n o t k n ow h ow to ¯ x th e b in d in g en erg y. O f co u rse, n o o n e a ssu red y o u th a t if y o u so lv e a d i® eren tia l eq u a tio n w ith a n in p u t fu n ctio n V (x ;¾ ) w h ich d ep en d s o n a p a ra m eter ¾ a n d ta k e a (so m ew h a t d u b io u s) lim it o f ¾ ! 0 , th en th e so lu tio n s w ill a lso h av e a sen sib le lim it. S o o n e ca n say th a t th e p ro b lem is ill-d e¯ n ed . R a th er th a n leav in g it a t th a t, w e w a n t to a ttem p t h ere so m eth in g sim ila r to w h a t w e d id in th e electro sta tic ca se. L et u s eva lu a te th e in teg ra l w ith a cu t-o ® a t so m e va lu e k m a x = ¤ w ith ¤ 2 À E . T h en w e g et µ ¶ E 1 1 ; = ¡ ln (1 2 ) ¸ 4¼ ¤2 w h ich ca n b e in v erted to g iv e th e b in d in g en erg y to b e: E = ¤ 2 ex p (¡ 4 ¼ = ¸ );
(1 3 )
w h ere th e sca le is ¯ x ed b y th e cu t-o ® p a ra m eter. O f co u rse th is is w h a t w e w o u ld h av e g o t if w e a ctu a lly u sed a p o ten tia l w ith a len g th sca le. T h ere is a w a y o f in terp retin g th is resu lt ta k in g a cu e fro m w h a t is d o n e in q u a n tu m ¯ eld th eo ry. T h e essen tia l id ea is to a ccep t u p fro n t th a t th e th eo ry req u ires a n ex tra sca le w ith p ro p er d im en sio n s fo r its in terp reta tio n a n d trea t th e co u p lin g co n sta n t a s a fu n ctio n o f th e sca le a t w h ich w e p ro b e th e sy stem . H av in g d o n e th a t w e a rra n g e m a tters so th a t th e o b serv ed resu lts a re a ctu a lly in d ep en d en t o f th e sca le w e h av e in tro d u ced . In th is ca se, w e w ill d e¯ n e a p h y sica l co u p lin g co n sta n t b y µ ¶ E 1 1 ¡1 ¡1 2 2 ; (1 4 ) ¸ p h y (¹ ) = ¸ ¡ ln (¤ = ¹ ) = ¡ ln ¹2 4¼ 4¼
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w h ere ¹ is a n a rb itra ry b u t ¯ n ite sca le. O b v io u sly ¸ p h y (¹ ) is in d ep en d en t o f th e cu t-o ® p a ra m eter ¤ . T h e b in d in g en erg y is n ow g iv en b y E = ¹ 2 ex p (¡ 4 ¼ = ¸ p h y (¹ ))
(1 5 )
w h ich , in sp ite o f a p p ea ra n ce, is in d ep en d en t o f th e sca le ¹ . T h is is sim ila r to o u r eq u a tio n (5 ) in th e electro sta tic p ro b lem , in w h ich w e in tro d u ced a sca le a b u t Á (x ) w a s in d ep en d en t o f a .
The breaking down of naive scaling arguments and the appearance of logarithms are rather ubiquitous in such a case.
In q u a n tu m ¯ eld th eo ry a resu lt lik e th is w ill b e in terp reted a s fo llow s: S u p p o se o n e p erfo rm s a n ex p erim en t to m ea su re so m e o b serva b le q u a n tity (lik e th e b in d in g en erg y ) o f th e sy stem a s w ell a s so m e o f th e p a ra m eters d escrib in g th e sy stem (like th e co u p lin g co n sta n t). If th e ex p erim en t is p erfo rm ed a t a sca le co rresp o n d in g to ¹ (w h ich , fo r ex a m p le, co u ld b e th e en erg y o f th e p a rticles in a sca tterin g cro ss-sectio n m ea su rem en t, say ), th en o n e w ill ¯ n d th a t th e co u p lin g co n sta n t th a t is m ea su red d ep en d s o n ¹ . B u t w h en o n e va ries ¹ in a n ex p ressio n lik e (1 5 ), th e va ria tio n o f ¸ p h y w ill b e su ch th a t o n e g ets th e sa m e va lu e fo r E . W h en y o u th in k a b o u t it, y o u w ill ¯ n d th a t it m a k es a lo t o f sen se. A fter a ll th e p a ram eters w e u se to d escrib e o u r p h y sica l sy stem (lik e ¸ p h y ) a s w ell a s so m e o f th e resu lts w e o b ta in (lik e th e b in d in g en erg y E o r a sca tterin g cro ss-sectio n ) n eed to b e d eterm in ed b y su ita b le ex p erim en ts. In th e q u a n tu m m ech a n ica l p ro b lem s a lso o n e ca n th in k o f sca tterin g of a p a rticle w ith m o m en tu m k (rep resen ted b y a n in cid en t p la n e w av e, say ) b y a p o ten tia l. T h e resu ltin g scatterin g cro ss-sectio n w ill co n ta in in fo rm a tio n a b o u t th e p o ten tia l, esp ecia lly th e co u p lin g co n sta n t ¸ . If th e scatterin g ex p erim en t in tro d u ces a (m o m en tu m o r len g th ) sca le ¹ , th en o n e ca n in d eed im a g in e th e m ea su red co u p lin g co n sta n t to b e d ep en d en t o n th a t sca le ¹ . B u t w e w o u ld ex p ect p h y sica l p red ictio n s o f th e th eo ry (lik e E ) to b e in d ep en d en t
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o f ¹ . T h is is p recisely w h a t h a p p en s in q u a n tu m ¯ eld th eo ry a n d th e toy m o d el a b ov e is a sim p le illu stra tio n . W e see fro m (7 ) th a t, in D = 1 , th e co u p lin g co n sta n t ¸ h a s th e d im en sio n s o f L ¡1 so th ere is n o d i± cu lty in o b ta in in g E / ¸ 2 . T h e o n e-d im en sio n a l in teg ra l co rresp o n d in g to (1 0 ) is co n v erg en t a n d y o u ca n ea sily w o rk th is o u t to ¯ x th e p ro p o rtio n a lity co n sta n t to b e 1 / 4 . T h e lo g a rith m ic d iv erg en ce o ccu rs in D = 2 , w h ich is k n ow n a s th e critica l d im en sio n fo r th is p ro b lem . T h e b rea k in g d ow n o f n a iv e sca lin g a rg u m en ts a n d th e a p p ea ra n ce o f lo g a rith m s a re ra th er u b iq u ito u s in su ch a ca se. (T h ere a re o th er fa scin a tin g issu es in D ¸ 3 a n d in sca tterin g b u t th a t is a n o th er sto ry.)
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
T h e ex a m p les d iscu ssed h ere a re a ll ex p lo red ex ten siv ely in th e litera tu re a n d a g o o d sta rtin g p o in t w ill b e th e referen ces [1 -5 ]. Suggested Reading [1] L R Mead and J Godines, Am. J. Phys., Vol.59, No.10, pp.935–937, 1991. [2] P Gosdzinsky and R Tarrach, Am. J. Phys., Vol.59, No.1, pp.70–74 1991. [3] B R Holstein, Am. J. Phys., Vol.61, No.2, pp.142–147, 1993. [4] A Cabo, J L Lucio and H Mercado, Am. J. Phys., Vol.66, No.3, pp.240– 246, 1998. [5] M Hans, Am. J. Phys., Vol.51, No.8, pp.694–698, 1983.
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Snippets of Physics 7. Thomas Precession T Padmanabhan
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Keywords Relativity, electron spin.
610
T h o m a s p r e c e ssio n is a c u r io u s e ® e c t in s p e c ia l r e la tiv ity w h ic h is p u r e ly k in e m a tic a l in o r ig in a n d illu str a te s so m e im p o r ta n t fe a tu r e s o f th e L o r e n tz tr a n sfo r m a tio n . It a lso h a s a b e a u tifu l g e o m e tr ic in te r p r e ta tio n . W e w ill e x p lo r e th e s e in th is a n d th e n e x t in s ta llm e n t. T h e sim p lest co n tex t in w h ich th e T h o m a s p recessio n a rises is w h en a n o b ject w ith a n in trin sic sp in (lik e a n electro n o r a g y ro sco p e) m ov es in a clo sed o rb it w ith va ria b le v elo city { a n ex a m p le b ein g th e electro n o rb itin g th e n u cleu s in a n a to m , trea ted a lo n g cla ssica l lin es. It tu rn s o u t th a t, d u e to T h o m a s p recessio n , th e e® ectiv e en erg y o f co u p lin g b etw een th e sp in a n d th e o rb ita l a n g u la r m o m en tu m o f th e electron p ick s u p a n ex tra fa cto r o f (1 / 2 ) w h ich , o f co u rse, h a s ex p erim en ta lly v eri¯ a b le co n seq u en ces. N a iv ely, y o u m ig h t h av e th o u g h t th a t a n y sp ecia l rela tiv istic e® ect sh o u ld lea d to a co rrectio n w h ich is o f th e o rd er o f (v = c)2 a n d h en ce w ill b e a v ery w ea k e® ect fo r a n electro n in a n a to m . T h is is in d eed tru e. B u t ex p erim en ta lly o b serva b le e® ects o f th e sp in o rb it in tera ctio n a re also rela tiv istic e® ects. T h ese a rise b eca u se, in th e in sta n ta n eo u s rest fra m e o f th e o rb itin g electro n , th e C o u lo m b ¯ eld (Z e 2 = r 2 ) o f th e n u cleu s g iv es rise to a m a g n etic ¯ eld (v = c)(Z e 2 = r 2 ). T h is m a g n etic ¯ eld co u p les to th e m a g n etic m o m en t (e~= 2 m e c) o f th e electro n . S o a n y o th er e® ect w h ich is o f th e o rd er o f O (v 2 = c 2 ) w ill ch a n g e th e o b serva b le co n seq u en ces b y o rd er u n ity fa cto rs. It tu rn s o u t th a t th is p recessio n a lso h a s a n in terestin g g eo m etrica l in terp reta tio n th a t a llow s o n e to rela te it to o th er { a p p a ren tly u n co n n ected { p h y sica l p h en o m en a
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lik e th e ro ta tio n o f th e p la n e o f th e F o u ca u lt p en d u lu m . In th is in sta llm en t I w ill p rov id e a stra ig h tfo rw a rd (a n d p o ssib ly n o t v ery in sp irin g ) d eriva tio n o f th e T h o m a s p recessio n . N ex t m o n th w e w ill ex p lo re th e F o u ca u lt p en d u lu m a n d th e g eo m etrical rela tio n sh ip . C o n sid er th e sta n d a rd L o ren tz tra n sfo rm a tio n eq u a tio n s b etw een tw o in ertia l fra m es w h ich a re in rela tiv e m o tio n a lo n g th e x -a x is w ith a sp eed v ´ c¯ . T h is is g iv en b y x = ° (x 0+ v t0);t = ° (t0+ v x 0= c 2 ), w h ere ° = (1 ¡ ¯ 2 )¡ 1 = 2 . W e k n ow th a t th e L o ren tz tra n sfo rm a tio n leav es th e q u a n tity s 2 ´ (¡ c 2 t2 + jx j2 ) in va ria n t. A q u a d ra tic ex p ressio n o f th is fo rm is sim ila r to th e len g th o f a v ecto r in th ree d im en sio n s w h ich is in va ria n t u n d er ro ta tio n o f th e co o rd in a te a x es. T h is su g gests th a t th e tra n sfo rm a tio n b etw een th e in ertia l fra m es ca n b e th o u g h t o f a s a ro ta tio n in fo u r-d im en sio n a l sp a ce. T h e ro ta tio n m u st b e in th e t{ x p la n e ch a ra cterized b y a p a ra m eter, say, à . In d eed , th e L o ren tz tra n sfo rm a tio n ca n b e eq u iva len tly w ritten a s x = x 0co sh à + ct0sin h à ;
ct = x 0sin h à + ct0co sh à : (1 )
w ith ta n h à = (V = c), w h ich d eterm in es th e p a ra m eter à (ca lled th e rapidity ) in term s o f th e rela tiv e v elo city b etw een th e tw o fra m es. E q u a tio n (1 ) ca n b e th o u g h t o f a s a ro ta tio n b y a co m p lex a n g le ià . T w o su ccessiv e L o ren tz tra n sfo rm a tio n s w ith v elo cities v 1 a n d v 2 , alon g the sam e direction x , w ill co rresp o n d to tw o su ccessiv e ro ta tio n s in th e t{ x p la n e b y a n g les, say, à 1 a n d à 2 . S in ce tw o ro ta tio n s in th e sa m e p la n e a b o u t th e sa m e o rig in co m m u te, it is o b v io u s th a t th ese tw o L o ren tz tra n sfo rm a tio n s co m m u te a n d a re eq u iva len t to a ro ta tio n b y a n a n g le à 1 + à 2 in th e t{ x p la n e. T h is resu lts in a sin g le L o ren tz tra n sfo rm a tio n w ith a v elo city p a ra m eter g iv en b y th e rela tiv istic su m o f th e tw o v elo cities v 1 a n d v 2 . N o te th a t th e ra p id ities sim p ly a d d w h ile th e v elo city a d d itio n fo rm u la is m o re co m p lica ted .
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The transformation between inertial frames can be thought of as a rotation in fourdimensional space.
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T h e situ a tio n , h ow ev er, ch a n g es in th e ca se o f L o ren tz tra n sfo rm a tio n s a lo n g tw o d i® eren t d irectio n s. T h ese w ill co rresp o n d to ro ta tio n s in tw o d i® eren t p la n es a n d it is w ell k n ow n th a t su ch ro ta tio n s w ill n o t co m m u te. T h e o rd er in w h ich th e L o ren tz tra n sfo rm a tio n s a re ca rried o u t is im p o rta n t if th ey a re alo n g d i® eren t d irectio n s. S u p p o se a fra m e S 1 is m ov in g w ith a v elo city v 1 = v 1 n 1 (w h ere n 1 is a u n it v ecto r) w ith resp ect to a referen ce fra m e S 0 a n d w e d o a L o ren tz b o o st to co n n ect th e co o rd in a tes o f th ese tw o fra m es. N ow su p p o se w e d o a n o th er L o ren tz b o o st w ith a v elo city v 2 = v 2 n 2 to g o fro m S 1 to S 2 . W e w a n t to k n ow w h a t k in d o f tra n sfo rm a tio n w ill n ow ta k e u s d irectly fro m S 0 to S 2 . If n 1 = n 2 , th en th e tw o L o ren tz tra n sfo rm a tio n a re a lo n g th e sa m e a x is a n d o n e ca n g o fro m S 0 to S 2 b y a sin g le L o ren tz tra n sfo rm a tio n . B u t if th e tw o d irectio n s n 1 a n d n 2 a re d i® eren t, th en th is is n o t p o ssib le. It tu rn s o u t th a t in a d d itio n to th e L o ren tz tra n sfo rm a tio n o n e a lso h a s to ro ta te th e sp a tia l co o rd in a tes b y a p a rticu la r a m o u n t.
The root cause of Thomas precession is this: when a body is accelerated, with its velocity vector changing continuously, the instantaneuous Lorentz frames are obtained by boosts along different directions, and there is an effective rotation of coordinate axes which occurs in the process.
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T h is is th e ro o t ca u se o f T h o m a s p recessio n . W h en a b o d y is m ov in g in a n a ccelera ted tra jecto ry w ith th e d irectio n o f v elo city v ecto r ch a n g in g co n tin u o u sly, th e in sta n ta n eo u s L o ren tz fra m es a re o b ta in ed b y b o o sts a lo n g d i® eren t d irectio n s a t ea ch in sta n t. S in ce su ch su ccessiv e b o o sts a re eq u iva len t to a b o o st p lu s a ro ta tio n o f sp a tia l a x es, th ere is a n e® ectiv e ro ta tio n o f th e co o rd in a te a x es w h ich o ccu rs in th e p ro cess. If th e b o d y ca rries a n in trin sic v ecto r (lik e sp in ) w ith it, th e o rien ta tio n o f th a t v ecto r w ill u n d erg o a sh ift. A fter a ll th a t E n g lish , let u s d o so m e m a th s to esta b lish th e id ea rig o ro u sly. T o d o th is w e n eed th e L o ren tz tra n sfo rm a tio n s co n n ectin g tw o d i® eren t fra m es o f referen ces, w h en o n e o f th em is m ov in g a lo n g a n a rb itra ry d irectio n n w ith sp eed V ´ ¯ c. T h e tim e co o rd in a tes a re rela ted b y th e o b v io u s fo rm u la x 00 = ° (x 0 ¡ ¯ ¢ x );
(2 )
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SERIES ⎜ ARTICLE
w h ere w e a re u sin g th e n o ta tio n x i = (x 0 ;x ) = (ct;x ) to d en o te th e fo u r-v ecto r co o rd in a tes. T o o b ta in th e tra n sfo rm a tio n o f th e sp a tia l co o rd in a te, w e ¯ rst w rite th e sp a tia l v ecto r x a s a su m o f tw o v ecto rs; x k = V (V ¢ x )= V 2 w h ich is p a ra llel to th e v elo city v ecto r a n d x ? = x ¡ x k w h ich is p erp en d icu la r to th e v elo city v ecto r. W e k n ow th a t, u n d er th e L o ren tz tra n sfo rm a tio n , w e h av e x 0? = x ? w h ile x 0k = ° (x k ¡ V t): E x p ressin g ev ery th in g a g a in in term s o f x a n d x 0, it is ea sy to sh ow th a t th e ¯ n a l resu lt ca n b e w ritten in th e v ecto ria l fo rm (w ith ¯ = ¯ n ) a s: x 0= x +
(° ¡ 1 ) (¯ ¢ x )¯ ¡ ° ¯ x 0 : ¯2
(3 )
E q u a tio n s (2 ) a n d (3 ) g iv e th e L o ren tz tra n sfo rm a tio n b etw een tw o fra m es m ov in g a lo n g a n a rb itra ry d irectio n . W e w a n t to u se th is resu lt to d eterm in e th e e® ect o f tw o co n secu tiv e L o ren tz tra n sfo rm a tio n s fo r th e ca se in w h ich b o th v 1 = v 1 n 1 a n d v 2 = v 2 n 2 a re sm a ll in th e sen se th a t v 1 ¿ c, v 2 ¿ c. L et th e ¯ rst L o ren tz tra n sfo rm a tio n ta k e th e fo u r v ecto r x b = (ct;x ) to x b1 a n d th e seco n d L o ren tz tra n sfo rm a tio n ta k e th is fu rth er to x a2 1 . P erfo rm in g th e sa m e tw o L o ren tz tra n sfo rm a tio n s in rev erse o rd er lea d s to th e v ecto r w h ich w e w ill d en o te b y x a1 2 . W e a re in terested in th e d i® eren ce ± x a ´ x a2 1 ¡ x a1 2 to th e low est n o n triv ia l o rd er in (v = c). S in ce th is in v o lv es p ro d u ct o f tw o L o ren tz tra n sfo rm a tio n s, w e n eed to co m p u te it k eep in g a ll term s u p to qu adratic o rd er in v 1 a n d v 2 . E x p licit co m p u ta tio n , u sin g , (3 ) a n d (2 ) n ow g iv es (try it o u t!) x 02 1 ¼ x 21 ¼
1 (¯ 2 + ¯ 1 )2 ]x 0 ¡ (¯ 2 + ¯ 1 ) ¢ x ; 2 x ¡ (¯ 2 + ¯ 1 )x 0 + [¯ 2 (¯ 2 ¢ x ) + ¯ 1 (¯ 1 ¢ x )] + ¯ 2 (¯ 1 ¢ x ); (4 ) [1 +
a ccu ra te to O (¯ 2 ). It is o b v io u s th a t th e term s w h ich a re sy m m etric u n d er th e ex ch a n g e o f 1 a n d 2 in th e
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a b ov e ex p ressio n w ill ca n cel o u t w h en w e co m p u te ± x a ´ x a2 1 ¡ x a1 2 . H en ce, w e im m ed ia tely g et ± x 0 = 0 to th is o rd er o f a ccu ra cy. In th e sp a tia l co m p o n en ts th e o n ly term w h ich su rv iv es is th e o n e a risin g fro m la st term in th e ex p ressio n fo r x 2 1 w h ich g iv es ± x = [¯ 2 (¯ 1 ¢ x ) ¡ ¯ 1 (¯ 2 ¢ x )] =
1 (v 1 £ v 2 ) £ x : (5 ) c2
C o m p a rin g th is w ith th e sta n d a rd resu lt fo r in ¯ n itesim a l ro ta tio n o f co o rd in a tes ± x = − £ x , w e ¯ n d th a t th e n et e® ect o f tw o L o ren tz tra n sfo rm a tio n s leav es a resid u a l spatial rotation a b o u t th e d irectio n v 1 £ v 2 . S in ce th is resu lt w a s o b ta in ed b y ta k in g th e d i® eren ce b etw een tw o su ccessiv e L o ren tz tra n sfo rm a tio n s, ±x ´ x 2 1 ¡ x 1 2 , w e ca n th in k o f ea ch o n e co n trib u tin g a n e® ectiv e ro ta tio n b y th e a m o u n t (1 = 2 )(v 1 £ v 2 )= c 2 . C o n sid er n ow a p a rticle w ith a sp in m ov in g a circu la r o rb it. (F o r ex a m p le, it co u ld b e a n electro n in a n a to m ; th e cla ssica l a n a ly sis co n tin u es to a p p ly m a in ly b eca u se th e e® ect is p u rely k in em a tic!). A t tw o in sta n ces in tim e t a n d t+ ± t, th e v elo city o f th e electro n w ill b e in d i® eren t d irectio n s v 1 a n d v 1 + a ± t, w h ere a is th e a ccelera tio n . T h is sh o u ld lea d to a ch a n g e in th e a n g le o f o rien ta tio n o f th e a x es b y th e a m o u n t ±− =
1 (v 1 £ a ) 1 (v 1 £ v 2 ) = ±t 2 2 c 2 c2
(6 )
co rresp o n d in g to th e a n g u la r v elo city ! = ±− = ± t = (1 = 2 )(v 1 £ a )= c 2 . T h is is in d eed th e co rrect ex p ressio n fo r th e T h o m a s p recessio n in th e n o n rela tiv istic lim it (sin ce w e h a d a ssu m ed v 1 ¿ c;v 2 ¿ c). L et m e n ow o u tlin e a rig o ro u s d eriva tio n o f th is e® ect, leav in g th e a lg eb ra ic d eta ils fo r y o u to ¯ g u re o u t! T o set th e sta g e, w e a g a in b eg in w ith th e ro ta tio n s in 3 d im en sio n a l sp a ce. A g iv en ro ta tio n ca n b e d e¯ n ed b y sp ecify in g th e u n it v ecto r n in th e d irectio n o f th e a x is o f ro ta tio n a n d th e a n g le μ th ro u g h w h ich th e a x es a re
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ro ta ted . W e a sso cia te w ith th is ro ta tio n a 2 £ 2 m a trix R (μ ) = co s(μ = 2 ) ¡ i(¾ ¢ n ) sin (μ = 2 ) = ex p ¡
iμ (¾ ¢ n ); 2 (7 )
w h ere ¾ ® a re th e sta n d a rd P a u li m a trices a n d th e co s(μ = 2 ) term is co n sid ered to b e m u ltip lied b y th e u n it m a trix th o u g h it is n o t ex p licitly in d ica ted . T h e eq u iva len ce o f th e tw o fo rm s { th e ex p o n en tia l a n d trig o n o m etric { o f R (μ ) in (7 ) ca n b e d em o n stra ted b y ex p a n d in g th e ex p o n en tia l in a p ow er series a n d u sin g th e ea sily p rov ed rela tio n (¾ ¢ n )2 = 1 . (In cid en ta lly, th e o ccu rren ce o f th e a n g le μ = 2 h a s a sim p le g eo m etrica l o rig in : A ro ta tio n th ro u g h a n a n g le μ a b o u t a g iv en a x is m ay b e v isu a lized a s th e co n seq u en ce o f su ccessiv e re° ectio n s in tw o p la n es w h ich m eet a lo n g th e a x is a t a n a n g le μ = 2 .) W e ca n a lso a sso cia te w ith a 3 -v ecto r x , th e 2 £ 2 m a trix X = x ¢ ¾ . T h e e® ect o f a n y ro ta tio n ca n n ow b e co n cisely d escrib ed b y th e m a trix rela tio n X 0 = R X R ¤. S in ce w e ca n th in k o f L o ren tz tra n sfo rm a tio n s a s ro ta tio n s b y a n im a g in a ry a n g le, a ll th ese resu lts g en era lize, in a n a tu ra l fa sh io n , to L o ren tz tra n sfo rm a tio n s. W e sh a ll a sso cia te w ith a L o ren tz tra n sfo rm a tio n in th e d irectio n n w ith th e sp eed v = c ta n h ® , th e 2 £ 2 m a trix L = co sh (® = 2 ) + (n ¢ ¾ ) sin h (® = 2 ) = ex p
1 (® ¢ ¾ ): 2 (8 )
T h e ch a n g e fro m trig o n o m etric fu n ctio n s to h y p erb o lic fu n ctio n s is in a cco rd a n ce w ith th e fa ct th a t L o ren tz tra n sfo rm a tio n s co rresp o n d to ro ta tio n b y a n im agin ary a n g le. J u st a s in th e ca se o f ro ta tio n s, w e ca n a sso cia te to a n y ev en t x i = (x 0 ;x ), a (2 £ 2 ) m a trix P ´ x i¾ i w h ere ¾ 0 is th e id en tity m a trix a n d ¾ ® a re th e P a u li m a trices. U n d er a L o ren tz tra n sfo rm a tio n a lo n g th e d i0 rectio n n^ w ith sp eed v , th e ev en t x i g o es to x i a n d P g o es P 0. (B y co n v en tio n ¾ i's d o n o t ch a n g e.) T h ey a re
RESONANCE ⎜ July 2008
The change from trigonometric functions to hyperbolic functions is in accordance with the fact that Lorentz transformations correspond to rotation by an imaginary angle.
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w ith P 0= L P L ¤
(9 )
w h ere L is g iv en b y (8 ). C o n sid er a fra m e S 0 w h ich is a n in ertia l, la b o ra to ry fra m e a n d let S (t) b e a L o ren tz fra m e co m ov in g w ith a p a rticle (w ith a sp in ) a t tim e t. T h ese tw o fra m es a re rela ted to ea ch o th er b y a L o ren tz tra n sfo rm a tio n w ith a v elo city v . C o n sid er a p u re L o ren tz b o o st in th e com ovin g fra m e o f th e p a rticle w h ich ch a n g es its v elo city rela tiv e to th e la b fra m e fro m v to v + d v . W e k n ow th a t th e resu ltin g ¯ n a l co n ¯ g u ra tio n ca n n o t b e rea ch ed fro m S 0 b y a p u re b o o st a n d w e req u ire a ro ta tio n b y so m e a n g le ± μ = ! d t fo llow ed b y a sim p le b o o st. T h is lea d s to th e rela tio n , in term s o f th e 2 £ 2 m a trices co rresp o n d in g to th e ro ta tio n a n d L o ren tz tra n sfo rm a tio n s, a s: L (v + d v )R (! d t) = L co m
ov (d v
)L (v ):
(1 0 )
T h e rig h t-h a n d sid e rep resen ts, in m a trix fo rm , tw o L o ren tz tra n sfo rm a tio n s. T h e left-h a n d sid e rep resen ts th e sa m e e® ect in term s o f o n e L o ren tz tra n sfo rm a tio n a n d o n e ro ta tio n { th e p a ra m eters o f w h ich a re a t p resen t u n k n ow n . In th e rig h t-h a n d sid e o f (1 0 ) th e m a trix L co m o v (d v ) h a s a su b scrip t `co m ov in g ' to stress th e fa ct th a t th is o p era tio n co rresp o n d s to a p u re b o o st on ly in th e co m ov in g fra m e a n d n ot in th e la b fra m e. T o ta k e ca re o f th is, w e d o th e fo llow in g : W e ¯ rst b rin g th e p a rticle to rest b y a p p ly in g th e in v erse L o ren tz tra n sfo rm a tio n o p era to r L ¡1 (v ) = L (¡ v ). T h en w e a p p ly a b o o st L (a co m ov d ¿ ), w h ere a co m o v is th e a ccelera tio n o f th e sy stem in th e co m ov in g fra m e. S in ce th e o b ject w a s a t rest in itia lly, this operation ca n b e ch a ra cterized b y a p u re b o o st. F in a lly, w e tra n sfo rm b a ck fro m th e la b to th e m ov in g fra m e b y a p p ly in g L (v ). T h erefo re w e h av e th e rela tio n L
616
co m o v (d v
) = L (v )L (a co m
o v d ¿ )L
(¡ v ):
(1 1 )
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U sin g th is is in (1 0 ), w e g et L (v + d v )R (! d t) = L (v )L (a co m
o v d ¿ ):
In th is eq u a tio n , th e u n k n ow n s a re ! a n d a co m ov . M ov in g th e u n k n ow n term s to th e left-h a n d sid e, w e h av e th e eq u a tio n , R (! d t)L (¡ a co m
ov d ¿ )
= L (¡ [v + d v ])L (v );
(1 2 )
w h ich ca n b e so lv ed fo r ! a n d a co m o v . If w e d en o te th e ra p id ity p a ra m eters fo r th e tw o in ¯ n itesim a lly sep a ra ted L o ren tz b o o sts b y ® a n d ® 0 ´ ® + d ® a n d th e co rresp o n d in g d irectio n s b y n a n d n 0 ´ n + d n , th en th is m a trix eq u a tio n ca n b e ex p a n d ed to ¯ rst o rd er q u a n tities to g iv e ¾ 2 = [co sh (® 0= 2 ) ¡ (n 0¢¾ ) sin h (® 0= 2 )] [co sh (® = 2 ) ¡ (n ¢ ¾ ) sin h (® = 2 )]: 1 ¡ (i! d t + a d ¿ ) ¢
(1 3 )
P erfo rm in g th e n ecessa ry T ay lo r series ex p a n sio n in d ® a n d d n in th e rig h t-h a n d sid e a n d id en tify in g th e co rresp o n d in g term s o n b o th sid es, w e ¯ n d { a fter so m e a lg eb ra ! { th a t a co m ov = n^ (d ® = d ¿ ) + (sin h ® )(d n^ = d ¿ ) a n d m o re im p o rta n tly, μ ¶ d n^ ! = (co sh ® ¡ 1 ) £ n^ (1 4 ) dt w ith ta n h ® = (v = c). (T h is resu lt fo r ! d t h a s a n ice g eo m etrica l in terp reta tio n w h ich w e w ill d iscu ss n ex t m o n th .) E x p ressin g ev ery th in g in term s o f th e v elo city, it is ea sy to sh ow th a t th e ex p ressio n fo r ! is eq u iva len t to ! =
°2 a £ v (v £ a ) = (° ¡ 1 ) : 2 ° + 1 c v2
(1 5 )
In th e n o n rela tiv istic lim it (v < < c), th is g iv es a p recessio n a l a n g u la r v elo city ! »= (1 = 2 c 2 )(a £ v ) w h ich th e RESONANCE ⎜ July 2008
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Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
sp in w ill u n d erg o b eca u se o f th e n o n -co m m u ta tiv ity o f L o ren tz tra n sfo rm a tio n s in d i® eren t d irectio n s. W o rk in g o u t th e d eta ils o f th e d eriva tio n g iv en a b ov e is a w o rth w h ile ex ercise in sp ecia l rela tiv ity. Suggested Reading [1] C W Misner, K S Thorne and J A Wheeler, Gravitation, Chapter 41, (Freeman), 1973.
Information and Announcements Refresher Course on RS and GIS Applications for Water and Environmental Technologies The Centre for Water Resources, Institute of Science and Technology, JNTU, Hyderabad 25th August 2008 to 16th September 2008 The Centre for Water Resources, Institute of Science and Technology, JNTU, Hyderabad, Andhra Pradesh is organizing an UGC sponsored above refresher course for faculty members working in universities and colleges. This course is designed to help the participants upgrade their academic and research activities. The course will be further supplemented by field trips, hands on practical training on Arc GIS 9.1 and ERDAS 8.7 and interactions with industry/academia experts. Eligible candidates would be provided to and fro second class railway fare by the shortest route for attending the programme and each participant shall pay an amount of Rs.500/- towards registration fee. Free boarding and lodging will be provided for outstation participants. This course is planned as a residential programme and stay in the University Guest House at Kukatpally Campus of JNTU, Hyderabad, is compulsory for outstation participants. Selection will be based on a first-come-firstserved basis. Application form can be downloaded from the Centre for Water Resources web site www.cwr.co.in and also from the JNT University website www.jntu.ac.in. Last date for registration is 18th August, 2008. Dr. M V S S GIRIDHAR (Course Coordinator) , Assistant Professor, Centre for Water Resources Institute of Science and Technology, JNT University, Kukatpally, Hyderabad 500 085, AP Telephone: 09440590695, 040-23157220 (R), Email:[email protected]
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Snippets of Physics 8. Foucault Meets Thomas T Padmanabhan
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
1
Thomas Precession, Resonance, Vol.13, No.7, pp.610– 618, July 2008.
Keywords Spin, Thomas precession, Earth's rotation.
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T h e F o u c a u lt p e n d u lu m is a n e le g a n t d e v ic e th a t d e m o n str a te s th e r o ta tio n o f th e E a r th . A fte r d e sc r ib in g it, w e w ill e la b o r a te o n a n in te r e s tin g g e o m e tr ic a l r e la tio n sh ip b e tw e e n th e d y n a m ic s o f th e F o u c a u lt p e n d u lu m a n d T h o m a s p r e c e ssio n d isc u sse d in th e la st in sta llm e n t 1 . T h is w ill h e lp u s to u n d e r sta n d b o th p h e n o m e n a b e tte r . T h e ¯ rst titu la r p resid en t o f th e F ren ch rep u b lic, L o u isN a p o leo n B o n a p a rte, p erm itted F o u ca u lt to u se th e P a n th eo n in P a ris to g iv e a d em o n stra tio n o f h is p en d u lu m (w ith a 6 7 m eter w ire a n d a 2 8 k g p en d u lu m b o b ) o n 3 1 M a rch 1 8 5 1 . In th is im p ressive ex p erim en t, o n e co u ld see th e p la n e o f o scilla tio n o f th e p en d u lu m ro ta tin g in a clo ck w ise d irectio n (w h en v iew ed fro m th e to p ) w ith a freq u en cy ! = − co s μ , w h ere − is th e a n g u la r freq u en cy o f E a rth 's ro ta tio n a n d μ is th e co -la titu d e o f P a ris. (T h a t is, μ is th e sta n d a rd p o la r a n g le in sp h erica l p o la r co o rd in a tes w ith th e z-a x is b ein g th e a x is o f ro ta tio n o f E a rth . S o ¼ = 2 ¡ μ is th e g eo g ra p h ica l la titu d e). F o u ca u lt cla im ed , q u ite co rrectly, th a t th is e® ect a rises d u e to th e ro ta tio n o f th e E a rth a n d th u s sh ow ed th a t o n e ca n d em o n stra te th e ro ta tio n o f th e E a rth b y a n in situ ex p erim en t w ith o u t lo o k in g a t celestia l o b jects. T h is resu lt is q u ite ea sy to u n d ersta n d if th e ex p erim en t w a s p erfo rm ed a t th e p o les o r eq u a to r (in stea d o f P a ris!). T h e situ a tio n a t th e N o rth P o le is a s sh ow n in F igu re 1 . H ere w e see th e E a rth a s ro ta tin g (fro m w est to ea st, in th e co u n ter-clo ck w ise d irectio n w h en v iew ed fro m th e to p ) u n d ern ea th th e p en d u lu m , m a k in g o n e fu ll tu rn in 2 4 h o u rs. It seem s rea so n ab le to d ed u ce fro m th is
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Figure 1.
th a t, a s v iew ed fro m E a rth , th e p la n e o f o scilla tio n o f th e p en d u lu m w ill m a k e o n e fu ll ro ta tio n in 2 4 h o u rs; so th e a n g u la r freq u en cy ! o f th e ro ta tio n o f th e p la n e o f th e F o u ca u lt p en d u lu m is ju st ! = − . (T h ro u g h o u t th e d iscu ssio n w e a re co n cern ed w ith th e ro ta tio n o f th e p la n e o f o scilla tio n o f th e p en d u lu m ; n o t th e p erio d o f th e p en d u lu m 2 ¼ = º , w h ich { o f co u rse { is g iv en b y th e sta n d a rd fo rm u la in v o lv in g th e len g th o f th e su sp en sio n w ire, etc.). A t th e eq u a to r, o n th e o th er h a n d , th e p la n e o f o scilla tio n d o es n ot ro ta te. S o th e fo rm u la , ! = − co s μ , ca p tu res b o th th e resu lts co rrectly. It is triv ia l to w rite d ow n th e eq u a tio n s o f m o tio n fo r th e p en d u lu m b o b in th e ro ta tin g fra m e o f th e E a rth a n d so lv e th em to o b ta in th is resu lt [1 , 2 ] a t th e lin ea r o rd er in − . E ssen tia lly, th e F o u ca u lt p en d u lu m e® ect a rises d u e to th e C o rio lis fo rce in th e ro ta tin g fra m e o f th e E a rth w h ich lea d s to a n a ccelera tio n 2 v £ − , w h ere v , th e v elo city o f th e p en d u lu m b o b , is d irected ta n g en tia l to th e E a rth 's su rfa ce to a g o o d a p p rox im a tio n . If w e ch o o se a lo ca l co o rd in a te sy stem w ith th e Z -a x is p o in tin g n o rm a l to th e su rfa ce o f th e E a rth a n d th e X ;Y co o rd in a tes in th e ta n g en t p la n e a t th e lo ca tio n , th en it is ea sy to sh ow th a t th e eq u a tio n s o f m o tio n fo r th e
RESONANCE ⎜ August 2008
Jean Bernard Lèon Foucault (1819 –1868) was a French physicist, famous for the demonstration of Earth's rotation with his pendulum. Although Earth's rotation was not unknown then, but this easy-to-see experiment caught everyone's imagination.
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p en d u lu m b o b a re w ell a p p rox im a ted b y XÄ + º 2 X = 2 − z Y_ ;
YÄ + º 2 Y = ¡ 2 − z X_ ;
(1 )
w h ere º is th e p erio d o f o scilla tio n o f th e p en d u lu m a n d − z = − co s μ is th e n o rm a l co m p o n en t o f th e E a rth 's a n g u la r v elo city. In a rriv in g a t th ese eq u a tio n s w e h av e ig n o red th e term s q u a d ra tic in − a n d th e v ertica l d isp la cem en t o f th e p en d u lu m . T h e so lu tio n to th is eq u a tio n is o b ta in ed ea sily b y in tro d u cin g th e va ria b le q (t) ´ X (t) + iY (t): T h is sa tis¯ es th e eq u a tio n qÄ + 2 i− z q_ + º 2 q = 0 :
(2 )
T h e so lu tio n , to th e o rd er o f a ccu ra cy w e a re w o rk in g w ith , is g iv en b y q = X (t) + iY (t) = (X 0 (t) + iY 0 (t)) ex p (¡ i− z t); (3 ) w h ere X 0 (t);Y 0 (t) is th e tra jecto ry o f th e p en d u lu m in th e a b sen ce o f E a rth 's ro ta tio n . It is clea r th a t th e n et e® ect o f ro ta tio n is to ca u se a sh ift in th e p la n e o f ro ta tio n a t th e ra te − z = − co s μ . B a sed o n th is k n ow led g e a n d th e resu lts fo r th e p o le a n d th e eq u a to r o n e ca n g iv e a `p la in E n g lish ' d eriva tio n o f th e resu lt fo r in term ed ia te la titu d es b y say in g so m eth in g lik e: \ O b v io u sly, it is th e co m p o n en t o f − n o rm a l to th e E a rth a t th e lo ca tio n o f th e p en d u lu m w h ich m a tters a n d h en ce ! = − co s μ ." T h e ¯ rst-p rin cip le a p p ro a ch , b a sed o n (1 ), o f co u rse h a s th e a d va n ta g e o f b ein g rig o ro u s a n d a lg o rith m ic; fo r ex a m p le, if y o u w a n t to ta k e in to a cco u n t th e e® ects o f ellip ticity o f E a rth , y o u ca n d o th a t if y o u w o rk w ith th e eq u a tio n s o f m o tio n . B u t it d o es n o t g iv e y o u a n in tu itiv e u n d ersta n d in g o f w h a t is g o in g o n , a n d m u ch less a u n i¯ ed v iew o f a ll rela ted p ro b lem s h av in g th e sa m e stru ctu re. W e sh a ll n ow d escrib e a n a p p ro a ch to th is p ro b lem w h ich h a s th e a d va n ta g e o f p rov id in g a clea r g eo m etrica l p ictu re a n d co n n ectin g it u p { so m ew h a t q u ite su rp risin g ly { w ith T h o m a s p recessio n d iscu ssed in th e la st in sta llm en t.
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O n e p o in t w h ich ca u ses so m e co n fu sio n a s reg a rd s th e F o u ca u lt p en d u lu m is th e fo llow in g . W h ile a n a ly zin g th e b eh av io r o f th e p en d u lu m a t th e p o le, o n e a ssu m es th a t th e p la n e o f ro ta tio n rem a in s ¯ x ed w h ile th e E a rth ro ta tes u n d ern ea th it. If w e m a k e th e sa m e cla im fo r a p en d u lu m ex p erim en t d o n e a t a n in term ed ia te la titu d e, { i.e., if w e say th a t th e p la n e o f o scilla tio n rem a in s in va ria n t w ith resp ect to , say, th e \ ¯ x ed sta rs" a n d th e E a rth ro ta tes u n d ern ea th it { it seem s n a tu ra l th a t th e p erio d o f ro ta tio n o f th e p en d u lu m p la n e sh o u ld a lw ay s b e 2 4 h o u rs irresp ectiv e o f th e lo ca tio n ! T h is, o f co u rse, is n o t tru e a n d it is a lso in tu itiv ely o b v io u s th a t n o th in g h a p p en s to th e p la n e o f ro ta tio n a t th e eq u a to r. In th is w ay o f a p p ro a ch in g th e p ro b lem , it is n o t v ery clea r h ow ex a ctly th e E a rth 's ro ta tio n in ° u en ces th e m o tio n o f th e p en d u lu m . T o p rov id e a g eo m etrica l a p p ro a ch to th is p ro b lem , w e w ill rep h ra se it a s fo llow s [3 , 4 ]. T h e p la n e o f o scilla tio n o f th e p en d u lu m ca n b e ch a ra cterized b y a v ecto r n o rm a l to it o r eq u iva len tly b y a v ecto r w h ich is ly in g in th e p la n e an d ta n g en tia l to th e E a rth 's su rfa ce. L et u s n ow in tro d u ce a co n e w h ich is co a x ia l w ith th e a x is o f ro ta tio n o f th e E a rth a n d w ith its su rfa ce ta n g en tia l to th e E a rth a t th e la titu d e o f th e p en d u lu m (see F igu re 2 ). T h e b a se ra d iu s o f su ch a co n e w ill b e R sin μ , w h ere R is th e ra d iu s o f th e E a rth a n d th e sla n t h eig h t o f th e co n e w ill b e R ta n μ . S u ch a co n e ca n b e b u ilt o u t o f a secto r o f a circle (a s sh ow n in F igu re 3 ) h av in g th e circu m feren ce 2 ¼ R sin μ a n d ra d iu s R ta n μ b y id en tify in g th e lin es O A a n d O B . T h e `d e¯ cit a n g les' o f th e co n e, ® a n d ¯ ´ 2 ¼ ¡ ® , sa tisfy th e rela tio n s: (2 ¼ ¡ ® )R ta n μ = 2 ¼ R sin μ
A
Figure 2.
Figure 3.
C
(4 ) O
w h ich g iv es ® = 2 ¼ (1 ¡ co s μ );
¯ = 2 ¼ co s μ :
(5 ) A
B
T h e b eh av io r o f th e p la n e o f th e F o u ca u lt p en d u lu m
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The plane of oscillation of the pendulum will rotate with respect to a coordinate system fixed on the Earth, but it will always coincide with the lines drawn on the cone which remain fixed relative to the fixed stars. (Figures 2,3)
ca n b e u n d ersto o d v ery ea sily in term s o f th is co n e. In itia lly, th e F o u ca u lt p en d u lu m is sta rted o u t o scilla tin g in so m e a rb itra ry d irectio n a t th e p o in t A , say. T h e d irectio n o f o scilla tio n ca n b e in d ica ted b y so m e stra ig h t lin e d raw n a lo n g th e su rfa ce o f th e co n e (lik e A C in F igu re 3 ). W h ile th e p la n e o f o scilla tio n o f th e p en d u lu m w ill ro ta te w ith resp ect to a co o rd in a te sy stem ¯ x ed o n th e E a rth , it w ill a lw ay s co in cid e w ith th e lin es d raw n o n th e co n e w h ich rem a in ¯ x ed rela tiv e to th e ¯ x ed sta rs. W h en th e E a rth m a k es o n e ro ta tio n , w e m ov e fro m A to B in th e ° a tten ed o u t co n e in F igu re 3 . P h y sica lly, o f co u rse, w e id en tify th e tw o p o in ts A a n d B w ith th e sa m e lo ca tio n o n th e su rfa ce o f th e E a rth . B u t w h en a v ecto r is m ov ed a ro u n d a cu rv e a lo n g th e lin es d escrib ed a b ov e, o n th e cu rv ed su rfa ce o f E a rth , its o rien ta tio n d o es n o t retu rn to th e o rig in a l va lu e. It is o b v io u s fro m F igu re 3 th a t th e o rien ta tio n o f th e p lan e o f ro ta tio n (in d ica ted b y a v ecto r in th e p la n e o f ro ta tio n a n d ta n g en tia l to th e E a rth 's su rfa ce a t B ) w ill b e d i® eren t fro m th e co rresp o n d in g v ecto r a t A . (T h is p ro cess is ca lled p a ra llel tra n sp o rt a n d th e fa ct th a t a vecto r ch a n g es o n p a ra llel tra n sp o rt a ro u n d a n a rb itra ry clo sed cu rv e o n a cu rv ed su rfa ce is a w ell-k n ow n resu lt in d i® eren tia l g eo m etry a n d g en era l rela tiv ity.) C lea rly, th e o rien ta tio n o f th e vecto r ch a n g es b y a n a n g le ¯ = 2 ¼ co s μ d u rin g o n e ro ta tion o f E a rth w ith p erio d T . S in ce th e ra te o f ch a n g e is u n ifo rm th ro u g h o u t b eca u se o f th e stea d y sta te n a tu re o f th e p ro b lem , th e a n g u la r v elo city o f th e ro ta tio n o f th e p en d u lu m p la n e is g iv en by ! =
¯ 2¼ = co s μ = − co s μ : T T
(6 )
T h is is p recisely th e resu lt w e w ere a fter. T h e k ey g eo m etrica l id ea w a s to rela te th e ro ta tio n o f th e p la n e o f th e F o u ca u lt p en d u lu m to th e p a ra llel tra n sp o rt o f a v ecto r ch a ra cterizin g th e p la n e, a ro u n d a clo sed cu rv e o n th e su rfa ce o f E a rth . W h en th is clo sed cu rv e is n o t
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a g eo d esic { a n d w e k n ow th a t a cu rv e o f co n sta n t la titu d e is n o t a g eo d esic { th e o rien ta tio n o f th is v ecto r ch a n g es w h en it co m p letes o n e lo o p . T h ere a re m o re so p h istica ted w ay s o f ca lcu la tin g h ow m u ch th e o rien ta tio n ch a n g es fo r a g iv en cu rv e o n a cu rv ed su rfa ce. B u t in th e ca se o f a sp h ere, th e trick o f a n en v elo p in g co n e p rov id es a sim p le p ro ced u re. (W h en th e p en d u lu m is lo ca ted in th e eq u a to r, th e clo sed cu rv e is th e eq u a to r itself; th is, b ein g a g rea t circle is a g eo d esic o n th e sp h ere a n d th e v ecto r d o es n o t g et `d iso rien ted ' o n g o in g a ro u n d it. S o th e p la n e o f th e p en d u lu m d o es n o t ro ta te in th is ca se.)
This derivation also allows one to understand the Thomas precession of the spin of a particle.
T h is is g o o d , b u t a s I sa id , th in g s g et b etter. O n e ca n sh ow th a t a n a lm o st id en tica l a p p ro a ch a llow s o n e to d eterm in e th e T h o m a s p recessio n o f th e sp in o f a p a rticle (say, a n electro n ) m ov in g in a circu la r o rb it a ro u n d a n u cleu s [5 ]. W e saw in th e la st in sta llm en t [6 ] th a t th e ra te o f T h o m a s p recessio n is g iv en , in g en era l, b y a n ex p ressio n o f th e fo rm ! d t = (co sh  ¡ 1 ) (d n^ £ n^ ) ;
(7 )
w h ere ta n h  = v = c a n d v is th e v elo city o f th e p a rticle. In th e ca se o f a p a rticle m ov in g o n a circu la r tra jecto ry, th e m a g n itu d e o f th e v elo city rem a in s co n sta n t a n d w e ca n in teg ra te th is ex p ressio n to o b ta in th e n et a n g le o f p recessio n d u rin g o n e o rb it. F o r a circu la r o rb it, d n^ is a lw ay s p erp en d icu la r to n^ so th a t n^ £ d n^ is essen tia lly d μ w h ich in teg ra tes to g iv e a fa cto r 2 ¼ . H en ce th e n et a n g le o f T h o m a s p recessio n d u rin g o n e o rb it is g iv en b y © = 2 ¼ (co sh  ¡ 1 ):
(8 )
T h e sim ila rity b etw een th e n et a n g le o f tu rn o f th e F o u ca u lt p en d u lu m a n d th e n et T h o m a s p recessio n a n g le is n ow o b v io u s w h en w e co m p a re (8 ) w ith (5 ). W e k n ow th a t in th e ca se o f L o ren tz tra n sfo rm a tio n s, o n e rep la ces
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It will turn out that the sphere and the cone we introduced in the real space, to study the Foucault pendulum, have to be introduced in the velocity space to analyze Thomas precession.
rea l a n g les b y im a g in a ry a n g les w h ich a cco u n ts fo r th e d i® eren ce b etw een th e co s a n d co sh fa cto rs. W h a t w e n eed to d o is to m a k e th is a n a lo g y m a th em a tica lly p recise w h ich w ill b e o u r n ex t ta sk . It w ill tu rn o u t th a t th e sp h ere a n d th e co n e w e in tro d u ced in th e rea l sp a ce, to stu d y th e F o u ca u lt p en d u lu m , h av e to b e in tro d u ced in th e v elo city sp a ce to a n a ly ze T h o m a s p recessio n . A s a w a rm -u p to ex p lo rin g th e rela tiv istic v elo city sp a ce, let u s sta rt b y a sk in g th e fo llow in g q u estio n : C o n sid er tw o fra m es S 1 a n d S 2 w h ich m ov e w ith v elo cities v 1 a n d v 2 w ith resp ect to a th ird in ertia l fra m e S 0 . W h a t is th e m a g n itu d e o f th e rela tiv e v elo city b etw een th e tw o fra m es? T h is is m o st ea sily d o n e u sin g L o ren tz in va ria n ce a n d fo u r-v ecto rs (a n d to sim p lify n o ta tio n w e w ill u se u n its w ith c = 1 ). W e ca n a sso cia te w ith th e 3 v elo cities v 1 a n d v 2 , th e co rresp o n d in g fo u r-v elo cities, g iv en b y u i1 = (° 1 ;° 1 v 1 ) a n d u i2 = (° 2 ;° 2 v 2 ) w ith a ll th e co m p o n en ts b ein g m ea su red in S 0 . O n th e o th er h a n d , w ith resp ect to S 1 , th is fo u r-v ecto r w ill h av e th e co m p o n en ts u i1 = (1 ;0 ) a n d u i2 = (° ;° v ), w h ere v (b y d efin itio n ) is th e rela tiv e v elo city b etw een th e fra m es. T o d eterm in e th e m a g n itu d e o f th is q u a n tity, w e n o te th a t in th is fra m e S 1 w e ca n w rite ° = ¡ u 1 iu i2 . B u t sin ce th is ex p ressio n is L o ren tz in va ria n t, w e ca n eva lu a te it in a n y in ertia l fra m e. In S 0 , w ith u i1 = (° 1 ;° 1 v 1 );u i2 = (° 2 ;° 2 v 2 ) th is h a s th e va lu e ° = (1 ¡ v 2 )¡ 1 = 2 = ° 1 ° 2 ¡ ° 1 ° 2 v 1 ¢ v 2 :
(9 )
S im p lify in g th is ex p ressio n w e g et v2 =
(1 ¡ v 1 ¢ v 2 )2 ¡ (1 ¡ v 12 )(1 ¡ v 22 ) (1 ¡ v 1 ¢ v 2 )2 =
(v 1 ¡ v 2 )2 ¡ (v 1 £ v 2 )2 : (1 ¡ v 1 ¢ v 2 )2
(1 0 )
L et u s n ex t co n sid er a 3 -d im en sio n a l a b stra ct sp a ce in w h ich ea ch p o in t rep resen ts a v elo city o f a L o ren tz fra m e
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m ea su red w ith resp ect to so m e ¯ d u cia l fra m e. W e a re in terested in d e¯ n in g th e n o tio n o f `d ista n ce' b etw een tw o p o in ts in th is v elo city sp a ce. C o n sid er tw o n ea rb y p o in ts w h ich co rresp o n d to v elo cities v a n d v + d v th a t d i® er b y a n in ¯ n itesim a l q u a n tity. B y a n a lo g y w ith th e u su a l 3 -d im en sio n a l ° a t sp a ce, o n e w o u ld h av e a ssu m ed th a t th e `d ista n ce' b etw een th ese tw o p o in ts is ju st jd v j2 = d v x2 + d v y2 + d v z2 = d v 2 + v 2 (d μ 2 + sin 2 μ d Á 2 ); (1 1 ) w h ere v = jv j a n d (μ ;Á ) d en o te th e d irectio n o f v . In n o n -rela tiv istic p h y sics, th is d ista n ce a lso co rresp o n d s to th e m a g n itu d e o f th e rela tiv e v elo city b etw een th e tw o fra m es. H ow ev er, w e h av e ju st seen th a t th e rela tiv e v elo city b etw een tw o fra m es in rela tiv istic m ech a n ics is d i® eren t a n d g iv en b y (1 0 ). It is m o re n a tu ra l to d e¯ n e th e d ista n ce b etw een th e tw o p o in ts in th e v elo city sp a ce to b e th e rela tiv e v elo city b etw een th e resp ectiv e fra m es. In th a t ca se, th e in ¯ n itesim a l `d ista n ce' b etw een th e tw o p o in ts in th e v elo city sp a ce w ill b e g iv en b y (1 0 ) w ith v 1 = v an d v 2 = v + d v . S o d lv2 =
(d v )2 ¡ (v £ d v )2 : (1 ¡ v 2 )2
(1 2 )
U sin g th e rela tio n s (v £ d v )2 = v 2 (d v )2 ¡ (v ¢ d v )2 ;
(v ¢ d v )2 = v 2 (d v )2 (1 3 )
a n d u sin g (1 1 ) w h ere μ , Á a re th e p o la r a n d a zim u th a l a n g les o f th e d irectio n o f v , w e g et d lv2 =
dv2 v2 + (d μ 2 + sin 2 μ d Á 2 ): 2 2 2 (1 ¡ v ) 1¡ v
(1 4 )
If w e u se th e ra p id ity  in p la ce o f v th ro u g h th e eq u a tio n v = ta n h  , th e lin e elem en t in (1 4 ) b eco m es: d lv2 = d  2 + sin h 2  (d μ 2 + sin 2 μ d Á 2 ):
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(1 5 )
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T h is is a n ex a m p le o f a cu rved sp a ce w ith in th e co n tex t o f sp ecia l rela tiv ity. T h is p a rticu la r sp a ce is ca lled (th ree-d im en sio n a l) L o b a ch ev sk y sp a ce. If w e n ow ch a n g e fro m rea l a n g les to th e im a g in a ry o n es, b y w ritin g  = i´ , th e lin e elem en t b eco m es ¡ d lv2 = d ´ 2 + sin 2 ´ (d μ 2 + sin 2 μ d Á 2 );
(1 6 )
w h ich (ex cep t fo r a n ov era ll sig n w h ich is irreleva n t) rep resen ts th e d ista n ces o n a 3 -sp h ere h av in g th e th ree a n g les ´ ;μ a n d Á a s its co o rd in a tes. O f th ese th ree a n g les, μ a n d Á d en o te th e d irectio n o f v elo city in th e rea l sp a ce a s w ell. W h en a p a rticle m ov es in th e x ¡ y p la n e in th e rea l sp a ce, its v elo city v ecto r lies in th e μ = ¼ = 2 p la n e a n d th e releva n t p a rt o f th e m etric red u ces to d L 2v = d ´ 2 + sin 2 ´ d Á 2
(1 7 )
w h ich is ju st a m etric o n th e 2 -sp h ere. F u rth er, if th e p a rticle is m ov in g o n a circu la r o rb it w ith co n sta n t m a g n itu d e fo r th e v elo city, th en it fo llow s a cu rv e o f ´ = co n sta n t o n th is 2 -sp h ere. T h e a n a lo g y w ith th e F o u ca u lt p en d u lu m , w h ich m ov es o n a co n sta n t la titu d e cu rv e, is n ow co m p lete. If th e p a rticle ca rries a sp in , th e o rb it w ill tra n sp o rt th e sp in v ecto r a lo n g th is circu la r o rb it. A s w e h av e seen ea rlier, th e o rien ta tio n o f th e v ecto r w ill n o t co in cid e w ith th e o rig in a l o n e w h en th e o rb it is co m p leted a n d w e ex p ect a d i® eren ce o f 2 ¼ (1 ¡ co s ´ ) = 2 ¼ (1 ¡ co sh  ). S o th e m a g n itu d e o f th e T h o m a s p recessio n , ov er o n e p erio d is g iv en p recisely b y (8 ). I w ill let y o u w o rk o u t th e d eta ils ex a ctly in a n a lo g y w ith th e F o u ca u lt p en d u lu m a n d co n v in ce y o u rself th a t th ey h av e th e sa m e g eo m etrica l in terp reta tio n . W h en o n e m ov es a lo n g a cu rv e in th e v elo city sp a ce, o n e is sa m p lin g d i® eren t (in sta n ta n eo u sly ) co -m ov in g L o ren tz fra m es o b ta in ed b y L o ren tz b o o sts a lo n g d i® eren t d irectio n s. A s w e d escrib ed in th e la st in sta llm en t,
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L o ren tz b o o sts a lo n g d i® eren t d irectio n s d o n o t, in g en era l, co m m u te. T h is lea d s to th e resu lt th a t if w e m ov e a lo n g a clo sed cu rv e in th e v elo city sp a ce (trea ted a s rep resen tin g d i® eren t L o ren tz b o o sts) th e o rien ta tio n o f th e sp a tia l a x es w o u ld h av e ch a n g ed w h en w e co m p lete th e lo o p . It tu rn s o u t th a t th e id ea s d escrib ed a b ov e a re a ctu a lly o f fa r m o re g en era l va lid ity. W h en ev er a v ecto r is tra n sp o rted a ro u n d a clo sed cu rv e o n th e su rfa ce o f a sp h ere, th e n et ch a n g e in its o rien ta tio n ca n b e rela ted to th e so lid a n g le su b ten d ed b y th e a rea en clo sed b y th e cu rv e. In th e ca se o f th e F o u ca u lt p en d u lu m , th e releva n t v ecto r d escrib es th e o rien ta tio n o f th e p la n e o f th e p en d u lu m a n d th e tra n sp o rt is a ro u n d a circle o n th e su rfa ce o f th e E a rth . In th e ca se o f T h o m a s p recessio n , th e releva n t v ecto r is th e sp in o f th e p a rticle a n d th e tra n sp o rt o ccu rs in th e v elo city sp a ce. U ltim a tely, b o th th e e® ects { th e F o u ca u lt p en d u lu m a n d T h o m a s p recessio n { a rise b eca u se th e sp a ce in w h ich o n e is p a ra llel tra n sp o rtin g th e v ecto r (su rfa ce o f E a rth , rela tiv istic v elo city sp a ce) is cu rv ed . Suggested Reading [1] [2] [3]
L D Landau and E M Lifshitz, Mechanics, Pergamon Press, p.129, 1976. T Padmanabhan, Theoretical Astrophysics: Astrophysical Processes, Cambridge University Press, Vol.1, p.50, 2000. J B Hart et al., Am. J. Phys, Vol.55, No.1, p.67, 1987.
[4]
Jens von Bergmann, Gauss–Bonnet and Foucault, URL: http://www.nd.edu/~mathclub/foucault.pdf, 2007.
[5] [6]
M I Krivoruchenko, arXiv:0805.1136v1 [nucl-th], 2008. T Padmanabhan, Resonance, Vol.13, No.7, p.610, July 2008.
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Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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Snippets of Physics 9. Ambiguities in Fluid Flow T Padmanabhan
T h e id e a liz e d ° o w o f ° u id a ro u n d a sp h e r ic a l b o d y is a c la ssic te x tb o o k p r o b le m in ° u id m e c h a n ic s. In te r e stin g ly e n o u g h , it le a d s to so m e c u r io u s tw ists a n d tu rn s a n d c o n c e p tu a l c o n u n d r u m s. T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
T h e ° ow in d u ced in a ° u id w h en a b o d y m ov es th ro u g h it is o f trem en d o u s p ra ctica l im p o rta n ce { w ith th e a irp la n e w in g s p rov id in g ju st o n e ex a m p le. In g en era l, n o b o d y u n d ersta n d s th e ° ow o f real ° u id s a n d w e h av e to reso rt to sca led m o d els (e.g ., in w in d tu n n els) o r to n u m erica l sim u la tio n s to m a k e p ro g ress. B u t th ere a re so m e idealized m o d els th a t o n e ca n so lv e a n a ly tica lly w h ich co rresp o n d s, b ro a d ly sp ea k in g , to th e m y th ica l ° u id so m etim es ca lled \ d ry w a ter" . T h ese p ro b lem s a re su p p o sed ly w ell-u n d ersto o d a n d w e w ill see, in th is in sta llm en t, th a t ev en th e sim p lest o f th em ca n lea d to su rp rises. W h en a b o d y m ov es th ro u g h th e h y p o th etica l ° u id , th e resu ltin g ° ow sa tis¯ es th e fo llow in g co n d itio n s: F irst, th e ° u id ° ow is in co m p ressib le w ith th e d en sity b ein g a co n sta n t. T h en , th e co n serva tio n o f m a ss, ex p ressed in th e fo rm o f a co n tin u ity eq u a tio n
Keywords Fluid mechanics, electrostatics.
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@½ + r ¢ (½ v ) = 0 (1 ) @t (in w h ich ½ is th e d en sity a n d v is th e ° u id v elo city ) red u ces to th e sim p le co n d itio n r ¢ v = 0 . S eco n d , w e w ill a ssu m e th a t th e ° ow is irro ta tio n a l (r £ v = 0 ) a llow in g fo r th e v elo city to b e ex p ressed a s a g ra d ien t o f a sca la r p o ten tia l v = r Á . F in a lly w e w ill ig n o re a ll th e p ro p erties o f rea l ° u id s lik e v isco sity, su rfa ce ten sio n ,
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etc., a n d w ill trea t th e p ro b lem a s o n e o f ¯ n d in g th e so lu tio n s to th e tw o eq u a tio n s r ¢ v = 0 a n d r £ v = 0 su b ject to certa in b o u n d a ry co n d itio n s. E q u iva len tly, w e ¯ n d th a t th e p o ten tia l sa tis¯ es L a p la ce's eq u a tio n r 2 Á = 0 . S o th e p ro b lem red u ces to so lv in g th e L a p la ce eq u a tio n w ith v sa tisfy in g th e b o u n d a ry co n d itio n s { w h ich is th e o n ly n o n triv ia l fea tu re o f th e p ro b lem ! L et u s co n sid er a b o d y o f a n a rb itra ry sh a p e m ov in g w ith a v elo city u th o u g h th e ° u id . T h en w e n eed to so lv e th e L a p la ce eq u a tio n su b ject to th e b o u n d a ry co n d itio n n ¢v = n ¢u a t th e su rfa ce, w h ere n is th e n o rm a l to th e su rfa ce. W e w o u ld ex p ect th e ° u id ° ow n ea r th e b o d y to b e a ® ected b y its m o tio n , b u t a t su ± cien tly la rg e d ista n ce th is e® ect sh o u ld b e n eg lig ib le. H en ce th e ° u id v elo city v w ill b e zero a t sp a tia l in ¯ n ity. In terestin g ly en o u g h th e g en era l fo rm o f th e ° u id v elo city a t la rg e d ista n ces fro m th e b o d y (o f a rb itra ry sh a p e) ca n b e d eterm in ed b y th e fo llo w in g a rg u m en t. W e k n ow th a t th e fu n ctio n 1 = r sa tis¯ es th e L a p la ce eq u a tio n . F u rth er, if Á sa tis¯ es th e L a p la ce eq u a tio n , th e sp a tia l d eriv a tiv es o f Á a lso sa tisfy th e sa m e eq u a tio n . T h erefo re, th e d irectio n a l d eriva tiv e o f 1 = r , a lo n g so m e d irectio n sp eci¯ ed b y a n a rb itra ry v ecto r A w ill a lso sa tisfy th e L a p la ce eq u a tio n . S u ch a d irectio n a l d eriva tiv e is g iv en b y A ¢ r (1 = r ) a n d w ill fa ll a s 1 = r 2 a t la rg e d ista n ces. H en ce, a t la rg e d ista n ces fro m th e b o d y, w e ca n ta k e th e lea d in g o rd er term s in th e p o ten tia l to b e µ ¶ q 1 Á = ¡ + A ¢r + O (1 = r 3 ): (2 ) r r Y o u w ill, o f co u rse reco g n ize a ll th ese to b e electro sta tics in d isg u ise a n d th e ex p a n sio n in (2 ) to b e ju st th e la rg e d ista n ce ex p a n sio n o f th e p o ten tia l d u e to th e d istrib u tio n o f ch a rg es. T h e ¯ rst term is th e m o n o p o le co u lo m b term a n d th e seco n d o n e is th e d ip o le term . (In cid en ta lly, th e d ip o le term is ju st th e d i® eren ce in th e p o ten tia l d u e to tw o ch a rg es k ep t sep a ra ted b y a
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d ista n ce A ; clea rly, th e n et p o ten tia l w ill b e th e d irectio n a l d eriva tiv e a lo n g A . T h is is th e q u ick est w ay to g et th e d ip o le p o ten tia l.) A t su ± cien tly la rg e d ista n ces w e ig n o re fu rth er term s, o b ta in ed b y ta k in g th e seco n d , th ird , ... d eriva tiv es o f 1 = r . T h e v elo city ¯ eld w ill th en b e th e a n a lo g u e o f th e electric ¯ eld in electro sta tics. F ro m G a u ss' law w e k n ow th a t th e ° u x o f th e electric ¯ eld a t la rg e d ista n ces is p ro p o rtio n a l to th e `to ta l ch a rg e' q . S in ce w e ca n n o t h av e a n o n zero ° u x o f v elo city a t la rg e d ista n ce in o u r p ro b lem , it fo llow s th a t q = 0 a n d th e a sy m p to tic fo rm o f th e p o ten tia l m u st h av e th e fo rm : µ ¶ A ¢n 1 ; Á = A ¢r = ¡ (3 ) r r2 w h ere n is th e u n it v ecto r in th e ra d ia l d irectio n . T a k in g th e g ra d ien t, w e g et th e v elo city ¯ eld to b e µ ¶ 3 (A ¢ n )n ¡ A 1 : v = (A ¢ r )r = (4 ) r r3
It is interesting that the flow at large distances is fixed entirely in terms of a single vector A. In fluid mechanics, it is a bit of a surprise but in electrostatics it is not.
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(T h ese m a n ip u la tio n s a re m o st e± cien tly d o n e u sin g in d ex n o ta tio n a n d su m m a tio n co n v en tio n , w ith @ ® r = (1 = 2 r )@ ® r 2 = x ® = r u sed rep ea ted ly.) T h e a ctu a l fo rm o f A n eed s to b e d eterm in ed u sin g th e co n d itio n s n ea r th e b o d y (w h ich w ill b e a m ess fo r a b o d y o f a rb itra ry sh a p e) b u t it is in terestin g th a t th e ° ow a t la rg e d ista n ces is ¯ x ed en tirely in term s o f a sin g le v ecto r A . In ° u id m ech a n ics, it is a b it o f a su rp rise b u t in electro sta tics it is n o t. If th e m o n o p o le va n ish es, y o u w o u ld ex p ect th e d ip o le m o m en t to d eterm in e th e b eh av io u r o f th e electric ¯ eld a t la rg e d ista n ces. T h e rea l su rp rise co m es w h en w e try to ca lcu la te th e to ta l k in etic en erg y a sso cia ted w ith th e ° u id ° ow g iv en by Z 1 K la b = ½ d 3 x v 2 ; (5 ) 2
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w h ere th e in teg ra l is ov er a ll sp a ce o u tsid e th e b o d y a n d th e su b scrip t `la b ' sta n d s fo r th e la b fra m e in w h ich th e b o d y is m ov in g w ith a v elo city u . (T h e fa ct th a t th e b o d y is m ov in g is irreleva n t sin ce it o n ly sh ifts th e o rig in b y u t w h ich is a co n sta n t a s fa r a s th e sp a tia l in teg ra tio n is co n cern ed .) W h ile th e ° u id ° ow a t la rg e d ista n ces ca n b e ex p ressed en tirely in term s o f a sin g le v ecto r A , th e ° ow clo ser to th e b o d y ca n b e ex trem ely co m p lica ted . O n e m ig h t h av e th o u g h t th a t, in su ch a g en era l ca se, o n e ca n n o t say a n y th in g a b o u t th e to ta l k in etic en erg y o f th e ° u id . B u t it is in d eed p o ssib le to ex p ress th e to ta l k in etic en erg y o f th e ° u id ° ow en tirely in term s o f th e sin g le v ecto r A ev en th o u g h th e ° u id ° ow ev ery w h ere ca n n o t b e ex p ressed in term s o f A a lo n e. (T h is resu lt, a s w ell a s eq u a tio n s (8 ) a n d (1 8 ) b elow , a re d eriv ed in [1 ] b u t n o t d iscu ssed in d eta il in a n y o th er b o o k , a s fa r a s I k n ow .) T o o b ta in th is resu lt, w e u se th e id en tity v 2 = u 2 + (v + u ) ¢ (v ¡ u ). If w e in teg ra te b o th sid es o f th is eq u a tio n ov er a la rg e v o lu m e V , th e ¯ rst term o n th e rig h t w ill g iv e a co n trib u tio n p ro p o rtio n a l to (V ¡ V b o d y ). In th e seco n d term , w e w rite (v + u ) = r (Á + u ¢r ). N ow u sin g r ¢ v = 0 ; a n d r ¢ u = 0 , w e ca n w rite th e seco n d term a s a to ta l d iv erg en ce r ¢[(Á + u ¢r )(v ¡ u )]: In teg ra tin g th is ov er th e w h o le sp a ce, th e seco n d term b eco m es a su rfa ce in teg ra l ov er b o th th e su rfa ce o f th e b o d y a n d a su rfa ce a t la rg e d ista n ce. T h a t is, w e h av e p rov ed : Z I 2 2 v d V = u (V ¡ V 0 ) + (Á + u ¢ r )(v ¡ u ) ¢ n d S ; S+ S0
(6 ) w h ere S is a su rfa ce b o u n d in g th e v o lu m e V a t la rg e d ista n ce a n d S 0 is th e su rfa ce o f th e b o d y. T h e su rfa ce in teg ra l is ta k en ov er b o th . T h e m ira cle is n o w in sig h t. O n th e su rfa ce o f th e b o d y, (v ¡ u )¢n va n ish es d u e to th e b o u n d a ry co n d itio n s a n d h en ce w e g et n o co n trib u tio n s fro m th ere! T h is is g o o d sin ce w e h a v e n o clu e a b o u t th e
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p a ttern o f v elo city ° ow n ea r th e b o d y. A t la rg e d ista n ces fro m th e b o d y, w e ca n u se th e a sy m p to tic fo rm o f th e v elo city ¯ eld g iv en in (4 ) a n d d o th e in teg ra l ta k in g th e su rfa ce to b e a sp h ere o f la rg e ra d iu s R . S in ce d S = R 2 d - in crea ses a s R 2 w h ile v fa lls a s 1 = R 3 a n d Á fa lls a s 1 = R 2 w e ca n a p p rox im a te Á (v ¡ u ) ¢n ¼ ¡ Á u ¢ n o n S . H en ce th e su rfa ce in teg ra l in (6 ) o n S b eco m es th e su m I I 2 ¡ Á u ¢n R d - + (u ¢ n )(v ¢ n ) R 3 d S SI ¡ (u ¢ n )2 R 3 d - : (7 ) S
T h e in teg ra tio n ov er a n g u la r co o rd in a tes ca n b e d o n e u sin g th e ea sily p rov ed rela tio n < (A ¢ n )(B ¢ n ) > = (1 = 3 )A ¢ B w h ere < ::: > d en o tes th e a n g u la r average w h ich is 1 = 4 ¼ tim es th e in teg ra l ov er d - . U sin g th is, w e see th a t th e in teg ra l ov er ¡ (u ¢n )2 R 3 g iv es ¡ u 2 V w h ich p recisely ca n cels th e u 2 V in th e ¯ rst term in (6 ). U sin g (3 ) a n d (4 ) w e g et th e ¯ n a l a n sw er to b e: K
1
The electrostatic analogue is the following. You are given a distribution of charges with qtot = 0 and dipole moment p in a region bounded by a surface S0. You are also given a constant vector E0 and told that the normal component of the electric field normal to S0 is given by n . E 0 . Then the electrostatic energy is proportional to (4 p.E0–V0 E02) where V0 is the volume of the region bounded by S0.
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la b
=
1 ½ (4 ¼ A ¢ u ¡ V 0 u 2 ): 2
(8 )
T h u s, if w e k n ow th e m o tio n o f th e ° u id a t v ery la rg e d ista n ces fro m th e b o d y, w e ca n co m p u te th e to ta l k in etic en erg y o f th e ° u id ° ow w ith o u t ev er k n ow in g th e v elo city ¯ eld clo se to th e b o d y !1 O n e ca n u se th is to o b ta in a n o th er cu rio u s resu lt. T o d o th is, w e n o te th a t th e K la b ca n a lso b e ex p ressed in a d i® eren t fo rm o f su rfa ce in teg ra l. W ritin g v = r Á , th e ex p ressio n fo r k in etic en erg y red u ces to Z Z 1 1 3 2 K = ½ d x (r Á ) = ½ d 3 x r ¢ (Á r Á ); (9 ) 2 V 2 V
w h ere w e h av e u sed r 2 Á = 0 . U sin g G a u ss' th eo rem , th is ex p ressio n ca n b e co n v erted to a su rfa ce in teg ra l
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ov er th e b o d y a n d ov er a su rfa ce a t la rg e d ista n ce. T h e seco n d o n e va n ish es, g iv in g I I 1 1 K la b = ¡ ½ d S (n ¢ v )Á = ¡ ½ d S (n ¢ u )Á ; 2 S0 2 S0 (1 0 )
w h ere w e h av e u sed n ¢ v = n ¢ u sin ce w e k n ow w h a t K la b is, th is a n in teg ra l ov er th e su rfa ce o f th e ex p ressio n , in th e fo rm I ¡ d S (n ¢ u )Á = (4 ¼ A
a t th e su rfa ce. B u t a llow s u s to o b ta in b o d y o f a p a rticu la r ¢u ¡ V 0u 2)
(1 1 )
S0
ev en th o u g h w e d o n o t k n ow eith er th e sh a p e o f th e b o d y o r th e v elo city p o ten tia l o n th e su rfa ce! L et u s n ow sp ecia lize to th e sim p lest o f a ll p o ssib le sh a p es fo r th e b o d y : a sp h ere o f ra d iu s a . In th is ca se, th e d ip o le p o ten tia l h a p p en s to b e th e exact so lu tio n a t a ll d ista n ces o u tsid e th e sp h ere. T h is is n o t d i± cu lt to u n d ersta n d . G iv en th e sp h erica l sy m m etry, th e o n ly v ecto r th a t ca n a p p ea r in th e so lu tio n is th e v elo city o f th e b o d y u . L in ea rity o f th e L a p la ce eq u a tio n (a n d th e b o u n d a ry co n d itio n ) tells u s th a t th e p o ten tia l m u st b e lin ea r in th is v ecto r u . H en ce th e so lu tio n m u st h av e th e fo rm in eq u a tio n (3 ) w ith A / u . U sin g th e b o u n d a ry co n d itio n n ¢ v = n ¢u a t th e su rfa ce, it is ea sy to sh ow th a t A =
1 3 a u 2
(1 2 )
w h ich co m p letely so lv es th e p ro b lem . W e w ill n ow p lay a ro u n d w ith th is so lu tio n . G iv en th e ° u id ° ow p a ttern ev ery w h ere, w e ca n ex p licitly co m p u te th e to ta l k in etic en erg y ca rried b y th e ° ow u sin g a n y o f th e ex p ressio n s d eriv ed a b ov e. W e g et
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K
la b
1 = ¡ ½ 2
Z
µ ¶ 1 a d - ¡ 2 (A ¢ n )(u ¢ n ) a 2
1 1 1 ½ (4 ¼ ) (A ¢ u ) = m d isp u 2 ; (1 3 ) 2 3 4 w h ere m d isp is th e m a ss o f th e ° u id d isp la ced b y th e sp h ere. S o th e to ta l k in etic en erg y is (1 = 2 )[m b o d y + (1 = 2 )m d isp ]u 2 a n d th e ° u id a d d s (1 = 2 )m d isp to th e effectiv e m a ss o f th e sp h ere. O f co u rse, o u r g en era l ex p ressio n , eq u a tio n (8 ) lea d s to th e sa m e resu lt w h en w e u se (1 2 ) a n d ev ery th in g seem s ¯ n e. =
L et u s n ex t co n sid er th e to ta l m o m en tu m P ca rried b y th e ° u id w h ich is th e in teg ra l ov er a ll sp a ce o f ½ v . In a rea so n a b le w o rld , w e w o u ld h av e ex p ected it to b e (1 = 2 )m d isp u b u t w e a re in fo r a ru d e sh o ck . B y sy m m etry, th e v ecto r P h a s to b e in th e d irectio n o f u so w e o n ly n eed to co m p u te th e sca la r P ¢u . B u t sin ce v fa lls a s 1 = r 3 a n d th e v o lu m e g row s a s r 3 w e a re in tro u b le! (T h is d id n o t h a p p en fo r th e k in etic en erg y sin ce w e w ere in teg ra tin g v 2 / 1 = r 6 ov er a ll sp a ce.). E x p licitly, w e h av e, Z 1 P la b ¢ u = ½ d 3 x 3 [3 (A ¢ n )(u ¢ n ) ¡ A ¢ u ] r Z Z1 dr d - [3 (A ¢ n )(u ¢ n ) ¡ A ¢ u ]: = ½ r a (1 4 ) Whoever would have guessed that the simplest problem in fluid flow past a body will actually lead to a product of zero and infinity!
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O b v io u sly, o u r p ow er co u n tin g a rg u m en t is co rrect a n d th e r -in teg ra l d iv erg es lo g a rith m ica lly a t la rg e d ista n ces! O n th e o th er h a n d th e a n g u la r in teg ra tio n ov er sp h erica l su rfa ces g iv es zero b eca u se < 3 (A ¢ n )(u ¢ n ) > = A ¢ u ca n cels th e seco n d term . W h o ev er w o u ld h av e g u essed th a t th e sim p lest p ro b lem in ° u id ° ow p a st a b o d y w ill a ctu a lly lea d to a p ro d u ct o f zero a n d in ¯ n ity ! If w e d o th e in teg ra l b etw een tw o sp h eres o f ra d ii r = a a n d r = R cen tred o n th e m ov in g sp h ere a t a n y g iv en
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in sta n t o f tim e, th en th e a n sw er is in d eed zero b eca u se th e a n g u la r av era g e g iv es zero . T h is w o u ld h av e b een a n a ccep ta b le resu lt, ex cep t fo r tw o rea so n s. F irst, th e resu lt d ep en d s o n ta k in g th e o u ter b o u n d a ry to b e a sp h ere. If w e ch o o se so m e o th er sh a p e, say, a cy lin d er co a x ia l w ith th e d irectio n o f m o tio n o f th e sp h ere, th e resu lt ca n b e d i® eren t. O n e feels u n ea sy a b o u t th e resu lt b ein g d ep en d en t o n w h a t o n e is d o in g a t in ¯ n ity esp ecia lly sin ce th e d irectio n o f u b rea k s th e sp h erica l sy m m etry.
One feels uneasy about the result being dependent on what one is doing at infinity especially since the direction of u breaks the spherical symmetry.
S eco n d , o n e ca n a rg u e th a t, if y o u p u sh th e sp h ere fro m rest to let it a cq u ire a v elo city u , th en { in th e p ro cess { y o u im p a rt so m e m o m en tu m to th e ° u id . T o d o th is co m p u ta tio n o n e n eed s to k n ow th e p ressu re w h ich a cts o n th e sp h ere w h en u is a fu n ctio n o f tim e [2 ]. L et m e b rie° y in d ica te h ow th is ca n b e o b ta in ed . T h e sta rtin g p o in t is th e E u ler eq u a tio n @ v = @ t + (v ¢ r )v = r p = ½ . W h en v = r Á (t;x ), y o u ca n m a n ip u la te th is eq u a tio n to sh o w th a t r [p + (1 = 2 )½ v 2 + ½ (@ Á = @ t)] = 0 , so th a t th e p ressu re ca n b e ex p ressed in th e fo rm p = p1 ¡
@Á 1 2 ½v ¡ ½ ; @t 2
(1 5 )
w h ere p 1 is th e p ressu re a t in ¯ n ity. (T h is is ju st a tim e d ep en d en t v ersio n o f B ern o u lli's eq u a tio n .) W e a re in terested in th e n et fo rce in th e d irectio n o f m o tio n o f th e sp h ere, ta k en to b e th e z-a x is, w h ich ca n b e o b ta in ed b y in teg ra tin g p co s µ ov er th e su rfa ce o f th e sp h ere. F ro m (4 ) w e see th a t v 2 w ill b e a fu n ctio n o f co s2 µ so th e co n trib u tio n fro m th e ¯ rst tw o term s w ill va n ish o n in teg ra tio n ov er a sp h ere. T h e o n ly su rv iv in g co n trib u tio n co m es fro m th e la st term w h ich ca n b e ea sily eva lu a ted to g iv e ¸ · Z¼ 1 duz 1 2 duz 2 Fz = ¡ ½ a co s µ : 2 ¼ a sin µ d µ = m d isp 2 dt 2 dt 0 (1 6 )
RESONANCE September 2008
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C lea rly, th e to ta l m o m en tu m im p a rted is Z 1 F z d t = m d isp u z ; 2
(1 7 )
w h ich m a k es sen se w h en w e rem em b er th a t th e k in etic en erg y co m es w ith th e e® ectiv e m a ss (1 = 2 )m d isp . S o , th is is a n o th er p u rely lo ca l rea so n to b eliev e th e to ta l m o m en tu m o f th e ° u id ° ow is n o n -zero . In fa ct, o n e ca n g en era lize th is a rg u m en t a n d o b ta in a ¯ n ite ex p ressio n fo r th e m o m en tu m fo r a n y b o d y m ov in g th ro u g h a ° u id [1 ]. O n ce a g a in th e resu lt ca n b e ex p ressed en tirely in term s o f th e v ecto r A fo r a b o d y o f arbitrary sh a p e. T o o b ta in th is resu lt, w e ca n u se a trick w h ich rela tes th e in ¯ n itesim a l ch a n g es in th e en erg y a n d m o m en tu m b y th e rela tio n d E = u ¢d P a n d u se th e resu lt in (8 ). T o p rov e th is rela tio n , let u s a ssu m e th a t th e b o d y is a ccelera ted b y so m e ex tern a l fo rce F ca u sin g th e m o m en tu m o f ° u id ° ow to in crea se b y a n a m o u n t d P in a tim e in terva l d t. F ro m th e rela tio n d P = F d t w e im m ed ia tely g et u ¢ d P = F ¢ u d t = d E . G iv en th e fo rm o f E , it is n ow a n elem en ta ry m a tter to v erify th a t th e to ta l m o m en tu m o f th e ° u id ° ow is g iv en b y P = 4¼ ½A ¡ ½V 0u :
The need for regularizing the problem by
(1 8 )
W e see th a t th is is, in g en era l, n o n -zero . In th e ca se o f th e sp h ere it d o es g iv e (1 = 2 )m d isp u ; th is is w h a t w e w o u ld h av e n a iv ely ex p ected . O f co u rse, th e a rg u m en t is d esig n ed to g iv e th is.
introducing a very large but finite volume for the total fluid becomes more apparent when we study the same result in the rest frame of the sphere.
810
T h e n eed fo r reg u la rizin g th e p ro b lem b y in tro d u cin g a v ery la rg e b u t ¯ n ite v o lu m e fo r th e to ta l ° u id b eco m es m o re a p p a ren t w h en w e stu d y th e sa m e resu lt in th e rest fra m e o f th e sp h ere. In th is fra m e, w e h av e a sp h ere o f ra d iu s a lo ca ted a ro u n d th e o rig in a n d th e ° u id is ° ow in g p a st it. T h e b o u n d a ry co n d itio n a t in ¯ n ity is n ow d i® eren t a n d w e ex p ect th e ° u id v elo city to rea ch a co n sta n t va lu e ¡ u a t la rg e d ista n ces. T h is is ea sily
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SERIES ARTICLE
a ch iev ed b y a d d in g a co n sta n t electric ¯ eld to a d ip o le in th e electro sta tic ca se. T h is lea d s to a v elo city p o ten tia l o f th e fo rm à = ¡ r ¢u + Á = ¡ r ¢u ¡
A ¢n : r2
(1 9 )
W e d en o te th e v elo city p o ten tia l in th e rest fra m e b y à to d istin g u ish it fro m th e v elo city p o ten tia l in th e la b fra m e Á . L et u s n ow a sk w h a t is th e k in etic en erg y o f th e ° u id in th is fra m e in w h ich th e b o d y is a t rest. T h e ° u id v elo city n ow is V = v ¡ u . T h e k in etic en erg y in th e rest fra m e w ill b e Z Z £ ¤ 1 2 3 1 K rest = d x ½ V = ½ d 3 x v 2 + u 2 ¡ 2v ¢u 2 2 Z 1 ½ d 3 x u 2 ¡ u ¢ P la b + K la b : = (2 0 ) 2 W e see th a t th e la st term is th e k in etic en erg y in th e la b fra m e co m p u ted a b ov e, w h ich is q u ite w ell-d e¯ n ed . T h e seco n d term is a m b ig u o u s a n d va n ish es if w e u se sp h erica l reg u la riza tio n w h ile is g iv en b y (1 8 ) if w e u se lo ca l en erg y co n serva tio n a rg u m en ts. In th e la tter ca se, K la b ¡ u ¢ P la b = ¡ (1 = 4 )m d isp u 2 is n egative. T h e ¯ rst term , h ow ev er, w ill b e d iv erg en t if w e ta k e th e v o lu m e o f th e ° u id to b e in ¯ n ite a n d is p o sitiv e. T h is d iv erg en ce a rises b eca u se if th e ° u id ex ten d s a ll th e w ay to in ¯ n ity th en m o st o f it w ill b e m ov in g w ith a v elo city ¡ u in th e rest fra m e o f th e sp h ere. T h is w ill co n trib u te a n in ¯ n ite a m o u n t o f k in etic en erg y. W h ile q u ite u n d ersta n d a b le, it sh ow s th a t G a lilea n in va ria n ce n eed s to b e u sed w ith ca re in th e p resen ce o f a n ex tern a l m ed iu m . Suggested Reading
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind
[1]
L D Landau, E M Lifshitz, Fluid Mechanics, Section 10,11, Pergamon, 1989.
[2]
T E Faber, Fluid Mechanics for Physicists, Section 4.8, Cambridge University Press, 1995.
RESONANCE September 2008
Pune 411 007, India. Email: [email protected] [email protected]
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Snippets of Physics 10. Thermodynamics of Self-Gravitating Particles T Padmanabhan
T h e sta tistic a l m e ch a n ic s o f a sy ste m o f p a rtic le s in te ra c tin g th ro u g h g ra v ity le a d s to se v e ra l c o u n te r-in tu itiv e fe a tu re s. W e e x p lo re o n e o f th e m , c a lle d A n to n o v in sta b ility , in th is in sta llm e n t. S u p p o se w e p u t a la rg e n u m b er (N ) o f p a rticles, ea ch o f m a ss m a n d in tera ctin g th ro u g h a tw o -b o d y p o ten tia l U (x ¡ y ) in to a sp h erica l b ox o f ra d iu s R . W e w ill a rra n g e m a tters su ch th a t th e p a rticles m ov e ra n d o m ly to sta rt w ith a n d b o u n ce o ® th e su rfa ce o f th e sp h ere ela stica lly. L et th e to ta l en erg y o f th e sy stem b e E w h ich , o f co u rse, w ill rem a in a co n sta n t. W e a re in terested in th e b eh av io u r o f th e sy stem a t la te tim es, w h en th e p a rticles w ill h av e h a d su ± cien t tim e to in tera ct w ith ea ch o th er a n d ex ch a n g e en erg y.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
T h e resu lt w ill clea rly d ep en d o n th e n a tu re o f th e in tera ctio n , sp eci¯ ed b y U (x ¡ y ) a s w ell a s th e o th er p a ra m eters. If U (x ¡ y ) is a sh o rt ra n g e p o ten tia l rep resen tin g in term o lecu la r fo rces a n d if E is su ± cien tly h ig h , th en th e sy stem w ill rela x tow a rd s a M a x w ellia n d istrib u tio n o f v elo cities a n d n ea rly u n ifo rm d en sity in sp a ce. (T h e v elo city d istrib u tio n w ill h av e a ch a ra cteristic tem p era tu re T ' 2 E = 3 N a n d w e a re a ssu m in g th a t th is is h ig h er th a n th e `b o ilin g p o in t' o f th e `liq u id ' m a d e o f th ese p a rticles. If n o t, th e ev en tu a l eq u ilib riu m sta te w ill b e a m ix tu re o f m a tter in liq u id a n d va p o u r sta te. A lso n o te th a t w e u se u n its w ith k B = 1 th ro u g h o u t.) A ll th is is p a rt o f sta n d a rd lo re in sta tistica l m ech a n ics. W h a t h a p p en s if th e U (x ¡ y ) is d u e to g rav ita tio n a l in tera ctio n o f th e p a rticles? W h a t a re th e d i® eren t p h a ses
RESONANCE October 2008
Keywords Thermodynamics, gravitation. .
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in w h ich m a tter ca n ex ist in su ch a ca se? I w ill d iscu ss so m e o f th e p ecu lia r e® ects th a t a rise in th is co n tex t. T o d o th is, let u s b eg in b y q u ick ly rev iew in g th e w ay o n e in tro d u ces th e eq u ilib riu m co n ¯ g u ra tio n in sta tistica l m ech a n ics. C o n sid er a sy stem d escrib ed b y a d istrib u tio n fu n ctio n f (x ;p ;t) su ch th a t f d 3 x d 3 p d en o tes th e to ta l m a ss in a sm a ll p h a se sp a ce v o lu m e. W e a ssu m e th a t th e ev o lu tio n o f th e d istrib u tio n fu n ctio n is g iv en b y so m e eq u a tio n (u su a lly ca lled th e B o ltzm a n n eq u a tio n ) o f th e fo rm d f = d t = C (f ). T h e term C (f ) o n th e rig h t h a n d sid e d escrib es th e e® ect o f co llisio n s. W h ile th e p recise fo rm o f C (f ) ca n b e q u ite co m p lica ted , w e ca n u su a lly a ssu m e th a t th e co llisio n a l ev o lu tio n o f f , d riv en b y C (f ), sa tis¯ es tw o rea so n a b le co n d itio n s: (a ) T h e to ta l m a ss a n d en erg y o f th e sy stem a re co n serv ed a n d (b ) th e m ea n ¯ eld en tro p y, d e¯ n ed b y S = ¡ 1
For those who are unfamiliar with this expression, here is a recap: In the standard derivation of Boltzmann distribution, one extremises the function S = – ni ln ni of the occupation numbers ni subject to the constraint on total energy and number. In the continuuum limit one works with f rather than ni and the summation over i becomes an integral over the phase space leading to (1).
Z
f ln f d 3 x d 3 p
(1 )
d o es n o t d ecrea se (a n d in g en era l in crea ses).1 F o r a n y su ch sy stem , w e ca n o b ta in th e eq u ilib riu m fo rm o f f b y ex trem isin g th e en tro p y w h ile k eep in g th e to ta l en erg y a n d m a ss co n sta n t u sin g tw o L a g ra n g e m u ltip liers. T h is is a sta n d a rd ex ercise in sta tistica l m ech a n ics a n d th e resu ltin g d istrib u tio n fu n ctio n is th e u su a l B o ltzm a n n d istrib u tio n g ov ern ed b y : ·
f (x ;v ) / ex p ¡ ¯
µ
1 2 v + Á 2
¶¸
; Á (x ) =
Z
d 3 y U (x ;y )½ (y ):
(2 ) In teg ra tin g ov er v elo cities, w e g et th e clo sed sy stem o f eq u a tio n s fo r th e d en sity d istrib u tio n : ½ (x ) = Á (x ) =
Z Z
d 3 v f = A ex p (¡ ¯ Á (x )); d 3 y U (x ;y )½ (y ):
(3 )
T h e ¯ n a l resu lt is q u ite u n d ersta n d a b le: It is ju st th e B o ltzm a n n fa cto r fo r th e d en sity d istrib u tio n : ½ / ex p (¡ ¯ V ), 942
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SERIES ARTICLE
w h ere V is th e p o ten tia l en erg y a t a g iv en lo ca tio n d u e to th e d istrib u tio n o f p a rticles. O n e co u ld h av e a lm o st w ritten th is d ow n `b y in sp ectio n '! (S ee A p p en d ix fo r m o re d eta ils)
One could have almost written down equation (3) 'by inspection'.
E v ery th in g th a t w e h av e sa id so fa r is in d ep en d en t o f th e n a tu re o f th e p o ten tia l U (ex cep t fo r o n e im p o rta n t cav ea t w h ich w e w ill d iscu ss rig h t a t th e en d ). In th e ca se o f g rav ita tio n a l in tera ctio n , (3 ) b eco m es: ½ (x ) = A ex p (¡ ¯ Á (x )); Á (x ) = ¡ G
Z
½ (y )d 3 y : jx ¡ y j
(4 )
T h e in teg ra l eq u a tio n (4 ) fo r ½ (x ) ca n b e ea sily co n v erted to a d i® eren tia l eq u a tio n fo r Á (x ) b y ta k in g th e L a p la cia n o f th e seco n d eq u atio n { lea d in g to r 2 Á = 4 ¼ G ½ { a n d u sin g th e ¯ rst eq u a tio n . W e th en g et, fo r th e sp h erica lly sy m m etric ca se, th e iso th erm a l sp h ere eq u a tio n : 1 d r Á = 2 r dr 2
Ã
dÁ r dr 2
!
= 4 ¼ G ½ c e ¡¯ [Á (r )¡Á (0 )]:
(5 )
T h e co n sta n ts ¯ a n d ½ c (th e cen tra l d en sity ) h av e to b e ¯ x ed in term s o f th e to ta l n u m b er (o r m a ss) o f th e p a rticles a n d th e to ta l en erg y. G iv en th e so lu tio n to th is eq u a tio n , w h ich rep resen ts a n ex trem u m o f th e en tro p y, a ll o th er q u a n tities ca n b e d eterm in ed . A s w e sh a ll see, th is sy stem sh ow s sev era l p ecu lia rities. T o a n a ly se (5 ), it is co n v en ien t to in tro d u ce len g th , m a ss a n d en erg y sca les b y th e d e¯ n itio n s G M 0 L0 (6 ) A ll o th er p h y sica l va ria b les ca n b e ex p ressed in term s o f th e d im en sio n less q u a n tities x ´ (r = L 0 ); n ´ (½ = ½ c ); m = (M (r ) =M 0 ); y ´ ¯ [Á ¡ Á (0 ))]; w h ere M (r) is th e m a ss in sid e a sp h ere o f ra d iu s r. T h ese va ria b les sa tisfy L 0 ´ (4 ¼ G ½ c ¯ )1 = 2 ;
M
RESONANCE October 2008
0
= 4 ¼ ½ c L 30 ;
Á 0 ´ ¯ ¡1 =
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SERIES ARTICLE
th e eq u a tio n s: y 0= m = x 2 ;
m
0
n 0= ¡ m n = x 2 :
= n x2;
(7 )
In term s o f y (x ) th e iso th erm a l eq u a tio n , (5 ), b eco m es 1 d x2 dx 2
We have assumed that the system is spherically symmetric; it turns out that this is indeed the extremal solution.
Ã
dy x dx 2
!
= e¡ y
(8 )
w ith th e b o u n d a ry co n d itio n y (0 ) = y 0(0 ) = 0 .2 L et u s co n sid er th e n a tu re o f th e so lu tio n s to th is eq u a tio n . B y d irect su b stitu tio n , w e see th a t n = (2 = x 2 ) ;m = 2 x ;y = 2 ln x sa tisfy (7 ) a n d (8 ). T h is so lu tio n , h ow ev er, is sin g u la r a t th e o rig in a n d h en ce is n o t p h y sica lly a d m issib le. T h e im p o rta n ce o f th is so lu tio n lies in th e fa ct th a t { a s w e w ill see { a ll o th er (p h y sica lly a d m issib le) so lu tio n s ten d to th is so lu tio n [1 , 2 ] fo r la rg e va lu es o f x . T h is a sy m p to tic b eh av io r o f a ll so lu tio n s sh ow s th a t th e d en sity d ecrea ses a s (1 = r 2 ) fo r la rg e r im p ly in g th a t th e m a ss co n ta in ed in sid e a sp h ere o f ra d iu s r in crea ses a s M (r ) / r a t la rg e r . O f co u rse, in o u r ca se, th e sy stem is en clo sed in a sp h erica l b ox o f ra d iu s R w ith a g iv en m a ss M . E q u a tio n (8 ) is in va ria n t u n d er th e tra n sfo rm a tio n y ! y + a ; x ! k x w ith k 2 = e a . T h is in va ria n ce im p lies th a t, g iv en a so lu tio n w ith so m e va lu e o f y (0 ), w e ca n o b ta in th e so lu tio n w ith a n y o th er va lu e o f y (0 ) b y sim p le resca lin g . T h erefo re, o n ly o n e o f th e tw o in teg ra tio n co n sta n ts n eed ed in th e so lu tio n to (8 ) is rea lly n o n triv ia l. H en ce it m u st b e p o ssib le to red u ce th e d eg ree o f th e eq u a tio n fro m tw o to o n e b y a ju d icio u s ch o ice o f va ria b les. O n e su ch set o f va ria b les is: v ´
m ; x
nx3 nx2 u ´ : = m v
(9 )
In term s o f v a n d u , (5 ) b eco m es u dv (u ¡ 1 ) : = ¡ v du (u + v ¡ 3 ) 944
(1 0 )
RESONANCE October 2008
SERIES ARTICLE
T h e b o u n d a ry co n d itio n s y (0 ) = y 0(0 ) = 0 tra n sla te in to th e fo llow in g : v is zero a t u = 3 , a n d (d v = d u ) = ¡ 5 = 3 a t (3 ,0 ). (Y o u ca n p rov e th is b y ex a m in in g th e b eh av io u r o f (7 ) n ea r x = 0 reta in in g u p to n ecessa ry o rd er in x ; try it o u t!) T h e so lu tio n v (u ) to eq u a tio n (1 0 ) ca n b e ea sily o b ta in ed n u m erica lly : it is p lo tted in F igu re 1 a s th e sp ira llin g cu rv e. T h e sin g u la r p o in ts o f th is d i® eren tia l eq u a tio n a re g iv en b y th e lo ca tio n in th e u v p la n e a t w h ich b o th th e n u m era to r a n d d en o m in a to r o f th e rig h t h a n d sid e o f (1 0 ) v an ish . S o lv in g u = 1 a n d u + v = 3 sim u lta n eo u sly, w e g et th e sin g u la r p o in t to b e u s = 1 , v s = 2 . U sin g (9 ), w e ¯ n d th a t th is p o in t co rresp o n d s to th e a sy m p to tic so lu tio n n = (2 = x 2 );m = 2 x . It is o b v io u s fro m th e n a tu re o f th e eq u a tio n th a t th e so lu tio n cu rv e w ill sp ira l a ro u n d th e sin g u la r p o in t a sy m p to tica lly a p p ro a ch in g th e n = 2 = x 2 so lu tio n a t la rg e x . T h e n a tu re o f th e so lu tio n sh ow n in F igu re 1 a llow s u s to p u t in terestin g b o u n d s o n va rio u s p h y sica l q u a n tities
RESONANCE October 2008
Figure 1. Bound on RE/GM2 for the isothermal sphere.
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in clu d in g en erg y. T o see th is, w e sh a ll co m p u te th e to ta l en erg y E o f th e iso th erm a l sp h ere. T h e p o ten tia l a n d k in etic en erg ies a re U
=
K
=
G M 02 Z x 0 G M (r) d M m n xdx; dr = ¡ r L0 0 dr 0 G M 02 3 Z x 0 2 3 G M 02 3M m (x 0 ) = n x dx; = L0 2 0 2 ¯ 2 L0 (1 1 )
¡
ZR
w h ere x 0 = R = L 0 is th e b o u n d a ry a n d th e ex p ressio n fo r K fo llow s fro m th e v elo city d ep en d en ce o f f in (2 ). T h e to ta l en erg y is, th erefo re, E
= =
G M 02 Z x 0 K + U = d x (3 n x 2 ¡ 2 m n x ) 2L 0 0 G M 02 Z x 0 d d x f2n x 3 ¡ 3m g 2L 0 0 dx ¾ 2 ½ G M 0 3 3 n 0x 0 ¡ m 0 ; = L0 2
(1 2 )
w h ere n 0 = n (x 0 ) a n d m 0 = m (x 0 ). T h e d im en sio n less q u a n tity (R E = G M 2 ) is g iv en b y RE ¸ ´ G M
2
1 = v0
½
3 u0 ¡ 2
¾
(1 3 )
:
N ote that the com bin ation (R E = G M 2 ) is a fu n ction on ly of (u ;v ) at the bou n dary. L et u s n ow co n sid er th e co n stra in ts o n ¸ . S u p p o se w e sp ecify so m e va lu e fo r ¸ b y sp ecify in g R ;E a n d M . T h en su ch a n iso th erm a l sp h ere m u st lie o n th e cu rv e v =
1 ¸
µ
u ¡
¶
3 ; 2
¸ ´
R E G M
2
(1 4 )
w h ich is a stra ig h t lin e th ro u g h th e p o in t (1 :5 ;0 ) w ith a slo p e ¸ ¡1 . O n th e o th er h a n d , sin ce all iso th erm a l sp h eres m u st lie o n th e u ¡ v cu rv e, an isotherm al sphere 946
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SERIES ARTICLE
can exist on ly if the lin e in equ ation (14) in tersects the u ¡ v cu rve. F o r la rg e p o sitiv e ¸ (p o sitiv e E ) th ere is ju st o n e in tersectio n . W h en ¸ = 0 , (zero en erg y ) w e still h av e a u n iq u e iso th erm a l sp h ere. (F or ¸ = 0 , (1 4 ) rep resen ts a v ertica l lin e th ro u g h u = 3= 2 .) W h en ¸ is n eg a tiv e (n eg a tiv e E ), th e lin e ca n cu t th e u ¡ v cu rv e a t m o re th a n o n e p o in t; th u s m o re th a n o n e iso th erm a l sp h ere ca n ex ist w ith a g iv en va lu e o f ¸ . (O f co u rse, sp ecify in g M ;R ;E in d iv id u a lly w ill rem ov e th is n o n -u n iq u en ess). B u t a s w e d ecrea se ¸ (m o re an d m o re n eg a tiv e E ) th e lin e in (1 4 ) w ill slo p e m o re a n d m o re to th e left; a n d w h en ¸ is sm a ller th a n a critica l va lu e ¸ c , th e in tersectio n w ill cea se to ex ist. S o w e rea ch th e k ey co n clu sio n th a t n o isotherm al sphere can exist if (R E = G M 2 ) is below a critical valu e ¸ c . T h is fa ct3 fo llow s im m ed ia tely fro m th e n a tu re o f th e u ¡ v cu rv e a n d (1 4 ). T h e v a lu e o f ¸ c ca n b e fo u n d fro m th e n u m erica l so lu tio n a n d tu rn s o u t to b e a b o u t ¡ 0 :3 3 5 . W h a t d o es th is resu lt m ea n ? T o u n d ersta n d its im p lica tio n s, co n sid er co n stru ctin g su ch a sy stem w ith a g iv en m a ss M , ra d iu s R a n d a n en erg y E = ¡ jE j w h ich is n eg a tiv e. (T h e la st co n d itio n m ea n s th a t th e sy stem is g rav ita tio n a lly b o u n d .) In th is ca se, ¸ = R E = G M 2 = ¡ R jE j= G M 2 is a n eg a tiv e n u m b er b u t let u s a ssu m e th a t it is a b ov e th e critica l va lu e; th a t is, ¸ > ¸ c . T h en w e k n ow th a t a n iso th erm a l sp h ere so lu tio n ex ists fo r th e g iv en p a ra m eter va lu es. B y co n stru ctio n , th is so lu tio n is th e lo ca l ex trem u m o f th e en tro p y a n d co u ld rep resen t a n eq u ilib riu m co n ¯ g u ra tio n if it is a lso a g lo b a l m a x im u m o f en tro p y. B u t fo r th e sy stem w e a re co n sid erin g , it is a ctu a lly q u ite ea sy to see th a t th ere is n o g lo b a l m a x im u m fo r en tro p y. T h is is b eca u se, fo r a sy stem o f p o in t p a rticles in tera ctin g v ia N ew to n ia n p oten tia l, th ere is n o low er b o u n d to th e g rav ita tio n a l p oten tia l en erg y. If w e ta k e
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3
This derivation is due to the author [3,1]. It is surprising that Chandrasekhar, who has worked out the isothermal sphere in u–v coordinates as early as 1939, missed discovering the energy bound shown in Figure 1. Chandrasekhar [2] has the u–v curve but does not over-plot lines of constant . If he had done that, he would have discovered Antonov instability decades before Antonov did [4].
But for the system we are considering, it is actually quite easy to see that there is no global maximum for entropy.
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Suggested Reading [1]
T Padmanabhan, Physics Reports, Vol.188, p.285, 1990.
[2]
S Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover ,1939.
[3]
T Padmanabhan, Astrophys. Jour. Supp., Vol.71, p.651, 1989.
[4]
V A Antonov, Vest. Leningrad Univ., Vol.7, p.135, 1962. Translation: IAU Symposium, Vol.113, p.525, 1985.
a n a m o u n t o f m a ss m < M a n d fo rm a co m p a ct co re o f ra d iu s r in sid e th e sp h erica l cav ity, th en b y d ecrea sin g r o n e ca n su p p ly a rb itra rily la rg e a m o u n t o f en erg y to th e rest o f th e p a rticles. V ery so o n , th e rem a in in g p a rticles w ill h av e v ery la rg e k in etic en erg y co m p a red to th eir g rav ita tio n a l p o ten tia l en erg y a n d w ill essen tia lly b o u n ce a ro u n d in sid e th e sp h erica l cav ity lik e a n o n in tera ctin g g a s o f p a rticles. T h e co m p a ct co re in th e cen ter w ill co n tin u e to sh rin k th ereb y su p p ly in g en erg y to th e rest o f th e p a rticles. It is ea sy to see th a t su ch a co re{ h a lo co n ¯ g u ra tio n ca n h av e a rb itra rily h ig h va lu es fo r th e en tro p y. A ll th is g o es to sh ow th a t th e iso th erm a l sp h ere ca n n o t b e a global m a x im u m fo r th e en tro p y. (T h is w a s th e cav ea t in th e ca lcu la tio n w e p erfo rm ed to d eriv e th e iso th erm a l sp h ere eq u a tio n ; w e ta citly a ssu m ed th a t th e ex trem u m co n d itio n ca n b e sa tis¯ ed fo r a ¯ n ite va lu e o f en tro p y.) If w e in crea se th e ra d iu s o f th e sp h erica l b ox (w ith so m e ¯ x ed va lu e fo r E = ¡ jE j), th e p a ra m eter ¸ w ill b eco m e m o re a n d m o re n eg a tiv e a n d fo r su ± cien tly la rg e R , w e w ill h av e a situ a tio n w ith ¸ < ¸ c . N ow th e situ a tio n g ets w o rse. T h e sy stem d o es n o t ev en h av e a lo ca l ex trem u m fo r th e en tro p y a n d w ill ev o lv e d irectly tow a rd s a co re{ h a lo co n ¯ g u ra tio n . T h is is clo sely rela ted to a p h en o m en o n ca lled A n to n ov in sta b ility [4 , 3 ].
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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In rea l life, o f co u rse, th ere is a lw ay s so m e sh o rt d ista n ce cu t-o ® b eca u se o f w h ich th e co re ca n n o t sh rin k to a n a rb itra rily sm a ll ra d iu s. In su ch a ca se, th ere is a g lo b a l m a x im u m fo r en tro p y a ch iev ed b y th e (¯ n ite) co re{ h a lo co n ¯ g u ra tio n w h ich co u ld b e th o u g h t o f a s th e ¯ n a l sta te in th e ev o lu tio n o f su ch a sy stem . It w ill b e h ig h ly in h o m o g en eo u s a n d , in fa ct, is v ery sim ila r to a sy stem w h ich ex ists a s a m ix tu re o f tw o p h a ses. T h is is o n e k ey p ecu lia rity in tro d u ced b y lo n g ra n g e a ttra ctiv e in tera ctio n s in sta tistica l m ech a n ics.
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A p p e n d ix In th e tex t, I d eriv ed eq u a tio n (3 ) fro m th e ex p ressio n fo r en tro p y in eq u a tio n (1 ). G iv en th e p ecu lia rities o f g rav ita tin g sy stem s o n e m ay w o n d er h ow tru stw o rth y th is a p p ro a ch is. H ere I d escrib e b rie° y a m o re b a sic d eriva tio n o f th e ex p ressio n in (3 ). T h e eq u ilib riu m sta te is th e o n e th a t m a x im izes th e en tro p y o f th e sy stem . W h en w e stu d y th e sy stem in th e m icro ca n o n ica l en sem b le, th is en tro p y S is th e lo g a rith m o f th e v o lu m e g (E ) o f th e p h a se sp a ce ava ila b le to th e sy stem if th e to ta l en erg y is E . T h a t is: Z
1 e = g (E ) = d 3 N x d 3 N p ± (E ¡ H ); (1 5 ) N ! w h ere H is th e H a m ilto n ia n fo r th e sy stem o f N p a rticles g iv en b y th e su m o f th e k in etic en erg ies (p 2i = 2 m ) (i = 1 ;2 ;:::;N ) a n d th e p o ten tia l en erg y o f p a irw ise in tera ctio n . T h e D ira c d elta fu n ctio n tells u s th a t th e sy stem is co n ¯ n ed to th e b o u n d a ry o f a 3 N d im en sio n a l sp h ere in m o m en tu m sp a ce g iv en b y th e eq u a tio n S
XN
i= 1
2
3
1X p = 2 m 4E ¡ U (x i;x j )5 ´ l2 : 2 i6= j 2 i
(1 6 )
O b v io u sly, th e m o m en tu m in teg ra tio n in (15 ) w ill lea d to a term p ro p o rtio n a l to l3 N ¡1 ; so , p erfo rm in g th e m o m en tu m in teg ra tio n s a n d u sin g N À 1 , w e g et 1 e S = g (E ) / N !
Z
d 3N
2
33 N
2 1X x 4E ¡ U (x i ;x j )5 : 2 i6= j
(1 7 )
T h e in teg ra l in (1 7 ) is im p o ssib le to eva lu a te fo r a n y rea listic p o ten tia l b u t th ere is a sta n d a rd a p p rox im a tio n u sin g w h ich w e ca n m a p th is p ro b lem to a m o re tra cta b le o n e. L et u s d iv id e th e th e sp a tia l v o lu m e V in to J (w ith J ¿ N ) cells o f eq u a l size, la rg e en o u g h to co n ta in m a n y p a rticles b u t sm a ll en o u g h fo r th e p o ten tia l to b e trea ted a s a co n sta n t w ith in ea ch cell. (W e w ill a ssu m e th a t su ch a n in term ed ia te sca le ex ists, w h ich u su a lly d o es.) In stea d o f in tegratin g ov er a ll th e p a rticle co o rd in a tes (x 1 ;x 2 ;:::;x N ) w e sh a ll su m over th e n u m b er o f p a rticles n a in th e cell cen tered a t x a (w h ere a = 1 ;2 ;:::;J ) th ereb y a p p rox im a tin g th e in teg ra l b y
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a d iscrete su m . U sin g th e o b v io u s fa ct th a t th e in teg ra tio n ov er (N !)¡ 1 d 3 N x ca n b e rep la ced b y 1 X
n 1= 1
µ
1 n 1!
¶ X1
µ
n 2= 1
1
¶
n 2!
:::
X1
nJ= 1
P
µ
1
¶ µ
nJ!
V J
¶N
(1 8 )
(su b ject to th e co n stra in t th a t n a = N ) a n d a p p rox im a tin g th e n ! s b y S tirlin g 's a p p rox im a tio n : ln n ! = n ln (n = e) w e g et, a fter so m e stra ig h tfo rw a rd a lg eb ra th e resu lt: 1 1 1 eS ¼
w h ere
X
X
n 1= 1 n 2= 1
2
1 XJ
3N S [f n a g ] = ln 4E ¡ 2 2
a6 = b
:::
X
nJ= 1
ex p S [f n a g]; 3
n a U (x a ;x b )n b 5 ¡
XJ
(1 9 )
n a ln
a= 1
µ
n aJ eV
¶
:
(2 0 )
If th e n u m b er o f p a rticles in ea ch cell is su ± cien tly la rg e { a s u su a lly is th e ca se { th is is n o t a d ra stic a p p rox im a tio n . W e a re, o f co u rse, in terested in th e co n ¯ g u ra tio n o f f n a g fo r w h ich th e su m m a n d in (1 9 ) rea ch es th e m a x im u m va lu e, su b ject to th e co n stra in t o n th e to ta l n u m b er. T h is w o rk s in sta n d a rd sta tistica l m ech a n ics b eca u se, in m o st ca ses o f in terest, th e la rg est term a ctu a lly d o m in a tes th e su m a n d th e erro r in v o lv ed in ig n o rin g th e rest is sm a ll. T h a t is, to a h ig h o rd er o f a ccu ra cy S = S [n a ;m a x ], w h ere n a ;m a x is th e so lu tio n to th e va ria tio n a l p ro b lem (±S = ± n a ) = 0 w ith th e su m o f p a rticle n u m b ers n a b ein g k ep t eq u a l to N . Im p o sin g th is co n stra in t b y a L a g ra n g e m u ltip lier a n d u sin g th e ex p ressio n (2 0 ) fo r S , w e o b ta in th e eq u a tio n sa tis¯ ed b y n a ;m a x : 1 XJ U (x a ;x b )n b;m T b= 1
ax
+ ln
µ
n a ;m a x J V
¶
= co n sta n t;
(2 1 )
w h ere w e h av e de¯ n ed th e tem p era tu re T th ro u g h th e rela tio n : 0
1 ¡1
1 3N @ 1 XJ E ¡ n a U (x a ;x b )n b A = T 2 2 a 6= b
= ¯
(2 2 )
w ith n a = n a ;m a x . T o see th a t th is is n o t a s stra n g e a s it lo o k s, y o u o n ly n eed to n o te, fro m (2 0 ) th a t th is ¯ is a lso eq u a l to (@ S = @ E ); th erefo re o n e ca n th in k o f T a s th e co rrect th erm o d y n a m ic tem p era tu re. W e ca n n o w retu rn b a ck to th e co n tin u u m lim it o f (2 1 ) b y w ritin g n a ;m a x (J = V ) = ½ (x a ) a n d rep la cin g th e su m ov er p a rticles b y in teg ra tio n w ith th e m ea su re J = V . In th is co n tin u u m lim it, th e ex trem u m so lu tio n in (2 1 ) is g iv en p recisely b y (3 ).
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Snippets of Physics 11. Isochronous Potentials T Padmanabhan
O sc illa to ry m o tio n o f a p a rtic le in a o n e d im e n sio n a l p o te n tia l b e lo n g s to a c la ss o f e x a c tly so lv a b le p ro b le m s in c la ssic a l m e ch a n ic s. In th is in sta llm e n t, w e e x a m in e so m e le sse r k n o w n a sp e c ts o f th e o sc illa tio n s in so m e p o te n tia ls. T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
T h e m o tio n o f a p a rticle o f m ass m in o n e d im en sio n u n d er th e a ctio n o f a p o ten tia l V (x ) is th e sim p lest p ro b lem w h ich o n e stu d ies in cla ssica l m ech a n ics. In fa ct, a fo rm a list w ill co n sid er th is a s a so lv ed p ro b lem , in th e sen se th a t th e d i® eren tia l eq u a tio n g ov ern in g th e m o tio n ca n b e red u ced to a q u a d ra tu re; ie., th e tra jecto ry o f th e p a rticle ca n b e ex p ressed a s a n in d e¯ n ite in teg ra l. In sp ite o f th is a p p a ren t triv ia lity o f th e p ro b lem th ere a re so m e in terestin g su rp rises o n e en co u n ters in th eir stu d y. U sin g th e co n sta n cy o f th e to ta l en erg y, E = (1 = 2 )m x_2 + V (x ), o n e ca n w rite d ow n th e eq u a tio n d eterm in g th e tra jecto ry o f th e p a rticle x (t) in th e fo rm o f th e in teg ra l r Zx m dx p t(x ) = : (1 ) 2 E ¡ V (x )
Keywords Classical mechanics, oscillation.
T h is d eterm in es th e in v erse fu n ctio n t(x ) fo r a g iv en V (x ) a n d th e p ro b lem is co m p letely so lv ed . In th is in sta llm en t, w e a re in terested in th e ca se o f b o u n d ed o scilla tio n s o f a p a rticle in a p o ten tia l w ell V (x ) w h ich h a s th e g en era l sh a p e sh ow n in F igu re 1 . T h e p o ten tia l h a s a sin g le m in im u m a n d in crea ses w ith o u t b o u n d a s jx j ! 1 . F o r a g iv en va lu e o f en erg y E , th e p a rticle w ill o scilla te b etw een th e tw o tu rn in g p o in ts x 1 (E ) a n d x 2 (E ) w h ich a re th e ro o ts o f th e eq u a tio n V (x ) = E . T h e p erio d o f o scilla tio n ca n b e im m ed ia tely w ritten
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Figure 1. A one-dimensional potential with a single minimum which supports oscillations.
d ow n u sin g eq u a tio n (1 ). W e g et r Z x 2 (E ) m dx p : T (E ) = 2 x 1 (E ) E ¡ V (x )
(2 )
F o r a g en era l p o ten tia l V (x ), th e resu lt o f in teg ra tio n o n th e rig h t h a n d sid e w ill d ep en d o n th e va lu e o f th e en erg y E . In o th er w o rd s, th e p erio d o f o scilla tio n w ill d ep en d o n th e en erg y o f th e p a rticle; eq u iva len tly, if o n e im a g in es relea sin g th e p a rticle fro m th e lo ca tio n x = x 1 , say, o n e m ig h t say th a t th e p erio d d ep en d s o n th e a m p litu d e o f o scilla tio n . F o r a sim p le cla ss o f p o ten tia ls, it is q u ite ea sy to d eterm in e th e sca lin g o f th e p erio d T w ith th e en erg y E . C o n sid er, fo r ex a m p le, a cla ss o f p o ten tia ls o f th e fo rm V (x ) = k x 2 n w h ere n is a n in teg er. T h ese p o ten tia ls a re sy m m etric in th e x -a x is a n d h av e a m in im u m a t x = 0 w ith th e m in im u m va lu e b ein g V m in = 0 . In th is ca se, b y in tro d u cin g a va ria b le q su ch th a t q = (k = E )1 = 2 n x , th e en erg y d ep en d en ce o f th e in teg ra l in (2 ) ca n b e ea sily id en ti¯ ed to g iv e Z1 1 1¡ n 1 dq 1=2n p T (E ) / p E / E 2 ( n ) : (3 ) 2 n E 1¡ q 0
W e ¯ n d th a t, fo r a ll va lu es o f n o th er th a n n = 1 , th e p erio d T h a s a n o n -triv ia l d ep en d en ce o n th e en erg y. W h en n = 1 , w h ich co rresp o n d s to th e h a rm o n ic o scilla to r p o ten tia l, V (x ) = k x 2 , w e ¯ n d th a t th e p erio d
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We find that, for all values of n other than n = 1, the period T has a non-trivial dependence on the energy.
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is in d ep en d en t o f th e en erg y. T h is, o f co u rse, is th e w ellk n ow n resu lt th a t th e p erio d o f a h a rm o n ic o scilla to r d o es n o t d ep en d o n th e a m p litu d e o f th e o scilla to r. T h e a b ov e a n a ly sis a lso sh ow s th a t a m o n g st a ll th e sym m etric p o ten tia ls o f th e fo rm V (x ) / x 2 n , o n ly th e h a rm o n ic o scilla to r h a s th is p ro p erty. L et u s n ow co n sid er th e in v erse p ro b lem . S u p p o se w e a re g iv en th e fu n ctio n T (E ). Is it th en p o ssib le fo r u s to d eterm in e th e p o ten tia l V (x )? F o r ex a m p le, if w e a re to ld th a t th e p erio d is in d ep en d en t o f th e a m p litu d e, w h a t ca n o n e say a b o u t th e form o f th e p o ten tia l V (x )? S h o u ld it n ecessa rily b e a h a rm o n ic o scilla to r p o ten tia l o r ca n it b e m o re g en era l? B efo re la u n ch in g in to a m a th em a tica l a n a ly sis, let m e d escrib e a sim p le ex a m p le w h ich d eserv es to b e b etter k n ow n th a n it is. C o n sid er a p o ten tia l o f th e fo rm V (x ) = a x 2 +
It turns out that the period of oscillation in this potential is independent of the amplitude just as in the case of a harmonic oscillator potential! So clearly harmonic oscillator is not unique in having this property.
1000
b x2
(4 )
in th e reg io n x > 0 . In th is reg io n , th is p o ten tia l h a s a d istin ct m in im u m a t x m in = (b=pa )1 = 4 w ith th e m in im u m va lu e o f th e p o ten tia l b ein g 2 a b. (B ein g sy m m etric in x , th e p o ten tia l h a s tw o m in im a in th e fu ll ra n g e ¡ 1 < x < 1 , b u t w e sh a ll co n ¯ n e o u r a tten tio n to th e ra n g e x > 0 . B y sh iftin g th e o rig in su ita b ly w e ca n m a k e th e p o ten tia l in th is ra n g e to lo o k lik e th e o n e in F igu re 1 ). F o r a n y ¯ n ite en ergy, a p a rticle w ill ex ecu te p erio d ic o scilla tio n s in th is p o ten tia l. It tu rn s o u t th a t th e p erio d o f o scilla tio n in th is p o ten tia l is in d ep en d en t o f th e a m p litu d e ju st a s in th e ca se o f a h a rm o n ic o scilla to r p o ten tia l! S o clea rly h a rm o n ic o scilla to r is n o t u n iq u e in h av in g th is p ro p erty. T h ere a re sev era l w ay s to p rove th is resu lt, th e h a rd est ro u te b ein g to eva lu a te th e in teg ra l in (2 ) w ith V (x ) g iv en b y (4 ); th e cu test p ro ced u re is p ro b a b ly th e fo llow in g . C o n sid er a p a rticle m ov in g n o t in o n e d im en sio n
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b u t in tw o (say in th e x y p la n e) u n d er th e a ctio n o f th e tw o -d im en sio n a l h a rm o n ic o scilla to r p o ten tia l 1 m ! 2 (x 2 + y 2 ): (5 ) 2 C lea rly, su ch a p a rticle w ill o scilla te w ith a p erio d w h ich is in d ep en d en t o f its en erg y. N ow co n sid er th e sa m e p ro b lem in p o la r co o rd in a tes in stea d o f C a rtesia n co o rd in a tes. T h e co n serva tio n o f en erg y n ow b eco m es V (x ;y ) =
1 1 m (x_2 + y_2 ) + m ! 2 (x 2 + y 2 ) 2 2 1 1 2 2 _2 m (r_ + r µ ) + m ! 2 r 2 : = (6 ) 2 2 U sin g th e fa ct th a t fo r su ch a m o tio n { u n d er th e cen tra l fo rce V (r ) / r 2 { th e a n g u la r m o m en tu m J = m r 2 µ_ is co n serv ed , th e en erg y ca n b e ex p ressed in th e fo rm E
E =
=
B 1 1 1 J2 1 m r_2 + m ! 2 r 2 + = m r_2 + A r 2 + 2 2 r 2 2 2m r 2 (7 )
w ith A = (1 = 2 )m ! 2 ;B = J 2 = 2 m . W e n ow see th a t, m a th em a tica lly, th is is id en tical to th e p ro b lem o f a p a rticle m ov in g in o n e d im en sio n u n d er th e a ctio n o f a p o ten tia l o f th e fo rm in (4 ). B u t w e k n ow b y co n stru ctio n th a t th e p erio d o f o scilla tio n d o es n o t d ep en d o n th e co n serv ed en erg y E in th e case o f (7 ). It fo llow s th a t th e p o ten tia l in (4 ) m u st h av e th is p ro p erty. T h e a ctu a l freq u en cy o f o scilla tio n is ! 0 = (8 a = m )1 = 2 w h ich is m o st ea sily fo u n d b y u sin g th e fa ct th a t th e freq u en cy m u st b e th e sa m e a s th a t fo r v ery sm a ll o scilla tio n s n ea r th e m in im u m . O n e m ay th in k th at sin ce ! 0 is in d ep en d en t o f b, it m u st b e (2 a = m )1 = 2 fo r b = 0 . T h is is, h ow ev er, n o t tru e b eca u se h ow ev er sm a ll b m ay b e, th e p o ten tia l d o es rise to in ¯ n ity a t x = 0 th ereb y d o u b lin g th e freq u en cy. P o ten tia ls lik e th a t o f th e h a rm o n ic o scilla to r o r th e o n e in (4 ) a re ca lled `iso ch ro n o u s p o ten tia ls', th e term referrin g to th e p ro p erty th a t th e p erio d is in d ep en d en t o f
RESONANCE November 2008
Potentials like that of the harmonic oscillator or the one in equation (4) are called ‘isochronous potentials’, the term referring to the property that the period is independent of the amplitude.
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In fact, for every function T(E), one can construct an infinite number of potentials V(x) such that equation (4) holds.
th e a m p litu d e. It is n o t d i± cu lt to see th a t th ere a re a ctu a lly a n in ¯ n ite n u m b er o f su ch p o ten tia ls. In fa ct, fo r ev ery fu n ctio n T (E ), o n e ca n co n stru ct a n in ¯ n ite n u m b er o f p o ten tia ls V (x ) su ch th a t eq u a tio n (4 ) h o ld s. T h ere is a n elem en ta ry w ay b y w h ich o n e ca n co n stru ct th em w h ich w e w ill n ow d escrib e [1 ]. N o te th a t th e p erio d T (E ) is d eterm in ed b y th e in teg ra l in (2 ) w h ich is essen tia lly th e a rea u n d er th e cu rv e (E ¡ V (x ))¡ 1 = 2 . S u p p o se w e a re g iv en a p o ten tia l V 1 (x ) fo r w h ich th e en erg y d ep en d en ce o f th e p erio d is g iv en b y a fu n ctio n T (E ). L et u s n ow co n stru ct a n o th er p o ten tia l V 2 (x ) b y `sh ea rin g ' th e o rig in a l p o ten tia l V 1 (x ) p a ra llel to th e x -a x is. T h is is d o n e b y sh iftin g th e p o ten tia l cu rv e h o rizo n ta lly b y a n a m o u n t ¢ (V ) a t ev ery va lu e o f V u sin g so m e a rb itra ry fu n ctio n ¢ (V ). T h e o n ly restrictio n o n th e fu n ctio n ¢ (V ) is th a t th e resu ltin g p o ten tia l sh o u ld b e sin g le va lu ed ev ery w h ere. A m o m en t o f th o u g h t sh ow s th a t su ch a sh ift leav es th e a rea u n d er th e cu rv e in va ria n t a n d h en ce T (E ) d o es n o t ch a n g e. In o th er w o rd s, g iv en a n y p o ten tia l V (x ), th ere a re a n in ¯ n ite n u m b er o f o th er p o ten tia ls fo r w h ich w e w ill g et th e sa m e p erio d { en erg y d ep en d en ce T (E ); ea ch o f th ese p o ten tia ls is d eterm in ed b y th e fo rm o f th e fu n ctio n ¢ (V ). In th e ca se o f th e h a rm o n ic o scilla to r p o ten tia l, th e d ista n ce h (V p) b etw een th e tw o tu rn in g p o in ts (`w id th ') va ries a s V w h en th e p o ten tia l is m ea su red fro m its m in im a . S in ce (4 ) h a s th e iso ch ro n o u s p ro p erty, w e w o u ld su sp ect th a t it is o b ta in ed fro m th e h a rm o n ic o scilla to r p o ten tia l b y a sh ea rin g m o tio n k eep in g th e w id th h (V ) va ry in g a s (V ¡ V m in )1 = 2 . T h is is in d eed tru e a n d ca n b e d em o n stra ted a s fo llow s. F ro m (4 ), w e ca n d eterm in e th e in v erse, d o u b le-va lu ed fu n ctio n x (V ) th ro u g h th e eq u a tio n a x 4 + b ¡ V x 2 = 0:
(8 )
If th e ro o ts o f th is eq u a tio n a re x 21 a n d x 22 , w e im m ed i-
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a tely h av e x 21 + x 22 = V = a a n d x 21 x 22 = b= a . E lem en ta ry a lg eb ra n ow g iv es r V b h (V )2 = (x 1 ¡ x 2 )2 = ¡ 2 : (9 ) a a O r, eq u iva len tly, h (V ) = p
1 a
(V ¡ V m in )1 = 2 :
(1 0 )
For those of you who do not like such a geometric argument, here is a more algebraic derivation of the same result.
T h is sh ow s th a t th e p o ten tia l in (4 ) is in d eed o b ta in ed b y a sh ea rin g o f th e h a rm o n ic o scilla to r p o ten tia l. F o r th o se o f y o u w h o d o n o t lik e su ch a g eo m etric a rg u m en t, h ere is a m o re a lg eb ra ic d eriv a tio n o f th e sa m e resu lt [2 ]. L et u s su p p o se th a t w e a re g iv en th e fu n ctio n T (E ) a n d a re a sk ed to d eterm in e th e p o ten tia l V (x ) w h ich is a ssu m ed to h av e a sin g le m in im u m a n d a sh a p e ro u g h ly lik e th e o n e in F igu re 1 . W e ca n a lw ay s a rra n g e th e co o rd in a tes su ch th a t th e m in im u m o f th e p o ten tia l lies a t th e o rig in o f th e co o rd in a te sy stem . T h e sh a p e o f th e cu rv e in th e reg io n s x > 0 a n d x < 0 w ill, o f co u rse, b e d i® eren t. In o rd er to m a in ta in sin g le va lu ed n ess o f th e in v erse fu n ctio n x (V ), w e w ill d en o te th e fu n ctio n a s x 1 (V ) in th e reg io n x < 0 a n d x 2 (V ) in th e reg io n x > 0 . O n ce th is is d o n e, w e ca n rep la ce d x in th e in teg ra l in (2 ) b y (d x = d V )d V . T h is a llow s u s to w rite ¸ ZE · p dx2 dx1 dV p T (E ) = ¡ : 2m (1 1 ) dV dV E ¡ V 0 T h is is a n in teg ra l eq u a tio n w h ich , fo rtu n a tely, ca n b e in v erted b y a sta n d a rd trick . W e d iv id e b o th sid es o f th e eq u a tio n b y (z ¡ E )1 = 2 , w h ere z is a p a ra m eter a n d in teg ra te w ith resp ect to E fro m 0 to z . T h is g iv es Zz T (E ) d E p = z¡ E 0 p
¸ Zz ZE · dx2 dx1 dV dE p : (1 2 ) ¡ 2m dV dV [(z ¡ E )(E ¡ V )] 0 0
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The family of curves which has the same width will give rise to the same T(E) and vice versa.
In th e rig h t h a n d sid e w e ca n ch a n g e th e o rd er o f in teg ra tio n a n d u se th e fa ct th a t th e in teg ra l ov er E ju st g iv es ¼ . T h e rest is triv ia l a n d w e o b ta in Zz p T (E ) d E p = ¼ 2 m [x 2 (z ) ¡ x 1 (z )]: (1 3 ) z¡ E 0 T h is is a n im p licit eq u a tio n va lid fo r a n y z . C a llin g th e va ria b le z a s V , g iv es th e fu n ctio n a l fo rm o f x 2 (V ) ¡ x 1 (V ). W e g et th e ¯ n a l resu lt ZV T (E ) d E 1 p x 2 (V ) ¡ x 1 (V ) = p : (1 4 ) V ¡ E ¼ 2m 0 T h is resu lt sh ow s ex p licitly th a t th e fu n ctio n T (E ) ca n o n ly d eterm in e fo r u s th e `w id th ' o f th e cu rv e x 2 (V ) ¡ x 1 (V ). T h e fa m ily o f cu rv es w h ich h a s th e sa m e w id th w ill g iv e rise to th e sa m e T (E ) a n d v ice v ersa . T h e sh ea rin g m o tio n b y w h ich w e tra n sfo rm o n e p o ten tia l to a n o th er p reserv es th is w id th a n d h en ce th e fu n ctio n a l fo rm o f T (E ). O n e ca n a lso o b ta in so m e in terestin g rela tio n s in q u a n tu m m ech a n ics fo r th e co rresp o n d in g sy stem s. In q u a n tu m th eo ry, th e p o ten tia ls lik e th e o n e in F igu re 1 w ill h av e a set o f d iscrete en erg y lev els E n . F o rm a lly in v ertin g th e fu n ctio n E (n ) { w h ich is o rig in a lly d e¯ n ed o n ly fo r in teg ra l va lu es o f n { o n e ca n o b ta in th e in v erse fu n ctio n n (E ) fo r th is sy stem . T h is fu n ctio n essen tia lly p lay s th e ro le a n a lo g o u s to T (E ) in th e ca se o f q u a n tu m th eo ry. W e ca n n ow a sk w h eth er o n e ca n d eterm in e th e p o ten tia l V (x ) g iv en th e en erg y lev els E n o r, eq u iva len tly, th e fu n ctio n n (E ). It tu rn s o u t th a t o n e ca n d o th is fa irly ea sily in th e sem i-cla ssica l lim it co rresp o n d in g to la rg e n . T o see th is, reca ll th a t th e en erg y E n o f th e n -th lev el o f a q u a n tu m m ech a n ica l sy stem is g iv en b y th e B o h r q u a n tiza tio n co n d itio n r Z Z x2 p 1 x2 2m n (E ) ' pdx = E ¡ V d x : (1 5 ) ~ x1 ~2 x 1
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SERIES ARTICLE
T rea tin g x a s a fu n ctio n o f V , w e ca n tra n sfo rm th is rela tio n to g iv e r Z x2 p 2m n (E ) ' E ¡ V dx ~2 x 1 r ZE p 2m dx E ¡ V dV = 2 ~ dV r ZE m = (E ¡ V )¡ 1 = 2 x (V ) d V . (1 6 ) 2 ~2 H ere w e h av e d o n e a n in teg ra tio n b y p a rts a n d h av e trea ted th e in teg ra l a s a fu n ctio n o f th e u p p er lim it. T h is in teg ra l eq u a tio n ca n a g a in b e so lv ed b y ex a ctly th e sa m e trick w h ich w e u sed in th e ca se o f (1 1 ). T h is w ill lea d to th e resu lt r Z 2 ~2 n (V ) dn p x (V ) = : (1 7 ) m V ¡ E (n )
w h ich d eterm in es th e fo rm o f th e p o ten tia l V (x ) { in term s o f th e in v erse fu n ctio n x (V ) { su ch th a t in th e sem i-cla ssica l lim it it w ill h av e th e en erg y lev els g iv en b y th e fu n ctio n E (n ).
E v en th o u g h w e w o rk ed it o u t fo r a o n e-d im en sio n a l m o tio n w ith a C a rtesia n x -a x is, it is o b v io u s th a t th e sa m e fo rm u la sh o u ld b e a p p lica b le fo r en erg y lev els in a sp h erica lly sy m m etric p o ten tia l V (r) p rov id ed w e o n ly co n sid er th e zero a n g u la r m o m en tu m q u a n tu m sta tes. A s a cu rio sity, co n sid er th e p o ten tia l w h ich w ill rep ro d u ce th e en erg y lev els (w h ich w e k n ow is th e o n e a risin g in th e ca se o f th e C o u lo m b p ro b lem ) g iv en b y
The formula given in equation (17) should be applicable for energy levels in a spherically symmetric
4
E
n
= ¡
2
m e Z : 2 ~2 n 2
(1 8 )
T h is g iv es n (V ) = (¡ 2 ~2 V = m e 4 Z 2 )¡1 = 2 so th a t a n elem en ta ry in teg ra tio n u sin g (1 7 ) w ith a su ita b le ch o ice fo r
RESONANCE November 2008
potential V(r) provided we only consider the zero angular momentum quantum states.
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th e co n sta n t o f in teg ra tio n g ives r · µ 4 2 ¶¸¡ 1 = 2 Z m e Z Z e2 2 ~2 n (V ) V + n ¡ r= d = m V 2 ~2 n 2 (1 9 ) th ereb y lea d in g to V (r ) = ¡ Z e 2 = r w h ich , o f co u rse, w e k n ow is ex a ct. (T h is is o n e o f th e m a n y cu rio sities in th e C o u lo m b p ro b lem w h ich w e w ill tu rn to in a fu tu re in sta llm en t.) W e w ill n ow d escrib e a n o th er in terestin g fea tu re in q u a n tu m th eo ry rela ted to iso ch ro n o u s p o ten tia ls. It is w ell k n ow n th a t w h en w e m ov e fro m cla ssica l to q u a n tu m m ech a n ics, th e h a rm o n ic o scilla to r p o ten tia l lea d s to eq u id ista n t en erg y lev els. C u rio u sly en o u g h , a ll th e iso ch ro n o u s p o ten tia ls h av e th is p ro p erty in the sem i-classical lim it. T h is is m o st ea sily seen b y d i® eren tia tin g (1 5 ) w ith resp ect to E a n d u sin g (2 ) so a s to o b ta in r Z x2 m T (E ) dn dx p : = = (2 0 ) 2 ~ dE 2~ x 1 E ¡ V In o th er w o rd s, th e q u a n tu m n u m b ers a re g iv en b y th e eq u iva len t fo rm u la Z 1 T (E )d E n (E ) ' (2 1 ) ~ In the case of the potential in equation (4), something more surprising happens: The exact solution to the Schrödinger equation itself has equally spaced energy levels!
1006
w h ich n icely co m p lem en ts th e ¯ rst eq u a tio n in (1 5 ). If th e p o ten tia l is iso ch ro n o u s, th en T (E ) = T 0 is a co n sta n t in d ep en d en t o f E a n d th e in teg ra l im m ed ia tely g iv es th e lin ea r rela tio n b etw een E a n d n o f th e fo rm E = ® n + ¯ , w h ere ® = (~= T 0 ). C lea rly, th ese en erg y lev els a re eq u a lly sp a ced ju st as in th e ca se o f h a rm o n ic o scilla to rs. In th e ca se o f th e p o ten tia l in (4 ), so m eth in g m o re su rp risin g h a p p en s: T h e ex a ct so lu tio n to th e S ch rÄo d in g er eq u a tio n itself h a s eq u a lly sp a ced en erg y lev els! I w ill
RESONANCE November 2008
SERIES ARTICLE
in d ica te b rie° y h ow th is a n a ly sis p ro ceed s leav in g th e d eta ils fo r y o u to w o rk o u t. (Y o u ca n n o t rea ch th is co n clu sio n b y th e tw o -d im en sio n a l trick u sed ea rlier in cla ssica l p h y sics.) T o b eg in w ith , w e ca n red e¯ n e th e p o ten tia l to th e fo rm · ¸2 B V (x ) = A x ¡ A 2 ´ a; B 2 ´ b ; (2 2 ) x b y a d d in g a co n sta n t so th a t th e m in im u m va lu e o f th e p o ten tia l is zero a t x = (B =A )1 = 2 . T h e freq u en cy o f o scilla tio n s in th is p o ten tia l is ! 0 = (8 a = m )1 = 2 . T o stu d y th e S ch rÄo d in g er eq u a tio n fo r th e p o ten tia l in (2 2 ), it is co n v en ien t to in tro d u ce th e u su a l d im en sio n less va ria b les » = (m ! 0 = ~)1 = 2 x ; ² = 2 E = (~! 0 ) a n d ¯ = B (2 m )1 = 2 = ~, in term s o f w h ich th e S ch rÄo d in g er eq u a tio n ta k es th e fo rm : " µ ¶2 # ¯ 1 à 00+ ² ¡ »¡ à = 0: (2 3 ) » 2 A s » ! 1 , th e ¯ = » term b eco m es n eg lig ib le a n d { a s in th e ca se o f th e sta n d a rd h a rm o n ic o scilla to r { th e w av efu n ctio n s w ill d ie a s ex p [¡ (1 = 4 )» 2 ]. N ea r th e o rig in , th e S ch rÄo d in g er eq u a tio n ca n b e a p p rox im a ted a s » 2 à 00 ¼ ¯ 2 à w h ich h a s so lu tio n s o f th e fo rm à / » s w ith s b ein g th e p o sitiv e ro o t o f s (s ¡ 1 ) = ¯ 2 . W e n ow fo llow th e sta n d a rd p ro ced u re a n d w rite th e w av efu n ctio n in th e fo rm à = Á (» )[» s ex p (¡ (1 = 4 )» 2 )] a n d lo o k fo r a p ow er-law ex p a n sio n fo r Á o f th e fo rm Á (» ) =
X1
cn » n :
(2 4 )
n= 0
S u b stitu tin g th is fo rm in to th e S ch rÄo d in g er eq u a tio n w ill lea d , a fter so m e a lg eb ra , to th e recu rren ce rela tio n cn + 2 n + s ¡ ² ¡ ¯ + (1 = 2 ) : = cn (n + 2 )(n + 2 s + 1 )
RESONANCE November 2008
(2 5 )
1007
SERIES ARTICLE
Do all isochronous potentials lead to evenly spaced energy levels as exact solutions to Schrödinger equation rather than only in the asymptotic limit?
A sy m p to tica lly, th is w ill lea d to th e b eh av io u r c n + 2 = c n ' (1 = n ) so th a t Á (» ) ' ex p [(1 = 2 )» 2 ] m a k in g à d iv erg e u n less th e series term in a tes. S o , ² m u st b e so ch o sen th a t th e n u m era to r o f (2 5 ) va n ish es fo r so m e va lu e o f n . C lea rly, o n ly ev en p ow ers o f » a p p ea r in Á (» ) a llow in g u s to w rite n = 2 k , w h ere k is a n in teg er. P u ttin g ev ery th in g b a ck , th e en erg y o f th e k -th lev el ca n b e w ritten in th e fo rm " ¶1 = 2 # µ 1 1 E k = (k + C )~! 0 ; C = 1 ¡ ¯ + ¯2 + 2 4 (2 6 ) sh ow in g th a t th e en erg y lev els a re eq u a lly sp a ced w ith th e w id th ~! 0 b u t w ith C rep la cin g (1 = 2 ) in th e ca se o f th e h a rm o n ic o scilla to r. Y o u ca n co n v in ce y o u rself th a t a ll th e lim itin g b eh av io u r is co rrectly rep ro d u ced .
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus
D o a ll iso ch ro n o u s p o ten tia ls lea d to ev en ly sp a ced en erg y lev els a s ex a ct so lu tio n s to S ch rÄo d in g er eq u a tio n ra th er th a n o n ly in th e a sy m p to tic lim it? T h e a n sw er is \ n o " . T h e sim p le co u n ter-ex a m p le is p rov id ed b y tw o p a ra b o lic w ells co n n ected to g eth er sm o o th ly a t th e m in im a w ith V (x ) = (1 = 2 )m ! R2 x 2 fo r x ¸ 0 a n d V (x ) = (1 = 2 )m ! L2 x 2 fo r x · 0 . It is o b v io u s th a t th is p o ten tia l is iso ch ro n o u s cla ssica lly. S o lv in g th e S ch rÄo d in g er eq u a tio n req u ires a b it o f e® o rt b eca u se y o u n eed to en su re co n tin u ity o f à a n d à 0 a t th e o rig in . T h is lea d s to a set o f en erg y lev els w h ich n eed to b e so lv ed fo r n u m erica lly. O n e ¯ n d s th a t th e en erg y lev els a re n o t eq u a lly sp a ced b u t th e d ep a rtu re fro m ev en sp a cin g is su rp risin g ly sm a ll. T o th e ex ten t I k n ow , th ere is n o sim p le ch a ra cteriza tio n o f p o ten tia ls w h ich lea d to ev en ly sp a ced en erg y lev els in q u a n tu m th eo ry.
Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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Suggested Reading [1]
A B Pippard, The Physics of Vibration, Omnibus Edition, Cambridge University Press, p.15, 1989.
[2]
L D Landau and E M Lifshitz, Mechanics, Pergammon Press, 1960.
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SERIES ARTICLE
Snippets of Physics 12. Paraxial Optics and Lenses T Padmanabhan
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Keywords Optics, waves.
. 1098
D isc o v e rin g u n e x p e c te d c o n n e c tio n s b e tw e e n c o m p le te ly d i® e re n t p h e n o m e n a is a lw a y s a d e lig h t in p h y sic s. In th is a n d th e n e x t in sta llm e n t, w e w ill lo o k a t o n e su ch c o n n e c tio n b e tw e e n tw o u n lik e ly p h e n o m e n a : p ro p a g a tio n o f lig h t a n d p a th in te g ra l a p p ro a c h to q u a n tu m m e c h a n ic s! T h e p ro p a g a tio n o f lig h t { w h ich is ju st a n electro m a g n etic w av e { is g ov ern ed b y a w av e eq u a tio n . T h e electric ¯ eld a n d th e m a g n etic ¯ eld o b ey a w av e eq u a tio n , th e so lu tio n to w h ich d escrib es th e p ro p a g a tio n o f lig h t in a n y sp eci¯ c co n tex t. In th is in sta llm en t w e lo o k a t th e w av e n a tu re o f lig h t fro m a p a rticu la r p o in t o f v iew w h ich w e w ill co n n ect u p w ith a seem in g ly d i® eren t p h en o m en o n in th e n ex t in sta llm en t. F o r o u r p u rp o se th e v ecto r n a tu re o f th e electro m a g n etic ¯ eld is n o t releva n t (sin ce w e w ill n o t b e in terested , e.g ., in th e p o la riza tio n o f th e lig h t.) H en ce w e w ill ju st d ea l w ith o n e co m p o n en t o f th e releva n t v ecto r ¯ eld { let u s ca ll it A (t;x ) { w h ich sa tis¯ es th e w av e eq u a tio n . T h e b a sic so lu tio n to th e w av e eq u a tio n ¤ A = 0 is d escrib ed b y th e (rea l a n d im a g in a ry p a rts o f th e) fu n ctio n ex p i[k ¢ x ¡ ! t]. H ere k d en o tes th e d irectio n o f p ro p a g a tio n o f th e w av e w h ich a lso d eterm in es its freq u en cy th ro u g h th e d isp ersio n rela tio n ! = jk jc. S in ce th e w av e eq u a tio n is lin ea r in A , w e ca n su p erp o se th e so lu tio n s w ith d i® eren t va lu es o f k , ea ch w ith a n a m p litu d e F 1 (k ), say. T h is lea d s to a so lu tio n o f th e fo rm : Z d 3k A (t;x ) = F 1 (k )e ik ¢x e ¡ i! t : (1 ) (2 ¼ )3 W e n o w w a n t to sp ecia lize to a situ a tio n w h ich a rises
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in th e stu d y o f o p tica l p h en o m en a q u ite o ften w h ere w e a re co n cern ed w ith w av es w h ich a re p ro p a g a tin g , b y a n d la rg e, in so m e g iv en d irectio n , say a lo n g th e p o sitiv e z a x is. (F o r ex a m p le, co n sid er th e stu d y o f d i® ra ctio n b y a circu la r h o le in a screen w h ich is lo ca ted in th e z = 0 p la n e. W e w ill co n sid er, in su ch a co n tex t, lig h t in cid en t o n th e screen fro m th e left a n d g ettin g d i® ra cted .) M a th em a tica lly, th is m ea n s th a t th e fu n ctio n F 1 (k ) is n o n zero o n ly fo r w av e v ecto rs w ith k z > 0 . F u rth er, sin ce th e w av e h a s a d e¯ n ite freq u en cy ! , th e m a g n itu d e o f th e w av e v ecto r is ¯ x ed a t th e va lu e ! = c. It fo llow s th a t o n e o f th e co m p o n en ts o f th e w av e v ecto r, say k z , ca n b e ex p ressed in term s o f th e o th er th ree. S o , th e fu n ctio n F 1 h a s th e stru ctu re µ ¶ q 2 2 2 F 1 (k z ;k ? ) = 2 ¼ f (k ? )± D k z ¡ ! = c ¡ k ? ; (2 ) w h ere th e su b scrip t ? d en o tes th e co m p o n en ts o f th e v ecto r in th e tra n sv erse x ¡ y p la npe. N o te th a t, in g en era l, w e co u ld h av e h a d k z = § ! 2 = c 2 ¡ k 2? a n d w e h av e co n scio u sly p ick ed o u t o n e w ith k z > 0 . S u b stitu tin g th is ex p ressio n in (1 ) w e ¯ n d th a t A (t;z ;x ? ) ca n b e w ritten in th e fo rm a (z ;x ? )e ¡ i! t (in w h ich w e h av e sep a ra ted o u t th e o scilla tio n s in tim e) w h ere a (z ;x ? ) =
Z
· q ¸ iz d2k ? ik ? ¢x ? 2 2 2 f (k ? )e ! ¡ c k? : ex p c (2 ¼ )2
(3 )
S in ce th e tim e v a ria tio n o f a m o n o ch ro m a tic w av e is a lw ay s ex p (¡ i! t), w e sh a ll ig n o re th is fa cto r a n d co n cen tra te o n th e sp a tia l d ep en d en ce o f th e a m p litu d e, a (z ;x ? ). T o p ro ceed fu rth er w e sh a ll con sid er th e ca se in w h ich a ll th e co m p o n en ts b u ild in g u p th e w av e a re trav ellin g
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essen tia lly a lo n g th e p o sitiv e z-a x is w ith a sm a ll tra n sv erse sp rea d . F o r su ch a w av e trav ellin g , b y a n d la rg e, a lo n g th e z d irectio n , th e tra n sv erse co m p o n en ts o f k a re sm a ll co m p a red to its m a g n itu d e; th a t is, c 2 k ?2 ¿ ! 2 . U sin g th e T ay lo r series ¶ µ q 2 2 c k 1 c 2 k ?2 1 ? 2 2 2 » ! ¡ c k? = ! 1 ¡ ; (4 ) = ! ¡ 2 !2 2 ! in (3 ), w e ¯ n d th a t a (z ;x ? ) ´ e
i! z = c
Z
£¡ ¢¤ d2k ? f (k ? ) ex p i k ? ¢ x ? ¡ (c= 2 ! )k ?2 z : 2 (2 ¼ ) (5 )
T h is eq u a tio n d escrib es th e p ro p a g a tio n o f a w av e a lo n g th e p o sitiv e z -a x is w ith a sm a ll sp rea d in th e tra n sv erse d irectio n . T h e fu n ctio n f (k ? ) ca n b e d eterm in ed b y a sim p le F o u rier tra n sfo rm if th e a m p litu d e a (z 0;x 0? ) is g iv en a t so m e lo ca tio n z 0. D o in g th is, w e ca n rela te th e a m p litu d es o f th e w av e a t tw o p la n es w ith co o rd in a tes z a n d z 0b y a (z ;x ? ) = e
i! (z ¡ z 0)= c
Z
d 2 x 0? a (z 0;x 0? ) G (z ¡ z 0;x ? ¡ x 0? ) ; (6 )
w h ere G (z ¡ z 0;x ? ¡ x 0? ) Z 2 d k ? ik ? ¢(x ? ¡x 0? ) ¡ (ic= 2 ! )k ?2 (z ¡z 0) e e = (2 ¼ )2 " # ³ ! ´ 1 i! (x ? ¡ x 0? )2 : = ex p 2 ¼ ic jz ¡ z 0j 2 c (z ¡ z 0)
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T h e fu n ctio n G m ay b e th o u g h t o f a s a p ro p a g a to r w h ich p ro p a g a tes th e a m p litu d e fro m th e lo ca tio n (z 0;x 0? ) to 0 th e lo ca tio n (z ;x ? ). T h e fa cto r e i! (z ¡z )= c in (6 ) d o es n o t co n trib u te to th e in ten sity a n d w e w ill d ro p it w h en n o t n ecessa ry.
The actual form of
A little th o u g h t sh ow s th a t w e h av e a ch iev ed so m eth in g q u ite in terestin g . W e k n ow th a t th e a m p litu d e sa tis¯ es a seco n d o rd er d i® eren tia l eq u a tio n (v iz. th e w av e eq u a tio n ) a n d h en ce its ev o lu tio n ca n n o t b e d eterm in ed b y ju st k n ow in g th e a m p litu d e (ie., o n e sin g le fu n ctio n , a (z 0;x 0? )) a t a g iv en lo ca tio n (z 0;x 0? ). T h is co u ld b e d o n e in (6 ) o n ly b eca u se o f th e a ssu m p tio n th a t th e w av e is trav ellin g essen tia lly fo rw a rd in th e z d irectio n . T h e a ctu a l fo rm o f th e p ro p a g a to r d ep en d s o n th e a ssu m p tio n th a t th e tra n sv erse co m p o n en ts o f th e w av e v ecto r a re sm a ll co m p a red to k z . T h e stu d y o f w av e p ro p a g a tio n u n d er th ese a p p rox im a tio n s is ca lled paraxial optics. (W e sh a ll see in th e n ex t in sta llm en t th a t a ll th ese ex p ressio n s h av e in terestin g co n n ectio n s w ith th e p a th in teg ra l p ro p a g a to r in q u a n tu m m ech a n ics { w h ich w ill em erg e a s th e p a ra x ia l o p tics o f rela tiv istic ¯ eld th eo ry !)
components of the
the propagator depends on the assumption that the transverse wave vector are small compared to kz. The study of wave propagation under these approximations is called paraxial optics.
L et u s ta k e a clo ser lo o k a t th e stru ctu re o f th e p ro p a g a to r G . It in tro d u ces a fa cto r jz ¡ z 0j¡1 to th e a m p litu d e a n d , m o re im p o rta n tly, co n trib u tes a n a m o u n t ! (x ? ¡ x 0? )2 Á = 2 c (z ¡ z 0)
(8 )
to th e p h a se. T h e ch a n g e in th e a m p litu d e m erely re° ects r ¡2 fa ll-o ® o f th e in ten sity (w h ich is p ro p o rtio n a l to th e sq u a re o f th e a m p litu d e) o f th e w av e. B u t w h a t d o es th e p h a se fa cto r m ea n ? T o u n d ersta n d th e o rig in o f th e ch a n g e in p h a se, n o te th a t a p a th d i® eren ce ¢ s b etw een tw o p o in ts in sp a ce w ill in tro d u ce a p h a se d ifferen ce o f k ¢ s in a p ro p a g a tin g w av e. In o u r ca se, it is
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clea r th a t th e p h a se d i® eren ce is ·q ¸ ! 2 0 0 2 k¢ s = (x ? ¡ x ? ) + (z ¡ z 0) ¡ (z ¡ z ) c " # 0 2 »= ! 1 (x ? ¡ x ? ) ; (9 ) c 2 (z ¡ z 0) p rov id ed th e tra n sv erse d isp la cem en ts a re sm a ll co m p a red to th e lo n g itu d in a l d ista n ce { a n a ssu m p tio n w h ich is cen tra l to p a ra x ia l o p tics. W ith h in d sig h t w e co u ld h av e g u essed th e fo rm o f G w ith o u t d o in g a n y a lg eb ra ! In p a ra x ia l o p tics, it in tro d u ces a p h a se co rresp o n d in g to th e p a th d i® eren ce a n d d ecrea ses th e a m p litu d e to ta k e in to a cco u n t th e n o rm a l sp rea d o f th e w av e. E q u a tio n (6 ) a llow s o n e to co m p u te th e w av e a m p litu d e a t a n y lo ca tio n o n th e p la n e z = z 2 , if th e a m p litu d e o n a p la n e z = z 1 < z 2 is g iv en . T o see it in a ctio n , let u s a p p ly it to a sta n d a rd situ a tio n , w h ich a rises q u ite o ften in o p tics. A w av e fro n t p ro p a g a tes freely u p to a p la n e z = z 1 w h ere it p a sses th ro u g h a n o p tica l sy stem (say a len s, screen w ith a h o le, a tm o sp h ere, etc.) w h ich m o d i¯ es th e w av e in a p a rticu la r fa sh io n . T h e o p tica l sy stem ex ten d s fro m z = z 1 to z = z 2 a n d th e w av e p ro p a g a tes freely fo r z > z 2 . W e w ill b e in terested in th e a m p litu d e a t z > z 2 , g iv en th e a m p litu d e a t z < z 1 . It is clea r th a t o u r eq u a tio n (6 ) ca n b e u sed to p ro p a g a te th e a m p litu d e fro m so m e in itia l p la n e z = z O < z 1 to z = z 1 a n d fro m z = z 2 to so m e ¯ n a l p la n e z = z I > z 2 . (T h e su b scrip ts O a n d I sta n d fo r o b ject a n d im a g e, b a sed o n th e id ea o f th e o p tica l sy stem b ein g a len s). T h e p ro p a g a tio n o f w av e fro m z 1 to z 2 d ep en d s en tirely o n th e o p tica l sy stem a n d , in fa ct, d e¯ n es th e p a rticu la r o p tica l sy stem . A n o p tica l sy stem is ca lled lin ear if th e o u tp u t is lin ea r in in p u t. In su ch a ca se, th e a m p litu d es a t th e ex it p o in t o f th e o p tica l sy stem is rela ted to th e a m p litu d e a t th e en tra n ce p o in t b y a rela tio n o f th e k in d Z a (z 2 ;x 2 ) = d 2 x 1 P (z 2 ;z 1 ;x 2 ;x 1 ) a (z 1 ;x 1 ) ; (1 0 )
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w h ere th e fu n ctio n a l fo rm o f P d e¯ n es th e k in d o f o p tica l sy stem . (H ere a n d in w h a t fo llow s, w e sh a ll o m it th e su b scrip t ? w ith th e u n d ersta n d in g th a t th e v ecto r x is in th e tra n sv erse p la n e a n d is tw o d im en sio n a l.) In th is ca se, th e a m p litu d e a t th e im a g e p la n e ca n b e ex p ressed in term s o f th e a m p litu d e a t th e o b ject p la n e b y th e rela tio n Z a (z I;x I ) = d 2 x O G (z I ;z O ;x I ;x O ) a (z O ;x O ) ; (1 1 ) w h ere G (z I ;z O ;x I;x O ) Z = d 2 x 2 d 2 x 1 G (z I ¡ z 2 ;x I ¡ x 2 ) P (z 2 ;z 1 ;x 2 ;x 1 ) £ G (z 1 ¡ z O ;x 1 ¡ x O ) :
(1 2 )
G iv en th e p ro p erties o f a n y lin ea r o p tica l sy stem , o n e ca n co m p u te th e q u a n tity P , a n d th u s eva lu a te G a n d d eterm in e th e p ro p erties o f w av e p ro p a g a tio n . A s a sim p le ex a m p le let u s ¯ n d o u t th e fo rm o f th e fu n ctio n P fo r a co n v ex len s. If th e len s is su ± cien tly th in , P w ill b e n o n zero o n ly a t th e p la n e o f th e len s z 2 = z 1 = z L . S in ce th e len s d o es n o t a b so rb ra d ia tio n , it ca n n o t ch a n g e th e a m p litu d e ja (z L ;x L )j o f th e in cid en t w av e a n d ca n o n ly m o d ify th e p h a se. T h erefo re, P m u st h av e th e fo rm P = ex p [iµ (x L )]. T h en th e a m p litu d e a t th e im a g e p la n e is g iv en b y a (z I ;x I ) Z = d 2 x L a (z L ;x L ) P (z L ;x L ) G (z I ¡ z L ;x I ¡ x L ) Z = a d 2 x L e iµ (z L ;x L ) G (z I ¡ z L ;x I ¡ x L ) ; (1 3 ) w h ere w e h a v e u sed th e fa ct th at th e a m p litu d e a (z L ;x L ) o n th e len s p la n e is co n sta n t fo r a p la n e w av e in cid en t fro m a la rg e d ista n ce. T o d eterm in e th e fo rm o f µ (x L ),
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w e u se th e b a sic d e¯ n in g p ro p erty o f a len s o f fo ca l len g th f : If a p la n e w av efro n t o f co n sta n t in ten sity is in cid en t o n th e len s p la n e z = z L , th e ray s w ill b e fo cu sed a t a p o in t z I = z L + f , w h en th e w av e n a tu re o f th e lig h t is ig n o red . In th e lim it o f zero w av elen g th fo r th e w av e, m o st o f th e co n trib u tio n to th e in teg ra l in (1 3 ) co m es fro m p o in ts a t w h ich th e p h a se o f th e in teg ra n d is sta tio n a ry. S in ce th e p h a se o f G is (k = 2 )[(¢ x )2 = ¢ z ], th e p rin cip le o f sta tio n a ry p h a se g iv es th e eq u a tio n @µ k = (x I ¡ x L ) ; @xL f
(1 4 )
w h ere f = z I ¡ z L . F o r th e im a g e to b e fo rm ed a lo n g th e z -a x is, th is eq u a tio n sh o u ld b e sa tis¯ ed fo r x I = 0 . S ettin g x I = 0 a n d in teg ra tin g th is eq u a tio n w e ¯ n d th a t µ = (¡ k x 2L = 2 f ) a n d ¶ µ ik 2 x : P (x L ) = ex p ¡ (1 5 ) 2f L T h u s th e e® ect o f a len s is to in tro d u ce a p h a se va ria tio n w h ich is q u a d ra tic in th e tra n sv erse co o rd in a tes. S u ch a len s w ill fo cu s th e lig h t to a p o in t o n th e z -a x is, in th e lim it o f zero w av elen g th .
The effect of a lens is to introduce a phase variation which is quadratic in the transverse coordinates. Such a lens will focus the light to a point on the z-axis, in the limit of zero wavelength.
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A g eo m etrica l in terp reta tio n o f th is resu lt is g iv en in F igu re 1 . T h e co n sta n t p h a se su rfa ces a re p la n es to th e left o f th e len s a n d th ey a re a rcs o f circles (cen tered o n th e fo cu s F ) o n th e rig h t sid e o f th e len s. C h a n g in g th e p la n e to a circle (o f ra d iu s f ) a t z = z L in tro d u ces a p a th d i® eren ce o f ¢ l = [f ¡ (f 2 ¡ x 2L )1 = 2 ] ' (x 2L = 2 f ) a t a tra n sv erse d ista n ce x L . T h is co rresp o n d s to a p h a se d i® eren ce k ¢ l = (k x 2L = 2 f ) = µ in tro d u ced b y th e len s. L et u s n ex t co n sid er th e e® ect o f th is len s o n a p o in t so u rce o f ra d ia tio n a lo n g th e z -a x is a t z = z O . [T h a t is, w e ta k e th e in itia l a m p litu d e to b e a (z O ;x O ) / ± D (x O ).] T h is ca n b e o b ta in ed b y ¯ rst p ro p a g a tin g th e ¯ eld fro m z O to z L , m o d ify in g th e p h a se d u e to th e len s a t z =
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z L a n d p ro p a g a tin g it fu rth er to so m e p o in t z w ith th e tra n sv erse co o rd in a te set to zero . T h e n et resu lt is g iv en by µ ¶ Z k2 ik 2 2 a (z ;0 ) = ¡ x ¢ d x L ex p ¡ 4¼ 2u v 2f L · 2 ¸ ik x 2L ik x L ; + ex p (1 6 ) 2u 2v
Figure 1. The focusing action of a convex lens in terms of the phase change of wave fronts.
w h ere u = z L ¡ z O a n d v = z ¡ z L . In th e lim it o f zero w av elen g th (ca lled ray optics), th e m a x im u m co n trib u tio n to th is in teg ra l ca n a g a in b e o b ta in ed b y settin g th e va ria tio n o f th e p h a se to zero . T h is g iv es ¡
k k k x L + x L + x L = 0; f u v
(1 7 )
or 1 1 1 + = ; u v f
(1 8 )
w h ich is a fa m ilia r fo rm u la in th e th eo ry o f len ses. T h e a b ov e resu lt w a s o b ta in ed in th e lim it o f ray o p tics. T o stu d y th e w av e p ro p a g a tio n th ro u g h th e len s w e n o te
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th a t th e a ctio n o f a len s o n th e p h a se o f a n in itia l in ten sity d istrib u tio n is g ov ern ed b y th e in teg ra l µ ¶ Z ik 2 2 a (z ;x ) / x d x L a (z L ;x L ) ex p ¡ 2f L ¸ · ik 2 £ ex p (x ¡ x L ) ; (1 9 ) 2 (z ¡ z L )
w h ere a (z L ;x L ) is th e in cid en t a m p litu d e o n th e len s, th e ¯ rst ex p o n en tia l g iv es th e d isto rtio n in p h a se p ro d u ced b y th e len s a n d th e seco n d ex p o n en tia l g iv es th e p ro p a g a tio n a m p litu d e z L to z . O n th e fo ca l p la n e, w h ich is a p la n e lo ca ted a t a d ista n ce f fro m th e len s, a t z = z L + f , th e seco n d ex p o n en tia l ch a ra cterizin g th e p ro p a g a tio n b eco m es ¢ ik (x ¡ x L )2 ik ¡ 2 x + x 2L ¡ 2 x ¢ x L : (2 0 ) ex p = ex p 2 (z ¡ z L ) 2f T h e q u a d ra tic term (ik x 2L = 2 f ) in th e p ro p a g a tio n a m p litu d e is n ow p recisely ca n celled b y th e p h a se d isto rtio n in tro d u ced b y th e len s, so th a t th e resu lta n t a m p litu d e ca n b e w ritten a s ¶Z µ ik 2 x a (z L + f ;x ) / ex p d 2 x L a (z L ;x L ) 2f µ ¶ ik £ ex p x ¢ x L : (2 1 ) f
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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T h e in ten sity a t th e fo ca l p la n e is g iv en b y ja (z L + f ;x )j2 in w h ich th e p h a se fa cto r ex p [ik x 2 = 2 f ] d o es n o t co n trib u te. W e th en g et th e ra th er cu te resu lt th a t th e len s essen tia lly p ro d u ces { o n th e fo ca l p la n e { th e tw o d im en sio n a l F o u rier tra n sfo rm o f th e in cid en t a m p litu d e! Suggested Reading [1]
Several textbooks in optics and electromagnetic theory describe these aspects; see, for example, T Padmanabhan, Theoretical Astrophysics – Vol.I (Astrophysical Processes), Chapter 3, Cambridge University Press, 2000.
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Snippets of Physics 13. The Optics of Particles T Padmanabhan
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
T h e p ro b a b ility a m p litu d e fo r a p a rtic le to p ro p a g a te fro m e v e n t to e v e n t in sp a c e tim e sh o w s so m e n ic e sim ila ritie s w ith th e c o rre sp o n d in g p ro p a g a to r fo r th e e le c tro m a g n e tic w a v e a m p litu d e d isc u sse d in th e la st in sta llm e n t. In fa c t, th is a n a lo g y p ro v id e s a n in te re stin g in sig h t in to th e tr a n sitio n fro m q u a n tu m ¯ e ld th e o ry to q u a n tu m m e c h a n ic s! In cla ssica l m ech a n ics, th e m o tio n o f a p a rticle w ith p o sitio n x (t) u n d er th e a ctio n o f a (tim e in d ep en d en t) p o ten tia l V (x ) is d eterm in ed th ro u g h th e N ew to n 's law o f m o tio n m xÄ = ¡ r V . G iv en th e in itia l p o sitio n x (0 ) a n d v elo city v (0 ) a t t = 0 w e ca n in teg ra te th is eq u a tio n to d eterm in e th e tra jecto ry. A ll o th er p h y sica l o b serv a b les in cla ssica l m ech a n ics ca n b e o b ta in ed fro m th e tra jecto ry x (t). W h a t is th e co rresp o n d in g situ a tio n in q u a n tu m m ech a n ics? H ere th e w av efu n ctio n o f th e p a rticle à (t;x ) co n ta in s co m p lete in fo rm a tio n a b o u t th e sta te o f th e sy stem a n d sa tis¯ es th e S ch rÄo d in g er eq u a tio n @à ~2 2 i~ r à + V à ´ H à : = ¡ @t 2m
Keywords Optics, waves.
. 8
(1 )
G iv en th e w av efu n ctio n à (0 ;x ) a t t = 0 w e ca n in teg ra te th is eq u a tio n a n d o b ta in th e w av efu n ctio n a t a n y la ter tim e. B u t, u n lik e in th e ca se o f cla ssica l m ech a n ics, th ere is a n ice w ay o f sep a ra tin g th e d y n a m ica l ev o lu tio n fro m th e in itia l co n d itio n in th e ca se o f q u a n tu m m ech a n ics w h ich w e sh a ll ¯ rst d escrib e. T o k eep th e d iscu ssio n so m ew h a t g en era l w e w ill a ssu m e th a t th e
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sp a ce h a s D d im en sio n s w ith th e u su a l ch o ices b ein g D = 1 ;2 ;3 . W e k n ow th a t w h en th e p o ten tia l is in d ep en d en t o f tim e, eq u a tio n (1 ) h a s en erg y eig en sta tes w h ich sa tisfy th e eig en va lu e eq u a tio n H Á n (x ) = E n Á n (x ). U sin g th ese eig en fu n ctio n s w e ca n ex p a n d th e in itia l w av efu n ctio n à (0 ;x ) in term s o f th e en erg y eig en fu n ctio n s a s Z X à (0 ;x ) = c n Á n (x ); c n = d y à (0 ;y )Á ¤n (y ); (2 ) n
w h ere th e ex p ressio n fo r c n fo llow s fro m th e o rth o n o rm a lity o f th e en erg y eig en fu n ctio n s a n d th e sp a tia l in teg ra tio n s a re ov er th e D -d im en sio n a l sp a ce. S in ce th e en erg y eig en fu n ctio n ev o lv es in tim e w ith a p h a se fa cto r ex p (¡ iE n t= ~), it fo llow s th a t th e w av efu n ctio n a t tim e t is g iv en b y X Ã (t;x ) = c n Á n (x )e ¡ iE n t= ~ ; (3 ) n
w h ich , in p rin cip le, so lv es th e p ro b lem . A ctu a lly, w e ca n d o b etter b y ex p ressin g th e c n s in (3 ) in term s o f à (0 ;x ) u sin g th e seco n d rela tio n in (2). T h is g iv es Z X à (t;x ) = d y à (0 ;y ) Á n (x )Á ¤n (y )e ¡ iE n t= ~ ´
Z
n
d y K (t;x ;0;y )Ã (0 ;y );
(4 )
w h ere w e h av e d e¯ n ed th e fu n ctio n { u su a lly ca lled th e p ro p a g a to r o r k ern el { b y X K (t;x ;0 ;y ) = Á ¤n (y )Á n (x )e ¡iE n t= ~ : (5 ) n
It is o b v io u s th a t th e p ro p a g a to r K (t;x ;0 ;y ) ca rries co m p lete in fo rm a tio n a b o u t th e d y n a m ica l ev o lu tio n o f th e sy stem a n d ca n b e co m p u ted o n ce th e H a m ilto n ia n
RESONANCE January 2009
It is obvious that the propagator K(t,x;0,y) carries complete information about the dynamical evolution of the system.
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Curiously enough, such a separation has no direct analog in the case of classical mechanics.
H is k n ow n , in term s o f its eig en fu n ctio n s a n d th e eig en va lu es. W h a t is m o re, eq u a tio n (4 ) n icely sep a ra tes th e d y n a m ics, en co d ed in K (t;x ;0 ;y ), fro m th e in itia l co n d itio n en co d ed in à (0 ;y ). C u rio u sly en o u g h , su ch a sep a ra tio n h a s n o d irect a n a lo g in th e ca se o f cla ssica l m ech a n ics. S in ce à (0 ;y ) g iv es th e a m p litu d e to ¯ n d th e p a rticle a ro u n d y a t t = 0 , it fo llow s th a t K (t;x ;0 ;y ) ca n b e th o u g h t o f a s th e p ro b a b ility a m p litu d e fo r a q u a n tu m p a rticle to p ro p a g a te fro m th e ev en t (0 ;y ) to th e ev en t (t;x ). O f co u rse, sin ce th e p o ten tia l is in d ep en d en t o f tim e, w e ca n u se tim e tra n sla tio n in va ria n ce to w rite a n ex p ressio n fo r K (t2 ;x 2 ;t1 ;x 1 ) b y rep la cin g E n t= ~ in (5 ) b y (E n = ~)(t2 ¡ t1 ). F o r th is in terp reta tio n to b e va lid th e p ro p a g a to r m u st sa tisfy th e in teg ra l co n d itio n : Z K (t3 ;x 3 ;t1 ;x 1 ) = d x 2 K (t3 ;x 3 ;t2 ;x 2 )K (t2 ;x 2 ;t1 ;x 1 ): (6 )
U sin g th e d e¯ n itio n in (5 ) a n d th e o rth o n o rm a lity o f eig en fu n ctio n s, y o u ca n p rov e th a t th is is in d eed tru e. N o te th a t th is is a n o n triv ia l co n d itio n : A t th e in term ed ia te ev en t w e in teg ra te o n ly ov er x 2 leav in g t2 a lo n e; n ev erth eless, th e ¯ n a l resu lt { a n d th e left h a n d sid e { is in d ep en d en t o f t2 . S o th e p ro p a g a to r a ctu a lly p ro p a g a tes th e p a rticle fro m ev en t to ev en t. Note that equation (6) is a nontrivial condition: At the intermediate event we integrate only over x2 leaving t2 alone; nevertheless, the final result – and the left hand side – is independent of t2.
10
S in ce Á n s a re en erg y eig en fu n ctio n s, it is a lso stra ig h tfo rw a rd to v erify th a t th e p ro p a g a to r sa tis¯ es th e S ch rÄo d in g er eq u a tio n µ ¶ @ i~ ¡ H K (t;x ;0 ;y ) = 0 ; (7 ) @t w ith th e sp ecia l in itia l co n d itio n lim K (t;x ;0 ;y ) = ± D (x ¡ y ): t! 0
(8 )
T h is co n d itio n ca n a lso b e o b ta in ed ea sily fro m (4 ) b y ta k in g th e lim it o f t ! 0 . RESONANCE January 2009
SERIES ARTICLE
A fter th is p rea m b le a b o u t th e p ro p a g a to r, w e w ill tu rn to th e k ey to p ic o f th is in sta llm en t. T o in tro d u ce it, let u s w o rk o u t th e p ro p a g a to r for a free p a rticle (V = 0 ) u sin g th e ex p ressio n in (5 ). In th e ca se o f a free p a rticle th e en erg y eig en fu n ctio n s a n d eig en va lu es ca n b e ta k en to b e la b elled b y a w av en u m b er p in stea d o f a d iscrete in d ex n w ith Á p (x ) =
1 (2 ¼ )D
=2
ex p i(p ¢ x );
E
p
=
~2 p 2 : 2m
(9 )
T h e n o rm a liza tio n o f Á p (x ) is so m ew h a t a rb itra ry b u t w e u se th e co n v en tio n th a t m o m en tu m sp a ce in teg ra ls co m e w ith a m ea su re d p so th at th e o rth o n o rm a lity co n d itio n rea d s a s Z d p Á p (x )Á ¤p (y ) = ±(x ¡ y ): (1 0 ) T h e p ro p a g a to r is n ow g iv en b y a n ex p ressio n sim ila r to (5 ) b u t w ith a n in teg ra l ov er p ra th er th a n a su m ov er th e d iscrete in d ex n . H en ce w e g et Z d p ip ¢(x ¡y )= ~ ¡ip 2 ~t= 2 m K (t;x ;0 ;y ) = e e (2 ¼ )D · ¸ ³ m ´D = 2 im (x ¡ y )2 = ex p (;1 1 ) ~ 2 ¼ i~t 2t w h ere D is th e d im en sio n o f sp a ce (1 , 2 o r 3 ) in w h ich th e p a rticle is m ov in g . (T h e in teg ra l is ju st th e D d im en sio n a l F o u rier tra n sfo rm o f a G a u ssia n w h ich sep a ra tes o u t in ea ch o f th e d im en sio n s.) W e ca n v erify d irectly th a t K (t;x ;0 ;y ) sa tis¯ es (7 ) a n d (8 ). It is a lso o b v io u s th a t it sa tis¯ es th e \ n o rm a liza tio n co n d itio n " Z d x K (t;x ;0 ;y ) = 1 : (1 2 ) U su a lly in q u a n tu m m ech a n ics w e n o rm a lize th e p ro b a b ilities a n d n o t th e p ro b a b ility a m p litu d es. B u t th is is a n ex cep tio n to th e ru le in w h ich p ro b a b ility a m p litu d e
RESONANCE January 2009
Equation (12) is an exception to the rule in which probability amplitude for propagation from event to event comes out normalized to unity.
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fo r p ro p a g a tio n fro m ev en t to ev en t co m es o u t n o rm a lized to u n ity. T h ere is a co n v en tio n a l in terp reta tio n o f th e p h a se o f th e p ro p a g a to r in (1 1 ) w h ich w e w ill n ow d escrib e. T o d o th is, let u s co n sid er th e a ctio n fo r th e free p a rticle in cla ssica l m ech a n ics g iv en b y Zt m A (t;x ;0 ;y ) = x j2 ; d t j_ (1 3 ) 2 0 w h ich is d e¯ n ed fo r a n y tra jectory x (t) th a t co n n ects th e tw o en d p o in ts. In p a rticu la r, w e ca n d eterm in e th e cla ssica l tra jecto ry x c (t) fro m ex trem isin g th e a ctio n a n d eva lu a te th e cla ssica l va lu e o f th e a ctio n A c (t;x ;0 ;y ) fo r th is p a rticu la r cla ssica l tra jecto ry. F o r th e free p a rticle, th is is triv ia l to eva lu a te a n d is g iv en b y A c (t;x ;0 ;y ) =
m (x ¡ y )2 ; t 2
(1 4 )
so th a t th e p ro p a g a to r in (1 1 ) ca n b e ex p ressed in th e fo rm · ¸ i K (t;x ;0 ;y ) = N (t) ex p A c (t;x ;0 ;y ) : (1 5 ) ~ W e see th a t th e p h a se o f th e p ro p a g a to r ca n b e in terp reted a s ju st th e cla ssica l va lu e o f th e a ctio n d iv id ed b y ~.
We see that the phase of the propagator can be interpreted as just the classical value of the action divided by ~).
12
T h e situ a tio n is a ctu a lly b etter th a n th is. L et u s co n sid er, in stea d o f th e cla ssica l p a th , a n y a rb itra ry p a th co n n ectin g th e tw o ev en ts (la b elled 1 a n d 2 ) w e a re in terested in . S in ce o n e ca n n o t m ea su re p o sitio n a n d v elo city o f a p a rticle sim u lta n eo u sly in q u a n tu m m ech a n ics, it d o es n o t m a k e sen se to say th a t th e p a rticle w en t fro m o n e p o in t to a n o th er a lo n g a p a rticu la r tra jecto ry. T h e b est w e ca n say is th a t th ere is so m e a m p litu d e P (2 ;1 jx (t)) ´ P (t2 ;x 2 ;t1 ;x 1 jx (t)) fo r th e p a rticle to ch o o se a p a rticu la r p a th x (t). W e n ow p o stu la te, fo llo w in g D ira c a n d F ey n m a n , th a t th is a m p litu d e is g iv en b y
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P (2 ;1 jx (t)) = ex p (iA [2 ;1 jx (t)]= ~). T h en th e to ta l a m p litu d e fo r p ro p a g a tio n fro m ev en t 1 to ev en t 2 (w h ich , o f co u rse, is o u r p ro p a g a to r) m u st b e g iv en b y X X K (2 ;1 ) = P (2 ;1 jx (t)) = ex p (iA [2 ;1 jx (t)]= ~); p a th s
x (t)
(1 6 )
w h ere th e su m m a tio n sy m b o l in d ica tes th a t w e h av e to su m ov er a ll p a th s x (t) co n n ectin g th e tw o ev en ts. In g en era l, it is n o t ea sy to d e¯ n e a n d eva lu a te th is su m b u t so m eth in g in terestin g h a p p en s if th e a ctio n co n ta in s n o term s w h ich a re m o re th a n q u a d ra tic in v elo city o r p o sitio n . In th ese ca ses, w e b eg in b y w ritin g a n y a rb itra ry p a th { ov er w h ich w e h av e to su m { in term s o f th e cla ssica l p a th x c (t) p lu s a d ev ia tio n fro m it: th a t is, w e w rite x (t) = x c (t) + r (t). S u m m in g ov er a ll x (t) is th e sa m e a s su m m in g ov er a ll r(t) b u t th e `b o u n d a ry co n d itio n s' o n r (t) a re ea sier to h a n d le. S in ce b o th th e cla ssica l p a th x c (t) a n d a n y a rb itra ry p a th x (t) co n n ect th e sa m e en d p o in ts, th e r (t) m u st va n ish a t th e en d p o in ts. F u rth er, if th e a ctio n h a s o n ly u p to q u a d ra tic term s in x (t), th en it sp lits u p a s th e su m : A [x (t)] = A [x c (t) + r (t)] = A [x c (t)] + A lin [x c (t);r(t)] + A
q u a d [r(t)];
In general, it is not easy to define and evaluate the sum in (16) but something interesting happens if the action contains no terms which are more than quadratic in velocity or position.
(1 7 )
w h ere A lin is lin ea r in x (t) a n d r(t) w h ile A q u a d is q u a d ra tic in r (t). (T h is is essen tially p a ra p h ra sin g th e fo rm u la fo r (a + b)2 !). B u t reca ll th a t th e cla ssica l p a th is a n ex trem u m o f th e a ctio n ; so th e ch a n g e in a ctio n w h en th e p a th ch a n g es b y a d ev ia tio n r(t) h a s to b e to q u a d ra tic o rd er in r . T h erefo re, A lin = 0 a n d th e su m ov er p a th s in (1 6 ) ca n b e w ritten a s: X K (2 ;1 ) = ex p (iA [2 ;1 jx (t)]= ~) x (t)
=
ex p (iA [2 ;1 jx c (t)]= ~)
RESONANCE January 2009
X
ex p (iA
q u a d [r(t)]= ~):(1 8 )
r(t)
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The sum over r(t) in (18) (which we haven’t even defined properly!) can only be a function of t2,t1. We get the neat result that for all actions which have only up to quadratic terms in their variables, the
T h e cru cia l p o in t is th a t th e su m ov er r(t) is d o n e w ith th e co n d itio n r(t2 ) = r(t1 ) = 0 a n d h en ce it d o es n o t d ep en d o n x 1 ;x 2 . S o th e su m ov er r(t) in (1 8 ) (w h ich w e h av en 't ev en d e¯ n ed p ro p erly !) ca n o n ly b e a fu n ctio n o f t2 ;t1 . W e g et th e n ea t resu lt th a t fo r a ll a ctio n s w h ich h av e o n ly u p to q u a d ra tic term s in th eir va ria b les, th e p ro p a g a to r h a s th e fo rm in (1 5 ). (In fa ct, o n e ca n a lso d eterm in e N (t2 ;t1 ) u sin g th e co n d itio n (6 ) b u t w e w o n 't g o in to th a t.). F o r a ll th ese ca ses, o f w h ich th e free p a rticle is a sp ecia l ca se, th e p h a se o f th e p ro p a g a to r is ju st th e cla ssica l a ctio n .
propagator has the form in (15).
1
See equation (7) in Resonance, Vol.13, p.1100, December 2008.
T h is is a ll n ice b u t in n o n -rela tiv istic m ech a n ics th e a ctio n fu n ctio n a l in (1 3 ) h a s n o sim p le g eo m etrica l in terp reta tio n . W e w ill n ow p rov id e a n a ltern a tiv e p ersp ectiv e o n th e p h a se o f th e p ro p a g a to r w h ich w ill lea d to a g eo m etric in sig h t. T h e m o st rem a rka b le fea tu re a b o u t th e p ro p a g a to r in (1 1 ), fro m th is a ltern a tiv e p ersp ectiv e, is th a t w e h av e a lrea d y seen th is ex p ressio n in th e la st in sta llm en t in co n n ectio n w ith th e p ro p a g a tio n o f electro m a g n etic w a v es a lo n g th e z -d irectio n ! T h ere w e h a d th e ex p ressio n 1 fo r a p ro p a g a to r w h ich is rep ro d u ced h ere fo r y o u r co n v en ien ce: G (z ¡ z 0;x ? ¡ x 0? ) = ³ ! ´ 1 ex p 2 ¼ ic jz ¡ z 0j
" # i! (x ? ¡ x 0? )2 : 2 c (z ¡ z 0)
(1 9 )
C o m p a rin g (1 9 ) w ith (1 1 ) w e see th e fo llow in g co rresp o n d en ce. T h e (z ¡ z 0)= c, w h ich is th e tim e o f lig h t trav el a lo n g th e z -a x is (a lo n g w h ich th e w av e is p ro p a g a tin g ) is a n a lo g o u s to tim e t in q u a n tu m m ech a n ics. T h e tw o tra n sv erse sp a tia l d irectio n s in th e ca se o f electro m a g n etic w av e p ro p a g a tio n a re a n a lo g o u s to th e sp a tia l co o rd in a tes in q u a n tu m m ech a n ics in 2 -d im en sio n s; so w e ca n set D = 2 in (1 1 ). T h e freq u en cy sh o u ld g et m a p p ed to th e rela tio n ~! = m c 2 w h ich is essen tia lly
14
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th e freq u en cy a sso cia ted w ith th e C o m p to n w av elen g th o f th e p a rticle. T h is w ill m a k e th e p ro p a g a to rs id en tica l! O b v io u sly, th is d eserv es fu rth er p ro b in g esp ecia lly sin ce th e co rresp o n d en ce b rin g s in a c fa cto r w h en w e th o u g h t w e a re d o in g n o n rela tiv istic q u a n tu m m ech a n ics.
Obviously, this
In th e ca se o f th e p ro p a g a tio n o f electro m a g n etic w av e a m p litu d e, w e w ere p ro p a g a tin g it a lo n g th e p o sitiv e z -d irectio n w ith x a n d y a ctin g a s tw o tra n sv erse d irectio n s. In th e ca se o f q u an tu m m ech a n ics, w e a re p ro p a g a tin g th e a m p litu d e fo r a p a rticle a lo n g th e p o sitiv e t-d irectio n w ith a ll th e sp a tia l co o rd in a tes a ctin g a s `tra n sv erse d irectio n s'. In th e la n g u a g e o f p a ra x ia l o p tics, th e sp ecia l a x is is a lo n g th e tim e d irectio n in q u a n tu m m ech a n ics.
brings in a c factor
deserves further probing especially since the correspondence when we thought we are doing nonrelativistic quantum mechanics.
B u t w e k n ow th a t p a ra x ia l op tics is ju st a n a p p rox im a tio n to a m o re ex a ct p ro p a g a tio n in term s o f th e w av e eq u a tio n . In th e w av e eq u a tio n fo r th e electro m a g n etic w av e, th e th ree co o rd in ates (x ;y ;z ) a p p ea r q u ite sy m m etrica lly a n d to o b ta in p a ra x ia l lim it, w e ch o o se o n e a x is (z -a x is) a s sp ecia l a n d p ro p a g a te th e a m p litu d e a lo n g th e p o sitiv e d irectio n . T h is is w h y th e p ro p a g a to r in (1 9 ) h a s th e x ;y co o rd in a tes a p p ea rin g d i® eren tly co m p a red to th e z -a x is. D o in g a b it o f rev erse en g in eerin g w e ca n a sk th e q u estio n : If th e q u a n tu m m ech a n ica l p ro p a g a to r is so m e k in d o f p a ra x ia l o p tics lim it o f a m o re ex a ct th eo ry, w h a t w ill it b e? A n o b v io u s w ay to ex p lo re th e situ a tio n is to resto re th e sy m m etry b etw een z a n d x ;y in o p tics a n d , sim ila rly, resto re th e sy m m etry b etw een t a n d x in q u a n tu m m ech a n ics. W e ca n d o th is if w e reca ll th e in terp reta tio n o f th e p h a se a s d u e to th e p a th d i® eren ce in th e ca se o f electro m a g n etic w av e. T h e releva n t eq u a tio n 2 is a g a in rep ro d u ced b elow : ¸ ·q ! 2 0 0 2 0 k¢ s = (x ? ¡ x ? ) + (z ¡ z ) ¡ (z ¡ z ) c
RESONANCE January 2009
2 See equation (9) in Resonance, Vol.13, p.1103, December 2008.
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»= ! c
" # 1 (x ? ¡ x 0? )2 : 2 (z ¡ z 0)
(2 0 )
W e u se th e fa ct th a t a p a th d i® eren ce ¢ s b etw een tw o p o in ts in sp a ce w ill in tro d u ce a p h a se d i® eren ce o f k ¢ s in a p ro p a g a tin g w av e. T h e p a ra x ia l o p tics resu lts w h en th e tra n sv erse d isp la cem en ts a re sm a ll co m p a red to th e lo n g itu d in a l d ista n ce. T a k in g a cu e fro m th is, let u s co n stru ct th e q u a n tity o ¤ l(t;x ;0 ;y ) ¡ ct m c n£ 2 2 2 1=2 ´ c t ¡ (x ¡ y ) ¡ ct ; ¸ ~ (2 1 ) w h ere l(t;x ;0 ;y ) is th e sp ecia l rela tiv istic sp a cetim e in terva l b etw een th e tw o ev en ts. W e a re su b tra ctin g fro m it th e `p a ra x ia l d ista n ce' ct a lo n g th e tim e d irectio n a n d d iv id in g b y ¸ ´ (~= m c) w h ich is th e C o m p to n w av elen g th o f th e p a rticle. T h is is ex a ctly th e co n stru ctio n su g g ested b y th e co rresp o n d en ce b etw een (1 9 ) a n d (1 1 ), d iscu ssed p rev io u sly, ex cep t fo r u sin g th e sp ecia l rela tiv istic lin e in terva l, w ith a m in u s sig n b etw een sp a ce a n d tim e. T h e p a ra x ia l lim it n ow a rises a s th e n o n rela tiv istic lim it o f th is ex p ressio n in (2 1 ) w h en c ! 1 ; th is is g iv en b y l ¡ ct » m (x ¡ y )2 ; = ¡ ¸ ~t 2 We are subtracting from l(t, x; 0, y) the ‘paraxial distance’ ct along the time direction and dividing by
(~ / mc) which is the Compton wavelength of the particle.
16
(2 2 )
w h ich is p recisely th e p h a se o f th e p ro p a g a to r in (1 1 ) ex cep t fo r a sig n . S o th e p ro p a g a to r ca n b e th o u g h t o f a s th e n o n rela tiv istic lim it o f th e fu n ctio n µ · ¸¶ l(t;x ;0 ;y ) i(m c 2 = ~)t K (t;x ;0 ;y ) = N (t)e : ex p ¡ i ¸ (2 3 ) S o th e p h a se o f th e p ro p a g a to r is ju st th e p ro p er d ista n ce b etw een th e tw o ev en ts, in u n its o f th e C o m p to n
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w av elen g th , ju st a s th e p h a se in th e ca se o f th e electro m a g n etic w av e p ro p a g a to r is th e p a th len g th in u n its o f th e w av elen g th . (T h e ex tra fa cto r (m c 2 = ~)t d o es n o t co n trib u te to th e p ro p a g a tio n in teg ra l in (4 ) a n d g o es fo r a rid e; w e ca n ig n o re it b u t if y o u w a n t y o u ca n a lso th in k o f it a s a risin g fro m th e en erg y b ein g sh ifted b y m c 2 .). W e ca n th in k o f th e p a th d i® eren ce b etw een a stra ig h t p a th a lo n g th e tim e d irectio n (w ith x = y ) a n d a n o th er sp eci¯ ed p a th a s co n trib u tin g a p h a se l= ¸ to th e p ro p a g a to r. T h is g eo m etric in terp reta tio n is lo st fo r th e p h a se in th e p a ra x ia l lim it (in th e ca se o f electro m a g n etic th eo ry ) a n d in th e n o n rela tiv istic lim it (in th e ca se o f a p a rticle).
This geometric interpretation is lost for the phase in the paraxial limit (in the case of electromagnetic theory) and in the nonrelativistic limit (in the case of a particle).
T h is ex ten sio n su g g ests th a t th e p h a se in th e rela tiv istic ca se ca n b e rela ted to th e co rresp o n d in g a ctio n . T h e a ctio n fo r a free p a rticle in sp ecia l rela tiv ity is g iv en b y ¶1 = 2 Zt µ v2 2 A R (t;x ;0 ;y ) = ¡ m c : dt 1 ¡ 2 (2 4 ) c 0 O n ce a g a in , ev a lu a tin g th is fo r a rela tiv istic cla ssica l tra jecto ry w e g et · ¸1 = 2 (x ¡ y )2 c 2 A R (t;x ;0 ;y ) = ¡ m c t 1 ¡ c 2 t2 £ ¤1 = 2 ; (2 5 ) = ¡ m c c 2 t2 ¡ (x ¡ y )2
w h ich is essen tia lly th e in terva l b etw een th e tw o ev en ts in th e sp a cetim e. T h is su g g ests ex p ressin g th e p ro p a g a to r fo r th e rela tiv istic free p a rticle in th e fo rm µ c ¶ iA R im c 2 t K (t;x ;0 ;y ) = N (t) ex p : + (2 6 ) ~ ~
T h is resu lt is tru e b u t o n ly in a n a p p rox im a te sen se to th e lea d in g o rd er; th e a ctu a l p ro p a g a to r fo r a p a rticle in rela tiv istic q u a n tu m th eo ry tu rn s o u t to b e m o re co m p lica ted . T h is is b eca u se th e a ctio n in (2 4 ) fo r th e rela tiv istic p a rticle is n o t q u a d ra tic a n d o u r p rev io u s resu lt
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It is the second interpretation which makes contact with optics so clear and is lacking when we do non-relativistic quantum mechanics.
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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in (1 8 ) d o es n o t h o ld . B u t, to th e lea d in g o rd er, a ll o f it h a n g s to g eth er v ery n icely. T h e p h a se o f th e p ro p a g a to r is in d eed th e va lu e o f th e cla ssica l a ctio n d iv id ed b y ~ a n d it is a lso g iv en b y th e ra tio o f th e sp a cetim e in terva l b etw een th e ev en ts a n d th e C o m p to n w av elen g th . It is th e seco n d in terp reta tio n w h ich m a k es co n ta ct w ith o p tics so clea r a n d is la ck in g w h en w e d o n o n -rela tiv istic q u a n tu m m ech a n ics. T h ere is a ctu a lly a va lid m a th em a tica l rea so n fo r th is to h a p p en w h ich ca n b e d escrib ed q u a lita tiv ely a s fo llow s: T h e S ch rÄo d in g er eq u a tio n d escrib in g th e n o n -rela tiv istic p a rticle in v o lv es ¯ rst d eriva tive w ith resp ect to tim e b u t seco n d d eriva tiv e w ith resp ect to sp a tia l co o rd in a tes. T h is w o rk s in n o n -rela tiv istic m ech a n ics in w h ich tim e is sp ecia l a n d a b so lu te. In co n tra st, in rela tiv istic th eo ries, w e trea t tim e a n d sp a ce a t a m o re sy m m etric fo o tin g a n d u se a w av e eq u a tio n in w h ich seco n d d eriva tiv es w ith resp ect to tim e a lso a p p ea r. T h e so lu tio n s to su ch a n eq u a tio n w ith a llow p ro p a g a tio n o f a m p litu d es b o th fo rw a rd a n d b a ck w a rd in tim e co o rd in a te ju st a s it a llo w s p ro p a g a tio n b o th fo rw a rd s a n d b a ck w a rd s in sp a tia l co o rd in a tes. W hen on e takes the n on -relativistic lim it of the ¯ eld theory, w e select ou t the m odes w hich on ly propagate forw ard in tim e. T h is is ex a ctly in a n a lo g y w ith p a ra x ia l o p tics w e stu d ied in th e la st in sta llm en t. T h e b a sic eq u a tio n fo r electro m a g n etic w av e w ill a llow p ro p a g a tio n in b o th p o sitiv e z -d irectio n a s w ell a s n eg a tiv e z -d irectio n . B u t, w h en w e co n sid er a sp eci¯ c co n tex t o f p a ra x ia l o p tics (fo r ex a m p le, a b ea m o f lig h t h ittin g a co u p le o f slits in a screen a n d fo rm in g a n in terferen ce p a ttern o r lig h t p ro p a g a tin g th ro u g h a len s a n d g ettin g fo cu sed ), w e select o u t th e m o d es w h ich a re p ro p a g a tin g in th e p o sitiv e z -d irectio n . It is th erefo re n o w o n d er th a t th e p ro p a g a to r in n o n -rela tiv istic q u a n tu m m ech a n ics is m a th em a tica lly id en tica l to th a t in p a ra x ia l o p tics!
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Snippets of Physics 14. The Power of Nothing T Padmanabhan
T h e v a c u u m sta te o f th e e le c tr o m a g n e tic ¯ e ld is fa r fr o m tr iv ia l. A m o n g st o th e r th in g s it c a n e x e rt fo r c e s th a t a r e m e a su r a b le in th e la b , in a c u r io u s p h e n o m e n o n k n o w n a s C a sim ir e ® ec t. W e a ll k n ow th a t cla ssica l electro m a g n etic ¯ eld s ca n ex ert fo rces o n ch a rg ed p a rticles. T h e sta n d a rd ex p ressio n fo r th e cla ssica l fo rce is q (E + v £ B ) w h ich , o f co u rse, va n ish es w h en E = B = 0 . T h a t so u n d s em in en tly rea so n a b le. B u t th en , w e k n ow th a t th e rea l w o rld is q u a n tu m m ech a n ica l in ch a ra cter a n d n o t cla ssica l, w h ich im p lies th a t w e n eed to trea t th e electro m a g n etic ¯ eld a s a q u a n tu m en tity. W h en w e d o th a t, p h o to n s em erg e a s th e q u a n ta o f electro m a g n etic ¯ eld . A s in th e ca se o f a n y o th er q u a n tu m sy stem , e.g ., th e h y d ro g en a to m , o n e ca n d escrib e th e p h y sics in term s o f th e q u a n tu m sta tes o f th e electro m a g n etic ¯ eld . In th is la n g u a g e, o n e ca n a lso d e¯ n e a sta te w ith zero p h o to n s w h ich co u ld b e th o u g h t o f a s th e va cu u m sta te o f th e electro m a g n etic ¯ eld . O n e w o u ld h av e im a g in ed th a t, if th ere a re n o p h o to n s, th en th ere w ill b e n o m ea su ra b le p h y sica l e® ects d u e to th e electro m a g n etic ¯ eld . W h ile th is is m o re o r less tru e { w h ich is ra th er rea ssu rin g { th ere a re in d eed in terestin g situ a tio n s in w h ich it is n ot tru e! W e w ill d escrib e o n e su ch co n tex t, ca lled C a sim ir e® ect, in th is in sta llm en t. T h e sim p lest, th o u g h a b it id ea lized , d escrip tio n o f C a sim ir e® ect is th e fo llow in g . S u p p o se y o u k eep tw o p a ra llel, p erfectly co n d u ctin g p la tes in th e o th erw ise em p ty sp a ce, sep a ra ted b y a d ista n ce L . T h en , th ey w ill a ttra ct
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T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Keywords Harmonic oscillator, quantum mechanics, electromagnetism.
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Remember that there are no net charges put on the plates; we are not talking about a charged parallel plate capacitor. The force acts between two plates kept in the vacuum.
To minimize the total energy, we need to allow for some amount of fluctuation in both q and p that is commensurate with the uncertainty principle Δp Δq &
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~.
ea ch o th er w ith a fo rce ¼ 2 ~c A; (1 ) 240 L 4 w h ere A is th e cro ss-sectio n a l a rea o f th e p la tes!! R em em b er th a t th ere a re n o n et ch a rg es p u t o n th e p la tes; w e a re n o t ta lk in g a b o u t a charged p a ra llel p la te ca p a cito r. T h e fo rce a cts b etw een tw o p la tes k ep t in th e va cu u m . T h is e® ect w a s p red icted [1 ] b y th e D u tch p h y sicist H en d rick C a sim ir in 1 9 4 8 a n d h a s a ctu a lly b een m ea su red in th e la b [2 ]. O n e n ice w ay o f u n d ersta n d in g th is resu lt is a s a ta n g ib le fo rce ex erted b y th e electro m a g n etic va cu u m . L et u s see h ow . F = ¡
B efo re la u n ch in g in to m a th em a tics, let m e ex p la in th e b a sic rea so n fo r th is p h en o m en o n in q u a lita tiv e term s. C o n sid er th e fa m ilia r ex a m p le o f a h a rm o n ic o scilla to r, w ith th e H a m ilto n ia n 1 H (p ;q ) = [p 2 + ! 2 q 2 ]: (2 ) 2 W e h av e set th e m a ss o f th e p a rticle to u n ity fo r sim p licity. C la ssica lly, th e m in im u m en erg y fo r su ch a sy stem is zero (E cla ss = 0 ), w h ich is a ch iev ed w h en q = p = 0 . W e k n ow , h ow ev er, th a t th is is n o t p o ssib le in q u a n tu m th eo ry, essen tia lly b eca u se o f th e u n certa in ty p rin cip le. T o m in im ize th e p o ten tia l en erg y, w e n eed to set q = 0 ; b u t if w e k n ow th e p o sitio n to su ch in ¯ n ite p recisio n , th e m o m en tu m w ill b e in ¯ n itely u n certa in a n d w e ca n n o t g u a ra n tee a low va lu e fo r p 2 = 2 ! S o to m in im ize th e to ta l en erg y, w e n eed to a llow fo r so m e a m o u n t o f ° u ctu a tio n in b o th q a n d p th a t is co m m en su ra te w ith th e u n certa in ty p rin cip le ¢ p ¢ q & ~. T h e resu ltin g g ro u n d sta te, a s w e k n ow , h a s a n o n -zero en erg y E q u a n t = (1= 2 )~! . S u p p o se w e co n sid er a d i® eren t p h y sica l sy stem w ith th e H a m ilto n ia n H n ew = H (p ;q) ¡ (1 = 2 )~! w h ere H (p ;q) is g iv en b y (2 ). S in ce th e su b tra ctio n o f a co n sta n t fro m th e H a m ilto n ia n d o es n o t ch a n g e th e eq u a tio n s o f m o tio n , w e still a g a in h av e a h a rm o n ic o scilla to r b u t w ith
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SERIES ⎜ ARTICLE
a sh ift in th e en erg y. C la ssica lly, th e m in im u m en erg y sta te w ill still co rresp o n d to q = p = 0 b u t w ith en erg y E cla ss = ¡ (1= 2 )~! . B u t q u a n tu m m ech a n ica lly, th e g ro u n d sta te w ill ex h ib it ° u ctu a tio n s in q a n d p b u t w ill h av e zero en erg y ; E q u a n t = 0 ! T h is is th e cru cia l p o in t. Q u a n tu m m ech a n ics a llow s y o u to h av e a sta te fo r th e h a rm o n ic o scilla to r w ith th e H a m ilto n ia n H n ew (p ;q) su ch th a t E q u a n t = 0 w h ich ca n h o st ° u ctu a tio n s in th e d y n a m ica l va ria b les q a n d p .
Quantum mechanics
S o m eth in g v ery a n a lo g o u s h a p p en s in th e ca se o f a n electro m a g n etic ¯ eld . A s w e sh a ll see th e electro m a g n etic ¯ eld ca n b e th o u g h t o f a s a b u n ch o f h a rm o n ic o scilla to rs. T h e g ro u n d sta te w ill co rresp o n d to a sta te o f zero p h o to n s a n d o n e ca n a rra n g e m a tters su ch th a t it h a s zero en erg y. B u t th e electric a n d m a g n etic ¯ eld s w ill p lay th e ro le a n a lo g o u s to p a n d q o f th e o scilla to r a n d th ey w ill ex h ib it ° u ctu a tio n s { u su a lly ca lled va cu u m ° u ctu a tio n s { in th e g ro u n d sta te. T h erefo re o n e ca n n o t rea lly say th a t th e electro m a g n etic ¯ eld s va n ish in th e va cu u m sta te ev en th o u g h w e ca n m a k e its en erg y va n ish . O n ce w e reco g n ize th is fa ct, it is n o t su rp risin g th a t th e electro m a g n etic va cu u m ca n ex ert fo rces o n b o d ies. A ctu a lly th e situ a tio n is a little b it m o re co m p lica ted b eca u se th e p ro ced u re a n a lo g o u s to th e su b tra ctio n o f (1= 2 )~! is m o re n o n triv ia l in th is ca se b u t th e essen tia l id ea is th e sa m e.
and p.
L et u s n ow try to u n d ersta n d th is in a m o re m a th em a tica l la n g u a g e a n d in a so m ew h a t b ro a d er co n tex t. A s it tu rn s o u t, th e essen tia l id ea ca n b e illu stra ted b y ig n o rin g tw o co m p lica tio n s o f th e rea l w o rld . F irst is th e v ecto r n a tu re o f electro m a g n etism a n d th e seco n d is th e fa ct th a t sp a ce is th ree d im en sio n a l. W e w ill w o rk o u t ¯ rst a sim p ler p ictu re u sin g ju st a sca la r ¯ eld w ith o n e d eg ree o f freed o m (ra th er th a n w ith th e electro m a g n etic ¯ eld ) a n d a lso ig n o rin g th e tw o tra n sv erse d irectio n s a n d trea tin g sp a ce a s o n e-d im en sio n a l. A fter w e w o rk o u t th e sim p li¯ ed p ictu re, w e w ill d escrib e h ow it
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allows you to have a state for the harmonic oscillator with the Hamiltonian Hnew (p,q) such that Equant= 0 which can host fluctuations in the dynamical variables q
One cannot really say that the electromagnetic fields vanish in the vacuum state even though we can make its energy vanish.
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g en era lizes to th e rea l life. O n ce w e ig n o re th e v ecto r n a tu re o f th e electro m a g n etic ¯ eld , w e ca n w o rk w ith a sin g le sca la r ¯ eld Á (t;x ) w h ich is a fu n ctio n o f o n e sp a ce d im en sio n a n d tim e. (If y o u w a n t, y o u ca n th in k o f it a s a n a lo g o u s to a n y o n e co m p o n en t o f th e electro m a g n etic ¯ eld .) In th e a b sen ce o f so u rces, w e k n ow th a t ea ch co m p o n en t o f th e electro m a g n etic ¯ eld sa tis¯ es th e w av e eq u a tio n ; so w e w ill a ssu m e th a t o u r sca la r ¯ eld sa tis¯ es th e eq u a tio n : @ 2Á @ 2Á ¡ = 0: @ t2 @x2
(3 )
(W e h av e ch o sen u n its w ith c = 1 . In 3 -d im en sio n s, th e seco n d term w o u ld h av e b een ¡ r 2 Á w h ich b eco m es ¡ (@ 2 Á = @ x 2 ) w h en w e ig n o re tw o sp a tia l co o rd in a tes.) T h is eq u a tio n ca n b e sim p li¯ ed b y in tro d u cin g th e sp a tia l F o u rier tra n sfo rm Q k o f Á (t;x ) b y Z1 dk Á (t;x ) = Q k (t) ex p (ik x ): (4 ) ¡1 (2¼ )
The field φ (t,x) is completely specified by the function Qk(t) so that we can think of Qk(t) as the dynamical variables describing our system.
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S u b stitu tin g th is in (3 ) w e ¯ n d th a t Q k (t) sa tis¯ es th e eq u a tio n QÄ k + k 2 Q k = 0: T h e ¯ eld Á (t;x ) is co m p letely sp eci¯ ed b y th e fu n ctio n Q k (t) so th a t w e ca n th in k o f Q k (t) a s th e d y n a m ica l va ria b les d escrib in g o u r sy stem . T h e fa ct th a t w e a re d ea lin g w ith th e ¯ eld tra n sla tes in to th e fa ct th a t w e n ow h av e a n in ¯ n ite n u m b er o f d y n a m ica l va ria b les, o n e fo r ea ch va lu e o f k . O th er th a n th a t, w e ca n w o rk d irectly w ith Q k (t) in stea d o f th e o rig in a l ¯ eld Á (t;x ). O n e m in o r p ro b lem w ith Q k (t) is th a t it is a co m p lex n u m b er (sin ce Á (t;x ) is rea l) a n d w e w o u ld lik e to w o rk w ith d y n a m ica l va ria b les th a t a re rea l. T h is is ea sily ta k en ca re o f. A s Q k is co m p lex , w e h a v e tw o d eg rees o f freed o m co rresp o n d in g to th e rea l a n d im a g in a ry p a rts o f Q k fo r ea ch k w ith th e co n stra in t Q ¤k = Q ¡k . If w e w rite Q k = (A k + iB k ), th en , sin ce Á is a rea l sca la r
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SERIES ⎜ ARTICLE
¯ eld , w e ca n rela te th e va ria b les fo r k to th a t fo r ¡ k a s A k = A ¡k a n d B k = ¡ B ¡ k . E v id en tly, o n ly h a lf th e m o d es co n stitu te in d ep en d en t d eg rees o f freed o m . T h erefo re, w e ca n w o rk w ith a n ew set o f rea l m o d es q k , d e¯ n ed fo r a ll va lu es o f k w ith a su ita b le red e¯ n itio n , say, b y ta k in g q k = A k fo r o n e h a lf o f k a n d q ¡k = B k fo r th e o th er h a lf. T h is w ill, o f co u rse, lea d to th e sa m e eq u a tio n b u t fo r th e real va ria b le q k (t). qÄk + k 2 q k = 0:
(5 )
In particular, we can quantize the field by quantizing each of the harmonic oscillators qk(t). (In fact, that is the essence of quantum field theory of noninteracting fields; rest is just detail.)
T h a t is, th e d y n a m ica l va ria b le q k (t) sa tis¯ es th e h a rm o n ic o scilla to r eq u a tio n w ith freq u en cy ! = jk j, fo r ea ch va lu e o f k . O u r ¯ eld is m a th em a tica lly th e sa m e a s a n in ¯ n ite n u m b er o f h a rm o n ic o scilla to rs, o n e fo r ea ch k . It fo llow s th a t ev ery th in g w e k n ow a b o u t h a rm o n ic o scilla to rs ca n n ow b e a p p lied to th is sy stem . In p a rticu la r, w e ca n q u a n tize th e ¯ eld b y q u a n tizin g ea ch o f th e h a rm o n ic o scilla to rs q k (t). (In fa ct, th a t is th e essen ce o f q u a n tu m ¯ eld th eo ry o f n o n -in tera ctin g ¯ eld s; rest is ju st d eta il.) C la ssica lly, w e ca n n ow co n stru ct th e g ro u n d sta te b y ta k in g q k = 0 fo r a ll va lu es o f k . T h is w ill, o f co u rse, m a k e th e ¯ eld va n ish a lo n g w ith its en erg y, a s to b e ex p ected fro m a sen sib le g ro u n d sta te. B u t, a s w e d iscu ssed ea rlier, th is d o es n o t h o ld fo r th e q u a n tu m g ro u n d sta te. T h e g ro u n d sta te o f th e h a rm o n ic o scilla to r fo r a g iv en va lu e o f k is d escrib ed b y th e g ro u n d sta te en erg y eig en fu n ctio n µ ¶ ³! ´1 = 4 1 k 2 Ã (q k ) = ex p ¡ ! k q k : (6 ) ¼ 2 W e a re u sin g u n its w ith ~ = 1 fo r sim p lify in g th e ex p ressio n s. T h e g ro u n d sta te w av e fu n ctio n fo r th e fu ll sy stem , m a d e o f a b u n ch o f in d ep en d en t o scilla to rs, ca n b e d escrib ed b y th e p ro d u ct o f th e g ro u n d sta te w av e
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This expression can be interpreted along similar lines as we interpret a harmonic oscillator wave function in usual quantum mechanics.
fu n ctio n s o f ea ch o f th e o scilla to rs: µ ¶ Y ³! k ´1 = 4 1 2 ª [Á (x )] = ex p ¡ ! k q k ¼ 2 k ¸ · Z1 1 d k 2 ´ N¹ ex p ¡ ! k qk : 2 ¡1 (2¼ )
T h is ex p ressio n ca n b e in terp reted a lo n g sim ila r lin es a s w e in terp ret a h a rm o n ic o scilla to r w av e fu n ctio n in u su a l q u a n tu m m ech a n ics. S u p p o se w e h av e a h a rm o n ic o scilla to r in th e g ro u n d sta te a n d w e m ea su re th e p o sitio n q . T h en th e rela tiv e p ro b a b ility th a t w e w ill g et a va lu e q = a co m p a red to a va lu e q = b is g iv en b y R =
For any choice of f1(x) and f2(x) the number R can be computed, allowing us to determine the probabilities of different field configurations in the vacuum state.
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(7 )
¡ ¢ jà (a )j2 2 2 = ex p ¡ ! [a ¡ b ] : jà (b)j2
(8 )
N ow su p p o se w e h av e a qu an tu m ¯ eld w h ich is in th e g ro u n d sta te a n d w e m ea su re th e ¯ eld ev ery w h ere a t, say, t = 0 . T h en , th ere is so m e p ro b a b ility th a t w e w ill g et a ¯ eld co n ¯ g u ra tio n d escrib ed b y th e fu n ctio n Á (0;x ) = f 1 (x ) a n d so m e o th er p ro b a b ility th a t ¯ eld co n ¯ g u ra tio n is d escrib ed b y th e fu n ctio n Á (0;x ) = f 2 (x ). J u st a s in th e p rev io u s ca se, w e w a n t to k n ow th e rela tiv e p ro b a b ility o f g ettin g o n e co n ¯ g u ra tio n co m p a red to a n o th er. T o ¯ n d th is, w e ¯ rst o b ta in th e sp a tia l F o u rier tra n sfo rm s o f f 1 (x ) a n d f 2 (x ) a n d ca ll th em a k a n d b k . T h en th e rela tiv e p ro b a b ility is g iv en b y µ Z1 ¶ jª (f 1 (x ))j2 dk 2 2 ! k [a k ¡ b k ] : (9 ) R = = ex p ¡ jª (f 2 (x ))j2 ¡1 (2¼ ) F o r a n y ch o ice o f f 1 (x ) a n d f 2 (x ) th e a b ov e n u m b er ca n b e co m p u ted , a llow in g u s to d eterm in e th e p ro b a b ilities o f d i® eren t ¯ eld co n ¯ g u ra tio n s in th e va cu u m sta te. Y o u w o u ld h av e n o ticed th a t w e p ro b a b ilities ra th er th a n a b so lu te d iscu ssio n . F o r a sin g le h a rm o n ic h av e sa id th a t jà (q )j2 d q g iv es th e
sw itch ed to rela tiv e p ro b a b ilities in th is o scilla to r, o n e co u ld a b so lu te p ro b a b ility
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o f ¯ n d in g th e p a rticle in th e in terva l (q ;q + d q ). W h en w e h av e in ¯ n ite n u m b er o f o scilla to rs, th e n o rm a liza tio n fa cto r N¹ in (7 ) in v o lv es a n in ¯ n ite p ro d u ct w h ich is h a rd to d e¯ n e rig o ro u sly. W e g et a ro u n d th is b y ta lk in g a b o u t rela tiv e p ro b a b ilities in w h ich th e n o rm a liza tio n fa cto r ca n cels o u t.
We get around this by talking about relative probabilities in which the normalization factor cancels out.
B efo re w e p ro ceed fu rth er, let m e m en tio n th e co rresp o n d in g resu lt in th ree sp a tia l d im en sio n s. In th is ca se (7 ) h a s th e o b v io u s g en era liza tio n to : µ ¶ Y ³! k ´1 = 4 1 2 ª [Á (x )] = ex p ¡ ! k jq k j ¼ 2 k · Z 3 ¸ 1 d k 2 ¹ = N ex p ¡ ! k jq k j : (1 0 ) 2 (2¼ )3 In fa ct, in th is ca se it is n icer to ex h ib it th e resu lt in term s o f th e ¯ eld co n ¯ g u ra tio n itself b y u sin g ! k = jk j a n d ! k jq k j2 = k 2 jq k j2 = jk j. S in ce ik q k is essen tia lly th e F o u rier sp a tia l tra n sfo rm o f r Á , w e ca n ea sily o b ta in Z 3 Z 3 d k d k jk j2 jq k j2 2 ! jq j = k k (2 ¼ 3 ) (2¼ 3 ) jk j 1 = 2¼ 2
Z
3
d x
Z
½ 3
d y
r x Á ¢r y Á jx ¡ y j2
¾ :
(1 1 )
(P rov e th is!) S u b stitu tin g th is in to (1 0 ) a n d ta k in g th e m o d u lu s, w e g et th e p ro b a b ility d istrib u tio n in th e g ro u n d sta te to b e P [Á (x )] = jª [Á (x )]j2 ½ ¾ ZZ 1 3 3 r x Á ¢r y Á ; = N ex p ¡ d x d y 2¼ 2 jx ¡ y j2
shows clearly that the vacuum state of
(1 2 )
w ith N = jN¹ j2 . O n ce a g a in , th is ex p ressio n sh o w s clea rly th a t th e va cu u m sta te o f th e ¯ eld ca n h o st, w h a t
RESONANCE ⎜ February 2009
Once again, expression (12)
the field can host, what is usually called, zero point fluctuations of the field variable φ.
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Let us now ask what happens if we introduce two perfectly conducting, parallel plates into the vacuum.
is u su a lly ca lled , zero p o in t ° u ctu a tio n s o f th e ¯ eld va ria b le Á . T h e p ro b a b ility th a t o n e d etects a p a rticu la r ¯ eld co n ¯ g u ra tio n Á (x ) w h en th e ¯ eld is in th e va cu u m sta te ca n b e o b ta in ed b y eva lu a tin g th e va lu e o f P fo r th is p a rticu la r fu n ctio n a l fo rm Á (x ). T h e resu lt is in d ep en d en t o f tim e b eca u se o f th e sta tio n a rity o f th e va cu u m sta te. G iv en th e a m b ig u ity in th e ov era ll n o rm a liza tio n fa cto r N th is p ro b a b ility sh o u ld a g a in b e in terp reted a s a rela tiv e p ro b a b ility. T h a t is, th e ra tio P 1 = P 2 w ill g iv e th e rela tiv e p ro b a b ility b etw een tw o ¯ eld co n ¯ g u ra tio n s ch a ra cterized b y th e fu n ctio n s Á 1 (x ) a n d Á 2 (x ). L et u s n ow a sk w h a t h a p p en s if w e in tro d u ce tw o p erfectly co n d u ctin g , p a ra llel p la tes in to th e va cu u m . T h e fa ct th a t th e p la tes a re p erfectly co n d u ctin g im p lies th a t th e electro m a g n etic ¯ eld { fo r w h ich o u r Á (t;x ) is a p rox y { m u st sa tisfy so m e n o n -triv ia l b o u n d a ry co n d itio n s a t x = 0 a n d x = L w h ere th e p la tes a re lo ca ted . F o r th e sca la r ¯ eld , w e ca n ta k e th e b o u n d a ry co n d itio n to b e th a t th e ¯ eld va n ish es a t th e p la tes: Á (t;0 ) = Á (t;L ) = 0 in o n e sp a tia l d im en sio n . Y o u ca n n o t d escrib e a ¯ eld sa tisfy in g su ch a b o u n d a ry co n d itio n u sin g th e F o u rier in teg ra l in (4 ) w ith k ta k in g a ll p o ssib le va lu es in ¡ 1 < k < 1 . In stea d w e ca n restrict it to a d iscrete, th o u g h in ¯ n ite, set o f va lu es g iv en b y k = n (¼ = L ) w ith n = 1;2;:::: a n d w rite Á (t;x ) =
X1 n= 1
h ¼x i q n (t) sin n L
(1 3 )
so th a t th e b o u n d a ry co n d itio n s a t x = 0 a n d x = L a re sa tis¯ ed . W e still h av e to d ea l w ith a n in ¯ n ite n u m b er o f o scilla to rs b u t th eir freq u en cies a re n ow g iv en b y ! n = k n = n (¼ = L ). If w e n ow w o rk o u t th e co rresp o n d in g g ro u n d sta te, it w ill clea rly b e d i® eren t fro m th e o n e d escrib ed b y (7 ) b eca u se th e in teg ra l ov er k w ill b e rep la ced b y th e su m
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ov er n . T h is is n eed ed b eca u se, o u r b o u n d a ry co n d itio n tells u s th a t th e g ro u n d sta te sh o u ld n ow h av e zero p ro b a b ility fo r ¯ eld co n ¯ g u ra tio n s w h ich d o n o t va n ish a t th e p la tes. T h e in tro d u ctio n o f th e p la tes, th ro u g h ch a n g in g th e b o u n d a ry co n d itio n , h a s ch a n g ed th e g ro u n d sta te. W h a t a b o u t th e en erg y o f th e g ro u n d sta te w ith a n d w ith o u t th e p la tes? T h ey a re a lso d i® eren t. In th e a b sen ce o f p la tes, ea ch h a rm o n ic o scilla to r co n trib u tes a n en erg y (1= 2 )~! = (1= 2 )~jk j. S o th e to ta l g ro u n d sta te en erg y, p er u n it len g th o f sp a ce, is g iv en b y a n in teg ra l ov er a ll k o f (1 = 2 )~jk j. T h erefo re, th e en erg y in a reg io n o f len g th L w ill b e: Z1 Z1 L 1 L E0= d k ~jk j = d k ~k : (1 4 ) (2¼ ) ¡1 2 (2 ¼ ) 0
The introduction of the plates, through changing the boundary condition, has changed the ground state.
T h is is m a n ifested ly in ¯ n ite, essen tia lly b eca u se th ere a re a n in ¯ n ite n u m b er o f h a rm o n ic o scilla to rs. W h a t a b o u t th e g ro u n d sta te en erg y in th e p resen ce o f th e p la tes? T h is is g iv en b y th e su m E
0 0 =
1 1 1X 1X ~! n = ~(n ¼ = L ) 2 n= 0 2 n= 0
(1 5 )
w h ich is a lso in ¯ n ite, essen tia lly b ein g th e su m o f a ll p o sitiv e in teg ers. T h ese in ¯ n ities a re b a d n ew s b u t th ere is a trick to g et a ro u n d th em . A s w e sa id b efo re, th e eq u a tio n o f m o tio n fo r th e k -th o scilla to r w ill n o t ch a n g e if w e su b stra ct fro m th e H a m ilto n ia n (1= 2 )~! k b u t it w ill `reg u la rize th e g ro u n d sta te en erg y to zero . T h is is eq u iva len t to lo o k in g a t th e d i® eren ce (E 00 ¡ E 0 ) a s th e p h y sica lly releva n t q u a n tity. T o stu d y th is, it is co n v en ien t to in tro d u ce in (1 4 ) a co n tin u o u s va ria b le n v ia th e eq u a tio n k = (¼ = L )n . T h en w e g et fro m (1 4 ) a n d (1 5 ): "1 # Z1 X ~¼ (E 00 ¡ E 0 ) = n ¡ dn n : (1 6 ) 2L n = 0 0
RESONANCE ⎜ February 2009
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Both the expressions in (17) as well as their difference are now finite and the idea is to first compute the difference as a function of λ and then take the limit of λ → 0 hoping for the best.
Y o u m ay th in k th a t th is is n o t m u ch h elp b eca u se th is is o f th e fo rm (1 ¡ 1 ) w h ich d o es n o t h av e a p recise m ea n in g . T h a t is tru e b u t th ere a re w ay s o f g iv in g m ea n in g to su ch ex p ressio n s in a fa irly sy stem a tic m a n n er. T h e sim p lest p ro ced u re is to co n sid er, in stea d o f th e ex p ressio n in (1 6 ), th e ex p ressio n : E 00(¸ ) ¡ E 0 (¸ ) # "1 Z1 ~¼ X n ex p (¡ n ¸ ) ¡ d n n ex p (¡ n ¸ ) : (1 7 ) ´ 2L n = 0 0 H ere w e h a v e m u ltip lied b o th th e ex p ressio n s b y a `reg u la to r fu n ctio n ' ex p (¡ n ¸ ) w h ere ¸ is ju st a p a ra m eter. B o th th e ex p ressio n s a s w ell a s th eir d i® eren ce a re n ow ¯ n ite a n d th e id ea is to ¯ rst co m p u te th e d i® eren ce a s a fu n ctio n o f ¸ a n d th en ta k e th e lim it o f ¸ ! 0 h o p in g fo r th e b est. T h a t is, w e in terp ret th e ex p ressio n in (1 6 ) a s th e lim it o f th e ex p ressio n in (1 7 ) w h en ¸ ! 0 . I w ill let y o u w o rk o u t th e ex p ressio n s. Y o u sh o u ld ¯ rst g et: X1
n ex p (¡ n ¸ ) =
n= 0
e ¡¸ 1 1 ¸2 = ¡ + + O (¸ 4 ) (1 ¡ e ¡¸ )2 ¸2 12 240 (1 8 )
w h ich d iv erg es w h en ¸ ! 0 a s to b e ex p ected . S im ila rly Z1 1 d n n ex p (¡ n ¸ ) = 2 (1 9 ) ¸ 0 w h ich a lso d iv erg es w h en ¸ ! 0 . B u t, a sto n ish in g ly en o u g h , th e d i® eren ce b etw een (1 8 ) a n d (1 9 ) rem a in s ¯ n ite a s ¸ ! 0 : Z1 X1 n ex p (¡ n ¸ ) ¡ d n n ex p (¡ n ¸ ) 0
n= 0
= ¡
188
1 1 + O (¸ 2 ) = ¡ 12 12
(2 0 )
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w h en ¸ ! 0 . T h is a llow s u s to o b ta in th e fo llow in g rem a rka b le resu lt: E (L ) ´ (E 00 ¡ E 0 ) ´ lim (E 00(¸ ) ¡ E 0 (¸ )) = ¡ ¸! 0
¼~ : 24L (2 1 )
S o w e see th a t th e g ro u n d sta te en erg y o f th e sy stem w ith th e p la tes { w h en reg u la rized b y su b tra ctin g aw ay th e en erg y in th e a b sen ce o f th e p la tes { is a n eg a tiv e n u m b er1 a n d is in v ersely p ro p o rtio n a l to th e sep a ra tio n b etw een th e p la tes! C lea rly, th is w ill lea d to a n a ttra ctiv e fo rce F = ¡ (d E = d L ) / L ¡2 b etw een th e p la tes, sin ce red u cin g th e sep a ra tio n b etw een th e p la tes lea d s to th e low erin g o f th e en erg y. A m o re p h y sica l w ay o f th in k in g a b o u t th is resu lt is a s fo llow s. If w e ch a n g e th e sep a ra tio n b etw een th e p la tes b y a n a m o u n t ¢ L , th e en erg y o f th e co n ¯ g u ra tio n w ill ch a n g e b y (d E = d L )¢ L w h ich m u st b e a cco u n ted b y th e w o rk d o n e b y th e a g en cy sep a ra tin g th e p la tes a ctin g a g a in st th e a ttra ctiv e fo rce F . E q u a tin g it to ¡ F ¢ L , w e ¯ n d th a t F = ¡ d E = d L . In th e m y th ica l w o rld o f o n e sp a tia l d im en sio n , th e p la tes a re zero -d im en sio n a l p o in ts w h ich is n o t o f m u ch u se. T h e co rresp o n d in g ca lcu la tio n fo r electro m a g n etic ¯ eld in 3 -d im en sio n s is m o re co m p lica ted a lg eb ra ica lly b u t a ll th e co n cep ts rem a in th e sa m e. T h e ¯ n a l resu lt in th is ca se is a n ex p ressio n fo r en erg y p er u n it tra n sv erse a rea o f th e p la tes, g iv en b y : (E 00 ¡ E 0 ) ¼ 2 ~c = ¡ ; A 720 L 3
1
With this `regularization', quantum ¯eld theorists often conclude that the sum of all positive integers is not only ¯nite but is a negative fraction (¡ 1= 12)! If you are familiar with Riemann-zeta function, ³ (x ) P = 1n = 1 n ¡ x , you will recognize that the sum of all positive integers is formally the same as ³ (¡ 1). One can de¯ne this quantity by analytic continuation in the complex plane and one does recover the result ³ (¡ 1) = ¡ 1= 12. Of course, this does not make one any wiser as to what is going on.
(2 2 )
w h ere w e h av e re-in tro d u ced th e c fa cto r. T h e fo rce p er u n it a rea a ctin g b etw een th e p la tes is g iv en b y F d (E 00 ¡ E 0 ) ¼ 2 ~c = ¡ = ¡ : A dL A 240 L 4
(2 3 )
T h is tin y fo rce h a s a ctu a lly b een m ea su red in th e la b ! N o te th a t, th o u g h th e resu lt is electro m a g n etic b y n a tu re, it is in d ep en d en t o f th e electro n ic ch a rg e e. T h e
RESONANCE ⎜ February 2009
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The essential lesson is that the pattern of quantum fluctuations is sensitive to the boundary conditions we impose, both mathematically and practically.
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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electro m a g n etism o n ly en ters th ro u g h th e b o u n d a ry co n d itio n o n th e p erfect co n d u cto rs, w h ich is o n e rea so n w e co u ld m im ic it w ith a sca la r ¯ eld . T h e w h o le p h en o m en o n is q u ite b ew ild erin g a n d if y o u a re sh a k in g y o u r h ea d in d isb elief, I w ill n o t b la m e y o u ! B u t th e rea lity o f th is e® ect is b ey o n d d isp u te a n d it h a s b een d eriv ed fro m sev era l d i® eren t p ersp ectiv es ov er y ea rs. T h e essen tia l lesso n is th a t th e p a ttern o f q u a n tu m ° u ctu a tio n s is sen sitiv e to th e b o u n d a ry co n d itio n s w e im p o se, b o th m a th em a tica lly a n d p ra ctica lly. T h e g ro u n d sta te o f th e electro m a g n etic ¯ eld in th e p resen ce o f tw o p a ra llel, co n d u ctin g , p la tes is q u ite d i® eren t fro m th e g ro u n d sta te in th e a b sen ce o f th e p la tes. T h is m u ch a lo n e is ea sy to u n d ersta n d b eca u se th e g ro u n d sta te in th e p resen ce o f th e p la tes m u st en su re th a t, th e ¯ eld co n ¯ g u ra tio n s w h ich d o n o t sa tisfy th e b o u n d a ry co n d itio n s a t th e p la tes, h av e zero p ro b a b ility fo r th eir ex isten ce. B u t w h a t is ra th er cu rio u s is th a t th is g ro u n d sta te h a s a n en erg y w h ich d i® ers fro m th a t in th e a b sen ce o f th e p la tes b y a ¯ n ite a m o u n t. T h ere is n o sim p le ex p la n a tio n fo r th is fa ct, w h ich m a k es C a sim ir e® ect a ll th e m o re fa scin a tin g . Suggested Reading [1] H B G Casimir, Proc. Kon. Ned. Akad. Wetensch., Vol.B51, p.793, 1948. [2] S K Lamoreaux, Phys. Rev. Lett., Vol.78, p.58 1997. G Bressi, G Carugno, R Onofrio, G Ruoso, Phys. Rev. Lett., Vol.88, p.041804, 2002.
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SERIES ⎜ ARTICLE
Snippets of Physics 15. Hubble Expansion for Pedestrians T Padmanabhan
M a n y fe a tu r e s o f th e e x p a n d in g u n iv e r se , w h ic h sh o u ld b e le g itim a te ly d isc u sse d u sin g g e n e r a l r e la tiv ity , c a n b e sn e a k e d in b y u sin g N e w to n ia n p h y sic s in a n e x p a n d in g c o o r d in a te sy ste m . In th is sp e c ia l issu e o n H u b b le , I d e sc r ib e s e v e r a l o f th e se fe a tu r e s a lo n g w ith so m e c a u tio n a r y c o m m e n ts. S in ce th e la rg e-sca le d y n a m ics o f th e u n iv erse is essen tia lly g ov ern ed b y g rav ity, a n y th eo retica l m o d el fo r g rav ity w ill h av e im p lica tio n s fo r th e la rg e-sca le p h y sics o f th e u n iv erse. T h is w a s k n ow n , o f co u rse, ev en to N ew to n w h o d id a ttem p t to d escrib e th e u n iv erse u sin g h is id ea s o f g rav ity. W e n ow k n ow , h ow ev er, th a t th e p ro p er d escrip tio n o f g rav ity sh o u ld b e b a sed o n E in stein 's g en era l rela tiv ity ra th er th a n o n N ew to n ia n id ea s. It w a s m en tio n ed in a p rev io u s in sta llm en t th a t th e d escrip tio n o f g rav ity in E in stein 's th eo ry is b a sed o n th e n o tio n o f cu rv ed sp a cetim e. In th e ca se o f th e u n iv erse, th is w ill in v o lv e trea tin g it a s a cu rv ed sp a cetim e w ith a g eo m etry d eterm in ed b y th e d istrib u tio n o f m a tter. T h e sim p lest o f su ch m o d els trea ts th e d istrib u tio n o f m a tter in th e u n iv erse a s u n ifo rm a n d iso tro p ic (a t su ± cien tly la rg e sca les) a n d tries to u n d ersta n d th e p ro p erties o f th e cu rv ed sp a cetim e p ro d u ced b y su ch a m a tter d istrib u tio n . W h ile a ll th ese m ig h t so u n d co m p lica ted , a su rp risin g fea tu re a b o u t su ch a m o d el o f th e u n iv erse is th a t m u ch o f its d y n a m ics ca n b e u n d ersto o d fa irly ea sily { w ith o u t in tro d u cin g co m p lica ted n o tio n s fro m g en era l rela tiv ity. N eed less to say, su ch a n a p p ro a ch is ¯ lled w ith p itfa lls a n d o n e n eed s to co n sta n tly v erify th a t o n e is n o t g ettin g ca rried aw ay b y th e sim p lify in g
RESONANCE ⎜ March 2009
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Keywords Cosmology, expansion of the universe, structure formation.
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It is not possible to introduce a set of coordinates on the surface of a sphere such that the 'line interval' reduces to the Pythagorean form. This is the difference between curved and flat space.
° av o u r o f N ew to n ia n p h y sics. In th is in sta llm en t, I w ill d escrib e h ow th ese id ea s w o rk . T h e d i® eren ce b etw een a ° a t sp a ce a n d a cu rv ed sp a ce ca n b e en co d ed in th e g en era liza tio n o f P y th a g o ra s th eo rem fo r in ¯ n itesim a lly sep a ra ted p o in ts. F o r ex a m p le, a ° a t 2 -d im en sio n a l su rfa ce (say, a p la in sh eet o f p a p er) a llow s u s to in tro d u ce sta n d a rd C a rtesia n co o rd in a tes (x ;y ) su ch th a t th e d ista n ce b etw een in ¯ n itesim a lly sep a ra ted p o in ts ca n b e ex p ressed in th e fo rm d l2 = d x 2 + d y 2 w h ich , o f co u rse, is ju st th e sta n d a rd P y th a g o ra s th eo rem . In co n tra st, co n sid er th e tw o -d im en sio n a l su rfa ce o n a sp h ere o f ra d iu s r o n w h ich w e h av e in tro d u ced tw o a n g u la r co o rd in a tes (µ ;Á ). T h e co rresp o n d in g fo rm u la w ill n ow rea d d l2 = r 2 d µ 2 + r 2 sin 2 µ d Á 2 . It is n o t p o ssib le to in tro d u ce a n y o th er set o f co o rd in a tes o n th e su rfa ce o f a sp h ere su ch th a t th is ex p ressio n { u su a lly ca lled th e `lin e in terva l' { red u ces to th e P y th a g o rea n fo rm . T h is is th e d i® eren ce b etw een a cu rv ed sp a ce a n d ° a t sp a ce. M ov e o n fro m sp a ce to sp a cetim e a n d fro m p o in ts to ev en ts. In ° a t sp a cetim e, w h ich w e u se in sp ecia l rela tiv ity, th e `P y th a g o ra s th eo rem ' g en era lizes to th e fo rm d s 2 = ¡ c 2 d t2 + d x 2 + d y 2 + d z 2 :
(1 )
T h e sp a tia l co o rd in a tes a p p ea r in th e sta n d a rd fo rm a n d th e in clu sio n o f tim e in tro d u ces th e a ll im p o rta n t m in u s sig n . B u t o n e ca n liv e w ith it an d trea t it a s a g en era liza tio n o f th e fo rm u la d l2 = d x 2 + d y 2 to 4 -d im en sio n s (w ith a n ex tra m in u s sig n ). B u t in a cu rv ed sp a cetim e, th is ex p ressio n w ill n o t h o ld a n d th e co o rd in a te d i® eren tia ls lik e c 2 d t2 ;d x 2 , etc., in th e in terva l w ill g et m u ltip lied b y fu n ctio n s o f sp a ce a n d tim e. T h is is ju st lik e o u r u sin g sin 2 µ d Á 2 ra th er th a n ju st d Á 2 to d escrib e th e cu rv ed 2 -d im en sio n a l su rfa ce o f a sp h ere. T h e p recise m a n n er in w h ich su ch a m o d i¯ ca tio n o ccu rs is d eterm in ed b y E in stein 's eq u a tio n a n d d ep en d s o n th e d istrib u tio n o f m a tter in sp a cetim e.
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W h ile th is ca n lea d to p retty co m p lica ted sp a cetim es in g en era l, th e la rg e-sca le u n iv erse tu rn s o u t to b e d escrib ed b y a rem a rka b ly sim p le g en era liza tio n o f th e lin e in terva l in (1 ). W e o n ly n eed to m o d ify it in to th e fo rm £ ¤ d s 2 = ¡ c 2 d t2 + a 2 (t) d x 2 + d y 2 + d z 2 ; (2 ) w h ere th e fu n ctio n a (t) is ca lled th e `ex p a n sio n fa cto r'. A ll th e in fo rm a tio n a b o u t th e b eh a v io u r o f th e u n iv erse is co n ta in ed in th is sin g le fu n ctio n w h ich { in tu rn { ca n b e d eterm in ed b y E in stein 's eq u a tio n if w e k n ow th e co n ten ts o f th e u n iv erse 1 . H ow ev er, ev en w ith o u t k n ow in g th e ex p licit fo rm o f a (t), o n e ca n ¯ g u re o u t a lo t o f th in g s a b o u t su ch a u n iv erse, a s w e sh a ll see. T h e k ey trick is to n o tice th a t, a t a n y g iv en tim e t, o n e ca n in tro d u ce a n ew sp a tia l co o rd in a te r (t) ´ a (t)x so th a t, a t th is in sta n t o f tim e, th e sp a ce lo o k s ju st lik e w h a t w e a re a ccu sto m ed to in sp ecia l rela tiv ity. T h e r is ca lled `p ro p er co o rd in a te' w h ile x is ca lled th e `co m ov in g co o rd in a te'. S in ce th e sp a ce lo o k s fa m ilia r in term s o f r , o n e u ses sta n d a rd law s o f p h y sics in term s o f th ese p ro p er co o rd in a tes a n d th en tra n sla tes th em b a ck to x , h o p in g fo r th e b est. A m a zin g ly, it w o rk s fo r m o st p u rp o ses.
1
As usual, we are simplifying the universe a little bit; it turns out that you actually need one more number characterizing the curvature of the space to describe the universe completely. But observations show that this number is quite close to zero and hence I will ignore it.
T o b eg in w ith , co n sid er tw o p a rticles lo ca ted a t x 1 a n d x 2 = x 1 + ±x w h ich a re in ¯ n itesim a lly sep a ra ted . T h e co o rd in a te d ista n ce b etw een th ese tw o p a rticles is j± x j w h ile th e p ro p er d ista n ce is j± l(t)j = a (t)j±x j. W e n ow n o te th a t, ev en if th e p a rticles d o n o t m ov e in term s o f x -co o rd in a te (i.e., ea ch p a rticle h a s a ¯ x ed x -co o rd in a te w h ich d o es n o t ch a n g e w ith tim e) th eir proper sep a ra tio n ch a n g es w ith tim e b eca u se o f th e a (t) fa cto r. T h e rela tiv e v elo city a t w h ich th ese p a rticles a re m ov in g fro m ea ch o th er is g iv en b y d ±l a_ = a_± x = ± l : (3 ) ±v = dt a T h is resu lt, a s w e ca n see, is essen tia lly H u b b le's law ! It sh ow s th a t th e tw o p a rticles a re m ov in g aw ay fro m
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ea ch o th er w ith a sp eed p ro p o rtio n a l to th eir sep a ra tio n w h en H (t) ´ (a_= a ) is p o sitiv e. G iv en th is resu lt, o n e ca n o b ta in sev era l o th er in terestin g co n seq u en ces. S u p p o se a n a rrow p en cil o f (n ea rly ) m o n o ch ro m a tic electro m a g n etic ra d ia tio n cro sses th ese tw o co m ov in g o b serv ers lo ca ted a t x 1 a n d x 2 = x 1 + ± x . W e w a n t to k n ow w h a t freq u en cy th ese tw o o b serv ers w ill a ttrib u te to th e electro m a g n etic ra d ia tio n . T h e tim e fo r th e electro m a g n etic ra d ia tio n to trav erse th e d ista n ce ± l w ill b e ± t = ± l= c. L et th e freq u en cy o f th e ra d ia tio n m ea su red b y th e ¯ rst o b serv er b e ! . S in ce th e ¯ rst o b serv er sees th e seco n d o n e to b e recedin g w ith v elo city ± v , sh e w ill ex p ect th e seco n d o b serv er to m ea su re a D o p p ler sh ifted freq u en cy (! + ± ! ), w h ere ±! ±v a_ ± l a_ ±a = ¡ = ¡ = ¡ ±t = ¡ : ! c a c a a
(4 )
H ow d o es o n e in terp ret th is rela tio n ? It sh ow s th a t th e freq u en cy o f electro m a g n etic ra d ia tio n a s m ea su red b y th e co m ov in g o b serv ers ch a n g es w h en th e ex p a n sio n fa cto r a (t) ch a n g es w ith tim e. If ± a is p o sitiv e (i.e., if th e u n iv erse is ex p a n d in g ), ± ! is n eg a tiv e in d ica tin g a red sh ift in th e freq u en cy o f ra d ia tio n . In fa ct, th e a b ov e eq u a tio n ca n b e im m ed ia tely in teg ra ted to g iv e ! (t)a (t) = co n sta n t :
The frequency of electromagnetic radiation changes due to expansion of the universe according to the law ω ∝ a–1.
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(5 )
W e th u s co n clu d e th a t th e freq u en cy o f electro m a g n etic ra d ia tio n ch a n g es d u e to ex p a n sio n o f th e u n iv erse a cco rd in g to th e law ! / a ¡1 . T h is a p p ro a ch w o rk s b eca u se, in a n in ¯ n itesim a l reg io n a ro u n d a n ev en t, o n e ca n a lw ay s u se th e law s o f sp ecia l rela tiv ity. (O n e ca n th in k o f it a s a va ria n t o f th e so -ca lled prin ciple of equ ivalen ce w h ich essen tia lly tells y o u th a t th e g en u in e e® ects o f sp a cetim e cu rva tu re a re seco n d o rd er in th e sep a ra tio n b etw een clo se ev en ts.) T h is is tru e in a n y sp a cetim e b u t w e w ill n o t u su a lly b e a b le to in teg ra te th e lo ca l resu lt a n d o b ta in a g lo b a l law in a g en era l sp a cetim e w h en
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th e g eo m etry va ries fro m p o in t to p o in t in sp a ce. W e co u ld a ch iev e it in th is p a rticu la r ca se b eca u se th e m o d i¯ ca tio n o f sp a cetim e in terva l fro m th e o n e in (1 ) to th e o n e in (2 ) d id n o t in v o lv e a n y fu n ctio n th a t d ep en d ed o n th e spatial co o rd in a tes. In fa ct, o n e ca n o b ta in a sim ila r resu lt fo r a n y p a rticle, n o t ju st p h o to n s. T o d o th is, let u s co n sid er a m a teria l p a rticle w h ich p a sses th e ¯ rst o b serv er w ith v elo city v . W h en it h a s cro ssed th e p ro p er d ista n ce ± l (in a tim e in terva l ± t), it p a sses th e seco n d o b serv er w h o se v elo city (rela tiv e to th e ¯ rst o n e) is ±u =
a_ a_ ±a ±l = vd t = v : a a a
(6 )
T h e v elo city a ttrib u ted to th is p a rticle b y th e seco n d o b serv er ca n b e o b ta in ed b y u sin g th e sp ecia l rela tiv istic law fo r th e a d d itio n o f v elo cities. T h is g iv es · ¸ v ¡ ±u (± u )2 v2 0 v = = v ¡ (1 ¡ 2 )± u + O 1 ¡ (v ± u = c 2 ) c c2 = v ¡ (1 ¡
v 2 ±a )v : c2 a
(7 )
v 2 ±a ) c2 a
(8 )
R ew ritin g th is eq u a tio n a s ± v = ¡ v (1 ¡ a n d in teg ra tin g , w e g et p = p
v co n sta n t = : a 1 ¡ (v 2 = c 2 )
(9 )
In o th er w o rd s, th e m a g n itu d e o f th e 3 -m o m en tu m d ecrea ses a s a ¡ 1 d u e to th e ex p a n sio n . If th e p a rticle is n o n -rela tiv istic, th en v / p a n d v elo city itself d eca y s a s a ¡1 .
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O n ce w e h av e th e sca lin g o f th e m o m en tu m o f p a rticles a n d p h o to n s, o n e ca n p ro ceed fu rth er a n d u n d ersta n d h ow th e en erg y d en sity o f ra d ia tio n ch a n g es in a n ex p a n d in g u n iv erse. T o d o th is, let u s co n sid er th e d escrip tio n o f a b u n ch o f p h o to n s in term s o f a d istrib u tio n fu n ctio n f (r ;p ;t) in p h a se sp a ce. A s u su a l, d N = f (r ;p ;t)d 3 rd 3 p g iv es th e n u m b er o f p h o to n s in a sm a ll p h a se v o lu m e. In ex p a n d in g co o rd in a tes, th e sp a tia l v o lu m e in crea ses a s a 3 w h ile, fro m o u r p rev io u s resu lt, w e k n ow th a t th e m o m en tu m sp a ce v o lu m e d ecrea ses a s a ¡3 . S o th e p h a se sp a ce v o lu m e elem en t is in va ria n t a n d { sin ce d N is in va ria n t { th e d istrib u tio n fu n ctio n f rem a in s in va ria n t a s th e u n iv erse ex p a n d s. E x p ressin g th e m o m en tu m o f th e p h o to n a s p = (~! = c)^p , w h ere p^ is a u n it v ecto r in th e d irectio n o f p ro p a g a tio n , w e ¯ n d th a t th e m o m en tu m sp a ce v o lu m e is p ro p o rtio n a l to ! 2 d ! d − w h ere d − d en o tes th e so lid a n g le in th e d irectio n o f th e m o m en tu m p^ . S o w e ca n a lso w rite d N = f (r;p ;t)d 3 rd 3 p / f (r;! ; p^ ;t)! 2 d 3 r d ! d − : (1 0 )
Using our previous result that the distribution function remains invariant, we conclude that ρ (ω) /ω3 remains invariant as the universe expands.
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F u rth er, b eca u se f g iv es th e n u m ber d en sity o f p h o to n s p er u n it p h a se v o lu m e, th e co rresp o n d in g en erg y d en sity is g iv en b y (~! )f . It fo llow s th a t th e en erg y d en sity o f ra d ia tio n p er u n it ra n g e o f freq u en cy is p ro p o rtio n a l to ½ / ! 3 f . U sin g o u r p rev io u s resu lt th a t th e d istrib u tio n fu n ctio n rem a in s in va ria n t, w e co n clu d e th a t ½ (! )= ! 3 rem a in s in va ria n t a s th e u n iv erse ex p a n d s. T h is h a s a v ery in terestin g co n seq u en ce. S u p p o se th e u n iv erse is ¯ lled w ith a ra d ia tio n b a th , th e en erg y d en sity o f w h ich h a s th e fo rm ½ (! ) = ! 3 F (! = ® ), w h ere ® is so m e p a ra m eter a n d F is so m e a rb itra ry fu n ctio n o f its a rg u m en t. A s th e u n iv erse ex p a n d s, ½ = ! 3 rem a in s in va ria n t w h ile ! itself ch a n g es a s ! / a (t)¡1 . If w e d en o te th e va lu es m ea su red to d ay b y a su b scrip t 0 , th en ! 0 = ! (t)a (t)= a 0 . It fo llow s th a t fo r th e fa cto r ! 0 = ® , w e
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ca n w rite !0 ! (t)(a (t)= a 0 ) ! (t) = = : ® ® ® (a 0 = a (t))
As the universe
(1 1 )
T h is sh ow s th a t th e red sh iftin g o f th e freq u en cy ca n b e eq u iva len tly th o u g h t o f a s resca lin g o f th e p a ra m eter ® a s th e u n iv erse ex p a n d s w ith ® (t) / 1 = a (t). S o th e ra d ia tio n en erg y d en sity o f th e fo rm ½ (! ) = ! 3 F (! = ® ) reta in s its sh a p e a s th e u n iv erse ex p a n d s, ex cep t fo r a n ov era ll sca lin g .
expands, a Planck spectrum remains a Planck spectrum with the temperature redshifting according to the law T ∝ a–1.
T h e P la n ck sp ectru m o f ra d ia tio n is o n e sp ecia l ca se in w h ich en erg y d en sity h a s th e a b o v e-m en tio n ed fu n ctio n a l fo rm w ith ½ / ! 3 [ex p (~! = k T ) ¡ 1 ]¡ 1 ´ ! 3 F (! = T ) :
(1 2 )
T h e releva n t p a ra m eter n ow is th e tem p era tu re o f th e ra d ia tio n . T h erefo re, a s th e u n iv erse ex p a n d s, a P la n ck sp ectru m rem a in s a P la n ck sp ectru m w ith th e tem p era tu re red sh iftin g a cco rd in g to th e law T / a ¡1 . It sh o u ld b e stressed th a t th is resu lt h a s n o th in g to d o w ith th erm a l eq u ilib riu m ! In fa ct, th e situ a tio n w e a re co n sid erin g is p recisely th e o th er ex trem e o f th erm a l eq u ilib riu m in w h ich th e ra d ia tio n h a s co m p letely d eco u p led fro m m a tter. T o cla rify th is p o in t, let m e b rie° y d escrib e w h a t h a p p en s in o u r rea l u n iv erse. V ery ea rly in th e ev o lu tio n o f th e u n iv erse, ch a rg ed p a rticles a n d p h o to n s w ere stro n g ly co u p led to ea ch o th er a n d ex isted in th e fo rm o f p la sm a in real th erm o d y n a m ic eq u ilib riu m . S u ch a stro n g co u p lin g im p lies th a t th e ra d ia tio n w ill b e th erm a lized a n d its sp ectra l d istrib u tio n w o u ld h av e th e P la n ck ia n fo rm . L et u s a ssu m e th a t a t so m e in sta n t o f tim e, w e sw itch o ® a ll th e in tera ctio n b etw een ra d ia tio n a n d m a tter (in o u r u n iv erse th is h a p p en ed w h en it w a s a b o u t o n e-th o u sa n d th o f th e p resen t size). F ro m th a t ep o ch o n w a rd s, ea ch o f th e p h o to n s h a s b een p ro p a g a tin g in th e ex p a n d in g u n iv erse w ith its freq u en cy red sh iftin g a cco rd in g to th e law ! / a ¡ 1 . T h e p h o to n s a re
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In Wheelerian language, we get “thermal equilibrium without thermal equilibrium”.
n o t in tera ctin g w ith m a tter: th ere is n o ex ch a n g e o f en erg y a n d th ere is n o p ro cess w h ich `m a in ta in s' th erm a l eq u ilib riu m . N ev erth eless, a t p resen t, th e p h o to n s w ill b e d escrib ed b y a P la n ck d istrib u tio n w ith a red sh ifted tem p era tu re, b eca u se th e fo rm o f th e P la n ck sp ectru m w ill b e p reserv ed w ith th e tem p era tu re o f th e ra d ia tio n d ecrea sin g a s T / a ¡1 u n d er co sm ic ex p a n sio n . T h is is a p u rely k in em a tic e® ect o ccu rrin g fo r p h o to n s w h ich a re p ro p a g a tin g freely th ro u g h th e u n iv erse. In W h eeleria n la n g u a g e, w e g et \ th erm a l eq u ilib riu m w ith o u t th erm a l eq u ilib riu m " . L et u s n ex t ta k e a clo ser lo o k a t th e d y n a m ics o f n o n rela tiv istic p a rticles in su ch a n ex p a n d in g u n iv erse. C o n sid er a p a rticle lo ca ted a t th e co m ov in g co o rd in a te x co rresp o n d in g to th e p ro p er co o rd in a te r (t) = a (t)x . E v en if x d o es n o t ch a n g e w ith tim e { i.e., ev en if th e p a rticle h a s co n sta n t co m o v in g co o rd in a te { its p ro p er co o rd in a te r w ill ch a n g e w ith tim e d u e to a (t). T h is w ill in d u ce a n a ccelera tio n o n th e p a rticle g iv en b y Är = (Äa = a )r. G iv en o u r u su a l p reju d ice th a t a ccelera tio n s a rise d u e to fo rces, it seem s n a tu ra l to a ttrib u te th is a ccelera tio n to th e ex isten ce o f a g lo b a l \ co sm ic p o ten tia l" © = ¡ (1= 2 )(Äa = a )r 2 , so th a t w e ca n w rite Är = ¡ r r © . (T h e su b scrip t r in r r is to rem in d o u rselv es th a t th e g ra d ien t is w ith resp ect to r a n d n o t x ; n o te th a t r r = a ¡ 1 r x .) W ith in th e co n tex t o f su ch N ew to n ia n co n sid era tio n s, w e ca n a ttrib u te th is p o ten tia l to a m a ss d en sity ½ b g (t) su ch th a t r 2r © = 4¼ G ½ b g (t). S im p le d ifferen tia tio n o f © g iv es aÄ 4¼ G = ¡ ½ b g (t) a 3
(1 3)
w h ich rela tes th e ex p a n sio n fa cto r a (t) to a u n ifo rm b a ck g ro u n d d en sity ½ b g (t) o f m a tter in th e u n iv erse. It a ll seem s n a tu ra l to a ssu m e th a t th e to ta l n u m b er o f p a rticles w ith in a p ro p er v o lu m e sh o u ld n o t ch a n g e a s th e u n iv erse ex p a n d s, th ereb y su g g estin g ½ b g (t) / a ¡ 3 . If y o u su b stitu te th is in to (1 3), th en it is ea sy to sh ow
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th a t a (t) / t2 = 3 ev en th o u g h w e w ill n o t req u ire th is resu lt in o u r d iscu ssio n . In th is a p p ro a ch , w e th in k o f a la rg e n u m b er o f p a rticles b ein g d istrib u ted u n ifo rm ly th ro u g h o u t th e u n iv erse a n d a ttrib u te © to su ch a co llectio n o f p a rticles. T h is is, o f co u rse, co n cep tu a lly a d u b io u s p ro ced u re w ith in th e co n tex t o f N ew to n ia n g rav ity b u t it tu rn s o u t th a t { fo r th e sp eci¯ c ca se u n d er d iscu ssio n { g en era l rela tiv ity lea d s to th e sa m e resu lt. G iv en th is in terp reta tio n , o n e ca n a sk w h a t h a p p en s if w e p ertu rb th e d en sity in th e u n iv erse in a sp a ced ep en d en t m a n n er so th a t ½ b g (t) ! ½ b g (t)[1 + ± (t;x )]. T h e p o ten tia l w ill ch a n g e to ª ´ © + Á w ith th e ex tra b it Á p ro d u ced b y th e ex tra d en sity ½ b g (t)±(t;x ); th a t is: r 2r Á =
1 2 r Á = 4 ¼ G ½ b g (t)± (t;x ) : a2 x
(1 4 )
W e w o u ld lik e to in terp ret th is situ a tio n in term s o f a sy stem o f a la rg e n u m b er o f p a rticles w ith co m ov in g p o sitio n s x i, w h ere i = 1;2;::: la b els ea ch o f th e p a rticles. A s lo n g a s a ll th e p a rticles h av e co n sta n t va lu es fo r x i , a ll o f th em w ill m ov e aw ay fro m ea ch o th er d u e to th e ex p a n sio n w h ich w e a ttrib u te to th e p o ten tia l © . T h is is th e sp a tia lly u n ifo rm d en sity situ a tio n . B u t if th e p a rticles a re d istu rb ed fro m th eir co n sta n t x i va lu es, th en th e m a tter d en sity w ill b eco m e n o n u n ifo rm { w ith ± (t;x ) 6= 0 { a n d th e p o ten tia l © is m o d i¯ ed to ª ´ © + Á . O f co u rse, ea ch o f th e p a rticles w ill n ow feel th e fo rce d u e to th is to ta l p o ten tia l ª . T h e a ccelera tio n o f th e j -th p a rticle, n ow g iv en b y d 2 rj = aÄ x j + 2 aÄ x_ j + a xÄ j d t2
(1 5 )
a rises d u e to th e g ra d ien t o f th e m o d i¯ ed p o ten tia l ª ´ © + Á . W e n o te th a t th e ¯ rst term in th e rig h t-h a n d sid e o f (1 5 ) is (Äa = a )r w h ich is ju st ¡ r r © . T h erefo re, r r Á (= a ¡1 r x Á ) sh o u ld lea d to th e o th er tw o term s in
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Equation (16) is the key to understanding gravitational clustering in an expanding background.
(1 5 ). W ith so m e rea rra n g em en t, th is lea d s to a_ 1 Äx j + 2 x_ j = ¡ 2 r x Á : (1 6 ) a a T h is eq u a tio n tells y o u h ow th e co m ov in g co o rd in a tes o f th e p a rticles ch a n g e th ereb y m a k in g th e d en sity d istrib u tio n o f p a rticles in th e u n iv erse n o n -u n ifo rm w h ich , in tu rn , g iv es rise to r x Á . T h e p o ten tia l Á ca n b e th o u g h t o f a s b ein g g en era ted b y th e p ertu rb a tio n s fro m th e u n ifo rm d en sity o f p a rticles. T h is eq u a tio n is th e k ey to u n d ersta n d in g g rav ita tio n a l clu sterin g in a n ex p a n d in g b a ck g ro u n d . E a ch o f th e term s in th is eq u a tio n h a s a n in terestin g in terp reta tio n . T h e ¯ rst term xÄ j is th e a ccelera tio n co rresp o n d in g to th e co m ov in g p o sitio n o f th e p a rticle, w h ich a rises ov er a n d a b ov e th e a ccelera tio n d u e to th e b a ck g ro u n d ex p a n sio n (th e ¯ rst term in (1 5 ) w h ich w e h av e a lrea d y a cco u n ted fo r b y th e g ra d ien t o f co sm ic p o ten tia l). T h e seco n d term is a d a m p in g (frictio n ) term w h ich tries to d ecrea se th e sp eed o f th e p a rticle. In fa ct, if th e rig h t-h a n d sid e o f (1 6 ) is zero , th e eq u a tio n ca n b e in teg ra ted to g iv e a 2 x_ = co n sta n t o r, a ltern a tiv ely a x_ = 1= a . T h is ca n b e rew ritten in th e fo rm 1 : (1 7 ) a T h e left-h a n d sid e is th e d ev ia tio n o f th e p ro p er v elo city r_ fro m th e H u b b le ex p a n sio n v elo city H r_. T h e q u a n tity a x_ is so m etim es ca lled `p ecu lia r v elo city ' fo r n o g o o d rea so n . T h e rig h t-h a n d sid e sh ow s th a t th is d i® eren ce d ecay s d ow n a 1= a so th a t { in th e a b sen ce o f th e r x Á term o n th e rig h t h a n d sid e o f (1 6 ) { p a rticles w ill ten d to a p p ro a ch th e co sm ic ex p a n sio n v elo city. T h is is th e k ey e® ect o f th e `frictio n ' term . a x_ = r_ ¡ H r =
F in a lly, th e g ra d ien t in th e rig h t h a n d sid e o f (1 6 ) is w h a t is k eep in g th e a ccelera tio n a liv e a n d lea d s to g rav ita tio n a l clu sterin g . H ere to o , th ere is o n e cu rio u s fea tu re. T h e p ertu rb a tio n o f th e b a ck g ro u n d d en sity, ±(t;x ),
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ca n b e eith er p o sitiv e o r n eg a tiv e in a n y g iv en reg io n (ex cep t fo r th e co n d itio n th a t ± > ¡ 1 to k eep ½ > 0 ). T h erefo re, th e so u rce fo r th e p o ten tia l Á in (1 4 ) ca n b e p o sitiv e d en sity o r n eg a tiv e d en sity ! (T h is is so m ew h a t lik e electro sta tics in w h ich th e ch a rg e d en sity ca n b e p o sitiv e o r n eg a tiv e.) S o th e g rav ita tio n a l fo rce p ro d u ced b y th is d istrib u tio n ca n b e a ttra ctiv e o r rep u lsiv e in th e rig h t-h a n d sid e o f (1 6 )! In fa ct a n u n d erd en se reg io n o f th e u n iv erse { u su a lly ca lled a `v o id ' in th e d istrib u tio n o f g a la x ies { ex erts a n e® ectiv e rep u lsiv e fo rce o n th e su rro u n d in g m a tter.
An underdense region of the universe – usually called a ‘void’ in the distribution of galaxies – exerts an effective repulsive force on the surrounding matter.
B y m u ltip ly in g (1 6 ) th ro u g h o u t b y a 2 w e ca n reca st it in th e fo rm d 2 (a x_ j ) = ¡ r x Á : dt
(1 8 )
O b v io u sly, th is eq u a tio n fo r ea ch o f th e p a rticles ca n b e o b ta in ed fro m a n ex p licitly tim e-d ep en d en t L a g ra n g ia n o f th e fo rm ¶ X µ1 2 2 L = m a x_j ¡ m Á ´ K ¡ U ; (1 9 ) 2 j w h ere w e h av e su m m ed ov er a ll th e p a rticles w ith th e u n d ersta n d in g th a t Á a t th e lo ca tio n o f ea ch p a rticle is p ro d u ced b y th e rest o f th e p a rticles. T h is L a g ra n g ia n , in tu rn , w ill lea d to a H a m ilto n ia n o f th e fo rm ¶ X µ p 2j H = + m Á ´ K + U: (2 0 ) 2 2m a j T h is a llow s u s to o b ta in a ra th er in terestin g a n d cu rio u s resu lt. W e ¯ rst reca ll th a t w h en ev er a H a m ilto n ia n d ep en d s ex p licitly o n tim e, w e h av e th e resu lt (d H = d t) = (@ H = @ t) w h ere th e rig h t-h a n d sid e is eva lu a ted k eep in g th e co o rd in a tes a n d m o m en ta co n sta n t. F ro m th e fo rm o f H , w e im m ed ia tely g et (@ K = @ t) = ¡ 2 (a_= a )K . O n th e o th er h a n d , th e p o ten tia l en erg y b etw een a n y tw o p a rticles va ries a s jr i ¡ r j j¡ 1 = a ¡1 jx i ¡ x j j¡1 . T h is im p lies
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th a t (@ U = @ t) = ¡ (a_= a )U . P u ttin g a ll th ese to g eth er, w e g et th e resu lt a_ d (K + U ) = ¡ (2K + U ): dt a
(2 1 )
T h is rela tio n g o es u n d er th e n a m e `C o sm ic V iria l th eo rem ' (o r `C o sm ic E n erg y eq u a tio n ') a n d is o n e o f th e few ex a ct resu lts y o u ca n o b ta in ra th er ea sily in th is p a rticu la r co n tex t. T h e left-h a n d sid e o f th e eq u a tio n rep resen ts th e ra te o f ch a n g e o f to ta l en erg y o f a co llectio n o f p a rticles. T h e rig h t-h a n d sid e tells y o u th a t (K + U ) is n o t co n serv ed fo r su ch a sy stem o f p a rticles ex cep t in tw o d i® eren t co n tex ts. T h e ¯ rst o n e is th e ra th er triv ia l ca se o f a_ = 0 w h ich is ju st sta n d a rd cla ssica l m ech a n ics w ith o u t a n y b a ck g ro u n d ex p a n sio n a n d th e to ta l en erg y is, o f co u rse, co n serv ed . T h e seco n d { a n d m o re cu rio u s situ a tio n { co rresp o n d s to 2 K + U = 0 w h ich y o u w ill reco g n ize is th e sta n d a rd v iria l eq u ilib riu m co n d itio n fo r a set o f p a rticles in tera ctin g v ia N ew to n ia n g rav ity. In p ra ctica l term s, th is resu lt im p lies th e fo llow in g . S u p p o se d u rin g th e ev o lu tio n o f th e u n iv erse a b u n ch o f p a rticles co m e to g eth er a n d fo rm a v iria lized self-g rav ita tin g clu ster. T h en , to th e ex ten t w e ca n ig n o re th e in tera ctio n o f th is clu ster w ith th e rest o f th e p a rticles in th e u n iv erse, its en erg y w ill b e co n serv ed . R o u g h ly sp ea k in g , su ch v iria lized clu sters d o n o t p a rticip a te in th e co sm ic d y n a m ics. F in a lly I w a n t to d escrib e a n in terestin g a n d ex a ct b o u n d o n th e k in etic en erg y o f p a rticles in su ch a clu ster fo rm ed in th e ex p a n d in g u n iv erse, w h ich ca n b e o b ta in ed fro m th e co sm ic v iria l th eo rem . T o d o th is, w e ¯ rst n o te th a t (2 1 ) ca n a lso b e w ritten a s: Virialized clusters do not participate in the cosmic dynamics.
270
d a (K + U ) = ¡ K a_ < 0 dt
(2 2 )
S o w e k n ow th a t a (K + U ) is a d ecrea sin g fu n ctio n o f tim e. V ery ea rly o n , w h en n o sig n i¯ ca n t clu sterin g h a s
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o ccu rred , w e w ill h av e K ¼ 0 a n d U < 0 , m a k in g (K + U ) n eg a tiv e v ery ea rly in th e ev o lu tio n . It th en fo llo w s th a t w e m u st h av e K < ¡ U o r ra th er K < jU j a t a ll tim es. N ex t, n o te th a t (2 1 ) ca n a lso b e w ritten a s 1 d 2 a2 dU a (K + U ) = ¡ : dt 2 2 dt
(2 3 )
A s stru ctu res d ev elo p in th e u n iv erse, p o ten tia l w ells w ill g et d eep er a n d d eep er a n d h en ce d (¡ U )= d t > 0 m a k in g th e left-h a n d sid e p o sitiv e. In th e ea rly sta g es, sin ce U ¼ 0 , w e h av e (K + U = 2 ) > 0 . H en ce w e co n clu d e th a t, a t a n y la ter tim e K > ¡ (1= 2 )U o r K > (1 = 2 )jU j. C o m b in in g w ith th e p rev io u s resu lt, w e g et 1 jU j < K < jU j : 2
(2 4 )
T h is is a ra th er n ea t resu lt o n th e rela tio n sh ip b etw een k in etic a n d p o ten tia l en erg ies o f stru ctu res fo rm ed b y g rav ita tio n a l clu sterin g , w h ich m u st h o ld in d ep en d en t o f th e d eta ils o f th e p ro cess! Suggested Reading [1]
P J E Peebles, Large scale structure of the universe, Princeton University Press, 1980, section 24 .
[2]
T Padmanabhan, Structure formation in the universe, Cambridge University Press, 1992, Chapter 4.
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Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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Snippets of Physics 16. Lagrange has (more than) a Point! T Padmanabhan
A so lu tio n to th e 3 -b o d y p ro b le m in g ra v ity , d u e to L a g ra n g e , h a s se v e ra l re m a rk a b le fe a tu r e s. In p a r tic u la r, it d e sc rib e s a situ a tio n in w h ich a p a r tic le , lo c a te d a t th e m a x im a o f a p o te n tia l, c a n r e m a in sta b le a g a in st sm a ll p e rtu rb a tio n s. T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Keywords Classical mechanics, gravitation, rotation.
318
M o tio n o f b o d ies u n d er th eir m u tu a l g rav ita tio n a l a ttra ctio n is o f h isto rica l, th eoretica l a n d ev en p ra ctica l (th a n k s to th e sp a ce-a g e a n d sa tellites) im p o rta n ce. T h e sim p lest ca se o f tw o b o d ies { co rresp o n d in g to th e so ca lled K ep ler p ro b lem { a lread y p o ssesses sev era l in terestin g fea tu res, lik e th e ex isten ce o f a n ex tra in teg ra l a n d th e fa ct th a t th e tra jecto ry o f th e p a rticle in th e v elo city sp a ce is a circle (see, fo r ex a m p le, [1 ]). T h e situ a tio n b eco m es m o re in terestin g , b u t a lso terrib ly co m p lica ted , w h en w e a d d a th ird p a rticle to th e fray. T h e 3 -b o d y p ro b lem , a s it is ca lled , h a s attra cted th e a tten tio n o f sev era l d y n a m icists a n d a stro n o m ers b u t, u n fo rtu n a tely, it d o es n o t p o ssess a clo sed solu tio n . W h en a n ex a ct p ro b lem ca n n o t b e so lv ed , p h y sicists lo o k a ro u n d fo r a sim p ler v ersio n o f th e p ro b lem w h ich w ill a t lea st ca p tu re so m e featu res o f th e o rig in a l o n e. O n e su ch ca se co rresp o n d s to w h a t is k n ow n a s th e restricted three-body problem w h ich co u ld b e d escrib ed a s fo llow s. C o n sid er tw o p a rticles o f m a sses m 1 a n d m 2 w h ich o rb it a ro u n d th eir co m m o n cen tre o f m a ss, ex a ctly a s in th e ca se o f th e sta n d a rd K ep ler p ro b lem . W e n ow co n sid er a th ird p a rticle of m a ss m 3 w ith m 3 ¿ m 1 a n d m 3 ¿ m 2 w h ich is m ov in g in th e g rav ita tio n a l ¯ eld o f th e tw o p a rticles m 1 a n d m 2 . S in ce it is fa r less m a ssiv e th a n th e o th er tw o p a rticles, w e w ill a ssu m e th a t it b eh av es lik e a test p a rticle a n d d o es n o t a ® ect th e
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o rig in a l m o tio n o f m 1 a n d m 2 . Y o u ca n see th a t th is is eq u iva len t to stu d y in g th e m o tio n o f m 3 in a tim ed ep en d en t ex tern a l g rav ita tio n a l p o ten tia l p ro d u ced b y m 1 a n d m 2 . G iv en th e fa ct th a t w e lo se b o th th e tim e tra n sla tio n in va ria n ce a n d a x ia l sy m m etry, a n y h o p e fo r sim p le a n a ly tic so lu tio n s is m isp la ced . B u t th ere is a sp ecia l ca se { d escrib ed in th is in sta llm en t { fo r w h ich a b ea u tifu l so lu tio n ca n b e o b ta in ed .
Traditionally, the maxima of a potential have bad press due to their tendency to induce instability.
T h is co rresp o n d s to a situ a tio n in w h ich a ll th e th ree p a rticles m a in ta in th eir rela tiv e p o sitio n s w ith resp ect to o n e a n o th er b u t ro ta te rig id ly in sp a ce w ith a n a n g u la r v elo city ! ! In fa ct, th e th ree p a rticles a re lo ca ted a t th e v ertices o f a n eq u ila tera l tria n g le irresp ectiv e o f th e ra tio o f th e m a sses m 1 = m 2 . If y o u th in k a b o u t it, y o u w ill ¯ n d th a t th is so lu tio n , ¯ rst fo u n d b y L a g ra n g e, is q u ite eleg a n t a n d so m ew h a t co u n ter-in tu itiv e. H ow d o y o u b a la n ce th e fo rces, w h ich d ep en d o n m a ss ra tio s, w ith o u t a d ju stin g th e d ista n ce ra tio s b u t a lw ay s m a in ta in in g th e eq u ila tera l co n ¯ g u ra tio n ? W h a t is m o re, th e lo ca tio n o f m 3 h a p p en s to b e a t th e lo ca l m axim u m o f th e e® ectiv e p o ten tia l in th e fra m e co -ro ta tin g w ith th e sy stem . T ra d itio n a lly, th e m a x im a o f a p o ten tia l h av e b a d p ress d u e to th eir ten d en cy to in d u ce in sta b ility. It tu rn s o u t th a t, in th is so lu tio n , sta b ility ca n b e m a in ta in ed (fo r a rea so n a b le ra n g e o f p a ra m eters) b eca u se o f th e ex isten ce o f C o rio lis fo rce { w h ich is o n e o f th e th in g s m a n y stu d en ts d o n o t h av e a n in tu itiv e g ra sp o f. I w ill n ow o b ta in th is so lu tio n a n d d escrib e its p ro p erties leav in g (a s u su a l!) th e d eta iled a lg eb ra fo r y o u to w o rk o u t. If th e sep a ra tio n b etw een m 1 a n d m 2 is a , th e sta n d a rd K ep ler so lu tio n tells u s th a t th ey ca n ro ta te in circu la r o rb its a ro u n d th e cen tre o f m a ss w ith th e a n g u la r v elo city g iv en b y !2 =
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G (m
1
+ m 2)
a3
:
(1 )
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The Coriolis force has the form identical to the force exerted by a magnetic field (2m/q)on a particle of charge q.
S in ce L a g ra n g e h a s sh ow n th a t a rig id ly ro ta tin g so lu tio n ex ists w ith th e th ird b o d y, w e w ill sav e w o rk b y stu d y in g th e p ro b lem in th e co o rd in a te sy stem co ro ta tin g w ith th e m a sses, in w h ich th e th ree b o d ies w ill b e a t rest. W e w ill ¯ rst w o rk o u t th e eq u a tio n s o f m o tio n in a ro ta tin g fra m e b efo re p ro ceed in g fu rth er. T h is is m o st ea sily d o n e b y sta rtin g fro m th e L a g ra n g ia n fo r a p a rticle L (x ; x_ ) = (1 = 2 )m x_ 2 ¡ V (x ) a n d tra n sfo rm in g to a ro ta tin g fra m e b y u sin g th e tra n sfo rm a tio n law v in ertia l = v ro t + ! £ x . T h is lea d s to th e L a g ra n g ia n o f th e fo rm 1 1 L = m v 2 + m v ¢ (! £ x ) + m (! £ x )2 ¡ V (x ) (2 ) 2 2 a n d eq u a tio n s o f m o tio n @V dv = ¡ + 2 m v £ ! + m ! £ (x £ ! ) (3 ) dt @x W e see th a t th e tra n sfo rm a tio n to a ro ta tin g fra m e in tro d u ces tw o a d d itio n a l fo rce term s in th e rig h t-h a n d sid e o f (3 ) o f w h ich th e 2 m (v £ ! ) is ca lled th e C o rio lis fo rce a n d m ! £ (x £ ! ) is th e m o re fa m ilia r cen trifu g a l fo rce. T h e C o rio lis fo rce h a s th e fo rm id en tica l to th e fo rce ex erted b y a m a g n etic ¯ eld (2 m = q )! o n a p a rticle o f ch a rg e q. It fo llow s th a t th is fo rce ca n n o t d o a n y w o rk o n th e p a rticle sin ce it is a lw ay s o rth o g o n a l to th e v elo city. T h e cen trifu g a l fo rce, o n th e o th er h a n d , ca n b e o b ta in ed a s th e g ra d ien t o f a n e® ectiv e p o ten tia l w h ich is th e th ird term in th e rig h t-h a n d sid e o f (2 ). m
W e a re n ow rea d y to ¯ n d th e rig id ly ro ta tin g so lu tio n in w h ich a ll th e th ree p a rticles a re a t rest in th e ro ta tin g fra m e in w h ich (3 ) h o ld s. W e w ill ch o o se a co o rd in a te sy stem in w h ich the test particle is at the origin a n d d en o te b y r 1 ;r 2 th e p o sitio n v ecto rs o f m a sses m 1 a n d m 2 . T h e p o sitio n o f th e cen tre o f m a ss o f m 1 a n d m 2 w ill b e d en o ted b y r so th a t (m
320
1
+ m 2 )r = m 1 r 1 + m 2 r 2 :
(4 )
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In th e so lu tio n w e a re lo o k in g fo r, a ll th ese th ree v ecto rs a re in d ep en d en t o f tim e in th e ro ta tin g fra m e a n d th e C o rio lis fo rce v a n ish es b eca u se v = 0 . S in ce m 1 a n d m 2 a re a lrea d y ta k en ca re o f (a n d a re a ssu m ed to b e o b liv io u s to m 3 ), w e o n ly n eed to sa tisfy th e eq u a tio n o f m o tio n fo r m 3 w h ich d em a n d s: G m 1 G m 2 r1 + r 2 = ! 2 r: 3 3 r1 r2
(5 )
To work out the exact position of equilibrium, one has to solve a fifthorder equation which will lead to three real roots.
Y o u sh o u ld n ow b e a b le to see th e eq u ila tera l tria n g le em erg in g . If w e a ssu m e r 1 = r 2 , a n d ta k e n o te o f (4 ), th e left-h a n d sid e o f (5 ) ca n b e red u ced to (G = r 13 )(m 1 + m 2 )r w h ich is in th e d irectio n o f r. If w e n ex t set r 1 = a , th is eq u a tio n is id en tica lly sa tis¯ ed , th a n k s to (1 ). (T h e co g n o scen ti w o u ld h av e rea lized th a t m a k in g th e lo ca tio n o f th e test p a rticle th e o rig in is a n a lg eb ra ica lly clev er th in g to d o .) T h is a n a ly sis clea rly sh ow s h ow th e m a ss ra tio s g o aw ay th ro u g h th e p ro p o rtio n a lity o f b o th sid es to th e ra d iu s v ecto r b etw een th e cen tre o f m a ss a n d th e test p a rticle. T o m a k e su re w e ca tch all th e eq u ilib riu m so lu tio n s, w e ca n d o th is a b it m o re fo rm a lly. W e d e¯ n e th e v ecto r q b y th e rela tio n m 1 r 1 ¡ m 2 r 2 = (m 1 + m 2 )q . A little b it o f a lg eb ra ic m a n ip u la tio n a llow s u s to w rite (5 ) a s: ¤ G (m 1 + m 2 ) G (m 1 + m 2 ) £ 3 r: (r 1 + r 23 )r + (r 23 ¡ r 13 )q = 3 3 a3 2 r1 r2 (6 ) F o r th is eq u a tio n to h o ld , a ll th e v ecto rs a p p ea rin g in it m u st b e co llin ea r. O n e p o ssib ility is to h av e r a n d q to b e in th e sa m e d irectio n . It th en fo llow s th a t r 1 ;r 2 a n d r a re a ll co llin ea r a n d th e th ree p a rticles a re in a stra ig h t lin e. T h e eq u ilib riu m co n d itio n ca n b e m a in ta in ed a t th ree lo ca tio n s u su a lly ca lled L 1 ;L 2 a n d L 3 . T o w o rk o u t th e ex a ct p o sitio n o f eq u ilib riu m , o n e h a s to so lv e a ¯ fth -o rd er eq u a tio n w h ich w ill lea d to th ree rea l ro o ts. W e a re, h ow ev er, n o t in terested in th ese (a t lea st, n o t in
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th is in sta llm en t!) th o u g h L 2 o f th e S u n { E a rth sy stem h a s lo ts o f p ra ctica l a p p lica tio n s. If w e d o n ot w a n t r a n d q to b e p a ra llel to ea ch o th er, th en th e o n ly w a y to sa tisfy (6) is to m a k e th e co e± cien t o f q va n ish w h ich req u ires r 1 = r 2 . S u b stitu tin g b a ck , w e ¯ n d th a t ea ch sh o u ld b e eq u a l to a . S o w e g et th e rig id ly ro ta tin g eq u ila tera l co n ¯ g u ra tio n o f th ree m a sses w ith : r1 = r2 = a :
(7 )
O b v io u sly, th ere a re tw o su ch co n ¯ g u ra tio n s co rresp o n d in g to th e tw o eq u ila tera l tria n g les w e ca n d raw w ith th e lin e jo in in g m 1 a n d m 2 a s o n e sid e. T h e lo ca tio n s o f th e m 3 co rresp o n d in g to th ese tw o so lu tio n s a re ca lled L 4 a n d L 5 , g iv in g L a g ra n g e a to ta l o f ¯ v e p o in ts. In cid en ta lly, th ere a re sev era l ex a m p les in th e so la r sy stem in w h ich n a tu re u ses L a g ra n g e's in sig h t. T h e m o st fa m o u s a m o n g th em is th e co llectio n o f m o re th a n a th o u sa n d a stero id s ca lled T ro ja n s w h ich a re lo ca ted a t th e v ertex o f a n eq u ila tera l tria n g le, th e b a se o f w h ich is fo rm ed b y S u n a n d J u p iter { th e tw o la rg est g rav ita tin g b o d ies in th e so la r sy stem . S im ila r, b u t less d ra m a tic, fea tu res a re fo u n d in th e L 5 p o in t o f S u n { M a rs sy stem a n d in th e sa tellites o f S a tu rn . T h e en tire co n ¯ g u ra tio n g o es a ro u n d in rig id ro ta tio n sin ce th e o rb it o f J u p iter is a p p ro x im a tely circu la r. The existence of real life solutions tells us that the equilateral solution must be stable in the sense that if we displace m3 from the equilibrium position L5 slightly, it will come back to it.
322
T h e ex isten ce o f su ch rea l life so lu tio n s tells u s th a t th e eq u ila tera l so lu tio n m u st b e sta b le in th e sen se th a t if w e d isp la ce m 3 fro m th e eq u ilib riu m p o sitio n L 5 slig h tly, it w ill co m e b a ck to it. (It tu rn s o u t th a t th e o th er th ree p o in ts L 1 ;L 2 ;L 3 a re n o t.) O u r n ex t jo b is to stu d y th is sta b ility ; fo r th is a d i® eren t co o rd in a te sy stem is b etter. It w ill a lso h elp to resca le va ria b les to sim p lify life. W e w ill n ow ta k e th e o rig in o f th e ro ta tin g co o rd in a te sy stem to b e at the location of the cen tre of m ass o f m 1 a n d m 2 w ith th e x -a x is p a ssin g th ro u g h th e tw o m a sses
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a n d th e m o tio n co n ¯ n ed to th e x { y p la n e. M ea su rin g a ll m a sses in term s o f th e to ta l m a ss m 1 + m 2 , w e ca n d en o te th e sm a ller m a ss b y ¹ a n d th e la rg er b y (1 ¡ ¹ ). S im ila rly, w e w ill m easu re a ll d ista n ces in term s o f th e sep a ra tio n a b etw een th e tw o p rim a ry m a sses a n d ch o o se th e u n it o f tim e su ch th a t ! = 1 . (If th ese a p p ea r stra n g e fo r y o u , ju st w rite d ow n th e eq u a tio n s in n o rm a l u n its a n d re-sca le th em ; su ch trick s a re w o rth lea rn in g .) T h e p o sitio n o f m 3 is (x ;y ) a n d r 1 a n d r 2 w ill d en o te th e (sca la r) d ista n ces to m 3 fro m th e m a sses (1 ¡ ¹ ) a n d ¹ resp ectiv ely. (N o te th a t th ese a re n ot th e d ista n ces to m 3 fro m th e o rig in .) It is n ow ea sy to see th a t th e eq u a tio n s o f m o tio n in (3 ) red u ce to th e set: xÄ ¡ 2 y_ = ¡
@© ; @x
yÄ + 2 x_ = ¡
@© ; @y
(8 )
w h ere © = ¡
¹ 1 2 (1 ¡ ¹ ) ¡ (x + y 2 ) ¡ r1 r2 2
(9 )
is th e e® ectiv e p o ten tia l in th e ro ta tin g fra m e w h ich in clu d es a term fro m th e cen trifu g a l fo rce. T h e o n ly k n ow n in teg ra l o f m o tio n is th e ra th er o b v io u s o n e co rresp o n d in g to th e en erg y fu n ction (1 = 2 )v 2 + © = co n sta n t. A little th o u g h t sh ow s th a t r © = 0 a t L 4 a n d L 5 , co n ¯ rm in g th e ex isten ce o f a sta tio n a ry so lu tio n . T o stu d y th e sta b ility, w e n o rm a lly w ou ld h av e ch eck ed w h eth er th ese co rresp o n d to m a x im a o r m in im a o f th e p o ten tia l. A s w e sh a ll see (F igu re 1 ), it tu rn s o u t th a t L 4 a n d L 5 a ctu a lly co rresp o n d to m a x im a , so if th a t is th e w h o le sto ry L 4 a n d L 5 sh o u ld b e u n sta b le. B u t, o f co u rse, th a t is n o t th e w h o le sto ry sin ce w e n eed to ta k e in to a cco u n t th e C o rio lis fo rce term co rresp o n d in g to (2 y_;¡ 2 x_) in (8 ). T o see th e e® ect o f th is term clea rly, w e w ill ta k e th e C o rio lis fo rce term to b e (C y_;¡ C x_) so th a t th e rea l p ro b lem co rresp o n d s to C = 2 . B u t th is trick a llow s u s to stu d y th e sta b ility
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Figure 1. A contour plot of the potential (x,y) when = 0.3. The L4 and L5 are at the potential maxima. One can also see the saddle points L1, L2, L3 along the line joining the two primary masses.
fo r a n y va lu e o f C , in p a rticu la r fo r C = 0 , to see w h a t h a p p en s if th ere is n o C o rio lis fo rce. W e n ow h av e to d o a T ay lo r series ex p a n sio n o f th e term s in (8 ) in th e fo rm x (t) = x 0 + ¢ x (t); y (t) = y 0 + ¢ y (t) w h ere th e p o in t (x 0 ;y 0 ) co rresp o n d s to th e L 5 p o in t w ith y 0 > 0 . W e a lso n eed to ex p a n d © u p to q u a d ra tic o rd er in ¢ x a n d ¢ y to g et th e eq u a tio n s g ov ern in g th e sm a ll p ertu rb a tio n s a ro u n d th e eq u ilib riu m p o sitio n . T h is is stra ig h tfo rw a rd b u t a b it ted io u s. If y o u w o rk it th ro u g h , y o u w ill g et th e eq u a tio n s à p ! d2 d 3 3 3 ¢ x = ¢ x + (1 ¡ 2 ¹ ) ¢ y + C ¢ y ; 2 dt dt 4 4 (1 0 ) d2 9 ¢ y = ¢ y+ 2 dt 4
324
Ã
p ! d 3 3 (1 ¡ 2 ¹ ) ¢ x ¡ C ¢ x : (1 1 ) dt 4
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T o ch eck fo r sta b ility, w e try so lu tio n s o f th e fo rm ¢ x = A ex p (¸ t); ¢ y = B ex p (¸ t) a n d so lv e fo r ¸ . A n elem en ta ry ca lcu la tio n g iv es ¸2 =
3¡ C
2
2 2
1=2
§ [(3 ¡ C ) ¡ 2 7 ¹ (1 ¡ ¹ )] 3
:
(1 2 )
S ta b ility req u ires th a t w e sh o u ld n o t h av e a p o sitiv e rea l p a rt to ¸ ; th a t is, ¸ 2 m u st b e rea l a n d n eg a tiv e. F o r ¸ 2 to b e rea l, th e term in (1 2 ) co n ta in in g th e sq u a re ro o t sh o u ld h av e a p o sitiv e a rg u m en t w h ich req u ires (C
2
¡ 3 )2 > 2 7 ¹ (1 ¡ ¹ ):
We conclude that the motion is unstable if
p
C
3p. T h u s w e co n clu d e th a t th e m o tio n is u n sta b le if C < 3 ; in p a rticu la r, in th e a b sen ce o f th e C o rio lis fo rce (C = 0 ), th e m o tio n is u n sta b le b eca u se th e pp o ten tia l a t L 5 is a ctu a lly a m a x im u m . B u t w h en C > 3 a n d in p a rticu la r fo r th e rea l ca se w e a re in terested in w ith C = 2 , th e m o tio n is sta b le w h en co n d itio n in (1 3 ) is sa tis¯ ed . U sin g C = 2 w e ca n red u ce th is co n d itio n to ¹ (1 ¡ ¹ ) < (1 = 2 7 ). T h is lea d s to à ! r 1 23 ¹ < ¼ 0 :0 3 8 5 : ¡ (1 4 ) 2 108 T h is criterio n is m et b y th e S u n { J u p iter sy stem w ith ¹ ¼ 0 :0 0 1 a n d b y th e E a rth { M o o n sy stem w ith ¹ ¼ 0 :0 1 2 . S ta b ility o f T ro ja n s is a ssu red . In fa ct, th e L 5 s a n d L 4 s a re th e fav o u rites o f scien ce ¯ ctio n w riters a n d so m e N A S A scien tists fo r settin g u p sp a ce co lo n ies. (T h ere is ev en a U S -b a sed so ciety ca lled th e `L 5 so ciety ', w h ich w a s k een o n sp a ce co lo n iza tio n b a sed o n L 5 !) S o h ow d o es C o rio lis fo rce a ctu a lly sta b ilize th e m o tio n ? W h en th e p a rticle w a n d ers o f th e m a x im a , it a cq u ires a n o n -zero v elo city a n d th e C o rio lis fo rce in d u ces a n
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There is even a USbased society called the ‘L5 society’, which was keen on space colonization based on L5!
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B o x 1 . G e o m e tric a l P ro o f o f L a g r a n g e 's E q u ila te ra l S o lu tio n If you like matters geometrical, you might ¯nd the following proof interesting. In F igu re A, the triangle ABL 5 is Lagrange's equilateral triangle of unit side with mass ¹ located at A, mass (1 ¡ ¹ ) located at B, and the test particle located at L 5 . The centre of mass of the primary bodies is C and all the three masses rotate rigidly around C. We need to prove that the resultant of the gravitational attraction along L 5 A and L 5 B will be precisely along L 5 C and will have a magnitude equal to the (outward) centrifugal force acting on L 5 . With our choice of units, ! 2 = 1 and the centrifugal force is numerically the same as the length L 5 C. To prove this, draw DE perpendicular to CL 5 and drop perpendiculars AD and BE as shown. Also draw a perpendicular from A to BE (shown by dashed line) . If AD is equal to x and EB is equal to x + y , it follows from elementary geometry that L 5 F= x and FC= y (1 ¡ ¹ ) . We can also write DL 5 and L 5 E as (1 ¡ ¹ ) l and ¹ l respectively for some l. To prove that forces match at L 5 , we need to show that the component of the gravitational force along L 5 D due to the mass ¹ is balanced by the component of the gravitational force along L 5 E due to the mass (1 ¡ ¹ ) . This leads to the condition ¹ (1 ¡ ¹ ) l = (1 ¡ ¹ ) ¹ l which is true. (While taking cosines and sines of angles, recall that the equilateral triangle has unit side. ) Next consider the component of the force along L 5 C. The sum of the two gravitational forces along L 5 C is given by ¹ x + (1 ¡ ¹ ) (x + y ) which should balance the outward centrifugal force equal to the length of L 5 C, viz., x + y (1 ¡ ¹ ) . Since ¹ x + (1 ¡ ¹ ) (x + y ) = x + y (1 ¡ ¹ ) , we are again through with the proof. This proves that one can achieve force balance in the equilateral con¯guration for any value of ¹ . The fact that C divides AB in the inverse ration of the masses is, of course, crucial.
Figure A
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a ccelera tio n in th e d irectio n p erp en d icu la r to th e v elo city. A s w e n o ted b efo re, th is is ju st lik e th e m o tio n in a m a g n etic ¯ eld a n d th e p a rticle ju st g o es a ro u n d L 5 . T h e id ea th a t a fo rce w h ich d o es n o t d o w o rk ca n still h elp in m a in ta in in g th e sta b ility m ay a p p ea r a b it stra n g e b u t is co m p letely p la u sib le. In fa ct, th e a n a lo g y b etw een C o rio lis a n d m a g n etic fo rces tells y o u th a t o n e m ay b e a b le to a ch iev e sim ila r resu lts w ith m a g n etic ¯ eld s to o . T h is is tru e a n d o n e ex a m p le is th e so -ca lled `P en n in g tra p ', w h ich y o u m ig h t lik e to rea d a b o u t w ith th e cu rren t in sig h t. T o b e a b so lu tely co rrect a n d fo r th e sa k e o f ex p erts w h o m ay b e rea d in g th is, I sh o u ld a d d a co m m en t reg a rd in g a n o th er p ecu lia rity w h ich th is sy stem p o ssesses. A m o re p recise sta tem en t o f o u r resu lt o n sta b ility is th a t, w h en (1 4 ) is sa tis¯ ed , th e so lu tio n s a re n ot lin early u n stable. T h e ch a ra cteriza tio n \n o t u n sta b le" is q u a li¯ ed b y say in g th a t th is is a resu lt in lin ea r p ertu rb a tio n th eo ry. A fa irly co m p lex p h en o m en o n (w h ich is to o so p h istica ted to b e d iscu ssed h ere, b u t see [2 ] if y o u a re in terested ) m a k es th e sy stem u n sta b le fo r tw o p recise va lu es o f ¹ w h ich pd o sa tisfy (1 4 ). T h ese va p lu es h a p p en to b e (1 = 3 0 )[1 5 ¡ 2 1 3 ] a n d (1 = 9 0 )[4 5 ¡ 1 8 3 3 ]. (Y es, b u t I sa id th e p h en o m en o n is co m p lex !) W h ile o f g rea t th eo retica l va lu e, th is is n o t of m u ch p ra ctica l releva n ce sin ce o n e ca n n o t ¯ n e-tu n e m asses to a n y p recise va lu es. Suggested Reading
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India
[1] [2]
T Padmanabhan, Planets move in circles, Resonance, Vol.1, No.9, pp.34–40, 1996. D Boccaletti and G Pucacco, Theory of Orbits, Springer, Vol. 1, p. 271, 1996.
RESONANCE April 2009
Email: [email protected] [email protected]
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Snippets of Physics 17. Why does an Accelerated Charge Radiate? T Padmanabhan
T h e fa c t t h a t a n a c c e le ra te d c h a rg e ra d ia t e s e n e r g y is c o n sid e r e d a n e le m e n ta ry t e x tb o o k r e su lt in e le c tro m a g n e tism . N e v e r th e le ss, th is p ro c e s s o f ra d ia tio n (a n d its re a c tio n o n t h e ch a r g e d p a rt ic le ) ra is e s se v e ra l c o n u n d ru m s a b o u t w h ich te c h n ic a l p a p e rs a r e w ritte n e v e n t o d a y . I n th is in sta llm e n t, w e w ill tr y to u n d e rs ta n d w h y a n a c c e le r a te d c h a r g e r a d ia te s in a sim p le , y e t r ig o r o u s, m a n n e r . T h e elec tric ¯ eld o f a p o in t ch a rg e a t rest at th e orig in fa lls as (1 = r 2 ) a n d is d irected ra d ia lly o u tw a rd fro m th e ch a rg e. If th e ch a rg e m ov es w ith a u n ifo rm v elo city v , th e ¯ eld is g iv en b y qr (1 ¡ v 2 = c 2 ) E = 3 ¡ ; r 1 ¡ (v 2 = c 2 ) sin 2 µ ¢3 = 2
B =
1 v£ E; c
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
(1 )
w h e re µ is th e a n gle b etw een th e d irectio n o f m o tion a n d th e ra d iu s v ec to r r w h ich h a s th e c o m p o n en ts (x ¡ V t;y ; z ). T h is ex p ressio n { w h ich is m o st e a sily ob ta in ed b y tran sfo rm in g th e C o u lo m b ¯ eld from th e re st fra m e o f th e ch a rge to a m o v in g fra m e u sin g th e L o ren tz tra n sfo rm a tio n p ro p erties o f th e elec trom a g n etic ¯ e ld s { is m o re c o m p lica ted b u t still p o ssesses tw o k ey p ro p ertie s o f th e static ch a rg e. It falls a s (1 = r 2 ) a t larg e d ista n ce s an d it is radially directed from the instan tan eous position o f th e ch a rg e . T h e en e rg y ° o w c orre sp o n d in g to th e ele ctro m a g n etic ¯ eld sc ales a s th e sq u a re o f th e e lectro m a g n etic ¯ eld . If th e ¯ e ld fa lls a s (1= r 2 ), th e en ergy ° u x w ill fa ll a s (1 = r 4 ) a n d , sin ce th e a re a o f a sp h erical su rfa ce sca le s a s r 2 , th e to ta l en ergy ° o w in g th ro u g h a sp h e re a t la rg e d ista n ces
RESONANCE May 2009
Keywords Electromagnetism, theory of relativity, radiation.
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When a charge is accelerating, something dramatic happens.
from th e ch a rg e fa lls a s r 2 £ (1 = r 4 ) = (1 =r 2 ). T h ere fo re, on e ca n n o t tra n sfer en erg y to larg e d ista n ce s in th is k in d of ¯ e ld . T h is is u n d e rsta n d a b le b eca u se su ch a tran sfer ca n n o t ta k e p la ce in th e re st fram e o f th e ch a rg e { in w h ich w e o n ly h a v e a sta tic C o u lo m b ¯ eld { a n d sin c e w e ex p ect su ch a p h y sica l p ro ce ss to b e L oren tz in v aria n t it sh o u ld n o t h a p p en fo r a ch a rg e m o v in g w ith u n ifo rm ve lo city eith er. B u t w h e n th e ch a rge is a cce lera tin g, so m eth in g d ra m a tic h a p p en s. T h e e lectric ¯ eld , say , p ick s u p a n ad d ition a l term w h ich fa lls o n ly a s (1 = r ) a t la rg e d ista n ces. T h e ch a n g e fro m th e (1= r 2 ) d ep en d en ce to th e (1 = r) d e p e n d en ce m a k es trem en d o u s d i® eren ce (a n d m u ch o f m o d ern tec h n o log y ow es its e x iste n ce to th is fa ct). W h en th e ¯ eld fa lls a s (1 = r ) a t la rg e d ista n c es, th e en erg y ° u x w ill fa ll a s (1= r 2 ) a n d th e tota l en erg y ° ow in g th ro u g h a sp h ere at la rg e d ista n ces from th e ch a rg e, r 2 £ (1= r 2 ), is a co n sta n t! T h erefo re, th e ¯ eld s a risin g fro m a n a ccele rated ch a rge are c ap a b le o f tra n sm ittin g en erg y to larg e d ista n ces fro m th e ch a rge . C lea rly, it w o u ld b e n ice to u n d ersta n d b ette r h o w ac celera tio n lea d s to su ch a sh ift from (1 = r 2 ) to (1 = r ) d ep en d en ce { w h ich ch an g es th e ca terp illa r to a b u tter° y.
In the case of an accelerated motion, the electric field picks up a transverse component which is perpendicular to the radial direction.
500
T h e re is a lso a n o th er p ec u liar fea tu re th a t a rises w h en th e ch a rg e u n d e rg o es a n a cc elera te d m o tion . T h e C o u lo m b ¯ eld o f a ch a rg e a t rest, a n d th a t o f a ch a rg e m o v in g w ith a u n ifo rm v elo city , is rad ia l. T h e elec tric ¯ eld ve cto r in th ese ca ses p oin ts ra d ia lly ou tw a rd fro m th e ch a rg e. B u t in th e ca se of a n a cce lerated m o tio n , th e electric ¯ eld p ick s u p a tra n sv e rse c o m p o n en t w h ich is p e rp en d ic u la r to th e ra d ial d irectio n . S in ce a p ro p a g a tin g electro m a g n e tic p la n e w a v e, fo r ex am p le, w ill h a v e an electric ¯ e ld th a t is tra n sv erse to th e d irec tio n of p rop a g a tio n o f th e w a v e, th is fa ct is c ru cia l fo r id en tify in g th e ¯ eld ge n era te d b y th e a cce lera tio n w ith e lectro m a g n e tic ra d ia tio n .
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SERIES ARTICLE
It tu rn s o u t th a t th ere is a rem a rk ab ly eleg a n t a n d sim p le w a y o f u n d ersta n d in g b oth th ese fe atu res, o rig in a lly d u e to J J T h o m son [1 ], w h ich d eserv es to b e k n ow n b etter a n d p ossib ly co u ld rep la ce th e u n im a g in a tiv e d e riv a tio n u sin g L ien ard { W ie ch ert p o ten tia ls in th e cla ssro o m s!. (T h is d eriva tio n is d iscu ssed , fo r ex a m p le, in [2 ] a n d a lso a p p ea rs in th e sta n d a rd tex tb o o k s [3 , 4 ] th o u g h in th ese tex tb o o k s a n im p ression is crea ted th a t th e resu lt is va lid o n ly fo r n o n -rela tiv istic m o tion .) I w ill d e sc rib e th is a p p ro a ch an d its essen tia l fea tu re s. T o b e gin w ith , le t u s co n sid er a fe w elem en tary fa cts a b ou t M a x w ell's eq u atio n s w h ich co n n ect th e electro m a gn etic ¯ e ld s to th e m o tio n o f th e sou rc e. S in ce th e ele ctric ¯ eld is E = ¡ (1 =c)(@ A = @ t) ¡ r Á , w e see th a t th e electric ¯ eld h as a co m p o n e n t w h ich d ep en d s lin ea rly o n (@ A = @ t). It is a lso w ell k n o w n th a t th e so u rce for th e v ecto r p o ten tia l A is th e c u rren t j in th e sen se th a t ¤ A / j. T h erefo re (@ A = @ t) w ill h a v e a so u rce th a t d e p e n d s o n (@ j= @ t). S in ce j is lin ea r in th e v elo c ity o f th e ch arg e, w e co n clu d e th a t th e ele ctric ¯ eld w ill h av e a so u rce term w h ich is lin ea r in th e tim e d eriva tiv e o f th e v elo c ity, v iz., th e a ccelera tio n a .
Since j is linear in the velocity of the charge, we conclude that the electric field will have a source term which is linear in the time derivative of the velocity, viz., the acceleration a.
A n a lte rn a tiv e w a y of u n d e rsta n d in g th is re su lt is a s follow s: A ch a rg e q m o v in g w ith u n iform v elo city v is eq u iv alen t to a c u rren t j = qv . T h is cu rren t w ill p ro d u ce a m a g n etic ¯ eld (in a d d itio n to th e e lectric ¯ eld ) w h ich sc a les in p ro p ortio n to j. If a = v_ 6= 0 , it w ill p ro d u c e a n o n zero (@ j= @ t) a n d h e n ce a n o n z ero (@ B = @ t). T h rou g h F ara d a y 's la w , th e (@ B = @ t) w ill in d u ce a n electric ¯ eld w h ich sca les as (@ j= @ t). (T h at is, if (@ j= @ t) ch a n ge s b y fa cto r 2 , th e electric ¯ eld w ill ch a n g e b y fa cto r 2.) It fo llo w s th at a n a ccele ra ted ch a rg e w ill p ro d u ce a n ele ctric ¯ eld w h ich is lin e a r in (@ j= @ t) = q a . (T h is, o f co u rse, is in a d d itio n to th e u su a l C o u lo m b term w h ich is in d ep en d en t o f a a n d fa lls a s r ¡2 .) F u rth er, sin ce th e w a ve eq u a tion ¤ A / j p ro p a g a te s
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Before we do more sophisticated mathematics, let us try a bit of dimensional analysis to determine the electric field which arises from the acceleration.
in fo rm a tio n a t th e sp ee d of lig h t, w e a lso k n o w th a t th e ele ctric ¯ eld a t an ev en t (t;x ) is d eterm in ed en tirely b y th e b eh av io u r of th e so u rce a t th e ev en t (tR ;x 0), w h ere t ¡ tR = (1 = c)jx ¡ x 0j ´ (r= c). It is u su a l to c a ll tR th e `reta rd ed tim e'. B efo re w e d o m o re so p h istica ted m a th em a tics, let u s try a b it o f d im en sio n al a n a ly sis to d e te rm in e th e elec tric ¯ e ld w h ich a rises fro m th e a cc elera tio n . W e k n ow th a t th e e lectric ¯ e ld h a s to b e d eterm in ed b y th e ch a rg e o f th e p a rtic le q , ve lo city o f ligh t c, a cce lera tio n a a n d th e d ista n ce r (w ith a a n d r ca lcu la ted a t th e reta rd ed tim e). In g en e ra l, th e ¯ eld w ill a lso d e p e n d o n th e v elo city o f th e p a rticle a t th e reta rd ed tim e b u t w e w ill ch o o se a L o ren tz fra m e in w h ich th e ch arg e w as a t rest at the retarded tim e th e re b y e lim in a tin g a n y v d ep en d en c e. W e n e x t u se th e fa ct th a t th e electric ¯ eld , w h ich is lin ea r in @ j= @ t, sh ou ld b e lin ea r in b oth q a n d a to w rite: ³ q ´³ a ´ qa E = C (µ) n m = C (µ ) 2 ; (2 ) c r r cn r m ¡ 2 w h ere C is a d im en sio n le ss fac to r, d ep en d in g o n ly o n th e a n g le µ b etw e en r a n d a , a n d n a n d m n eed to b e d ete rm in ed . (S in c e v = 0 in th e in sta n ta n eo u s rest fra m e, th e ¯ e ld ca n n o t d ep en d on th e v elo city.) F rom d im e n sio n a l a n aly sis, n o tin g th a t E h a s th e d im en sio n s o f q= r 2 (G a u ssia n u n its, so rry !) it im m ed ia tely fo llow s th a t (a = c n r m ¡2 ) m u st b e d im en sio n less, le ad in g to n = 2 ;m = 1 . S o w e g et th e resu lt: E = C (µ )
Dimensional analysis plus the fact that E must be linear in q and a, implies the r–1 dependence for the radiation term.
502
qa : c2 r
(3 )
T h u s, d im en sion a l a n a ly sis p lu s th e fac t th at E m u st b e lin ea r in q a n d a , im p lies th e r ¡1 d ep e n d en ce fo r th e ra d ia tion term . W h ile th is re su lt sh ow s w h y a term lin ea r in a ccelera tio n w ill also h a v e a (1 =r ) d ep en d en ce, it d o es n o t rea lly tell u s h ow ex a ctly it co m es a b o u t. M o reov e r, d im e n sio n a l
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SERIES ARTICLE
an a ly sis ca n n ot d eterm in e th e n a tu re o f th e d im en sio n less fu n ctio n C (µ ). T h e a rg u m en t d u e to J J T h o m so n [1 ] d o es b o th o f th ese in an e leg a n t w a y a n d I w ill d escrib e a slig h tly m o d i¯ ed v ersio n o f th e sam e. L et u s co n sid er a ch a rge d p a rtic le A m o v in g a lo n g so m e arb itra ry tra jectory z (t). W e are in te rested in th e electric ¯ eld , say , p ro d u ced at an e ve n t P (t;x ) b y th is ch a rg e. S in ce th e ch ara cte ristics o f th e w a ve eq u a tio n sh o w s th at in fo rm a tio n p ro p ag a tes a t th e sp eed o f lig h t fro m th e so u rce p o in t to th e ¯ e ld p o in t, w e a lre a d y k n o w th at th e ¯ eld a t P w ill b e d eterm in e d b y th e p ro p e rties of th e tra jectory a t th e reta rd ed tim e tR . F u rth e r, th e electric ¯ e ld c an o n ly d ep en d o n th e p o sitio n z (tR ), v elo city z_(tR ) a n d th e a ccelera tio n zÄ (tR ) a t th e reta rd e d tim e b u t n ot o n h ig h er tim e d eriva tiv es. (T h is fo llo w s from th e fa ct th at th e sou rc e fo r electro m a g n e tic ¯ eld on ly in v o lv es u p to th e ¯ rst tim e d eriva tiv e of th e cu rren t w h ich w ill b e p ro p o rtio n al to th e a c celera tio n .) W e w ill n o w ch o o se o u r L oren tz fra m e su ch th a t th e ch arg e w as at rest a t th e orig in o f th e sp a ce tim e co o rd in a te s at th e re ta rd ed tim e tR = 0. L et th e a cceleratio n o f th e ch a rg e b e a = Äz ( tR ) a t th is in sta n t. W e w ill ro ta te th e co o rd in a te sy stem so th a t a is a lo n g th e x -a x is. W e n o w co n sid er another ch a rg ed p a rticle B w h ich w a s at re st, a t th e o rig in , fro m t = ¡ 1 to t = 0 a n d u n d e rgo es co n stan t a ccele ra tio n a alo n g th e x -a x is fo r a sh o rt tim e ¢ t. F o r t > ¢ t, it m ov es w ith co n sta n t ve lo city v = a ¢ t a lo n g th e x -a x is. L et u s stu d y th e electric ¯ eld p ro d u ced b y th is ch a rge B a t so m e tim e t À ¢ t. S in ce ¢ t is a rb itra rily sm a ll, w e h a ve a ¢ t ¿ c an d w e ca n u se th e n o n -re la tiv istic a p p ro x im a tion th ro u gh o u t. Since the trajectory of this charge m atches iden tically in position , velocity an d acceleration w ith the trajectory of the charged particle w e are origin ally interested in , it follow s that both of them w ill produce iden tical electric ¯elds at P . T h is w a s th e k ey in sig h t of T h om so n . A s w e sh a ll se e, th e ¯ eld p ro d u ced b y B is fa irly triv ia l to
RESONANCE May 2009
Since the trajectory of this charge matches identically in position, velocity and acceleration with the trajectory of the charged particle we are originally interested in, it follows that both of them will produce identical electric fields at
P .
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Figure 1.
(b)
(a)
c alcu late a n d h e n ce w e ca n o b ta in th e ¯ eld d u e to A . T h e `n ew s', th a t th e ch a rg e w as a cce lerated a t t = 0, c ou ld h a ve o n ly tra v elle d to a d ista n ce r = ct in tim e t. T h u s, a t r > ct, th e e lectric ¯ eld sh o u ld b e th at d u e to a ch a rg e lo c ated a t th e o rigin a s sh ow n in F igu re 1 a : E =
q ^r r2
(fo r r > ct):
(4 )
A t r . ct, th e ¯ eld is th a t d u e to a ch a rg e m ov in g w ith v elo city v a lo n g th e x -a x is, g iv en b y (1 ). T h e k ey p o in t is th a t th is ¯ eld is ra d ia lly d irec ted from th e in sta n ta n eo u s p o sitio n o f th e ch a rg ed p a rticle. W h en v ¿ c, w h ich is th e situ a tio n w e a re in terested in , th is is a g a in a C ou lo m b ¯ e ld ra d ia lly d irecte d fro m th e instan tan eous p o sition o f th e ch a rg ed p a rtic le (see F igure 1 b ) : The key point is
E =
q 0 ^r r 02
(for r < ct):
(5 )
that this field is radially directed from the instantaneous position of the charged particle.
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A ro u n d r = ct, th e re ex ists a sm a ll sh ell o f th ick n e ss (c¢ t) in w h ich n eith er re su lt h o ld s g o o d . It is clea r th a t th e electric ¯ eld in th e tra n sitio n re gion sh o u ld in te rp o la te b etw een th e tw o C o u lo m b ¯ eld s. T h e c ru cia l q u estio n is h ow to d o th is m ak in g su re th a t th e ° u x o f th e
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Figure 2.
e lectric ¯ eld v ecto r th ro u g h a n y sm a ll b o x in th is re gion v a n ish es, a s it sh o u ld to sa tisfy M a x w ell's eq u a tio n s. A s w e sh allsee b elow , it tu rn s ou t th a t th is req u ires th e ¯ eld lin e s to a p p ea r so m ew h a t a s sh o w n in F igure 2 . O n e c an e x p licitly w o rk o u t th is c o n d itio n a n d p rov e th a t ta n µ = ° ta n Á , w h ere ° = (1 ¡ v 2 = c 2 ) ¡1 = 2 . (Y o u sh o u ld try th is o u t; it is d o n e in d eta il in [3 ].) In th e n o n -rela tiv istic lim it th a t w e a re co n sid erin g, µ ¼ Á m ak in g th e ¯ eld lin e s p ara lle l to ea ch o th er in th e in sid e an d o u tsid e reg io n s; th a t is, Q P is p a ra llel to R S . (T h is is ea sy to u n d ersta n d b e ca u se th e ra d ia l ¯ e ld is ju st th e C o u lo m b ¯ eld b o th in th e o u tsid e an d in th e in sid e reg ion . F or th e ° u x to b e co n serv ed , th ese tw o ¯ eld lin es sh o u ld b e p a ra llel to ea ch o th er.) W h a t is rea lly in terestin g is th a t w e n ow n ee d a p ie ce o f ele ctric ¯ eld lin e P R in te rp o la tin g b etw e en th e tw o C o u lo m b ¯ eld s. T h is is c lea rly tran sve rse to th e ra d ia l d ire ctio n a n d a ll th a t w e n eed to d o is to p rov e th a t its m a gn itu d e va ries a s 1 = r . L et u s see h ow th is c o m e s a b o u t. T h e situ a tio n is d e sc rib ed in d eta il in F igure 3 w h ich is self-ex p la n a to ry. L et E k a n d E ? b e th e m a g n itu d es of RESONANCE May 2009
What is really interesting is that we now need a piece of electric field line PR interpolating between the two Coulomb fields.
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Figure 3. (a) The electric field due to a charged particle which was accelerated for a small time interval t. For t > t, the particle is moving with a uniform non-relativistic velocity along the x-axis. At r > ct, the field is that of a charge at rest in the origin. At r < c(t t), the field is directed towards the instantaneous position of the particle. The radiation field connects these two Coulomb fields in a small region of thickness ct (b) Pill box construction to relate the normal component of the electric field around the radiation zone.
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th e electric ¯ eld p a ra llel an d p erp en d icu lar to th e d irec tio n ^r . F ro m th e g eo m e try, w e h av e E? v t = ? : Ek c¢ t
(6 )
B u t v ? = a ? ¢ t a n d t = (r = c) g iv in g ³r ´ E? (a ? ¢ t) (r = c) = = a? 2 : Ek c¢ t c
(7 )
T h e va lu e o f E k c an b e d eterm in ed b y u sin g G a u ss th e o re m to a sm a ll p ill b ox , as sh o w n in th e sm a ll in set in F igure 3. T h is g iv es E k = E r = (q = r 2 ); th u s w e ¯ n d th a t ³r ´ q q ³a ? ´ E ? = a? 2 : 2 = 2 : (8 ) c r c r RESONANCE May 2009
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T h is is th e rad ia tio n ¯ eld lo cated in a sh ell a t r = ct, w h ich is p ro p ag a tin g o u tw a rd w ith a v elo c ity c. T h e a b ov e a rg u m en t clea rly sh o w s th a t th e orig in of th e r ¡ 1 d ep en d en ce lies in th e n ec essity to in terp o la te b etw een th e tw o C o u lo m b ¯ eld s. W e h a ve th u s d eterm in ed th e electric ¯ eld g en erated d u e to th e a cceleratio n o f th e ch a rg e an d h av e sh ow n th a t it is tran sve rse an d a lso fa lls as (1= r )! W e ca n ex p ress th is resu lt m o re co n cisely in th e v ec to r n ota tio n a s 1 1 E ra d (t;r ) = 2 [ n^ £ (^n £ a )]re t ; (9 ) c r w h ere n^ = (r = r) a n d th e su b scrip t `ret' im p lies th a t th e ex p ressio n in sq u a re b ra ck ets sh o u ld b e e va lu a ted a t t 0= t ¡ r =c. C o m p a riso n w ith eq u a tion (3) sh o w s th a t C (µ ) = sin µ . T h e fu ll e lectric ¯ eld in the fram e in w hich the charge is in stantan eou sly at rest is E = E c o u l + E r a d . W e em p h asise th a t th is resu lt is ex ac t in th e L o re n tz fra m e in w h ich th e ch arg e w a s a t rest a t th e reta rd ed tim e. (O n e d o e s n o t h a v e to m a k e a n o n -rela tiv istic `a p p ro x im a tion ' b ec au se v = 0 a u to m atica lly ta ke s ca re o f it!). If w e n o w m a ke a L oren tz tra n sform a tio n to a fra m e in w h ich th e p article w a s m o v in g w ith so m e v elo city v = z_(tR ) a t th e reta rd e d tim e, th en w e ca n ob ta in th e sta n d a rd , fu lly rela tiv istic, ex p ression w ith th e v elo city d ep en d en ce. T h is is a lg eb ra ica lly a little c om p lica ted b eca u se o n e n e ed s to m a k e a L oren tz tra n sfo rm a tion in a n a rb itra ry d irectio n sin ce v an d a w ill n o t, in g en e ra l, b e in th e sa m e d irectio n . (T h is is d o n e in [5 ] if y ou a re in terested .) T h u s J J T h o m so n 's id ea is q u ite ca p a b le o f g iv in g u s th e c o m p lete so lu tio n to th e p ro b lem .
RESONANCE May 2009
Suggested Reading [1]
[2]
[3]
[4]
[5]
J J Thomson, Electricity and Matter, Archibald Constable, London, Chapter III, 1907. T Padmanabhan, Cosmology and Astrophysics through Problems, Cambridge University Press, 1996. E M Purcell, Electricity and Magnetism, The Berkeley Physics Course, Vol.2, 2nd ed. McGrawHill, New York, 2008. F S Crawford, Waves, The Berkeley Physics Course, Vol.3, 2nd ed. McGraw-Hill, New York, 2008. Hamsa Padmanabhan, A simple derivation of the electromagnetic field of an arbitrarily moving charge, Am. J. Phys., Vol.77, pp.151–155, 2009, [arXiv: 0810. 4246]
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India Email: [email protected] [email protected]
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Sni ppe t so fPhy s i c s 18.Per t ur bi ngCoul ombt oAv oi dAc c i dent s ! TPadmanabhan
TPadmanabhanwor ksat I UCAA,Puneandi s i nt e r e s t e di nal lar e as oft he or e t i c alphys i c s , e s pe c i al l yt hos ewhi c h haves ome t hi ngt odowi t h gr avi t y.
TheCoul ombpr obl em,whi c hcor r es pondst omo¡1 t i oni napot ent i alt hatvar i esasr ,hasapecul i ars ymmet r ywhi c hl eadst oaphenomenon knownas` ac ci dent al 'degener acy.Thi scur i ous f eat ur eexi s t sbot hi nt he cl as s i c aland quant um domai n and i sbes tunder s t ood by s t udyalpot ent i alandobt ai ni ngt he i ngamor egener Coul ombpr obl em asal i mi t i ngcas e. r Themot i o nofapar t i c l ei na nat t r ac t i v e¡( 1= )po t e nt i ali so fhi s t o r i c ala ndt he or e t i c a li mpo r t a nc e .Thef a c t t ha tt hec l a s s i c a lbo undo r bi t si ns uc hapot e nt i a la r e e l l i ps e spl a ye dac r uc i a lr o l ei nt hehi s t o r i c als t udyo f pl a ne t a r ymot i o nandg r a vi t y .TheCoul ombpot e nt i a l , o nt heo t he rhand,pl a y e dac r uc i a lr o l ei nt hee ar l yda ys o fquant um t he o r yi nt hes t udyofhydr og e ns pe c t r a .I n =r bot ht hec a s e s ,i twa ss oonr e al i z e dt hatt he( 1 )pot e nt i a lhass omeve r ys pe c i a lf e a t ur e snots har e dbya g e ne r i cc e nt r a lpo t e nt i a l .I nt hi si ns t a l l me nt ,wewi l l i nve s t i g at es e ve r alas pe c t so ft hi spr o bl e mf r o m bo t h c l as s i c alandqua nt um pe r s pe c t i v e s . I tt ur nso utt ha tani c ewa yofunde r s andi ngt hepe c ul i a rf e a t ur e so ft heCo ul o mbpr o bl e mi st os t ar twi t ha o th s l i ght l ymor ege ne r alpo t e nt i a l{whi c hdoe sn a v e t he s epe c ul i arf e a t ur e s{a ndt r e a tt heCo ul o mbpr obl e ma sas pe c i a lc as eo ft hi smo r eg e ne r a ls i t uat i on.Thi s c a nbedo nei nma nydi ®e r e ntwa ysa ndIwi l lc hoo s et o s t udyt hedyna mi c sunde rt hea c t i o noft hepo t e nt i a l g i v e nby
Keywor ds Coul ombpr obl em, Runge–Lenz vect or ,acci dent aldegener acy.
622
® ¯ U( r )=¡ + 2 ; r r
( 1 )
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whi c h,ofc o ur s e ,r e duc e st ot hea t t r a c t i v eCoul ombpot e nt i a lwhe n¯! 0+.For t hes ak eo fde ¯ni t e ne s s ,Iwi l lt a k e®>0a nd¯¸0t ho ug hmo s to ft heana l ys i s c a nbeg e ne r a l i z e dt oot he rc as e s . Cl a s s i c a lmo t i o no fapa r t i c l eo fmas sm,i n3 di me ns i o ns ,unde rt hea c t i onof U( r )i ss t r a i ght f or wa r dt oa na l yz eus i ngt hes t a ndar dt e xt bookde s c r i pt i ono fa c e nt r a lf or c epr o bl e m.Jus tf o rf un,Iwi l ldoi ti nas l i ght l ydi ®e r e ntma nne r .We kno wt ha t ,aswi t ha nyc e nt r a lf o r c epr o bl e m,ang ul a rmo me nt umJi sc ons e r v e d, c o n¯ni ngt hemot i ont oapl anewhi c hwewi l lt a k et obeµ= ¼=2 .Us i ngJ= 2_ mr Á,t hee ne r g yo ft hepa r t i c l ec anbee xpr e s s e das ¶ µ ® ¯ J2 1 2 E= m r _+ 2 2 ¡ + 2 : ( 2) mr r r 2 2 2 =r Combi ni ngt het wot e r mswi t h( 1 )de pe nde nc ei nt oC2=r ,whe r eC2= J2=2 m)+¯andc ( o mpl e t i ngt hes quar e ,weg e tt her e l a t i on µ ¶ ® 2 C ®2 1 2 ¡ E+ 2 = m_ r+ ´E2 : r 2 C 4C 2
t Thi ss ug g e s t si nt r oduc i ngaf unc t i o nf( )v i at hee quat i o ns r µ ¶ C ® m r t ¡ t _=Es i nf( ); =Ec osf( ): r 2 C 2
( 3)
( 4)
Di ®e r e nt i a t i ngt hes e c o nde quat i onwi t hr e s pe c tt ot i meandus i ngt he¯r s te qua 2 _ _ J=mr t i onwi l lg i v eyo ua ne xpr e s s i o nf o rf .Di v i di ngt hi se xpr e s s i o nbyÁ= 1=2 mC2=J2) l e adst ot hes i mpl er e l at i on( df= dÁ)=( 2 .He nc e ,fi sal i ne arf unc t i onofÁa ndf r omt hes e c onde qua t i oni n( 4 )wege tt hee qua t i o nf o rt het r aj e c t or y t obe µ ¶ C2=®) EC ( 2 2 !Á); c os ( ( 5) =1+ r ® whe r e
µ ¶ m 2 m¯ 2 2 ! = 2 C = 1+ 2 : J J 2
( 6)
No wt ha tweha ves o l v e dt hepr obl e mc o mpl e t e l y ,l e tusl oo katt hepr o pe r t i e sof t hes o l ut i on.Tobe g i nwi t h,l e tusa s kwha tki ndo fo r bi twewo ul de xpe c tgi ve n
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t hekno wns ymme t r i e so ft hepr o bl e m.Apar t i c l emo vi ngi n3s pac edi me ns i o ns ha sapha s es pac ewhi c hi s6di me ns i o nal . Fo ra nyt i me i nde pe nde ntc e nt r a l f o r c e ,weha v ec o ns t a nc yo fe ne r g yEa nda ngul armo me nt umJ.Co ns e r v at i onof E; Jx; Jy; Jz)c t he s ef ourquant i t i e s( on¯ne st hemo t i o nt oar e g i o no f6¡4=2 di me ns i o ns .Thepr oj e c t i onoft hi sphas es pa c et r aj e c t or yont oxypl a newi l l ,i n g e ne r a l ,¯l lat wo di me ns i o na lr e gi onofs pac e .Soyo uwo ul de xpe c tt heo r bi tt o ¯l la¯ni t et wodi me ns i o nalr e g i o no ft hi spl ane ,i ft he r ea r enoot he rc o ns e r ve d qua nt i t i e s .Thi si spr e c i s e l ywha thappe nsf o rag e ne r i cv a l ueo ft hec o ns e r ve d ¼,t qua nt i t i e sJa ndE.Be c aus e!wi l lno tbea ni nt e g e r ,whe nÁc ha nge sby2 he c o s i nef a c t o rwi l lpi c kupat e r mc o s ( 2¼!)whi c hwi l lno tbeuni t y .I nge ne r a l , t heor bi twi l l¯l la2 di me ns i ona lr e gi o ni nt hepl a nebe t we e nt wor adi ir a n dr . 1 2 Wec a nno ws e eho wt heCoul ombpr obl e m be c o me sr at he rs pe c i al .I nt hi sc a s e , weha ve¯= 0ma ki ng!= 1 .Thec ur v ei n( 5 )c l o s e so ni t s e l ff oranyv a l ue ®) o fJa ndE and,i nf ac t ,be c ome sane l l i ps ewi t ht hel at us r e c t um p=( 2C2= EC= ®) a nde c c e nt r i c i t ye=( 2 .Yo us houl dv e r i f yt ha tt hi si si nde e dt hes t anda r d ;! = 1,t ot heKe pl e rpr o bl e m. Sowhe n¯ = 0 heo r bi t t e xt books o l ut i ont c l os e sa ndbe c o me saone di me ns i o nalc ur v er at he rt ha n¯l l i nga2 di me ns i o na l r e g i on.Thi sa na l ys i ss ho wsho wt ur ni ngo nano nz e r o¯c ompl e t e l yc hang e st he t o po l o g i c a lc har a c t e ro ft heo r bi t . I nt hear g ume ntgi v e nabo v e ,wel i nke dt henat ur eoft heor bi tt ot henumbe r o fc o ns e r v e dqua nt i t i e sf o rt hemo t i o n.Gi v e nt hef ac tt hat¯= 0r e duc e st he di me ns i o no ft heo r bi t als pac ebyone ,wee xpe c tt oha veo nemo r ec o ns e r ve d qua nt i t yi nt hepr obl e mwhe n¯=0butno tot he r wi s e .Todi s c o ve rt hi sc ons t a nt , ^ p£J)i r r c o ns i de rt het i mede r i v a t i v eoft hequant i t y( nanyc e nt r a lf o r c ef( ) . Weha v e f( r d ) p£J) = p£J= _ r£( r£m_ r ( ) r dt mf( r ) 2 r r¢r _ r r = [ ( )¡_ ] r ³´ 2d r r r : = ¡mf( ) ( 7) dt r Thati s , ^ r d 2d p£J)=¡mf( r r ; ( ) ( 8) dt dt ri whe r e^ st heuni tve c t o ri nt her a di a ldi r e c t i on.Themi r ac l eofi nv e r s es quar e 2 r r f o r c ei sno wi ns i g ht :Whe nf( ) =c o ns t ant= ¡®,we¯ndt ha tt heve c t or
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e {Le nzv e c t ort ( c a l l e dRung ho ughi twa so r i g i na l l ydi s c o v e r e dbyHami l t on! ) : r A ´p£J¡®m^
( 9)
i sc ons e r v e d.Butwene e de donl yo nec o ns t anto fmot i onwhi l eweno wha v eg ot 3c o mpo ne nt so fA whi c hwi l lpr e v e ntt hepar t i c l ef r o m mo vi ngatal l !Suc ha n o v e r ki l li sa vo i de dbe c aus eA s at i s ¯e st hef ol l o wi ngt wo,e as i l yv e r i ¯e d,r e l at i ons : A2=2 mJ2E+®2m2; A: J=0;
( 1 0)
2 = m¡®=ri whe r eE=p 2 st hec ons e r v e de ne r g yf o rt hemot i on.The¯r s tr e l a t i o n t e l l sy out ha tt hema gni t udeofA i s¯xe di nt e r mso fo t he rc ons t a nt sofmo t i o n a ndt hes e c ondo nes ho wst hatA l i e si nt heor bi t alpl a ne .The s et woc o ns t r a i nt s r e duc et henumbe ro fi nde pe nde ntc o ns t a nt si nA f r om 3t o1 ,e xa c t l ywha twe ne e de d.I ti st hi se xt r ac o ns t a ntt hatk e e pst hepl a ne tonac l o s e dor bi t .
Theul t i mat et e s tofo ura na l ys i si swhe t he rwec an¯ndt heor bi ti n( 5 )f o rt he c a s eo f¯= 0wi t houti nt e gr a t i nga nydi ®e r e nt i a le quat i o n.Thi si s ,o fc o ur s e , t r ue .To¯ndt heo r bi t ,weo nl yha vet ot a ket hedo tpr oduc tof( 9)wi t ht he : p£J)=J: r£p)=J2.Thi r a di usve c t orra ndus et hei de nt i t yr ( ( sgi ve s A: r=Arc osÁ=J2¡®mr;
( 1 1)
o r ,i namor ef ami l i arf o r m,t hec oni cs e c t i on: J2=®m) A ( =1+ c o sÁ: r ®m
( 1 2)
Asabo nuswes e et ha tA i si nt hedi r e c t i o no ft hemaj o raxi so ft hee l l i ps ea nd A = ® m.F i t sma gni t udei se s s e nt i al l yt hee c c e nt r i c i t yo ft heo r bi t :e= o rt hi s m®i r e a s on,A= sc al l e dt hee c c e nt r i c i t yve c t or . r Ha vi ngde ve l o pe da l lt he s ef or mal i s mss t ar t i ngf r o m U( )i n( 1 )wec anc l os et he 2 rt c i r c l ebya s ki ngwha thappe nst ot hee c c e nt r i c i t yve c t o rwhe nwea dda¯= e r m. 3 = r Obvi o us l y ,i fy oua dda1 c o mpone ntt ot hef or c e ,( whi c hc a nar i s e ,f o re xa mpl e , f r o mt hege ne r alr e l a t i vi s t i cc o r r e c t i o nst oNe wt on' sl a wofg r a vi t a t i o norbe c a us e t heSuni snots phe r i c a la ndhasas ma l lquadr upol emome nt )JandE a r es t i l l c o ns e r v e dbutnotA.I ft hepe r t ur bat i oni ss mal l ,i twi l lma ket hedi r e c t i o nofA s l o wl yc ha ngei ns pa c ea ndwewi l lg e ta` pr e c e s s i ng'e l l i ps e ,whi c hwi l lo fc our s e ¯l la2 di me ns i ona lr e g i o n.Fo rt hepo t e nt i a li n( 1 )we¯nd,us i ng( 8 ) ,t ha tt he _=¡( r r =r r a t eo fc ha ngeofRunge {Le nzve c t o ri sno wg i v e nbyA 2¯m= ) ( d=dt ) ( ) .
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_ to T) Thec hang e¢A pe ro r bi ti so bt a i ne dbyi nt e g r a t i ngAd v e rt her a nge( 0; , whe r eTi st hepe r i odo ft heo r i g i na lo r bi t .Do i ngonei nt e g r a t i o nbypar t sa nd c hang i ngt hev a r i a bl eo fi nt e g r at i onf r o m tt ot hepo l a ra ng l eÁ,wege t¢A pe r o r bi tt obe Z2¼ ¯ rdr ¯ ¯m ¢A orbit=¡2 dÁ: ( 1 3) 3 Á 0 rd Le tust a ket hec oo r di na t es ys t e ms uc ht ha tt heunpe r t ur be do r bi tor i gi nal l y ha dA poi nt i ngal ongt hexa xi s .Af t e ro neo r bi t ,a¢Ay c ompo ne ntwi l lbe g e ne r a t e dandt hemaj o ra xi soft hee l l i ps ewoul dha vepr e c e s s e dbyanamount A.The¢Ay c ¢Á=¢Ay= a nbee as i l yo bt a i ne df r o m( 1 3 )byus i ngy=rs i nÁ, = r c o nv e r t i ngt hede pe nde ntv a r i a bl ef r o mrt ou=( 1 )a nds ubs t i t ut i ng( du= dÁ)= ¡( A=J2)s i nÁ( whi c hc o me sf r om ( 1 2) ) .Thi sgi ve st hea ng l eo fpr e c e s s i onpe r o r bi tt obe Z 2¼¯m du ¢Ay 2¯m 2¼ = s i nÁ dÁ=¡ 2 : ( 1 4) ¢Á= A A 0 J dÁ Si nc eweha vet hee xa c ts ol ut i o ni n( 5 ) ,y ouc a ne as i l yv e r i f yt ha tt hi si si nde e d 2 ri t hepr e c e s s i o no ft heor bi twhe n¯= st r e at e da sape r t ur bat i on.TheRung e { =r Le nzve c t o rno to nl ya l l o wsust os ol vet he( 1 )pr o bl e m,bute ve nt e l l susho w ¡2 a nr pe r t ur ba t i o nma ke st heo r bi tpr e c e s s ! Wewi l lne xtc ons i de rt hequa nt um v e r s i ono ft hes a mepr o bl e m.I nt hi sc a s e ,we ¯r s tne e dt os o l v et heSc hr o Ä di ng e re quat i onf o rt hepo t e nt i ali n( 1) .I tt ur nso ut t ha tt hi si si nde e dpo s s i bl ea ndt hea na l ys i spr oc e e dse xa c t l yasi nt hec as eo fno r ma lhydr og e nat o mpr obl e m.Onc et heang ul a rde pe nde nc ei ss e par a t e doutus i ng µ ; Á) t hes t anda r ds phe r i c alhar moni c sY`m( ,t her a di a lpar toft hewa v e f unc t i o n R( r )wi l ls at i s f yt hedi ®e r e nt i a le qua t i o n ½ ¾ m ¯ ® ~2 0 0 2 0 2 R + R + 2 E¡ ` `+1 R=0; ( )¡ 2 + ( 1 5) 2 r ~ r r 2mr = v 0 e ¡ ® t a n d d isp ersio n < v 2 > ¡ < v > 2 = (¾ 2 = ® )(1 ¡ e ¡2 ® t ). A t la te tim es (t ! 1 ), th e m ea n v elo city < v > g o es to zero w h ile th e v elo city d isp ersio n b eco m es (¾ 2 = ® ). T h u s th e eq u ilib riu m co n ¯ g u ra tio n is a M a x w ellia n d istrib u tio n o f v elo cities w ith th is p a rticu la r d isp ersio n , fo r w h ich @ f = @ t = 0 . T o see th e e® ect o f th e tw o term s in d iv id u a lly o n th e in itia l d istrib u tio n f (v ;0 ) = ± (v ¡ v 0 ), w e ca n set ® o r ¾ to zero . W h en ® = 0 , w e g et p u re d i® u sio n : µ ¶1 = 2 ½ ¾ 1 (v ¡ v 0 )2 f ® = 0 (v ;t) = : (1 4 ) ex p ¡ 2¼ ¾ 2 t 2¾ 2 t
The equilibrium configuration is a Maxwellian distribution of velocities with this particular dispersion, for which f/ t = 0.
N o th in g h a p p en s to th e stea d y v elo city v 0 ; b u t th e v elo city d isp ersio n in crea ses in p ro p o rtio n to t rep resen tin g a ra n d o m w a lk in th e v elo city sp a ce. O n th e o th er h a n d , if w e set ¾ = 0 , th en w e g et f ¾ = 0 (v ;t) = ± (v ¡ v 0 e ¡® t ):
(1 5 )
N ow th ere is n o sp rea d in g in v elo city sp a ce (n o d i® u sio n ); in stea d th e frictio n stea d ily d ecrea ses < v > . G o in g b a ck to th e d iscrete ca se, w e ca n m a k e a n o th er u sefu l g en era liza tio n o f (5 ) b y a ssu m in g th a t p (¢ y ) itself d ep en d s o n N so th a t th e fu n d a m en ta l eq u a tio n b eco m es Z P N (x ) = d D y P N ¡ 1 (x ¡ ¢ y )p N (¢ y ) : (1 6 ) T h is eq u a tio n , w h ich is a co n v o lu tio n in teg ra l, is triv ia l to so lv e in F o u rier sp a ce in w h ich th e co n v o lu tio n in teg ra l b eco m es a p ro d u ct. If w e d en o te b y P N (k ) a n d p N (k ) th e F o u rier tra n sfo rm s o f P N (x ) a n d p N (¢ y ) th en
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Once again, it is possible to make some general comments if the
th is eq u a tio n b eco m es P N (k ) = P N ¡ 1 (k )p N (k ). Itera tin g th is N tim es a n d n o rm a lizin g th e in itia l p ro b a b ility b y a ssu m in g th e p a rticle w a s a t th e o rig in w e im m ed ia tely g et
individual probability distributions pn ( y)
P N (k ) =
satisfy some reasonable conditions.
YN
p n (k ) :
(1 7 )
n= 1
D o in g a n in v erse F o u rier tra n sfo rm w e ¯ n d th e so lu tio n to o u r p ro b lem to b e P N (x ) =
Z
N d D k ik ¢x Y e p n (k ) : (2 ¼ )D n= 1
(1 8 )
O n ce a g a in , it is p o ssib le to m a k e so m e g en era l co m m en ts if th e in d iv id u a l p ro b a b ility d istrib u tio n s p n (¢ y ) sa tisfy so m e rea so n a b le co n d itio n s. S u p p o se, fo r sim p licity, th a t p n (¢ y ) is p ea k ed a t th e o rig in a n d d ies d ow n sm o o th ly a n d m o n o to n ica lly fo r la rg e j¢ y j. T h en , its F o u rier tra n sfo rm w ill a lso b e p ea k ed a ro u n d th e o rig in in k -sp a ce a n d w ill d ie d ow n fo r la rg e va lu es o f jk j. F u rth er, b eca u se th e p ro b a b ility is n o rm a lized , w e h av e th e co n d itio n p n (k = 0 ) = 1 . W h en w e ta k e a p ro d u ct o f N su ch fu n ctio n s, th e resu ltin g fu n ctio n w ill a g a in h av e th e va lu e u n ity a t th e o rig in . B u t a s w e g o aw ay fro m th e o rig in , w e a re ta k in g th e p ro d u ct o f N n u m b ers ea ch o f w h ich is less th a n u n ity. S o clea rly w h en N ! 1 , th e p ro d u ct o f p n (k ) w ill h av e sig n i¯ ca n t su p p o rt o n ly clo se to th e o rig in . T h e n o n triv ia l a ssu m p tio n w e w ill n ow m a k e is th a t p n (k ) h a s a sm o o th cu rva tu re a t th e o rig in o f th e F o u rier sp a ce a n d is n o t `cu sp y '. T h en , n ea r th e o rig in in F o u rier sp a ce, w e ca n a p p rox im a te p n (k ) ' 1 ¡
646
1 2 2 2 2 ® n k ' e ¡ (1 = 2 )® n k 2
(1 9 )
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SERIES ARTICLE
w ith so m e co n sta n t ® n . H en ce th e p ro d u ct b eco m es YN
n=
N N 1 2X 2 p n (k ) = ex p ¡ k ® n ´ ex p ¡ ¾ 2 k 2 ; 2 n= 1 2 1
An observant reader would have noticed
(2 0 )
that we have essentially proved a variant of the central
w h ere w e h av e d e¯ n ed
limit theorem for the N 1 X 2 2 ¾ = ® : N n= 1 n
(2 1 )
sum x1 + x2 + ... + xN.
In th is lim it, th e ¯ n a l F o u rier tra n sfo rm in (1 8 ) is triv ia l a n d w ill g iv e a G a u ssia n in x w ith hx 2 i / N . A n o b serva n t rea d er w o u ld h av e n o ticed th a t w e h av e essen tia lly p rov ed a va ria n t o f th e cen tra l lim it th eo rem fo r th e su m (x 1 + x 2 + ::: + x N ) o f N in d ep en d en tly d istrib u ted ra n d o m va ria b les ea ch h av in g its ow n p ro b a b ility d istrib u tio n p n (x n ). In fa ct, th e jo in t p ro b a b ility fo r th ese v a ria b les to b e in so m e g iv en in terv a l is g iv en b y th e p ro d u ct o f p n (x n )d D x n ov er a ll n = 1 ;2 ;:::N . T h e p ro b a b ility fo r th eir su m to b e x is g iv en b y Z YN
P N (x ) =
p n (x n )d D x n ± D
n= 1
³ ´ X x ¡ xn ;
(2 2 )
w h ere th e D ira c d elta fu n ctio n en su res th a t th e su m o f th e ra n d o m va ria b les is x . W ritin g th e D ira c d elta fu n ctio n in F o u rier sp a ce, w e im m ed ia tely g et P N (x ) = =
Z Z
N Z d D k ik ¢x Y e d D x n p n (x n )e ¡ik ¢x n (2 ¼ )D n= 1 N d D k ik ¢x Y e p n (k ); (2 ¼ )D n= 1
(2 3 )
w h ich is id en tica l to th e resu lt w e o b ta in ed ea rlier in (1 8 ). A cla ssic ex a m p le in w h ich o u r a n a ly sis (a n d cen tra l lim it th eo rem ) fails is g iv en b y th e ca se in w h ich ea ch
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The key reason for the central limit
o f th e p ro b a b ility d istrib u tio n s p n (¢ y ) is g iv en b y a L o ren tzia n
theorem to fail in
p n (¢ y ) =
this case is that the Lorentzian distribution has a diverging second moment.
(¯ = ¼ ) : (¢ y )2 + ¯ 2
(2 4 )
T h e F o u rier tra n sfo rm n ow g iv es p n (k ) = ex p (¡ ¯ jk j). C lea rly o u r a p p rox im a tio n in (1 9 ) fa ils fo r th is fu n ctio n sin ce it is `cu sp y ' d u e to a lin ea r term in jk j n ea r th e o rig in . W e ca n , o f co u rse, ca rry o u t th e a n a ly sis in (1 8 ) to g et Z D d k ik ¢x ¡N ¯ jk j (N ¯ = ¼ ) P N (x ) = e e : (2 5 ) = jx j2 + (N 2 ¯ 2 ) (2 ¼ )D W e h av e th e resu lt th a t th e p ro b a b ility d istrib u tio n fo r th e ¯ n a l d isp la cem en t is id en tica l to th e p ro b a b ility d istrib u tio n o f in d iv id u a l step s w h en th e la tter is a L o ren tzia n { ex cep t fo r th e (ex p ected ) sca lin g o f th e w id th . T h e k ey rea so n fo r th e cen tral lim it th eo rem to fa il in th is ca se is th a t th e L o ren tzia n d istrib u tio n h a s a d iv erg in g seco n d m o m en t. Y o u sh o u ld rem em b er th is th e n ex t tim e y o u th in k o f fu ll w id th a t h a lf m a x im u m o f a L o ren tzia n a s `sim ila r to ' th e w id th o f a G a u ssia n ! T h ere a re p h y sica l situ a tio n s, (e.g ., o n e ca lled a n o m a lo u s d iffu sio n ), w h ich ca n b e m o d elled a lo n g th ese lin es. T h ey a re ch a ra cterized b y ra n d o m w a lk s in w h ich ev ery o n ce in a w h ile th e p a rticle ta k es a la rg e step b eca u se o f th e slow d ecrea se in th e p ro b a b ility p (¢ y ). V ery o ften o n e co n sid ers ra n d om w a lk o n a la ttice o f sp eci¯ c sh a p e, th e sim p lest b ein g th e D -d im en sio n a l cu b e. H ere th e p a rticle h o p s fro m o n e site o f th e la ttice to a n o th er n ea rb y site a lo n g a n y o n e o f th e a x es w ith th e la ttice sp a cin g ta k en to b e u n ity fo r sim p licity. In th is ca se th e F o u rier in teg ra ls in (1 8 ) w ill b eco m e F o u rier series a n d w e g et: P N (x ) =
Z¼
¡¼
648
YN dD k ¢x p n (k ) ; [co s(k )] (2 ¼ )D n= 1
(2 6 )
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SERIES ARTICLE
w h ere a ll th e in teg ra ls a re in th e ra n g e (¡ ¼ ;¼ ) a n d x is a v ecto r w ith in teg er va lu ed co m p o n en ts. If p n (k ) is in d ep en d en t o f n a n d h o p s in a ll d irectio n s fro m a n y site a re eq u a lly lik ely, th en p (k ) = (1 = D )(co s k 1 + co s k 2 + ¢¢¢co s k D ) a n d w e g et P N (x ) =
Z¼
¡¼
Ã
D d k 1 X [co s(k ¢ x )] co s k j D j= 1 (2 ¼ )D D
!N
1 2N
N
C
nL
=
[1]
:
N ! 1 : 2 N ((1 = 2 )(N + J ))!((1 = 2 )(N ¡ J ))! (2 8 )
An entertaining discussion of history is available in B Hughes, Random Walks and Random Environments, Vol.1, Oxford, 1965. Also see E W Montroll and M F Shlesinger, On the wonderful world of random walks, in Studies in Statistical Mechanics, edited by J L Lebowitz and E W Montroll, North-Holland, Vol.11, Amsterdam, 1984.
(2 7 )
A s a test, w e ca n rep ro d u ce th e sta n d a rd resu lt fo r o n ed im en sio n a l la ttice u sin g (2 7 ). In th is ca se x = J , w ith J b ein g a p o sitiv e o r n eg a tiv e in teg er. A fter N step s w h en th e p a rticle h a s ta k en n L step s to th e left o f o rig in a n d n R step s to th e rig h t, w e h a v e n L + n R = N a n d n R ¡ n L = J . S o lv in g , w e g et n R = (1 = 2 )(N + J ); n L = (1 = 2 )(N ¡ J ). T h e p ro b a b ility th a t o u t o f N step s n L w ere to th e left a n d n R w ere to th e rig h t is th e sa m e a s g ettin g , say, n L h ea d s w h ile to ssin g N co in s a n d is g iv en by P N (J ) =
Suggested Reading
[2]
Joseph Rudnick and George Gaspari, Elements of the Random Walk, Cambridge University Press, 2004.
[3]
There is an interesting history associated with this issue, involving S Chandrasekhar; see T Padmanabhan, Stellar Dynamics and Chandra, Current Science, Vol.70, p.784, 1996.
Y o u ca n a m u se y o u rself b y p rov in g th a t th is is a lso g iv en b y th e in teg ra l in (2 7 ) fo r D = 1 , Z¼ dk1 P N (J ) = [co s(k 1 J )](co s k 1 )N (2 9 ) ¼ (2 ) ¡¼ a s it sh o u ld . T h e resu lt in (2 7 ) w ill b e u sefu l in th e n ex t in sta llm en t w h en w e a d d ress so m e in terestin g d im en sio n d ep en d en t p ro p erties o f ra n d o m w a lk s (a n d a n u n ex p ected co n n ectio n w ith electrica l n etw o rk s!).
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected], [email protected]
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Snippets of Physics 20. Random Walk Through Random Walks – II T Padmanabhan
W e c o n tin u e o u r e x p lo ra tio n o f ra n d o m w a lk s w ith so m e m o r e c u r io u s r e su lts. W e d isc u ss th e d im e n sio n d e p e n d e n c e o f so m e o f th e fe a tu r e s o f th e ra n d o m w a lk , d e sc r ib e a n u n e x p e c te d c o n n e c tio n b e tw e e n ra n d o m w a lk s a n d e le c tric a l n e tw o r k s a n d ¯ n a lly d isc u ss so m e re m a r k a b le fe a tu re s o f r a n d o m w a lk w ith g e o m e tr ic a lly d e c re a sin g ste p -le n g th . In th e la st in sta lm en t, w e lo o k ed a t sev era l elem en ta ry fea tu res o f ra n d o m w a lk a n d , in p a rticu la r, o b ta in ed a g en era l fo rm u la fo r th e p ro b a b ility P N (x ), fo r th e p a rticle to b e fo u n d a t p o sitio n x a fter N step s. T h is resu lt, in th e ca se o f ra n d o m w a lk in a cu b ic la ttice ca n b e w ritten a s th e in teg ra l P N (x ) =
Z¼
¡¼
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
à !N D dD k 1 X : (1 ) [co s(k ¢ x )] co s k j D (2 ¼ )D j= 1
T h is resu lt { fo r a n a rb itra ry d im en sio n D { m ig h t d eciev e y o u in to b eliev in g th a t th e b eh av io u r o f ra n d o m w a lk in , say, D = 1 ;2 ;3 is a ll essen tia lly th e sa m e. In fa ct th ey a re n o t, a s ca n b e illu stra ted b y stu d y in g th e p h en o m en o n k n ow n a s recu rren ce. R ecu rren ce refers to th e p ro b a b ility fo r th e ra n d o m w a lk in g p a rtcle to co m e b a ck to th e o rig in , w h ere it sta rted fro m , in th e co u rse o f its p era m b u la tio n , w h en w e w a it fo rev er. L et u n d en o te th e p ro b a b ility th a t a p a rticle retu rn s to th e o rig in o n th e n th step a n d let R b e th e ex p ected n u m b er o f tim es it retu rn s to th e o rig in .
RESONANCE August 2009
Keywords Random walk, decreasing steps, recurrence, Watson integrals, grid of resistors.
799
SERIES ARTICLE
C lea rly, R =
X1
(2 )
un :
n= 0
W e ca n n ow d istin g u ish b etw een tw o d i® eren t scen a rio s. If th e series in (2 ) d iv erg es, th en th e m ea n n u m b er o f retu rn s to th e o rig in is in ¯ n ite a n d w e say th a t th e ra n d o m w a lk is recu rren t. If th e series is co n v erg en t, lea d in g to a ¯ n ite R , th en w e say th a t th e ra n d o m w a lk is tra n sien t. T h is id ea ca n b e rein fo rced b y th e fo llow in g a ltern a tiv e in terp reta tio n o f R . S u p p o se u is th e p ro b a b ility fo r th e p a rticle to retu rn to th e o rig in . T h en th e n o rm a lized p ro b a b ility fo r it to retu rn ex a ctly k tim es is u k (1 ¡ u ). T h e m ea n n u m b er o f retu rn s to th e o rig in is, th erefo re, R =
X1
k= 1
k u k ¡ 1 (1 ¡ u ) = (1 ¡ u )¡1 :
(3 )
O b v io u sly, if R = 1 , th en u = 1 , sh ow in g th a t th e p a rticle w ill d e¯ n itely retu rn to th e o rig in . B u t if R < 1 , th en u < 1 a n d w e ca n 't b e certa in th a t th e p a rticle w ill ev er co m e b a ck h o m e. L et u s co m p u te u n a n d R fo r ra n d o m w a lk s in D = 1 ;2 ;3 d im en sio n s w ith th e la ttice sp a cin g set to u n ity fo r sim p licity. F ro m (1 ), settin g x = 0 , w e h av e: Ã !n Z¼ D dD k 1 X u n (x ) = : co s k j (4 ) D D j= 1 ¡ ¼ (2 ¼ ) D o in g th e su m in (2 ) w e g et R =
X1
n= 0
un =
Z¼
¡¼
dD k (2 ¼ )D
Ã
D 1 X 1¡ co s k j D j= 1
!¡ 1
:
(5 )
W e w a n t to k n ow w h eth er th is in teg ra l is ¯ n ite o r d iv erg en t. C lea rly, th e d iv erg en ce, if a n y, ca n o n ly a rise d u e to its b eh av io u r n ea r th e o rig in in k -sp a ce. U sin g
800
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SERIES ARTICLE
th e T ay lo r series ex p a n sio n o f th e co sin e fu n ctio n , w e see th a t, n ea r th e o rig in w e h av e th e b eh av io u r: Z Z ¢ k D ¡1 d k d k 1 d k 2 :::d k D ¡ 2 2D 2 2 ¡1 R ¼ 2D k k ¢¢¢ k / : (6 ) + + + 1 2 D (2 ¼ )D (2 ¼ )D k ¼ 0 k 2 k ¼0 T h e d im en sio n d ep en d en ce is n ow o b v io u s. In D = 1 ;2 th e in teg ra l is d iv erg en t a n d R = 1 ; so w e co n clu d e th a t th e ra n d o m w a lk in D = 1 ;2 is recu rren t a n d th e p a rticle w ill d e¯ n itely retu rn to th e o rig in if it w a lk s fo rev er. B u t in D = 3 , th e R is ¯ n ite a n d th e w a lk is n o n -recu rren t. T h ere is a ¯ n ite p ro b a b iltity th a t th e p a rticle w ill co m e b a ck to th e o rig in b u t th ere is a lso a ¯ n ite p ro b a b ility th a t it w ill n o t. A d ru n k en m a n w ill d e¯ n itely co m e h o m e (g iv en en o u g h tim e) b u t a d ru n k en b ird o n ° ig h t m ay o r m ay n o t! T h e m ea n n u m b er o f recu rren ces in D = 3 is g iv en b y th e W a tso n in teg ra l Z¼ Z¼ Z¼ 3 R = dk1 dk2 d k 3 [1 ¡ (co s k 1 + co s k 2 + co s k 3 )]¡1 ; 3 (2 ¼ ) ¡ ¼ ¡¼ ¡¼
(7 )
w h ich is n o to rio u sly d i± cu lt to eva lu a te a n a ly tica lly [1 ]. S in ce th e a n sw er h a p p en s to b e p µ ¶ µ ¶ µ ¶ µ ¶ 6 1 5 7 11 R = ; ¡ ¡ ¡ ¡ (8 ) 3 32¼ 24 24 24 24 y o u a n y w ay n eed to lo o k it u p in a ta b le so o n e m ig h t a s w ell d o th e in teg ra l n u m erica lly w h ich is triv ia l in M athem atica. (O f co u rse, if y o u lik e th e ch a llen g e o f a d e¯ n ite in teg ra l, try it o u t. I d o n o t k n ow o f a sim p le w ay o f d o in g it; n eith er d o th e ex p erts in ra n d o m w a lk I h av e ta lk ed to !) T h e resu lt is R ¼ 1 :5 1 6 4 g iv in g th e retu rn p ro b a b ility u ¼ 0 :3 4 0 5 . In th e ca se o f D = 1 ;2 it is a lso ea sy to o b ta in u n ex p licitly b y co m b in a to rics. In 1 -d im en sio n , th e p a rticle ca n retu rn to th e o rig in o n ly if it h a s ta k en a n ev en n u m b er o f step s, h a lf to th e rig h t a n d h a lf to th e left. T h e p ro b a b ility fo r th is is clea rly u 2n =
RESONANCE August 2009
2n
C
1 n
22n
:
(9 )
801
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Fp o r su ± cien tly la rg e n , w e capn u se S terlin g 's a p p rox im a tio n fo r fa cto ria ls (n ! ¼ 2 ¼ n e ¡ n n n ) to g et u 2 n ¼ 1 = ¼ n . T h e series in (2 ) in v o lv es th e a sy m p to tic su m w h ich is d iv erg en t: m =
X n
u 2n ¼
X n
p
1 = 1 : ¼n
(1 0 )
O b v io u sly, th e 1 -d im en sio n a l ra n d o m w a lk is recu rren t. In terestin g ly en o u g h th e resu lt fo r D = 2 is ju st th e sq u a re o f th e resu lt fo r D = 1 . T h e in teg ra l in (4 ) b eco m es fo r D = 2 : Z¼ Z¼ 1 1 u n (x ) = dk1 d k 2 (co s k 1 + co s k 2 )n : (1 1 ) (2 ¼ )D 2 n ¡¼ ¡¼ If y o u n ow ch a n g e va ria b les o f in teg ra tio n to (k 1 + k 2 ) a n d (k 1 ¡ k 2 ) it is ea sy to sh ow th a t th is in teg ra l b eco m es th e p ro d u ct o f tw o in teg ra ls g iv in g · ¸2 1 2n u 2n = Cn ; (1 2 ) 22n w h ich is th e sq u a re o f th e resu lt fo r D = 1 . N ow th e series in (2 ) w ill b e d o m in a ted a sy m p to tica lly b y m ¼
X n
1 = 1 ¼n
;
(1 3 )
m a k in g th e D = 2 ra n d o m w a lk recu rren t a g a in . Y o u m ig h t g u ess a t th is sta g e, th a t in 3 -D , th e a sy m p to tic series w ill in v o lv e su m ov er n ¡3 = 2 (a n d h en ce w ill co n v erg e) m a k in g th e 3 -D ra n d o m w a lk n o n -recu rren t. T h is is p a rtia lly tru e a n d th e 3 -d im en sio n a l series is b o u n d ed fro m a b ov e b y th e su m ov er n ¡3 = 2 . B u t th e 3 -d im en sio n a l ca se is n ot th e p ro d u ct o f th ree 1 -d im en sio n a l ca ses. W e n ow tu rn o u r a tten tio n to a n o th er cu rio u s resu lt. S u m m in g P N (x ) ov er a ll N o n e ca n co n stru ct th e q u a n tity P (x ) w h ich is th e p ro b a b ility o f rea ch in g x . U sin g (1 ) a n d d o in g th e g eo m etric su m , w e ¯ n d in D = 2 , th is q u a n tity to b e: µ ¶¡1 Z¼ Z¼ d k 1d k 2 1 P (x ) = : [co s(k ¢ x )] 1 ¡ (co s k 1 + co s k 2 ) (1 4 ) 2 2 ¡¼ ¡¼ (2 ¼ ) 802
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SERIES ARTICLE
C o n sid er n o w th e ex p ressio n Z¼ Z¼
1 R = (P (x ) ¡ P (0 )) = 2
¡¼
¡¼
dk1dk2 [1 ¡ co s(k ¢ x )] : 8¼ 2 [1 ¡ 12 (co s k 1 + co s k 2 ]
(1 5 )
In cred ib ly en o u g h th is p rov id es th e so lu tio n to a co m p letely d i® eren t p ro b lem ! It g iv es th e e® ectiv e resista n ce b etw een a la ttice p o in t x a n d th e o rig in in a n in ¯ n ite g rid o f 1 -o h m resisto rs co n n ected b etw een th e la ttice sites. L et u s see h ow th is co m es a b o u t b y a n a ly sin g th e g rid o f resisto rs. L et a n o d e x in th e in ¯ n ite p la n a r sq u a re la ttice b e d en o ted b y tw o in teg ers (m ;n ) a n d let a cu rren t I m ;n b e in jected a t th a t n o d e. T h e ° ow o f cu rren t w ill in d u ce a v o lta g e a t ea ch n o d e a n d , co m b in in g K irch o ® 's a n d O h m 's law s fo r th e 1 -o h m resisto rs, w e ca n w rite th e rela tio n : Im
;n
= (V m
;n
¡ Vm
+ 1 ;n
) + (V m
= 4V m
;n
¡ Vm
+ 1 ;n
¡ Vm
¡ Vm
;n
¡1 ;n
¡ Vm
¡1 ;n
) + (V m
;n + 1
¡ Vm
;n
¡ Vm
;n + 1 )
+ (V m
;n
¡ Vm
;n ¡1 )
(1 6 )
;n ¡1 ;
w h ere V m ;n is th e p o ten tia l a t th e n o d e (m ;n ) d u e to th e cu rren t. T h is eq u a tio n ca n a g a in b e so lv ed b y in tro d u cin g th e F o u rier tra n sfo rm o n th e d iscrete la ttice. If w e w rite Z¼Z¼ 1 I m ;n = d k 1 d k 2 I (k 1 ;k 2 )e i(m k 1 + n k 2 ) ; (1 7 ) 2 4 ¼ ¡¼ ¡ ¼ Z¼Z¼ 1 V m ;n = d k 1 d k 2 V (k 1 ;k 2 )e i(m k 1 + n k 2 ) ; (1 8 ) 2 4 ¼ ¡¼ ¡ ¼ th en o n e ca n o b ta in fro m (1 6 ) th e resu lt in th e F o u rier sp a ce: I (k 1 ;k 2 ) = 2 V (k 1 ;k 2 ) [2 ¡ co s(k 1 ) ¡ co s(k 2 )] :
(1 9 )
S u p p o se w e n ow in ject a cu rren t o f 1 a m p at (0 ,0 ) a n d ¡ 1 a m p a t (N ;M ). T h en I m ;n = ± m ;n ¡ ± m ¡ M ;n ¡ N , lea d in g to I (k 1 ;k 2 ) = 1 ¡ e ¡i(M
k1+ N k2)
;
(2 0 )
so th a t (1 9 ) g iv es th e v o lta g e to b e V (k 1 ;k 2 ) =
RESONANCE August 2009
1 1 ¡ e ¡i(M k 1 + N k 2 ) £ : 2 2 ¡ co s(k 1 ) ¡ co s(k 2 )
(2 1 )
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T h e eq u iva len t resista n ce b etw een n o d es (0 ,0 ) a n d (M ,N ) w ith a ° ow o f u n it cu rren t is ju st th e v o lta g e d i® eren ce b etw een th e n o d es: R
M ;N
= V 0 ;0 ¡ V M ;N Z¼ Z¼ £ ¤ 1 i(M k 1 + N k 2 ) k k V ;k ¡ e = d d (k ) 1 1 2 1 2 4 ¼ 2 ¡¼ ¡¼ Z¼ Z¼ 1 1 (1 ¡ e ¡ i(M k 1 + N k 2 ) )(1 ¡ e i(M k 1 + N k k = d d 1 2 4 ¼ 2 ¡ ¼ ¡¼ 2 2 ¡ co s(k 1 ) ¡ co s(k 2 ) Z¼ Z¼ 1 1 ¡ co s(M k 1 + N k 2 ) ; = dk1 dk2 2 4 ¼ ¡ ¼ ¡¼ 2 ¡ co s(k 1 ) ¡ co s(k 2 )
k2)
)
(2 2 )
w h ich is ex a ctly th e sa m e a s th e in teg ra l in (1 5 )! T h e in ¯ n ite g rid o f sq u a re la ttice resisto rs is a cla ssic p ro b lem a n d th e e® ectiv e resista n ce b etw een tw o a d ja cen t n o d es is a `trick q u estio n ' th a t is a fav o u rite o f ex a m in ers. T h e a n sw er (0 .5 o h m ) ca n b e fo u n d b y triv ia l su p erp o sitio n b u t su ch trick s a re u seless to ¯ n d th e e® ectiv e resista n ce b etw een a rb itra ry n o d es. In fa ct, th e e® ectiv e resista n ce b etw een tw o d ia g o n a l n o d es o f th e b a sic sq u a re { th e (0 ,0 a n d (1 ,1 ), say { is g iv en b y th e in teg ra l Z¼ Z¼ 1 1 ¡ co s(k 1 + k 2 ) R 1 ;1 = : dk1 dk2 (2 3 ) 2 4 ¼ ¡¼ ¡¼ 2 ¡ co s(k 1 ) ¡ co s(k 2 ) T h is is d o a b le, b u t n o t ex a ctly ea sy, a n d th e a n sw er is 2 = ¼ . (N ex t tim e so m eo n e lectu res y o u a b o u t th e p ow er o f clev er a rg u m en ts, a sk h er to u se th em to g et th is a n sw er, w h ich h a s a ¼ in it!) B u t w h y d o es th is w o rk ? W h a t h a s ra n d o m w a lk o n a la ttice to d o w ith resisto r n etw o rk s? T h ere a re d i® eren t lev els o f so p h istica tio n a t w h ich o n e ca n a n sw er th is q u estio n a n d a n en tire b o o k [2 ] d ea lin g w ith th is su b ject ex ists. T h e m a th em a tica l rea so n h a s to d o w ith th e fa ct th a t b o th th e ra n d o m w a lk p ro b a b ility to v isit a n o d e a n d th e v o lta g e o n a n o d e (w h ich d o es n o t h a v e a n y cu rren t in jected o r rem ov ed ) a re h a rm o n ic fu n ctio n s. T h ese a re fu n ctio n s w h o se va lu e a t a n y g iv en n o d e is g iv en b y th e av era g e o f th e va lu e o f th e fu n ctio n o n th e a d ja cen t la ttice sites. T h is is o b v io u s in th e ca se o f ra n d o m w a lk b eca u se a p a rticle w h ich rea ch es th e n o d e (m ;n ) m u st h av e h o p p ed to th a t n o d e w ith eq u a l p ro b a b ility fro m o n e o f th e n eig h b o u rin g n o d es (m § 1 ;n § 1 ). In th e ca se o f a resisto r n etw o rk , th e sa m e resu lt is o b ta in ed fro m (1 6 ) w h en I m n = 0 . If y o u n ow in ject th e v o lta g es 1
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a n d 0 a t tw o sp eci¯ c n o d es A a n d B , th en th e v o lta g e a t a n y o th er n o d e X ca n b e in terp reted a s th e p ro b a b ility th a t a ra n d o m w a lk er sta rtin g a t X w ill g et to A b efo re B . O n e ca n th en u se th is in terp reta tio n to m a k e a fo rm a l co n n ectio n b etw een v o lta g e d istrib u tio n in a n electric n etw o rk a n d a ra n d o m w a lk p ro b lem . T h e in terested rea d er ca n ¯ n d m o re in th e b o o k [2 ] referred to a b ov e. H av in g d o n e a ll th ese in th e la ttice, w e n ow g o b a ck to th e ra n d o m w a lk in fu ll sp a ce fo r w h ich w e h a d o b ta in ed th e resu lt in th e la st in sta llm en t, w h ich , sp ecia lised to o n e d im en sio n , is g iv en b y : P N (x ) =
Z1
¡1
N d k ik x Y p n (k ) : e (2 ¼ ) n= 1
(2 4 )
W e w a n t to co n sid er a situ a tio n in w h ich th e step s a re ra n d o m a n d u n co rrela ted b u t th eir len g th s a re d ecrea sin g m o n o to n ica lly. (T h is is w h a t w ill h a p p en if th e d ru n ka rd g ets tired a s h e w a lk s!) In p a rticu la r, w e w ill a ssu m e th a t ea ch step len g th is a fra ctio n ¸ o f th e p rev io u s o n e w ith ¸ < 1 a n d th e ¯ rst step is o f u n it len g th . It is clea r th a t P N (x ) is n ow g iv en b y P N (x ) =
Z1
¡1
N d k ik x Y e co s(k ¸ n ) : (2 ¼ ) n= 1
(2 5 )
W e ca n n ow stu d y th e lim it o f N ! 1 a n d a sk h ow th e p ro b a b ility P (x ) ´ P 1 (x ) is d istrib u ted . T h is fu ctio n sh ow s in cred ib ly d iv erse p ro p erties d ep en d in g o n th e va lu e o f ¸ a n d is still n o t co m p letely a n a ly sed . I w ill co n ¯ n e m y self to a sim p le situ a tio n , referrin g y o u to th e litera tu re if y o u a re in terested [3 ]. L et u s ¯ rst co n sid er th e ca se w h en ¸ = 1 = 2 . In th is ca se th e releva n t in ¯ n ite p ro d u ct is g iv en b y 1 Y
n= 1
co s
k sin k : = n k 2
(2 6 )
(T h is is a cu te resu lt w h ich y o u m ig h t w a n t to p rov e fo r y o u rself; A ll y o u n eed to d o is to w rite co s(k = 2 n ) = (1 = 2 )[sin (k = 2 n ¡ 1 )= sin (k = 2 n )], ta k e a p ro d u ct o f N term s ca n cellin g o u t th e sin es a n d th en ta k e th e lim it N ! 1 .) S in ce th e fo u rier tra n sfo rm o f (sin k = k ) is ju st a u n ifo rm d istrib u tio n , w e g et th e ta n ta lisin g resu lt th a t P (x ) is ju st a u n ifo rm d istrib u tio n in th e in terva l (¡ 1 ;1 ) a n d zero elsew h ere!
RESONANCE August 2009
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SERIES ARTICLE
T h is trick w h ich h a s b een u sed to g et (2 6 ) a lso w o rk s fo r ¸ = 2 ¡1 = 2 ;2 ¡ 1 = 4 :::; etc. F o r ex a m p le, w h en ¸ = 2 ¡1 = 2 th e in ¯ n ite p ro d u ct is p 1 Y sin k sin 2 k n =2 p : co s(k = 2 ) = (2 7 ) k k 2 n= 1 T h e F o u rier tra n sfo rm o f th is in v o lv es a co n v o lu tio n o f tw o recta n g u la r d istrib u tio n s a n d is ea sily seen to b e a tria n g u la r p ro b a b ility d istrib u tio n . (I w ill let y o u ex p lo re th e g en era l ca se o f ¸ = 2 ¡1 = k .) It tu rn s o u t th a t th e in ¯ n ite p ro d u ct o f co s(k ¸ n ) is ex tra o rd in a rily sen sitiv e to ch a n g es in ¸ . F o r a lm o st ev ery ¸ in th e in terva l (0 :5 ;1 ) th is p ro d u ct is sq u a re in teg ra b le. B u t o n ce in a w h ile, it is n o t so . F u rth er, if ¸ < (1 = 2 ), th e su p p o rt fo r P (x ) h a p p en s to b e th e Cp a n to r set. T h e rea lly b iza rre b eh av io u r o ccu rs w h en ¸ is th e g o ld en ra tio g = ( 5 ¡ 1 )= 2 . C lea rly, th ere a re en o u g h su rp rises in sto re in th e stu d y o f ra n d o m w a lk s. Suggested Reading [1] This integral was first evaluated by Watson in terms of elliptic integrals and a “simpler” result was obtained by Glasser and Zucker: G N Watson, Three triple integrals, Quarterly J. Math., Vol.10, p.266, 1939; M L Glasser and I J Zucker, Extended Watson integrals for the cubic latice, Proc. Natl. Acad. Sci., USA,Vol.74, p.1800, 1977. [2] There is a large literature on this subject, most of which can be found from the references in the book: P G Doyle and J Laurie Snell, Random Walks and Electric Networks, Mathematical Association of America, Oberlin, OH, 1984. The book is available on the web as a pdf file at: http://math.dartmouth.edu/~doyle docs/walks/walks.pdf. [3] This interesting topic does not seem to have been explored extensively. A good discussion is available in the paper: P L Krapivsky and S Redner, Am. J. Phys., Vol.72(5), p.591, 2004. Also see, K E Morrison, Random Walks with Decreasing Steps, available at: http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf
Address for Correspondence T Padmanabhan, IUCAA, Post Bag 4, Pune University Campus ,Ganeshkhind, Pune 411 007, India. Email: [email protected], [email protected]
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SERIES ARTICLE
Snippets of Physics 21. Extreme Physics T Padmanabhan
V a ria tio n a l p rin c ip le s p la y a c e n tr a l ro le in th e o re tic a l p h y sic s in m a n y g u ise s. W e w ill d isc u ss , in th is in sta lm e n t, so m e c u r io u s fe a tu r e s a sso c ia te d w ith a c o u p le o f v a r ia tio n a l p r o b le m s. In H erm a n M elv ille's 1 8 5 1 cla ssic M oby D ick th ere is a ch a p ter ca lled \ T h e T ry -W o rk s" w h ich d escrib es h ow th e try -p o ts o f th e sh ip P equ od a re clea n ed . (In ca se y o u h av en 't rea d th e b o o k , a try -p o t is a la rg e ca u ld ro n , u su a lly m a d e o f iro n , w h ich is u sed to o b ta in liq u id o il fro m w h a le b lu b b er.) In th a t is th e p a ssa g e: \ It w a s in th e left h a n d try -p o t o f th e P eq u o d ..... th a t I w a s ¯ rst in d irectly stru ck b y th e rem a rka b le fa ct, th a t in g eo m etry a ll b o d ies g lid in g a lo n g th e cy clo id , m y so a p sto n e fo r ex a m p le, w ill d escen d fro m a n y p o in t in p recisely th e sa m e tim e." T h e rem a rka b le fa ct M elv ille w rites a b o u t is rela ted to w h a t is k n ow n a s th e b ra ch isto ch ro n e p ro b lem (brachistos m ea n in g sh o rtest a n d chron os referrin g to tim e) w h ich req u ires u s to ¯ n d a cu rv e co n n ectin g tw o p o in ts A a n d B in a v ertica l p la n e su ch th a t a p a rticle, slid in g u n d er th e a ctio n o f g rav ity, w ill trav el fro m A to B in th e sh o rtest p o ssib le tim e. It w a s k n ow n to J o h a n n B ern o u lli (a n d to sev era l o th ers; see B ox 1 fo r a ta ste o f h isto ry ) th a t th is cu rv e is (a p a rt o f) a cy clo id if w e ta k e th e E a rth 's g rav ita tio n a l ¯ eld to b e co n sta n t. T h e cy clo id a l p a th a lso h a s th e p ro p erty th a t th e tim e ta k en fo r a p a rticle to ro ll fro m a n y p o in t to th e m in im u m o f th e cu rv e is in d ep en d en t o f w h ere it sta rted fro m { w h ich is w h a t M elv ille w a s ta lk in g a b o u t. In o th er w o rd s, a p a rticle ex ecu tin g o scilla tio n s in a cy clo id a l tra ck u n d er th e a ctio n o f g rav ity w ill m a in ta in a p erio d w h ich is
RESONANCE September 2009
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Keywords Brachistochrone, cycloid, extremum problem, Bernoulli.
907
SERIES ARTICLE
in d ep en d en t o f a m p litu d e. T h is is q u ite va lu a b le in th e co n stru ctio n o f p en d u lu m clo ck s a n d th e ea rly clo ck m a k ers k n ew th is w ell. (T h is ea rn ed cy clo id th e n a m es iso ch ro n e a n d ta u to ch ro n e a s if b ra ch isto ch ro n e is n o t en o u g h !) T h e cy clo id is th e cu rv e tra ced b y a p o in t o n th e circu m feren ce o f a w h eel w h ich is ro llin g w ith o u t slip p in g a lo n g a stra ig h t lin e. F ro m th is it is ea sy to sh ow (see F igu re 1 ) th a t th e p a ra m etric eq u a tio n (x = x (µ ); y = y (µ )) to a cy clo id h a s th e fo rm x = a (µ ¡ sin µ );
y = a (1 ¡ co s µ ) ;
(1 )
w h ere a is th e ra d iu s o f th e ro llin g circle. W e sh a ll n ow ta k e a clo ser lo o k a t th is resu lt. W h ile th e in itia l so lu tio n to th e b ra ch isto ch ro n e p ro b lem in v o lv ed so m e o f th e in tellectu a l g ia n ts o f th e sev en teen th cen tu ry, it is n ow w ith in th e g ra sp o f a n u n d erg ra d u a te stu d en t. L et y (x ) d en o te th e eq u a tio n to th e cu rv e w h ich is th e so lu tio n to th e b ra ch isto ch ro n e p ro b lem w ith th e co o rd in a tes ch o sen su ch th a t x is h o rizo n ta l a n d y is m ea su red v ertica lly d ow n w a rd s a s in F igu re 1 . L et th e p a rticle b eg in its slid e fro m th e o rig in w ith zero v elo city. If th e in ¯ n itesim a l a rc len g th a lo n g th e cu rv e a ro u n d th e p o in t P (x ;y ) is d s = (1 + y 02 )1 =p2 d x , w h ere y 0 = (d y = d x ), th en th e p a rticle ta k es tim e d t = d s= v , w h ere v = 2 g y is its sp eed a t P . T o d eterm in e th e cu rv e w e o n ly n eed to ¯ n d th e ex trem u m o f th e in teg ra l ov er d t w h ich is a stra ig h tfo rw a rd p ro b lem in th e ca lcu lu s o f va ria tio n . W e w ill, h ow ev er, a n a ly se it fro m tw o slig h tly d i® eren t a p p ro a ch es. In th e ¯ rst a p p ro a ch , w e sh a ll m a k e a co o rd in a te tra n sfo rm a tio n w h ich sim p li¯ es th e p ro b lem co n sid era b ly. L et u s in tro d u ce tw o n ew co o rd in a tes ® a n d ¯ in p la ce o f th e sta n d a rd C a rtesia n co o rd in a tes (x ;y ) in th e ¯ rst q u a d ra n t b y th e rela tio n s
Figure 1.
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x = ®
2
µ
¯ ¯ ¡ sin ® ®
¶
µ ¶ ¯ y = ® ; 1 ¡ co s ® 2
;
(2 )
w h ere ® > 0 a n d 0 · ¯ · 2 ¼ ® . O b v io u sly, fo r a ¯ x ed ® , th e cu rv e x (¯ );y (¯ ) is a cy clo id (w h ich tells y o u th a t w e a re ch ea tin g a little b it u sin g o u r k n ow led g e o f th e ¯ n a l so lu tio n !). T h e sq u a re o f th e v elo city o f th e p a rticle v 2 = 2 g y = x_2 + y_2 ;
(3 )
w h ere ov erd o ts d en o te d i® eren tia tio n w ith resp ect to tim e, ca n n ow b e ex p ressed in term s o f ¯_ a n d ®_ b y stra ig h tfo rw a rd a lg eb ra . T h is g iv es th e rela tio n µ ¶2 ¯ ¯ 2 ®_ 2 : ¡ ¯ co s 2 g y = 2 y ¯_ + 4 2 ® sin (4 ) 2® 2®
p g. T h e term in v o lv in g ®_ 2 is n o n -n eg a tiv e; fu rth er, sin ce y > 0 w e h av e ¯_ · In teg ra tin g th is rela tio n b etw een t = 0 a n d t = T , w h ere T is th e tim e o f d escen t, w e g et ZT ZT p p _ ¯ (T ) = ¯ dt · g dt = g T : (5 ) 0
0
It fo llow s th a t th e tim e o f d escen t is b o u n d ed fro m b elow b y th e eq u a lity T = p p ¯ (T )= g . T h e b est w e ca n d o is to set ¯_ = g a n d ®_ = 0 to sa tisfy (4 ) a n d h it th e low er b o u n d in (5 ). S in ce th e req u ired cu rv e h a s ® = co n sta n t, it is o b v io u sly a cy clo id p a ra m eterized b y ¯ . T h e a n g u la r p a ra m eter µ = ¯ = ® o f th e cy clo id va ries w ith tim e a t a co n sta n t ra te p µ_ = ¯_= ® = g = ® . It is clea r fro m th e p a ra m eteriza tio n in (2 ) th a t th e ra d iu s a o f 2 th e circle w h ich ro ta tes to g en era te th e cy clo id is rela pted to ® b y a = ® . H en ce g = a . If th e p a rticle m ov es th e a n g u la r v elo city o f th e ro llin g circle is ! = µ_ = a ll th e w ay to th e o th er en d o f th e cy clo id a t a h o rizo n ta l d ista n ce L = 2 ¼ a , th en th e tim e o f ° ig h t w ill b e T = 2 ¼ = ! = (2 ¼ L = g )1 = 2 . If L is th e 1 0 0 m , th en w ith g = 9 :8 m s¡ 2 w e g et T ¼ 8 sec w h ich is b etter th a n th e w o rld reco rd fo r 1 0 0 m d a sh ! G rav ity seem s to d o q u ite w ell. T h ere is a n o th er in d irect w ay o f a rriv in g a t th e cy clo id a l so lu tio n th a t is o f so m e in terest. T h is a p p ro a ch u ses th e co n cep t o f h o d o g ra p h w h ich is th e cu rv e tra ced b y a p a rticle in th e v elo city sp a ce [1 ]. L et u s try to d eterm in e th e h o d o g ra p h
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co rresp o n d in g to th e m o tio n o f sw iftest d escen t. F o r sim p licity, co n sid er th e fu ll tra n sit o f th e p a rticle fro m a p o in t A to a p o in t B in th e sa m e h o rizo n ta l a x is y = 0 . L et th e sp eed o f th e p a rticle b e v w h en th e v elo city v ecto r m a k es a n a n g le µ w ith resp ect to th e v x ¡ a x is in th e v elo city sp a ce. T h en th e h o d o g ra p h is g iv en b y so m e cu rv e u (µ ) w h ich w e a re try in g to d eterm in e. U sin g x_ = v co s µ ; y_ = v sin µ ;y = v 2 = 2 g , w e ca n w rite th e rela tio n s dt =
dv ; g sin µ
dx =
vdv co t µ : g
(6 )
W e a re n ow req u ired to m in im ize th e in tegra l ov er d t w h ile k eep in g th e in teg ra l ov er d x ¯ x ed . In co rp o ra tin g th e la tter co n stra in t b y a L a g ra n g e m u ltip lier (¡ ¸ ), w e see th a t w e n eed to m in im ize th e fo llo w in g in teg ra l µ ¶ Z dv 1 I = ¡ ¸ v co t µ : (7 ) g sin µ T h e m in im iza tio n is triv ia l sin ce n o d eriva tiv es o f th e fu n ctio n s a re in v o lv ed a n d lea d s to th e rela tio n v = (1 = ¸ ) co s µ w ith ¡ ¼ = 2 < µ < ¼ = 2 . W e ca n n ow tra d e o ® th e L a g ra n g e m u ltip lier ¸ fo r th e to tal h o rizo n ta l d ista n ce L (o b ta in ed b y in teg ra tin g d x ) a n d o b ta in ¸ 2 = ¼ = 2 g L . H en ce, o u r h o d o g ra p h is g iv en b y th e eq u a tio n r 2gL v (µ ) = co s µ ´ 2 R 0 co s µ : (8 ) ¼ T h is is ju st th e p o la r eq u a tio n fo r a circle o f ra d iu s R w ith th e left-m o st p o in t o f th e circle.
0
w ith th e o rig in co in cid in g
H ow d o w e g et to th e cu rv e in rea l sp a ce fro m th e h o d o g ra p h in th e v elo city sp a ce? In th is p a rticu la r ca se, it is q u ite ea sy. S u p p o se w e sh ift th e circu la r h o d o g ra p h h o rizo n ta lly to th e left b y a d ista n ce R 0 . T h is req u ires su b tra ctin g a h o rizo n ta l v elo city w h ich is n u m erica lly eq u a l to th e ra d iu s o f th e h o d o g ra p h . A fter th e sh ift, w e o b ta in th e h o d o g ra p h o f a u n ifo rm circu la r m o tio n w h ich is, o f co u rse, a circu la r h o d o g ra p h w ith th e o rig in a t its cen tre. H en ce th e m o tio n th a t m in im izes th e tim e o f d escen t is ju st u n ifo rm circu la r m o tio n a d d ed to a rectilin ea r u n ifo rm m o tio n w ith a v elo city eq u a l to th a t o f circu la r m o tio n . T h is is, o f co u rse, th e p a th tra ced b y a p o in t on a circle th a t ro lls o n a h o rizo n ta l su rfa ce, w h ich is a cy clo id . T h is a p p ro a ch h a s th e a d va n ta g e th a t y o u o b ta in th e cy clo id n o t in term s o f eq u a tio n s b u t in term s o f its g eo m etrica l d e¯ n itio n .
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T h ere is a n in terestin g g en era liza tio n o f th e b ra ch isto ch ro n e p ro b lem w h ich h a s n o t receiv ed m u ch a tten tio n . T h e cy clo id so lu tio n w a s o b ta in ed u n d er th e a ssu m p tio n o f a u n ifo rm , co n sta n t g ra v ita tio n a l ¯ eld o f a ° a t E a rth . In rea lity, o f co u rse, th e g rav ita tio n a l ¯ eld va ries a s (1 = r 2 ) a ro u n d a sp h erica l o b ject. T h e q u estio n a rises a s to h ow th e cu rv e o f sw iftest d escen t g ets m o d i¯ ed w h en w e w o rk w ith th e (1 = r 2 ) fo rce. T o ta ck le th is p ro b lem , it is co n v en ien t to u se p o la r co o rd in a tes in th e p la n e o f m o tio n a n d a p p rox im a te th e g rav ita tio n a l so u rce a s a p o in t p a rticle o f m a ss M a t th e o rig in . W e a re in terested in d eterm in in g th e cu rv e r(µ ) su ch th a t a p a rticle sta rtin g fro m a p o in t A (w ith co o rd in a tes r = R a n d µ = 0 ) w ill rea ch a p o in t B (w ith co o rd in a tes r = r f ; µ = µ f ) in th e sh o rtest p o ssib le tim e. A s u su a l, so m e n ew cu rio sities creep in . T h e m a th em a tica l fo rm u la tio n o f th e va ria tio n a l p rin cip le is triv ia l. If v (r) is th e sp eed o f th e p a rticle w h en it is a t th e ra d ia l d ista n ce r , th en µ ¶ µ ¶ 1 1 1 2 2 v = 2G M ¡ ¡ 1 ; = C (9 ) r R x w h ere x = r= R a n d C 2 = 2 G M = R . T h e va ria tio n a l p rin cip le req u ires u s to m in im ize th e in teg ra l ov er d s = v w h ere d s = R d µ (x 02 + x 2 )1 = 2 is th e a rc len g th a lo n g th e cu rv e w ith x 0 = d x = d µ . T h is, in tu rn , req u ires d eterm in in g th e ex trem u m o f th e in teg ra l µ 02 ¶ Z Z R x + x 2 1=2 T = L (x 0;x )d µ : ´ dµ (1 0 ) C (1 = x ) ¡ 1 T h e E u ler{ L a g ra n g e eq u a tio n w ill a s u su a l lea d to a seco n d o rd er d i® eren tia l eq u a tio n in v o lv in g x 00(µ ). B u t sin ce th e in teg ra n d is in d ep en d en t o f µ (tim e), w e k n ow th a t x 0(@ L = @ x 0) ¡ L is co n serv ed (en erg y ). E q u a tin g it to a co n sta n t K g iv es a ¯ rst in teg ra l th ereb y a llow in g th e p ro b lem to b e red u ced to q u a d ra tu re. F a irly stra ig h tfo rw a rd a lg eb ra th en lea d s to th e fo rm o f th e fu n ctio n µ (x ) g iv en b y th e in teg ra l r Zx d x¹ 1 ¡ x¹ µ (x ) = ; (1 1 ) x¹ ¸ x¹ 3 + x¹ ¡ 1 1 w h ere ¸ ´ (R = K C ). U n fo rtu n a tely, th is is a n ellip tic in teg ra l (a n d a p retty b a d o n e a t th a t) w h ich m a k es fu rth er a n a ly tic p ro g ress d i± cu lt. W o rk in g th in g s o u t n u m erica lly, o n e
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Figure 2.
ca n p lo t th e releva n t cu rv es w h ich sh ow a v ery in terestin g b eh av io u r (see F igu re 2 ). T o b eg in w ith , o n e n o tices th a t ea ch cu rv e h a s a tu rn in g p o in t x = l, say, w h ere (d x = d µ ) = 0 . T h is is a p o in t o f m in im u m a p p ro a ch rela ted to ¸ b y ¸ = l¡ 3 (1 ¡ l). W h a t is cu rio u s is th e a sy m p to tic b eh av io u r o f th e cu rv e after it tu rn s a ro u n d . It is clea r fro m F igu re 2 th a t th e cu rv es n ev er en ter th e `fo rb id d en reg io n ' b etw een µ = ¡ 2 ¼ = 3 a n d µ = + 2 ¼ = 3 . W ith so m e h a rd w o rk , o n e ca n a ctu a lly p rov e th is resu lt a n a ly tica lly fro m th e fo rm o f th e in teg ra l (1 1 ). (T ry it o u t y o u rself; d eterm in e th e a n g le µ (l) a t th e p o in t o f m in im u m a p p ro a ch fro m (1 1 ) a n d th en ca refu lly eva lu a te th e l ! 0 lim it o f µ (l)). In fa ct, th e 3 in (2 ¼ = 3 ) o f th e fo rb id d en zo n e co m es fro m th e p ow er law in d ex o f th e fo rce. F o r th e b ra ch isto ch ro n e p ro b lem in r ¡ n fo rce law , th e fo rb id d en zo n e is g iv en b y ¡ 2 ¼ = (n + 1 ) < µ < 2 ¼ = (n + 1 ). T h e p a th o f q u ick est d escen t fro m r = R ;µ = 0 to a n y p o in t in th e fo rb id d en zo n e m u st n ecessa rily p a ss th ro u g h th e sin g u la rity a t th e o rig in . H av in g d escrib ed th e cla ssic va ria tio n a l p ro b lem w h ich sta rted it a ll, I w a n t to d iscu ss a n o th er o n e w h ich d o es n o t ev en seem to h av e a resp ecta b le n a m e. T h is p ro b lem [2 ] ca n b e sta ted a s fo llow s. C o n sid er a p la n et o f a g iv en m a ss M a n d v o lu m e V a n d a co n sta n t d en sity ½ = M = V . W e a re a sk ed to va ry th e sh a p e o f
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Figure 3.
th e p la n et so a s to m a k e th e g rav ita tio n a l fo rce ex erted b y th e p la n et o n a g iv en p o in t a t its su rfa ce th e m a x im u m p o ssib le va lu e. W h a t is th e resu ltin g sh a p e? M o st p eo p le w o u ld g u ess th a t th e sh a p e is eith er a sp h ere o r so m eth in g lik e th e a p ex o f a co n e. T h e seco n d g u ess is ea sy to refu te sin ce it p u ts a fa ir a m o u n t o f th e m a ss aw ay fro m th e ch o sen p o in t; b u t a sp h ere rem a in s a n in trig u in g p o ssib ility. T h e co rrect a n sw er, h ow ev er, is q u ite stra n g e a n d ca n b e o b ta in ed a s fo llow s. L et th e ch o sen p o in t b e a t th e o rig in a n d let z -a x is b e a lo n g th e d irectio n o f th e m a x im a l fo rce a ctin g o n a test p a rticle a t th e o rig in . It is o b v io u s th a t th is z -a x is m u st b e a n a x is o f sy m m etry fo r th e p la n et; if it is n o t, th en o n e ca n in crea se th e z -co m p o n en t o f th e n et fo rce b y m ov in g m a teria l fro m la rg er to sm a ller tra n sv erse d ista n ce u n til th e p la n et is a x ia lly sy m m etric. T h u s o u r p ro b lem red u ces to d eterm in in g th e cu rv e x = x (z ) (w ith 0 < z < z 0 , say ) w h ich , o n rev o lu tio n a ro u n d th e z -a x is g en era tes th e su rfa ce o f th e p la n et. (T h e so lu tio n is p lo tted a s a th ick u n b ro k en cu rv e in F igu re 3 .) T h e ea siest w a y to ca lcu la te th e z -co m p o n en t o f th e g rav ita tio n a l fo rce a ctin g o n th e o rig in is to d iv id e th e p la n et in to circu la r d iscs ea ch o f th ick n ess d z , lo ca ted p erp en d icu la r to th e z -a x is. T o g et th e fo rce ex erted o n a test p a rticle o f m a ss m b y a n y sin g le d isc, w e fu rth er d iv id e it in to a n n u la r rin g s o f in n er ra d ii x a n d o u ter ra d ii x + d x . T h e fo rce a lo n g th e z -a x is b y a n y o n e su ch rin g w ill b e g iv en by d F = G m (½ 2 ¼ x d x ) d z
z 1 p : 2 x + z x 2 + z2 2
(1 2 )
H en ce th e to ta l fo rce is g iv en b y
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F
=
2¼ G m ½
=
3G M m 2a3
Z z0
Z0 z 0 0
Z x (z )
z (x + z 2 )3 = 2 µ0 ¶ z : dz 1 ¡ (x 2 (z ) + z 2 )1 = 2
dz
x dx
2
(1 3 )
In a rriv in g a t th e la st ex p ressio n w e h av e ex p ressed th e d en sity a s ½ = 3 M = 4 ¼ a 3 so th a t th e v o lu m e o f th e p la n et is co n stra in ed b y th e co n d itio n Z z0 4¼ a 3 2 V = ¼ : d z x (z ) = (1 4 ) 3 0 Im p o sin g th is co n d itio n b y a L a g ra n g e m u ltip lier (¡ ¸ ), w e see th a t w e h av e to essen tia lly ¯ n d th e ex trem u m o f th e in teg ra l ov er th e fu n ctio n z L = 1¡ ¡ ¸x2 : (1 5 ) 2 2 1 = 2 (x + z ) T h is is stra ig h tfo rw a rd a n d w e g et z 1 ¸ ; = 2 = z 02 (x 2 + z 2 )3 = 2
(1 6 )
w h ere th e la st eq u a lity d eterm in in g th e L a g ra n g e m u ltip lier fo llow s fro m th e co n d itio n th a t x (z 0 ) = 0 . O u r co n stra in t o n th e to ta l v o lu m e b y (1 4 ) im p lies th a t z 03 = 5 a 3 th ereb y co m p letely so lv in g th e p ro b lem . T h e p o la r eq u a tio n to th e cu rv e is r 2 = 5 2 = 3 a 2 co s µ ; fo r co m p a riso n , a sp h ere w ith th e sa m e v o lu m e w ill co rresp o n d to r = 2 a co s µ . T h e sh a p e o f o u r w eird p la n et is sh ow n in F igu re 3 b y a th ick u n b ro k en cu rv e (a lo n g w ith th a t co rresp o n d in g to a sp h ere o f sa m e v o lu m e). It h a s n o cu sp s a t th e p o les a n d I a m n o t aw a re o f a n y sp eci¯ c n a m e fo r th is su rfa ce. T h e to ta l fo rce ex erted b y th is p la n et a t th e o rig in h a p p en s to b e µ ¶1 = 3 GM m G M m 27 F = ¼ 1 :0 3 (1 7 ) 2 a a2 25 w h ich is n o t to o m u ch o f a g a in ov er a sp h ere. B u t th en , a s th ey say, it is th e p rin cip le th a t m a tters. T h ere is a m in o r su b tlety w e g lo ssed ov er w h ile d o in g th e va ria tio n in th is p ro b lem . U n lik e th e u su a l va ria tio n a l p ro b lem s, th e en d p o in t z 0 is n o t g iv en to u s a s ¯ x ed w h ile d o in g th e va ria tio n o f th e in teg ra ls in (1 3 ) a n d (1 4 ). It is p o ssib le to ta k e
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B o x 1 . A B it o f H isto r y One of the early investigations about the time of descent along a curve was by Galileo. He, like many others, was interested in the time taken by a particle to perform an oscillation on a circular track which, of course, is what a simple pendulum of length L hanging from the ceiling will do. Today we could write down this period of oscillation as s Z ¼ =2 L dµ p T = ; (1) g 0 1 ¡ k 2 sin2 µ
where k is related to the angular amplitude of the swing. Of course, in the days before calculus, the expression would not have meant anything! Instead, Galileo used an ingenious geometrical argument and { in fact { thought that he has proved the circle to be the curve of fastest descent. It was, however, known to mathematicians of the 17th century that Galileo's argument did not establish such a result. The maj or development came when Bernoulli threw a challenge in 1697 in the form of the brachistochrone problem to the mathematicians of that day with the interesting announcement: \I, Johann Bernoulli, greet the most clever mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to earn the gratitude of the entire scienti¯c community by placing before the ¯nest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall the publicly declare him worthy of praise" . Bernoulli, of course, knew the answer and the problem was also solved by his brother Jakob Bernoulli, Leibniz, Newton and L' Hospital. Newton is said to have received Bernoulli's challenge at the Royal Society of London one afternoon and (according to second hand sources, like John Conduit { the husband of Newton's niece) , Newton solved the problem by night-fall. The \solution" , which was simply a description of how to construct the relevent cycloid, was published anonymously in the P h ilo so p h ica l T ra n sa ctio n s o f th e R o ya l S ociety of January 1697 (back dated by the editor Edmund Halley) . Newton actually read aloud his solution in a Royal Society meeting only on 24 February 1697. Legend has it that Bernoulli had immediately recognized Newton's style and exclaimed \ta n qu a m ex u n gu e leo n em " meaning `the lion is known by its claw' !
th is in to a cco u n t b y a slig h tly m o re so p h istica ted trea tm en t b u t it w ill lea d to th e sa m e resu lt in th is p a rticu la r ca se. Suggested Reading [1]
T Padmanabhan, See e.g., Planets move in circles! A different view of orbits, Resonance, Vol. 1, No.9, p.34, 1996 for a description of hodograph in the case of Kepler problem. [2] W D Macmillan, Theory of the Potential, Dover Publications, 1958. Address for Correspondence : T Padmanabhan, IUCAA, Post Bag 4, Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected], [email protected]
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Snippets of Physics 22. Wigner's Function and Semi-Classical Limit T Padmanabhan
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
O b ta in in g th e c la ssic a l lim it o f q u a n tu m m e c h a n ic s tu rn s o u t to b e c o n c e p tu a lly a n d o p e r a tio n a lly n o n -triv ia l a n d , e v e n to d a y , so m e o f th e e x p e r ts c o n sid e r th is issu e to b e u n se ttle d . T h e re is a fu n c tio n , o rig in a lly d e v ise d b y W ig n e r, w h ich p la y s a k e y r o le in th is a sp e c t a n d th ro w s so m e lig h t o n th e w a y th e c la ssic a l w o rld e m e rg e s fro m th e q u a n tu m d e sc r ip tio n . Q u a n tu m p h y sics is n o th in g lik e cla ssica l p h y sics a n d it is p ro b a b ly n o t a n ex a g g era tion to say th a t w e ju st g et u sed to q u a n tu m p h y sics, w ith o u t rea lly u n d ersta n d in g it, a s w e lea rn m o re a b o u t it! T h ere a re sev era l co n cep tu a l a n d tech n ica l p ro b lem s in v o lv ed in ta k in g th e cla ssica l lim it o f a q u a n tu m m ech a n ica l d escrip tio n a n d w e w ill co n cen tra te o n o n e p a rticu la r a sp ect in th is in sta llm en t. W e w ill n o t w o rry to o m u ch a b o u t th e co n cep tu a l issu es { h o w ev er in terestin g th ey a re { b u t w ill, in stea d , co n cen tra te o n certa in tech n ica l a sp ects. L et u s b eg in w ith a sim p le o n e-d im en sio n a l p ro b lem in q u a n tu m m ech a n ics in w h ich a p a rticle o f m a ss M ev o lv es u n d er th e in ° u en ce o f a p o ten tia l V (Q ). C la ssica lly, su ch a sy stem is d escrib ed b y th e a ctio n fu n ctio n a l Z 1 A = L d t; L = M Q_ 2 ¡ V (Q ) : (1 ) 2
Keywords Wigner function, semi-classical limit, phase space.
934
T h e eq u a tio n s o f m o tio n ca n b e o b ta in ed b y va ry in g th is a ctio n w ith resp ect to th e co o rd in a te a n d w e g et QÄ + V 0(Q ) = 0 . W e a lso k n o w th a t th e sy stem ca n b e eq u iva len tly d escrib ed u sin g th e H a m ilto n ia n H (P ;Q ) = (P 2 = 2 M ) + V a n d th e H a m ilton { J a co b i eq u a tio n fo r th e
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sy stem is g iv en b y @A + H @t
· ¸ @A ;Q = 0 @Q
(2 )
w h ich is so lv ed b y th e a ctio n , trea ted a s a fu n ctio n o f th e va ria b les a t u p p er lim it o f in teg ra tio n in (1 ). W h en w e so lv e th e eq u a tio n s o f m o tio n w e ty p ica lly o b ta in th e tra jecto ry o f th e p a rticle Q (t) fro m w h ich w e ca n o b ta in th e m o m en tu m P (t) = M Q_ (t). G iv en Q (t) a n d P (t) w e ca n d eterm in e th e fu n ctio n a l fo rm P = P (Q ) th ereb y o b ta in in g th e tra jecto ry o f th e p a rticle in th e p h a se sp a ce. (T h is is, o f co u rse, u n iq u e o n ly lo ca lly, sin ce in g en era l, e.g ., fo r p erio d ic m o tio n s, o n e w ill b e led to m u ltip le- va lu ed fu n ctio n s.) T h e tra jecto ry in th e p h a se sp a ce tells y o u th a t y o u ca n a ssig n to th e p a rticle a p o sitio n Q a n d m o m en tu m P sim u lta n eo u sly. L et u s m ov e o n to q u a n tu m th eo ry. S in ce u n certa in ty p rin cip le p rev en ts o u r a ssig n in g sim u lta n eo u sly th e p o sitio n a n d m o m en tu m to a p a rticle, w e ca n n o lo n g er d escrib e th e sy stem in term s o f a tra jecto ry eith er in rea l sp a ce o r in p h a se sp a ce. In stea d w e h av e to in v o k e a p ro b a b ilistic in terp reta tio n a n d d escrib e th e q u a n tu m sta te o f th e sy stem in term s o f a w a v e fu n ctio n à . T h is w av e fu n ctio n sa tis¯ es th e sta n d a rd S ch ro d in g er eq u a tio n i~à _ = ¡
~2 @ 2 Ã + V (Q )Ã = E Ã ; 2M @ Q 2
(3 )
w h ere th e seco n d eq u a lity h o ld s if w e a re in terested in sta tio n a ry sta tes w ith th e tim e ev o lu tio n d escrib ed b y th e fa cto r ex p (¡ iE t= ~). F o r th e sa k e o f sim p licity, w e sh a ll a ssu m e th a t th is is th e ca se. W e a lso k n ow th a t th e cla ssica l b eh av io u r { tra jecto ries a n d a ll { h a s to em erg e fro m th is eq u a tio n in th e lim it o f ~ ! 0 . T h e q u estio n is: H ow d o w e g o a b o u t ta k in g th is lim it? It is w o rth th in k in g a b o u t th is issu e a little b it b efo re ju m p in g o n to th e sta n d a rd tex t b o o k d escrip tio n . T h e S ch rÄo d in g er eq u a tio n in (3 ) is a d i® eren tia l eq u a tio n w ith ~ a p p ea rin g a s a p a ra m eter. If y o u h av en 't rea d tex tb o o k s, y o u m ig h t h av e th o u g h t th a t o n e w o u ld ex p a n d à in a T ay lo r series in ~ lik e à = à 0 + ~à 1 + ~2 à 2 ..., p lu g it in to th e eq u a tio n a n d try to so lv e it o rd er b y o rd er in ~. T h e à 0 ;à 1 ::: w ill a ll h av e w eird d im en sio n s sin ce ~ is n o t d im en sio n less; th is, h ow ev er, is n o t a serio u s issu e. T h e k ey p o in t is th a t, in su ch a n ex p a n sio n , w e a re a ssu m in g à to b e a n a ly tic in ~ w ith à 0 d escrib in g th e cla ssica l lim it. T h is id ea , h ow ev er, d o es n o t w o rk , a s y o u ca n ea sily v erify. In fa ct, w e w o u ld h av e b een in a b it o f tro u b le
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if it h a d w o rk ed sin ce w e w ill th en h av e to in terp ret à 0 a s so m e k in d o f `cla ssica l' w av e fu n ctio n . T h e w ay o n e o b ta in s cla ssica l lim it is fa irly n o n -triv ia l w h ich w e w ill n ow d escrib e. W e w ill b eg in b y w ritin g th e w av e fu n ctio n in th e fo rm à (Q ) = R (Q ) ex p [iS (Q )= ~]
(4 )
w h ich is ju st th e sta n d a rd rep resen ta tio n o f a co m p lex n u m b er in term s o f th e a m p litu d e a n d p h a se. S u b stitu tin g in to (3 ) a n d eq u a tin g th e rea l a n d im a g in a ry p a rts, w e g et th e tw o eq u a tio n s (R 2 S 0)0= 0 ;
(5 )
S 02 ~2 R 00 : + V (Q ) ¡ E = 2M 2M R
(6 )
an d
T h ese tw o eq u a tio n s ca n b e m a n ip u la ted to g iv e a sin g le eq u a tio n fo r S (w h en S 06= 0 ). W e g et · ¸ p S 02 ~2 p 0 d 2 0 S + V (Q ) ¡ E = [1 = S ] : (7 ) 2M 2M dQ 2 T h e S ch rÄo d in g er eq u a tio n is co m p letely eq u iva len t to th e tw o rea l eq u a tio n s in (5 ) a n d (6 ). A n y th in g y o u ca n d o w ith a co m p lex w av e fu n ctio n à y o u ca n a lso d o w ith tw o rea l fu n ctio n s R a n d S . B u t, o f co u rse, S ch rÄo d in g er eq u a tio n is lin ea r in à w h ile eq u a tio n s (5 ) a n d (6 ) a re n o n lin ea r, th ereb y h id in g th e p rin cip le o f su p erp o sitio n o f q u a n tu m sta te, w h ich is a co rn ersto n e o f q u a n tu m d escrip tio n . E q u a tio n (7 ) su g g ests a n a ltern a te ro u te fo r d o in g th e T ay lo r series ex p a n sio n in ~. W e ca n n ow try to in terp ret th e left-h a n d sid e o f (7 ) a s p u rely cla ssica l a n d th e rig h t-h a n d sid e a s g iv in g `q u a n tu m co rrectio n s'. In su ch a ca se, w e ca n a ttem p t a T ay lo r series ex p a n sio n in th e fo rm S (Q ) = S 0 (Q ) + ~2 S 1 (Q ) + ¢¢¢ :
(8 )
T h is m ea n s th a t th e lea d in g b eh av io u r o f th e w av e fu n ctio n is g iv en b y ex p (iS 0 = ~) w h ich is n on -an alytic in ~. It d o es n o t h av e a T ay lo r series ex p a n sio n in p ow ers o f ~ w h ich is a d i® eren t k ettle o f ¯ sh w h en it co m es to series ex p a n sio n in term s o f a p a ra m eter in a d i® eren tia l eq u a tio n . A lso n o te th a t th e tim e-in depen den t
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S ch rÄo d in g er eq u a tio n d ep en d s o n ly o n ~2 a n d n o t o n ~; so th e seco n d term in th e T ay lo r series sta rts w ith ~2 a n d n o t w ith ~. W h y d o es th is a p p ro a ch w o rk w h ile à = à 0 + ~à 1 + ~2 à 2 ..., d o es n o t lea d to sen sib le resu lts? T h e rea so n essen tia lly h a s to d o w ith th e fa ct th a t, in p ro ceed in g fro m q u a n tu m p h y sics to cla ssica l p h y sics, w e a re d o in g so m eth in g a n a lo g o u s to o b ta in in g th e ray o p tics fro m electro m a g n etic w av es. O n e k n ow s th a t th is ca n co m e a b o u t o n ly w h en th e p h a se o f th e w av e is n o n -a n a ly tic in th e ex p a n sio n p a ra m eter, w h ich is essen tia lly th e w av elen g th in th e ca se o f lig h t p ro p a g a tio n . S o y o u n eed to b rin g in so m e ex tra p h y sical in sig h t to o b ta in th e co rrect lim it. W h ile à is n o n -a n a ly tic in ~, w e h av e n ow tra n sla ted th e p ro b lem in term s o f R a n d S w h ich a re (a ssu m ed to b e) a n a ly tic in ~ so th a t th e sta n d a rd p ro ced u re w o rk s. T o th e lea d in g o rd er, w e w ill ig n o re th e rig h t-h a n d sid e o f (7 ) a n d o b ta in th e eq u a tio n S 002 + V (Q ) ¡ E = 0 : 2M
(9 )
(T h is m ig h t seem p retty o b v io u s b u t th ere is a su b tlety lu rk in g h ere w h ich w e w ill co m m en t o n la ter.) T o th e sa m e o rd er o f a ccu ra cy, w e ¯ n d th a t R (Q ) / jS 00(Q )j¡1 = 2 . P u ttin g it to g eth er a n d n o tin g th e fa ct th e tw o in d ep en d en t so lu tio n s w ill in v o lv e § S 00, w e ca n w rite th e so lu tio n to th e S ch rÄo d in g er eq u a tio n to th is o rd er o f a ccu ra cy b y · · ¸ · ¸¸ i i 1 (0 ) à E = p 0 C 1 ex p S 0 (Q ) + C 2 ex p ¡ S 0 (Q ) ; (1 0 ) ~ ~ jS 0 j w h ere C 1 a n d C 2 a re a rb itra ry co n sta n ts. T o th is o rd er o f a ccu ra cy, th e (9 ) is ju st th e H a m ilto n { J a co b i eq u a tio n fo r th e actio n A = S 0 so th a t w e ca n id en tify th e p h a se o f th e w av efu n ctio n w ith S 0 = ~. T h e co n d itio n o f va lid ity fo r th is W K B a p p rox im a tio n is n o t d i± cu lt to d eterm in e b y co m p a rin g th e term s w h ich w ere ig n o red w ith th o se w h ich w ere reta in ed . W e ¯ n d th a t th is co n d itio n is eq u iva len t to ¯ 00¯ ¯ µ ¶¯ ¯S ¯ ¯d ~ ¯ ¯= ¯ ¯ ~¯ (1 1 ) ¯S 02 ¯ ¯d x S 0 ¯¿ 1 w h ich tra n sla tes to
2 M ~jV 0j ¿ (2 M [E ¡ V (Q )])3 = 2 :
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S o , a s lo n g a s w e a re fa r aw ay fro m th e tu rn in g p o in ts in th e p o ten tia l (w h ere E = V (Q )), o n e ca n sa tisfy th is co n d itio n . T h o u g h w e a p p ro a ch ed th is resu lt fro m a d esire to o b ta in th e cla ssica l lim it, m a th em a tica lly sp ea k in g , it is ju st a n a p p rox im a tio n to th e d i® eren tia l eq u a tio n u su a lly k n ow n a s W K B a p p rox im a tio n . T h is fa ct is strik in g ly ev id en t in th e co n tex t o f q u a n tu m m ech a n ica l tu n n elin g w h ich , o f co u rse, h a s n o cla ssica l a n a lo g y. N ev erth eless, w e ca n g et a rea so n a b le a p p rox im a tio n to th e w av e fu n ctio n in a cla ssica lly fo rb id d en fo rm b y ta k in g E < V (Q ) in (9 ). In th is ra n g e, say, a < Q < b in w h ich E < V (Q ), th e S 0 b eco m es p u rely im a g in a ry a n d is g iv en b y Zpb p p Z bp S 0 = 2M E ¡ V (Q ) d Q = i 2 M V (Q )¡ E d Q : (1 3 ) a
a
T h e w av e fu n ctio n in (1 0 ) b eco m es ex p o n en tia lly d ecrea sin g (o r in crea sin g ) { w ith o u t o scilla to ry b eh av io u r { in th is cla ssica lly fo rb id d en ra n g e. T h is is va lid , a g a in , a s lo n g a s w e a re aw ay fro m th e tu rn in g p o in t. L et u s n ow g et b a ck to th e q u estio n w e sta rted w ith , v iz., h ow to g et th e cla ssica l lim it. T o d o th is w e n eed to u n d ersta n d w h y th e w av e fu n ctio n in (1 0 ) h a s a n y th in g to d o w ith th e cla ssica l lim it. T h e co n v en tio n a l a n sw er is a s fo llow s: L et u s co n sid er fo r sim p licity a situ a tio n w ith C 2 = 0 . In th a t ca se th e p ro b a b ility d istrib u tio n a sso cia ted w ith th e w av e fu n ctio n va ries a s P ´ jà j2 /
1 ; P (Q )
(1 4 )
w h ere S 00(Q ) = P (Q ) is th e classical m o m en tu m o f th e p a rticle a t Q . If w e n ow in terp ret th e p ro b a b ility to ca tch a p a rticle in th e in terva l (Q ;Q + d Q ) a s p ro p o rtio n a l to th e tim e in terva l d t = d Q = V (Q ) (w h ere V (Q ) is th e v elo city o f th e p a rticle w h en it is a t Q ) th en th e ex p ressio n in (1 4 ) ca n b e g iv en so m e k in d o f a cla ssica l in terp reta tio n . T h is is, h ow ev er, n o t co m p letely sa tisfa cto ry b eca u se, a s w e sa id ea rlier, w e a sso cia te th e cla ssica l lim it w ith a d eterm in istic tra jecto ry in p h a se sp a ce. T h ere is a w ay o f o b ta in in g th is resu lt w h ich b rin g s u s to th e d iscu ssio n o f th e W ig n er fu n ctio n . T h e W ig n er fu n ctio n F (Q ;p ;t) co rresp o n d in g to a w av e fu n ctio n à (Q ;t) (w h ich co u ld , in g en era l, b e tim e d ep en d en t) is d e¯ n ed b y th e rela tio n µ ¶ µ ¶ Z1 1 1 ¤ ¡ip u F (Q ;p ;t)= Q ¡ ~u ;t e à Q + ~u ;t : duà (1 5 ) 2 2 ¡1
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T h e in teg ra n d m ea su res th e co rrela tio n b etw een à a n d à ¤ in a F o u rier tra n sfo rm ed sp a ce w ith va ria b le p . T h is fu n ctio n h a s sev era l rem a rka b le p ro p erties w h ich w e w ill n ow d iscu ss. T h e b a sic id ea is to see w h eth er o n e ca n th in k o f F a s a p ro b a b ility d istrib u tio n fu n ctio n in th e p h a se sp a ce w ith p o sitio n (Q ) a n d m o m en tu m (p ) a s co o rd in a tes. T o b eg in w ith , if y o u in teg ra te F ov er th e m o m en tu m va ria b le p , a n d u se th e fa ct th a t th e in teg ra l o f ex p (ip u ) ov er p is a D ira c d elta fu n ctio n in u , y o u g et Z1 d p F (Q ;p ;t) = jà (Q ;t)j2 : (1 6 ) ¡1
T h is sh ow s th a t w h en m a rg in a lized ov er p , w e d o g et th e p ro b a b ility d istrib u tio n Q w h ich is q u ite n ice. F u rth er, it is a lso ea sy to see th a t if y o u in teg ra te F ov er Q y o u g et th e resu lt Z1 d Q F (Q ;p ;t) = jÁ (p ;t)j2 ; (1 7 ) ¡1
w h ere Á (p ;t) is th e F o u rier tra n sfo rm o f à (Q ;t). F ro m th e sta n d a rd ru les o f q u a n tu m m ech a n ics, w e k n ow th a t Á (p ;t) g iv es th e p ro b a b ility a m p litu d e in th e m o m en tu m sp a ce. T h erefo re (1 7 ) tells u s th a t { w h en m a rg in a lized ov er th e co o rd in a te Q { th e W ig n er fu n ctio n F g iv es th e p ro b a b ility d istrib u tio n in th e m o m en tu m sp a ce. S o clea rly, F sa tis¯ es tw o n ice p ro p erties w e w o u ld h av e ex p ected o u t o f a p ro b a b ility d istrib u tio n . It sim u lta n eo u sly en co d es b o th co o rd in a te sp a ce a n d m o m en tu m sp a ce p ro b a b ilities in a sta te rep resen ted b y à . It is a lso p o ssib le to o b ta in a n eq u a tio n sa tis¯ ed b y F w h ich is sim ila r to th e co n tin u ity eq u a tio n th a t w e ex p ect p ro b a b ility d istrib u tio n s to sa tisfy. D irect d i® eren tia tio n o f (1 5 ) a n d so m e clev er m a n ip u la tio n w ill a llow y o u to o b ta in a n eq u a tio n o f th e fo rm @F p @F ~2 d 3 V @ 3 F dV @F ¡ + = + ¢¢¢ ; @t M @Q dQ @p 24 d Q 3 @ p3
(1 8 )
w h ere ¢¢¢ d en o tes term s w h ich a re h ig h er o rd er in ~. T h is eq u a tio n a llow s y o u to d raw tw o in terestin g co n clu sio n s. T o b eg in w ith , if th e p o ten tia l is a t m o st q u a d ra tic in co o rd in a tes, th e rig h t-h a n d sid e va n ish es a n d w e g et ex a ctly th e co n tin u ity eq u a tio n in th e p h a se sp a ce w ith th e sem i-cla ssica l id en ti¯ ca tio n s
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Figure 1.
Q_ = p = m a n d p_ = ¡ V 0. N ex t, th is in terp reta tio n h o ld s ev en fo r arbitrary p o ten tia ls u p to lin ea r o rd er in ~. If w e ca n ig n o re th e ~2 in th e rig h t-h a n d sid e w e a g a in g et th e co n tin u ity eq u a tio n in p h a se sp a ce. B efo re w e rejo ice, o n e h a s to fa ce u p to a ra th er d a m a g in g p ro p erty o f F w h ich p rev en ts u s fro m m a k in g th e p ro b a b ilistic in terp reta tio n rig o ro u s. T h e k ey tro u b le is th a t F is n o t p o sitiv e d e¯ n ite a n d sin ce w e d o n o t k n ow h ow to in terp ret n eg a tiv e p ro b a b ilities, w e ca n n o t u se F a s a p ro b a b ility d istrib u tio n in p h a se sp a ce. O n e sim p le w ay to see th a t F ca n b eco m e n eg a tiv e is to co m p u te it fo r so m e w ell-ch o sen sta te. F o r ex a m p le, F igu re 1 g iv es th e W ig n er fu n ctio n co rresp o n d in g to th e ¯ rst ex cited sta te o f a h a rm o n ic o scilla to r. W e h av e, in su ita b ly ch o sen u n its, th e resu lts: Ã 1 (Q ) = F (Q ;p ) =
µ ¶1 = 4 4 2 Q e ¡ (Q = 2 ) ; ¼ µ ¶ 1 2 2 2 e ¡(p + Q 4 Q + p ¡ 2
(1 9 ) 2)
:
(2 0 )
It is clea r th a t F ca n g o n eg a tiv e. T h is d o es n o t, h ow ev er, p rev en t u s fro m u sin g th e W ig n er fu n ctio n in su ita b le lim its a s a n a p p rox im a tio n to cla ssica l p ro b a b ility d istrib u tio n . In p a rticu la r, th e W ig n er fu n ctio n co rresp o n d in g to th e sem i-cla ssica l w av e fu n ctio n in (1 0 ) is q u ite ea sy to in terp ret. L et u s ¯ rst co n sid er th e ca se w ith C 2 = 0 w h en th e w av e fu n ctio n b eco m es à ! C1 p 0 ex p (iS 0 = ~) : à (Q ) = (2 1 ) S0 940
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S u b stitu tin g th is in to (1 5 ) a n d eva lu a tin g th e in teg ra ls { reta in in g u p to th e co rrect o rd er, w h ich is n ecessa ry sin ce à itself is a p p rox im a te { w e ca n ea sily sh ow th a t µ ¶ jC 1 j2 @S0 F (Q ;p ) = ± p¡ + O (~2 ) : (2 2 ) jS 00(Q )j @Q T h is resu lt, w h en w e th in k o f F a s a p ro b a b ility d istrib u tio n , h a s a n ice in terp reta tio n . T h e D ira c d elta fu n ctio n tells y ou th a t w h en th e p a rticle is a t Q , its m o m en tu m is sh a rp ly p ea k ed a t @ S 0 = @ Q w h ich is ex a ctly w h a t w e w o u ld h av e ex p ected if th e p a rticle w a s m ov in g a lo n g a cla ssica l tra jecto ry. F u rth er, th e p ro b a b ility to ¯ n d th e p a rticle a ro u n d Q is p ro p o rtio n a l to (1 = S 0(Q )) w h ich a g a in ca n b e in terp reted in term s o f th e tim e d t = d Q = V (Q ) w h ich th e p a rticle sp en d s in th e in terva l (Q ;Q + d Q ). T h e k ey p o in t is th a t, fo r th e sem i-cla ssica l w av e fu n ctio n w e d eterm in ed in (1 0 ), th e W ig n er fu n ctio n g iv es stro n g ly co rrelated p ro b a b ility d istrib u tio n in p h a se sp a ce. In fa ct, if y o u ta k e th e D ira c d elta fu n ctio n litera lly, it g iv es a u n iq u e p fo r ev ery Q . T h is is th e ch a ra cteristic o f a cla ssica l tra jecto ry a n d , fro m th is p o in t o f v iew , th e W ig n er fu n ctio n p rov id es a n a tu ra l in terp reta tio n fo r th e sem i-cla ssica l w av e fu n ctio n w e h av e o b ta in ed . N o te th at th e p ro b a b ility d istrib u tio n is n o t p ea k ed a ro u n d a n y sin g le tra jecto ry b u t o n ce y o u p ick a Q , it g iv es y o u a u n iq u e p . T h is co rrela tio n b etw een m o m en tu m a n d p o sitio n is th e k ey fea tu re o f cla ssica l lim it. T h is in terp reta tio n co n tin u es to h o ld ev en w h en w e k eep C 2 6= 0 . In th is ca se w e g et µ ¶ µ ¶ jC 1 j2 @S0 jC 2 j2 @S0 F (Q ;p ) = ± p¡ ± p+ + + O (~2 ) : (2 3 ) jS 00(Q )j @Q jS 00(Q )j @Q T h e W ig n er fu n ctio n h a s a term w h ich rep resen ts in terferen ce b etw een th e tw o in d ep en d en t so lu tio n s b u t th is term is O (~2 ) a n d d o es n o t co n trib u te a t th e lea d in g o rd er. T h is W ig n er fu n ctio n is n ow p ea k ed a t tw o d i® eren t va lu es o f m o m en ta p = § @ S 0 = @ Q a n d co rresp o n d s to m o tio n a lo n g fo rw a rd a n d b a ck w a rd d irectio n in th e co o rd in a te sp a ce. In th e p h a se sp a ce, F w ill n ow b e p ea k ed o n tw o fa m ilies o f tra jecto ries. T h ese p ro p erties a re o b v io u sly sp ecia l to th e sem i-cla ssica l w a v e fu n ctio n w e h av e ch o sen . If y o u ta k e a cla ssica lly fo rb id d en reg io n in w h ich th e w av e fu n ctio n is ex p o n en tia lly d a m p ed , ra th er th a n o scilla to ry, y o u w ill ¯ n d a co m p letely d i® eren t b eh av io u r fo r th e W ig n er fu n ctio n . In fa ct, in su ch a `p u rely q u a n tu m m ech a n ica l'
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situ a tio n , y o u w ill ¯ n d th a t th e W ig n er fu n ctio n fa cto rizes in to a p ro d u ct o f tw o fu n ctio n s, o n e d ep en d en t o n Q a n d th e o th er d ep en d en t o n p w ith F (Q ;p ) = F 1 (Q )F 2 (p ). T h is sh ow s th a t th e m o m en tu m a n d p o sitio n a re to ta lly u n co rrela ted in su ch a sta te w h ich clea rly is th e o th er ex trem e o f th e sem i-cla ssica l sta te in w h ich th e m o m en tu m is co m p letely co rrelated w ith p o sitio n . T h e sa m e d eco u p lin g o f m o m en tu m a n d p o sitio n d ep en d en ce o ccu rs fo r m a n y o th er sta tes. O n e sim p le ex a m p le is th e g ro u n d sta te o f th e h a rm o n ic o scilla to r fo r w h ich y o u w ill ¯ n d th a t th e W ig n er fu n ctio n fa cto rizes in to tw o p ro d u cts, b o th G a u ssia n in p o sitio n a n d m o m en tu m . S o th e g ro u n d sta te o f th e h a rm o n ic o scilla to r is a s n o n -cla ssica l a s a sta te co u ld g et in th is in terp reta tio n . F in a lly, let u s g et b a ck to th e su b tlety w h ich I m en tio n ed ea rlier in ig n o rin g th e rig h t-h a n d sid e o f (6 ) w h ich is a clo sely rela ted issu e. F o r th is a p p rox im a tio n to b e va lid , w e m u st h av e ~2 R 00 lim = 0 : ~! 0 2 M R
(2 4 )
It is ea sy to co n stru ct sta tes fo r w h ich th is co n d itio n is v io la ted ! A s a sim p le ex a m p le, co n sid er th e g ro u n d sta te o f a sy stem in a b o u n d ed p o ten tia l w h ich w ill b e d escrib ed b y a rea l w av e fu n ctio n . In th is ca se à = R a n d S = 0 . F ro m (6 ) w e n ow see th a t ~2 R 00 = V (Q ) ¡ E : 2M R
(2 5 )
T h e lim it in (2 4 ) ca n n o t n ow h o ld , in g en era l. C lea rly o u r a n a ly sis fa ils fo r th e g ro u n d sta te o f a q u a n tu m sy stem . T o see th is ex p licitly, co n sid er a g a in th e g ro u n d sta te o f a h a rm o n ic o scilla to r: · ¸ M ! 2 Ã (Q ) = N ex p ¡ Q : (2 6 ) 2~ B ein g a n ex a ct so lu tio n to th e S ch rÄo d in g er eq u a tio n th e a m p litu d e a n d p h a se (w h ich is zero ) o f th is w av e fu n ctio n sa tis¯ es (5 ) a n d (6 ). A stra ig h tfo rw a rd co m p u ta tio n n ow sh ow s { n o t su rp risin g ly { th a t ~2 R 00 1 = M ! 2Q 2M R 2
2
¡
1 ~! : 2
(2 7 )
W h en w e ta k e th e lim it ~ ! 0 , th e seco n d term o n th e rig h t-h a n d sid e va n ish es b u t n o t th e ¯ rst term ! T h is m ea n s th ere a re q u a n tu m sta tes fo r w h ich w e ca n n o t
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n a iv ely ta k e th e rig h t-h a n d sid e o f (6 ) to b e zero a n d d eterm in e th e cla ssica l lim it. In terestin g ly en o u g h , th is is a lso tru e fo r th e tim e-d ep en d en t, co h eren t sta tes o f th e o scilla to r. Y o u m ay w a n t to a m u se y ou rself b y a n a ly zin g th is situ a tio n in g rea ter d eta il. Suggested Reading [1]
Some of the pedagogical details regarding Wigner functions can be found in the article, W B Case, Am. J. Phys., Vol.76, p.937, 2008.
Address for Correspondence: T Padmanabhan, IUCAA, Post Bag 4, Pune, University Campus , Ganeshkhind, Pune 411 007, India. Email: [email protected], [email protected]
*****
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Snippets of Physics 23. Real Effects from Imaginary Time T Padmanabhan
S o m e o f th e c u rio u s e ® e c ts in q u a n tu m th e o ry a n d sta tistic a l m e ch a n ic s c a n b e in te rp re te d b y a n a ly tic a lly c o n tin u in g th e tim e c o o rd in a te to p u r e ly im a g in a ry v a lu e s. W e e x p lo r e so m e o f th e se issu e s in th is in sta lm e n t. T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
In o n e o f th e p rev io u s in sta lm en ts [1 ], w e d iscu ssed h ow o n e ca n stu d y th e tim e ev o lu tio n o f a q u a n tu m w av e fu n ctio n u sin g a p a th in teg ra l p ro p a g a to r g iv en b y X K (q 2 ;t2 ;q 1 ;t1 ) = ex p iA [p a th ] ; (1 ) p a th s
w h ere A is th e cla ssica l a ctio n eva lu a ted a lo n g a p a th co n n ectin g (q 1 ;t1 ) w ith (q 2 ;t2 ) a n d w e a re u sin g u n its w ith ~ = 1 . T h is p a th in teg ra l k ern el a llow s y o u to d eterm in e th e w av e fu n ctio n a t tim e t2 if it is k n ow n a t tim e t1 th ro u g h th e in teg ra l Z+ 1 Ã (q 2 ;t2 ) = d q 1 K (q 2 ;t2 ;q 1 ;t1 )Ã (q 1 ;t1 ) : (2 ) ¡1
Keywords Imaginary time, density matrix, tunneling, black, hole temperature, Schwinger effect.
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T h ese ex p ressio n s a re q u ite g en era l. B u t w h en th e H a m ilto n ia n H d escrib in g th e sy stem is tim e in d ep en d en t, w e ca n in tro d u ce th e en erg y eig en fu n ctio n s th ro u g h th e eq u a tio n H à n = E n à n . W e h av e a lso seen in th e ea rlier in sta lm en t [1 ] th a t th e k ern el ca n b e ex p ressed in term s o f en erg y eig en fu n ctio n s th ro u g h th e fo rm u la X K (T ;q 2 ;0 ;q 1 ) = à n (q 2 )à n¤(q 1 ) ex p (¡ iE n T ) : (3 ) n
S o , if th e en erg y eig en fu n ctio n s a n d eig en va lu es a re g iv en o n e ca n d eterm in e th e k ern el.
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d irectly b y eva lu a tin g o r a p p rox im a tin g th e p a th in teg ra l. T h e q u estio n a rises a s to w h eth er o n e ca n d eterm in e th e en erg y eig en fu n ctio n s a n d eig en va lu es b y `in v ertin g ' th e a b ov e rela tio n . In p a rticu la r, o n e is o ften in terested in th e g ro u n d sta te eig en fu n ctio n a n d th e g ro u n d sta te en erg y o f th e sy stem . C a n o n e ¯ n d th is if th e k ern el is k n ow n ? It ca n b e d o n e u sin g a n in terestin g trick w h ich v ery o ften tu rn s o u t to b e m o re th a n ju st a trick , h av in g a ra th er p erp lex in g d o m a in o f va lid ity. T o a ch iev e th is, let u s d o th e u n im a g in a b le b y a ssu m in g th a t tim e is a ctu a lly co m p lex a n d a n a ly tica lly co n tin u e fro m th e rea l va lu es o f tim e t to p u rely im a g in a ry va lu es ¿ = it. In sp ecia l rela tiv ity su ch a n a n a ly tic co n tin u a tio n w ill ch a n g e th e lin e in terva l fro m L o ren tzia n to E u clid ea n fo rm th ro u g h d s 2 = ¡ d t2 + d x 2 ! d ¿ 2 + d x 2 :
(4 )
B eca u se o f th is rea so n , o n e o ften ca lls q u a n tities eva lu a ted w ith a n a ly tic co n tin u a tio n to im a g in a ry va lu es o f tim e a s `E u clid ea n ' q u a n tities a n d o ften d en o tes th em w ith a su b scrip t `E ' (w h ich sh o u ld n o t b e co n fu sed w ith en erg y !). If w e n ow d o th e a n a ly tic co n tin u a tio n o f th e k ern el in (3 ) w e g et th e resu lt X K E (T E ;q 2 ;0 ;q 1 ) = Ã n (q 2 )Ã n¤(q 1 ) ex p (¡ E n T E ) : (5 ) n
L et u s co n sid er th e fo rm o f th is ex p ressio n in th e lim it o f T E ! 1 . If th e en erg y eig en va lu es a re o rd ered a s E 0 < E 1 < :::: th en , in th is lim it, o n ly th e term w ith th e g ro u n d sta te en erg y w ill m a k e th e d o m in a n t co n trib u tio n a n d rem em b erin g th a t g ro u n d sta te w av e fu n ctio n is rea l fo r th e sy stem s w e a re in terested in , w e g et, K
E
(T E ;q 2 ;0 ;q 1 ) ¼ Ã 0 (q 2 )Ã 0 (q 1 ) ex p (¡ E 0 T E ); (T E ! 1 ) :
(6 )
S u p p o se w e n ow p u t q 2 = q 1 = 0 , ta k e th e lo g a rith m o f b o th sid es a n d d iv id e b y T E , th en in th e lim it o f T E ! 1 , w e g et a fo rm u la fo r th e g ro u n d sta te en erg y : · ¸ 1 ¡ E 0 = lim ln K E (T E ;0 ;0 ;0 ) : (7 ) TE! 1 TE S o if w e ca n d eterm in e th e k ern el b y so m e m eth o d w e w ill k n ow th e g ro u n d sta te en erg y o f th e sy stem . O n ce th e g ro u n d sta te en erg y is k n ow n w e ca n p lu g it b a ck in to th e a sy m p to tic ex p a n sio n in (6 ) a n d d eterm in e th e g ro u n d sta te w av e fu n ctio n .
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V ery o ften , w e w o u ld h av e a rra n g ed m a tters su ch th a t th e g ro u n d sta te en erg y o f th e sy stem is a ctu a lly zero . W h en E 0 = 0 th ere is a n icer w ay o f d eterm in in g th e w av e fu n ctio n fro m th e k ern el b y n o tin g th a t lim K (T ;0 ;0 ;q) ¼ Ã 0 (0 )Ã 0 (q ) / Ã 0 (q ) :
T!1
(8 )
S o th e in ¯ n ite tim e lim it o f th e k ern el { o n ce w e h av e in tro d u ced th e im a g in a ry tim e { a llow s d eterm in a tio n o f b o th th e g ro u n d sta te w av e fu n ctio n a s w ell a s th e g ro u n d sta te en erg y. T h e p ro p o rtio n a lity co n sta n t o f à 0 ca n b e ¯ x ed b y n o rm a lisin g th e w av e fu n ctio n . O f co u rse, th ese id ea s a re u sefu l o n ly if w e ca n co m p u te th e k ern el w ith o u t k n ow in g th e w av e fu n ctio n s in th e ¯ rst p la ce. T h is is p o ssib le { a s w e d iscu ssed in [1 ] { w h en ev er th e a ctio n is q u a d ra tic in th e d y n a m ica l va ria b le. In th a t ca se, th e k ern el in rea l tim e ca n b e ex p ressed in th e fo rm K (t2 ;q 2 ;t1 ;q 1 ) = N (t1 ;t2 ) ex p [iA c (t2 ;q 2 ;t1 ;q 1 )] ;
(9 )
w h ere A c is th e a ctio n eva lu a ted fo r a cla ssica l tra jecto ry a n d N (t2 ;t1 ) is a n o rm a liza tio n fa cto r. T h e sa m e id ea s w ill w o rk ev en w h en w e ca n approxim ate th e k ern el b y th e a b ov e ex p ressio n . W e saw in th e la st in sta lm en t th a t in th e sem i-cla ssica l lim it th e w av e fu n ctio n s ca n b e ex p ressed in term s o f th e cla ssica l a ctio n . It fo llow s th a t th e k ern el ca n b e w ritten in th e a b ov e fo rm in th e sa m e sem i-cla ssica l lim it. If w e n ow a n a ly tica lly co n tin u e th is ex p ressio n to im a g in a ry va lu es o f tim e, th en u sin g th e resu lt in (8 ) w e g et a sim p le fo rm u la fo r th e g ro u n d sta te w av e fu n ctio n in term s o f th e E u clid ea n a ctio n (th a t is, th e a ctio n fo r a cla ssica l tra jecto ry o b ta in ed a fter a n a ly tic co n tin u a tio n to im a g in a ry v a lu e o f tim e): Ã 0 (q ) / /
ex p [¡ A ex p [¡ A
E E
(T E = 1 ;0 ;T E = 0 ;q )] (1 ;0 ;0 ;q)] :
(1 0 )
A s a n a p p lica tio n o f th ese resu lts, co n sid er a sim p le h a rm o n ic o scilla to r w ith th e L a g ra n g ia n L = (1 = 2 )(q_2 ¡ ! 2 q 2 ). T h e cla ssica l a ctio n w ith th e b o u n d a ry co n d itio n s q (0 ) = q i a n d q (T ) = q f is g iv en b y A
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c
=
£ 2 ¤ ! (q i + q f2 ) co s ! T ¡ 2 q iq f : 2 sin ! T
(1 1 )
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T h e a n a ly tic co n tin u a tio n w ill g iv e th e E u clid ea n a ctio n co rresp o n d in g to iA b e ¡ A E w h ere A
E
=
£ 2 ¤ ! (q i + q f2 ) co sh ! T ¡ 2 q iq f : 2 sin h ! T
c
to
(1 2 )
U sin g th is in (1 0 ) w e ¯ n d th a t th e g ro u n d sta te w av e fu n ctio n h a s th e fo rm à 0 (q) / ex p ¡ [(! = 2 )q 2 ]
(1 3 )
w h ich , o f co u rse, is th e sta n d a rd resu lt. Y o u ca n a lso o b ta in th e g ro u n d sta te en erg y (1 = 2 )~ ! b y u sin g (7 ). W h a t is a m a zin g , w h en y o u th in k a b o u t it, is th a t th e E u clid ea n k ern el in th e lim it o f in ¯ n ite tim e in terva l h a s in fo rm a tio n a b o u t th e g ro u n d sta te o f th e q u a n tu m sy stem . T h is is th e ¯ rst ex a m p le in w h ich im a g in a ry tim e lea d s to a rea l resu lt! T h e a n a ly tic co n tin u a tio n to im a g in a ry va lu es o f tim e a lso h a s clo se m a th em a tica l co n n ectio n s w ith th e d escrip tio n o f sy stem s in th erm a l b a th . T o see th is, co n sid er th e m ea n va lu e o f so m e o b serva b le O (q ) o f a q u a n tu m m ech a n ica l sy stem . If th e sy stem is in a n en erg y eig en sta te d escrib ed b y th e w av e fu n ctio n à n (q ), th en th e ex p ecta tio n va lu e o f O (q ) ca n b e o b ta in ed b y in teg ra tin g O (q)jà n (q)j2 ov er q . If th e sy stem is in a th erm a l b a th a t tem p era tu re ¯ ¡1 , d escrib ed b y a ca n o n ica l en sem b le, th en th e m ea n va lu e h a s to b e co m p u ted b y av era g in g ov er a ll th e en erg y eig en sta tes as w ell w ith a w eig h ta g e ex p (¡ ¯ E n ). In th is ca se, th e m ea n va lu e ca n b e ex p ressed a s Z 1 X hO i = d q à n (q)O (q )à n¤(q) e ¡ ¯ E n Z Zn 1 ´ d q ½ (q ;q )O (q ) ; (1 4 ) Z w h ere Z is th e p a rtitio n fu n ctio n a n d w e h av e d e¯ n ed a den sity m atrix ½ (q ;q 0) by X ½ (q ;q 0) ´ à n (q )à n¤(q 0) e ¡¯ E n (1 5 ) n
in term s o f w h ich w e ca n rew rite (1 4 ) a s hO i =
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T r (½ O ) ; T r (½ )
(1 6 )
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w h ere th e tra ce o p era tio n in v o lv es settin g q = q 0 a n d in teg ra tin g ov er q . T h is sta n d a rd resu lt sh ow s h ow ½ (q ;q 0) co n ta in s in fo rm a tio n a b o u t b o th th erm a l a n d q u a n tu m m ech a n ica l av era g in g . In fa ct, th e ex p ressio n fo r th e d en sity m a trix in (1 5 ) is ju st th e co o rd in a te b a sis rep resen ta tio n o f th e m a trix co rresp o n d in g to th e o p era to r ½ = ex p (¡ ¯ H ). T h a t is, ½ (q;q 0) = hqje ¡¯ H jq 0i :
(1 7 )
B u t w h a t is in terestin g is th a t w e ca n n ow rela te th e d en sity m a trix o f a sy stem in ¯ n ite tem p era tu re { so m eth in g v ery rea l a n d p h y sica l { to th e p a th in teg ra l k ern el in im a g in a ry tim e. T h is is o b v io u s fro m co m p a rin g (1 5 ) w ith (3 ). W e ¯ n d th a t th e d en sity m a trix ca n b e im m ed ia tely o b ta in ed fro m th e E u clid ea n k ern el by: ½ (q;q 0) = K E (¯ ;q ;0 ;q 0) :
(1 8 )
W h a t is su rp risin g n ow is th a t th e im a g in a ry tim e is b ein g id en ti¯ ed w ith th e in v erse tem p era tu re. V ery cru d ely, th is id en ti¯ ca tio n a rises fro m th e fa ct th a t th erm o d y n a m ics in ca n o n ica l en sem b le u ses e ¡¯ H w h ile th e sta n d a rd tim e ev o lu tio n in q u a n tu m m ech a n ics u ses e ¡itH . B u t b ey o n d th a t, it is d i± cu lt to u n d ersta n d in p u rely p h y sica l term s w h y im a g in a ry tim e a n d rea l tem p era tu re sh o u ld h av e a n y th in g to d o w ith ea ch o th er. In o b ta in in g th e ex p ecta tio n va lu es o f o p era to rs w h ich d ep en d o n ly o n q { lik e th e o n es u sed in (1 4 ) { w e o n ly n eed to k n ow th e d ia g o n a l elem en ts ½ (q ;q ) = K E (¯ ;q; 0 ;q ). T h e k ern el in th e rig h t h a n d sid e ca n b e th o u g h t o f a s th e o n e co rresp o n d in g to a p erio d ic m o tio n in w h ich a p a rticle sta rts a n d en d s a t q in a tim e in terva l ¯ . In o th er w o rd s, p erio d icity in im a g in a ry tim e is n ow lin k ed to ¯ n ite tem p era tu re. B eliev e it o r n o t, m o st o f th e resu lts in b la ck h o le th erm o d y n a m ics ca n b e o b ta in ed fro m th is sin g le fa ct b y n o tin g th a t th e sp a cetim es rep resen tin g a b la ck h o le, fo r ex a m p le, h av e th e a p p ro p ria te p erio d icity in im a g in a ry tim e. C o n sid erin g th e eleg a n ce o f th is resu lt, let u s p a u se fo r a m o m en t a n d see h ow it co m es a b o u t. C o n sid er a cu rv ed sp a cetim e in g en era l rela tiv ity w h ich h a s a lin e in terva l d s 2 = ¡ f (r)d t2 +
d r2 + dL f (r)
2 ?
;
(1 9 )
w h ere d L 2? rep resen ts m etric in tw o tra n sv erse d irectio n s. F o r ex a m p le, w e sa w in a p rev io u s in sta lm en t [2 ] th a t th e S ch w a rzsch ild m etric rep resen tin g a b la ck h o le h a s th is fo rm w ith f (r ) = 1 ¡ (r g = r ), w h ere r g = (2 G M = c 2 ) = 2 M (in u n its w ith 1064
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G = c = 1 ) a n d d L 2? rep resen ts th e sta n d a rd m etric o n a tw o sp h ere. T h e o n ly p ro p erty w e w ill a ctu a lly n eed is th a t f (r) h a s a sim p le zero a t so m e r = a w ith f 0(a ) ´ 2 · b ein g so m e co n sta n t. In th e ca se o f th e b la ck h o le m etric, · = (1 = 2 r g ). W h en w e co n sid er th e m etric n ea r th e h o rizo n r ¼ a , w e ca n ex p a n d f (r) in a T ay lo r series a n d red u ce it to th e fo rm d s 2 = ¡ 2 · ld t2 +
d l2 + dL 2· l
2 ?
;
(2 0 )
w h ere l ´ (r ¡ a ) is th e d ista n ce fro m th e h o rizo n . If w e n ow m a k e a co o rd in a te tra n sfo rm a tio n fro m l to a n o th er sp a tia l co o rd in a te x su ch th a t (· x )2 = 2 · l, th e m etric b eco m es d s 2 = ¡ · 2 x 2 d t2 + d x 2 + d L
2 ?
:
(2 1 )
T h is rep resen ts th e m etric n ea r th e h o rizo n o f a b la ck h o le. S o fa r w e h av e n o t d o n e a n y th in g n o n -triv ia l. N ow w e sh a ll a n a ly tica lly co n tin u e to im a g in a ry va lu es o f tim e w ith it = ¿ a n d d en o te · ¿ = µ . T h en th e co rresp o n d in g a n a ly tica lly co n tin u ed m etric b eco m es d s 2 = x 2 d µ 2 + d x 2 + d L 2? :
(2 2 )
B u t d x 2 + x 2 d µ 2 is ju st th e m etric o n a tw o -d im en sio n a l p la n e in p o la r co o rd in a tes a n d if it h a s to b e w ell b eh av ed a t x = 0 , th e co o rd in a te µ m u st b e p erio d ic w ith p erio d 2 ¼ . S in ce µ = · ¿ , it fo llow s th a t th e im a g in a ry tim e ¿ m u st b e p erio d ic w ith p erio d 2 ¼ = · a s fa r a s a n y p h y sica l p h en o m en o n is co n cern ed . B u t w e saw ea rlier th a t su ch a p erio d icity o f th e im a g in a ry tim e is m a th em a tica lly id en tica l to w o rk in g in ¯ n ite tem p era tu re w ith th e tem p era tu re ¯ ¡1 =
· 1 ~c 3 = = 2¼ 4 ¼ rg 8¼ G M
;
(2 3 )
w h ere th e ¯ rst eq u a lity is va lid fo r a g en era l cla ss o f m etrics (w ith su ita b ly d e¯ n ed · b y T ay lo r ex p a n sio n ) w h ile th e la st tw o resu lts a re fo r th e S ch w a rzsch ild m etric a n d in th e ¯ n a l ex p ressio n w e h av e rev erted b a ck to n o rm a l u n its. T h is is p recisely th e H aw k in g tem p era tu re o f a b la ck h o le o f m a ss M w h ich w e o b ta in ed b y a d i® eren t m eth o d in a p rev io u s in sta lm en t [3 ]. H ere w e co u ld d o th a t ju st b y lo o k in g a t th e fo rm o f th e m etric n ea r th e h o rizo n a n d u sin g th e rela tio n b etw een p erio d icity in im a g in a ry tim e a n d tem p era tu re. W h ile th ese resu lts h av e b een
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v eri¯ ed b y sev era l o th er m eth o d s in th e co n tex t o f g en era l rela tiv ity, a tra n sp a ren t p h y sica l u n d ersta n d in g is still la ck in g . T h e im a g in a ry tim e a n d E u clid ea n a ctio n also p lay a n in terestin g ro le in th e ca se o f tu n n elin g . T o see th is, let u s sta rt w ith th e ex p ressio n fo r th e cla ssica l a ctio n w ritten in a slig h tly d i® eren t fo rm : Z Z Z Z A = pdq ¡ H dt : d tL = d t(p q_ ¡ H ) = (2 4 ) W h ile u sin g th e a ctio n p rin cip le, w e u su a lly co n cen tra te o n tra jecto ries w ith ¯ x ed en d p o in ts (t1 ;q 1 ) a n d (t2 ;q 2 ). W h en th e H a m ilto n ia n is in d ep en d en t o f tim e, w e ca n a lso stu d y cla ssica l tra jecto ries o f p a rticles w ith a ¯ x ed va lu e fo r en erg y E . In th is ca se, th e seco n d term in (2 4 ) b eco m es ju st E t apn d th e n o n -triv ia l va ria tio n a ctu a lly co m es fro m th e ¯ rst term . E x p ressin g p a s 2 m (E ¡ V ) fo r a p a rticle o f m a ss m m ov in g in a p o ten tia l V , w e g et a n a ctio n { clo sely rela ted to w h a t is ca lled `J a co b i a ctio n ' { g iv en b y Z Zp S = pd q = 2 m (E ¡ V )d q : (2 5 ) A s lo n g a s E w h en E < V w ith E = 0 m ov in g in a a n d is g iv en
> V , th is w ill lea d to a rea l va lu e fo r S . T u n n elin g o ccu rs, h ow ev er, . T o sim p lify m a tters a little b it, let u s co n sid er th e ca se o f a p a rticle (w h ich ca n a lw ay s b e a ch iev ed b y a co n sta n t to th e H a m ilto n ia n ) p o ten tia l V > 0 . In th a t ca se th e a ctio n b eco m es p u re im a g in a ry by Zp S = i 2m V d q ; (2 6 )
a n d th e co rresp o n d in g b ra n ch o f th e sem i cla ssica l w av e fu n ctio n (stu d ied in th e la st in sta lm en t) w ill b e ex p o n en tia lly d a m p ed : µZ p ¶ Ã / ex p (iS ) = ex p ¡ 2m V d q : (2 7 ) T h is rep resen ts th e fa ct th a t y o u ca n n o t h av e a cla ssica l tra jecto ry w ith E = 0 in a reg io n in w h ich V > 0 . It is h ow ev er p o ssib le to h av e su ch a tra jecto ry if w e a n a ly tica lly co n tin u e to im a g in a ry va lu es o f tim e. In rea l tim e, th e co n serva tio n o f en erg y fo r a p a rticle w ith E = 0 g iv es (1 = 2 )m (d q = d t)2 = ¡ V (q) w h ich ca n n o t h av e rea l so lu tio n s w h en 1066
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V > 0 . B u t w h en w e set t = ¡ i¿ th is eq u a tio n b eco m es (1 = 2 )m (d q = d ¿ )2 = V (q ) w h ich , o f co u rse, h a s p erfectly va lid so lu tio n s w h en V > 0 . S o th e tu n n elin g th ro u g h a p o ten tia l b a rrier ca n b e in terp reted a s a p a rticle m ov in g o ® to im a g in a ry va lu es o f tim e a s fa r a s th e m a th em atics g o es. T h e E u clid ea n a ctio n w ill n ow b e Zp SE = 2m V d q : (2 8 ) A ll th a t w e n eed to d o to o b ta in th e tu n n elin g a m p litu d e is to rep la ce iS b y ¡ S E in th e a rg u m en t o f th e releva n t ex p o n en tia l so th a t th e w av e fu n ctio n in (2 7 ) b eco m es: µZ p ¶ Ã / ex p iS = ex p ¡ 2 m V d q = ex p (¡ S E ) : (2 9 ) S o w e ¯ n d th a t th e tu n n ellin g a m p litu d e a cro ss th e p o ten tia l ca n a lso b e rela ted to a n a ly tic co n tin u a tio n in th e im a g in a ry tim e a n d a n d th e E u clid ea n a ctio n . F in a lly w e w ill u se th ese id ea s to o b ta in a rea lly n o n -triv ia l p h en o m en o n in q u a n tu m electro d y n a m ics, ca lled th e S ch w in g er e® ect, n a m ed a fter J u lia n S ch w in g er w h o w a s o n e o f th e crea to rs o f q u a n tu m electro d y n a m ics a n d receiv ed a N o b el P rize fo r th e sa m e. In sim p lest term s, th is e® ect ca n b e sta ted a s fo llow s. C o n sid er a reg io n o f sp a ce in w h ich th ere ex ists a co n sta n t, u n ifo rm electric ¯ eld . O n e w ay to d o th is is to set-u p tw o la rg e, p a ra llel, co n d u ctin g p la tes sep a ra ted b y so m e d ista n ce L a n d co n n ect th em to th e o p p o site p o les o f a b a ttery. T h is ch a rg es th e p la tes a n d p ro d u ces a co n sta n t electric ¯ eld b etw een th em . S ch w in g er sh ow ed th a t, in su ch a co n ¯ g u ra tio n , electro n s a n d p o sitro n s w ill sp o n ta n eo u sly a p p ea r in th e reg io n b etw een th e p la tes th ro u g h a p ro cess w h ich is ca lled p a ir p ro d u ctio n fro m th e va cu u m . T h e ¯ rst q u estio n o n e w o u ld a sk is h ow p a rticles ca n a p p ea r o u t o f n ow h ere. T h is is n a tu ra l sin ce w e h av en 't seen ten n is b a lls o r ch a irs a p p ea r o u t o f th e va cu u m sp o n ta n eo u sly. In q u a n tu m ¯ eld th eo ry, w h a t w e ca ll `va cu u m ' is a ctu a lly b ristlin g w ith q u a n tu m ° u ctu a tio n s o f th e ¯ eld s w h ich ca n b e in terp reted in term s o f v irtu a l p a rticle-a n tip a rticle p a irs (see [4 ]). U n d er n o rm a l circu m sta n ce, su ch a v irtu a l electro n -p o sitro n p a ir w ill b e d escrib ed b y th e situ a tio n in th e left fra m e o f F igu re 1 . W e th in k o f a n electro n a n d p o sitro n b ein g crea ted a t th e ev en t A a n d th en g ettin g a n n ih ila ted a t th e ev en t B . In th e a b sen ce o f a n y ex tern a l ¯ eld s, th ere is n o fo rce a ctin g o n th ese v irtu a l p a irs a n d th ey co n tin u o u sly a p p ea r a n d d isa p p ea r q u ite ra n d o m ly in th e sp a cetim e.
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Figure 1. In the vacuum there will exist virtual electron-positron pairs which are constantly created and annihilated as shown in the left frame (a) An electronpositron pair is created at A and annihilated at B with the positron being interpreted as an electron going backward in time. The right frame (b) shows how, in the presence of an electric field, this virtual process can lead to creation of real electrons and positrons.
C o n sid er n ow w h a t h a p p en s if th ere is a electric ¯ eld p resen t in th is reg io n o f sp a ce. T h e electric ¯ eld w ill p u ll th e electro n in o n e d irectio n a n d p u sh th e p o sitro n in th e o p p o site d irectio n sin ce th e electro n s a n d p o sitro n s ca rry o p p o site ch a rg es. In th e p ro cess th e electric ¯ eld w ill d o w o rk o n th e v irtu a l p a rticlea n tip a rticle p a ir a n d h en ce w ill su p p ly en erg y to th em . If th e ¯ eld is stro n g en o u g h , it ca n su p p ly a n en erg y g rea ter th a n th e rest en erg y o f th e tw o ch a rg ed p a rticles w h ich is ju st 2 £ m c 2 w h ere m is th e m a ss o f th e p a rticle. T h is a llow s th e v irtu a l p a rticles to b eco m e rea l. T h a t is h ow th e co n sta n t electric ¯ eld b etw een tw o co n d u ctin g p a ra llel p la tes p ro d u ces p a rticles o u t o f th e va cu u m . It essen tia lly d o es w o rk o n th e v irtu a l electro n -p o sitro n p a irs w h ich a re p resen t in th e sp a cetim e a n d co n v erts th em in to rea l p a rticles a s sh o w n in th e rig h t fra m e o f F igu re 1 (b ). O n e w ay to m o d el th is is to a ssu m e th a t th e th e p a rticle tu n n els fro m th e tra jecto ry o n th e left to th e o n e o n th e rig h t th ro u g h th e sem icircu la r p a th in th e low er h a lf. T h e tra jecto ries o n th e left a n d rig h t a re rea l tra jecto ries fo r th e ch a rg ed p a rticle b u t th e sem icircle is a `fo rb id d en ' q u a n tu m p ro cess. W e w ill n ow see h ow im a g in a ry tim e m a k es th is p o ssib le. T o d o th is, w e b eg in w ith th e tra jecto ry in rea l tim e w h ich w ill co rresp o n d to rela tiv istic m o tio n w ith u n ifo rm a ccelera tio n g = qE = m . W e h av e w o rk ed th is o u t in a p rev io u s in sta lm en t [3 ] a n d th e resu lt is g iv en { w ith su ita b le ch o ice o f in itia l co n d itio n s { b y x = (1 = g ) co sh (g ¿ ); t = (1 = g ) sin h (g ¿ ); x 2 ¡ t2 = 1 = g 2 :
(3 0 )
T h e tra jecto ry is a (p a ir o f) h y p erb o la in th e t ¡ x p la n e sh ow n in F igu re 1 (b ). If w e n ow a n a ly tica lly co n tin u e to im a g in a ry va lu es o f ¿ a n d t, th e tra jecto ry 2
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2
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b eco m es a circle x 2 + t2E = 1 = g 2 o f ra d iu s (1 = g ) a n d th e p a ra m etric eq u a tio n s b eco m e x = (1 = g ) co s µ ;
t = (1 = g ) sin µ ;
µ = g ¿E :
(3 1 )
B y g o in g fro m µ = ¼ to µ = 2 ¼ , say, w e ca n g et th is to b e a sem icircle co n n ectin g th e tw o h y p erb o la s. T o o b ta in th e a m p litu d e fo r th is p ro cess w e h a v e to eva lu a te th e va lu e o f th e E u clid ea n a ctio n fo r th e sem icircu la r tra ck . T h e a ctio n fo r a p a rticle o f ch a rg e q in a co n sta n t electric ¯ eld E rep resen ted b y a sca la r p o ten tia l Á = ¡ E x is g iv en by Z Z A = ¡m xdt; d ¿ + qE (3 2 ) w h ere ¿ is th e p ro p er tim e o f th e p a rticle. S o , o n a n a ly tic co n tin u a tio n w e g et Z Z iA = ¡ im xdt d ¿ + iqE Z Z x d tE ´ ¡ A E : ! ¡m d ¿ E + qE (3 3 ) T h e E u clid ea n a ctio n A E in (3 3 ) ca n b e ea sily tra n sfo rm ed to a n in teg ra l ov er µ a n d n o tin g th a t th e in teg ra l ov er x d tE is essen tia lly th e a rea en clo sed b y th e cu rv e, w h ich is a sem icircle o f ra d iu s (1 = g ), w e g et Z 2¼ Z 2¼ ¡ A E = ¡ (m = g ) d µ + (m = 2 g ) d µ = ¡ (m ¼ = 2 g ) : (3 4 ) ¼
¼
T h e lim its o f th e in teg ra tio n a re so ch o sen th a t th e p a th in th e im a g in a ry tim e co n n ects x = ¡ (1 = g ) w ith x = (1 = g ) th ereb y a llow in g a v irtu a l sem i-circu la r lo o p to b e fo rm ed a s sh ow n in F igu re 1 (b ). H en ce th e ¯ n a l resu lt fo r th e E u clid ea n a ctio n fo r th is cla ssica lly fo rb id d en p ro cess is A
E
=
¼m ¼m 2 : = 2g 2qE
(3 5 )
W ith th e u su a l ru le th a t a p ro cess w ith ex p iA g ets rep la ced b y ex p (¡ A E ) w h en it is cla ssica lly fo rb id d en , w e ¯ n d th e a m p litu d e fo r th is p ro cess to ta k e p la ce to b e A / ex p (¡ A E ). T h e co rresp o n d in g p ro b a b ility P = jA j2 is g iv en b y P ¼ ex p ¡ (¼ m 2 = q E ) :
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(3 6 )
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T h is is th e lea d in g term fo r th e p ro b a b ility w h ich S ch w in g er o b ta in ed fo r th e p a ir crea tio n p ro cess. (In fa ct, o n e ca n ev en o b ta in th e su b -lea d in g term s b y tra n sferin g p a th s w h ich w in d a ro u n d sev era l tim es in th e circle b u t w e w ill n o t g o in to it; if y o u a re in terested , ta k e a lo o k a t ref.[5 ]). O n ce a g a in th e m o ra l is clea r. W h a t is fo rb id d en in rea l tim e is a llow ed in im a g in a ry tim e! Suggested Reading [1]
T Padmanabhan, The Optics of Particles, Resonance, Vol.14, pp.8–18, 2009.
[2] [3]
T Padmanabhan, Schwarzschild Metric at a discounted price, Resonance, Vol.13, pp.312–318, 2008. T Padmanabhan, Why are black holes hot? Resonance,Vol.13, pp.412–419, 2008.
[4]
T Padmanabhan, The Power of Nothing Resonance,Vol.14, pp.179–190, 2009.
[5]
K Srinivasan and T Padmanabhan, Particle Production and Complex Path Analysis, Phys. Rev. D, Vol.60, p.24007, 1999.
Address for Correspondence: T Padmanabhan, IUCAA, Post Bag 4, Pune, University Campus, Ganeshkhind, Pune 411 007, India. Email: [email protected], [email protected]
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Snippets of Physics 24. Kepler and his Problem T Padmanabhan
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
T h e m a jo r c o n tr ib u tio n o f K e p le r w a s in d isc o v e r in g h is la w s o f p la n e ta r y m o tio n , o n e o f w h ic h sta te s th a t p la n e ts m o v e in e llip tic a l o rb its a r o u n d th e S u n w h ich is lo c a te d a t o n e o f th e fo c i o f th e e llip s e . T h e m o tio n o f a te st p a rtic le in a n in v e rse sq u a re fo rc e la w is u su a lly c a lle d th e K e p le r p ro b le m . W e w ill e x a m in e se v e r a l a sp e c ts o f th is m o tio n in th is la st in sta llm e n t o f S n ip p e ts in P h y sic s. It is a tex tb o o k ex ercise to sh ow th a t th e b o u n d o rb its in N ew to n ia n g rav ity u n d er th e in ° u en ce o f a n in v erse sq u a re law fo rce a re ellip ses. W e h av e a lrea d y d iscu ssed sev era l fea tu res o f su ch a m o tio n in p rev io u s a rticles [1 , 2 ]. In p a rticu la r, th is m o tio n h a s th e fo llow in g fea tu res, w h ich a re sp ecia l to th is p ro b lem a n d d isa p p ea r if y o u ch a n g e th e p ro b lem ev er so slig h tly ! ² T h e tra jecto ry o f th e p a rticle in th e v elo city sp a ce (u su a lly ca lled a h o d o g ra p h ) u n d erg o in g su ch a m o tio n , is a circle [1 ]. T h e v elo city v o f th e p a rticle ca n b e ex p ressed in th e fo rm v (t) = v 0 + u (t) w h ere v 0 is a v ecto r co n sta n t in m a g n itu d e a n d d irectio n w h ile u (t) h a s co n sta n t m a g n itu d e b u t va ry in g d irectio n .
Keywords Kepler problem, Coulomb field, precession, general relativity.
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² In a d d itio n to th e a n g u la r m o m en tu m a n d en erg y, w h ich w o u ld b e co n serv ed in a n y sta tic cen tra l fo rce p ro b lem , w e h av e a n ex tra co n serv ed q u a n tity ca lled th e R u n g e{ L en z v ecto r. T h e m a g n itu d e o f th is v ecto r ca n b e a rra n g ed to g iv e th e eccen tricity o f th e o rb it w h ile its d irectio n is a lo n g th e m a jo r a x is o f th e ellip se. T h e co n serva tio n o f th is v ecto r en su res th a t th e d irectio n o f th e m a jo r a x is d o es n o t ch a n g e a n d th e ellip se stay s in p la ce. RESONANCE December 2009
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T h e ¯ rst ca su a lty w h en w e m o d ify th e p ro b lem is th e R u n g e{ L en z v ecto r. T h is m ea n s th a t ev en w h en th e m o d i¯ ca tio n is sm a ll, th e d irectio n o f th e ellip se is n o t p ro tected b y a co n serva tio n law . T h is ca u ses th e ellip se to p recess a n d it is o ften o f in terest to co m p u te th e ra te o f th is p recessio n . W e sh a ll n ow d iscu ss h ow th is co m es a b o u t in d i® eren t co n tex ts a n d w h a t it im p lies. T h e stu d y o f o rb its in ex tern a l ¯ eld s is m o st eco n o m ica lly d o n e u sin g th e H a m ilto n { J a co b i eq u a tio n . S o lv in g th e releva n t H a m ilto n { J a co b i eq u a tio n in th e co n tex t o f a cen tra l fo rce p ro b lem lea d s to a n a ctio n w h ich ca n b e ex p ressed in th e fo rm A (t;r ;E ;L ) = ¡ E t + L µ + S (r;E ;L );
(1 )
w h ere (r;µ ) a re th e sta n d a rd p o la r co o rd in a tes in th e p la n e o f o rb it, L is th e a n g u la r m o m en tu m , E is th e en erg y a n d S (r ;E ;L ) h a s to b e d eterm in ed b y in teg ra tin g th e H a m ilto n { J a co b i eq u a tio n . T h e o rb ita l eq u a tio n r = r (µ ) ca n b e o b ta in ed b y d i® eren tia tin g A w ith resp ect to L a n d eq u a tin g it to a co n sta n t: @S = µ 0 = co n sta n t : @L
µ+
(2 )
T h e d i® eren t co n tex ts w e w o u ld b e in terested in o n ly d i® ers in th e n a tu re o f H a m ilto n { J a co b i eq u a tio n ; o n ce w e o b ta in th e o rb ita l eq u a tio n in (2 ) o n e ca n co m p a re th e d i® eren t m o d els fa irly ea sily. L et u s sta rt w ith th e sta n d a rd N ew to n ia n th eo ry. If th e p a rticle is m ov in g in a cen tra l p o ten tia l V (r ), th en fro m th e H a m ilto n { J a co b i eq u a tio n @ A = @ t + H = 0 , it is ea sy to sh ow th a t S sa tis¯ es th e eq u a tio n Ã
dS dr
!2
= 2 m (E ¡ V ) ¡ (L 2 = r 2 ):
(3 )
T h is, in tu rn , a llow s u s to w rite th e o rb ita l eq u a tio n in (2 ) in th e fo rm µ ¡ µ0 =
Z
d r(L 2 = r 2 ) [2 m (E ¡ V ) ¡ (L 2 = r 2 )]1 = 2
:
(4 )
C o n v ertin g th is in to a n eq u a tio n fo r d r = d µ a n d in tro d u cin g th e sta n d a rd su b stitu tio n u ´ (1 = r ) w e ca n o b ta in th e d i® eren tia l eq u a tio n sa tis¯ ed b y u (µ ):
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u 00+ u = ¡
m dV ; L 2 du
(5 )
w h ere th e p rim e d en o tes d i® eren tia tio n w ith resp ect to µ . In th e sta n d a rd K ep ler p ro b lem , V = ¡ G M m = r = ¡ G M m u so th a t th e rig h t-h a n d sid e o f (5 ) b eco m es a co n sta n t a n d w e g et th e so lu tio n u = ® + ¯ co s µ w h ich rep resen ts a n ellip se. L et u s n ow a sk w h a t h a p p en s to th e K ep ler p ro b lem w h en w e in tro d u ce p h y sica lly releva n t m o d i¯ ca tio n s. T h e ¯ rst g en era liza tio n th a t y o u m ig h t th in k o f w ill b e to in tro d u ce th e e® ects o f sp ecia l rela tiv ity. T h is tu rn s o u t to b e m o re n o n -triv ia l th a n o n e m ig h t h av e im a g in ed fo r th e fo llow in g rea so n . In th e n o n rela tiv istic co n tex t, th e m o tio n o f a p a rticle u n d er th e a ctio n o f a p o ten tia l V is g ov ern ed b y th e eq u a tio n d p ® = d t = ¡ @ ® V , w h ere ® = 1 ;2 ;3 d en o tes th e th ree sp a tia l co m p o n en ts o f th e m o m en tu m p a n d @ ® d en o tes th e d eriva tiv e w ith resp ect to th e co o rd in a te x ® . O n e m ig h t h av e th o u g h t th a t th e n a tu ra l g en era liza tio n o f th is N ew to n ia n resu lt in to th e sp ecia l rela tiv istic d o m a in w o u ld in v o lv e th e fo llow in g rep la cem en ts: C h a n g e th e th ree m o m en tu m p ® to th e fo u r m o m en tu m p i (w ith i = 0 ;1 ;2 ;3 ), th e co o rd in a te tim e t in to th e p ro p er tim e ¿ o f th e p a rticle a n d th e th ree-d im en sio n a l g ra d ien t @ ® to th e fo u r-g ra d ien t @ i. T h is w o u ld h av e led to th e eq u a tio n d p i = d ¿ = ¡ @ iV . U n fo rtu n a tely, th ere is a p ro b lem w ith th is `g en era liza tio n '. T h e fo u r-v elo city u i sa tis¯ es th e co n stra in t: u iu i =
d x id x i d¿2 = ¡ = ¡1 ; d¿d¿ d¿2
(6 )
w h ere w e h av e in tro d u ced th e su m m a tio n co n v en tio n w h ich req u ires u s to su m a ll rep ea ted in d ices ov er th e ra n g e i = 0 ;1 ;2 ;3 a n d u i = ´ ij u j w ith ´ ij = d ia [¡ 1 ;1 ;1 ;1 ]. S in ce th e fo u r-m o m en tu m p i = m u i is p ro p o rtio n a l to fo u rv elo city u i, w e h av e th e co n stra in t ui
m d d pi dui = m ui = (u iu i) = 0 ; d¿ d¿ 2 d¿
(7 )
w h ere w e h av e u sed (6 ). T h is im p lies th a t o u r p o ten tia l V h a s to sa tisfy th e co n stra in ts u i@ iV = 0 ; th a t is, th e p o ten tia l sh o u ld n o t ch a n g e a lo n g th e w o rld lin e o f th e p a rticle w h ich is n o t p o ssib le in g en era l. T h is is o n e rea so n w h y v elo city -in d ep en d en t fo rces lik e @ iV ca n n o t b e in tro d u ced in sp ecia l rela tiv ity.
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SERIES ARTICLE
S o th e g en era liza tio n to sp ecia l rela tiv ity h a s to co m e fro m so m e o th er d irectio n . O n e p o ssib ility is to n o te th a t th e K ep ler p ro b lem a lso a rises in electro d y n a m ics w h en w e co n sid er th e m o tio n o f a test ch a rg e in th e C o u lo m b ¯ eld o f a n o th er ch a rg e. S in ce w e h av e a fu lly sp ecia l rela tiv istic fo rm u la tio n o f electro d y n a m ics, w e ca n a ttem p t to stu d y th e m o tio n o f a p a rticle u n d er th e a ctio n o f a fo u r-v ecto r p o ten tia l A i = (V (r );0 ;0 ;0 ) w h ich w o u ld co rresp o n d to a cen tra lly sy m m etric electro sta tic p o ten tia l. W e ca n o n ce a g a in w rite d ow n th e H a m ilto n { J a co b i eq u a tio n fo r th is ca se a n d o b ta in , in a n a lo g y w ith (3 ), th e d i® eren tia l eq u a tio n fo r S g iv en b y Ã
dS dr
!2
=
L2 1 2 ¡ V ¡ ¡ m 2 c2 : (E ) c2 r2
(8 )
It is fa irly stra ig h tfo rw a rd to sh ow th a t, in th is ca se, (5 ) g ets m o d i¯ ed to th e fo rm (E ¡ V ) u + u = ¡ L 2 c2 00
Ã
dV du
!
E = c2 d V 1 1 dV 2 : = ¡ + L 2 du 2 L 2 c2 d u
(9 )
C o m p a rin g (9 ) w ith (5 ) w e see th a t th e ¯ rst term in v o lv es rep la cem en t o f m b y E = c 2 w h ich , o f co u rse, m a k es sen se th o u g h it b rin g s in a v elo city d ep en d en ce; th e seco n d term sh ow s th a t th e p o ten tia l p ick s u p a V 2 term a s a co rrectio n w h ich ca n b e tra ced b a ck to th e fa ct th a t w h ile p 2 / E in th e n o n -rela tiv istic ca se, p 2 = (E = c)2 ¡ m 2 c 2 in sp ecia l rela tiv ity. M o re fo rm a lly, w e ca n a ttem p t to d e¯ n e a N ew to n ia n e® ectiv e p o ten tia l V e® in w h ich w e w ill o b ta in th e sa m e eq u a tio n o f m o tio n . In th e ca se o f m o tio n in a C o u lo m b ¯ eld w ith V (r ) = ¡ ® = r = ¡ ® u , th is req u ires u s to sa tisfy th e co n d itio n m d V e® ®E ®2 ¡ ¡ u = L 2 du L 2 c2 L 2 c2
(1 0 )
w h ich in teg ra tes to g iv e V e® = ¡
µ
E m c2
¶
®u ¡
® 2 u2 : m c2 2
(1 1 )
S in ce E = m c 2 is ° , w e ca n th in k o f th e ¯ rst term a s th e o rig in a l p o ten tia l tra n sfo rm ed to th e rest fra m e o f th e m ov in g b o d y. T h e seco n d term is a p u rely rela tiv istic co rrectio n .
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SERIES ARTICLE
In th is ca se o f rela tiv istic m o tio n in a C o u lo m b ¯ eld , th e o rb it eq u a tio n b eco m es: u 00+ ! 2 u =
®E ; L 2 c2
®2 ; L 2 c2
!2 ´ 1¡
(1 2 )
w h ich is a g a in essen tia lly a h a rm o n ic o scilla to r eq u a tio n . T h e tra jecto ry o b ta in ed b y so lv in g (1 2 ) ca n b e ex p ressed in th e fo rm E ® 1 1 = co s(! µ ) + 2 2 2 ; r R c L !
(1 3 )
w h ere L!2 R ´ m c
"µ
E m c2
¶2
®2 ¡ 1+ 2 2 c L
#¡ 1 = 2
(1 4 )
is a co n sta n t. In a m o re fa m ilia r fo rm , th e tra jecto ry is l= r = (1 + e co s ! µ ) w ith c2 J L 2 ! 2 l= ; E j® j
"
L 2 c2 m 2 c4 ! 2 e = ¡ 1 ®2 E 2 2
#
:
(1 5 )
It is ea sy to v erify th a t, w h en c ! 1 , th is red u ces to th e sta n d a rd eq u a tio n fo r a n ellip se in th e K ep ler p ro b lem . In term s o f th e n o n -rela tiv istic en erg y E n r = E ¡ m c 2 , w e g et, to lea d in g o rd er, ! ¼ 1 ;l ¼ L 2 = m j® j a n d e 2 ¼ 1 + (2 E n r L 2 = m ® 2 ), w h ich a re th e sta n d a rd resu lts. In th e fu lly rela tiv istic ca se, a ll th ese ex p ressio n s ch a n g e b u t th e k ey n ew e® ect a rises fro m th e fa ct th a t ! 6= 1 . D u e to th is fa cto r, th e tra jecto ry is n o t clo sed a n d th e ellip se p recesses. W h en ! 6= 1 , th e r in (1 3 ) d o es n o t retu rn to th e va lu e a t µ = 0 w h en µ = 2 ¼ ; in stea d , w e n eed a fu rth er tu rn b y ¢ µ (th e `a n g le o f p recessio n ') fo r r to retu rn to th e o rig in a l va lu e. T h is is d eterm in ed b y th e co n d itio n (2 ¼ + ¢ µ )! = 2 ¼ . F ro m E q . (1 3 ) w e ¯ n d th a t th e o rb it p recesses b y th e a n g le 2Ã ¢ µ = 2¼ 4 1 ¡
®2 c2 L 2
!¡1 = 2
3
¡ 15'
¼®2 c2 L 2
(1 6 )
p er o rb it w h ere th e seco n d ex p ressio n is va lid fo r ® 2 ¿ c 2 L 2 . T h is is a p u rely rela tiv istic e® ect a n d va n ish es w h en c ! 1 . 1148
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SERIES ARTICLE
T h ere is a n o th er p ecu lia r fea tu re th a t a rises in th e sp ecia l rela tiv istic ca se w h ich h a s n o N ew to n ia n a n a lo g u e. Y o u w o u ld h av e n o ticed th a t ! 2 in (1 2 ) h a s tw o term s o f o p p o site sig n a n d in o b ta in in g o u r resu lt in (1 3 ), w e h av e ta citly a ssu m ed th a t ! 2 > 0 . B u t in p rin cip le, o n e ca n h av e a situ a tio n w ith v ery low b u t n o n -zero a n g u la r m o m en tu m m a k in g ! 2 < 0 . T h is is a fea tu re w h ich th e n o n -rela tiv istic K ep ler p ro b lem sim p ly d o es n o t h av e a n d { u n d er su ch d ra stic ch a n g e o f circu m sta n ces { o n e ca n n o lo n g er th in k in term s o f p recessin g ellip ses. E q u a tio n (1 2 ) n ow h a s th e so lu tio n ³
® 2 ¡ c2 L
2
´1
r
q
2
0 s
= § c (L E ) + m 2 c 2 (® 2 ¡ L 2 c 2 ) co sh @ µ
1
®2 ¡ 1 A ¡ E ® : (1 7 ) c2 L 2
In th is ex p ressio n , w e ta k e th e p o sitiv e ro o t fo r ® > 0 a n d th e n eg a tiv e ro o t fo r ® < 0 . It is o b v io u s th a t, a s µ in crea ses, (1 = r ) k eep s in crea sin g in th e ca se o f a ttra ctiv e m o tio n so th a t th e test p a rticle sp ira ls to th e o rig in . T h is d o es n o t h a p p en in th e K ep ler p ro b lem in N ew to n ia n p h y sics. A s is w ell k n ow n , th e a n g u la r m o m en tu m term g iv es a rep u lsive L 2 = r 2 co n trib u tio n to th e e® ectiv e p o ten tia l in a n y cen tra l fo rce p ro b lem . In th e ca se o f ¡ (1 = r) p o ten tia l, th e a n g u la r m o m en tu m term p rev en ts a n y p a rticle w ith n o n -zero L fro m rea ch in g th e o rig in . T h is is n o t th e ca se in sp ecia l rela tiv istic m o tio n u n d er a ttra ctiv e C o u lo m b ¯ eld . If th e a n g u la r m o m en tu m is less th a n a critica l va lu e, ® = c, th en th e p a rticle sp ira ls d ow n to th e o rig in . Y ou w ill ¯ n d it u sefu l to re-a n a ly se th is situ a tio n in term s o f su ita b le e® ectiv e p o ten tia ls. If w e th in k o f ® a s G M m , th e seco n d term in (1 2 ) g iv es a co rrectio n to th e p o ten tia l (¡ G 2 M 2 = 2 c 2 )(m = r 2 ). T h is term w ill lea d to a p recessio n o f th e ellip se b u t th e m o d el is to ta lly w ro n g . O n e ca n n o t rep resen t g rav ity u sin g a v ecto r p o ten tia l; in su ch a th eo ry, lik e ch a rg es rep el w h ile th e g rav ity h a s to b e a ttra ctiv e. T h e p ro p er w ay o f g en era lizin g th e g rav ita tio n a l K ep ler p ro b lem , ta k in g in to a cco u n t th e e® ects o f rela tiv ity, is o f co u rse to u se g en era l rela tiv ity to d escrib e th e g rav ita tio n a l ¯ eld [3 ]. In th e ca se o f a test p a rticle m ov in g in th e g rav ita tio n a l ¯ eld o f a p o in t m a ss a t th e o rig in , w e h av e to ta k e in to a cco u n t th e e® ect o f th e cu rva tu re o f th e sp a ce w h ich is d escrib ed b y th e m o d i¯ ca tio n to th e lin e in terva l in th e fo rm d s 2 = ¡ f (r )c 2 d t2 +
RESONANCE December 2009
³ ´ d r2 + r 2 d µ 2 + sin 2 µ d µ 2 ; f (r )
(1 8 )
1149
SERIES ARTICLE
w h ere f (r ) = [1 ¡ (2 G M = c 2 r )]. W e h a d a lrea d y d iscu ssed th is in a p rev io u s in sta llm en t [3 ] w h ere it w a s p o in ted o u t th a t o n e ca n o b ta in th e co rrect eq u a tio n b y rep la cin g co o rd in a te in terva ls b y th e p ro p er g eo m etrica l q u a n tities. F o r a rela tiv istic free p a rticle, th e rela tio n sh ip b etw een en erg y a n d m o m en tu m is g iv en b y E 2 = p 2 c 2 + m 2 c 4 . W h en ex p ressed in p o la r co o rd in a tes, th is ta k es th e fo rm E
2
= m 2 c4 +
µ
E c
¶2 Ã
dr dt
!2
+
c2 L 2 : r2
(1 9 )
p r= f ); d t b y th e W e n ow h av pe to rep la ce in (1 9 ), d r b y th e p ro p er len g th (d p p ro p er tim e f d t a n d th e en erg y b y th e red sh ifted o n e E = f . T h e ex p ressio n 2 fo r co n serv p ed a n g u la r m o m en tu m w ill a lso ch a n g e fro m L = m r (d µ = d t) to L = 2 m r (d µ = f d t). M a n ip u la tin g th ese eq u a tio n s, it is ea sy to o b ta in a n ex p ressio n fo r d r= d µ , d i® eren tia tin g w h ich w e w ill g et th e eq u a tio n fo r th e o rb it in th e sta n d a rd fo rm : G M m d 2u u + = L2 d µ2
2
+
3G M 2 u : c2
(2 0 )
T h e ¯ rst term o n th e rig h t-h a n d sid e is p u rely N ew to n ia n a n d th e seco n d term is th e co rrectio n fro m g en era l rela tiv ity. T h e ra tio o f th ese tw o term s is (L = m rc)2 ¼ (v = c)2 , w h ere r a n d v a re th e ty p ica l ra d iu s a n d sp eed o f th e p a rticle. T h is co rrectio n term ch a n g es th e n a tu re o f th e o rb its in tw o w ay s. F irst, it ch a n g es th e rela tio n sh ip b etw een th e p a ra m eters o f th e o rb it a n d th e en erg y a n d a n g u la r m o m en tu m o f th e p a rticle. M o re im p o rta n tly, it m a k es th e ellip tica l o rb it o f N ew to n ia n g rav ity to p recess slow ly w h ich is o f g rea ter o b serva tio n a l im p o rta n ce. T h e ex a ct so lu tio n to (2 0 ) ca n b e g iv en o n ly in term s o f ellip tic fu n ctio n s a n d h en ce is n o t v ery u sefu l. A n a p p rox im a te so lu tio n to (2 0 ), h ow ev er, ca n b e o b ta in ed fa irly ea sily w h en th e o rb it h a s a v ery low eccen tricity a n d is n ea rly circu la r (w h ich is th e ca se fo r m o st p la n eta ry o rb its). T h en th e low est o rd er so lu tio n w ill b e u = u 0 = co n sta n t a n d o n e ca n ¯ n d th e n ex t o rd er co rrectio n b y p ertu rb a tio n s th eo ry. T h is ca n b e d o n e w ithou t a ssu m in g th a t 2 G M u 0 = c 2 = 2 G M = c 2 r 0 is sm a ll, so th a t th e resu lt is va lid ev en fo r o rb its clo se to th e S ch w a rzsch ild ra d iu s, a s lo n g a s th e o rb it is n ea rly circu la r. L et th e ra d iu s o f th e circu la r o rb it b e r 0 fo r w h ich u = (1 = r 0 ) ´ k 0 . F o r th e a ctu a l
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o rb it u = k 0 + u 1 , w h ere w e ex p ect th e seco n d term to b e a sm a ll co rrectio n . C h a n g in g th e va ria b les fro m u to u 1 , w h ere u 1 = u ¡ k 0 , eq u a tio n (2 0 ) ca n b e w ritten a s u 010+ u 1 + k 0 =
G M m L2
´ 3G M ³ 2 2 u k u k : + + 2 1 0 1 0 c2
2
+
(2 1 )
W e n ow ch o o se k 0 to sa tisfy th e co n d itio n k0 =
G M m L2
2
+ k 02
3G M ; c2
(2 2 )
w h ich d eterm in es th e ra d iu s r 0 = 1 = k 0 o f th e o rig in a l circu la r o rb it in term s o f th e o th er p a ra m eters. N ow th e eq u a tio n fo r u 1 b eco m es u
00 1 +
Ã
6k0G M 1¡ c2
!
u1 =
3G M 2 u 1: c2
(2 3 )
T h is eq u a tio n is ex a ct. W e sh a ll n ow u se th e fa ct th a t th e d ev ia tio n fro m circu la r o rb it, ch a ra cterized b y u 1 is sm a ll a n d ig n o re th e rig h t-h a n d sid e o f (2 3 ). S o lv in g (2 3 ), w ith th e rig h t-h a n d sid e set to zero , w e g et u 1 »= A co s
"µ
6G M 1¡ 2 c r0
¶1 = 2 #
µ :
(2 4 )
W e see th a t r d o es n o t retu rn to its o rig in a l va lu e a t µ = 0 w h en µ = 2 ¼ in d ica tin g a p recessio n o f th e o rb it. W e en co u n tered th e sa m e p h en o m en o n in th e ca se o f m o tio n in a C o u lo m b ¯ eld a s w ell. A s d escrib ed in th a t co n tex t, th e a rg u m en t o f th e co sin e fu n ctio n b eco m es 2 ¼ w h en µ c ¼ 2 ¼ [1 ¡ (6 G M = c 2 r 0 )]¡1 = 2
(2 5 )
w h ich g iv es th e p recessio n (µ c ¡ 2 ¼ ) p er o rb it. W e ca n m a k e a n a iv e co m p a riso n b etw een th is p recessio n ra te a n d th e co rresp o n d in g o n e in th e C o u lo m b p ro b lem b y n o ticin g th a t, in th e la tter ca se, w e ca n su b stitu te ® = G M m a n d L 2 ¼ G M m 2 r 0 (w h ich fo llow s fro m (2 2 ) a t th e low est o rd er) to o b ta in
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2 ! el ! 1¡
G M c2 r 0
(2 6 )
w h ich d i® ers b y a fa cto r 6 in th e co rresp o n d in g term in g en era l rela tiv ity. If w e a ttem p t to rep ro d u ce th e g en era l rela tiv istic resu lts b y a n e® ectiv e N ew to n ia n p o ten tia l, w e n eed to ¯ n d a V e® w h ich sa tis¯ es th e eq u a tio n m d V e® GM m ¡ 2 = L du L2
2
+
3G M u2 2 c
(2 7 )
w h ich in teg ra tes to g iv e V e® = ¡
G M m G M L2 1 ¡ : r m c2 r 3
(2 8 )
T h e tro u b le w ith th is e® ectiv e p o ten tia l is th a t th e co rrectio n term d ep en d s o n th e a n g u la r m o m en tu m L o f th e p a rticle w h ich is so m ew h a t d i± cu lt to m o tiva te p h y sica lly. B u t if y o u a re w illin g to liv e w ith it, th en o n e ca n in tro d u ce a p seu d o N ew to n ia n d escrip tio n o f th e g en era l rela tiv istic K ep ler p ro b lem b y ta k in g th e eq u a tio n s o f m o tio n to b e m (d 2 r= d ¿ 2 ) = F w ith GM m F = ¡ ^r r2
Ã
3 (^r £ u )2 1+ c2
!
;
u =
dr ; d¿
(2 9 )
w h ere ¿ is th e p ro p er tim e a n d ^r is a u n it v ecto r in th e ra d ia l d irectio n . Y o u ca n co n v in ce y o u rself th a t th e co n serv ed a n g u la r m o m en tu m n ow is L = m r £ u w h ich w ill en su re th a t th e a b ov e fo rce rep ro d u ces th e co rrect rela tiv istic o rb it eq u a tio n . U n fo rtu n a tely, th is fo rce law d o es n o t seem to lea d to a n y o th er u sefu l co n cep t. Suggested Reading [1]
T Padmanabhan, Planets Move in Circles!, Resonance, Vol.1, No.9, pp.34–40, 1996.
[2]
T Padmanabhan, Perturbing Coulomb to Avoid Accidents!, Resonance, Vol.14, No.6, pp.622–629, 2009.
[3]
T Padmanabhan, Schwarzschild Metric at a discounted price, Resonance, Vol.13, No.4, pp.312–318, 2008.
Address for Correspondence: T Padmanabhan, IUCAA, Post Bag 4, Pune University Campus, Ganeshkhind, Pune 411 007, India. Email: [email protected], [email protected]
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Dawn of Science 1. The First Tottering Steps T Padmanabhan
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
In mankind’s quest for knowledge, spanning the last four thousand years, certain developments stand out as milestones in the progress of science. This new series, intended for the general reader, will highlight such key events in the growth of mathematics, physics, chemistry, biology and engineering in an approximately chronological manner. You can never really say when science was born – probably when the pre-historic hunter devised the perfect bow – or with the proverbial fire and the wheel. The true beginnings will never be known. Several little acts of logic and imagination are lost to us for lack of recorded history. Even where some records exist, it is never easy to distinguish true science from mythology, magic and mysticism. Every fable and fairy tale, myth and legend, epic and adventure story contains very imaginative novelties. Did the creators of these works produce science fiction and flights of imagination or did they have first-hand experience with certain magnificent inventions? We believe it is the former and that the ancient literary fables cannot be taken too seriously in determining the early evolution of science.
Keywords Pyramids, Imhotep, Moscow Papyrus, Thales of Miletus.
498
With such a conservative stand, the earliest achievement of humanity requiring mammoth scientific and technological skill is definitely the pyramids in Egypt. (See Box 1). These constructions, intended to ensure the safe passage of the rulers or Pharaohs (and certain other privileged ones) to the other world, certainly needed elaborate technical skill. Such a skill could not have been achieved without the understanding of the basics of what we today call ‘science’. It is rather difficult to identify ‘the scientist’ involved with
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Box 1. The Pyramids of Egypt The Egyptian pyramids, built around 2500 BC were essentially funeral edifices. But their size, structure, artistic and technical perfection shows the remarkable engineering skills of this ancient civilisation.
Section of the Great Pyramid of Cheops at Giza
Belt Stars of Orion (Osiris) Sirius (Isis)
Narrow Shaft
Narrow Shaft
The earliest pyramid was the mastaba, a
King’s Chamber flat-topped rectangular structure of mudQueen’s brick or stone with a shaft descending to Chamber the burial chamber far below it. The original mastaba, built by Djoser has a base measuring 120 m by 108 m and a height of 60 m. Some of the earlier pyramids
Grand Gallery Rock
Lower Chamber
Descending Passage
have a step-like structure. Usually the builders start with a step pyramid and add packing of stone to form a continuous slope covered with smooth packing of limestone. The biggest of these pyramids is the one at Giza, erected by the Pharaohs Khufu, Khafre and Menkure. It measures a 230 m square base and a towering height of 146 m and is made of 2,300,000 blocks of stone; the King’s chamber in this pyramid alone measures 105 m by 5.2 m.
pyramids. Many people must have contributed. All the same, an Egyptian scholar by name Imhotep who lived around 2960 BC has been named the architect of the ‘step pyramid’ at Saqqara in Egypt; this is said to be the first major Egyptian structure in stone. In addition to being an architect, he was also said to be a good vizier, mathematician and a medical man. There are also some ancient manuscripts, which suggest that Imhotep counseled Zoser (the Pharaoh of step pyramid) on the seven-year famine in the Nile – a story that parallels the legends of Joseph in the Old Testament. Later, the Greeks identified Imhotep, calledimouthes in Greek, with Asklepios – the Greek god of medicine. This identification has made several historians doubt whether Imhotep really existed; probably he is no more real than Viswakarma of Indian epics. But the man who designed the pyramids does deserve a mention while describing the dawn of science. Moving lightly over a thousand years, we come across the next milestone – the golden age of Egyptian mathematics. There is a
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Imhotep, the man who designed the pyramids does deserve a mention while describing the dawn of science.
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The intuitive leap involved in arriving at the formula in Box 2 is considered an important achievement by historians of mathematics.
scribe, Ahmose, mentioned in a mathematical treatise of ancient Egypt. He was a copyist who lived around 1000 BC and copied a mathematical treatise dealing with simple equations, fractions and some of the rudimentary mathematical details from the earlier, scattered works of anonymous origin. Though some of the mathematical writings of Egyptians show remarkable ingenuity (see for example, Box 2), they never generalised their methods and did not bother to develop mathematics as a scientific and logical discipline. That had to wait for yet another thousand years – for Thales of Miletus (640–546 BC). He probably deserves to be called the founder of Greek science, mathematics and philosophy. Thales was born in Miletus, which is on the western coast of Turkey. He definitely traveled around and visited Egypt and Babylonia. He learnt a lot from the Babylonians (who had by then developed a science of astronomy and had worked out detailed rules for Box 2. The Moscow Papyrus
The MoscowPapyrus is an ancient Egyptian mathematical text, dating back to about 1850 BC, now kept in the museum in Moscow. It contains a discussion of 25 problems in different areas of mathematics. Of these, the discussion in problem 14 is of special importance. This problem attempts to calculate the volume of a section (frustum) of a square pyramid of base areas x and y and height h (Figure A). Both the Babylonians and the Egyptians knew the answer to the ‘corresponding’ problem in two dimensions, namely, the area of a section of a triangle with parallel sides a and b and height h (Figure B). This is given by the formula (½) h (a + b). Working by ‘analogy’, the
(A)
(B) Top area: y
Babylonians took the volume of the frustum of the pyramid to be (½) h (x + y) which is wrong. On the other hand, the Moscow Papyrus shows that the Egyptians ‘guessed’ the correct formula: (1/3) h(x + (xy) + y)! The intuitive leap involved in arriving at this formula is considered an important achievement by historians of mathematics.
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a Height: h h
Base area: x
b
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WHERE
WHEN
calculating the eclipses and other astronomical phenomena) and Figure 1. (left) put it to good use. There is a story that Thales predicted a solar Figure 2. (right) eclipse which is supposed to have occurred on the day the armies of Medes and Lydians were about to clash. The eclipse scared them into signing a treaty of peace. (Calculating back we can now know that this event must have taken place on 28 May 585 BC!) Thales also borrowed concepts from Egyptian mathematics but he went ahead and developed it in a rather axiomatic way. He had the vision to see the necessity of proof and had proceeded to develop step-by-step logic leading to conclusions from given premises. This feat is truly incredible because, as far as we know, several other major civilisations like the Egyptians, Babylonians, Indians and the Chinese didn’t seem to have worried about the rigours of logical proof, the cornerstones of modern mathematics. Thales is credited with the following five elementary theorems: (1) The circle is bisected by the diameter; (2) the base angles of an isosceles triangle are equal; (3) vertically opposite angles of two intersecting straight lines are equal; (4) two triangles are congruent if they have two angles and the corresponding side respectively equal; and (5) an angle inscribed in a semi-circle is a right
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There is a story that Thales predicted a solar eclipse which is supposed to have occurred on the day the armies of Medes and Lydians were about to clash.
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x
y z
Figure 3.
When the Greeks listed the ‘seven wise men’, Thales was always the first.
Address for Correspondence T Padmanabhan
angle. These results were known long before Thales but were accepted as true based on direct, empirical measurement. Thales’s genius lies in supporting each of these theorems by logical reasoning, starting from certain basic axioms. For example, he proved (3) in the following manner: “In Figure 3, x is the supplementary angle to y; but y is also a supplementary angle to z. Since things equal to the same thing are equal to one another, x = z.” Thales also worried about the larger questions of Nature like, “What is the Universe made of?” It is irrelevant that he didn’t get the right answer. What is important was that he asked the question and, probably, for the first time in history, attempted to answer it without invoking gods, demons and other mythological objects, which were available in plenty at the time. He influenced his contemporaries and the coming generations in no small measure. So much so that when the Greeks listed the ‘seven wise men’, Thales was always the first.
IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007 India.
Suggested Reading [1]
Email: [email protected] [email protected]
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[2]
Howard Eves, Great Moments in Mathematics Before 1650, (Dolciani Mathematical Expositions No 5), Mathematical Association of America, 1983. Isaac Asimov, Asimov’s Biographical Encyclopedia of Science and Technology, Doubleday, 1982.
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Dawn of Science 2. The Athens Factor T Padmanabhan The tradition started by Thales was ably continued by the later Greeks, making Athens and Alexandria shine as scientific capitals around the Mediterranean. Prominent among the Athenians were Pythagoras (582–497 BC), Plato (427–347 BC) and Aristotle (384–324 BC) . T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Part 1. Resonance, Vol.15, No.6, pp.498–502, 2010.
Pythagoras, born at Samos, an Aegean Island emigrated to Croton in southern Italy in 529 BC. By that time, the coasts of southern Italy and eastern Sicily had been colonised by the Greeks and had adopted the Greek way of life. Pythagoras founded a school in Croton, thereby extending the philosophic tradition of Thales – prevalent in eastern Greece – to the far west. The members of the school debated mathematics, philosophy and theology in great detail but, unfortunately, maintained a code of secrecy over the entire exercise. This earned the school the disrepute of being a mystery cult of dangerous values and brought it under active persecution even during Pythagoras’s lifetime. In fact, Pythagoras had to flee the city and live his last ten years under voluntary exile. This secrecy over their transactions had prevented later historians from judging the Pythagorean contribution in an objective manner. None of the writings by Pythagoras has survived and his contributions have to be ascertained by references made by later thinkers. The following two contributions definitely deserve to be highlighted.
Keywords Pythagoras, Kepler, Plato, Academy at Athens.
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The first one was the series of experiments that Pythagoras conducted on the production of sound by studying the notes emitted by plucking a stretched string. He realised that sounds pleasant to the human ear are invariably associated with rational steps in the scale of notes. He also related the note to the length of
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the plucked string. For example, if one string was twice as long as the other, the note it emitted was just one octave lower. These were probably the first sensible systematic experiments conducted in any branch of physics.
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Figure 1.
These results led Pythagoreans to think that the entire world can be constructed using simple, rational numbers. (So enamoured were they with this idea that they attributed several mystical properties to natural numbers!) Though such ideas were too simplistic, it opened up the study of numbers – an active branch of modern mathematics. (See Box 1.) One can easily imagine their shock when they realised that there existed numbers which were not rational; that is, numbers which could not be expressed
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Box 1. Proving The Pythagoras Theorem The ‘standard’ proof of this famous theorem is pretty complicated. But it is possible to see the validity of this result without much ado if suitable figures are drawn. The figure here illustrates a few ‘proofs without words’ of this theorem. The left one was known to the ancient Chinese; the right one was given by Bhaskara with the comment ‘Behold!’. (a)
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(b)
Figure 2.
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Pythagoras was the first man to introduce the atomistic view of matter.
as the ratio between two natural numbers. If the side of a square is one unit, then the length of the diagonal of the square represents an irrational number. Legend has it that Pythagoreans tried hard to keep the existence of such numbers a secret but the information slipped out – through some unfaithful ones! Pythagoras was also probably the first who taught that the Earth was spherical and that the orbits of the Sun, the Moon and planets were different from that of stars. He also guessed that the Morning Star (Phosphorus) and the Evening Star (Hesperus) were in fact the same; he named it Aphrodite, which we now call Venus. Pythagorean notions influenced several later thinkers like Anaxagoras (500–428 BC), who taught at Athens for nearly 30 years and tried to put Pythagorean ideas on a more rational basis, and Democritus (470–380 BC). He was the first man to introduce the atomistic view of matter.
Figure 3. Platonic solids.
The most famous among those who hailed from Athens is, however, Plato who was more a philosopher than scientist. (Incidentally, it is through him that we come to know of most of the thoughts of Socrates.) Plato did, however, have a fascination for mathematics, as the “purest form of philosophy” and tried to apply mathematical ideas to describe the heavens. He knew that there were five, and only five, regular solids. (A regular solid is the one with identical faces with all the lines and angles formed by the faces equal, see Figure 3.) These are the four-sided tetrahedron, the six-sided cube, the eight-sided octahedron, the twelve-sided dodecahedron and the twenty-sided icosahedron. Plato tried to fit the heavens in a model based on these ‘perfect’ solids. This insistence that heavens should reflect our ideas of perfection held sway in the ages to follow (see Box 2 ) . Plato established the famous Academy at Athens which influenced the philosophical thinking of people around the Mediterranean for
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Box 2. Platonic Solids And The Music Of The Sphere. The Pythagoreans and later Plato believed that the five regular polyhedra play a crucial role in nature. The ‘atoms’ of the four elements – fire, earth, air and water – were supposed to be in the shapes of a tetrahedron, cube, octahedron and icosahedron, respectively. (In the post-Aristotelian days, aether was identified with dodecahedron.) Centuries later, Kepler (1571–1630 AD) even attempted to model the orbits of the six planets known in those days, using the Platonic solids. He imagined a sphere with radius that of the orbit of Saturn, and inscribed a cube in it. He next put a sphere inside the cube, representing Jupiter’s orbit, a tetrahedron inside it and a Martian sphere, a dodecahedron inside it with the terrestrial sphere, an icosahedron with the sphere of Venus, and finally, an octahedron with a sphere inside it representing Mercury’s orbit. Interestingly enough, the radii of the
Kepler’s platonic solid solar system
orbits of the planet – calculated in this form – matched fairly well with the observed radii of orbits!
years. (In fact, it remained the stronghold of paganism in a Christian world in the coming centuries, and was ordered to be closed by Emperor Justinian in 529 AD.) Aristotle was the giant among the thinkers produced by the Academy and was, in fact, considered the “intelligence” of the Academy. Plato, however, named someone else as his successor, and Aristotle quit the school (probably) in protest! He was called to Macedon to tutor Alexander which he did for about six years. He returned to Athens later and formed his own school, the Lyceum, where he lectured for nearly 12 years. Aristotle’s lectures at this school form virtually a one-man encyclopaedia of knowledge running over 50 volumes. His best contribution was in the field of biology where he made a careful and meticulous classification of animal species and arranged over five hundred animal species in different hierarchies. His classification scheme and ideas were truly ‘modern’. For example, he classified dolphin with the beasts of the land because dolphins nourished the foetus by a placenta! (Later workers put dolphin back in the sea and it took nearly 2,000 years for biolo-
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Plato established the famous Academy at Athens which influenced the philosophical thinking of people around the Mediterranean for years.
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gists to set the record right.) He also made careful observations of the developing embryo of the chicken and the stomach structure of the cow.
Plato and Aristotle as potrayed in Raphael’s fresco ‘The School of Athens’ (1509–1510).
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus
Aristotle’s attempts in ‘natural philosophy’ – what we now call physics – do not seem to be so successful. For some strange reason, he did not use the experimental and observational approach in these attempts. He tried to explain natural phenomena using the properties of five elements (earth, water, fire, air and aether – the last of which was his own innovation), by attributing a ‘natural place’ for each element. He believed, for example, that a heavier stone will fall faster than a lighter one, and, it seems, never bothered to check it. (The fact that he was wrong is to have momentous consequences, as we shall see in a later installment.) Ironically enough, Aristotle was not as influential during his times as, for example, Plato. His works were published only after his death, and, soon after the fall of Rome were lost to Europe. These volumes, however, survived among the Arabs, who valued them dearly. Much later, in the 12th and 13th centuries, Christian Europe regained the Arabic texts and produced Latin translations. This led to Aristotle becoming the most influential ancient philosopher in medieval Europe.
Ganeshkhind Pune 411 007 India. Email:
Suggested Reading
[email protected]
[1]
Arthur Koestler, The Sleepwalkers, Penguin Books, 1959.
[email protected]
[2]
Isaac Asimov, Asimov’s Biographical Encyclopedia of Science and Technology, Doubleday, 1982.
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Dawn of Science 3. ‘Ishango Bone’ to Euclid T Padmanabhan From primitive counting and tally marks, mathematics progressed rather rapidly.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498, p.590.
Figure 1. Ishango bone.
Keywords Euclid, Ishango bone, Sulvasutras, Chiu Chang.
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Even a prehistoric tribe would have needed the notion of counting – to make sure that all the cattle came home, or that the tribe outnumbered its enemies. The primitive form of counting involved setting up a correspondence between the objects to be counted and some other convenient set of objects like, for example, the fingers in one’s hand. Even today, some primitive African hunters keep count of the number of wild boars they kill by collecting the tusks of each animal, and young girls in the Masai tribe – who live on the slopes of Mt. Kilimanjaro – wear brass rings around their necks in numbers equal to their ages. This process of counting soon evolved into a more sophisticated process of keeping ‘tally marks’ on a bone or a stone. One such bone (Ishango bone, Figure 1) belonging to the period from 9000 to 6500 BC was found at the fishing village of Ishango (on the shores of Lake Edward in Congo) in 1962. If the markings on this bone are in fact tally marks, this is probably the earliest available record of a mathematical activity. From such humble beginnings, mathematics progressed rather rapidly. Counting made one realise that ‘one cow and one cow make two cows’ just as ‘one spear and one spear make two spears’. To extract from such a concrete experience the abstract idea that ‘one and one make two’ was the first major breakthrough in mathematical thought. This idea, which appears so obvious to us today, involves thinking of ‘one’ and ‘two’ as independent abstract entities existing on their own. (This abstraction has not been achieved by some tribal societies even today. For example, Fiji tribals distinguish ten boats (called bole) from ten coconuts
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(koro) and call thousand coconuts by a separate name, saloro!) In practical terms, this idea requires words in a language to describe these numbers as separate nouns. All ancient civilisations – Egyptia n, Chinese, Babylonian and Indian developed such verbal descriptions for numbers at some stage in their development. In the early stages, words for numbers often originated from parts of the body, names of fingers, etc., and covered only small numbers. In fact, words originally existed only for 1, 2, rarely for 3 and for ‘many’. For example, Egyptian and Chinese writings often identify 3 with many. The Egyptian word for water was just the word for wave repeated three times; in Chinese, ‘forest’ is ‘3 trees’, ‘fur’ is ‘3 hair’ and – quite chauvinistically – ‘all’ is ‘3 men’ and ‘gossip’ ‘3 women’!
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Figure 2. Figure 3.
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The written forms of these ‘number words’ formed the earliest of mathematical notations. Very soon, they were condensed into forms, which were more compact and useful. The most primitive and complicated among them which has, surprisingly enough, survived till today are the Roman numerals: I, V, X, L, C, etc. The most useful, of course, are the Arabic numerals, which we will discuss fully in a later installment. Other civilisations had their own symbols (Box 1).
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Box 1. Positional Notation and Sunya. Expressing arbitrarily large numbers using a small set of symbols required great ingenuity, involving three distinct ideas: (i) The positional notation in which the value of a symbol depends on its location, for example, we interpret 23 as ‘2 tens’ and ‘3 ones’ and 32 as ‘3 tens’ and ‘2 ones’, (ii) a convenient choice for the ‘base’ in the positional notation; we normally use 10 as the base so that 467 will stand for 4 102 + 6 101 + 7 100, and (iii) a symbol for ‘nothing’ (0) which allows us to distinguish 203 from 23. Different civilisations achieved varying degrees of success in this task. The Babylonians, the Egyptians, the Chinese, and the Indians all had the idea of positional notations. The Babylonians developed, as early as 3000– 2000 BC, a positional system with a base of 60! However, to avoid having to use separate symbols for 1 and 59, they also used a grouping scheme based on 10. The Indian Kharosti numerals needed the use of three different groupings of 4, 10 and 20. The Chinese, whose language provides a separate pictographic character for each idea, used a notation which completely spells out the value of the number. The last crucial step, a symbol for nothing, came much later. The earliest known written form of zero occurs in a text inscribed on the wall of a small temple near Gwalior in AD 870. It lists, among the gifts given by the king to the temple, a tract of land “…270 royal hastas long and 187 wide, for a flower garden.” The sunya of India reached the West through the Arabs, becoming zero in the process.
Next to counting, the ancients were preoccupied with sizes and shapes, which naturally led to the development of what we now call geometry. The Greeks, especially Thales and Pythagoras, contributed significantly to the development of geometry. But there is one name, which stands out above the rest and has exerted a lasting influence over the centuries. That was Euclid of Alexandria. After the death of Alexander in 323 BC, the Macedonian Empire was divided into three, and Egypt came under the rule of Ptolemy
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Soter (whose dynasty continued for nearly 250 years ending with Cleopatra). He chose Alexandria as his capital and opened the gates of the University of Alexandria to scholars from all over, making this city the centre of academic activity for centuries. One of the scholars was Euclid1, the mathematician. Very little is known about his life; he lived around 300 BC, taught for a few years at the university and what is most important, compiled the monumental 13-part treatise, Elements. This work, which has dominated teaching of and thinking in geometry for the past two thousand years, definitely earns him a special mention in the annals of science. We do not have a copy of Elements from Euclid’s own time. All modern editions are based either on a version by Theon of Alexandria (a Greek commentator who lived 700 years after Euclid) or on an anonymous compilation found in the Vatican library. Greek commentaries on Euclid were translated by three Arabic scholars at different times in the Middle Ages (Figure 4). From the Arabic, it was translated to Latin; the first Latin translation was made in AD 1120 by Adelard of Bath (who had to travel disguised as a Muslim student in Spain to obtain an Arabic copy!). In all these versions, Elements comprises 13 books with a total of 465 theorems.
Elements, which has dominated teaching of and thinking in geometry for the past two thousand years, definitely earns him a special mention in the annals of science.
1
Resonance, Vol.12, No.4,
2007.
Figure 4. Al-Tusl’s Arabic rendering (AD 1258) of Euclid’s proof of the Pythagoras theorem.
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Box 2. Sulvasutras and Chiu Chang There are two other ancient mathematical texts of Oriental origin, which contain several interesting results. These are the Indian treatise called Sulvasutras dated by various historians, anywhere from 800 to 100 BC and the Chinese work Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art) written probably around 250 BC. The Sulvasutras contain, among other things (1) an explicit statement of Pythagoras theorem (in terms of length, breadth and diagonal of a rectangle), (2) several examples of Pythagorian triples – integers (a, b, c) satisfying the relation a2 + b2 = c2, and (3) construction for producing a square equal in area to a rectangle (a problem, which incidentally, arises in making a falcon-shaped altar for sacrifices!) What is probably more important than these results is a discussion in Apastamba Sulvasutra of the area of a trapezium with bases 24 and 30 units and width 36 units. The text not only calculates the area correctly, but also gives a purely geometrical (Euclid-like) proof for the result! The Chinese work Nine Chapters deals with a gamut of problems in elementary mathematics: operation of fractions, measurement of areas of rectangles, trapezia, triangles (which are done correctly), circle, circle segments and sectors (which are done approximately with pi taken as 3!), volumes of elementary solids (including the frustum of the pyramid), extraction of square and cube roots, and a system of linear equations.
Euclid enlarged upon the work by Theudius, Exodus and Hippocrates of Cos – all of whom had contributed to various aspects of geometry and number theory.
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Book 1 begins with the basic axioms and develops the theorems on the congruence of triangles, parallel lines and rectilinear figures. (Theorem 47, for example, is the Pythagoras theorem.) Book 2 deals with the algebraic results arising out of Pythagorean constructions while Book 3 deals with standard results on circles, tangents and secants. Books 4, 5 and 6 discuss geometrical constructions and similarity of figures and the last three books (11, 12 and 13) contain theorems on solid geometry. Probably the most remarkable and the least known volumes are Books 7 to 10. These discuss, not pure geometry, but elementary number theory! They contain some of the most fundamental results in this subject. This compilation, of course, was based on the earlier works of several people. Before Euclid, a compilation by Theudius was used at Alexandria. Euclid enlarged upon this work, drawing considerably on material developed by Theudius, Exodus and Hippocrates of Cos – all of whom had contributed to various aspects of geometry and number theory. In fact, historians have
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sometimes cynically commented that Euclid’s exposition excelled only in those parts in which he had excellent sources at his disposal. Even so, Elements is a notable achievement. The pattern of logic and order in this treatise was wholly due to Euclid, and this in itself was an outstanding contribution to the evolution of the subject. Euclid might not have been a first-rate mathematician but he was a first-rate teacher of mathematics, for his book remained in use, practically unchanged, for nearly 2,000 years!
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus
Suggested Reading
Ganeshkhind Pune 411 007 India.
[1]
Howard Eves, Great Moments in Mathematics Before 1650 (Dolciani
Email:
Mathematical Expositions No 5), Mathematical Association of America,
[email protected]
1983. [2]
[email protected]
Georges Ifrah, The Universal History of Numbers - I, Penguin, 2005.
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Dawn of Science 4. Archimedes T Padmanabhan His work ranged from optics, mechanics and hydrodynamics to pure mathematics.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity. Previous parts: Res onance, Vol.15: p.498; p.590; 684.
Figure 1. Tombstone of Archimedes. Courtesy: http://www.math.s unys b.edu/ ~ t on y/ wh at s n ew / c ol um n / ar c h i m ed es - 0 1 0 0 / archimedes3.html Keywords Archimedes, volume of sphere, lever, hero of Alexandria, value of pi.
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In 1965, while excavating for laying the foundation of a new hotel in Syracuse (Sicily), Italy, the steam-shovel unearthed a tombstone bearing the picture shown in Figure 1. This was the tombstone of Archimedes, one of the greatest scientists who ever lived and of whose equal there have been only two since then, Newton and Einstein. It is rather ironic that such a scientist did not come from the centre of intellectual activity at the time – Alexandria (see ‘Dawn of Science 3’, Resonance, August 2010, p.684). He was born in Syracuse, about 287 BC. His father was an astronomer of considerable talent and repute. However, Archimedes did spend some time in Alexandria training under Euclid’s students. He returned to his native town, possibly because of his close friendship with the King of Syracuse, Hieron II. Archimedes thrived in Syracuse. No other scientist of ancient times, not even Thales, had so many tales and legends told about him; the most famous, of course, is the one about ‘Eureka’ and the principle of buoyancy. Archimedes was responsible for several inventions and discoveries in the branches of mechanics, hydrodynamics, optics and in pure mathematics, the details of which are spelt out in nine Greek treatises which are available to us. He worked out the principle of lever in clear mathematical terms (even though several others before him had conjectured about it). “A small weight at larger distance from a fulcrum can balance a large weight nearer to the fulcrum.” This led to the science of statics and to the notion of centre of gravity of bodies. Two of the volumes by Archimedes – On the Equilibrium of Planes and On
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Floating Bodies – elaborate on the implications of these concepts. In these volumes, Archimedes spends considerable time establishing the position of equilibrium of floating bodies of various shapes; the results are of considerable importance in naval architecture. Archimedes used these principles in several practical devices. He is supposed to have perfected a hollow helical cylinder which, when rotated, served as a water pump (Figure 4). He also devised a heavenly globe and a model planetarium depicting the motion of the planets. His engineering tradition influenced several people (see Box 1).
Figure 2. (left) Figure 3. (right). Courtesy: www.boglewood.com/ sicily/vassal.html
Figure 4. Archimedes’ water pump. Courtesy: http://picsdigger.com/domain/ decodingtheheavens.com/
He was, however, a purist and did not really care too much for these applications. What he was most pleased with were the results he could obtain in the branch of pure mathematics – in the determination of areas and volumes of geometrical shapes. He came up with ingenious arguments – described in the treatises – On Sphere and Cylinder and Method – to show that the volume of a sphere is 4R3/3 and its surface area
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Box 1. Other Mechanical Inventions The engineering tradition of Archimedes influenced several contemporaries and future generations. One among them who lived in the second century BC was Ctesibius who is responsible for devising a sensible water clock. He improved upon the more ancient Egyptian ‘clepsydra’, in which water dripping into a container at a steady rate made a pointer move, indicating time. Ctesibius made the whole device practical, compact and accurate. In fact, his clock was as accurate as the timepieces of the Middle Ages run by falling weights. It was only after the invention of the pendulum that the accuracy of measuring time was improved, After about 120 BC Ptolemic Egypt fell into decadence and by 30 BC it was a Roman Province. Greek science was virtually over but for an occasional genius like Hero of Alexandria. His most famous invention was a hollow sphere with two tubes attached to it in which water could be boiled to make steam. The steam escaping through the tubes made the whole device spin. This was the first steam engine, though unfortunately it was only used in toys and by priests to deceive gullible believers. Hero also wrote extensively on mechanics elaborating on the principle of the lever and several simple machines, involving inclined planes, pulleys and levers. Almost at the same time as Hero was devising the levers, an unknown Chinese named Tsai Lun, made a breakthrough in China. Chinese historians credit him for inventing the product we now call ‘paper’ from tree bark and rags. This event took place around AD 105 and in the coming centuries, paper-making spread westwards. Baghdad had this technique by AD 800 and Europe inherited it after the Crusades (after the 13th century). In the nineteen centuries since Tsai Lun, this invention is yet to be improved upon!
4R2. To obtain these results he had to use the notion of a solid being made up of a large number of extremely small pieces. If only he had used more compact and consistent notations, he would have discovered integral calculus! Another of his contributions was in the development of a technique for the computation of which was used by several later workers as well (see Box 2).
If only Archimedes had used more compact and consistent notations, he would have discovered integral calculus!
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Archimedes could not, unfortunately, end his life in peace. The king of Syracuse, Hieron II, had a treaty of alliance with Rome. After his death, his grandson Hieronymus ruled Syracuse. During his reign Rome suffered a disastrous defeat by Carthage and seemed to be quite lost. Hieronymus, misjudging the situation, switched loyalties to the winning side, Carthage. The Romans did not like this a bit, and once they recovered, they set a fleet commanded by Marcellus, thereby laying siege to Syracuse. This started the strange three-year war between the mighty Roman fleet and virtually a single man – Archimedes. The mechanical
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Box 2. A Piece of Pi Every ancient civilisation, which built anything of significance, needed to know the length of the perimeter of a circle of a given diameter. They all knew that the ratio between the circumference and the diameter was a constant, roughly around 3. The question was to determine it exactly. Some civilisations (like the ancient Hebrews) were happy with a value of 3; while others had gone for more accurate values; the Egyptians, for instance, used 22/7 and the Chinese had the value 355/133. It was Archimedes who devised a systematic method, which allowed one to compute the value of to any desired accuracy. His idea was to inscribe and superscribe polygons around a circle and measure the perimeters of these polygons. When the number of sides of the polygon increased, the circle got crowded between the polygons (see Figure A) and the perimeters of the polygons offered a good approximation to the perimeter of the circle. With tremendous patience, Archimedes used polygons of 96 sides and got for the value of (3123/994) = 3.14185… which is off the correct value by
Figure A
only 1 part in 12,500!
inventions (Figure 5) Archimedes is supposed to have used in this war probably constituted the first massive application of superior technological knowledge in warfare. He is said to have constructed large mirrors and lenses to set Roman ships on fire and mechanical cranes to lift ships from the sea. (These details come from the description of Marcellus in the works of Plutarch, whose bias in favour of Greeks should not probably be overlooked!) The
Figure 5. A 17th century engraving shows the arrangement Archimedes is supposed to have used to burn Roman ships. Courtesy: www.corbisimages.com
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Figure 6 (left). Archimedes used his discovery that different objects with the same density and weight displace equal amounts of fluid to test the purity of the gold crown of the King of Syracuse. Courtesy:
http://neatorama.cachefly.net/images/2008-05/archimedes-eureka.jpg
Figure 7 (right). Death of Archimedes at the hands of a Roman soldier (a sixteenth century mosaic) Courtesy:
http://www.livius.org/a/1/greeks/archimedes_circles.JPG
city fell after three years, though, and Archimedes was killed (212 BC) by a Roman soldier – apparently much to the disappointment of Marcellus. It is said that Marcellus arranged a proper funeral with a tombstone as desired by Archimedes.
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind
Our knowledge about Archimedes and his times increased significantly in 2003 when historians of mathematics discovered long lost information in the form of an ancient parchment overwritten by monks nearly a thousand years ago (see [1]). This gives us information, among other things, about a curious puzzle called stomachion which involves fairly advanced concepts from combinatorics. The goal of the stomachion is to determine in how many ways a particular set of 14 pieces of varied planar figures can be put together to form a square. In 2003, mathematicians found that the answer is 17152!
Pune 411 007 India. Email:
Suggested Reading
[email protected] [email protected]
[1] R Netz and W Noel, The Archimedes Codex, Weidenfeld & Nicolson, 2007. [2] Petr Beckmann, A History of Pi, St. Martin’s Griffin, 1976.
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Dawn of Science 5.The Healing Art T Padmanabhan It was in the field of surgery that ancient Indian medical practice achieved the most.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498; p.590; p.684; p.774.
Keywords
Primitive tribes used a variety of plants and plant products for food and realised that some of these are poisonous and some curative. Those in the tribe who were quick to appreciate these effects could assert considerable power over others. These were the earliest among ‘witch doctors’. Healing soon got inextricably mixed with magical practices and with gods and demons. And medicine was in the hands of godmen and witch doctors as it is, to some extent, even today. To retain their special status, it was also necessary for them to shroud the details of their practices in magic and mystery. It is, therefore, rather surprising that the science of medicine developed to a high degree in two of the ancient civilisations – Indian and Greek – in spite of their abounding supernatural beliefs. The earliest concepts of Indian medicine are presented in one of the four Vedas, the Atharvaveda, which probably dates back to 2000 BC. Several diseases like fever, cough, diarrhoea, abscesses, etc., and their herbal remedies are mentioned in the Vedas. Unfortunately, the cures are mixed up with spurious magical practices to allow an objective evaluation. The golden age of Indian medicine, however, occurred in the post-vedic period, during 800 BC to AD 100. Two major medical treatises, Charaka Samhita and Sushruta Samhita, appeared during this period. These texts discuss several aspects of medicine: symptoms, diagnosis and classification of diseases, preparation of medicines from plants, diet and care of patients, etc.
Surgery, Charaka Samhita, Hippocrates, acupuncture, Galen.
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Indian medicine believed that diseases are caused by the imbalance among three vital entities acting in the body: air (vayu or
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Figure 1.
Figure 2.
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Figure 3. A noted physician in India, Sushruta, was known for a range of writings. Courtesy:http://www.blatner. c om /ad am / c on s c t r an s f / historyofmedicine/1-overview/ brief.html
vata), phlegm (kapha) and bile (pitta). The seven constituents of the body – blood, flesh, fat, bone, marrow, chyle end semen – are supposed to be produced by the action of these three entities. Most of the cures involved restoring the balance between these three entities by dietary and herbal means. Charaka lists 500 medicinal plants while Sushruta has a more extensive list of 760. In addition, several animal products and minerals were used. The Indian medical man could administer emetics, purgatives, enemas and also sneezing powders and herbal fumes. But it was in the field of surgery that Indian medicine achieved the most. By about AD 100, several surgical procedures were known and practised; these included excision of tumours, incision of abscesses, removal of fluids from parts of the body, probing of fistulas and stitching of open wounds. The classical texts give very detailed instructions about these operations and about the choice of instruments. Sushruta, for instance, describes 20 sharp and about 100 blunt instruments – knives of different kinds, scissors, trocars (for piercing tissues), saws, needles, forceps, levers and hooks. Most of the instruments were made of steel and the operations seem to have been performed using alcohol as an anaesthetic. Almost around the same time, medical art was thriving in Greece as well. Hippocrates (460 BC–370 BC) seems to be the first person to state categorically that diseases are due to natural causes and not curses from Gods. Very little is known about his life and work; historians think that most of the works attributed to
Figure 4. Hippocrates. Courtesy: http://www.blatner. c om /ad am / c on s c t r an s f / historyofmedicine/1-overview/ brief.html
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him could have been written by several other people who lived much later. (Since Hippocratas’ name carried much weight in Greece, people probably preferred to attribute their own ideas to Hippocrates.) Whatever the truth, the books that make up the collection Corpus Hippocraticum has earned him the title the ‘father of modern medicine’. Here he describes the symptoms and the courses of illnesses clearly and concisely. Hippocrates repeatedly emphasized the natural cause for illnesses and sought to cure them in a methodical way. He laid much stress on the effects of diet, occupation, climate and environment on health. His greatest legacy perhaps was the code of conduct for medical practitioners, the ‘Hippocratic Oath’, which is still used at the time of medical graduation. The first, formal medical school was established in Alexandria around 300 BC and thrived under the Greek anatomist, Herophilus. He was the first to dissect human bodies in public. (In the preChristian era, there was no taboo on dissection and the Greeks took advantage of it. In contrast ancient Indians desisted from cutting open bodies. Sushruta Samhita recommends an elaborate procedure for soaking a body in water so that parts can be removed without cutting!) Herophilus gave particularly detailed descriptions of the brain, parts of the eye, ovaries and uterus; he named the retina, the duodenum and the prostate gland. These investigations were continued by Erasistratus (304–250 BC) who also taught at Alexandria. By now medical science had detailed knowledge of the various organs of the body though it had very little idea of its functions.
Figure 5. Galen’s work around 140 AD in Rome ended up being authoritative in Europe until the 16th century! Here he is using the technique of ‘cupping’, creating small vacuums in heated cups to ‘draw the poisons out’. This technique continued in folk culture through the early 20th century. Courtesy: http://www.blatner. c om/adam/c ons ctransf / historyofmedicine/1-overview/brief.html
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Box 1. Chinese Medicine The Chinese system of medicine also has great antiquity. Several medical texts originated in China during the period 500 BC to AD 300. The most famous amongst them were the compilations Nei Ching and Mo Ching. Traditional medicine in China is based on the theory of Yin and Yang. Illnesses are supposed to be caused by the imbalance between these two active principles in the body. However, the Chinese understanding of anatomy was primitive – again because of the religious restrictions on the dissection of the body – but they made good progress in diagnostics. Mo Ching, for example, describes detailed rules for the interpretation of the pulse, which is to be measured not only at the wrist but also certain other parts of the body! The mysterious and typically Chinese concept in medicine is acupuncture. This consists of insertion of hot or cold metal needles into the body; the needles may range in lengths from 3 to 25 cm. These needles provide the cure, the practitioners claimed, by changing the distribution of Yin and Yang in the body and restoring the balance between them. Acupuncture dates from before 2500 BC and is strangely different from the widely practised concepts of modern (Western) medicine.
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4
Soon after Erasistratus, the study of anatomy declined because of religious objections to the dissection of the human body. Later workers, notably Galen (AD 130–200), had to rely on animal dissections to understand anatomy. In spite of such constraints, Galen could make progress. He noticed that arteries carried blood, which was set into a rhythmic motion by the pounding of the heart. He used the pulse as a diagnostic test but narrowly missed discovering the circulation of blood!
Pune University Campus Ganeshkhind
Suggested Reading
Pune 411 007 India. Email:
[1]
http://dodd.cmcvellore.ac.in/hom/thumbnails.html
[email protected] [email protected]
There are several articles on the history of medicine at
[2]
Isaac Asimov, Asimov’s Biographical Encyclopedia of Science and Technology, Doubleday, 1982.
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Dawn of Science 6.The Arab Legacy T Padmanabhan But for the phenomenal rise of the Arab civilisation, the earlier Greek scientific traditions could have been lost forever. In its declining years in the third century AD, the Roman empire was divided into two. Of these, the eastern part flourished and emerged as the Byzantine Empire in the course of time; but the western part drifted till it fell to barbarian conquests by the end of the fifth century. And with that began years of instability and anarchy in western Europe. The stage was set for the world to lose forever the earlier scientific contributions of the Greeks. And indeed, this is what would have happened – but for the phenomenal rise of the Arab civilization. In the years following the death of Prophet Mohammed in AD 632, the Arabs conquered Asia Minor, Persia, North Africa and Spain. From the ninth to the eleventh centuries, the Arab civilisation was the dominant force around the Mediterranean with Cordoba, Baghdad, Damascus and Samarkhand emerging as centres of learning and culture. The streets of Cordoba were paved and lit by lamps – amenities which were not available in London and Paris for another seven centuries! The Arabs absorbed every key idea of Greek science and added to it what they learnt from Persia, India and China. The Caliphs in Baghdad encouraged translations of every major scientific work into Arabic. The ‘western’ science of the later centuries originated from the study of these Arabic texts, retranslated into Latin. This Arabic reservoir of knowledge covered virtually all branches of science – medicine, chemistry, astronomy, mathematics and physics.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498; p.590; p.684; p.774. p.870.
Keywords Alchemy, paper-making, gunpowder, Almagest, Rhazes , Avicenna.
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Figure 1.
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In medicine, two names stand out from the Arabic world – Rhazes (Al-Razi) (850–923) and Avicenna (979–1037). Rhazes was born near Teheran and studied medicine in Baghdad. In addition to contributing his share of translation, he also wrote a voluminous treatise on medicine, Kitab-al-hawi, covering the subject in its entirety. Rhazes was also an excellent chemist; he was the first person to produce what we now call ‘plaster of paris’ and used it for setting broken bones. Avicenna, son of a wealthy tax collector, was a child prodigy who also had the advantage of the best education money could buy at the time. During his career, he wrote more than a hundred books, many of which were on medicine. His works were so authoritative that the Latin translations of these books were used in Europe until as late as 1650!
Figure 3 (left). European depiction of the Persian doctor Al-Razi (Rhazes) . Courtesy.: Wikipedia
Figure 4 (right). Colophon of Razi’s Book of Medicine. Courtesy: Wikipedia
Figure 5. Statue ofAvicenna in Dushanbe, Tajikistan. Courtesy: Wikipedia
Another important Arabic contribution to medicine was in the preparation of drugs. Many drugs which we now use are of Arab origin and so also are processes like distillation and sublimation. The Arabs, in collaboration with Nestorian Christians (an Eastern Church which was not affiliated with Constantinople), established a major hospital at Jundi Shahpur (in south-west Persia)
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which combined under one roof treatment of patients, medical education and translation of medical texts. The branch of science which we now call chemistry was very weak in ancient Greece and also in the Arab world. The origins of this discipline were to be found in the ancient Egyptian science called khem, which was used to preserve mummies. Very little is known of the original Greek and Egyptian works on the subject. Most of the information comes from a 28-volume encyclopaedia written by the Greek scientist, Zosimus (around AD 300); the contents of these volumes are unfortunately soaked in deep mysticism. Figure 6. An Arabic copy of Avicenna’s ‘Canon of Medicine’, dated 1593. Courtesy: Wikipedia
The primary occupation of most of the early and medieval alchemists (al is the Arab word for ‘the’; the word ‘chem’ arises from Egyptian khem) was to discover a method to convert base metals into gold. Medieval history is full of names of scientists who performed valuable experiments and made accurate observations, but who also wasted a lot of their time looking for a mysterious substance, al-iksir, which would transmute base metals into gold (al-iksir later became ‘elixir’ in Latin). One such scientist was Geber, aka Abu Musa Jabir ibn Hayyan (721–815), who successfully produced several chemical compounds like white lead, ammonium chloride, nitric acid and acetic acid but was (probably) really looking for the al-iksir! Unfortunately, later alchemists followed Geber’s mistaken theories into wilder morasses while ignoring his really important contributions. For all that, and amusingly, the Arabs excelled in practical ‘everyday’ chemistry. They were, for example, pioneers
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in making perfumes. In fact, perfume-making was a household art in upper class Arab families.
Figure 7. Paper making in ancient China. Courtesy:
The Arabs were also instrumental in transmitting two major innovations in chemical technology from the East to the West – the act of making paper and gunpowder. Paper was known in China as early as first century AD (see Part 4 of this series). The Arabs captured some of the Chinese craftsmen in the battle of Samarkhand (AD 751), and learnt the art from them. In another 50 years, the first Arab paper-makers had started their trade in Baghdad.
http://www. islamicspain.tv/ Islamic-Spain/Making_Paper. gif and http://www.chinatourphoto.com/ archives/4
The penetration of paper making into Europe was a slow process taking another four centuries. The case of gunpowder is more unclear. Invented by the Chinese, it was primarily used for ornamental fireworks. Arabs learnt this technique probably as early as AD 800, but full-scale use of gun powder in European weaponry probably started only in the 13th century.
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Figure 8. Earliest known written formula for gunpowder, from the Chinese Wujing Zongyao of 1044 AD. Courtesy: Wikipedia
Figure 9. Plotemy’s universe.
The Arabs also succeeded in preserving and refining earlier Greek works in astronomy, especially that of Ptolemy (AD 127– 151). Ptolemy drew extensively on the work of the previous Greek astronomer, Hipparchus (190–120 BC), and developed – what we now call – the Ptolemaic system of the universe. This scheme has the Earth at the centre of the universe with the Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn revolving around it. Ptolemy had improved on the original rules laid down by Hipparchus (‘epicycles’) and could compute the position of celestial objects with reasonable accuracy. (Ptolemy also made interesting contributions to trigonometry which we will discuss later.) Ptolemy wrote a detailed book, admiringly called by others, Megiste Mathematike Syntaxis (‘Great Mathematical Compositions’). The Arabs translated this as Almagest. This work, re-translated into Latin in 1175, dominated European astronomy throughout the Renaissance. One of the Arabian astronomers to study Ptolemy in detail was Albategnius (850–929). He could also improve Ptolemy’s work on a few points. He noticed, for instance that the location at which the Sun appeared the smallest – when the Earth is farthest from the Sun
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Box 1. Greek Fire Though chemistry was never a strong point of the Greeks, they did show remarkable ingenuity in this field in times of need. One such invention was the chemical mixture called ‘Greek fire’, attributed to the 7th century alchemist Callinicus of Greek origin. He is supposed to have fled Syria to Constantinople ahead of conquering Arabian armies, and here he invented the Greek fire to fight the Arabs. This mixture (probably) consisted of some inflammable petroleum compound, potassium nitrate to supply oxygen, and quick lime to supply further heat on reaction with water. It burned vigorously on water and hence could be used to destroy ships made mainly of wood. The Greeks of the Byzantine Empire used this in AD 673 to repel the Arab naval onslaught on Constantinople. It is possible that, but for this ‘surprise’ weapon, the Arabs would have taken Constantinople; and history would have been radically different.
Figure A. Depiction of Greek Fire in the Madrid Skylitzes manuscript. Courtesy: http://www.mlahanas.de/Greeks/Medieval/GreekFire.html
– is not fixed but moved slowly. He could also obtain more accurate values for the length of the year, the precise time of the equinox and the inclination of the Earth’s axis to the plane of revolution. It was the figure obtained by him for the length of the year which was used in the Gregorian reform of the Julian calendar seven centuries later. In the next issue, we will look at the Arab world’s contribution to the transmission and preservation of mathematics.
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4
Suggested Reading
Pune University Campus Ganeshkhind
[1]
Howard Eves, Great Moments in Mathematics before 1650 (Dolciani Mathematical Expositions No.5) Mathematical Association of America, 1983.
[2]
Isaac Asimov, Asimov’s Biographical Encyclopedia of Science and Tech-
Pune 411 007 India. Email: [email protected] [email protected]
nology, Doubleday, 1982.
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Dawn of Science 7. The Indo–Arabic Numerals T Padmanabhan It was traders who tilted the scale in favour of the new number system in the Arab world.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498; p.590; p.684; p.774. p.870, p.1009.
Keywords
In AD 773, at the height of Arab splendour, there appeared at the court of the Caliph, Al-Mansur, in Baghdad a man from distant India. This traveller had brought with him several volumes of writings from India. Al-Mansur promptly got them translated into Arabic and later several Arabic scholars assimilated their contents. One among them was Abu Jafar Mohammed ibn Musa AlKhowarizmi (which, freely translated, means ‘Mohammed, the father of Jafar and the son of Musa, the Khowarizmian’, the last word originating from the Persian province of Khoresem). This man, who lived between AD 780 and 850, was one of the greatest mathematicians of the Arab world and he quickly realised the importance of the number system used in the Indian writings. In fact, he wrote a small book explaining the use of these numerals around AD 820. The original of this book is lost but there is evidence to suggest that it reached Spain in about AD 1100; there it was translated into Latin by an Englishman, Robert of Chester. And this translation is probably the earliest known introduction of Indian numerals to the West. This manuscript begins with the words, Dixit Algoritmi: laudes deo rectori nostro atque defensori dicamus dignas (‘Algoritmi has spoken; praise be to God, our Lord and our Defender’, the Arab name Al-Khowarirmi having been transliterated into Algoritmi in Latin). In later years, careless readers of the book started attributing the calculational procedures described in the book to Algoritmi; that is how we got the term ‘algorithm’ for any computational procedure.
Indo–Arabic numberls , Al– Khowarizmi, Brahmagupta.
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Al-Khowsrizmi discusses in a systematic manner – among other things – the solution of algebraic equations up to the quadratic order.
The use of Indian numerals was picked up by many scholars and was taught in major cities. In particular, the use of zero became well established in these discussions. Al-Khowarizmi himself says explicitly: ‘When nothing remains...put down a small circle so that the place be not empty... and the number of places is not diminished and one number is mistaken for the other.’ However, the new system was not accepted by the average man easily; ultimately, what tilted the balance in its favour were not scholarly expositions but commercial considerations! For by the end of the first millennium, Italy had grown to be a major mercantile power around the Mediterranean. Italian ships were used for crusades, Italian bankers lent the money, and Venice, Genoa and Pisa rose as cities of prominence. The traders and merchants very quickly realised the advantages of the Indo–Arabic number system. Blessed by big business, the system stayed. For example, the Margarita Philosophica (the philosophic pearl), a beautifully illustrated encyclopaedia which was widely used as a university textbook in the early sixteenth century, authored by the monk Gregor Reisch (c.1467–1525) discusses arithmetic using Indo–Arabic numerals compared to the use of a counting board (Figures 3 and 4). Al-Khowarizimi also wrote another influential work called Aljabr-wa’l Muquabala, (which could be translated as ‘The science
Figure 3. The genealogy of modern numerals. Courtesy: V F Turchin, The Phenomenon of Science.
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of transposition and cancellation’). Here, he gives a detailed exposition of the fundamentals of the subject, which has come to be called ‘alegbra’. Al-Khowarizmi discusses in a systematic manner – among other things – the solution of algebraic equations up to the quadratic order. The clarity of discussion in this book has made later workers call Al-Khowarizmi the ‘father of algebra’. The synthesising power of Arabic civilisation also influenced trigonometry. This subject, well developed in both India and Greece due to the stimulus given by astronomical observations, attained a unified look in the hands of the Arabs. In Greece, it was developed by Aristarchus (310–230 BC), Hipparchus (around 140 BC) and most notably by Cladius Ptolemy (AD 85–165). In particular, Ptolemy constructed what he called a “table of chords” which is equivalent to the modern trigonometric table for the sine of an angle. He did this by using a very elegant geometrical procedure for all angles at half-degree intervals. This work, of course, was developed further by the Arabs. Abul-Wefa (AD 940–998), for example, produced the tables for sines and tangents at quarterdegree intervals; this table was used extensively by later scholars. Similar tables were constructed in the East by Aryabhata. Incidentally, there is an interesting story behind the term ‘sine’. In trigonometry, one associates with each angle certain ratios usually called sine, tangent and secant (three other ratios, cosine, cotangent and cosecant, arise as complements of these three ratios). Of these three, the terms ‘tangent’ and ‘secant’ have clear geometrical meaning and correspond to the standard definition in Euclidean geometry (see Figure 5). How did the word ‘sine’ originate? Surprisingly enough, it came from the Sanskrit term jya-ardha (‘half of chord’)! This is how it happened.
Figure 4. The title page of Gregor Reisch’s Margarita Philosophica (1503). The seven ‘liberal arts’ are around the three-headed figure in the centre with arithmetica, with a counting board, seated in the middle. Courtesy: Freiburg: Johann. Schott, 1503 [Rare Books Collection B765.R3 M2].
The Indian mathematician, Aryabhata (AD 475–550), used the term jya-ardha to denote what we now call sine. This term, abbreviated as jya, was converted phonetically as jiba by the Arabs. Following the standard Arabic practice of dropping the
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Box 1. Algebraic Symbolism The earliest discussions in algebra, both in the East and the West, were rhetorical. Questions and answers were given in the form of dialogues or discussions and no symbols were used. The first two mathematicians to realise the powers of symbolic manipulations were Diophantus in Greece (AD 250 ?) and Brahmagupta (AD 700 ?) in India. Diophantus had symbols to denote unknown quantities, various powers of an unknown quantity, reciprocals and equality. He also used Greek letters to denote numerals. The system followed by Brahmagupta was more elaborate. Addition was indicated by just placing the terms next to each other, subtraction by placing a dot over the term to be deducted, multiplication by writing bha (the first letter of bhavitha, the product), and square root by the prefix ka (from the word karana). The first unknown in the problem is denoted by ya and additional unknowns were indicated by the initial syllables of various colours. The various mathematical symbols we use today came into existence over the centuries. The ‘equal’ sign
Figure A. A page from Robert Recorde’s The Whetstone of Witte (1557), showing the much longer equality sign. Courtesty: http://www-groups. dcs.st-and.ac.uk/~history/Bookpages/ Recorde4.jpeg
(=) was due to Robert Recorde (Figure A). The ‘plus’ and ‘minus’ signs first appeared in print in an arithmetic text by John Widman published in 1489. The signs for multiplication and proportion were due to William Oughtred (1574–1660). It was Déscartés who introduced the present compact notation with indices a, a2, a3, etc. Finally, as a ratio between the circumference and the diameter of a circle was first used by the English writer, William Jones, in 1706.
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Figure 5. Geometrical meaning of trignometric ratios. In a circle of radius 1 unit, draw a tangent BE and secant BA from the external point B. The lengths of AB and BE give, respectively, the value of ‘tangent ’ (or tan ) and ‘secant ’ (or sec ). Similarly, the length of half-chord (‘jya-ardha’, in Sanskrit) CD is equal to sine .
vowels in the written version. this became just jb. Of course, the term jiba has no meaning in Arabic except in this technical context. Later writers, coming across jb as a shortened version for jiba (which appeared meaningless to them), decided to ‘correct’ it to jaib which is an Arabic word meaning ‘cove’ or ‘bay’. Still later, Gherardo of Cremona, while translating technical terms from Arabic to Latin, literally translated jaib to the Latin equivalent sinus. This, in English, became ‘sine’. That is how a cavity in our upper nose and a trigonometric ratio ended up having the same roots.
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4
Suggested Reading
Pune University Campus Ganeshkhind
[1]
V F Turchin, The Phenomenon of Science, Columbia University Press, 1977. Also available at: http://pespmc1.vub.ac.be/pos/default.html
[2]
G Ifrah, The Universal History of Numbers, Penguin, 2005.
[3]
Howard Eves, Great Moments in Mathematics, Vol. I, Mathematical
Pune 411 007 India. Email: [email protected] [email protected]
Association of America, 1983.
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Dawn of Science 8. The Printed Page T Padmanabhan The invention of printing spread literacy and spurred the Scientific Revolution.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498; p.590; p.684; p.774. p.870, p.1009; p.1062.
Keywords Gutenberg, printing, movable types, compass.
6
The first four centuries of this millennium, the period 1000–1400, witnessed instability and turmoil in much of Asia and Europe. The Arabs started fighting among themselves and their empire began to break up. From the remnants of the Western Empire, several smaller states sprang up in Europe but they were never at peace with each other; they were also bent on carrying on the ‘holy’ crusades against the Turks, often for political ends. This was also the period (about 1220–1250) which saw the rise of the Mongol power in Asia under Chengis Khan and his descendants who destroyed the cities of Bokhara, Samarkhand and Baghdad. To cap it all, there was an outbreak of the ‘Great Plague’ epidemic (around 1348) which spread through Europe, North Africa, Russia and even parts of China, wiping out entire populations. This was certainly not an atmosphere in which science could grow, and indeed science suffered. However, there came up at the turn of the 15th century an invention, which totally transformed the history of mankind. It was the invention of the movable printing process by Johannes Gutenberg (1398–1468) somewhere around the year 1430. This single invention exerted more influence on the Scientific Revolution than all the scholarly expositions of several medieval scientists. For all this, the story of the printing press is a tragic one, as far as the inventor is concerned. Gutenberg was the son of a patrician in the city of Mainz (now in Germany). There he was associated with the goldsmith’s guild and learnt several skills in metal work. Unfortunately, his colleagues who were envious of his growing skills and prosperity managed to get him exiled from Mainz.
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Books had been printed as early as AD 764 in Japan and China using primitive techniques – the text was cut out on wood and, by spreading ink on it the woodcut was transferred to paper.
Texts used to be reproduced mechanically much before Gutenberg’s time, but in a really primitive form. As early as the second century AD, the Chinese had the three essential elements of printing: paper, the manufacture of which was known to them; ink, the basic formula of which was known for centuries; and surfaces, on which letters and texts could be engraved conveniently. All the ancient civilisations had used some form of seals and insignia, which constituted the earliest form of printing. In fact, books had been printed as early as AD 764 in Japan and China using primitive techniques – the text was cut out on wood and, by spreading ink on it the woodcut was transferred to paper. But this process was so laborious that it never caught on in the Arab world or in Western Europe. In Gutenberg’s time, books in Europe were essentially reproduced by copying them laboriously by hand. This meant that books were few and expensive and only monasteries, universities and very rich people could possess them. What is more, every copy introduced the chances of errors creeping in which is unthinkable in the case of religious texts. (Jewish copyists of the Bible took the elaborate precaution of counting the total number of letters at the end of copying!)
Figure 3. Gutenberg’s printing press.
Gutenberg’s genius was in realising that by producing a series of small and durable metallic seals, each representing a single letter, printing could be made much more efficient. The letters can be assembled to form a page and once printed the page can be broken down and reassembled to make up the next page. So here was a possibility of printing unlimited copies of books using the same basic set of printer’s seals. Though the idea was simple, its execution was not. To be successful, Gutenberg needed to develop techniques, which would allow him form tiny letters out of metallic pieces with uniform quality. It was also necessary to produce good quality ink. All this took Gutenberg nearly 20 years.
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During these years, however, Gutenberg got involved in several lawsuits with his partners and financiers and unfortunately lost most of them. Gutenberg was naturally anxious to keep his project a secret to prevent others from cashing in on his idea. But the legal proceedings brought out into the open the nature of the project which he was working on and many of his financiers were quick to pounce on the possible profits. Notable among them was Johann Fust who won a crucial lawsuit against Gutenberg around 1450. As a result of this judgement – in which the court decided that Gutenberg had no reasonable means of paying back his creditors – Gutenberg had to hand over his entire printing press and the tools to Fust.
Gutenberg Bible is widely considered to be a commendable work of art and is one of the most expensive books in the world today.
This was just around the time when Gutenberg was getting ready for the printing of a beautiful version of the Bible – now well known as the Gutenberg Bible – printed in double columns with 42 lines in each page in Latin. It is widely considered to be a commendable work of art and is one of the most expensive books
Figure 4. A page from the Gutenberg Bible. Courtesy: Museo della Stampa, Genova.
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Box 1.
Adapted from: http://www.emersonkent.com/ map_archive/europe_ universities.htm
A printed page from the Chaucer.
10
Courtesy: http://www.britannica.com/EBchecked/topic-art/611830/3199
Figure 5. Medieval schools and universities in Western Europe.
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Box 2. The Mariner’s Compass Talking about inventions which changed the world, one should not ignore the development of the magnetic compass, which gave European navigators the confidence to sail the deep seas. The use of magnets in the form of lodestones was known to the Greeks as early as 500 BC. It is also probably true that for centuries Chinese navigators had used magnets to determine directions. But in Europe the idea caught on only around 1200 and the early compasses were very primitive. It was Peregrinus, an engineer in the army of Louis IX of France, who developed a workable compass. He put the compass needle on a pivot (rather then on a floating cork as was done earlier) which he placed in the middle of a graduated dial. This simple idea made the mariner’s compass a practical and very useful tool.
in the world today. Technically, of course, it was not Gutenberg who brought out this book, but Fust and his collaborators. Though Gutenberg died in debt, a broken man, his invention transformed the world. The technique of printing quickly spread all over Europe. Though initially only religious texts came out of the press, it wasn’t too long before scholars started using this medium to spread their good word. It is, for example, very unlikely that Martin Luther would have succeeded in his rebellion against the Church except for the fact that he could print and distribute large numbers of pamphlets. And since printing provided cheaper books, literacy rose and the ranks of the educated community swelled. This heralded the coming of the Scientific Revolution.
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus
Suggested Reading
Ganeshkhind Pune 411 007 India.
[1]
Joy Hakim, The Story of Science, Smithsonian Books, Vol.1, 2004.
[2]
James Dyson, History of Great Inventions, Edited by Robert Uhlig, Telegraph Group, UK, 2001.
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Email: [email protected] [email protected]
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Dawn of Science 9. The Conquest of the Seas T Padmanabhan The European voyages and the explorations of the sea and the distant lands laid the ground for the Copernican Revolution.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics,
By about AD 1260, Kublai Khan had set up a grand Mongol empire in China. It was during this Mongol supremacy that China came into closer contact with Europe. With a fairly stable Mongol Empire stretching across the plains of Asia, it was easy for European travellers to visit China and establish regular trade links.
especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498, p.590, p.684, p.774, p.870, p.1009, p.1062. Vol.16: p.6.
Keywords Marco Polo, Columbus, Ptolemy, Magellan.
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One among such traders from Venice who earned a place in history was Marco Polo. His father and an uncle made a trip to Kublai Khan’s China when Marco was still in his teens. The Mongol leader was fascinated by the Venetian merchants and sent them back to Europe to bring missionaries who would teach Christianity in China. The Polos could not persuade the papacy to send the clergy but Marco Polo accompanied his father and uncle on their second trip to China in 1275. He had cordial relationship with Kublai Khan and became the Emperor’s trusted diplomat. After spending nearly two decades in the eastern lands, Marco Polo finally returned to Venice in 1295. This was the first time Central Asia was observed so closely by the Europeans. Marco Polo, who held command in a Venetian fleet, was captured in a naval battle in 1298 and had to spend a year in a Genoese prison. There he wrote his travelogue (The Description of the World), setting down in detail the affairs of Asia and Far East. The book was popular but its contents were largely not believed. For, the Europeans did not quite like the idea of the existence of a high civilisation and riches in the Far East and even coined the term “Marco Milioni” (Marco Millions) to describe the way Marco Polo dealt with large numbers in his
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Figure 3. Marco Polo, on his return from China after serving the Mongols for nearly two decades, wrote a detailed account of what he saw in ‘The Description of the World’. It gave a lot of details about cities, canals, rivers, ports and industries in China and its neighbourhood. Courtesy: h t tp : / / www. b ri t an ni c a. c om / EBc hecked/topic-art/468139/ 119417.
descriptions of the East. There was, however, one man who believed every word of what Marco Polo said. This was Christopher Columbus, the Italian explorer, who felt strongly that the almighty had chosen him to achieve great deeds. He wanted to acquire the riches of the Indies and Cathay (the terms by which India and China were known) and thought he could do it by sailing westwards from Europe. It is a popular myth that Columbus believed the Earth was round while everyone else thought it was flat; the European scholars of the time had accepted the Earth was round and the sailors certainly knew this. What prompted Columbus on his voyage was an interesting calculation error.
What prompted Columbus on his voyage was an interesting calculation error.
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Around AD 100, Ptolemy had drawn up a map of the known world and gave estimates of the distances between various points on the globe. This map and the later versions of it claimed the length of one degree to be about 56.6 Italian miles, a mile being about 1,477 metres. Such a conversion made the equator about one quarter too small. A map produced along these lines by an Italian map-maker Toscanelli, came into Columbus’s possession. Columbus calculated the land distance between Spain and India to be 282 degrees and the distance over the sea about 78 degrees. According to the (wrong) conversion of degrees into miles, which he used, India should be about 3,900 miles from the Canary Islands in the Atlantic, which is more or less where America happens to be!
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Figure 4. Ptolemy’s map of the world. Courtesy: http://www.s 155239215. on l i n eh om e. u s / t u r k i c / b t n _ G eographyMaps /Ptol emy’s _ world_map.gif
Columbus tried to get several kings and nobles to finance his trip but met with a series of failures. In fact, it is interesting that the Portuguese king, John II, referred the project to geographers who pointed out that there must be something wrong with the maps Columbus was using. They felt that the surest way to Asia was around the southern tip of Africa and that, by going west, Columbus was going the wrong way. Of course, they were quite right but what they (or Columbus) did not know was that between Europe and Asia lay unknown continents (the American continent), roughly about 3,600 miles away. Columbus finally managed to get a subsidy from the Spanish royalty—Ferdinand and Isabella – and set sail on 3 August 1492. After several ups and downs, he reached the eastern end of the American continent on 12 October. He explored various regions and was given a hero’s welcome on his return. Columbus’s voyage caught the imagination of Europe and the age of exploration continued in full vigour. It is true that there were great sailors before Columbus – for instance, the Portuguese prince, Henry the Navigator, who even set up a school for navigation in Portugal – but Columbus’s voyage added the necessary glamour to explorations of the sea. That the new lands which he discovered contained vast mineral wealth and riches which could be plundered
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rather easily was another motive for the explorations; indeed this is what prompted the kings and nobles to finance the expeditions in the first place. Shortly after Columbus’s voyage, Spain and Portugal were at loggerheads on how the loot from the new lands was to be shared. The Pope had to intervene and negotiate a compromise. He drew a line, a hundred leagues west of Cape Verde islands in the Atlantic and ‘gifted’ all lands to the west of this line to Spain and those to the east to Portugal. This line, however, was not drawn completely round the Earth but only across the Atlantic Ocean. This fact was cleverly exploited by Ferdinand Magellan, the Portuguese explorer. Magellan was in the service of the Portuguese army but was dismissed in 1517 for some minor offence. Bitter at this treatment he joined the Spanish service and rose high. He pointed out to Charles V of Spain that if the Spaniards kept sailing westwards they will always be on the right side of the Papal line but can certainly reach the Indies, which, going by the papal decree, was left to the Portuguese to explore. Magellan was essentially repeating what Columbus did but was doing it right, having realised that the American sub-continent was not the Indies they were seeking. The Spanish monarch liked this idea and Magellan set sail on 10 August 1519 with five ships. The ships crossed the Atlantic,
Figure 5. Ferdinand Magellan, a Portuguese employed by the King of Spain, set out on a voyage from Spain in 1519. Courtesy: ht t p: / / epr ess . anu. edu. au/ spanish_ lake/ch02s07.htm
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Box 1. Marco Polo on Chinese Paper Money WHEN ready for use, it (a specially prepared paper) is cut into pieces of money of different sizes... The coinage of this paper money is authenticated with as much form and ceremony as if it were actually of pure gold or silver, for to each note a number of officers, specially appointed, not only subscribe their names, but affix their signets also; and when this has been done... the principal officer, deputed by his majesty, having dipped into vermilion the royal seal committed to his custody, stamps with it the piece of paper, so that the form of the seal remains impressed upon it, by which it receives full authenticity as current money, and the act of counterfeiting it is punished with death. When thus coined in large quantities, this paper currency is circulated in every part of the grand Khan’s domains, nor dares any person, at the peril of his life, refuse to accept it in payment All his subjects receive it without hesitation, because, wherever their business may call them, they can dispose of it again in the purchase of merchandise they may have occasion for, such as pearls, jewels, gold or silver. From: The Travels of Marco Polo, Everyman’s Library, New York and London, 1950.
found a small passage (now called the ‘Strait of Magellan’) at the southern end of South America into the Pacific Ocean and just managed to reach the island of Guam near the present Philippines on 6 March 1521 at the brink of starvation. Magellan was later killed in a squabble with the natives of Philippines but one of his five ships, Victoria, managed to make its way across the Indian Ocean, around the southern tip of Africa and back to Spain, arriving there on 8 September 1522. This circum-navigation of the globe proved beyond doubt three facts. First, the estimate of the size of the Earth by Ptolemy was wrong and the earlier estimates by Erosthemes were right. Second, there is one single ocean – not seven as thought by the Greeks – which girdled the Earth. And third, it showed that vast lands with new animals and plants, about which Aristotle and other ‘deep’ thinkers knew nothing, existed on the Earth. All this emphasised the insufficiency of the accumulated ancient knowledge and prepared the ground for the Copernican Revolution. Suggested Reading
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007 India.
[1]
The Illustrated Reference book of the Ages of Discovery, Ed. James Mitchell, Windward, UK, 1982.
[2]
John R Hale, Great Ages of Man: Age of Exploration, Time-Life Interna-
Email: [email protected] [email protected]
tional, Netherlands, 1966.
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Dawn of Science 10. The Rise of Modern Medicine T Padmanabhan Two persons contributed greatly to the ‘revival of reason’ in medical practice.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498, p.590, p.684, p.774, p.870, p.1009, p.1062. Vol.16: p.6, p.110.
Keywords Paracelsus, Vesalius, dissections, anatomy.
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The Dark and Middle Ages (AD 900 to around 1200) in Europe were marked by man’s preoccupation with theology and afterlife. Though Christian monastries did act as a reservoir of knowledge translating the ancient Greek medical works, views like disease is a punishment for sin and should be cured by prayer and repentance, ‘the human body is sacred and dissection of corpses is a sin’ and ‘any point of view contrary to the establishment’s views is blasphemous and must be referred to the inquisition’, did great harm to the advance of medical science. It was only by the end of the 13th century that there was a resurgence of interest in the works of ancient Greeks – both scientific and non-scientific. Scholars tried to follow their lines of thinking and out of this arose the modern Western science. An interest in humanities led to the study of human anatomy again and two major Italian schools of medicine came up at Salerno and Bologna. One of the earliest anatomists from Bologna was Mondino de Luzzi (1275–1326). Until then the dissection of corpses was considered an act beneath the dignity of a teacher. So the medical man would lecture from a high podium while some lowly servant dissected a corpse to illustrate the points. And it was impossible to coordinate the activities of the two; what was more, the servant – not being a man of medicine – could only make sloppy dissections. Mondino changed this practice and started dissecting. His lectures consequently were much more popular and earned him the title ‘Restorer of Anatomy’. In 1316, he wrote the first book in the history of medicine which was entirely devoted to anatomy.
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Unfortunately, the tradition set up by Mondino did not last. His successors reverted to older practices and continued to hold the earlier writings of Aristotle and Avicenna sacred and embodying the ultimate truth. It took early two centuries more for this stranglehold to loosen. The major contributors to this ‘revival of reason’ in medical practice were the Swiss physician and alchemist Paracelsus (1493–1541) and the Flemish anatomist Andreas Vesalius (1514–1564) – two men very different in temperament, talent and outlook.
Figure 2. Paracelsus. Courtesy: htt p:// en.wikip edia.org /wik i/ Paracelsus; Figure portrait by Quentin Massys.
Paracelsus’s original name was Theophratus Bombastus von Hohenheim. You can’t blame him for wanting to change that name! But the choice he made very well portrays his character: Paracelsus means ‘better than Celsus’. Celsus was the famous Roman physician whose works had been recently translated and had made a tremendous impact on contemporary thinkers. Paracelsus in fact thought that he was better than everybody else and never hesitated to say so. At a very early age, he acquired a good knowledge of mining, metallurgy and chemistry, and he wanted to pursue the alchemist’s goal of turning lead into gold. He also acquired medical education at several universities in Europe (as he moved from place to place, often following fights he invariably picked up at every place) and realized that chemistry could play a useful role in producing medicines for diseases. In a way, this marks the beginning of the transition from alchemy to chemistry. Paracelsus returned home in 1524 after nearly 10 years of wandering and was appointed the town physician and lecturer of medicine at University of Basel. He began his lecture series by publicly burning the works of Galen and Avicenna in front of the university. Though the authorities were infuriated, the students cheered the eccentric and charismatic teacher. His teachings were a mixture of deep insights and stupid blunders. For example, he produced a clear clinical description of syphyllis and treated cases with carefully measured doses of mercury compounds. He could understand that the ‘miner’s disease’ (silicosis) was caused by the inhalation of metal vapours and not by the anger of the
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mountain spirits. He also wrote the first book (Miners’ Sickness) on occupational diseases; cleanliness, he said, was crucial for health. He produced scores of medical compounds containing mercury, sulphur, iron and copper sulphate. Yet, while doing all this he firmly believed in the four elements of the Greeks and was assiduously searching for the elixir of life! Vesalius’s life was prosaic in comparison but more fruitful. Born in a wealthy and powerful family, he attended a medical school in Paris during 1533–36. He made a detailed study of Galen and Rhazes and at first could find no fault with their writings. On receiving the MD degree from the University of Padua, he was appointed a lecturer in surgery. While preparing for his lectures, Vesalius spent much of his time in dissecting corpses. It was very soon clear to him that Galen and others had not based their writings on the dissection of the human body (which was indeed banned in Roman times) but on extrapolations from studies of the structures of animal bodies. He had the courage to declare his conclusions and proceeded to write a textbook on human anatomy based on the correct knowledge. This book, De Humani Corporis Fabrica Libri Septem, (The Seven Books on the Structure of Human Body) was superbly illustrated – probably at the studio of the renaissance artist, Titian – and was printed in 1543. This book represented the culmination of several important trends in medicine, especially the revival of ancient learning and the growth of attention to the human body as an object of serious study. De Fabrica also contained an extensive and accurate description of the human body and provided anatomy with a new language for descriptions. Indeed, anatomy came of age with Vesalius. (Interestingly enough, De Fabrica was published in the same year as Copernicus published his book outlining the heliocentric theory – another milestone in science which we will describe next time.)
Figure 3. Vesalius. Courtesy: htt p:// en.wikip edia.org /wik i/ Andreas_Vesalius
Figure 4. An illustration from De Fabrica, showing the muscle groups of the human body.
Vesalius was extremely accurate as an anatomist, but his ideas on other branches of medicine were not always accurate. He did believe (quite correctly) that the brain and the nervous system
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Figure 5. (left) A woodcut from De Fabrica shows a skeleton meditating over a skull. Figure 6. (right) Before the days of Vesalius, the anatomy teacher never did the dissections; while the teacher only lectured, lowly servants, who had very little medical knowledge, dissected a human corpse. Courtesy: http://clendening.kumc.edu/dc/ rm/m_47p.jpg
represented the seat of emotions and he categorically denied the earlier Aristotlian view that these functions were governed by the heart. This idea fortunately got accepted since then. At the same time, he believed in Galen’s view on the circulation of blood and thought that blood must pass from one ventricle to another through some mysterious process. Success produces enemies and Vesalius met with the same fate. Some of his powerful enemies managed to bring a suit against him on the grounds of heresy, body-snatching and dissection. He would have been executed but for some royal connections he had; the death sentence was commuted to pilgrimage to the Holy Land. It did not really help; on the way back from the pilgrimage Vesalius died in a shipwreck off the Greek coast. Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India.
Suggested Reading [1]
Isaac Asimov, Asimov’s Biographical Encyclopedia of Science and Technology, Doubleday, 1982.
[2]
F E Udwadia, Man and medicine, Oxford University Press, 2001.
Email: [email protected] [email protected]
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Dawn of Science 11. The Copernican Revolution T Padmanabhan Slowly, through philosophic blunders and religious dogmas, to free-thinking.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498, p.590, p.684, p.774, p.870, p.1009, p.1062. Vol.16: p.6, p.110, p.274.
On 24 May 1543, Nicholas Copernicus was in bed, dying of brain haemorrhage. It is said that a copy of his book De Revolutionibus Orbium Coelestium (On the Revolution of the Heavenly Bodies) – the publication of which he had delayed by nearly 30 years – was brought to his death-bed so that he could have a last glimpse of it. In this book he had detailed a system of astronomy with the Sun at the centre and the planets going around it in fixed orbits. Copernicus, so to say, stopped the Sun and set the Earth in motion. Behind this event lies a fascinating story in the history of science – a story of extraordinary blunders, irresponsibilities and damaging effects of religious suppression of science. To see it in perspective, we have to go back over 2,000 years in history. Around 350 BC there lived a Greek astronomer, Aristarchus, who wrote a short treatise On the Sizes and Distances of the Sun and the Moon. In this treatise, he proclaimed that the Sun and not the Earth was at the centre of our world and that all planets revolved around the Sun. His book became a classic of antiquity and he was considered one of the foremost astronomers of his time. Both Archimedes and Plutarch knew about his work. “For, Aristarchus supposed that the fixed stars and the Sun are immovable but the Earth is carried around the Sun in a circle...” says Archimedes; “...that the heaven is at rest but the Earth revolves in an oblique orbit while it also rotates about its own axis,” is the reference in Plutarch to the ideas of the Greek astronomer.
Keywords Copernicus, Aristarchus, heliocentric, geocentric, retrograde
Incredibly enough, Aristarchus was forgotten! The geocentric system of Ptolemy, a much more complicated and aesthetically
motion.
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unappealing one, held sway even around the second and third centuries AD. From then on, throughout the Dark Ages, there was no hope of revival. Later, as Europe went through the Renaissance, one would have hoped for the right ideas to emerge. But the strong religious dogmas and theological interpretations of Aristotle’s outdated ideas suppressed the truth for centuries. Though it was only one step from Aristarchus to Copernicus (or, for that matter, from Hippocrates to Vesalius or even from Archimedes to Galileo), it took centuries in Europe for this step to be taken. As A N Whitehead, an English historian of science, remarks: “In the year 1500, Europe knew less about nature than Archimedes, who died in 212 BC, did.” Such was the effect of dogmatic ideas on the growth of science. Figure 3. In the pre-Copernican universe, the Earth is at the centre, and the Sun, the Moon, planets and the stars go around the Sun in concentric circles. Courtesy: htt p:// en.wikip edia.org /wik i/ Almagest
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The way Copernicus ended up writing his book is another interesting tale. Born in 1473 in Torun in Eastern Poland, Copernicus lost his father early. Thereafter, his uncle brought him up, and he gave Copernicus a very good education. In 1496, Copernicus travelled to Italy and studied medicine and canon law for 10 years. This was when he got interested in astronomy! In those days, the positions of the planets were calculated by the system evolved by Ptolemy. In spite of its complexity (and detailed mathematics), this system was cracking up. The predicted positions of planets were getting to be far away from the observed ones in spite of the several ad hoc corrections introduced by later astronomers. It occurred to Copernicus that the calculations could be considerably simplified if one adopted the heliocentric system. Copernicus’s genius – if you could call it that – was in putting this idea into practice and meticulously working out the details of the new model. He relied on the observations of others, though, especially because he was not good in this. (His instruments were less accurate than those used in Alexandria 2,000 years ago; and he made a
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mistake even in getting the terrestrial co-ordinates of his observatory right.) Copernicus realised that his model could explain several things, which Ptolemy could not. It must have certainly occurred to him that probably his was the true model – with a moving Earth and all that. But he hesitated to publish it knowing well that he could get into trouble with the Church. A private manuscript that he circulated created considerable interest among European scholars and finally the German mathematician, Georg Rheticus (1514–1576), who was a fan and student of Copernicus, persuaded Copernicus to publish the book. Rheticus also suggested that the book may be dedicated to Pope Paul III in order to pre-empt opposition from the Church. Copernicus entrusted Rheticus with this task.
Figure 4. Copernicus. Courtesy: htt p:// en.wikip edia.org /wik i/ Nicolaus_Copernicus
Unfortunately, Rheticus had to leave town, so he left a Lutheran minister, Osiander, in charge of the printing. Martin Luther (1483–1546) had earlier expressed himself strongly against Copernicus (which goes to show that religious reformists may not always do the right things for science) and Osiander decided to publish the book with the addition of a highly damaging preface.
Figure 4. The Copernican universe (as depicted by Thomas Digges in 1576), with the Sun at the centre and the Moon going around the Earth. The stars are not limited to a sphere but are spread infinitely. Courtesy: http://en.wikipedia.org/wiki/Universe
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Box 1. Clues, Cover-ups and the Climax There were several tell-tale signs in the behaviour of heavenly bodies, which suggested a heliocentric world. Unfortunately, Ptolemy and his followers were not brave enough to follow up these clues. To begin with, the estimate of the size of the Sun relative to the size of the Earth was available since the time of Aristarchus (about 300 BC). Though the actual figure he obtained was wrong, it was quite clear that the Sun was considerably bigger than the Earth. It was then rather strange to think that the Sun was going round the Earth instead of the other way around.. Further, the observation of the trajectories of planets revealed some strange anomalies. The planets Mercury and Venus were always seen close to the Sun, just after sunset or just before sunrise and never overhead at night. The other three planets, Mars, Jupiter and Saturn, showed irregular pattern of motion every once in a while. They travelled in one direction for some time, stopped in their tracks and then seemed to move backwards! These features are difficult to understand (in a natural manner) if the planets were moving around the Earth. Lastly, it was known that the brightness and the apparent size of the planet Venus varied periodically – something which should not happen if Venus was orbiting the Earth at a constant distance. All these problems disappear if we assume that the planets revolve around the Sun in the order Mercury, Venus, Earth, Mars, Jupiter and Saturn. Because the orbits of Mercury and Venus are closer to the Sun than that of the Earth, they will never appear overhead at night. The retrograde motion of the other three planets is also easy to understand: if the Earth revolves around the Sun at a faster rate than the outer planets, then the Earth will ‘overtake’ these planets every once in a while. Seen from the Earth, the outer planets will appear to go backwards. Also, since the distance between the Earth and Venus varies quite a lot, the appearance of this planet will be altered periodically. The Greeks had all the pieces of the puzzle but refused to put them together. So deep-rooted and dogmatic were Aristotle’s notions of circular motion that Ptolemy ended up saying: “We believe that the object of the astronomer ... is this: to demonstrate that all the phenomena in the sky are produced by means of uniform and circular motions.” To achieve this, Ptolemy had to invoke a complicated system of epicycles in which celestial objects moved in circles, whose centres themselves moved in other circles, etc. The Copernican idea could at one stroke resolve all the discrepancies mentioned earlier. The only place where Copernicus erred was in sticking to the circular motion. As a result, he also needed epicycles in his model (in fact, Copernicus used more epicycles than Ptolemy did!). And they remained, till Kepler later changed the paths to ellipses, laying to rest the last of the Aristotelian dogmas.
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The preface essentially conveyed the idea that the system described in the book was only a mathematical apparatus and may not represent reality. It explicitly stated that “... these hypotheses need not be true or even probable ...”. Historians are unsure whether Copernicus approved of this preface and the issue is still unsettled. (It shouldn’t be too surprising if he did, for old Nicholas always knew which side of the bread was buttered; for one thing, never in the book does he mention a note of thanks to Rheticus – a fact which deeply hurt Rheticus.) The book finally appeared with the preface and a dedication to the Pope.
Figure 5. The origin of the ‘retrograde’ motion of some planets, Mars in this case. Since the Earth revolves faster around the Sun than Mars, at times the Earth overtakes Mars in orbital motion. Seen from the Earth, Mars will appear to move ‘backwards’ in the sky relative to the Earth. Courtesy:
It was eminently unreadable and sold poorly. While several other contemporary books on planetary theory and astronomy were easily reaching 100th reprint in Germany, Copernicus’s book stopped with only one print. All the same, it was a milestone in science. It was immediately taken up by those who produced planetary tables and helped to set the heaven in order. What is more, it influenced at least a handful of later thinkers to come out of religious dogmas and think anew. It is this rebirth of freethinking which heralded the Scientific Revolution in Europe. Suggested Reading [1]
Isaac Asimov, Asimov’s Biographical Encyclopedia of Science and Technology, Doubleday, 1982.
[2]
Arthur Koestler, The Sleepwalkers, Penguin, 1990.
h tt p :/ / h is t or y.n as a. g ov/ SP 4212/p3a.html
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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Dawn of Science 12. Logarithms T Padmanabhan The invention of logarithms saved much time and effort in astronomical and mathematical calculations. Until the advent of electronic calculating machines, they remained the most effective tool for scientists. T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts:
“My Lord, I have undertaken this long journey purposely to see your person, and to learn by what engine of wit or ingenuity you came first to think of this most excellent help in astronomy.” This is how Henry Briggs (1561–1630) greeted the Scottish nobleman, John Napier (1550–1617), the inventor of logarithms, when they met first. And indeed it was an invention of great practical use not only in astronomy but also in other branches of computation. Until the advent of electronic calculating devices, logarithms remained the most effective tool for every scientist.
Res onance, Vol.15: p.498, p.590, p.684, p.774, p.870,
tion.
The inventor of such a device, John Napier, was quite an enigmatic persona. He was born in the Scottish aristocracy and travelled widely in Europe during his youth. This was the time when Europe was in total turmoil, split into warring camps by the Protestant reformation. His native country, Scotland, was fast turning into a Calvinist state. This influenced Napier and he became a very vocal Protestant gentleman. In 1593, he expressed his anti-Catholic views against the Church of Rome in a book entitled A Plaine Discovery of the Whole Revelation of Saint John. The book was an overnight bestseller and ran 21 editions of which at least 10 were during his lifetime! So engrossed was he in his religious sentiments that he definitely expected his claim to fame to rest on this book. (He also spent considerable amount of time thinking out different kinds of ingenious war machines to be used against Philip II of Spain, in case he should attack Scotland; none of them was needed.)
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p.1009, p.1062. Vol.16: p.6, p.110, p.274, p.304.
Keywords John Napier, logarithm, Henry Briggs, astronomical computa-
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Figure 3. John Napier. Courtesy: http://en.wikipedia. org/wiki/John_Napier
As Laplace said years later, the invention of logarithm effectively ‘doubled the life of the astronomer’.
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It is therefore rather surprising that he had the time and energy to ponder over mathematics. He was particularly concerned with the amount of labour involved in the multiplication and division of numbers. In fact, most scientists of his time spent a large part of their working day doing routine, boring mechanical calculations. This was especially so in astronomical computations involved in the preparation of planetary tables, etc. Napier’s logarithms changed the situation completely; it replaced multiplication by addition and division by subtraction. As Laplace said years later, the invention of logarithm effectively ‘doubled the life of the astronomer’. Incredibly enough, the idea behind logarithm is extremely simple. To understand the basic concept, let us consider the numbers: 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, 25 = 32... . Suppose we need to multiply the numbers 4 and 8. Since 4 is 22 and 8 is 23, the product can be written as 4 x 8 = 22+3 = 25. We know that 25 is 32, and this gives us the answer. The crucial point is that the multiplication of two numbers 4 and 8 was reduced to the addition of the superscripts 2 and 3. Now suppose we have a readymade table of all numbers expressed as various powers of 2. For example, the number 17 can be expressed as 24.087 to a very great accuracy and 19 can be written as 24.248 (we say that 4.087 is the ‘logarithm’ of 17 in ‘base’ 2, etc.). Therefore, to multiply 17 and 19 we only have to add the two powers (4.087 + 4.248) and get 8.335. The product is 28.335, which is 323. Of course, we would need a detailed table giving the power for each integer. Once such a table is prepared, any two numbers can be multiplied – or for that matter divided, which requires subtracting one index from the other, within seconds. For large numbers, this saves considerable amount of time and effort. In the above illustration, we used the number 2 to express all other integers. One could have used any other positive number in place of 2. Because of a rather complicated reason, Napier used a number which is about 0.3679 which is the reciprocal of a constant usually denoted by the letter ‘e’ = 2.718. This constant plays a crucial role in all branches of higher mathematics and is
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called ‘the base of natural logarithm’. While such a base may not look ‘natural’ at first sight, it does provide a simple geometrical interpretation (see Figure 4 ). Consider two line segments AB and DE with AB having unit length. C and F are two moving points on these line segments. They start simultaneously at A and D; F moves with uniform, unit, speed along DE while C moves with a speed which is numerically equal to the length CB. Then DF will give the magnitude of the natural logarithm of CB.
Figure 4.
Napier published his discussion in 1614 in a small brochure titled Mirifici Logarithmorum Canonis Descriptio (A Description of the Wonderful Law of Logarithms) which also contained a table giving the logarithms of sines of angles for successive minutes of arc. (This was the most important quantity needed in astronomical computations.) The book roused wide interest. The news travelled fast and in the next year Henry Briggs, a professor of mathematics at London, travelled all the way to Edinburgh (it was quite a distance to cover in those days!) to greet Napier. At their meeting, Briggs convinced Napier that it would be much more useful to use 10 as the base for logarithms. In other words, all numbers will be expressed as powers of 10, instead of 2. Since 100 = 102 and 1000 = 103, the logarithm of 100 will be 2 and that of 1000 will be 3; and that of any number between 100 and 1000 will be a number between 2 and 3. Briggs started the construction of such a table on his return to London and in 1624 published his Arithmetica Logarithmica containing 14-decimal place tables of logarithms for all numbers from 1 to 20,000 and from 90,000 to 100,000. The gap between 20,000 and 90,000 was filled in later by a Dutch bookseller, Adrian Vlacq (1600–1666). Incidentally, these tables remained in vogue for nearly three centuries and were superseded only around the 1940s when extensive 20-decimal place tables were calculated.
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Briggs convinced Napier that it would be much more useful to use 10 as the base for logarithms.
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Figure 5. The slide rule. Courtesy: http://en.wikipedia. org/wiki/Logarithm
Meanwhile, in the 1620s, the English mathematician William Oughtred, came to realise that even the process of looking up logarithmic tables can be eliminated by a simple mechanical device (Figure 5). It consisted of two sliding scales in which the numbers are marked in such a way that the distance from the left end of the scale is numerically equal to the logarithm of the number. Numbers could now be multiplied and divided by merely sliding the scale on one another. It is difficult to estimate how much the world of engineering and science owes to this gadget called the ‘slide rule’. Napier also made some contributions to other branches of mathematics – for instance, the perfection of the decimal notation which we now use so frequently. The idea of the decimal fraction had been worked out by the Dutch mathematician, Simon Stevin (1548–1620) earlier, but it was Napier who made the notation compact and convenient to use. Suggested Reading [1]
Address for Correspondence
Mathematical Expositions, Mathematical Association of America, No.5,
T Padmanabhan IUCAA, Post Bag 4 Pune University Campus
Howard Eves, Great Moments in Mathematics Before 1650, Dolciani 1983.
[2]
Isaac Asimov, Asimov’s Biographical Encyclopedia of Science and Technology, Doubleday, 1982.
Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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Dawn of Science 13. The Paths of Planets T Padmanabhan The confusing pathways in the heavens are finally charted.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498, p.590, p.684, p.774, p.870, p.1009, p.1062. Vol.16: p.6, p.110, p.274, p.304, p.446.
Brahe was a nobleman and governor of the Helsingborg castle in Denmark in the sixteenth century. Around AD 1540, he promised his brother, Joergen, that if he had a son, the latter could adopt him. But, when, in 1546, Brahe did have a son, he went back on his promise. Joergen waited till another son was born to Brahe and promptly kidnapped – and adopted – the first son. The son was Tycho Brahe (1546–1601) and he grew up to be the most accurate observational astronomer before the days of the telescope. Tycho’s foster-father died when Tycho was still very young, leaving him with a vast inheritance. Joergen jumped into a river to rescue Ferdinand II, the king of Denmark; though he succeeded in the attempt, he caught pneumonia and died. Tycho had an excellent education in law, but his heart was set on astronomy. Right from his student days, he kept a careful record of the night sky, day after day. The real turning point in his life probably came in August 1563 when he was observing the ‘conjunction’, or the close appearance, of Jupiter and Saturn. He discovered that all the almanacs were widely off the mark in predicting this event! This convinced him of the need for exact and accurate observations with good instruments – a task, which he set himself. He travelled all over Europe acquiring the instruments and set up a small observatory at Scania in 1571.
Keywords Tycho Brahe, Kepler, ellipse, epicycle.
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He had a chance, literally of a lifetime, on 11 November 1572, when he spotted a ‘new star’ near the constellation Cassiopeia.This new star was brighter than Venus. And Tycho’s careful observations showed that this star was too far away from the Earth, definitely farther than the Moon, and therefore among the fixed stars.
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This conclusion, set out in his book De Nova Stella in 1573, shattered prevailing dogmas. For, according to the accepted Aristotelian principles, all change and decay were confined to the Earth and the realms of the stars were immutable. The appearance of a new star was therefore a blow to this idea.
Figure 3. Tycho Brahe. Courtesy: http://en.wiki-pedia.org/wiki/ Tycho_Brahe
Figure 4. Tycho Brahe’s observatory. Courtesy:
This discovery brought Tycho royal patronage. Denmark’s king, Frederick II, gave him the island of Ven and also financial support to build an astronomical observatory. Tycho used it well; the observatory was a masterpiece of workmanship and was extremely accurate in making observations. (Ptolemy’s observations were correct to 10 minutes of arc while Tycho’s were exact up to two minutes of arc!) In 1577, a great comet appeared in the sky of which Tycho kept a careful track. His measurements again confirmed two facts: (i) the comet was much farther than the Moon, and (ii) its path was very different from a circle. These further damaged the credibility of Aristotle’s ideas. Tycho corrected every single astronomical measurement for the better. He observed the motions of the planets, especially that of Mars, with unprecedented accuracy. He also determined the length of the year to an accuracy of almost a second. This measurement had a bearing on calendar reforms.
h t t p : / / i m ag es . g o og l e. c o m / hosted/life/)
However, Tycho’s good fortunes declined with the death of Frederick II in 1588. Having picked fights with the new king, the nobility and the clergy, Tycho had to abandon his observatory and finally settle in Prague, under the patronage of Rudolf II. And here he made his most important discovery – Johann Kepler (1571–1630). Kepler was everything Tycho was not. He was born in Germany to a good-for-nothing mercenary soldier and a quarrelsome mother who, in her later years, almost got burnt at the stake as a witch; he was sickly and depressed, but managed a good education only because his superior intelligence was recognised by the local duke who gave him a scholarship. Kepler was training to be a Lutheran minister after his formal
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Figure 5. Brahe’s model of the solar system in which the Sun revolves around the Earth but the other planets revolve around the Sun. Courtesy: http://outreach.atnf.cs iro.au/ education/senior/cosmicengine/ renaissanceastro.html)
education at Tubingen University but by accident became a mathematics teacher in a Lutheran school at Graz in Austria (he was recommended for the job by the university on the death of the earlier teacher). As an astronomy student, Kepler was strongly influenced by Copernican concepts. Though his early attempts to fix planetary orbits based on platonic solids were not very successful, it brought him in contact with Tycho. In 1597, as religious disputes became intense in his hometown, Kepler accepted a position in Tycho’s observatory in Prague. With the death of Tycho in 1601, Kepler inherited the vast amount of astronomical observations which Tycho had recorded.
Figure 6. Johann Kepler. Courtesy: http://en.wiki-pedia.org/wiki/ Johannes_Kepler
Kepler started, somewhat erratically, to find simple rules describing the motion of the planets – especially that of Mars. The failure of simple models finally forced Kepler to assume that ‘the path of planets around the Sun are ellipses with the Sun at one focus’. The extraordinary accuracy of Tycho’s observations was instrumental in eliminating several other simpler models; some models differed from that of elliptical orbits only by eight minutes of arc in their predicted positions which would have been ignored in Ptolemy’s days.
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Box 1. The Ellipse The ellipse – the orbits taken by the planets – belongs to a class of curves called ‘conics’ first studied in detail by the Greek geometer Apollonius (third century BC). He showed that three different curves are generated when a plane intersects a cone. The ellipse, in particular, is the only closed curve out of the three. A more physical way of defining the ellipse will be the following. Think of a person walking in such a way that the sum of his distances from two different points F and S (see Figure A) remains constant; his path will be an ellipse. The fixed points will be called the ‘foci’ of the ellipse. When these two points coincide, the ellipse becomes a circle; the farther the points, the more elongated the ellipse will be from the circle. (The elongation is measured by a parameter called eccentricity which is zero for the circle.) According to Kepler’s laws, the planets move around the Sun in elliptical orbits with the Sun located at one of the foci. Notice that Kepler’s second law (Figure B) requires the planet to move faster while it is nearer the sun. (It takes the same time for the planet to move from P1 to P2 as it does to go from P3 to P4.)
Figure 1A
Figure 1B
Though Kepler replaced the circular paths by a less symmetrical elliptical path (Box 1), he could restore the symmetry in a different way. Based on Tycho’s meticulous observations again, Kepler could conclude that ‘the line joining the planet and the Sun traverses equal areas in equal amount of time’. These two laws, published in his Astronomia Nova in 1609, earn Kepler a place in the history of science. Ten years later, he published another book full of mysticism in the middle of which lies a gem: “The square of the period of revolution of a planet is proportional to the cube of its distance from the Sun”. This third law took a significant step in a new direction. By relating the orbital properties of various planets to the central agency, the Sun, it almost suggested that the Sun is the cause of planetary motion. This idea was very much present in Kepler’s
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Figure 7. The title page of Kepler’s New Astronomy. Courtesy: http://en.wiki-pedia.org/wiki/Astronomia_nova)
writings but it took the genius of Newton to form a workable law out of this suggestion. During the years 1620–1627, Kepler completed the new table of planetary motions based on Tycho’s observations and his theory of planetary orbits. In spite of severe financial difficulties, continuing war and religious unrest, these tables – called Rudolphine Tables in honour of Kepler’s first patron, and dedicated to the memory of Tycho – were published in 1627. Incidentally, it would have taken considerably longer, but for the use of logarithms by Kepler! The tables also contained a set of logarithms. Anyway, the pathways in the heavens were finally charted. Suggested Reading [1]
Arthur Koestler, Sleep walkers, Penguin, 1959.
[2]
David Layzer, Constructing the Universe, Scientific American Library, 1984. Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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Dawn of Science 14. The Galilean World T Padmanabhan What was remarkable about the most misunderstood medieval scientist’s discoveries was the way he arrived at them – by direct observation. Almost around the time when Kepler was perfecting the laws of planetary motion, another man was laying the foundations of theoretical mechanics. This was Galileo Galilei (1564–1642)1 – probably the most misunderstood of all medieval scientists. Contrary to popular belief, he did not invent the telescope or the thermometer or the pendulum clock; nor did he discover sunspots; and he never threw weights down from the tower of Pisa nor was he tortured by the inquisition. All these and more are attributed to Galileo from time to time because of historical convenience.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498,
Galileo was born on 15 February 1564 in Pisa, Italy. His father wanted him to do medicine – mainly because doctors used to earn considerably more than mathematicians even in those days. An accidental exposure to a lecture in geometry made Galileo turn to mathematics and later to physics. His first invention was a hydrostatic balance about which he wrote an essay in 1586. This contribution followed by a treatise on the centre of gravity of solids (published in 1589) won for Galileo the acclaim of Italian scholars, the sponsorship of the Duke of Tuscany, Ferdinand de Medici and, finally, the position of mathematics lecturer at the University of Pisa. Soon, in 1592, he became the professor of mathematics at the University of Padua where he remained for another 18 years.
p.590, p.684, p.774, p.870,
It was probably during this time that Galileo made his most fundamental contributions to mechanics regarding the nature and cause of motion (though he published these results much later).
Galelio, principle of relativity,
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p.1009, p.1062. Vol.16: p.6, p.110, p.274, p.304, p.446, p.582.
1
See Resonance, Vol.6, No.8,
2001.
Keywords inertia, telescope.
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Till then the ideas that prevailed about the motion of bodies were those of Aristotle. According to these doctrines, some force was necessary to keep a body in motion even at uniform speed. (Adherents of Aristotle used to argue that an arrow shot from a bow moves only because the air behind the arrow pushes it.) It was also believed that a body dropped from some height would fall with a steady velocity, that is, it will cover equal distances in equal periods of time. This speed was supposed to be larger for heavier bodies, making heavier bodies fall quicker than lighter ones. Galileo’s investigations proved all these to be incorrect. He realised that a body falling from a height acquires increasingly greater speeds as it drops. In fact, the distance it covers increases as the square of the time in flight. (If the body has travelled five metres in the first one second of its flight, it would have travelled 20 metres by the end of two seconds rather than ten metres.)
Figure 3. Galileo.
What was most remarkable about Galileo’s discovery was the way he arrived at it. He decided that the proper way to settle such a question was by direct observation! This was quite difficult to achieve in those days because accurate time-keeping instruments were not available. Galileo bypassed this problem by making bodies roll down a gently inclined plane rather than drop them vertically. The inclined plane slowed down the bodies considerably and Galileo could time them using his pulse. (Curiously enough, Galileo did know that a swinging pendulum is an instrument, which maintains good periodicity. Somehow it never occurred to him to design a clock using the pendulum.) Galileo also realised that all bodies would fall to the ground from a height in equal time, if air resistance was ignored. The fact that a steady force acting on a body increases its speed continuously, raises a question: is it at all necessary to have a force acting on a body to keep it moving at constant speed? Galileo thought – quite correctly – that motion in constant speed required no external agency. This principle – now known as the principle of inertia – played a crucial role in the later develop-
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Galileo decided that the proper way to settle such a question was by direct observation!
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ments of dynamics and the theory of relativity. Galileo also used this principle effectively in calculating the trajectories of projectiles thrown from the ground.
Figure 4. The telescope that Galileo designed and built. Courtesy: http://rockdalerm1.wordpress. com/2009/12
Called Io, Europa, Ganymede and Callisto, these satellites clearly showed that not all celestial objects went around the Earth.
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Yet, Galileo did not rise to prominence for these studies but for a very different reason. Around 1608, a spectacle-maker in Holland, Johann Lippershay, had invented an optical tube containing two lenses, which could make distant objects appear closer. Lippershay sold several of these models in the cities of Europe, and Galileo came to know of this invention in the spring of 1609. Galileo could easily make for himself a telescope with a magnifying power of about 30 and he turned the new invention towards the sky (Figure 4). Thus began the age of telescopic astronomy. Using his telescope, Galileo discovered several aspects of nature, which were until then hidden from the human eye. He found that the Moon had mountains and the Sun had dark spots, once again showing Aristotle to be wrong in assuming that only the Earth had irregularities and distortions. (To be sure, there were other astronomers who were exploring the skies with the telescope at the same time; the first reports on observations of sunspots, for example, came from Father Scheiner, a Jesuit astronomer. Galileo entered into a long and bitter controversy over priorities in these discoveries thereby making powerful enemies.) The stars and the planets appeared very different through the telescope and Galileo could see many more stars than were visible to the naked eye. All this made him conclude that stars were much farther away than the planets. More dramatically, Galileo found that Jupiter was attended by four subsidiary objects, which circled it regularly, and, within a few weeks of observation, he could work out the periods of each of these satellites. Called Io, Europa, Ganymede and Callisto, these satellites clearly showed that not all celestial objects went around the Earth. His telescope also revealed the phases of Venus and ring-like structures around Saturn. Galileo announced his initial discoveries in a periodical, which he called Sidereus Nuncius (The Starry Messenger). These announcements definitely caught the fancy of the public, especially RESONANCE July 2011
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because of the expository skills of the author. Peers in the scientific community were, however, initially a bit reluctant even to look through the telescope and were persuaded to come around only after the leading astronomer of the day, Johann Kepler, threw his weight behind the discoveries. In 1611, Galileo visited Rome and was treated with honour and delight. The Cardinal and Pope Paul VI gave him friendly audience and the Jesuit Roman College honoured him with various ceremonies which lasted a whole day. There were astronomers in that college who not only accepted Galileo’s discoveries but also improved on his observations, especially on the phases of Venus. At this stage, at least, there was no open animosity between the Church and Galileo. From such a friendly atmosphere, how a parting of ways between science and the Church arose is an interesting tale in the history of science. While the Church is universally condemned in this matter, a careful study of historical facts indicates that Galileo’s personality did go a long way in aggravating the situation. The way Galileo acted and wrote, created for him several enemies most of whom were powerful and influential. The personality of Galileo and the antagonism of his ‘scientific’ colleagues were as instrumental in bringing about his conflict with the Church as the Church itself (see Box 1).
Figure 5. Galileo viewing through the telescope. Courtesy: htt p:// en.wikip edia.org /wik i/ Galileo_Galilei
From such a friendly atmosphere, how a parting of ways between science and the Church arose is an interesting tale in the history of science.
Box 1. Galileo, Kepler and the Church The trial of Galileo has attracted much attention in the history of science. It is rather interesting to see how events actually unfolded. In 1597, Galileo received a copy of Kepler’s Cosmic Mystery, the preface to which contained detailed arguments in support of the Copernican theory. Galileo sent him a reply saying, “...I adopted the teaching of Copernicus many years ago ... I have written many arguments in support of him and in refutation of the opposite view – which, I dared not bring into public light... frightened by the fate of Copernicus ... who ... is, to a multitude an object of ridicule and derision.” Kepler replied, pleading with Galileo, that he should come out in the open in support of Copernican models. Galileo refrained from doing so, and stopped Box 1. Continued...
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Box 1. Continued... communication with Kepler. And for another 16 years (a period during which the Church did permit discussions on the Copernican model), Galileo was teaching Ptolemy’s ideas in his lectures! During these 16 years, Kepler had repeatedly communicated to Galileo about his findings and to none of which Galileo responded; in fact he always ignored Kepler’s scientific contributions (especially the elliptical nature of planetary orbits) and continued to use the old Copernican ideas of celestial bodies moving in circles and epicycles. In spite of this personal antipathy, Kepler treated Galileo with great generosity and wrote openly, in 1610, supporting the discoveries he made using the telescope. And this was the time when Galileo badly needed Kepler’s support. Even so, Galileo refused to reciprocate the friendship. In 1611, the head of the Jesuit Roman College, Cardinal Ballarmine, asked the Jesuit astronomers for their official opinion on the new discoveries. These astronomers – who had in fact improved upon the work of Galileo on the phases of Venus – had no hesitation in giving Galileo a clean bill and agreeing that at least Venus went around the Sun. The system of the world suggested by many Church astronomers of those days had planets orbiting the Sun with the Sun itself going around the Earth. In the years to follow, Galileo was forced to enter into controversies with jealous colleagues, Church astronomers, powerful members of the nobility and many others on whether the motion of the Earth around the Sun can be proved. The Church was willing to accept Copernican ideas as a mathematical hypothesis but demanded – quite correctly – incontrovertible proof
if
the scriptures are to be
reinterpreted. Galileo found himself at a loss in providing the “proof”, especially because he did not want to give credit to Kepler for the elliptical orbits; with circular orbits, Copernican models were as bad as Ptolemy’s and compromise models did much better. In 1614–15 Galileo wrote a few open letters, in which he supported the Copernican model, emphasised the scriptures had to be reinterpreted and even tried to argue that the burden of proof should rest with the Church. To top it all, he went for a direct showdown with the Pope based on what he considered to be the “proof” for the motion of the Earth – the proof was based on a completely incorrect theory of the origin of tides. From then on, things took an ugly turn. The Pope asked the Qualifiers of the Holy Office to take a clear stand on the matter and this they did on 23 February 1616 – categorically against the motion of the Earth around the Sun. Six days later, Galileo had an audience with the Pope and he was told not to exceed the limits set by the Church. (What precisely he was told is a matter of another major historical controversy.) The Holy Office put Galileo on trial in 1633, essentially on the charge that the contents of his book, Dialogue Concerning the Two Chief World Systems published in 1632, went against the decree of 1616. Found guilty, he was allowed to spend the rest of his life in house arrest.
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During the years 1611 to 1633, Galileo completed his masterpiece, Dialogue Concerning the Two Chief World Systems, in which he had two people, one representing Ptolemy and the other Copernicus, present their arguments before an intelligent layman. Needless to say, Galileo made the Copernican theory come out on top. Galileo died on 8 January 1642 while still remaining under house arrest in accordance with the verdict of the Church. His bones rest in the Pantheon of the Florentines, the Church of Santa Croce, next to those of Michelangelo and Machiavelli. His epitaph was written for him by posterity: Eppur si muove (nevertheless it moves).
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4
Suggested Reading
Pune University Campus Ganeshkhind
[1]
Arthur Koestler, Sleep walkers, Penguin, 1959.
[2]
Isaac Asimov, Asimov’s Biographical Encyclopedia of Science and Technology, Doubleday. 1982.
[3]
R Spangenburg and D K Moser, The Birth of Science Volume I: Ancient
Pune 411 007, India. Email: [email protected] [email protected]
Times to 1699, Viva Books Pvt. Ltd, 2006.
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Dawn of Science 15. The Affairs of the Heart T Padmanabhan From Nei Ching to Harvey, it took almost four thousand years to probe the mysteries of the heart and blood circulation.
T Padmanabhan works at IUCAA, Pune and is interested in all areas
As the Chinese text below shows, men of medicine, since antiquity, knew the heart played an important role in the human body and that it was ‘somehow’ connected with the flow of blood. Ignorance as to what this connection was remained for long a stumbling block in the advance of physiology.
of theoretical physics, especially those which have something to do with gravity.
“The blood current flows continuously in a circle and never stops. The heart regulates all the blood of the body. (The flow of blood) may be compared to a circle without beginning or end.” – Nei Ching, ancient Chinese medical treatise (about 2500 BC)
Previous parts: Res onance, Vol.15: p.498, p.590, p.684, p.774, p.870, p.1009, p.1062. Vol.16: p.6, p.110, p.274, p.304, p.446, p.582, p.663.
Early Greek scientists, with their blind devotion to Aristotle, believed that blood vessels contained blood and air. It was Galen, the second century Greek physician, who conclusively proved that arteries contained only blood. But he believed that air entered the right side of the heart (from the lungs) and that the flow of blood in the vessels was like tides in the sea – ebbs and flows moving back and forth. The force that prodded this flow came from the contractions of the arteries, the heart playing no significant role. Blood, it was thought, was produced in the liver from where it passed to the right auricle (one of the two upper chambers of the heart) and then to the right ventricle (one of the two lower chambers) and then somehow made its way to the left side where it met blood from the arteries which contained air coming from the lungs. These notions about blood circulation and the heart held sway for nearly fourteen centuries.
Keywords Galen, blood circulation, Vesalius, Fabricius, Harvey.
770
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The anatomical works of the Italian physician, Vesalius, and others showed that there were no holes in the partition of the heart, and how the blood flowed from the right side of the heart to the left became even more of a mystery.
in the sixteenth century. The anatomical works of the Italian physician, Vesalius, and others showed that there were no holes in the partition of the heart, and how the blood flowed from the right side of the heart to the left became even more of a mystery. The solution, though, existed in a book by an Arabic scholar, Ibnan-Nafis, published around AD 1242. There he had suggested that blood was pumped from the right ventricle into the arteries which led to the lungs; having picked up air from the lungs, the blood was brought back to the left ventricle from which it was further pumped to the entire body. (This circular flow of blood from the heart to the lungs and back is now called the ‘lesser circulation’.) But this book was not known to the later western scholars. So the ‘lesser circulation’ was to be independently discovered by a Spanish physician, Mignel Serveto (1511–1553), who published a book on anatomy and blood circulation in 1553. Unfortunately, the book also carried strong unitarian theological views which got Serveto into trouble with John Calvin, who preached a much more extreme version of Protestantism than even Martin Luther did. After publishing the book, Serveto had travelled to Geneva, which at the time was under Calvin’s rule. Calvin had Serveto arrested and burnt at the stake with all the copies of the book. However, the concept (of lesser circulation) surfaced again soon and spread when an Italian anatomist, Realdo Colombo (1516–1559), discovered it independently again and gave it prominence in his lectures. Meanwhile, these ‘new’ ideas as well as the fact that there were no perforations in the walls of the heart were proving difficult to reconcile with Galen’s views that dominated. And the dogmatists persisted. The next disturbing piece of evidence against Galen’s ideas came from the work of Fabricius (1537–1619), who was the professor of medicine at the University of Padua in Italy. Fabricius studied the veins in detail and found that they contained a series of valves, the function of which was unclear. (Fabricius was the student of
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Fallopius, who discovered the Fallopian tubes; Fallopius himself was the student of Vesalius, thereby maintaining the strong medical tradition of Padua.) Fabricius was close to discovering the circulation of blood but, being a strong Galenist, didn’t push his observations to their logical conclusion. That honour finally went to William Harvey (1578–1657), the English physician. Harvey was the son of a very prosperous businessman and had the best of education. He received his first degree in medicine in 1597 at the University of Cambridge. Determined to study medicine further, he went to the best school at the time, at the University of Padua. He spent about two years at Padua working with Fabricius. Returning to London, Harvey climbed very fast on the social and professional ladder. In quick succession, he became a fellow of the College of Physicians, a practising physician at St. Bartholomew Hospital, personal physician to King James I and later to King Charles I. Very courteous and dignified, he was held in affection and respect by his colleagues. He was successful in every way.
Fabricius was close to discovering the circulation of blood but, being a strong Galenist, didn’t push his observations to their logical conclusion.
In spite of active medical practice, Harvey found time to pursue scientific research over a long period (1604–1642). The two years with Fabricius had convinced Harvey that the flow of blood in the body was not well understood and he started a series of simple experiments and dissections to get to the bottom of this problem. By actual dissection, he noticed that there were valves separating the auricles from the ventricles so that blood could flow only from the auricle to the ventricle. He had earlier learnt from Fabricius about the valves in the veins. Combining these observations, he could easily arrive at the full and correct picture of blood circulation. That a fixed volume of blood circulated through the blood vessels and the heart in a definite direction was quite a revolutionary idea. Harvey also found simple and elegant arguments to demonstrate the fact that blood flowed only in a definite direction in the body. For example, he showed that when an artery was tied off, it bulged on the side towards the heart; when a vein was held the same way,
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Figure 3. William Harvey. Courtesy: htt p:// en.wikip edia.org /wik i/ W illiam_Harvey.
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Figure 4. Harvey demonstrating that blood flowed only one way in the veins by pressing blood up from a section of the vein and releasing his fingers. If he releases the upper finger, blood would not flow down. Courtesy: http://www.comptonhistory.com/ compton2/blood.htm
it bulged on the side away from the heart. He also found, by a simple calculation, that the quantity of blood pumped out by the heart in one hour was about three times the weight of a man. Harvey published these results in 1628 in a small book, Exerciratio de Motu Cordis er Sanguinis (On the motions of the heart and the blood), of about 72 pages. It was printed in Holland on very poor paper and was full of typographical errors. Nevertheless, the book became a scientific classic. The Galenists of course ridiculed Harvey but his forceful arguments convinced his colleagues of his new ideas.
Harvey also found, by a simple calculation, that the quantity of blood pumped out by the heart in one hour was about three times the weight of a man.
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The theory of the circulation of blood depended crucially on one question: how did blood flow from the arteries to the veins? Harvey, noting that the arteries and veins branched into finer and finer vessels, had the intuition to guess that the transfer occurred at the finest levels (capillaries) too small to see. This was later confirmed by the Italian physiologist, Marcelo Malpighi (1628– 1694), using the microscope. Harvey was a good friend of King Charles but stayed clear of politics. Because of this, he escaped the wrath of the parliamentary army when civil war started in England in 1642 and Charles I was arrested. The beheading of Charles, in 1649,
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Figure 5. The blood circulation system in the body. Courtesy: http://www.patient.co.uk/ diagram/Heart/lungcirculation.htm.
however, affected Harvey deeply. This loss, and the fact that Cromwell (the leader of the parliamentary forces) always treated Harvey as a suspect, made Harvey quite unhappy in his later years. He died in 1657 at the age of 80. Suggested Reading [1]
Isaac Asimov, Asimov’s Biographical Encyclopedia of Science and Technology, Doubleday, 1982.
[2]
I Harrison, (Compiled), The Book of Firsts, Cassell Illustrated, 2003.
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Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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Dawn of Science 15. The Invisible Weight T Padmanabhan Of vacuum, air pressure and some showmanship.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics,
Though air is a pervasive and powerful agent around, a serious understanding of how air (and gases in general) behaves developed only in the seventeenth century. This followed collective efforts of several persons especially that of Jan Baptista van Helmont (1579–1644) from Belgium, Otto von Guericke (1602– 1686) from Germany, Evangelista Torricelli (1608–1647) from Italy, and Blaise Pacal (1623–1662) from France.
especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498, p.590, p.684, p.774, p.870, p.1009, p.1062. Vol.16: p.6, p.110, p.274, p.304, p.446, p.582, p.663, p.770.
Keywords Otto von Guericke, Magdeburg, Torricelli, barometer.
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Of these, Helmont’s work is probably not so well known. Rather surprising (and a bit unfair), because it was he who first recognized the important fact that air was not a single entity and that more than one air-like substance could be produced by ordinary chemical processes. Like several others before him (notably Paracelsus), Helmont was a physician interested in chemistry. Most of his work revolved around finding a ‘philosopher’s stone’ and other alchemical endeavours and was worthless. In the midst of these, however, he discovered something important. He noticed that his experiments produced several ‘vapours’ which had no definite shape (and took the shape of the containers in which they were kept), behaved like air physically, but had very different chemical properties compared to normal air. In particular he made a detailed study of the vapour produced by burning wood. He called it ‘gas sylvestre’, Flemish for ‘gas from wood’. (Incidentally, he coined the term ‘gas’ which essentially meant ‘chaos’ in Flemish, but the term did not catch on for over a century and was reintroduced by Lavoisier!). He listed the properties of this gas and stressed the fact that it was different from air. In a way, this was the first time three states of matter (solid, liquid and gas) were recognized clearly.
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Figure 1.
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Figure 2.
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The life and work of Guericke was a lot more colourful than that of Helmont. An engineer by profession, he entered politics in 1627 in the midst of the Thirty Years’ War. His home town, Magdeberg, was ransacked by the Imperial Army in 1631 and Guericke barely managed to escape. He joined the army of Gustav II Adolphus of Sweden who turned the tide of the war. Guericke returned to Magdeburg in 1646 and became mayor of the city when it was rising from the ruins.
Figure 3. Otto von Guericke. Courtesy: htt p:// en.wikip edia.org /wik i/ Otto_von_Guericke
Meanwhile, Guericke got interested in the possibility of producing a vacuum. The prevailing view on the subject, again, was the one advocated by Aristotle: ‘Nature abhors a vacuum’. Aristotle had worked out a theory of motion in which a body would move faster if the surrounding medium became less dense. Bodies had to move with infinite speed in vacuum, which, Aristotle thought was impossible; hence, he concluded that a vacuum could not exist. Guericke was unimpressed by this argument and decided to settle it by direct experiment. He built an air pump, similar in design to the water pump which had been used for centuries, and used it to evacuate air from a closed chamber. With his flair for showmanship, Guericke put this pump to good use. He showed that candles would not burn in such a vessel and animals could not live in vacuum. After these simple demonstrations, he got more dramatic. He tied a rope to a piston and had 50 men pull the rope as he produced a vacuum on the other side with his pump. Very soon, the 50 men could not pull the piston out against ‘the force of the vacuum’.
Very soon, the 50 men could not pull the piston out against ‘the force of the vacuum’.
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He went on to do an even more dramatic experiment, involving the famous ‘Magdeburg hemispheres’. These were two hemispheres which fitted together along a greased edge. When the hemispheres were put together and the air inside was evacuated, the hemispheres could not be separated even by two teams of horses. And when he let air enter inside, they fell apart by themselves. Guericke, the showman that he was, arranged a demonstration in 1654 for the Emperor, Ferdinand III. The Emperor was highly impressed.
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Figure 4. Guericke demonstrates the effect of vacuum. Courtesy: h tt p: / /www. eoh t. in f o/ p hoto/ 1758074Guericke’s+piston+and+ cylinder+demonstration
Did Guericke understand why these demonstrations were successful? May be he did, but the clear enunciation of the basic principles came from Torricelli. A mathematician from the University of Rome, Torricelli was deeply attracted by Galileo’s books. So much so that he went to meet Galileo and volunteered to serve as a secretary during the last three months of Galileo’s life. Galileo pointed out to him a peculiar problem which needed explanation.
Figure 5. The Magdeburg experiment. Courtesy: htt p:// en.wikip edia.org /wik i/ Magdeburg_hemispheres)
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The problem was as follows. Pumps used to raise water from one level to another worked on pistons. How these pumps worked was explained on the basis of Aristotle’s principle: ‘Nature abhors a vacuum. So when the piston is raised, a vacuum will be created inside the pump unless the water level inside rises. To avoid the vacuum, water rushes in.’ There was however one problem. People who used these pumps knew that the pumps could raise water only to about 33 feet (10 metres). Even when longer pumps were used water just climbed up to about 33 feet and stopped (see Figure 6). Galileo found it strange that nature abhorred a vacuum to a certain limit but gave up after that. He suggested that Torricelli look into this strange behaviour.
Galileo found it strange that nature abhorred a vacuum to a certain limit but gave up after that.
Figure 6. (a) Water rises inside the tube when the piston is moved up. (b) Water does not rise above 33 feet even when the piston is raised higher.
(a)
It occurred to Torricelli that the entire phenomenon had a simple mechanical explanation. “Suppose air above us had some weight,” reasoned Torricelli, “then it will push on the water surface and make it rise inside the pump when the piston is raised. However, suppose the total weight of the air could only support about 33 feet of water. Then no pump with piston can raise the water higher. All this talk about ‘vacuum-abhorence’ is irrelevant.”
Torricelli went further. Since mercury was known to be about 13.5 times heavier than water, air could only support a column of mercury which is 13.5 times smaller compared to water which works out to about (33 feet / 13.5 ), that is, 30 inches (76 cm). Torricelli filled a four-foot long (b) tube (closed at one end) with mercury, put his thumb to close the other end and inverted the tube into a large dish of mercury. On releasing the thumb, mercury flowed from the tube into the dish, but not all of it. Nearly 30 inches of mercury remained in the tube, clearly proving Torricelli right. This was in 1643 (seven years before Guericke’s air pump) and must be considered a milestone in science. Not much for the intrinsic importance of the result, but for a clear and simple demonstration
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of the ‘method of science’. The first step was in realizing that a specific, unexplained fact existed (namely, water pumps did not work for more than 33 feet), and the second step was in postulating an explanation (that air had weight equivalent to that of 33 feet of water). Several other explanations could have been offered but what distinguished the scientific one from the others was the crucial step: a prediction was made using a hypothesis (mercury could be supported for nearly 30 inches by the weight of air). And finally, the prediction was tested and verified. The Aristotalian explanation, ‘nature abhors a vacuum’ was not only wrong but also deficient in predictive power.
This was the first human-made vacuum and is now called Torricelli vacuum in his honour.
The part above the mercury in the inverted tube was a vacuum (except small quantities of mercury vapour). This was the first human-made vacuum and is now called Torricelli vacuum in his honour. There was, however, a question: If the weight of the air supported by the mercury in the tube, then the mercury level should drop when a barometer (which was how Torricelli’s tubes were called) was taken up a mountain. This verification was designed by Pascal. Being a weak and chronically sick man, Pascal did not attempt the task himself but persuaded his brother-in-law to carry the barometers up the sides of Puy de Dome in France in 1646. The helpful relative climbed about a mile and found that the mercury had dropped by a few inches, as expected. Pascal also literally introduced true French spirit into these studies. He made a barometer using red wine (which is lighter than water) and verified that a column of about 14 meters could be supported!
Address for Correspondence T Padmanabhan
Suggested Reading
IUCAA, Post Bag 4 Pune University Campus
[1] Isaac Asimov, Asimov’s Biographical Encyclopedia of Science and Technology, Doubleday, 1982. [2] J Hakim, The Story of Science, Smithsonian Books, 2005.
Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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Dawn of Science 17. Geometry Without Figures T Padmanabhan Descartes discovered the link between algebra and geometry, which forms the cornerstone of applied geometry today.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics,
There is a story about Ptolemy Soter, the first king of Egypt and founder of the Alexandrian Museum, who studied geometry under Euclid. He found the subject rather difficult and apparently asked his teacher whether there is an easier way of learning geometry. To which he received the famous reply: “There is no royal road to geometry.”
especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498, p.590, p.684, p.774, p.870, p.1009, p.1062. Vol.16: p.6, p.110, p.274, p.304, p.446, p.582, p.663, p.770, p.854.
Keywords Descartes, analytic geometry, Fermat, W iles.
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In a sense, Euclid was wrong. There is a way of doing geometry using algebra which is considerably simpler and conceptually more straightforward. The discovery of this connection between algebra and geometry was definitely a milestone in science. One person who contributed most to this subject was René Descartes (1596–1650), the French philosopher and mathematician. (Descartes used to sign in the Latinized version of his name ‘Cartesins’, because of which both his systems of geometry and philosophy go under the name ‘Cartesian’.) Descartes was born in 1596 in France. He was a brilliant but very unhealthy student and obtained from his teachers the concession to remain in bed, as long as he wished, everyday – a habit which he continued to his adult years. His early Jesuit education made him extremely devout and faithful. In 1633, when he heard about Galileo’s fate, he abandoned the idea of writing a book in support of the Copernican theory. Instead, he came out with an astronomical model in which the Earth was in the centre of a ‘cosmic vertex’ with the vertex travelling around the Sun. Though many people accepted this compromise, it was worthless as an astronomical model.
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Figure 1.
WHEN
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Figure 2.
After his education, Descartes joined the French army. Fortunately, he was never exposed to actual warfare and hence had a lot of spare time to work out his ideas. It was during this time that Descartes seems to have stumbled upon an important discovery, from which originated the branch of ‘analytic geometry’. The
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Figure 3. Descartes. Courtesy: htt p:// en.wikip edia.org /wik i/ Rene_Descartes
basic idea was extremely simple. It was well known (right from the days when maps were used) that the position of a point on a two-dimensional surface could always be represented by two numbers. For example, the location of any city on the Earth could be specified by the latitude and the longitude. Similarly, the position of any point on this paper could be denoted by giving its distance from the bottom edge of the paper and from the left end. Descartes realized that this fact allowed any curve in geometry to be represented by an algebraic equation. For example, consider the equation y = x2. If we now give for x a series of values like 1, 1.5, 2, 2.5, etc., we will obtain for y the values 1, 2.25, 4, 6.25, etc., respectively. Each pair of these numbers – that is, (1, 1), (1.5, 2.25), (2, 4), (2.5, 6.25) – represents a point in the plane. By connecting these points, we obtain a smooth curve, which is unique. We have thus coded the information about the geometrical curve into the algebraic equation y = x2. We can now methodically translate all the basic geometrical relations concerning the plane figures into equivalent algebraic statements. Once this is done, any property of a geometrical figure can be
Figure 4. A correspondence can be established between any function, say, f = x2, and a curve in the plane. The function provides a set of pairs of numbers; in the case of f(x) = x2, f = 0 at x = 0; f =1 at x = 1; f = 4 at x = 2; f = 9 at x = 3, etc. These sets of points can be connected by a smooth curve. Similarly, any geometric curve can be expressed as an algebraic function.
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Figure 5. Any point on a plane can be specified by giving two numbers. We first draw reference lines (OY, OX) and mark them at equal intervals. Distance to the ‘left’ or ‘below’ are treated as negative. The point A can be specified by its distance from OY (2.4 units) and its distance from OX (1.5 units), in that order. Similarly, to denote a point below the OX axis (say B), we specify the corresponding distance with a negative number; as –2 in this case.
worked out using purely algebraic operations without ever drawing the figure. The connection described above is extremely powerful and forms the cornerstone of applied geometry today. Its power stems from two facts. First, this connection between algebra and geometry provides a systematic procedure for deriving and proving geometrical relations; for example, suppose we have to prove that the area of a triangle with sides a, b, c is the square root of the expression s(s – a)(s – b)(s – c), where s is half the perimeter of the triangle. In conventional geometry, there is no obvious way of proving such a result. One has to draw the figure of the triangle, introduce clever constructions and use appropriate theorems to arrive at this result. Analytic geometry provides a brute-force method for tackling the same problem. We can represent the three vertices of the triangle by three pairs of numbers : (x1, y1), (x2, y2), (x3, y3). By using the rules of analytic geometry – and without requiring much creativity or imagination – one can write down the expression for the area of a triangle and, comparing it with the given expression, one can easily prove the result. The second reason why analytic geometry is so powerful is that it
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The connection described here is extremely powerful and forms the cornerstone of applied geometry today.
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The second reason why analytic geometry is so powerful is that it extends enormously the range of geometrical curves and shapes that can be studied.
Suggested Reading [1]
E T Bell, Men of Mathematics,
(Touchstone)
Simon and Schuster, 2008. [2]
Ho war d Ev es, Gre at Moments in Mathemat-
extends enormously the range of geometrical curves and shapes that can be studied. Conventional Greek geometry was restricted to straight lines, circles, conic sections and some simple extensions of these. But now we see that every equation of the form y = f (x) represents some curve in the plane whose properties can be studied. In this sense, algebra has enriched the scope of geometry. Descartes was led to this discovery through his mechanistic view of the world. He tried to think of all phenomena in terms of the motion of mechanical gadgets; to him, a geometrical curve was essentially a path traced by an object moving in a particular way. Descartes published his results as the last of three appendices accompanying his treatise, Discours de la methode.... (Discourse on the method) which dealt with the proper way of conducting logical study in various branches of science. Most of the material in the treatise is trivial and useless; however, the third appendix contains the real gem. (It is rather amusing to note that there has been at least another occasion in mathematics when the appendix was more important than the text. In 1831, several key ideas in non-Euclidean geometry appeared as an appendix in a book by Bolyai!)
ics / Before 1650, The Mathematical Association of America, 1983.
In 1831, several key ideas in nonEuclidean geometry appeared as an appendix in a book by Bolyai!
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After his service in the army, Descartes returned to protestant Holland where he stayed almost till the end of his life. During this period Queen Christina ruled Sweden and she was one of the most eccentric queens who ever lived. It had become fashionable in Europe at this time for the royalty to invite intellectuals to their courts and pretend a keen interest in matters of the mind. Following this trait, Christina invited Descartes to the Swedish court, which he – unfortunately – accepted. Christina made Descartes call on her three times each week, at five in the morning, during one of the worst Swedish winters. It is not known whether the queen grew in intellect as a result of this exercise; poor Descartes, however, caught pneumonia and died. In protestant Sweden, he was buried in a cemetery reserved for unbaptized children, because he was a Catholic. In 1667, his remains were carried to Paris and buried again – but not with much pomp and splendour since it was thought that his views were ‘too hot’. Finally, he was
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Box 1. Fermat’s Last Theorem Fermat (1601–1665) was a contemporary of Descartes and has been often called the ‘prince among amateur mathematicians’. A counsellor at the French Parliament, he devoted much of his spare time to mathematics. He made contributions in developing the theories of probability and analytic geometry and had some notion of calculus. But what probably made him ‘immortal’ was his contributions to the theory of numbers – in particular, a note he scribbled on the margin of a book (Arithmetica of Diophantus). This marginal note said that an equation of the form xn + yn = zn cannot be satisfied with x, y, z and n being integers with n > 2. (When n = 2, there are several solutions like 32 + 42 = 52 and 52 + 122 = 132, etc.). Fermat added that “I have discovered a truly marvellous proof of this, which, however, the margin is not large enough to contain.” Many a later mathematician has wished that Arithmetica had a wider margin! Fermat had also stated several other ‘theorems’ in his correspondence and in all but one case, he was proved right (in one case he was proved wrong). Only the ‘last theorem’ quoted above remained for a long time without being proved right or wrong. A tremendous amount of research went into investigating this theorem and new vistas of mathematics have originated from these studies. This theorem also led to dozens of false proofs year after year. Since the statement of the theorem is so simple, it is accessible to anyone who has a basic education in algebra. Many amateurs try their hand at it without even appreciating the true problem. To be fair to them, it must be said that even some professional mathematicians have been guilty of publishing wrong proofs! In June 1993, Andrew Wiles presented what he considered to be the proof of Fermat’s Last Theorem using fairly sophisticated mathematical techniques. Interestingly enough, a critical portion of the proof contained an error which was caught by several mathematicians refereeing Wiles’ manuscript. Wiles, along with his former student Richard Taylor, had to spend nearly another year to take care of this difficulty and the final result was submitted to journals in the form of two papers (one by Wiles and the other co-authored with Taylor) in October 1994, about 358 years after Fermat first conjectured it!
disinterred during the French Revolution and was buried once again with other French thinkers in the Pantheon. Almost at the same time, there was another French man who developed parts of analytic geometry independently. This was Pierre de Fermat (1601–1665), a parliament counsellor who devoted only his spare time to mathematics. He worked in mathematics ‘for fun’ and developed the frustrating habit of stating theorems in the margins of books and in private correspondence. He developed analytic geometry in two and three dimensions around 1630 but never bothered to publish it. We only know of this fact from a letter he wrote to his friend in 1637.
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Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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Dawn of Science 18. The Questions of Life T Padmanabhan The idea of ‘spontaneous generation’ of life comes under rigorous scrutiny – with the help from an unexpected quarter, the advent of microscopy. In Shakespeare’s Antony and Cleopatra, there is a character, Lepidus, who, in a drunken mood, proclaims, “Your serpent of Egypt is bred now of your mud by the operation of your Sun; so is your crocodile.” Neither Shakespeare nor the Romans believed that snakes and crocodiles came out of mud spontaneously; they had definitely seen eggs being laid and hatched. However, people were not so sure, in the sixteenth century, about smaller insects. The prevailing idea was that small creatures like worms and vermins do come out of filth and mud spontaneously.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts:
Such a view accepted two different ways of creation of life. Every ancient civilization knew how horses and cattle, say, produced their offspring. Many of these civilizations also knew how to selectively breed these animals to enhance quality. It was never suggested that horses came out of mud or farmland. But ancient thinkers, including Aristotle, assumed that it was different when it came to much smaller creatures.
Res onance, Vol.15: p.498, p.590, p.684, p.774, p.870, p.1009, p.1062. Vol.16: p.6, p.110, p.274, p.304, p.446, p.582, p.663, p.770, p.854, p.950.
The reason for this dichotomy is not far to seek. It was almost an everyday experience to see maggots appearing in rotting meat. Bur nobody had ever seen a horse appearing out of earth. Thus arose the doctrine of ‘spontaneous generation’ which claimed that certain kinds of creatures could come up spontaneously. Doubts about this doctrine first arose in the seventeenth century following the work of Francesco Redi (1626–1697), a physician (and poet!) from Tuscany, Italy. Around the time Redi was born, William Harvey had published a book in which he had suggested
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Keywords Francesco Redi, Antony Van Leeuwenhoek, Marcello Malphigi, Lazzaro Spallanzani, John Ray, Carl Linnaeus, microscopes.
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Figure 1.
Figure 2.
WHEN
WHERE
Figure 3. Francesco Redi. Courtesy: htt p:// en.wikip edia.org /wik i/ Francesco_Redi
that probably small creatures came from eggs too tiny to be seen. Redi read it and thought of a simple experiment to settle this issue. Redi had noticed that decaying meat not only produced flies but also attracted them in large numbers. It occurred to him that a first generation of flies could be laying eggs from which the second generation originated. He prepared eight different flasks with a
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variety of meat in them. He sealed four of them airtight and left the other four open. The flies could now land only on meat in the open flasks and indeed only the four open flasks produced maggots. Meat in the closed flasks was as putrid and smelly but there were no living creatures. Redi even repeated the experiment by not sealing the flasks but covering them with gauze; this let in air but not the flies. And again no life form developed in the covered meat. This was probably the first clear biological experiment with a controlled sample. Redi’s experiment proved that flies originated from other flies and not spontaneously from decaying meat. However, this did not make people give up the older idea. Even Redi believed that there could be very small creatures, which actually came into life spontaneously even though flies were not born this way.
Figure 4. Antony van Leeuwenhoek. Courtesy: htt p:// en.wikip edia.org /wik i/ Antonie_van_Leeuwenhoek.
1
This idea, interestingly enough, gained support from the advent of microscopy allowing one to see very tiny creatures. Even when Galileo developed the telescope, he realised that there could also be an arrangement of lenses, which would magnify small objects. The theory of microscopes was developed by Kepler and Torricelli and by the mid-seventeenth century several investigators started using microscopes to study biological specimens.
See Resonance, Vol.16, No.1,
2011.
Figure 5. Leeuwenhoek’s microscope. Courtesy: htt p:// en.wikip edia.org /wik i/ Antonie_van_Leeuwenhoek.
The most famous among them was Leeuwenhoek1 (1632–1723) from Delft, Netherlands. Making microscopes was Leeuwenhoek’s passion and he used his lenses to observe virtually everything around him. His microscope consisted of a single lens which was ground to perfection so as to magnify objects as much as 200 times. In 1675, he discovered ‘living things’ in ordinary ditch water, which were too small to be seen by the naked eye. These ‘animalcules’ – now called protozoa – were as alive as an elephant or a man. He found that the yeast, used in making bread for ages, was actually made of tiny living creatures much smaller than even the ‘animalcules’. Finally, in 1683, Leeuwenhoek observed still smaller organisms, which we now call ‘bacteria’. (The first one to systematically study biological specimens using microscopes probably was Marcello Malphigi (1628–1694). He
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was an Italian physician who lectured in several Italian universities and especially in Bologna. He began biological investigations by studying the lungs of frogs and almost guessed the process of respiration. He also discovered very fine blood vessels allowing the transfer of blood from arteries to veins.)
Figure 6. Marcello Malphigi. Courtesy: htt p:// en.wikip edia.org /wik i/ Marcello_Malpighi
2
See Resonance, Vol.12, No.1,
2007.
Figure 7. Lazzaro Spallanzani. Courtesy: htt p:// en.wikip edia.org /wik i/ Lazzaro_Spallanzani
These tiny living organisms made people believe once again in spontaneous generation. Meanwhile, Leeuwenhoek learnt how to make a broth (by soaking pepper in water) in which the protozoa could multiply. In fact, such broths seemed to produce protozoa on their own. Even when a broth was boiled and filtered to eliminate any sign of protozoa, it showed signs of these organisms within a short time thereafter. And many took this to be a clear sign of generation of life from non-life. There were, however, a few sceptics. One among them, Lazzaro Spallanzani (1729–1799), repeated the experiment with the broth by sealing off the neck of the flask which contained the broth. He found the broth did not develop microscopic life; but the adherents of spontaneous generation maintained that the heating had removed the ‘vital spirit’ from the broth. The issue was not settled until much later, until the time of Louis Pasteur 2. Around the same time, biological understanding was growing on another front. The ancient world probably knew of only a few hundred species of living beings. Aristotle, the keenest observer among the Greeks, could list only about 500 species of animals and his student and famous ancient Greek botanist Theophrastus, knew of only about 500 species of plants. In the late medieval days, several naturalists attempted to enlarge the listing of animal species and also to produce a systematic classification of them. The classification of animals is far more difficult than one would imagine at first sight. Consider, for example, the definition of a ‘bird’. Calling it a ‘two-legged’ creature will make man a bird, while calling it a ‘winged creature’ will make the bat a bird. (Incidentally, an eighteenth century naturalist once told Voltaire that the briefest definition for a man would be a ‘featherless
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biped’ to which Voltaire retorted that biologists have made a man out of a plucked chicken!) Given such dangers, it was not surprising that the English biologist, John Ray (1627–1705), had to spend nearly a life-time in classifying plants and animals. In 1667, after many years of painstaking travel and observations, he published a catalogue of plants found in the British Isles. Later, during 1686 to 1704, he published a three-volume encyclopaedia describing over 18,600 plants. He also published, in 1693, the first logical classification of animals based chiefly on the hoofs, toes and teeth of the animals. This classification survived for a long time, until the Swedish naturalist, Linnaeus3 (1707–1778), produced a far more detailed classification.
Figure 8. John Ray. Courtesy: htt p:// en.wikip edia.org /wik i/ John_Ray
The greater understanding of the animal kingdom acquired in all these investigations paved the way for tackling much wider issues. The schemes of classification clearly showed two features: the number of species which inhabit the Earth was much larger than what the ancients had imagined, and there were some patterns of similarity between animals of very different kind when viewed from a fundamental basis. (One familiar example is between the domestic cat and the tiger.) It was not long before people started wondering where all these different species came from. Indeed, if different species produced only offsprings of their own kind, it followed that all the species must have existed from time immemorial. On the other hand, if species ‘changed’, it must have been at a very slow rate and it would have taken an incredibly long time to produce the kind of variety which we see. Questions of this nature were the ones which prompted Darwin to his remarkable discovery. Suggested Reading
3
See Resonance, Vol.5, No.6,
2000.
Figure 9. Carl Linnaeus. Courtesy: htt p:// en.wikip edia.org /wik i/ Carl_Linnaeus
Address for Correspondence T Padmanabhan
[1]
Isaac Asimov, Asimov’s Biographical Encyclopedia of Science and Technology, Doubleday, 1982.
[2]
R Spangenburg and D K Moser, The Birth of Science – Volume I, Viva Books Pvt. Ltd., 2006.
IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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Dawn of Science 19. Measuring the Heavens T Padmanabhan His improved telescope gave Huygens a better vision of the sky, but his ideas about light lay buried for a century.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498, p.590, p.684, p.774, p.870, p.1009, p.1062. Vol.16: p.6, p.110, p.274, p.304, p.446, p.582, p.663, p.770, p.854, p.950, p.1039.
Keywords Huygens, pendulum c lock, Saturn's rings, Titan, Cassini.
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After the Copernican Revolution, physics – or ‘natural philosophy’ as it was then called – was taking rapid strides with contributions from Descartes, Torricelli, Boyle, Pascal and many others. One of the important contributions to physics and astronomy in the later half of the 17th century came from Christian Huygens (1629–1695) from the Netherlands. Unfortunately, the importance of his work was overshadowed by the towering dominance of Newton’s work. Born in a wealthy family in The Hague, Huygens had an excellent education. Among his father’s friends were several intellectuals of Europe with whom he corresponded regularly. In 1645, Huygens went to the University of Leiden where he studied law and mathematics. His first breakthrough came in 1655. While helping his brother to make a telescope, Huygens stumbled upon a new and better method of grinding lenses, which enabled him to build telescopes with much higher resolutions. He promptly built an astronomical telescope, about 23 feet (7 metres) long, and started scanning the sky, coming up with several important discoveries. The first was a huge cloud of dust and gas in the constellation of Orion, which we now call the Orion nebula. Huygens also found a satellite of Saturn, which was as large as any of the Jovian satellites, and named it ‘Titan’. (In Greek mythology, Saturn is the leader of a group of gods called the Titans.) With this discovery, there were now an equal number of planets and satellites in the solar system (six each). Huygens fell into the natural trap and declared to the world that no more satellites or planets remained to be discovered; within his lifetime, four more
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WHEN
Figure 1.
WHERE
Figure 2.
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satellites were seen! Huygens’ telescope also showed that Saturn was surrounded by a thin ring, which did not touch the planet. These rings were something unique and attracted quite a lot of attention.
Figure 3. Christian Huygens. Courtesy: htt p:// en.wikip edia.org /wik i/ Christiaan_Huygens
Figure 4. Huygens pendulum clock. Courtesy: http://www.timekeepingsite.org /clock.htm
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Huygens also tried to make quantitative estimates in several areas of astronomy. He was the first to make a serious effort to determine the distances to the stars. Assuming that the brightest star Sirius was as bright as the Sun, he estimated its distance to be 2.5 trillion miles. (The actual distance turned out to be 20 times larger because Sirius is actually much brighter than the Sun. One shouldn’t blame Huygens for this. Errors of similar nature are, for similar reasons, routinely made in astronomy!) In the course of making these quantitative measurements, Huygens realised the need for accurate measuring devices. He developed two such instruments. The first was a micrometer with which he could measure angular separations as small as a few arc-seconds. The second was a pendulum clock, which brought Huygens fame and glory. The best clocks available in Huygens’ days were those designed in the Middle Ages. These were complicated mechanical devices powered by falling weights. Though such apparatus were adequate to produce fancy-looking decorative pieces for royal courts, their accuracy was limited to about a fraction of an hour; this made them useless for scientific studies. To build a more accurate clock, one needed a device which moved with a constant period. It was known from Galileo’s time that a pendulum did this fairly accurately. All that one needed to do was to connect the pendulum to suitable gear wheels and attach falling weights to supply the energy lost due to friction. This was basically what Huygens did and the first ‘grandfather clock’ was born. To improve accuracy, Huygens added one crucial refinement. He knew that the pendulum kept a constant period only approximately, so he adjusted the movement in such a way that the period would remain exactly constant. And his clock was accurate to a fraction of a minute.
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The pendulum clock spread Huygens’s name throughout Europe. He was elected a member of the Royal Society in 1663 and was invited to the court of Louis XIV in Paris in 1666. He did spend some time in Paris but returned to the Netherlands in 1681 when Louis XIV started promulgating ordinances against Protestants (Huygens was a Protestant). Yet, Huygens’s most significant contribution did not get any recognition during his lifetime. This related to the propagation of light. Huygens firmly believed that the propagation of light could be understood as a wave phenomenon. But this idea violently conflicted with the prevailing notion that light rays were corpuscular (that is, made of tiny particles) in nature. The strongest evidence against Huygens’s model came from everyday experience – that light always travelled in a straight line. This was contrasted with typical wave propagation like that of sound; while sound waves can ‘bend around’ obstacles, light cannot. (That is why you can hear someone around a bend even when you can’t see that person.) Huygens showed that there were indeed conditions under which light may appear to travel in a straight line even though it was fundamentally a wave. Unfortunately, no one took him seriously (partly because of the dominating influence of Newton’s concepts which assumed that light was made up of particles). And so Huygens’ ideas about light lay buried for another century. Meanwhile, astronomical discoveries similar to those of Huygens were also made by the French astronomer Giovanni Cassini (1625–1712) around the same time. Cassini began with a detailed study of Jupiter’s moons and measured Jupiter’s period of rotation. He then turned to Saturn and discovered four new satellites which were named Lapetus, Rhea, Dione and Tethys. In this process, he proved Huygens’ conjecture of ‘equal number of satellites and planets’ wrong· He also showed that the ring around Saturn was a double one; the gap between the rings is still called the Cassini division.
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Figure 4. Grandfather clock. Courtesy: htt p:// en.wikip edia.org /wik i/ Longcase_clock
Cassini’s most valuable contribution was probably the measurement of the distance to Mars.
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Figure 5. Saturn. Courtesy: http://en.wikipedia.org/wiki/ Saturn
Figure 6. Cassini Division.
Cassini calculated the distance between the Earth and the Sun to be about 87 million miles.
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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Cassini Division
Cassini’s most valuable contribution was probably the measurement of the distance to Mars. By comparing his own observations of the position of Mars with those made by the French astronomer, Jean Richer (1630–1696), he could arrive at the distance to Mars. The relative distances of the Sun and the planets were known accurately since the days of Kepler. Hence, knowing any one distance, every other distance could be determined. In particular, Cassini calculated the distance between the Earth and the Sun to be about 87 million miles. Only about 7 per cent lower than the actual distance, this was the first measurement which gave a value that was even nearly right. The previous estimates were in the range of five to 15 million miles. Suggested Reading [1]
Joy Hakin, The Story of Science, Smithsonian Books, 2005.
[2]
Isaac Asimov,
Asimov’s Biographical Encyclopedia of Science and
Technology, Doubleday, 1982.
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Dawn of Science 20. Calculus is Developed in Kerala T Padmanabhan
Foundations of calculus were developed by a school of mathematicians in Kerala during 1400{ 1600, years before similar developments took place in the west. T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which
An ancient text from Kerala has a verse, giving the circumference of a circle, the translation of which goes as: \Multiply the diameter by four. Subtract from it and add to it alternatively the quotients obtained by dividing four times the diameter to the odd numbers 3,5 etc. "
have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498, p.590, p.684, p.774, p.870, p.1009, p.1062; Vol.16: p.6, p.110, p.274, p.304, p.446, p.582, p.663, p.770, p.854, p.950, p.103; Vol.17: p.6.
Keywords Yuktibhasha, Madhava, tantrasamgrha, Kerala calculus.
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In modern notation this reduces to a remarkable in¯nite series expansion for ¼=4 given by ¼ 1 1 1 = 1 ¡ + ¡ + ¢¢¢ : 4 3 5 7
(1)
This result { usually attributed to Gregory (1638{1675) and others who were to be born more than a century after the text in question was authored { is one of the many gems to be found in the ancient Kerala texts (especially the one called Yuktibhasha) which laid the foundation for a branch of mathematics that we now call calculus. The discovery of the contributions from the Kerala school of mathematicians is comparatively recent and many details are still being probed extensively. This is inspite of the fact that some of these ancient Kerala texts have been referred to explicitly in an article by C M Whish who { having learnt Malayalam and collected palm leaf manuscripts from Kerala { found to his astonishment a \complete system of °uxions" in them. He published a paper in 1834 in the Transactions of the Royal Asiatic RESONANCE February 2012
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)
WHEN
Figure 1. Figure 2.
WHERE
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Virtually all these results were obtained by using techniques which can be thought of as belonging to calculus
Society of Great Britian and Ireland with a fairly explicit title `On the Hindu quadrature of the circle and the in¯nite series of the proportion of the circumference to the diameter exhibited in the four sastras, the Tantrasangraham, Yukti-Bhasha, Carana Padhati and Sadratnamala'. In addition to the result quoted above, these texts contain the in¯nite series expansion for tan¡1 (x) (from which the above result is but one step; you just put x = 1), the series expansion for sin and cos, and the inde¯nite integral of xn { just to list a few. Virtually all these results were obtained by using techniques which can be thought of as belonging to calculus.
The infinite series for sine, cosine, arctan as well as rudiments of integration are attributed to Madhava by many later sources.
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Who are these mathematicians and how did they develop these techniques? Though many of the details are still somewhat sketchy, historians of Indian mathematics have now put together the following picture. Most of these developments took place in villages around a river called Nila in the ancient days (and currently called river Bharatha, the second longest river in Kerala) during 1300{1600 or so. One of the key villages was called Sangama-grama in the ancient times and is thought of as referring to the village Irinhalakkuta (about 50 km to the south of Nila) in present day Kerala. (There are a few other candidates, like Kudalur and Tirunavaya, for which one could raise arguments in favour of them being the Sangama-grama; so this issue is not completely settled.) What is more certain was the existence of a remarkable lineage of mathematicians in Sangama-grama of which Madhava (» 1350{1420) seems to be the one who discovered many of the basic ideas of calculus. The in¯nite series for sine, cosine, arctan as well as rudiments of integration are attributed to Madhava by many later sources. He was strongly in°uenced, like many other Indian scholars of that period, by Aryabhata { which is no surprise since Aryabhateeyam (circa 500 AD) was a very in°uential text in India as well as (through its
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translations) in the Arab world and medieval Europe. Madhava founded a lineage which lasted possibly till the late seventeenth century. His student Parameshwara (» 1360{1455) was an astronomer-mathematician who had authored more than two dozen works and made astronomical observations for a very extended period of nearly ¯ve decades. It appears that his son Dhamodara was the teacher of another key ¯gure, Neelakanta Somayaji (» 1450{1550) who is the author of Tantrasangraha which contains extensive discussions on astronomy and related mathematics. Another student of Dhamodara was Jyeshtadeva (» 1500{1610), the author of Yuktibhasha which probably has the clearest exposition of some of the topics in calculus. The quotation in the beginning of this article is quoted in Yuktibhasha attributing it to Tantrasangraha which, in turn, attributes the result to Madhava. Similarly there is another detailed verse in Yuktibhasha which attributes to Madhava the in¯nite series expansion for arctan. The ¯rst sentence of Yuktibhasha states that it is going to explain all the mathematics \... useful in the motion of heavenly bodies following Tantrasangraha" but actually does much more as an independent treatise than work merely as a bhasya. (It is also remarkable for being written in Malayalam { not Sanskrit { and is in prose, not poetry.) Unlike Tantrasangraha, this work provides detailed argumentation for the results and is written in a style which is straightforward and unpretentious. (An English translation with detailed commentary is now available [1]). Here, I will concentrate on two speci¯c results in Yuktibhasha and highlight the role of calculus, as we call it today, in it. Let us begin with the series expansion for ¼=4. To understand what is involved in obtaining the in¯nite series expansion for ¼=4 given above, it is useful to recall the procedure in modern notation. If t = tan µ
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Another student of Dhamodara was Jyeshtadeva (~ 1500–1610), the author of Yuktibhasha which probably has the clearest exposition of some of the topics in calculus.
Unlike Tantrasangraha, Yuktibhasha provides detailed argumentation for the results and is written in a style which is straightforward and unpretentious.
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It is clear that the derivation involves the concepts of differentiation and integration in some form as well as the knowledge of indefinite integrals of powers.
with 0 µ (¼=4), then (dt=dµ) = 1 + t2 . Rewriting this equation by taking the reciprocals (which is actually a rather insightful step!) and using the geometric series expansion, one gets (dµ=dt) = 1 ¡ t2 + t4 + ¢ ¢ ¢ . Integrating both sides between 0 and 1 will give you tan¡1 (1) = ¼=4 as an in¯nite series. Of course, the inde¯nite integral also gives the in¯nite series expansion of µ in terms of tan µ. It is clear that the derivation involves the concepts of di®erentiation and integration in some form as well as the knowledge of inde¯nite integrals of powers. How did the ancient mathematicians do this? The basic idea was to consider the geometry of the circle and give meaning to all the relevant quantities in terms of suitably de¯ned geometrical constructions . In Figure 1, AB is a unit tangent at A, and P is an arbitrary point on AB so that the length AP gives t = tan µ. Our problem then reduces to ¯nding the length of the arc AP0 corresponding to the length AP which will give us µ as a function of t. Yuktibhasha begins by dividing the line AB into a large number n of equal segments marking out A0 = A; A1 ; A2; :::; An = B with the line OAi intersecting the arc at Qi . The key insight is to realize that as the half chord of a segment gets smaller and smaller, its length approaches the arclength. In modern language
Figure 3. Construction used in Yuktibhasha.
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language this corresponds to the realization sin(µ=n) ! µ=n as n ! 1. There is an explicit statement in Yuktibhasha to this e®ect: \If the segments of the side ... are very very small, these half chords will be almost the same as arc segments." With this realization, as well as some very innovative geometrical reasoning, one obtains the di®erential relationship ±µ = [±t=(1 + t2)] between ±t = (1=n) (given by Ai Ai+1) and ±µ (given by Qi Qi+1 ) which is the relevant arc length in modern notation.
“If the segments of the side ... are very very small, these half chords will be almost the same as arc segments.”
To get µ we now need to sum the series (1 + i2 =n2 )¡1 from i = 1 to n and ¯nally take the limit n ! 1. Yuktibhasha, of course, cannot do this directly and hence resorts to expanding the right-hand side into a series: ±µi =
1 i2 i4 ¡ 3 + 5 ¡ ¢¢¢ : n n n
(2)
(Interestingly enough, this is done by a powerful technique of recursive re¯ning { called samskaram, in Yuktibhasha.) This reduces the problem to one of ¯nding the sums of powers of integers. While the sum of ik was known already to Aryabhata for k = 1; 2; 3, one did not know the result for higher values of k. Once again Madhava uses the fact that he only needs the result for large n and any sub-dominant term can be discarded. This limiting process is done with great care in Yuktibhasha, ¯nally leading to the result (called samkalitam, or `discrete integration') which in modern notation gives lim
n!1
n 1 X
nk+1
i=1
ik =
1 : k+1 k
(3)
This result (equivalent to integrating x and obtaining xk+1 =(k + 1)) was obtained using a procedure which reduces the sum of k-th powers to the sum of (k ¡ 1)-th power, a technique we now call mathematical induction. From reading the relevant section of Yuktibhasha it is clear that its author was fully aware of the di®erence
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This limiting process is done with great care in Yuktibhasha, finally leading to the result (called samkalitam, or ‘discrete integration’)
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The clarity of the exposition regarding ideas like limiting process and infinitesimals shows how adept Madhava was at manipulating these to achieve his goal.
between this method of proof and the more traditional approaches usually based on geometrical reasoning; in fact, this is probably the ¯rst known instance of proof by mathematical induction in Indian mathematics. With this step completed, the result for the series can be obtained fairly easily. The clarity of the exposition regarding ideas like limiting process and in¯nitesimals shows how adept Madhava was at manipulating these to achieve his goal. Equally { if not more { innovative is the approach used to obtain the series for sines and cosines. The ¯rst part of the analysis is again geometrical and closely parallels the discussion given above. By similar reasoning one can obtain the di®erential (again in modern notation) d(sin µ) = cos µ dµ. In fact this is done with a clever geometrical construction leading to a discretized di®erence formula for sine and cosine of the form ±si ´ si+1 ¡ si¡1 = 2s1 ci ;
±ci ´ ci+1 ¡ ci¡1 = ¡2s1 si ; (4) where si ´ sin(iµ=2n) and ci ´ cos(iµ=2n) for an angle µ divided into 2n equal parts. Unlike the case of the arctan series, one cannot now integrate this equation because the derivative of sin µ itself is not a simple function to handle. The way Yuktibhasha handles this situation is absolutely ingenious.
This is a discretized version of the fundamental theorem of calculus and again requires careful handling of limits.
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It converts the two coupled ¯rst order di®erential equations for sine and cosine into a second order di®erence equation by introducing the quantity ±2 fi ´ ±fi+1 ¡ ±fi¡1 , where fi is si or ci and obtaining the result ±2 fi = ¡4s21 fi . This is, of course, nothing but the di®erential equation f 00 + f = 0 in discretized form. The next step is to solve the discretized equation by adding up the second di®erences to get ¯rst di®erences and adding up the ¯rst di®erences to get the function itself. This is a discretized version of the fundamental theorem of calculus and again requires careful handling of limits. The
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relevant sections of Yuktibhasha dwell on this important issue showing clearly that the author was aware of these subtleties. This leads to a samkalitam which, in the ¯rst instance, produces what we would now call an integral equation for sin µ in the discretised form. This equation is solved by another ¯nite samkalitam by repeated substitution of the equation into itself. which requires calculating a set of sums of sums (samkalitasamkalitam, as you might have guessed P the name should be!) de¯ned recursively i by Sk (i) ´ j=1 Sk¡1 (j) with appropriate boundary conditions. The result for these sums of sums is given in Yuktibhasha but this is one instant in which the proof is not provided. All this is done keeping a ¯nite n; in fact Yuktibhasha generally uses the technique of delaying the step of taking the limit as much as possible. After the relevant sums are done, the time is ripe to make n tend to in¯nity and the k-th term in the expansion of the sine series pops out to have the right value (¡1)k µ2k+1=(2k + 1)! { and one obtains the in¯nite series expansion for sine. In modern langauage what has been achieved is equivalent to converting the di®erential equation for sin µ into an integral equation of the form Z µ Z Á sin µ = µ ¡ dÁ d sin  (5) 0
0
and iterating it repeatedly to give µ3 + ¢¢¢+ 3! Z µ Z k (¡1) dÁ1 ¢ ¢ ¢
would now call an integral edsquation for sin in the discretised form.
After the relevant sums are done, the time is ripe to make n tend to infinity and
sine series pops out to have the right
Á2k¡1
dÁ2k sin Á2k :
(6)
0
During the entire process, Yuktibhasha does not lose sight of the need for taking n ! 1 limit in the end. The word used in this context is sunya-prayam which
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the first instance, produces what we
the k-th term in the expansion of the
sin µ = µ ¡
0
This leads to a samkalitam which, in
value of the sine serie ansion (¡1)kµ2k+1=(2k + 1)! – and one obtains the infinite series expansion for sine.
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Box 1. A Piece of Pi Estimating the circumference of a circle of given radius has intrigued all the ancient geometers and astronomers. Most of them used the procedure of inscribing and circumscribing the circle by a polygon of a large number of sides and approximating the circumference between the perimeters of the two polygons. This, of course, requires one to ¯nd the side of the polygon as a function of the radius and consequent development of trigonometry. While the approximate value of ¼ ' (22=7) was available since antiquity, better results were known to many ancient mathematicians. Archimedes (287{212 BC) used a polygon of 91 sides to get 3 10 < ¼ < 3 10 . Aryabhatiiya (500 AD) gives the \... circumference of 71 70 a circle of diameter 20000 is 62832" getting ¼ ¼ 3:1416 while Bhaskara (» 1114{1185) gives the circumference to be 3927 for a circle of diameter 1250 obtaining ¼ ¼ 3:14155. What is remarkable about Yuktibhasha is that it does not try to give yet another approximate value for ¼ but an exact in¯nite series expansion { which is a giant conceptual leap forward. In fact, in addition to the result described in the text, Yuktibhasha also gives an algebraic recursive method for ¯nding notation this corresponds ³p ¼. In modern ´ ¡1 2 to using the recursion relation xn+1 = xn 1 + xn ¡ 1 with x0 = 1 and obtaining ¼ as the limit ¼ = 4 lim 2n xn : n!1
(a)
This method, however, requires evaluation of square roots which is not an easy procedure while the series expansion given in the text has no square roots.
has a literal translation of `being similar to zero'; today, we would have called it in¯nitesimal! There is also the usage of the word yathestam meaning `as one wishes' while, for example, talking about dividing lines into arbitrarily large subdivisions. Yuktibhasha also obtained the surface area and volume of the sphere by integration of the in¯nitesimal elements. For example, surface area of a sphere of radius R is computed by considering a small strip between circles of latitudes µ and µ + dµ in, say, the upper hemisphere of the sphere. The area of this strip is shown to be equal to dA = (2¼R sin µ)(Rdµ) by very careful and meticulous reasoning. In the discretised version ±µ equals ¼=2n and µ = i¼=2n with i = 0; 1; 2; :::; 2n and the total area is obtained by summing over all strips labelled by i and
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then taking the limit of n ! 1. This is essentially the same as ¯nding the integral of sin µ as cos µ. Though it was known that ¯rst di®erence of cosine is sine, this summation is done by e®ectively going through the original steps once again. Historians are still unsure whether the developments on the banks of Nila in°uenced corresponding later developments in the west and { as far as we know { there is no direct evidence that it did. (Kerala, of course, had direct linkage with the west, for example through trading, from very ancient days.) It is also somewhat unclear how the Nila tradition died down after the seventeenth century though several obvious historical factors can be identi¯ed as causes for its demise.
Historians are still unsure whether the developments on the banks of Nila influenced corresponding later developments in the west and – as far as we know – there is no direct evidence that it did.
Acknowledgement I have bene¯tted signi¯cantly from several discussions with P P Divakaran on this subject. Suggested Reading Address for Correspondence [1]
K V Sarma, Ganita-Yukti-Bhasa of Jyesthadeva, Hindustan Book Agency, 2008.
[2]
[3]
P P Divakaran, Notes on Yuktibhasha: The birth of calculus, Indian
T Padmanabhan IUCAA, Post Bag 4 Pune University Campus
Journal of History of Science, 2011 [in press]; Calculus in India: The
Ganeshkhind
historical and mathematical context, Current Science, Vol.99, p.8, 2010.
Pune 411 007, India.
S G Rajeev, Neither Newton nor Leibnitz: The pre-history of calculus in
Email: [email protected]
medieval Kerala, Lectures at Canissius College, Buffalo, New York,
[email protected]
2005, available at http://www.river-valley.com/tug
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Dawn of Science 21. All Was Light – 1 T Padmanabhan Newton and his years of discovery.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics,
In the history of physics, there have been four men who belonged to a class of their own, far above the rest. They were Archimedes, Galileo, Newton and Einstein. Of the four, Isaac Newton (1642– 1721) belonged to a period neither too far in the past nor too close to the present; this left a trail of fables and (exaggerated) anecdotes about him. In this and the next instalment, we shall examine the life and work of Newton.
especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498, p.590, p.684, p.774, p.870, p.1009, p.1062; Vol.16: p.6, p.110, p.274, p.304, p.446, p.582, p.663, p.770, p.854, p.950, p.103; Vol.17: p.6, p.106.
Keywords Newton, Leibnitz, infinite series, optics.
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Newton was born on 4 January 1643, and was not really a Christmas baby as is popularly made out. This date is based on the modern Gregorian calendar which is in conformity with Nature. This calendar was being followed all over Europe at the time of Newton’s birth. England, however, was following the Julian calendar, according to which Newton was born on 25 December, 1642, thereby giving him the glamour of being a Christmas baby. His father was illiterate (though prosperous) and had died three months before Newton was born. His mother married again when he was three and left Newton with his grandmother for about nine years. (Much later, in 1662, Newton recorded that he had threatened his mother and stepfather “to burne them and the house over them”. This statement has provided much fodder for psychoanalysists to attrribute every trait of Newton to lack of motherly love in his childhood!). His mother returned to him after nine years when her second husband died, bringing with her the three children from the second marriage. Some attempt was then made to make Newton manage the family estate which would have been disastrous – both for the estate as well as Newton. Fortunately, Newton’s weakness was detected quite early and he was sent to the Free Grammar School of Grantham to prepare for entering the University of Cambridge.
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In June 1661, Newton joined the Trinity College, Cambridge, which was to be his home for most of the next 35 years. The official curriculum in Cambridge was distinctly Aristotelian, resting on the geocentric view of the universe and dealing with Nature in very qualitative terms. However, the scientific revolution was well advanced in Europe and the writings of Copernicus, Kepler, Galileo and Descartes were available to those who were interested. By around 1664, Newton had read and thought over many of these works. In particular, he was strongly influenced by the mechanistic philosophy advocated by Descartes in his works. In one of his notebooks, he had entered the slogan Amicus Plato, amicus Aristotle, magis amica veritas (Plato is my friend, Aristotle is my friend, but my best friend is truth). A scientist was thus born, in spite of the best efforts of a university. Around this time, Newton had mastered most of classical mathematics and was fast moving into new territory. The basic ideas of the binomial theorem and calculus were probably already in place though the world came to know of this much later. He received his bachelor’s degree (which was a formality, merely recording his completion of four years at Trinity) in 1665, one
Figure 1. Isaac Newton. Courtesy: htt p:// en.wikip edia.org /wik i/ Isaac_Newton)
Figure 2.
WHEN
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Figure 3.
WHERE
In particular, Newton was strongly influenced by the mechanistic philosophy advocated by Descartes in his works.
If we are to believe Newton’s own account of what took place in the next two years, then it must be concluded that no person has ever achieved so much in such a short span of time.
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year after he was elected a Scholar of Trinity. But his stay in Cambridge came to an abrupt end in 1665 when the university was closed because of the plague. Newton returned home and continued working and thinking for the next few years. If we are to believe Newton’s own account of what took place in the next two years (taken from RS Westfall’s biography Never at Rest, Cambridge Univesity Press, see Box 1), then it must be concluded that no person has ever achieved so much in such a short span of time. Though historians have raised doubts about Newton’s ‘recollection’, it is clear that he did lay the foundations for new areas in mathematics, optics and celestial dyamics around this time. During the same period, Newton also developed a keen interest in alchemy, something he pursued vigorously for the rest of his life. One trait in Newton’s character has puzzled historians. Newton had an abnormal fear of criticism and of even a healthy academic debate of his ideas. Time and again he refused to let the world know of his work lest there should be criticism and controversy. At the same time, he was very possessive of what he had discovered and could never tolerate even sharing the credit for these discoveries with anyone else. Much of Newton’s scientific career could have been different had he publised his results or collaborated with other scientists; this would have probably led to far greater scientific achievements for Newton and a more rapid growth of science itself.
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What finally prompted Newton to come out of his shell was the fact that in 1668 there appeared a book Logarithmotechnia by Nicholas Mercator in which the author had explained several results regarding the infinite series. Newton had worked out the same results, in much greater generality, a few years earlier and was horrified to see someone else taking the credit. He hastily wrote a treatise called De Analysi and asked his friend Isaac Barrow at Cambridge to communicate it to a select band of London mathematicians. Even at this stage, Newton asked Barrow to withhold the name of the author! Only when the treatise met with positive response did Newton reveal himself.
Figure 3. An example of the kind of hard work Newton was capable of. In this page of a manuscript, he has calculated the area under a hyperbola (which is essentially a logarithm) to 55 significant figures by adding the values from each term of an infinite series. Courtesy: http://www. departments. bucknell.edu/history/carnegie/ newton/logarithm.html
During the next few years, Newton perfected his early work on the branch of mathematics we now call calculus and wrote it up as
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Figure 5. A drawing by Newton, describing his crucial experiment involving the prism. The first prism separates white light into different colours and one of the colours is further refracted by a second prism without any change. Courtesy: http://www.astro.umontreal. ca/~paulchar/grps/histoire/newsite/sp/ great_moments_e.html.
Figure 6. The March 1672 issue of Philosophical Transactions of the Royal Society had an account of Newton’s telescope (marked Fig.1 in the picture). The two crowns show how an object 300 feet away will look through the telescope made by Newton (marked Fig.2) and through a more conventional 25-inch telescope (Fig.3). Courtesy: http://rstl.royalsocietypublishing .org/content/7/81-91.toc
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a short treatise. Though only a few knew it at that time, Newton had already become the foremost mathematician of his time. Shortly afterwards, Isaac Barrow resigned from the Lucasian Chair at Cambridge and recommended Newton for the job. Newton was appointed the Lucasian professor of mathematics at the young age of 26 and held this position for the next 32 years. In the first few years as the Lucasian professor, Newton lectured on some of his works in optics. At that time, there existed two conflicting viewpoints on light. The original idea of Aristotle treated light very qualitatively and considered phenomena like colours as arising out of the modification of light by materials. According to Aristotle, ‘pure’ light was colourless and homogeneous. An alternative point of view was advocated by Descartes who considered optical phenomena as inherently mechanical. Descates had made optics a quantitative science by stating clearly the laws governing reflection, refraction, etc. Newton accepted the mechanical view of light and, as usual, pushed it to its logical extreme. By sending a
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Box 1. Years of Discovery Here is Newton’s own account of his ideas and work during the two years, 1665–66 (From Never at Rest, by R S Westfall, [1]). “In the beginning of the year 1665 I found the Method of approximating series & the Rule for reducing any dignity of any Bionomial into such a series. The same year in May I found the method of Tangents of Gregory & Slusius, & in November had the direct method of fluxions & the next year in January had the Theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon & (having found out how to estimate the force with wch [a] globe revolving within a sphere presses the surface of the sphere) from Keplers rule of the periodic times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about wch they revolve: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly. All this was in the two plague years of 1665–1666. For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more then than at any time since.”
beam of white light through a prism and splitting it into varied colours, Newton demonstrated his point of view that white light could be thought of as a heterogeneous mixture of different colours. He also showed that these colours could be mixed together to produce white light again. In his view, material bodies only separated out the components in light rather than modified it. Using this idea, Newton could provide quantitative explanations for several optical phenomena including the rainbow.
Suggested Reading [1] Richard S Westfall, Never At Rest: A Biography of Isaac Newton, Cambridge University Press, New York, 1980. [2] Joy
Hakim, The Story of
Science – Newton at the Center,
Smithsonian Books,
2005.
Since lenses and prisms split white light into coloured bands, telescopes which use lenses suffer from a defect known as ‘chromatic aberration’. This aberration causes coloured fringes to appear around images viewed through a telescope. Newton believed (though wrongly) that this defect could never be eliminated in telescopes using lenses. To tackle this problem, he developed the first reflecting telescope using concave mirrors. This telescope caused a sensation when it reached London in late 1671 and ensured Newton’s election to the Royal Society.
[3] Also see: http://www. newton-project. sussex.ac.uk/.
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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Dawn of Science 22. All Was Light – II T Padmanabhan Principia, calculus and some bitter fights over priorities.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts: Res onance, Vol.15: p.498, p.590, p.684, p.774, p.870,
Newton presented his first paper on optics to the Royal Society in 1672 and a second in 1675. Both came under attack from Robert Hooke (1635–1703) who was then one of the leading figures of the Royal Society. Hooke wrote a condescending critique of the first paper and accused Newton of stealing his ideas in the second. Newton, who could never respond to criticism rationally, was deeply disturbed and the two scientists became sworn enemies. Around the same time, Newton was carrying on correspondence with a group of English Jesuits in Liege who were also raising objections against Newton’s theory of light. Their objections were very shallow (and arose from a mistaken notion about Newton’s experiments) but again Newton failed to react objectively. As a result, the correspondence dragged on for three years and ended with Newton suffering a severe nervous breakdown in 1678. What is more, these exchanges made Newton withdraw from the mainstream of intellectual life and become a recluse.
p.1009, p.1062; Vol.16: p.6, p.110, p.274, p.304, p.446, p.582, p.663, p.770, p.854, p.950, p.103; Vol.17: p.6, p.106, p.230.
Keywords Newton,
Hooke,
Princ ipia,
During these years, Newton turned to another passion of his – alchemy. He spent a considerable amount of time copying ancient texts by hand and trying to make sense out of the mystical imageries present in them. He failed in providing a scientific basis for chemistry – a task, which was achieved by people of far lesser genius in the next century showing that intellectual ability alone cannot precipitate a major scientific discovery. It is, however, possible that Newton’s dabbling with alchemy had another favourable influence: it made him think in terms of ‘attraction’ and ‘repulsion’ between particles and the mechanical notion of ‘force’ exerted by particles on one another.
Halley, Calculus, Leibniz.
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Around this time, Hooke tried to restart his correspondence with Newton on the topic of planetary motion, based on the notion that planets are influenced by the force exerted by a central agency. During this correspondence – which was actually quite brief, with Newton terminating it abruptly – they debated the question: what will be the path followed by a particle dropped from a tower to the Earth? Newton drew a figure for this path as a spiral ending at the centre of the Earth. This was wrong and Hooke immediately pointed it out. According to Hooke, the path would have been elliptical and the particle would return to its original position if the Earth was split along the path of the particle. Newton hated being proved wrong (especially by Hooke) but had to admit it this time. He, however, changed the elliptical curve drawn by Hooke by introducing the assumption that the gravitational force exerted by a body is a constant, independent of distance! Hooke immediately responded saying that he assumed the force of gravity decreases as the square of the distance from the body. Thereby hangs a tale of bitter fight over priorities. Hooke felt that he should also get credit as a co-discoverer of the law of gravitation and Newton – quite characteristically – refused to share the prize of recognition. The turn of events which led to this historic controversy are as follows. Sometime in 1684, Edmund Halley (1656–1742), the discoverer of Halley’s comet, got interested in the problem of planetary orbits and asked Hooke whether he knew what force could cause the elliptical orbits. Hooke gave the correct answer but could not produce any detailed justification. Later that year, Halley visited Newton in Cambridge and asked him the same question. Newton not only gave the correct answer but also could send Halley a proof of this claim in a short paper called De motu corporum in gyrum (On the Motion of Bodies in Orbits). The discussion with Halley convinced Newton of the importance of his work. By 1686, Newton had converted the nine-page De motu into the classic work Principia Mathematica. He sent this work to the Royal Society for publication. The society’s finances were in poor shape at this time (due to the earlier publication of the handsome book History of Fishes,
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Figure 1. Title page of Principia, first edition (1687). Courtesy: htt p:// en.wikip edia.org /wik i/ Philosophiæ_Naturalis_Principia _Mathematica
By 1686, Newton had converted the nine-page De motu into the classic work Principia Mathematica.
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This was probably the first time in the history of science that a fundamental force had been recognised and described as such.
which sold very poorly!), but Halley decided to publish the book at his own expense. As soon as Principia was submitted to the Royal Society, Hooke raised the cry of plagiarism. Newton’s response was typical. He went through his manuscript and eliminated almost all the references he had originally made to Robert Hooke. There is no question that Principia was a work of an intellectual giant; but in his reaction to Robert Hooke, Newton showed how small a man he was. The Principia is divided into an introduction and three books. Together, they provide a complete and systematic exposition of mechanics and a description of the system of the world. The introduction contains, among other things, the laws of motion, which allow mechanics to be formulated in a rigorous manner. Books 1 and 2 consider different hypothetical forces and motions under these forces. Finally, Book 3 applies the general theory developed in the previous two books to the study of planetary and terrestrial motions. Most of the proofs use geometrical techniques rather than algebraic or analytic methods, in accordance with the Box 1.
Birth of Newton
—
1642
Goes to Trinity College, Cambridge
—
1661
Royal Society founded
—
1662
Moves back to Lincolnshire because of plague
—
1665
Returns to Cambridge, elected Fellow of Trinity
—
1667
Elected Lucasian Professor of Mathematics
—
1669
Elected fellow of Royal Society
—
1672
—
1684
Halley’s visit leads to preparation of Principia Leibniz’s first paper on calculus Publication of Principia
326
}
—
1687
Member of Parliament for Cambridge University
—
1689
Master of the Mint
—
1700
Elected President of the Royal Society
—
1703
Knighted by Queen Anne
—
1705
Death of Isaac Newton
—
1727
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tradition of those days. The third book also contains the statement of the law of universal gravitation. This was probably the first time in the history of science that a fundamental force had been recognised and described as such. The Principia gave Newton international fame and rightly so. Young British scientists took Newton as a role model and within a generation, most of the salaried chairs in English universities were filled with Newtonians. In 1689, Newton was elected the Member of Parliament for Cambridge University. However, he suffered a nervous breakdown in 1693 and regained stability only after a couple of years. His creative life and scientific contributions slowly came to an end after this. Finally, in 1696, he took up the position of the Warden of the Mint and shifted his residence from Cambridge to London. Honours continued to be heaped on him: in 1703, he was elected president of the Royal Society and in 1705 was knighted. There is no doubt that Newton thoroughly enjoyed and took tremendous pride in his worldly success (which, of course, he fully deserved). On being knighted, Newton took the trouble to establish his pedigree and applied to the College of Heralds for his coat of arms. Also, based on his often quoted statement, “If I have seen further, it is by standing on ye shoulders of giants.” a
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Figure 2. These figures, on the windows of a cathedral built in Europe in 1227, relate to the theme “on the shoulders of giants”. It shows saints on the shoulders of earlier prophets. Compiled from figures in: http://www.therosewindow. com/pilot
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Box 2. Newton, Leibniz and Calculus One of the important mathematical contributions made by Newton was in systematising the branch of mathematics we now call calculus, though, as we saw in a previous installment, the basic ideas were developed in Kerala, India much earlier. He put together several ideas known earlier and added some significant innovations of his own, thereby developing both differential and integral calculus. In the simplest terms, these subjects deal with the manipulation of infinitesimally small quantities in a proper way. Newton’s approach to calculus was that of a mathematical physicist and he thought of calculus as a calculational tool. The rules of calculus were also developed
Figure 3. Leibniz.
independently in Europe by the German mathematician, G W Leibniz (1646–
Courtesy: http://en.wikipedia.org/ wiki/Gottfried_Leibniz
1716). As usual, Newton was dragged into another bitter fight over priorities – this time with national pride at stake. It has now been established with reasonable
certainty that Newton did develop calculus first. However, Leibniz arrived at it independently and published (in 1684) the results before Newton did. Both men stooped very low in their conduct of the controversy and probably Newton’s behaviour was the worse. At one stage, as President of the Royal Society, he appointed an ‘impartial’ committee to investigate this issue, clandestinely wrote the report published by the Society, and reviewed it anonymously. As a consequence of this dispute, British mathematicians were alienated from their European counterparts throughout the eighteenth century and, in fact, the standard of British mathematics fell behind that of continental Europe after Newton’s death.
sense of modesty is usually attributed to Newton. However, this may not be so. In his remarkable book, On the Shoulders of Giants, sociologist Robert Merton argues convincingly that this expression had achieved a very conventional meaning by Newton’s time. It was what the great and the noble were expected to say on certain occasions (more like ‘hello’ or ‘good morning’) without, of course, meaning it in its true sense. So familiar was this usage that themes based on it could be found decorating the windows of some cathedrals. Merton also lists several earlier and later uses of this expression by other famous people. Anyway, an objective historian studying the way Newton dealt with his fellow scientists would find it hard to accuse him of modesty!
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Newton died in 1727 in his eighty-fifth year – a death which triggered the display of pomp and pageantry, poems, statues and other commemorations. He was buried in Westminster Abbey like “a king who had done well by his subjects” as Voltaire put it. Suggested Reading
Address for Correspondence T Padmanabhan
[1]
S Richard Westfall, Never At Rest: A Biography of Isaac Newton, New York: Cambridge University Press, New York, 1980.
[2]
Joy Hakim, The Story of Science – Newton at the Center, Smithsonian Books, 2005.
[3]
Also see: http://www.newtonproject.sussex.ac.uk/.
[4]
K Robert Merton, On the Shoulders of Giants: A Shandean Postscript,
IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
University of Chicago Press, 1993.
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Dawn of Science 23. The Quest for Power T Padmanabhan The practical necessity for developing steam power came from the rapid deforestation in England in the seventeenth century.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts: Resonance, Vol.15: p.498, p.590, p.684, p.774, p.870, p.1009, p.1062; Vol.16: p.6, p.110, p.274, p.304, p.446, p.582, p.663, p.770, p.854, p.950, p.103; Vol.17: p.6, p.106, p.230, p.324.
Keywords James Watt, steam engine, industrial revolution, latent heat.
436
The earliest civilisations used human muscle power and animal power in their agricultural and domestic pursuits. The efficiency, which could be achieved in these forms, was necessarily very limited. Soon it was realised that water and wind could be harnessed to provide power. Though windmills and water wheels could be considered technological innovations, the real breakthrough came only with the use of steam. The story of steam forms an interesting chapter in the history of technology. The theoretical basis for using steam power came from the investigations of Otto Von Guericke (1602–1686) and Robert Boyle (1627–1691). The practical necessity, on the other hand, came from the rapid deforestation of England in the 17th century. The English Navy needed large quantities of wood for shipbuilding and consequently wood became scarce as a fuel. England, of course, had huge deposits of coal, which could serve as alternative fuel. But the coal-mines used to get repeatedly waterlogged rendering them unusable. The usual procedure was to pump out the water by hand or by using horses. This was quite complicated and slow. It occurred to an English engineer, Thomas Savery (1650–1715), that air pressure could be put to use to pump out water more efficiently. The idea essentially consisted of filling a vessel with steam and then condensing the steam; the vacuum produced inside the vessel would then suck the water up from the mine if a tube were connected between the vessel and the mine. This instrument, which was called the Miner’s Friend, was the first practical steam engine. Significant improvements in its design
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were introduced by Thomas Newcomen (1663–1729), a blacksmith, who made very robust structures and carefully polished pistons. The first machine was set up in Staffordshire in 1712.
Figure 3. James Watt. Courtesy: http://en.wikipedia.org/wiki/ James_Watt
1
See Resonance, Vol.14, No.6, 2009.
Figure 4. The Newcomen steam engine. Courtesy: http://en.wikipedia.org/wiki/ Steam_power_during_the_Industrial_Revolution
Though the Newcomen engine served its purpose, it consumed too much fuel. The next major breakthrough which allowed the economical use of the steam engine came from the Scottish engineer, James Watt1 (1736–1819). Watt was a sickly child and had an uneventful childhood. In his teens, he spent a year in London as an apprentice in a workshop learning the use of various tools. He had a very good aptitude for engineering. In 1757, he opened a shop at the University of Glasgow which made several mathematical instruments like quadrants, compasses, etc. There he met the Scottish chemist, Joseph Black (1728–1799), and learnt from him certain curious facts about heat. Black had conducted a series of experiments in thermodynamics, which made him realise that the quantity of the heat energy was not the same as the temperature. For instance, when he heated ice, it absorbed the heat energy and melted, though its temperature did not change. Similarly, he noticed that energy supplied to water at its boiling point went into converting it to steam without changing its temperature. These observations implied that there was more heat content in steam at 100 oC compared to boiling water at 100 oC. Black named it ‘latent heat’ and mentioned these results to James Watt. This knowledge was of crucial importance to Watt. In 1764, he was called upon to repair a Newcomen steam engine. In the course of doing so, Watt realised that it was the latent heat that was causing significant wastage of energy in these engines. To condense the steam, it was necessary to cool the vessel containing the steam; but then it had to be filled again with steam for the next cycle of operation. Most of the energy now went into merely heating the chamber back to a high temperature. Thus in every cycle of operation, a
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tremendous amount of energy was wasted because the vessel was repeatedly heated and cooled. Watt hit upon an ingeniously simple solution to this problem. He introduced a second chamber (called ‘condenser’) into which the steam could be led. It was now possible to keep the first chamber (called ‘cylinder’) always hot and the second chamber always cold. By 1769, Watt produced a steam engine, which had far greater fuel efficiency than a Newcomen engine. Further, it worked considerably faster since there was no pause between heating and cooling the chambers. In the following years Watt introduced several crucial improvements, making the steam engine truly versatile. For example, he arranged for the steam to enter alternately from two sides of a piston so that useful work could be extracted during both the pushing and pulling motions of the piston. He also devised mechanical attachments, which converted the back-and-forth movements of the piston into a rotary movement of the wheel or to any other form of motion. The steam engine thus became the first modern device, which could serve multiple purposes by using energy that occurred in nature (in the form of fuel) to run virtually any form of machinery. Watt entered into partnership with a businessman and produced steam engines commercially for sale. By 1800, his engines had totally replaced Newcomen engines. And there were over 500 of these working in England. The consequences of this invention were far-reaching. Steam engines powered by coal could deliver energy constantly at any spot. Manufacturing locations and factories no longer needed to be near streams or waterfalls. Large scale production of commodities became cheap with the availability of unlimited sources of power. Handicrafts became non-viable and the artisan was replaced by the factory worker. Cities grew along with industries, and so did urban life and all the benefits and evils of the factory system. In short, the industrial revolution began with the steam
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Figure 5. One of the first steam engines based on the double-acting principle, developed by Mathew Boulton and Watt. Courtesy: http://en.wikipedia.org/wiki/ James_Watt]
In the following years Watt introduced several crucial improvements, making the steam engine truly versatile.
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Figure 6. Mechanical arragements in Newcomen’s and Watt’s steam engines. Courtesy: Dickenson [1].
engine. One immediate use of mechanization was in the field of the textile industry, which was of prime importance to England. Richard Arkwright (1732–1792) and others invented machinery which would replace hard work in textile manufacture. They became the first group of ‘capitalists’.
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus
James Watt’s work was amply recognised in his time. He was elected Fellow of the Royal Society in 1785 and was made a foreign associate of the French Academy of Sciences in 1814. In addition, he earned enough wealth from his royalties. In one of his experiments, Watt noticed that a strong horse could raise a weight of a hundred pounds nearly four feet in about a second. He therefore coined the term ‘horsepower’, defining it as 550 footpounds per second. Today, however, the metric system measures power in the unit ‘Watt’ in honour of this inventor of the steam engine. One horsepower is 746 Watts. Suggested Reading [1]
Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
440
Henry Winram Dickinson, A short history of the steam engine, Cambridge University Press, 1939.
[2]
J Meidenbauer (Ed.), Discoveries and Inventions, Dumont Monte, 2002.
[3]
Isaac Asimov, Asimov’s Biographical Encyclopedia of Science and Technology, Doubleday, 1982.
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Dawn of Science 24. Chemistry Comes of Age T Padmanabhan
With the works of Cavendish, Priestley and Lavoisier, chemistry emerged as an exact science.
T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Previous parts: Resonance, Vol.15: p.498, p.590, p.684, p.774, 870, p.1009, p.1062; Vol.16: p.6, p.110, p.274, p.304, p.446, p.582, p.663, p.770, p.854, p.950, p.103; Vol.17: p.6, p.324, p.436.
p.106,
p.230,
Keywords Cavendish, Priestley, Lavoisier, Newton’s constant.
532
Yet another branch of science to come of age in the eighteenth century was chemistry. What helped its growth into an exact science was the preliminary work of Henry Cavendish (1731– 1810) and Joseph Priestley (1733–1804) from England and extensive contributions by Antoine Lavoisier (1743–1794) from France. Cavendish, descendant of two prestigious families of Dukes, had his early education in London followed by four years at Peterhouse College at Cambridge. Though he completed his studies, he never took the final degree for reasons which are quite unclear. After a short tour of the continent, he settled in London in 1755 with his father who was a skilled experimentalist. As an assistant to his father, Cavendish started doing his own experiments and was soon breaking new grounds, particularly in the study of properties of gases and electricity. When he was around 40, Cavendish inherited a large fortune from a relative and became a millionaire. (This made a contemporary scientist remark that Cavendish was the richest of all the learned men and most learned of all the rich!). But he was an eccentric genius. He dressed shabbily, spoke hesitantly and very little, never appeared in public and could not stand the sight of women. So much so that he communicated with his housekeeper by daily notes and ordered all female domestics to keep out of his sight. He usually wore a crumpled and faded suit and a three-coloured hat. He had disdain for public acclaim though he did accept fellowships of the Royal Society and Institut de France. He rarely
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published his results and consequently hindered the march of science to a certain extent. In terms of scientific calibre however, Cavendish was excellent. In 1766, he communicated some early results to the Royal Society describing his work on an inflammable gas produced by the action of acids on metals. Though this gas had been noticed before, Cavendish was the first to study its properties systematically. Twenty years later, Lavoisier named this gas hydrogen. Figure 3. Henry Cavendish (1731–1810). Courtesy: http://www.famousscientists. org/henry-cavendish
By careful measurement he could determine the density of hydrogen and discover that it was unusually lighter than air. One of the popular ideas in chemistry those days was the hypothetical substance called phlogiston (Greek, ‘to set on fire’). All combustible objects were supposed to contain large quantities of phlogiston and the process of combustion was assumed to involve loss of phlogiston. Thus, wood was supposed to contain phlogiston but not ash, which could explain the fact that wood can burn but ash cannot. The remarkable lightness of hydrogen and the fact that it helped combustion made Cavendish conclude (erroneously, of course) that he had isolated phlogiston. It took a few more years for the concept of phlogiston to be laid to rest. Cavendish, however, noticed that when a mixture of hydrogen and air was exploded by means of an electric spark, water was produced. Similar experiments had been performed earlier by Priestley and even James Watt. This result was of crucial significance because it gave the final blow to the medieval idea that water was a pure element; it became clear that water could be formed by a suitable chemical reaction.
Figure 4. Cavendish’s apparatus for making and collecting hydrogen. Courtesy: http://en.wikipedia.org/wiki/ Henry_Cavendish
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Box 1. Weighing The Earth According to Newton’s law of gravitation, two bodies of masses M and m separated by a distance r attract each other with a force given by F = GMm/r2. It follows from this law that the constant G can be computed if the force of attraction between any two bodies can be measured accurately. Once G is known, it would be possible to estimate the mass of the Earth by using the known value of the force by which Earth attracted all bodies. Figure A. Cavendish’s torsion balance instrument.
The difficulty, of course, is that the gravitational force F is extremely tiny at laboratory scales and requires a very sensitive Courtesy: http://en.wikipedia.org/wiki/ Cavendish_experiment experiment to measure it. One of Cavendish’s experiments was to design such an apparatus. His apparatus consisted of a light rod suspended by a wire at the middle. At each end of the rod was a light lead ball. The rod could twist freely about the wire and even a small force applied on the lead balls would make it twist. Cavendish brought two large balls near the light balls, one on either side. By measuring the twist, he could calculate the force of attraction and hence estimate the constant G. From the value of G, he estimated the mass of the Earth to be 6.6 x 1021 tons and the density to be about 5.5 g /cubic centimetre. The Cavendish balance has remained a valuable tool in the measurement of small forces.
Cavendish also used electric sparks to make nitrogen combine with air (with the oxygen in air, to use modern terminology) forming an oxide which he dissolved in water to produce nitric acid. He kept adding more air expecting to use up all the nitrogen. However, he noticed that a small bubble of gas, amounting to less than one per cent of the whole, remained uncombined. He speculated that normal air contained a small quantity of very inert gas. We now know that component to be essentially argon. Cavendish also made a significant contribution to the study of electrical phenomena and to the measurement of the gravitational constant (see Box 1).
Figure 4. Joseph Priestley. Courtesy: http://en.wikipedia.org/wiki/ Joseph_Priestley
Cavendish died in his 78th year. He left his large fortunes to his relatives and virtually nothing to science. This omission was later rectified by the Cavendish family when in 1875 they set up the Cavendish laboratory at Cambridge University which contributed greatly to the development of science in the next century. Another English chemist who lived during the same period was Joseph Priestley. He was the son of a non-conformist preacher
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Priestley used a new technique of collecting the gas over mercury rather than over water which was the practice earlier. In this way, he could collect several gases, which were soluble in water.
and was quite radical in his views on religion and politics. In his early days he studied languages, logic and philosophy and very little of science. He first worked as a teacher in a day school at Cheshire. During this time, he wrote several books dealing with English grammar, education and history. Though he never studied science formally, he was always curious about the several new discoveries occurring around him at the time. From 1765 onwards, he made it a point to spend a month every year in London where he could keep in touch with leading scientists. Influenced significantly by Benjamin Franklin (1706–1790), he decided to write a book, The History and Present State of Electricity, which earned him a place among scholars. Soon Priestley turned from physics to chemistry. His main interest was in the investigation of gases. Only three gases were known at the time – air, carbon dioxide (which was discovered by Black) and hydrogen which Cavendish had just discovered. Priestley went on to isolate and study several more gases such as ammonia and hydrogen chloride. He used a new technique of collecting the gas over mercury rather than over water which was the practice earlier. In this way, he could collect several gases, which were soluble in water. His major discovery, however, came in 1774. It was known that mercury when heated in air would form a brick-red-coloured ‘calx’ (which we now call mercuric oxide). Priestley found that when calx was heated in a test tube, it turned to mercury again but let out a gas with interesting properties. Combustibles burned brilliantly and more rapidly in this gas, mice were particularly frisky in that atmosphere and he himself felt ‘light and easy’ when he breathed it. Since Priestley believed in the phlogiston theory, he reasoned that the new gas must be particularly poor in phlogiston. He called it ‘dephlogisticated’ air; when Lavoisier heard about the discovery he could immediately recognise it for what it was and named it oxygen. Priestley also noted that plants restored the ‘used up air’ to its original freshness by supplying oxygen. In 1779, Priestley moved to Birmingham as minister of the New
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Meeting Congregation. Being a unitarian by faith, he rejected most of the fundamental doctrines of Christianity, including the Trinity, predestination and the divine inspiration of the Bible. In addition, he was a strong supporter and defender of the principles that inspired the French Revolution. This and his publications on these subjects made Priestley extremely unpopular in the local community. And when, on July 14, 1791, the second anniversary of the fall of the Bastille, the supporters of the French Revolution organised a meeting in Birmingham, the general public was not sympathetic, and decided to teach those gathered a lesson. In the mob violence that followed, Priestley’s house, laboratory and library were burnt down. Priestley managed to escape to London where he taught in a college.
Figure 5. Priestley Medal. Courtesy: http://en.wikipedia.org/wiki/ Joseph_Priestley
However, the progress of the French Revolution, the execution of Louis XVI in France, and the declaration of war between France and Britain made life even more difficult for him. In 1794, he left Britain forever and migrated to the United States where he spent his last ten years. He was probably the first scientist to go to the USA to escape local persecution and certainly not the last. To commemorate Priestley’s scientific achievements, the American Chemical Society named its highest honor the Priestley Medal in 1922. During one of his experiments, Priestley dissolved carbon dioxide in water and found that the solution tasted pleasant and refreshing. Though he didn’t make a commercial success of it, he probably deserves to be called the father of the soft drink industry.
Address for Correspondence T Padmanabhan IUCAA, Post Bag 4
Suggested Reading
Pune University Campus Ganeshkhind
[1]
Joy Hakim, The Story of Science – Newton at the center, Smithsonian
[2]
Isaac Asimov, Asimov’s Biographical Encyclopedia of Science and Tech-
Books, 2005.
Pune 411 007, India. Email: [email protected]
nology, Doubleday, 1982.
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