The Relativistic Theory of Gravitation

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A. Logunov and M. Mestvirishvili

:The r/(/fatzvzstzc :Theory 0/ (jravltatlon Mir Publishers Moscow

The Relativistic Theory ofGravitation

A. A.

n OryHOB,

M. A. Mecrea peme anu

OCHOBhI PEJI HT UBIICTCKOn TEOPUU rPABi1TAltiUl

ALogunov and M.Mestvirishvili

The Relativistic Theory ofGravitation

Q Mir Publishers Moscow

Tl'IUIaitd It1lCII Ilw a.......

by

t . ... Y, nb" 'y

Fi.. .... 11tMl 1981l lle'f'itId 1l'OIII t lw t l1l'ii R-.lu ••111100

TO THE READER

Mit Publllh"'" would be ,,,,,ltlu l fOt ' OUt ""m... ou OQ til . C. . ery tbouabtlea to.1f'" to this "ithout proper upuimental proo f. So f ir there is not. aiDa l, u peri melt....l fact l h. t , d irectly or indiTe&lly, ch. lIeapll !.he .. alid it y Dl conservation I,WI i " the macro- l od mkr&-ycrl cl.ll. Th ete is only (IrI1 c:oncl usioo t Aen: we most diteard GR , givi n, it credit u. I tage inlh. d. ...lnpmlll l 01 OU ' id us of gU 'fiut io ll. In Den iso.. an d Logu no\', 198Oa. 19110.., 1982 b. 1982t 1M W1idun o/ ' /u pJtpJcIU geometry If/' II,1lt the b.ue of phl/,fa but

w.

,_ try

1M ,rnutd IUlWumI dllll ll m /clll pro pD11a

of

mIllU, .

III our th lIOry . RT G, tile physica l geometry of . p.... l i.D. is dote'l'IIiued hot 011 th e buil of . t udi es of the propagalioD of lig ht alld the mev eeien ts 01 tes t bodies bllt on lilt bUie of geM " l dyn. mieal properli u of ma Uer. t he co nservaLion laws. wh ich I n no ~ onl}' of fundamen ta l i111 porllr.nc" bu t ca n be verified expe rimenla lly . I\eqlliremen l (a) Sl!b RT G ent irely a parl b orn the general t heory 01 rel aUvily. (h) A I ravi ta tiollal fleld ia described via a aym met rle seecnd-renk teneoe tI>'"" and con~ Li \u ll.'fI a real physica l field char ac terized by an energy- mumen tum densit y. a u ro lll!lt mass, an d . p le atll\lll 2 and O. Th ls aapllt t alao b...,lull y d iali n, uishM RT G from G R. (c) We Inlrod uc. the reom elrlll tl on principle, accordi nl 10 which the interact ion 01 ", a ra vlta llona l field ..ith maUeT is aeblev ed, in vie.. of l ite lUlivef'lali ty of th ie iDlera cUon, by "a ddi.lll" tlla ,tD v ila li oDal Ii. ld $ '"" to the metric te n_ y01 lh. Mink01ll:llk i .apae.II-tima in t he tegrang iaa della iLy of ma ile r auonI inl l.O rh. loll o. illl rula:

an d $ .. ara t he materi a l fielda. 8 y matta we mea n a ll of ib forlll5 euept gra v_ ita t iona l field.!. Aceord ill' to I hl genmet rilalioll prinei pll , mot ion of maUer under lhe . ction of a gravita tional field $ "" l u I h. Minkowski apace- li ma wlt ll a Dlelt lc y• • i. eq uinl l!flt to mot ion in .u eflectl ve Rie mull apaco-t ime wlrh a metric f· · , The metri c tenser y" . of the Minko....aki apaea· timo an d t he , rav it etl oul· lield tenser tI>"'" in th i.a . p"ct-t ime are pri mar)' concepts . wllile the R iowallll ~pa~e- l i me and i \.l metric Ii"" orOseconda ry ccncepte, OWillC" tlleir orl l/in to Ihe grni taUonaJ field and lUI u niversal ee tt on OIl meuee tbrough 4>.. . T hlt efleetiv. Riemann ~pa~o-ti mlt It litera lly of field oria lo . tha nka to tllll peesenee of the gTi vita tiona l field. E inst ein waa th e til'llt 10 ",uu est thlt th e ~paeo-l i m e ia Rie rua nnla n ra ther t han pseudo- Euc lid ean . He iden lifled gravil at ion wit h lh e metric telllOr 01 the Ril!ma nn a pa~o- li me. Bu\ Ih ia Itne of ..aSllDing as mucb 115 led 10 re jec tio n of the rravilati ona l field as a ph ys ical fie ld posse!Sinr an e nera y' momen tum de...ily a nd l.O the 1011 of l undam en la l toMervat! oll lawll. T he lIllelriu tion pri nciple, based on lhe Ilot.iona 01 tJw, lo! lnko" Y i apac.ti me u d • phr ti ce! rr... l ~tioDlI field . inlr o(hlCM lhe COIlcept of I n elleel!ve Rie lJllllD spacet ilne , Ind in th is Ei nstein '. Id.. of I Ril!maDn ia n reo metry linda ib indirec t eeDed ion. Aecord inr to lhe RT G id.eIDlY. aince the Minko.... k i lIpace-- l ime (~) lor llls its base, th ere ai'll ata od ard lempotal a nd s pati al sea l" lh l l do Dol up lici tl y depend 00 lhe , ra vila lional intenoctlon . I n ... le... of the geomeu iuUon pri nci ple , I he enll~

,DO-

depend_

of t he li M eleUlent In the eDective fti, me nn

sp.~ti lDe.

"

rUt - f,~ (~) d.i dr'. the I . av ita t ionel field Ii. in lhe metr~ eoel/ieluu ' rt (r ). T. l N!itioll to I lly olhlll' eoord iDltes in RI G, say , 1.0 the pro eoOl'lIi.u1lll. wil l resolt i n . shutioD in .. bleb the pro~ . pllCe-ti _ ...e n,bl ill depend botb 0Jl. the coord lo_"" J:in t b' lol inkofti sp_ t illlll an d on I.be grn ittltioDaI COIlllt.ant G. H eooe, propet UlIlll and 'Pf1"al ebaneles'Utiu ...HI depen d 011 lbe ....niholioll.l Beld. It is onl y in RTC that one Ga ll completely det«m ine tbe . Bed of. rr...itatioDal lil ld lllI th. p_ p of proper time .nd Oll tb, vlriat ion of the distlnc e between points. (d) The se. lar L..grl ngi'll density of .. V avil. Won. 1 field is • bilin ear form of the liJ'li1 con rin l derin li vee, with respect to the Minkowsk l metrle. Buic.Il}' there is no WI}' to ecnneuct • eealae LI I •• ngJan dens ity of such I form

chlu ",bo\~~. I k ..k'. a IoJr i' 01 n'til n' 0 ~l . ';~ ~r,wiilllinJI. It i. llk~ly Ihat ""m. ..1 the Ilh""O"'06' ...iII hi' delerDllned by tho cO"'JI'.rallvoly almpi. equation. in which \.b. ~()mpOll""tI 01cu.... "'alun or the world do llot . pp...r: Aueh "'l... ltono..~Il be Iba u ma for a ~urv" region al ln. a 0. 1 region. II ia to t heM Ih. 1 Ih. Principl, 01 Equlul,,,,,, applle" One ean nc t. however , assert ths t descript ions o{ pb~ica l p henomena in a grav Hs.tio nlll field and in a non tnertie l relerea ce fra me of tha pseudo-Euclidea n s pacetim e lIl'e fully equ iva lent, since (\15 put by E dding km ): ... tb... are m...... comple:o. pbanomena i ovomed by equation. In whith tbo cu' .... I....." 01 the IVf tho. COII yict illll IU t III • rn v; uotl ..., l fi~lcl ,U ill the ...... _ .. u III llIo . "-ce of ...... iuoll""'"'lli""~ bl l ill lId {th l_ ·.H..... _I ~ ' r - of -..a iDOtaI (equinll11OU II, .

Sinu ill th ose da)'ll. th anks to Minko"., ki ' . dQcovtry, it wu knolrll th a t to d iftt reot rtltteoce fu mn tha:re COITeIpond di ll_li t (pa«a ll, oll-d'lJronal) met· . ies of , pac. tim.. EhUlu ln a.nd GrtllIIImallll, t 913, concl uded t bat 1M nwtrle I4MOr 0/111.4 R ~n ' f'G""tllM IftU.d be tWn AStM ~ld ...,.Joble Itw" l /l.e cn>~"il4luJltld ~14 ud that thi, ~03Or iI d. &efUliDed by th , distr ibut ion nd mot ion oJ m.n... . h th ill "'a y tllere emerpd tho idea. of I. li nk botwean m. U« a nd t be po mlltry 01 s pac.,.ti mll. Proceed inl frOO1 thue _ umplians, EilUltllin a nd Gl'OIIlIml nn purelr intui tivel)' tlied to establi, h thll form of the equ a tions Unk illl thll components of t ba metric tensor of the Ri llmann 'PfoU""t hne with th e Iluetl)'-moman tum teeeer for maUeI'. Afler numereUf u4ltlC(,MSfu i attemplS lJUCh t qua t ioll' were fou nd by E in'l4Il n at th e en d 01 1915. Since eome".h, t ea rli er Hil bftt, 1915, arr ived . t the ,a rne eq u.tlOll.t

(hi~

reasoning was based on varialiona l prin ci ples), we will ca ll tbese equilions the Hil hert·Einllteln equa tion s, It lll~t be noted that t he metr ic tensor of the Hietnann sp aClI-time cann ot serv e II.! a char ac terist ic of t he grav itat ional field becaulle its Mympto lic behavior depends on the choice of the th roe-dimensional (apati al) system of coordi nates,

Chapter 2, Energy. Momeotum Pseudotenser s of tbe Gravitational Field in GR E iMlein believed th at in GR the grav itat iona l field together wilh matt er must obey II conset\'ation law of SOlUe ki nd (Ei nlltein, 1914); ...11 ", il h""t "' \""8 Ih. t we "'0'1 roq ui l'(l Ihal ....olUlr ond tlf uilllibnol lield li ke" tOBel ... ""H. ly . "ergy ond mamenlom eoo...",va!ioo lowo.

I"""

In his opinio n , thi s problem bad been full y w ived on the basil! 01 "eeeseeveucn laws" that. used the energy· moment um pseudotee scr as the energ)'- ffiomentum charac teris tic of the gravita tional lield . The eommon line 01 reason ing that leads to such "conservat ion laws" goes as follows (Landau an d Libh ill, t 975), 11 tile H ilbert- Ei nst ein equa tion.':! are "Ti tl en as

- ~ g [Rj~-+ g;~ R] _ _ gT'·,

(2.1)

where g = det gl ~' R ' ~ is the Ricc i tensor, and T l~ the energy-racmentuz» tenso r for matter, the n the l~rt-han d aide can be represooted as IIs um of two nonCQvatiant qu ~ n ti t ies:

(2.2)

where t l~ _ t~l is the gravita t ioul-field eMrgy·mo l1lentnm pseud oteneoe, an d 11,1"' _ _ hll. the sp in pseudotensoe. Thi s lransforlll!l the Hilbere-Elnsee tn equat ions (2. 1) into 1111 equiva l&Jlt lorm _gIT i" +TI"I _ _ '_ hl"l. (2.3)

,.'

I n view of the obvious lact t ha t

_·_·_ h l·' ~ O,

(2 A)

iJzJ> iJz'

th o Hilbert. Einstein I!qll alio ns (2.3) yield the foll owing d ifferent ial conserva tio n law: (2.,'1) which formally is similar to tba conservation law Ior energy-m omentu m ill etecreodynamics. Accordi ng to Gft , this law is ,'alid for any cboice tlf ceeedin a tes .0;", lor one thing, sp heetce! coord inates (/, T, e , 'l'). But in t be lntter case (2_ 5) will a lways lead to phys ically meaningless results. Th us, Eqs. (2, t ) in ar bitrllry eccedlna tes alwa ys lead to (2.5), whic h has no physical mean ing. I n accordance with thls Blulo g)', t ile grav ita tio nal energy "n ux" through the elem ental s urface ' rea dS~ is defined ln Gft thus : ill =c (- g) T""dS" .

"

T ,lliDI • I phUII of " dlUJ r .. tb ' surfac:e of inu. rltlon (IlIl, . cwrd lng .. the

Il leotano tonsor dOH nat or d Oell ".ni.h. Th i. ia I n I hoolut; ) ~~z(:>;)

-

11(ZI< )

" Thus, the geometry of spece-t tme delerm iile!l the possibili t)· 01

a :~l:l

oh ~a in i ng

J.

"integral

cODMrvati on Isws. In the ceee 01 lour dimensions (the physical sp.cort ime) only spaces ",i th consllln l curutul'fl posses.s 811 ten integral eOll.'lllrvation laws; ; n otber SPliCes lila number of these lews is lese.

Our analysis demcnsteates that if ""8 wish to have the great llllt number of to nserved qua ntities, we. must rejeet Rieuu,nnian goometry in its general form, tnd lor 811 fields, includ iog the grni ldi onal, we mIL'l l seillet one of the a bove-mentioned geometries of eonstant curvatu re as the natu ral one. Since the 8listiog 8l perimentai da\.B, on s trong, weak , a nd electrom agnet ic interactio ns suggesta lbat for the fi elds related 10 these interactions th e natu ra l geometry 01 space-time is pseudo-Euclidea n, we can assume At leest at the present level of our Ic nowledge that this golometry is the universal natural gwme tr y for all phyaieai prot_, including those Involving gravitation. Thill assertion consti tutes one of the main theees nf our approach to the theory of gravita tional interact ion, h obvinusly leads to the observance of all laws of cooservalion of energy·momentum and angular momentum and ensures the existence of all len integrals of moUon for a system consist ing of a gravitatioll8llield and oth ar mate rial lields, As we will shortly show, the gravita tional lield in cur framework, lIS all other phys i~ al fi elds, is cha racter ised by an anergy-momentum tenser that contr ibutes to t he total tensor of energj--momem urn of the system. Th is consti tutes tb main diflerenn between our apprnach and E j nst~in ' s. It must also he noted tht in th e peeudo-Euelid ean space-time th e integrat ion of tenser qllantiti85, in eddit ion to its geoeral simplicity, has a ",;elJ--ell'gel here u a etlta ia e lfec:U..e I lOmeky, .generated by the K t ion of a physk.al ar av ita t iooa l field ill lhe Minko..-kl apacati me Oil matter, But for lh is eontlr ueli oo of an al'l eeti va Rlemamniu met.-ie ill the .Minko"""k i apace-t ime varll bles to have physical mlllUl l!1i, we llllUt elUlu rl thl t th gravih lionl l-Aeld e-q ultiOIU eonta ln l he Mlnkowskl apae.-tl me mal rlc yl ~ . I II our l heory Iho t ensor tlI1k ;s t he fiold vadlbl e 01 th o grav! ta Uoou field, an d the physica l bound ary eondi tl ona m us~ be for mulated for ~h ill varla ble. We ....iII &Mu me !JuIt the er n l ta ll ooal 6.e1d in gelleral hu only eplne ~ a nd O. These p hYl i~ 1 fe!ltriel ioDl, AI sho wn ;0 Cha ple r " Iud io Galileao coord loa l" 10 the lollowi ng eq uationl lor th e lfr...ita tio na l field;

'(KI,

8,C1l'k "'"8,i'& _ 0.

(S.2)

1111 Ri ellllll oian po_try of apace-li me i.I da~rlllillad by 6.r.i"i Lbe met.-icleuor field (z) 10 a cerla lll 1)'ltl m oJ coorclllLlla ma ps. Although de Donder.

f,.

192 1. 19'26, IIIld Pock , t 939, 1957. 1959. WItd eOllditio!U of Ihe (8. 2) ty,. i n GR (l hey called them harmon ic condi ti ons), they W en! DoL able tl) I how In which ..! pl ee-time var;ables those condltl ona mll5t be ,,"Ilten, NeVet l helMl, I'ock, [0 d uerlbing problems 0 1 the i.sll nd typ e, u much lIS cons idered har monic cond itions 10 ~rlDl of II10bai Cat le:al,n coordlne t~. But whi re did he liod g loba l Cartosll o coordlna te$? Th,y hav e no placa in Rie man nia n geomeLty. Inl uit ively h, ml d e a cor1'Kt move. but ha could 00 1 eomrwvheod il!l.tillnilica nc:.. I I he hid eleatl y understood tha t Eqs. {8. 2} ne va lid only in an iDu Ual refereoct frame. lll Gamlllo eeerdlnates 01 the Mlnko.-sltl space-ti me, he l:.Ould have arr ived a t the l:.Oo« pt ion of &lIravlla t iooal hid u & pb,.lc.a l la nsar fi, ld i n tlle Minko""ki s p_tillla. Fock foeuaed apeeiallJ on the im porta nce of ba r.OIlIe eoord la ata l:.Oad ltiotlS for tblt IIlluti OD of isla n4 pro blema. For 1D511 nee, he WI'(lU (Fock, 1 ~ , p. 351); 'nIe Uoq ....... m lnr u.. pri"ilortd cht.n.cw 01 It,. "'.mODi< IJl I_ III _ . t iod of p roh l~ ;t; OlI 01 tbe of dl........ oboul.d no~ be omd \0 rero or lI.on·. .ro. which ..O\lld ba l", poal. bl, for . temot ll.hl. I• •Ull bot ~"".13W. ..G.,,I

.'G' ]

_1 .._ ..... _ I

Y _ c.....

Y ..... .. .

Heeee, ~( _ ': ' _ D. 6,... .("..

I LL I (DI6"' '')

"" _

I

tb

R...

(8.30)

wbore R",. is th e second -rank tensor of tho cur....t ore of th e Rie ma nn spece-Urne ;

R... _ D. G'_ _ D.G~ , + G ~ ..G~ _(f..,G:"' . Silltl iD view of (6. 6 b) a nd (6. tf l .".• hav e

2

',:M

a.,..

_~ ( T r - I

__

+,._r)

(8.31)

(8.32)

Eq. (8. 16) ylelda (8.3J) tha t la, ..... h ave arr ived a t !.he sys tem 01 Hi lbert-Ei oB18in oqua Lions , the one impor ta n t d ine renee be ing th ll. ~ eU lleld vllflo bl" In th e Hilb8J't- Ei ns le in eq ue110011 In our th oor y d"pc nd on u nivel'lla l s pot ial-temporal coord ina tes In the Mlnkowd l , pa ce-ti me. I n an 'lner lia l -ere enee fre me th... ulli vel'll. l cocmtln. te:s C' D bo chosen to be eamon. It must he empbasi-ed tha l tb a sy,te m of eq uatio... (8. 28) does not co inc ide with th e lI}', le m of H ll boert- E inMein eq u tiODll (8.33 ). Qnl y II the IOnuak ovu lan t equaLi ons (8 .29) hold true d oes th a S)'llte lD of Hil bnt- E iD· , lel D equ ati ons , lor m. lIy wr ittl ll i n G R in the ... rlIblq of the Min kcnnk l apac a·

,.

time, n duee to the 8Y8tem of equat iolQ (8..28) , and these depend eS!entially on the matri c tellsor of the Minkowsk i space-thne. I t h as long been know n (see Rosen , 1940 , 1963, a " d T olman, 1934) that Lag ra ngia" (8.26) leads to syste m (8.33). We have shown, however. that for a grav itat ioll. t field with spins 2 and 0 the gnvit ational-field Lag rang iall de nsity (8.22 ) is th e only one that leed s to a self-eonsiuen t system of equa tio ns for matter en d lield, (8.28) end (8.29). Th is means tha t the RTG eq uatio ns are the on ly simple!t second-order equations that ceo ex tet, In view of the Importa nce of th e equi vale nce of Eqs. (8.28) and (8 .33) in th e Millko\1ollki vari ables, we can give ano ther var ian t of the proof of the above . tU e· mont based on direct calculations of th e tenser dell.'l iUes till' an d 1,7tj, prov ided that (8.29) is vaHd. , II we takit formulas (6.17) an d the Lagrangian dens ity (8.22) a nd a llow for (8. t), we will lind that the grav iU t ional-field energy - momentu m te nser de nsity in the , Minkowskl space-t ime is

,-

~';.~ ... - 1~., J,,· ,::,, ~ { ,?.. ··t~ -'h"~,,"") R ~l'

(8.34)

We soo that the ~ ond -unk curvature tensor R p~ of the R iemenn space-time h as emerf9 d aut omati call )·. Sim ila rly , usin g form ulas (6. 17) and (8. 1) and the defin iti on (6.6a ) of the Hilbert-tenece delllli ty , we arr"'e lit the foll owing lormula lor the materi al energy-momentum tensor deJUity in t he Minkowski epee....ti "' e:

rl~)" ("fg)l/J ( ""'P""~-1: V"KY'~) Sub~ tit ut ing

(T ~~ _ ~ 'p~ T ).

(8 .35)

(8.M ) and (8.35) into the field eQ" uations (8.10), we get

( ,,"'PV~ - -} y"V )[ R ... -

:-, ( Tp~ - i-gPlT )] ""O,

which leads UlI to the system of equ ati ons for ' he gravita tiona l field in the form of (8.33), The complete system of eQ" uat ioJU for ma tter and gra vita ti ona l f'e ld , (8.28) a nd (8.29), is eQ"u ivalent t o t he lollowing system of equat ions:

V -r R... _ 8n (T"'. - +r",,, T)

(8.36)

D,:r· - O.

(8.37)

Th us, a lthough in RTC the complete system of equa tions (8.36l snd (8.3 7) does conta in the system of Hil bert- Eillstei ll equat ions, th a co nwnt 01 th e latter chs nges substant ia.lly, · since t he s pat id · tempora l variahles now coi ncide wit h the varia bles of the Minkowski s pace-time. We mua t aga in a mphasil e that Eqs. (8.37) are univeTS* l, s ince they are field equations deaeri bing gra vitat iolltl fields wit h sp ins 2 IIl1d 0; they unamb iguously se para te forces of inertia from grav it.lltional fields. Wi th in t he framework of GR th is Is ho possible to do in principle. The cho ice of the refere nce f.ame (or system of coordinates) is fixed by th e metric ten sor of the Minkowski ~ace- time, while Eq s. (8.37) lay n o '9$trictiollS on the choice of th e coord inate syale m.

."ull.

• Eq UOUOll1 (8 .;lG) do .."' ~ _ O.

+

(l1.2$)

We Bel! tlla' for a wea k ,...vHa li on. 1 field . onstant m is Ul. fTI" jl.Q n mus. Eq unti on.'l (It .9) are ga\lgo non invari . n l even i n !.he . bM... e of m.lta r, T·~ _ O. Th is meallS th el Ihe inlrou (If a In. SS term lift" the degene. acy wit h re,pac t to glo,e ~0 p"~oord i n . t8 tra nsformation s, olthou, h the free-grlYita tion Ol Ilie lll La8, .II,ia n dens ity (11.3) ....lisl\es th e raug. pri nciple form llla ~ ill t h is c.b( "'. ...+. ... I)

1

,, (, ')

• I

-

-

8";1".' , ,

-pr o

D,,') - pr, o

(12.81)

-

2.>

611'

, u(t')_ - p:;;iDiT + pt (6 + "" j - pi (6 + ..'J' Let 1M denote tile Car tes ian (Caill.an) coord inates in tb. M.inko...· ui .pace- ti lll' by '1"1 a nd I!lIUIl1e that tJle Kerr t:O:)~o .

(12 .8:!)

where r ;. (.>:) ill defined by (12.9). App lyiul tAe lrsMlor lIlatiolllaw to r;. ('1"),"(",1 fw th" t1bati t olion or t l(z) for Z', WIobta ifl rrom (12.82) lbe follow inl: r". . (..}t'" (z) _ -Ot z" _ O.

n _ O. I, 2, 3,

(t 2.83)

.. bUt 0t if ' h, coy,d. llt d'Al embert ian opel'ltor In lh, Kerr veeteb les

0,_ y

" lb

V [V -

- - ir{··sin

1

~' :

tlt) ,,"'ro ~.]

Ba: + 2(8' + " jiJ"IJ,

1.. .,

+ ~ ~.'ZB) a.a.+2Ba.+4~ +2(B - ...)a. :z." tB-_j a

.+ 01+_'

(A+ . ')"

+[ l& ~:')' ,

tl~~'

,

"

+-.f.rT ]a:+U:+ cot 04. } .

I

,

Hue , , ,,,,, 8' - 78 ' iJ' -lifo and ,). - , . . Wa lfill -X the v.riable z!>;:;:l I in the for m I _ 'I + 1 (8) .

(12.84)

SIIMtit" lion inLo (12.83 ) yi eltls th e foll o....in g equatloD for 1 (8): !l

d'Jj.ll)

+2( B -m) ~~hBl + 28 _0,

( 12.85)

.-hiGh een uel1y be in teva l.ed. T he solution I,

/ (B)_ - {B+ VOlt' l ..

G'

[B. In

8;,8, _ 8_ 1n

Bb_B-]},

(12 .86)

hE re .... h ive int r od uced the nota t ion

B", _m± V...'l - llf. W. ~ee tb" • r. al .soluti on for t ...

'I

+ I (8) elll,'" only

{12.87) if

8>8 ...

( \2.88)

T hus, th e Kerr vni,ble ~. _ B IId mita only tlu. lo ll ...wing It ... read ily noticed II" ,

• il.ll B_ B. at ' _ ....h.es: B ~ < D"'

:r _ to -

,?,)I\Ol'J B

(12.89)

iI . !'lliution to Eq. (12.83 ). What rem. ins 10 be l oua d i. "I and r . We ... ilI _k Ultlll in tbe for m :II:' ... Z (B ) cos ~ toW G a nd z* ... Z (B ) l i n " aie G. Eque-

iloa

(t 2.83) fot "I then b«;om~ (OS "

~' + 2 (B- m )"("'~')] T - (6 +_ " 1 -~ ..I. 2 Z [ ,) '7!'"

.

-aln" llo,.~ +_'

[" .-. z]... 0. 71f- d+_i

(t 2.90)

,tile lor 'z' Ibe equa tion is th, n llUl e;l;cept Lb,t ain " ia .ubstituted l or (Ol!! " end oes .. for - .ill qI. Slnu the t wo eq uat ions mu el be va lid (or any vl l" e 01 " . we wee-sufi! )' errive u (12.9 1)

so " d 4'Z li Z dd81+ 2 (B ~ m) "dF -(

."

(12.92)

( +lIin' 9 dV'),

wilh

T,

e, . nd 'f ... ry ing With in the Jollow-lna ruga:

1"",LI1' < ' ''';;;

00 ,

O :li;a ~n ,

OI!O; .. ! 312) are decelerated and 1I0nrelativist ic part icles ( I ,.; e' < 312) aNl accelera ted. As , hown by Kara but a nd Chugreev , 1987, th a mcuen 01 a charged test body in metr ic (12.109). whell charge e ill opp()/l ite in sign til charge Q, is Similar t.o the motion of a l.t'st body in melric (12.99). For example , when the tes t body a pproaches the hor ilQn (r--+ ";..;.), the componellta of veloc ity v" and aceelerat ton dv"ldt tend to eere . T he radi al component of accelerat ion, dUldt . is positi ve in the vicin ity of tile hortz cn (r ::>! ';:.;.), while fOr large sepat atiollS (' :::l>';.;.) it has the form

~ ~:-: (1- ~,)-~( et t _ I).

(13.62)

for relativilltic p4rUdes ( e' > ( ~ r > 3/2 ) , as demonstraLed by (13.61) , ther e is bot h grav it at ional and elKtrOlltatic d ecelera ~ion , whil e for llonrolativistlc pnti d es ( t < e (r , w)-

-,l;r a" [ lflOI (r,

w)-

+

~ (r, w)

(15.23)

l

(15.24)

Combining (15.23) , (15.24), and (15.9) with (t5.21), we arr ive at the fall OWing

expression for ¢I'll. [r , w) outasde mat ter :

¢I"'~( t, ",)_ ~ (r , w) - ~. (8"a,..S,d (r, w)+ aao"s"°(r, lol})

+ ~ yoaaoo,ss' (r, 01)+ wheTe we

h a~ e

z:.. a"ih.a,ss' (r, w),

(15.25)

introd uced t he notat ion

5" 1(r, ",) ... l1f'~ (r, ...) -+~~(r, (0) .

(15.26)

Herll y"1 ..on8 ti tu~ t he spa ti al par t of the Min kewsk! mel f i.. wit h _ I elements on the princi pal d iagonal. Note tbe S,,~ (r , t.l) is a traceless tenser, th at iI , y~ .S"~{r . }

= O.

(15.27)

T~. R.lol/vlm< ThLO'I

102

.f

a,..vjIGlI"~

Now lel us d~vota mOre at ten tion 10 solution (15. 15). Exp anding R-l in powers of r -' . where r 19 tho distance from the SOun:/l', eenle. to th e poin t wheu the field is deLeeled . and assuming that the lino at d imensions 1)1 t he source li fe much slu lle r th en r, we find tilet (15. 15) yields

lb~~{r,

varia nt divergence of the ten sor X'/..' , which is a ntisy mmetric in th e upper indices, and tha~ , theretcre , cont ributes nllthi Dg t o (15.60), - U we allow fo r (9. 13), formu la (15.60) ceu be rewr itten as follows:

D", (T:' + ..:) _ 0, where

..~• =- "L ~ I

(15.61)

+""'i&I ' [G-' +21 I -"",Ipq - G-'~I ] D-" ~g,

(15.52)

l"/

I n contrast to (t,) .48), t be left-ha nd side of (15. 61) ill a tr ue wnwI' aince it Is thtl co varian t d ivergence in the Minkowski met ri c of tbe tenser q ua nti ties T;;' an d T:, Hence, all intensi'y ca,lcula tion (or eelcalatlon of ot her ehuMter i! ti&.! of tb. grav itational lield) based on (15.6t ) will Dot depend on t he eheiee of t ile sys tem of coordi nal.es. Si nce in the Minkowski ' pace-time wo oao al\V8.YS select we Carte· sian (Galile an) l)"lw m nf coord inat~, (15.61) yields

am(r:' + ,:)_ o. • Term. .. bote « ,o""ntIOJl,

~ l n, g eft C (t) f!"(lm (16.75/, su bs tituting its value int o (16.86), and per formi ng rel alive ly s imple manipu ali ons. wiUi allow ance made lor t he defini ti on (16 .60) of the H ubb le functi on. we find t hat

~ +3 (I:v11P .."t 't'( R),

(16.87 )

wh ere V (RJ ",,3{1.j...3vl +3

---rnr-

It is easily noted th at

'V(R ) is

(j

-v) _ 3 (t+ ,,)

~

----z--.

poait ive if R < r. lam if R =l , Ileg>lti v(l if R > 1.

I

(t6. 88)

(16.89)

Si nce H ('T:) vanishes onl y at points R ('T:) = R." . and R (t) .,. R.,O', wh ile H (t ) *,0 for oUter values of 'R belongin g to the interval (R"" n, R.,u) . Bq. (16. 87) can be represented i n lhe form

-s

• With

t ..

(-}) +~=~'V( R).

0 we tilIociUe the vil ue R (0) .. RallO'

(16 .90)

'"

Since If (R) IS negati ve far t < R < R.. .., (16.90) yields the ineq uali ty dH -'fdT > 0; he nce, the Hubble functio n H (t ) monotonicall y decreases for R E(t, e,.. ). Prom (t6.90) i t read ily follows t ha t H (T) monot cnjca ll y decreesea lor R E (8" 1]. too, with R , the solution to We equa tion {

3~:

't_

{lin·1[(i-)'il Rl/I] _ !l n·' [( i- ) ' /~ R~ ]} .

(16,I ZS)

rc• and R_ ... < B' P this y ields

R (t)

Q/ ({-

r"

\iJt1l'J

(f Vit).

(16. 129)

lAt os esli ma" tb. mu illlum '0,1\18 of ,. 101' ..hLeb (IG.I29) SliU rem.ins valid . SilKll R (T) mly reach the ...h.1 R. _ l{lI2 _ a) HI''', (roUi (t6 .t2 !1) we pi

TOeo: 2 For t imes

1" >1:,

(f jl/: ~ ' ;0- ' (1_ eI).

(16.130)

we "' ril o (16. 124) in the form

~ /T "

T- 'r.o + V

"&1

~·"z

(8.... _ £4+ £ .

(16. 131)

1,1,1 '

We inlrod llCl! t he notation

R1 -

(T8)'",.... (.1.) U' '" .

.

(t6. 132)

h u n • ..,; Iy be damouslnlud lhu lor , II values of R (1") bt iongilll:' to th' lnlerul IR•• R,l we un di!ICud \h e t in the deno mi na tor of the integrand in (16. 13 1) fo nd write

_

• r Z(

T_T.+ V .. )

e,

r ' I' d"

Tlt e appfOJ:im. le \"111,11.'$ 01 lh fOOts 01 th e eql,l atioll B - 2.f' ll',~ •

,.

(16. 133)

(B_ :&>+3:z}',to

+ :b: ... 0 ano

... IF, "s.' -T ± -' l'3 ,- R•.

with

8)'" - (' I'" .

R. - ( T We can therelol"l!l wri te (16.133)

a.~

~

(16.134) (16. 135)

(ol1ows;

" [r (R,..r)Jil' [ (r+ R,I2)'"+ 3{R,I2)t ]'" . " ,

'r 1_ T.+ Via) Apply lll( lhe _

n-,..Ioe t lteo rem . we obta in

T_ l.o +*

•

~ Ir (A,~rn'1'

~

(16. 136)

,~

with

...- ((,+-11-)' +3 (i'-ll'"·

(t 6.137)

"",,- ~ is . numbe r beloDl ln, te tlte hILerva I IR•• R I. Allh ouJb . generally lJl'u.k· illl. t be lue or II. de pends on R. lb i, dependence is extremely weak , since fot . 11 t he Iues of R eonsider ed hOril the value of fl. ;s lim i ted by th e following inequalit ies;

' (

.

,

)

(t 6. 138)

}if l-T < 1'1< "'3 " In te, . U ion in (t 6.136) yil'lds

1 -1e+

l""k r$in- ' ( R,;;,UI )-1;11-' ( R,; ,ur

o

) ] .

(t6 .139)

Th ia Ma ble. ealcula\iog Ib' ap proximatt value of !.be U p'-nsiOJ] period 01 Ibe lIDiul'5e. Assu m ing t.hJIt R .. R, In (16.139) and Il kill( i nl.O l ocollnt (t G.t 26) and (16. 132), we 6nd that

1,:::0: 1. + .!l1 1"&( T') '" .

(t6.140)

Sllb:J ti t uti 01l: (16.130) and allowing for (16. 188) yields

' < (T' )''' (:\- (201'J ''] ;T

T,

< ("-)tIJ ;u ' J ;T

(16. 141)

where e i• • sufficiently sm. lI posi tive Dumber. w. note lb.t l be balf-period 01 the eyd ie evoluti on of Ihe universe. 1:" Is del.-mined by the \,. hlll of Ihe rrav il on ll'I ass, in . ddit ion to the v. lues of n e fund.men 1. 1 WIllI LaDI.a • • ed ~.

Let us .lIOW' e.t. bli$/\ lilt! upper bound for the rrav il.On mus. Bu illr our reason _ in, on t he wn dilio n thd the . ge of Ihe unive""', ..hich is determ ined vi. Hubble' , cOlUt. nt H ('t. ), must be ' ," 'ller t han th, cyclic b.lf -period,

,

_('a )"1m + . . "

'"

S.. _ y..IT'Il +...

S., = -

I

e' l

(

(17.32) (17.33)

,

)"'

T " ..T" + l'u Y-' - T Y.,)'o< ra' 1

12)

(2) ( I.

(Z) ( 0)

- T (l'••TM+ 'Id ' ..,TOO + f d f'lll)+ . . .

(17.31,)

S ubst It ut ing into Eq . (8.36) the u p.nsions (17.25)-(17.27) and (17.32H17 .3"g••: (4)

where

I

r

g.,-2lJl + 1ii O: J ¢I, =

-G

d'~ 'U (~ t)

I~

~ 1 ~~1

¢l1 ""G ~ 1 ~ !n"1

~ ; I - 4oIl, - 4¢l. - 2lDl

d',t',

dl.t ' ,

¢tt =G

¢l,=G )

- OO\ ,

~ ~lPx" I ~ !~'I

dJz'

(17.69)

(17.70)

Bra known u the geue..liled gra vitational pol.(lIltials. Sin ce (see Vladimi ro\', 19M ) I l' Ud' z' ""'E") 18- 8' 1

G

r (.

'Ids%,

r

dOz'

= -S J P !( , 3 \ 8- 1' 11&'- 8'1 _ _ oj p(,,· ,t)lx-x' ld'r ,

'"

the final expression for g. , proves \.0 be (I )

r

g""... 2CJ>. -~ .1 pIx', /) I x- x' I d'x ' _ 4$ , _ 411, _ 2 I

•

Ihil we find !.hat

In view of (8. t ).

4l-:::,, "";" lor i- we hav e

&5

I x l - r - .... .

(18.2)

the loll o_ in. lnr mu l.l:

;" ::.: 1+'; ' . On (),e other h'lld , if ....e de termine ;- fro m (11.1t). (17. 14), an d (17.55).... ee t

r-:l/ I + 4U• ...·ith U the Newtnoju po tenti.L Since (in th. I)'llt.ro. 01 v.eils . ·ben! C "" G _ 1) "... 1II. \·e U _ M lr far from the so uree . _ ith M . by definit ion. the p vi""ti onal mUl 01 the objec l. we . rri ve. t the foll owing iden ti ty: m, _ /of. (18.3. wh ich ill what ..... set ou t to pro ve. We IIOU in PUlIinl thai within the HTG Irtroework tbe quant ity P" =

J(tt:l + ~») d"z

(ISA)

n. is ..., ll-del\ned and ~ollstl tutes ~. 4-VK\Or 01 - v-moment um 01 the l)'Stem. S iella, l)', th e ' lI( u4. momentum of lb, system is ,Iso .. . lI-defioed in RTG Ill d is. te~ wiUl respect to ny eo«din . te tn nsform&tions III lb, low-di meuiOllI I Minh Wlllr.i I PI'ClO- t iml . I II GR, as 11O\.ed in Chaplet 3. th , .. Ilue of tiul l Dllrtl d mua dlpeolU 011 th e choice o' c:oonl. i.nltes io the tlu'M-dimlll!:iOlll.I space, I resa.h 1116ll.lliJlllt" Ito\Il. the sU ndpoint o' physie. . Now let ..,cmLSiderUtis problllD'l inlbe light of ~ rauluobtloinMl i ll Cbaplt'r 12. w. ",ilI shOll' thf:t i n .,i.... of l.h' a:rbllrario ess III the ebolce of 3OIuUons 1.0 lb,

Hil bert-Bin.Utill 4lqILl.UOll.S, the lnert i,l mass (u dellned in CRI may, gIlnlu lly .peak ing. aliIU llll Ill)' nhl8 I nd , t here fore, t h8l'e ea u be IlO eqWl lity between \h' inerti al aDd lP"lvil.atlODal m uses in GR . As in Chap in 3, '" base our calculaUoll! of th e inertial mus (in the syu , m 01 unl l.! in wbi eh

writ ten in

Illl'll'IlII

~

_ G _ I) 011 fonnuJ. (2.t1 )

of CarWllQ eoordiolles:

m, _~ h'o:. (:z:)dS",

(18.5)

(18.6)

In Ihe I.., u preuion " I IuI v. letrcd ueed lhe nolll illG ...... (z) _

SIIl(A..ith . .peet to

- u- (r l _~) .

IhreHl i ~i oM.1

r (>:) -D'r (z'). a.nd

r' -th.r'*t*"' " f orm

(18.7)

(spJlllaJ) ulnaforml ti oD.l'" is I x.lu,

wbeN O _del (

~;

).

is . lIKOnd-t lo k teaece , the u.llIiormllioll. I. ... for mS ' (z)

IUll lUM

(18.8) whar.

e,

If lor th. 7:''' we take the .pheriu l wordio. les z 'o _ r . ::1:" _ alld z" - If. then on !he bas" of (12A1) ..ilh A (t l _ 0 we . Tt'ive.t Ibe 101lo..il1f rorllluiu lor Ihe

• ." (z' ):

m" (z ') _ _ WZ siJll 8, Ill" (r)_ mD(z') _ -

...;~ (~ ~

)"Sill1e .

"'W"'~M (~r ·

......,,(r') _ 0 if

..

.,v.

(18. 10)

". It u n be ahO WD th ot

•

D _ ~,

ond.

be ~,

etJmbill;llI (18 .8) wi th (18. 10), "'ll it l

md(z) =

,

~

[ ];iT" ~ "'i7' '~ III..

(Z 1

+ ;;;or ~ I?I" ,~ '" " ( ~~ I

+ ~ ~. m"(%»).

118 . l t )

Henu , t akin&" illllI'ie coeflieie nlJ to be

,h.

, . (r)_U (rj _

t .. (r ) _

_ V(r j _ _

'u

vW'ti ' VW'o ( ' VWTo)' dr'" '

, I IV Cr)

ZGM@

(18.2.5)

_ W (r) , g.. (r , 9) - - IV (r) l in' 9.

(r ) -

Tll e~e

y'W"M - 2G''''0

form lila. t ran. form from (12.4 1) if we Jet A (r) _ 0, ". _ GM@,

(18.26)

....here Ms> I. the Sun ', act ive llravi h,t ionl l mal' and G til e gravil.lltioll BI con.tan t. The co ndIt ion A (r ) _ 0 doos 1l from the RTG resuh by 4(GM(1')"' r.. If Ille in _ uring the dtlleet ;oc of light aIld rad io . ifllils could be t a1Md to • leTel ,. which lIll&Ond-onler .fleets t Una Into pl. y , th b . mbignit y would become uprt'lllll nu, l1y varililble.

.eel''''':'

18.4 The SlIiU 01 Mertury'. Perihelion-

Su ppose th• • • partlcl, 01 mass III.,. 0 is movi og aloog • e lOlled curVI ' . ound lhl'l SUD. To des GM'i) ! f +;' 1 we ha ve ~he following ex pans ion 0)1 6qo in pewees of GMelr", to with in secon dor der term s: '

' )

6,,~ 3:o.GMe ( -;::-+ -;:-

51" (CMq)' +-,'.r.

+311 (GM0)~ (*-l)

t.r+*).

(18.8t )

The shilt g iven by (t 8.81) ceo be uprassed in te rms 01 the chMaCler istiea pe nd t

of th e tr a ject 'lry;

o W. > 0,> W. , wit h the result th a t th e integra (18.8G) can be represented thu s:

y

I (r o• r)

= (2GM@)' I , (r)

+ (2GMO)" I. (r ) + (2GMe) I , (r) + I . (r),

(18.87)

where I.(r) =

2

e )(W. - 2GM 0l I jlw. (W, w.nv x [(VW::-2GM I F(. )-(Vw- wl n(v (vW';- W,)(W.-2G.IIq) ) ] • e ,q . , ' e- , W,j( V w. 2GMe ) , q (V w ,

2GM

,

1. (r) ~ I VW, ( w. W.l I.,.. F Cv, q),

,

(18.8B)

(18.89 )

• So.. Lotun oy ud LOO kulOv, 1985a. t98Sb, 1118&, t 98lih, 1986., LC'guQOY, Loskul Ov, ud Chugft.v. 1986, sod ShapiJ'O. 19M, t919.

I , (r)_

2

IYW,(w,

W, )l' /t

x [W.F(v, q)+(l! w.- Wt) n(V, Y:::~~" I () ~ { • r ~ - ~p

'1)] .

(18.90)

2W, (VW,)'''' (1'14',

l ) (PY W.

tly w,- w,

x [Pl" : k- l F(v . q)

+ WIW,-Vw. V~ n ( ..., rf vi¥, W,

PW.

PVW.

1

I ' q

I]l

Pool '

(18.91)

whel'll q Is given by (18.62),

v,., aln-' [' (W.- W ,) (VWlTi- Viv,l] 111,

(VW. w,) (Vii'"f,i 14'.)

F (v, q) i!! an ellip tic integral of the flrst k tnd, and IT (v, of the t hird kind :

(I ,

(18.92)

q) is an ell iptic integral

•

n (~ , Q . q) = \

.fa

', \I + Oria' Il) Y t

.

q' .i,,' ..

Suppose thu in the $elected rererenee Irame the Earth has cccrdlna tes (r. , ltl.) and \.he reflectcr, coordin ates (r~ , lziV, I) • (y w.:- WoI lz VW:- I)

2

-

+ W,I~ Y W.-I)F (:tI2 .

1. (r, . rJ ...

( YW~ - w.) pc.-

_

,

+ 12Cm- Y W:) F(X,

_

2

, y'W;

a

x[lvw. - WJ n(l.

t"',rc-

iii- . • )

(y w.: - WJ (2GIn - YW.l

q)] . _

IVW_IVW': - WJJ'"

I Vw. (VW. - WJI'J'

(18.112d)

Wol IVW. (lI"W:- W.ll'I"

x [ IVW: - WJ n (l .

f, (r" ,_)_

o)]}... .

oj

(18.tl & ) F (X. q),

(18. t 131J)

'"

Hm

(YW: -VW:1W ,]'" , e» [ (YII' . _ w,.>yw_ 0 _

); _3; n. '

vw: )'" ( (Vw.."VW:-w.

(18.1I4b)

•

.,- - .,- ,- J'" [(J!W:-v'W:l ( v II'.

Wolhw,

) II' _)

( l8. t14e)

(V w.- W,)

If the pat h of motion of the !.est bod y is s uch th lll r~ > Cm and', :» Gm and , hence, y IV", > Gm and V ~ :> Gm, the n (18.111) yi81ds the following Ieemula lor the period 01 revolu ttcn (see Appendix 4): T':::!lI

(Vi¥.+V w.:),/' ( 1 + V2Gm

lIGm

= +

t

V i¥.+ yw_"

IrVW:-VW';IWW';-¥W:lI'l' YW. + ll w_

-i- [i-Sill-'( t-2 VII', y'W; - VW:: )]) . VII' _

(18.tl5)

Let us 1I0 W tak e the one-parameU!r fam il y Y W (tl ~ r + (1 + J.) Gm soluti on to the Hilberl·E illllt ain equatio ns. I n thi s ease, ob viously ,

"d

y W..... '''' +( I+).)Gm V'W;" - " + (1 +), ) 0"1

' (-'r.=- ) [

:=0: , _+(1 + ),j Gm+ ~

8 S lhe

(18. 11&)

3"'C.. {r.+ ,_)]%,

'.'_

(18. 116b)

wbeee we h ve allowed for th e approxima\.(l upMl8ll;on Jor " to within terms 01 th e order uf (Gm) ' , an expl'fl8llion tha t u n be obtai ned Irom (18 .107), (i8.1M..). en d (18.tMb). T hen for all v..l.,es QI p..ramete r A satillIying th e cQndition r~ > I t + AI Gm we QbU in. tf we comb ine (18 .115) with (18. 116a) and (18.116b), the lollowing formnl.. for the period of rev oluuon: T ~:I (, . + ,_W" { 1

J!:wm

+

3(1+I.lG.. +6 ~[, _

r.+,.

'.+'.

(-,=-)"Z ~] ). r.

z".

(18 .117) We see that the period of revol" tion 01 test bodies with in the GR Irimlework is "ot dete rmined una mbig uousl y beeaa ee t he value of ). Is not fixed in C R. On the othe r ha nd , in RT G ). = 0, so tha t the predic tion s 01 RT G concerning the period of re,·Q I.,t iQ n of a lllt;t body /lfe unambign ous:

T-::x:n. (r. +._)' /'

V2G..

{t + ~ [ , _ (-,=-) "Z ~ ]). r. +,_ '. 31'.

T he last lurmula reneou Kep ltlr's third law in RTC with ecreectto cs. 18.7 ShI...,ko,,·s Effect" Shiro kcv's efll'Ct is th at if a tes t par ticle moving in a spherica ll y sy mmetric gravita t ional field a lQng a closed orbtt is ..cted upon by Dweak pertu r bation , it will oscillate both rad ially and azimuthally. Th is problem is 01 interest to us from the • S,.. Sh;roko". 1973.

'"

mtlhodolOllcai ll'lgle . H ~ ...... will eonsider l si mpli r.td version; we IUSU me thd !.be U$t par ticle moves lUIi lorm1y al DOl' I Gin;:.la. r orbi t 01 r.d ius r _ ron!>L To det&m1 inl Ule il\flllil.aimI1 4-..eetc e ~ ' 01 a d. "i" Uon from th s gecdesic . ."., star l wi t h the deviation equ ation I)telenl to d' lennin. the 4' mOmetltu lD p . ( 19 .4, . Note \h it • rea l 1)'!5l.llm Un DO! be I t n ed y iso la ted beea ua of th e lllot ioo 01 iu co olIti tuen t partt. ...bieb ....... Inll.ioll of grui Utional .... ve., ,lid beo:a_ • re. 1 -r1l1l m u cha nps m ll.lt8r wilJl Olh.r " 'Il.ems i ll the form of eleet rol.U IlAet ie n d ie ' t ion, pertie' " _.to...., and th a like. Hence, ill pneR I. we ea nnot il llore th a eoettY no.l-u for matt er end I r, vi talion,l field. TIlere Ire es l ro ph ysi eal pro(J p8U (2$ 1

+

+ 2$. + lIlJ + 3Ib.) +

•

i'IJ

2p1'" "'1l

- t iJN" • +Py --;;;- + 4pvll(oaV" - i1"¥a) + pd'U + 2plPlJ&{JaU+ PlJ'1)"'U +0 (1:").

(19. t 8)

The form ulu ju st obta inlld , (19.1 7) and (19.18), ca n he $i m p lif,,~d if we take into eceount the cO lltinuity equ ation (17.49) and the Newtonian laws of motion of an elastic h ody , • J

";;'" + 2p :, (U....Hjw'-U+¥JiI"U

.

. •

I -

+ p en+ 2V+ 3p1p) d'fJ -~V. -y~. (N'" + Po'"12Cf\ + 2dl.+ 4l~ +3¢l.I + O (~ .

V'")

Subst itu ting th is into lbe rirllt·band side of (19.10), I II0w;lIl lo. Ln.

(t 9.2Z) id tn~it i e~

~ Pl n~,u+u.dl,l dV .O. ~ p tU8,v+ iJ.¢I. ldV . O,

~ [ p8~U + p".4J. l dV _ O • .~ p [I'2O.U+ 8.dl1IrlV _O,

~

=:

~ P"'d, (N"-Y"'ld V-O. d'U tlV . 2

Sp(II'"(;+ N")dY _O ,

I ,l"'dY _ _ \,.,.UdP, I p8P dY- 0, ~ lH'.o,V~ IW -

(19 .23)

Jfll',P.,\.... dV.O•

• nd u kiul inlo .ceoun ~ tb. taet \b at the volu me llller.,I, of sp. ti,1 div. ra; enee vani. h d lllr tu nsJorDilt ion lllln '\lrf.ee integ-rib , we obtain (19 .24\

{1!l.25}

w.

In der h'ini the Ian formul, employed lb. faeL tha t the loweriol . nd t. il iog of indica 10 (19.22) u n be achlaved by using the 1JI.U1e t ,,,!Of 01 Lbe MinkoWlki . peee-HIDI. The 1iD. 1 ' lIpreeiona ' merrl"g from (19.24) .nd (19.25) I re 1" _

~

p[t + n- yU-fll..»"]ev.

,. pa= rJ ['JMP (, l + fl - y" U- y ".,lI" ) + PCO- + ypN

(19.26)

'l dY .

(19.27)

Oia pte.r 20. Do Ext ended Objec ts Move Along Geodesies in tb e Riemann S pace-Time?

l.l,.

I" Cha plfr 17 _,Ive lorm uJu (t 7.7;'}-{11.TI ) for the md rle eoell>c ieoUl of Rle ma" " . pete -l lme In lectn.! of Soma of v arlo.... poer.Heed poten~; . 11 .... ith ten

ar bit ra r y coeffici en ts ~ . '\'. a , . r.tor,. dimeMi.OG' . Ihat ill. an ob jeet _hose . ra" it.t loaal self-e.neryy. elutle lIlr_ ell~, .nd t he Ilks .... 1I~li..;b l. if eom· pared with t be tot.1e6l!fJY of th e objec t. To deter m ina Ibe r.t; o of the ....Yit. ti ollal mass to the inert ll l oa. for a n exten ded objec t. ...·e mWlL ei t her drnt k .II)· r. ise t b. pree i. ion 01 ....Yi m.ttie e:rp" ri me Dtll inYolYior ob;ecu of I.botatory d imeMlollll (which is i mpcm ib le a l Ihe Prelle1lt 1. ...1 of t«hnoIOfr)') Dr elIrry out m. ..urem enU t hat ioyolve objec t.ll of grea le r di o' 81ll11011 , ....y plan . ts , for wh ieh t he rat io of the grav ltaUon . 1 llell.anergy to th e l otl l enerry ia cOMide rably hi gher th.ll for I)bjl:'C ts I)f la bor.tory di men! lons . But . ince grav iuUOlllI1 meU UNlmollt 01 Ihe r lli lio of t he paMive gravlt at lon,,1 man of ,,1\ ext ended object (. pleuet ) to Its inerti a l m Ull ill i mpQ!!1I1 blf~. we must look for phenomena in which t he difference betw een t h~ema !lSl!!lwll IDl .n l leal Itllell. O no Ie t be eflect of dev iati l)DI)f t he m o ~il) n 01 t he.cente r 01 m Ull 1)1a n ex tended I)bjtet from a geodllllic In t he m a mann apace-time . The flrllt \0 notice t his poe.ti bili ~y Wall Dir ke, 1962, who ,uggested t ha t t b• •at io 01 t he 8""a yita ti on. 1 m... 10 Ihl) inert l,l mus lor u tronomiul bod ies di ll'er SOmewha t fro m un ll y H t he 8""",ltall onlll 1Ml!f·uev of sue h bod i... vades under cha nge of t holr po$itlon In t he .,.",itet iona l fiel d of othet' bod in. Later t hb ellect Wall also studiod by Dicke. 1969. NOrd l Yt!ld t. 196&, .nd Will. t 97 h . Ha YIn, in m in d furth er appllut ion 01Ihe "",ul'" of hll u lcllll lil)l1$ll) ll. e ')'Ito m COllllbl i llf ollhe Sun alld a p lalloe\ i n Ihe w in lIystern. W i ll . I ln ta , dem oDlt r.ted on the bui. 1)1 Ihe PPN form a lism lh at the equa tlo... of mot ion of the eemee 01 m Ull of .n u te nded object (. planet) in lhe a .avitat i.ollli fI.ld of a point.like. 3OUl"CO (t he Sun) II resl a"lIm. lba form

m_

m,a" - "..rIIP.

where '" I ia t he i llert ial mallS of Ib, extended obi ec:t . m. ill lbe eeuve graYitatl onal m Ull 01 Ihe eecree, 4 '" are Ihe Cl)mponMI! of acce lerat ioo of lhe cen ter of maa 01 th e e1tlen ded obj ect, an d R la t ho dilltance between l he poillt-il ke &OU I'CiI of Rrulta tlon. 1fie.ld and the een tee of rna" of the ex tended objec:t. An um i n, th at th e veloc ity of a sphe.ica ll y ,ym met rie ex tended obiec:t i. eeec, W ill arr ived al th e. foll owing for mu la for yeotor p :

~ = - n" {I _ [ 4~ - v-3- o.t- et+ r.;- t to., H .-eJ] {)] ,

(20.3)

where n" _ n'"'1 R I. and t he lIpeeilie graYII.tion el u lf·e nl!f1'Y of t he n len ded objeel mealll Ih e. con di t ion

c _ ~~

I

,!!:' I d',:Id· ,:I'< I .

Will . 1971a , 6.l'5t defined t he pUli v. gnY itatiolla l m. . in aeeardauee wilh th e condlti oo /'" _ - "am ... Using Ihill de6.nition , ha . rrived at the r:onelusio n t h.ol t

~-

]a.

I - [ '4I-.,- 3 -a,. -t. +a..-T(4a +~-eJ

(2(U)

Wi Utin Ibis I pproaeh th. praleR«l of posl.-Newt.ollil R correct ion ~l1DS iD (2(H.) ..... i nler pt'6ted u I he l'e:lIuI\ of brtlakd o"m in aome Iheori n I)f graYih ti on at Ih.

posl-Newlon ian leve l of the equI Hly bH",-een Lhe pallll!"e i'tl vitat iona l mas• • nd the inerti, 1 m•• ol i n "teoded object. Il w.. , 1110 ata t'" t hat lb ........lity 01 tb. pus,"" r r. vih.tiona l mass a nd lb e inert i, l mass In Ih. posl.-Newton ian I p pro:timal ion W01l ld mu n 1b.1 the eee tee of mus oj the n llnded object mov• •long • eeoduic. lI ere it re important to note that tbe moll o" of • test bod y oeeurs. by utfinitlon , . long I geodH ie., ....hieh is determ ined from the principl e of INJlt aet ioD: th , funeUon.1 S _ ~ L dt I. kes th e least possible v, lue on • podesie ecrve. T he ~ u. l i on or th i. ClOn e is " yen by va rying th , L.;rang ian '1l0e tloll

L_ _

nl

("

U' UA )' " t -;U---;;;-

over Ib, ( oord ina lee of the pl rtl ele of mISS m plae td in the gra vitat ion. l field COrTe!pon ding to metrie Ilk in the Riem ann space-t ime. In eond ltio ns of • re. 1 nperioll nt it ;s d il licu lt to determi ne whe ther the center of mass 01 lin ex tended objec l moves along a geod..,le. Due a pproach to t his problem hn been to dete rm ine I~m exper imen ts the v.lullll of . 11 the req uired poetNewto ni an pars rueters a ud, using W(IJ' s Iormula (20.'1), lind the rat io of t ho pa8lli vo g ra vit at ion al m a~5 to u.e inerA i, 1 m ~ oJ a o ex ten ded obje.:t . nd l'5ta bJish the patt ei'll of molion of 1111 cent er of ml!l5 of t his objeel in rel. tion 10 ' s pec lr,c r eodHlic i n the R ieman n s pace-ti me. The Ilrst to ' \ligest al\ u pe,riment of th is lr pe "' 15 Nord"'edt. 1973. By ealcu b linr Ih a motion of the Elrt h-MO'Q n s)'lltem in the Sun's gr" vit. ti onll Ilald, he I Ua allll.ed the n istence of • number of ' Dom"li " in lb. Moon 's mOlion ""hose ob6e"" l tio n mi eht mah it ~i b le 10 measu re com bin ll iu!U of pcet-N e..... tolll. .. para Diele rs. On l sueb l oom"lf.is t he polariu tioll 01 tb. ),foon' . or bit in th e direcl ion oJ th e Su n wi th a n . mpli tudl 6r ::.: "I , where t is a COIll5U nl 01 th e ord ~ of leo meters , . nd

'l - ('~- Y - 3 - t, -a, +C;I- ~ (~ + !I., - fJ - ~;... T o dhicover thi s aRee.I, an a nal ysis .... n mad e of the da la obta ined from mu sllring t he E art ll·Moon M'pnat ion by 1_ ra nging. As II Fesult onl grou p oJ !tiearebers (~e. W illi am~ et III., 1976) conclud ed Ihat '1 _ O±O,03, whil e anoth er lound a etcee result : t'j _ _ 0.001± O,OI5. U ~i ng these cstimell1S and Will's form ula (21J. ~) fur lhe pesetve m. lI, a number (If re$('~ rc h ol'5 concluded t b ~ l lh~ relio (If the passiv e gravil al lon al m es~ of the EU ih 10 it. inerti al maM ~ h oil i d be close L(ln ll;ly: mplm, = 1± (1.5 x I O- II). TI' ''II, the dlltll Irom Illser ran lllnil of th e. Moon ....ou ld seem \ 0 5Ugl:1l5t (and Lbis ....as done hy She pirc , Coun ~~ lm ~Il, and K ing , 1976. and Wil Hams dill .. 1916) th s t in th o JlO$t-NeWlo nian a ppro)lim. t lon the paMi,,!"g~v i lation81 ma~ or an ex lended objre t is equa l to th e object 's iner lia l m U ll . nd lhat the center of ma..... mov e~ a long a llf'Od esic In t he Rieml nn spacel ime. ID ' 1. ler "'ork , Will . 1981, d id Il(ll assu m. that I h. ..eloeily 01' Ihe ex ten ded ob ject is u ro and wrote Ih.. equ a,i ollS of Ihe moUOJI of lhe center 01 man of Ihe e:rh ,nded object incorporat ed in I dou b le s)"lItem in Ih" form

'"ta- - -'";1'U +

""Il~ +

m"':",

(20.5)

...·hlJ'll lhe pPllive gra vit. ti on.1 energy m, of the spheric.lly aym llletric u lended obj«.1 15defined _ as before, via (20.4), Ib!! oz:t, are genera ted by lhe gravita t ion. 1 $! Jl.energy of Ihe extended ob}eet I nd for a totall y cOMervat iv!! metric theory .re u ro . U sta nda for Ihe Newlon ia n gravilatinna l pote nt i. l or lbe 5l'Cond ob;ee t in

th e !)'lItem, aud Ihe 1/.;. named iV·bod}' ac:.,..lerations by Will, cor uetn onl y " w. ] IrU.

(20.16)

+

T he sum 4"1~ lj" ll is s)'m met rie in i edtees a: and ~ , but s ince we "'ill DOl need il ln ,,'h at 0110'", we dQ Dot (i ve it upliei\ ly . I t U Ol be d emonstra led t h. t vector S- c.nno l be represented III tlte fann 01 • four-d illlensinn. 1 divercenee of • combin. tion of th e rener. li1ed ltMIv it.tiOfl. 1 pou. nliu. an d tbe cba racteri s liea of t be perfect Ould. Siau 10 metric lh eorl" of Jlllvl t.ti oD c:oolal.a.i0i ten cOAMlTl.tlon laws yec:lor S- mUllt be tqlL&1 10 zero , tb il Impli. thaI th e P PN pu ame-tees in these tb eories necessa ri ly nt isfy tbe

cond it ions ~.

~

_ O. 3t., +

2,. _O.

~I _

~ ••

t, + 2t. _

0,

(1. _

O.

(20.17)

is know. , If we requ ire Ihat lbe angv.lar momenlum be conserved. thell Ihe tensor 4:1 1t~1 mUllI be a)·mmu ric. Comparing (20. 13) ' Dd (20. 15). wa", l bat I i -U IT

+

n. t;; + t:':,.,.t:; + 1(":, only

if lit -a. - G.

01, _ _ 20,. G. _ O.

ThUll, l.Jl , nqllire ment th a l t h&«! be

polIl.-~e .. ~n; ...

(2O.t8)

conserv at ion laW!! luves

on ly three PP N !U'U metera independent : y, fl. In d t.- Th e other patlm , t.el1 Sill: " - _2~ . , t , - t ... to - -(V3) t ... a , - a . - a . ... ,. _ O. Th i. Is whal i!. known IIs lhe co mpletely conserv ati ve P PN Icrmalta m, an d withi n th Is formal ism we will opeu te . Note th .t , II t hese lim it at ions on the val uu of thl! P P N pa rameters were obtained by Lee, Light ", aD, I nd NI, 1974.....ho IUed t he p:lJlu dOleOSOl" . pproaeh. Th is approach , h owevee, is physical ly roe l nlnc 1ellS; • c.ril iea l review or It ce n be foulld in Dmisov and l.or\IDO.... 1982d. 20.2 The Eq_,t1en 01 Ml)lSon 01 !be e-ter of M_ E:dnOed O.jed .

01 a n

W. ~rt the deeintion of lila equation by defining lhe rad ius vect OC' ya of the eee ter of mi st of the u \.ended Objllel :

7II,Y" "" ~(~) +trll)) X"' dY . where m l is

~h

(20.19)

inerti al mass, or

m l _ ~ (tI.,+ttt.lld V . For tbe fund ion form:

q:) + ~ .... take t he u

pl'1!ll!lioll (20. 12) trlUW'orrned to 10e.1I

(20.20) "'h ich i5 llQi ly lntu preted ill. tha 10110"'; 11.' ma nner: ' p i , the df;lI.! it y of the rest and the df; lIJIit ies of the in ternal and pote nt ial energies. mn s of the objeet, rO!lpeeli "e ly, . nd _ {1/2} Pliov'" th e density of th e k ineti e energy. On the basis of the d illecen tial eonseC'll8tlo n Il w

pn

pU

• ('I Ii ) 0 hi tj"'J +GII -

... ani.,e I t I uniform ...d Noetili near 11I0tio ll. 01 th e eenter of mig of a doub le syst em: _~

~I

-K,Il'j11+ - Ktl'1n ..,

. '{,I+ - 'W

J (IWJ+ ':~J1' _eoo st. 5 ('(lIJ +t~4l'

(20.2 1)

In d dit ion to the inertll l ml$!l",e in trod uee the reer maM 01the e:tle nded ob ject, !oJ _ ~ p dV, whleh , in .,iew 01 the eont inuit y eq u.tion. i. t ime independent . At; shown in Foc:k, t 939. 1959, Ihe pee....Newtcn tan ".ri l~i on In the weigh ting luneuen in the deft n iUon 01 the een ter of maSl for extended objecU be! no efled on t ho equ l tiollJl of moll oo of aphotlc. lI)' aymmetric extended objeel.l in ~he lowest order in U R. _ hif:h Is th, rati o of t he charK lerlst ie . h , of sueh ob}ec: ts 10 th, ir Mptr. li oll. f or th is n.uoll.ll eal eul.Uo05 In.,ol.,11I1 the equat ion tbat dellCcibee Ibe mot ion of the emlel' 01 ma!lll of an u ttJIded objeel will b. bued 00 rathu Ih, D 0 11

lCl + tt\.}o

P

'f b, pollt_Newt oni an eq uiliODI 01 motion 01 • perfed Quid f;an be ohhined . foll owinlf f od" 1939 . 1959 , by writin, th eo'. .ri anl f;oDllU'luion eqllat lon V oT tA _ 0 I t It ... 101\.4.,) (rr- - ""r'd""Ue)}

"0

- "0[(,...,..,." -

+Jii"" ("- - 3Ie'"rtrl + N" + y

+i

Il"lljn. (.,...,..,.. -

Jrr

",I(ol~ ••- - ml"'" ""1f' + "'o T""'jjOn-

Sinu a point-Ilk. object mevee. by d efm l~l on. alon g a jlllOdl!Sic. ns iner ti al In d mus es ue equa l, ....i th th e NlSUII th at we un re wril . Eq . (20.33) In t il , form '

p '8Ii v8 Ira vi t . ~i on 8 1

/11M!

.

""-

0(11= - m/llOl""' ----,p- +.... ['~ TI't'It" (,-,-3- T

"L.) UIl{O)+N'" -]• (20.M)

beelUse /RIOI - "' ~ .) 10 lh i. u se. Th i. eq uat ioD forms ' basis for delia •." t he p. .l". p . ... it. tlon. 1 DlUil of I ' ext.ellded o bj«t. If we deflol i t in suc h. ma nnl r Ib.t the lid of Ita equi lity . itb the inerti . 1 miss InlllJorms the eqult ion 01 IIlolion of the center of miss in lo I geodHK mot ion equltloll, then for lbe Elrtb' . g.ulta l;ond mil . we hive , .,";tb d UB regard lor (20 .3"), t h" follo'lll'inr equation: • 1I(ll- _oO "'1$

".

"~""'1fr+

N. ( ") _ ] T-F"" 'Ii -3 --S-~" Ue (O)+N"

[ '

,

(20.35)

with ( ZO.3I1) Th U!, w" can s.y t hat t he dui. tlon 01 t h" motion 01 Ihe center 01 m.,. from th e motion , lonr a eeod er;ie i:t due enl lft:l y to Iht! dev iation from unit y 01 t he rati o of t he Euth 's pl.!l!ive eravi lat ione l mass to l he inerti el mL-"S. Let '" now give el p~loll.S for Ihe!e masses:

'"..- .... [ h· fle- De- T ~ + O (••)] , 1OI~-"'et- 1 +~ +

(3+ 1- 4JI + " jt.) a.

+ { ( 'Ii- 3 - ~ ~ ) Ufl(O )

(20.3 7)

-+ ~ +O(l')].

Fin all y, usi ng t he nu meerea l val ues 01 Ue (0) aed 'l e liste .f ~

P"

dl'

"" i1.11 6 (t _ r. ) tl,o cerumen E ucli dea n dcltll fun ction and ' . th e radi us veetor of Ihll .. 1.b ch'l'll"e, Ihen Eq " (20.oi5) for k _ 0 ..seu m,,",- Iho form

,.

t"A· _ -7jr AI +h

~

LJ

t.6 ( r - r. ) II

+ (3- , ) 1J"A.' iJ..U +O (A'E-) ,

+(1 -T)U I (20.' 7)

wi tb 9" _ a.a-. The solution to this eq uati on ca n be Ilian aceUfllt.y .. folIo.....s:

.

.

writ~n

....ith pot l.-Newl o-

A" -l: --"+ TI~ E t • Ir - r• l - ~ -.!±1 l: roUl.,) 1 . - r~1 IfI. - •• !

+

~[~ z L.J

•

, ' d'" Jr """'if.=7I 1.'--'••1 -

~

U (r) L.J

•

.' .

Tr'=f:i ] .

(20 .48)

We use th is solu tion to lind th e expresaicn for the law of molion 01 an e lectric t eet ch.rge in lh e electromlpetic a nd gravi ti t iona J fields. Vuy ing th, t..rnngi a n function (20.44 ) ov er lbe coo rdi nates of the cha rge Iu ds us to the foll owinl equ at ion of motion : (20 ..9 )

For I _ a t his eq uat ion transform " into 1f',I'-

. ..

..

(20 .50)

a" = djr_ lIill+ a..."

whet . "" is t he ra dius veete r of lhe teU charge re.ck unod frOID lhe Ear th '. ce nter o( man . and 11['",) ia the eon l ri hu Uoo to th e . eulen li on ce used by gnvltt tio n. l interae li oo a nd i , delined in (20.30). T he contri bution to the acee lerat ion ea ll~ by tbe Lon nu lorte and ill post- Newton i,n correc tions, which we b. ...e dtnolH by ~ , is givt o by t be followi ng form u l. ;

.a::. - ; {ag expression lor t he vec tor pot en ti" l ,4o.,

whe re I':: ;s th. " eJoe;ly of the lest eh,~ in th... l:",oe",ntric noo rol,tilll co mo,, ;nl reference I... me in which , II ea ltul. ti oll5 wi ll be ca rr ied ou t . Se lecling tb e cb""", ' od ualS!l of th e l.e!It. body so l bet l be eI..>dric ttll'U eompenH I"" for Ih",. c..e.le"'l ioo of gnv;t y I ...(\ rest ricti ng lb e . e.eun ey of ealcuil l ions by 10 -" (th o sens lli" it)" 01 th e CUlTl!!n l cro p of gr . vimete" i~ 10 -.°), WI obta in

(20.52) where we bov e Intrcd uced the loll owing noutlon for t be lial t! K tl nll" on the t~t charl:e ;

,. - . ...

E""_

3

~t, l '- '.~' • ll£'> _ - T

•

r •

(,CI - r:'l I I'C.k ("' - '~I· t,

1' - ', 1'

l:Iere we haY e ellcwed lo r the bel Ib' l, wilhin l he c:hO!en ll«ur,c:y, U (f.)'" U (r) nd V: ... YI~" Th . N_Ionl,n pol~tial U a~ l h $lIrfeca of lh~ E.rth ;1 defi.ned Ihus::

20.'1 Tn MfonnaUon to Pb)lllc:t;I Coordln.etf'S Time I . nd o;oordinal$"'. in Ierm, 01which lbe . 00.... n l. tioM have hem ....nll . n. que nti li ... They Ire rela ted 10 Ihe ph)-,iee l ohler-nobles t ,nd l'" th rough Ih, follow[ni formulas:

• !'to eoordi". I.

d11_ - (,..dt _

' -;::1 )d,..drf,

•• V-t.dl + V,.

(20.54.) (20,55)

dr>.

In t he poII t-N~w to n ian approxima t ion th ese formulu heeo me si mpler:

at' _ - 'd d,.. dt' ll +0($1)1,

(20.56) (20.57)

d1: _V =;';dll l+ O(r;I)].

NonllYedt , 1971, ulled (2'0.53) 10 inlegTate Eq. (20.56) &lid found !.bel (R il (ollltant)

U!l1I",ed

l" -( I +l~)" - f ~ ""rl"' + ? ~ ",e,,*,

(20.58 )

We ..ill use Ihis fnnnu!a la lrf . Noel', h01Je"u, Ih. l the . b5eIM:8 in il of. l.ecm responaible for Ihe o;onlrib"tioll of lb. Earth'. i ravitetion.l Field is dUl to Ihe lecl th. t Ih e {r) cPr _ Pill(J) cPl. Combilling th i' fao;l with (20.58 ) y[eldB qm(JfB (r) - fJUft (I)

+f .,hue

(t + 2;o .!:;-) - t nQ ~ 1,/jIUIII

/II

':;p-l"n.iJl'3. I · b _ - 1I. b· ;;lo O. Seceed, i n vi. .... of the s pheriu l symmetry in the distrib ut ion 0;>1 maUer in tlt l'! Ear th , we employ Iltll following u plnsioM or t.hl'! gra" ita lio"" t poUnl ;11 and their gradie nu lor I I l ;;lo Ie: if"lJe - ATel"/fS.

r4 - ~ 1Il"' ' ' -!.f- "*" ("""+3M"''''' I]. (2O.M) (p(j.~ _ ~ [3m""" "'. + .,..Jm.· +~m' - J e -}(m...,..,.+IIIllyG' + "'"V"" + 5"' ''''''11/1.") ]. Th!rd, we em ploy in our discusalun the Newton ian ... ,Iull of th e elec l rle field

8trenit h: (20.65) Fo urth , .... Dote th. t Will , 1981, Itas ana lyaoo lb . R- I· posl· Nt ...ton i,n eOG b ibuliOJl \.0 tbe ac:eel.... tilMl of gr.... ily. U1illl ibe dl ta obtailhld by War burt on and Goodk il1d, 1916, aha Melch ior, 1918) tIlrougb th e elOplny ment of l'Upen:OAdUeli ", grui met.er. b. establi5hed th . upper limit on .... PPN para meter ; . :

(r_) :>! Jt

-i-

5?(GM el- (

HI

[1+ 3G~0 (V'w, +~ V'w, )

' ' ) 51 w:+w:+T

~] vw.w,

+ O ((GM0)") '

(M. 1S)

A4.3 Time Delay of Radio Slgn .1s in the Gl'Ilvilationa l Fie ld o f the S un In ex perl ment s ecunu eted by I. I. Shapiro and hia gronp (Shapiro, 19(14, and Shepiro, 1979) the flrst measurements of the time delay of u dio aigna l, in the gravitational field of the Sun were perform ed. 1n t hese experiments th e eeneetoe was t he surface of Mercury in the superior conjun cti on. Hence. the ex peri mental cooditi mu were auch tha t (A4. t6 )

App.nJiz 4

wheu ro ill a dillt8nce 01 th e order 01 the Sun'lI radiUII , and r. a nd r ~ au disl.anCe!l from the center of the Sun to t he Eart h and Mucu ry, rellpect ively . Stuee beceuee of (A....16) we mUllt put W• • W (r.} > W. , W. ,"" W (r~» WI' (AId7 ) (M.t 8) and WI "" W (rol :>GMe' in formu lu (18.58HI8 .91) we ca n use ex pans ions in powers of GMQ. On th e buill of (A4.2) a nd (Ali .3) cernbined wit h (18.92) we can find the following appr(lximat e u pr&5S ions 10' \" '~ : v· ... ~

" 3CMQ I vii'; I 6M@ O( (~ )' ) T + 2yw; -"2 Yw•.• - "2 Vw.,. + yw,'

(A4 19)

.

whil e lor qt, which enters into (IB.88)-(18 .91), we alrea dy hav e the a pproximate express ion (:\ 4. 5). Note tha t the de nominator 01 (18 .58) conta ins the di fference IV, _ '1JJM0, which vanis hes if for W . "''& take expa nsion (A....3). Hence, to cal. culatc the u ymptllti e val ue of I. (' .,p) ' we must take an expansion 01 W, con ta ini ng the next higher-order term. J::q uat lon (18.58 ) yleld a W. ~2GM@ + 8

(GMr;)'

w,

(M.20)

+O(GM e) ' }.

II lor F (v. q) we use repreeentauou (Ali .5), for n (v, 0, q) with [ c 1llS int o (f 8 .1H ), bear ing in min d t hat

(

(VW:+Viii:HvW:

2Cm)( lfW:

2C", (V W.W_ 2Cm VW:

)"2_ { vw:+vw: )1 /1 (A4.39) 2 .. ' no hig her t hen th e first in Yam, W6 arr ive

U:m)

:w", yw:)

Mnd [ela in ing ter ms whose order is at th... foll owin g approx imat e ex press ion l or T: r~ n

Append ix 5 In RTG the grav it a tio llal field is a physica l flsld and acts on test par ticles and light. Hence , just li ke IIn y nIhil' field , i t does not take th e world ti ll C.'l 01 p• • tid"" an d l ight uut.'lide the clIU$lll ity C a . T he si tUitio n ch anges drutiully wben Ihe Vl.\'lklll. h.. . noa~eJ'O rest Ill.,.". It ,.,at sOOwo · thlt fo r 1lOl utl on (A5.'1) the ti nge wi th hl whith th e su i, I:Ict.or R etl may cbanp is aho finite :

+

R.. ,• '" R (/) .s;;; R N u

(A::'.7)

,

wit h R "". +- 0 and R ., u < 00 . Sine a R (t ) .c;; R N U ' by &e lecting an a ppro priate va l uB lor (I , ~ay I , we can easHy ll'uaU nleu tb at inequ alily (A::'.6) i3 Vllid. As we !lee, the e.'l.'lelltial thilli bere b that Lhet a ex.illu a fin ite R .... , whlcb . In t urn , Jollo ws from \.he a8lu mpti oll tb at the rav ito ll hu • nonzero IUt mllSll. Thu s• • ceordi ne to RT C .• hpmogeneo...-and i.:l l ropic uo in rse ill th e Minko ",,· aid I paca-time exla ts on ly If tlie rrav itoll hu a P ODuro rat mass. It mW!t be . mphaslzed th a t 'in RT C the aolut iona for (z) ha ve ph)'5ica l mea ning onl,. If they .....tisf, COftdiU OJ\ (AS. I) . Let Ull check the vali d i ty 01 (A5.1) lor lIevMal imporlaot !IOlnt ioos or RT C eq uations. AS.1. In Cha pte r 12 we lound th e so lution (12.71) to the RTC eqUlt iOllll for th e metr ic coefficients of th e e llflCti ve R iarM nll apar;e.-tl me GUuide a " pherica U)' "~' mmett i c and I ta tiC source. Following Pock, 1959. let Ull wr ite th ese coefficient" i n term" of Ca m ean coordlna t6ll:

n:.... +

I,..

,-.c

I. (.cl- t +"C ' ,.... (;I:)=> (t + -;

V1'... -

I ... (z) ~ 0, ~~: (~.

(AS.8)

;0.,,",.

+ it +

_here t =- V·~ ?,., and", is the mus of the source of gruilatio",,1 lield . Sub atitu U..... (AS.a) ;n\o (AS. t ), we obta in

.-..c _ 7+'iiiG Sl ~

,_MG

.+ ~

-

(. +-"'-): _~ .+ -e ~ (;o ,.u-O. • ~ • /.-

(.+ -;:.c ):-

2mC

2..c

( I"G) '

- "';':j:'ji;G- - . - - --;r-

.., 111:'9, TAl r/on ~ .J s,ory (OdON: P.rpm , I. I., C. C. COu.... lman , . nd n . W. King , lin 6, Plry•. H,,,. Leu, 36: :;:;5. Sha p"",. I. 1,. ,I . 1., 197210 , Ph~•. R. ". L.l" 28: 1594. S hi p!"", r. I. , " 01., 1972b, c.~. Rd. and (;ro r . 3: 1SS. S hirok , M. F.. 1970. lJ/rl. A ktld. /'I""k JS$ R 19.'\: Sit. S hire k , M. P., 1973. G.". Rt!. ,,"" G,,,,,. 4: 131. S hiMk o.., M. F. , and L. I. Su dk• . 1007. n,,/d . A k"" . N "d $ $ $ R 1, 2, 326. Sy nge, J . I•. , 1960, R,j"ll "II~: TIt. G. " .."l TIurI;'l/ (Am!ter,m,,,'on (Odcrd : Cluend"" P,... ). Ve.SllQt. R. I'. G.. l lIB4 , C""'e",p. PhYI. Z.'\: 355. VI.d;mi.., .., V. S. , 191'1' , £9 .011011' '''1 JIf",h,,,,,'I