Motion of an Electric Particle in a Riemann Space

Citation preview

VOiL. 9, 1923

PHYSICS: A. BRA MLEY

289

MOTION OF AN ELECTRIC PARTICLE IN A RIEMANN SPA&E BR ARTHUR BRAMLIIY DEPARTMZNT OF PHYSICS, PRINCITON UNIVERSITY Communicated June 17, 1923

Consider a Riemann space with linear element

dS2 = gij dx' dxj then the equations of motion of a charged particle can be written in the form (Eisenhart, THESZ PROCISDINGS, 9, pp. 175-S (1913) Id2XJ . dx dx~\ gij (d52 a+ -is- ds =i where where

a

- 2~~~~1 = 9 kj ( b9.k Xa x# + Jg,k

_g?.g,s aXk)

and the vector 4', is defined as the ponderomotive force of the electromagnetic field

ci = F,kj where Fij are the componentg of the electric and magnetic intensities and jk Po(dxl dx2 dx3 dx4 ds' ds' ds' ds Jk T0 is the current vector. This definition of 4i enables us to determine the q5's for any given distribution of charge density. Consider the case of a single positive charge, e.g., atomic nucleus. The fundamental form is evidently, =

_ds2 = (1

2e

dr2 + r2d02 + r2 sin2 Od2-(1 -2e)dt2

where e = charge constant. Since we can take the charge as fixed at the origin, we have dt dx2 dx3 dil ---=-= O~~ ds ds ds ds then

and-dt1;

gij r44

The only equation not zero, is glrl4'4=

= i

=

=

Fik k.

F14]4 = F14pO

P PROC. N.. A. S.

PHYSICS: A. BRA MLE Y

290

for Jk

=

po(O'

O,

0,

4i= gur4 =-r2

1),

-e (2e\ 1 =r\Po/r2 r'2

.-. ~~~F14 = F41=-p)

-

The law of force derived in this case is the well known Coulomb's law, according to which the'force varies inversely as the square of the distance the charge on the attracting particle is

-

e= 2e/po

Having determined the qb's and g's for a positive charge, the path of any charged particle under the action of the positive charge can be easily found. Let us consider the case of an infinitesimal charge, we have

fi

=

dt ~ Fik. Iky 41 = F14po

)4 = F41po

-

dr

the other equations being identically satisfied. Since 42 = 43 = 44 = 0

and

4

=

oF14 dt

=

dr ds

or dr

PF14

=

r1F

we have

dt

O

where po the charge on the central body. These equations have a direct bearing on the quantum theory. From dr/dt = 0, we see that the paths of a particle under the action of the inverse square law can be only circles, i.e., there can be no radiation of energy. For (Eddington, Report, p. 49) the equations of motion of the particle are (u= mass of particle), dr

(4)

(2)

(4)

2

1d-r

+

22e

v

(

] t

d20~xi (3) d2+ Vt d-r =0o d d '+ -2 2 ddf=° cdsOsnO\d/sdd d -Cos 0 sin 0 d+ + 2d-r

0d

29n1

PHYSICS: A. BRAMLEY

VOiL. 9, 1923

Since dr/ds = 0, we have from (2) d4/ds = const. = 0 (particular intial value) From (3).we see that dt/ds = const. = k From (1)

122eVX- v'k2

-

2

et-x '-M

which gives 2= l/u

From (4) dO

ds- = kor 0

1 = ,u2At + B

A and B constants.

whence the particle moves with uniform velocity in a circular orbit about the positive charge. The velocity is evidently ds v1 r const. the unit being the velocity of light. Thus the frequency depends on the ratio of the masses of the charges. The fact that an infinitesimal particle revolving about the atomic nucleus, must describe a. definite orbit with constant velocity is rather remarkable. Although the above considerations only apply to an infinitesimal charge, a similar occurrence will probably happen in the case of an electron revolving about- the nucleus. These considerations are in agreement with the quantum theory expounded by Bohr. In fact, he postulated as the fundamental hypothesis for interatomic mechanics the type of orbit we have just considered. In the case of radiating charges, we cannot say that the path is a geodesic in the space determined by the ,u and the of the central particle, so that the above method is not applicable to radiating electrons; but some other type of laws must be used other than the classical equations of electrodynamics. -