124 Dr. L. Silberstein on General Relativity 



6. Dynamics of a Particle. Example of a covariant 

 law of motion. 



Return to the general expression ds 2 =g i -du i du,, where 

 -u 1} ...m 4 are any coordinates. Since diii is a contravariant 

 four-vector and ds an invariant, diii/ds is again a contra- 

 variant vector ; similarly d 2 Ui/ds 2 , and so on. Such expres- 

 sions could, therefore, be employed for the construction of 

 generally covariant or contravariant laws of motion of a 

 particle, endowed, say, with some invariant " mass " or 

 inertia-coefficient. It seems more convenient, however, to 

 adopt another method. 



As was already mentioned, the equations of motion of 

 a free particle are contained in 6i^s = 0, the variational 

 equation of the world-geodesies. And the idea easily 

 suggests itself to derive possible laws of motion of a non- 

 free particle (or one " acted on by external forces ") from 

 similar variational equations after an appropriate amplifica- 

 tion of the integrand. The purpose of the present section 

 is to give only a very simple example of a generally covariant 

 law of motion obtainable by this method (but by no means 

 to develop the general dynamics of a particle or of a system 

 of particles). 



Let <I>, a function of all the ui, be a tensor of rank zero or 

 what is called a scalar, and therefore a general invariant. 

 Then <&ds will again be invariant, and the laws of motion 

 embodied in an equation of the form 



3j(l — 2<I>)^ = ...... (40) 



will obviously retain their form in any reference system 

 whatever, or will be generally covariant. Understanding 

 by ui the space-time coordinates of the particle in question, 

 and considering u t as fixed at the limits of the integral, 

 develop (40) by the usual methods. Then the result will be 

 a system of four differential equations, one of which is a 

 consequence of the others, 



|-r(l-2*)|il + (2*-l)|i = -|J,. . (41) 

 ds\_ K ^u K J Ou K Ou K 



where s 2 =gij'u i uj, the dot denoting the derivative^ with 

 respect to a parameter which is ultimately made to coincide 

 with s itself. Thus, for instance, if r, <j>, (9, t are used as 



