1 0 2 D O C . 9 1 R E L A T I V I T Y & E Q U A T I O N S O F M O T I O N The entire problem has now been reduced to the question: What characteristics must the have, so that exist that satisfy equations (21) and (18) and are reg- ular everywhere in the neighborhood of the singular point? From (18) and (21), immediately follows . (22) That this equation is fulfilled in the neighborhood of the singularity can be verified with the help of (15). It can further be shown that (22) is always fulfilled when the field equations are fulfilled to first approximation. Indeed, if we replace as abbre- viations , [31] then in terms of second order quantities we have precisely:1 . The field equations of gravitation are then given in first approximation by . They should be fulfilled. Furthermore, the electromagnetic equations to first approximation are fulfilled from which the vanishing of the divergence of to second order follows precisely. Since in addition, the divergence of vanishes identically, the vanishing of the (generally covariant) divergence of the second order quantity follows. In second order quantities, this diver- gence can be replaced precisely by , or, owing to the identical vanishing of , by . Fulfilling the field equations in first approximation thus re- sults in the vanishing of . Before we investigate the second approximation, we have to take into account a difficulty of principle. The approximation method according to equations (5) in any case presupposes the finiteness of the , etc. That condition is, however, violated in our problem, since at , there is a singularity. One could therefore think that the application of a series expansion as in (5) might not be generally permissible for treating our problem. is set equal to 1 for brevity. S i i x S i 0 = [p. 242] L i 1 2 -- i L by M i , and L i 1 2 -- i L by M i – – R i 1 2 --g i R T i + – M i M i S i + + = M i 0 = T i R i 1 2 g i R – M i S i + ---------------------- M i S i + x x M i x S i x S i g i g i x 1 x 2 x 3 0 = = =