D O C . 9 1 R E L AT I V I T Y & E Q U AT I O N S O F M O T I O N 9 5 It will be shown in the following that the field theory determines the law of mo- tion when the character of the singularity is postulated in the first approximation.[7] On the assumption of a static and centrally symmetric character for the singularity, one obtains the law of geodesics, extended to include the electromagnetic forces.[8] However, these results were derived by neglecting quantities of third order, so that one cannot be certain whether every solution possible with our results does in fact correspond to an exact solution.[9] Still, the method employed is in any case simple and transparent. §1. Fundamentals and Methods The field equations of general relativity are taken as fundamental, in the form , (1) where is the Riemannian tensor of second order, , (2) R is its scalar, and is the Maxwellian energy tensor of the electromagnetic field, . (3) In addition to these equations, we have the Maxwell field equations:[10] [11] (4) which, as is well known, are a result of (1).[12] Since, owing to (3), the scalar T is known to vanish, (1) can be replaced by . [13] (1a) We seek a solution of these equations that is a centrally symmetric, static solu- tion infinitely close to a pointlike singularity,[14] and that should correspond to the case of an electron in a weak external field. The solution sought should have the form (5) We shall assume the convergence of such a series expansion1[15] and limit 1 This form of series expansion presupposes the choice of an imaginary coordinate. [p. 236] R i 1 2 --g i R T i + 0 = R i R i x  i x  i i  i  + + = T i T i 1 4 g i   i  = x f i 0 = R i T i + 0 = g i i g i 2 g i + ... + + = i i 2 i + ... + = x 4
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