9 8 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 § 2. The First Approximation In the first approximation, the field with a singularity that we are seeking will be determined by the systems of equations (7) and (10). For their solution, the super- position principle holds. In addition, we can recognize that in this approximation, interactions between gravitational waves or inertial effects on the one hand, and electromagnetic effects on the other, are not to be expected. These properties im- mediately allow us to construct the gravitational field and the electromagnetic field additively from an “internal” field of the electron with a singularity and an “exter- nal” field without a singularity.[22] Then the internal field, as proposed, would be centrally symmetric (for a coordinate system moving with the electron) if the ex- ternal field did not exist. In the case that an external field does exist, deviations of the internal field from central symmetry can be expected. But such deviations will be of second order and need not be taken into account in the first approximation. We choose the coordinate system in such a way that its axis always coincides with the singularity, and so that when the internal gravitational field is left off, we have for that axis (11) where is the metric length when the internal field is neglected. In addition, we determine for the coordinate system that all the spacelike deriv- atives in the axis vanish, except for those of , and that the are all sup- posed to be equal to there.1 Finally, the coordinate system is chosen in such a way that within a finite region, the differ from the only by infinitely small amounts.[23] Under these circumstances, we can write the external gravitational field, i.e., the corresponding components of the , in the following form: 1 This can in fact be satisfied. If we suppose that the condition is indeed not fulfilled for a system of x(but, however, equation (11) is satisfied) then we can make a substitution: where a and b are spacelike indices only (1–3). In the calculation of the and the on the axis, the corresponding primed quantities occur only with the explicitly written coefficients c (which depend on ), along with their time derivatives i.e., 36 functions of time that we can use as desired. The coordinate condition given above, however, contains 9+27 = 36 constraints, which must be satisfied by an appropriate choice of the c functions. x 4 dx 4 ids (i 1) – = = ds [p. 239] x 4 g 44 g  –  g  –  g i x'  c a x a c ab x a x b + ... x' 4 x 4 c + 4a x a c 4ab x a x b + ... , + = + = g   x g  x 4 x 4