4 2 8 D O C . 4 4 3 G E N E R A L R E L A T I V I T Y A N D M O T I O N seems self-evident that one could advance from such an approximative solution by successive approximations to one differing very little from an exact solution. If this were the case, then a field solving the exact equations, given an arbitrary motion of the singularities, would be possible. Thus, the law of motion of the singularities would not be contained in the field equations. The fact that this cannot be the case follows from investigations of axially symmetrical static gravitational fields, for which we must thank Weyl, Levi-Civita and Bach.1) [14] We shall show this first, and only afterward treat the problem in a more general fashion. In this paper, we shall only investigate the purely gravitational field, even though the presence of electromagnetic fields does not pose any special difficulties. §1. Singularity in a Field (Axially Symmetrical Static Case) According to Weyl and Levi-Civita, in the axially symmetrical static case, by the introduction of “canonical cylinder coordinates,” can be written in the form ,[15] (1) where f and γ depend only on r and z. The same is true for the quantity ψ, which is related to f by the equation . (2) ψ satisfies the (Poisson) potential equation for cylinder coordinates , (3) where the indices stand for derivatives with respect to z and r. If ψ is known, then it determines γ by the equation , (4) where dγ is always a total derivative, because of (3). For the field to be regular at a point outside of the z-axis, regularity of ψ suffices.[16] For the metric field to also be regular on the z-axis, it has to be demanded in addition that γ = 0 on the z-axis. For if this were not the case, then the relation between the circumference and the diameter of an infinitely small circle around the point on the z-axis and perpendicular to the z-axis would differ from π, which would correspond to a singularity in the metric. This is easily concluded from (1). We now first consider the solution[17] 1) H. Weyl, Ann. d. Physik 54 (1918), pp. 117–145 Ann. d. Physik 59 (1919), pp. 185–188. Levi- Civita, ds2 einsteiniani in campi newtoniani VIII. Note, Rend. Acc. dei Lincei, 1919. R. Bach, Math. Zeitschr. Volume 13, Issue nos. 1–2, 1922. [p. 5] ds2 ds2 f 2dt2 dσ2 –= f 2dσ2 r2dϑ2 e2γ( dr2 dz2) + += f eψ = Δψ 1 r --§ - rψz) ∂( ∂z ---------------- rψr) ∂( ∂r ----------------· + © ¹ 0 = = dγ 2rψzψrdz r( ψr 2 ψz 2)dr – + =