D O C . 4 4 3 G E N E R A L R E L AT I V I T Y A N D M O T I O N 4 2 7 But even without any amendment, cannot be selected at will. The reason is that the (covariant) divergence of vanishes identically.[7] ( ) must hence satisfy the condition that the divergence of this tensor vanishes. If one as- sumes that matter is arranged along narrow “world tubes,” then by an elementary consideration one obtains from this the statement that the axes of those “world tubes” are geodesics (in the absence of electromagnetic fields). This means: The law of motion results as a consequence of the field law.[8] Therefore, it looks as if the general theory of relativity had already claimed vic- tory over that annoying dualism. This would indeed be the case, if we had already managed to describe matter by continuous fields, or if we could at least be con- vinced that we would succeed in doing so someday. However, this is not the case. All attempts of the last few years to explain elementary particles by continuous fields have failed. The suspicion that this is not the right way at all to conceive ma- terial particles has become very strong in us after very many futile attempts, which we shall not discuss here.[9] One is thus driven onto the path of conceiving elementary particles as singular points or, indeed, as singular world lines.[10] This is suggested also by the fact that the equations of the pure gravitational field, as well as the equations amended by the Maxwellian electromagnetic field ( = Maxwell’s energy tensor), have sim- ple, centrally symmetric solutions that contain a singularity.[11] We are thus led to a third perspective that permits, apart from the gravitational field and the electromagnetic field, no other field variables (with the exception per- haps of the “cosmological term”), but instead assumes singular world lines. If according to this perspective one had to formulate separate equations of motion for the singularities,[12] that is, equations that are logically independent of the field equations,—as is the case with Maxwell-Lorentz theory—this route would be rather less attractive. It turns out, though, to be probable that the law of motion[13] of singularities is fully determined by the field equations and by the character of the singularities, without additional assumptions being necessary. To show this is the goal of the present investigation. We had already thought much earlier about the possibility that the laws of mo- tion of the singularities might be contained in the gravitational field equations. However, the following argument seemed to speak against it and was discouraging. For cases occurring in the real world, the gravitational field equations can be ap- proximated very closely by a linear law. The linear equations, however, permit ar- bitrarily moving singularities similar to the electrodynamic equations. Now, it T i k R i k 1 2 --δikR - – T i k [p. 4] T i k