On the theory of point-particles

Abstract

It is deduced from the conservation of the energy-momentum tensor that if the flow of energy and momentum into a tube surrounding a time-like world-line, on which the field is singular, become singular as the size of the tube is contracted to zero, then the singular terms are necessarily perfect differentials of quantities on the world-line with respect to the proper time along the world-line. The same can be proved of any other tensor, as, for example, the angular-momentum tensor, which is conserved. It is proved from this that for any point-particle whatever having charge, spin or other properties, which need not be specified, it is always possible to deduce exact equations of motion which are finite. It is proved further that if the energy-momentum tensor is altered by the addition of partial K^{μ ν σ}/partial x^σ, where K^{μ ν σ} is any tensor antisymmetric in ν and σ , then the equations of motion are unaltered, but it is possible to choose K^{μ ν σ} in such a way as to make the flow of energy and momentum into a given tube non-singular.