The variational equation of relativistic dynamics

Abstract

If Tαβ is the energy-tensor, then, for any vector field ξα, the equationfollows from the energy equationSuppose first that the physical system considered is complete, i.e. that the energytensor vanishes beyond some world tube Z. Let L be a time-like world line running inside the tube. We integrate both sides of (a) over a portion of Z, and we transform an integral over a four-dimensional region into a linear integral over L. We obtain the variational equationin which the m's are tensors characteristic of the physical system. An essential feature of the m's is that they are symmetrical in their two last superscripts. Equation (c) has to be satisfied by every field ξα, provided that the ξ's vanish, with all their derivatives, at the ends of the integration path.