Non-Linear Invariants and the Problem of Motion

Abstract

The effect of adding quadratic invariants to the integrand of the usual variation principle in general relativity is considered in the light of a new approximation method. The case in which the quadratic invariant is a constant multiple of the square of the scalar curvature is treated up to the Newtonian approximation, insofar as the problem of motion is concerned. The Newtonian equations result for sufficiently large "distances" between particles regardless of the order of magnitude of the multiplying constant. The requirement that the "force function" be finite everywhere places restrictions on the number of "particles" $p$, comprising the system. When $p=1,2$, the requirement is fulfilled. The less stringent requirement that the "force function" be finite when all "particles" coincide restricts $p$ less severely. If the absolute values of the "masses" are not all equal there is no restriction on $p$, but if they are all equal $p$ is restricted to certain integral values.