"Gauge-Invariant" Variables in General Relativity

Abstract

Einstein's field equations for the gravitational field possess solutions having a large variety of topological properties; among them there are solutions whose curvature goes asymptotically to zero at spatial infinity. If we restrict ourselves to solutions that are asymptotically Minkowskian, then it is tempting to try to divide the effects of curvilinear coordinate transformations into those that correspond to a Lorentz transformation and those that represent "gauge-type" effects. In fact a number of authors have followed a variety of approaches toward a reformulation of general relativity that would make the theory resemble, to some extent, a conventional Lorentz-covariant field theory. In this paper we analyze the group-theoretical aspects of such schemes. Making a definite assumption concerning the group of curvilinear transformations that will preserve the asymptotic Minkowski character of the metric field, we come to the conclusion that the reduction to a Lorentz-covariant theory is in fact impossible. The course of the analysis suggests, however, that this negative result depends on the initial group of transformations adopted; it is conceivable that a slightly different invariance group would be compatible with a special-relativistic formulation of the theory.