Relativistic Conservation Laws on Curved Backgrounds and the Theory of Cosmological Perturbations

Abstract

We first consider the Lagrangian formulation of general relativity for perturbations with respect to a background spacetime. We show that by combining Noether's method with Belinfante's "symmetrization'' procedure we obtain conserved vectors that are independent of any divergence added to the perturbed Hilbert Lagrangian. We also show that the corresponding perturbed energy- momentum tensor is symmetrical and divergenceless but only on backgrounds that are "Einstein spaces" in the sense of A.Z. Petrov. de Sitter or anti-de Sitter and Einstein "spacetimes" are Einstein spaces but in general Friedmann-Robertson -Walker spacetimes are not. Each conserved vector is a divergence of an anti- symmetric tensor, a "superpotential". We find superpotentials which are a generalization of Papapetrou's superpotential and are rigorously linear, even for large perturbations, in terms of the inverse metric density components and their first order derivatives. The superpotentials give correct globally conserved quantities at spatial infinity. They resemble Abbott and Deser's superpotential, but give correctly the Bondi-Sachs total four-momentum at null infinity. Next we calculate conserved vectors and superpotentials for perturbations of a Friedmann-Robertson-Walker background associated with its 15 conformal Killing vectors given in a convenient form. The integral of each conserved vector in a finite volume V at a given conformal time is equal to a surface integral on the boundary of V of the superpotential. For given boundary conditions each such integral is part of a flux whose total through a closed hypersurface is equal to zero. For given boundary conditions on V, the integral can be considered as an "integral constraint" on data in the volume...