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

We first consider the Lagrangian formulation of general relativity for perturbations with respect to a background spacetime. We show that by combining Nœther’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 FriedmannRobertson-Walker spacetimes are not. Each conserved vector is a divergence of an antisymmetric 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. * A summary of this work has been presented at the Conference on “Fundamental Interactions: From Symmetries to Black Holes”, Brussels, March 25-27, 1999. It has been published in the Proceedings of the Conference under the title Conservation Laws for Large Perturbations on Curved Backgrounds with ref.: Fundamental Interactions: From Symmetries to Black Holes (eds.: J.M. J.M. Frére, M. Henneaux, A.Servin & Ph. Spindel), p.p. 147-157, Brussels: Université Libre de Bruxelles (1999), see also gr-qc/9905088. † E-mail: [email protected] ‡ E-mail: [email protected]