Relativistic conservation laws and integral constraints for large cosmological perturbations

Abstract

For every mapping of a perturbed spacetime onto a background and with any vector field $\xi$ we construct a conserved covariant vector density $I(\xi)$, which is the divergence of a covariant antisymmetric tensor density, a "superpotential". $ I(\xi)$ is linear in the energy-momentum tensor perturbations of matter, which may be large; $I(\xi)$ does not contain the second order derivatives of the perturbed metric. The superpotential is identically zero when perturbations are absent. By integrating conserved vectors over a part $\Si$ of a hypersurface $S$ of the background, which spans a two-surface $\di\Si$, we obtain integral relations between, on the one hand, initial data of the perturbed metric components and the energy-momentum perturbations on $\Si$ and, on the other hand, the boundary values on $\di\Si$. We show that there are as many such integral relations as there are different mappings, $\xi$'s, $\Si$'s and $\di\Si$'s. For given boundary values on $\di\Si$, the integral relations may be interpreted as integral constraints (e.g., those of Traschen) on local initial data including the energy-momentum perturbations. Conservation laws expressed in terms of Killing fields $\Bar\xi$ of the background become "physical" conservation laws. In cosmology, to each mapping of the time axis of a Robertson-Walker space on a de Sitter space with the same spatial topology there correspond ten conservation laws. The conformal mapping leads to a straightforward generalization of conservation laws in flat spacetimes. Other mappings are also considered. ...