Relativistic conservation laws and integral constraints for large cosmological perturbations

For every mapping of a perturbed spacetime onto a background and with any vector field ξ we construct a strongly, identically conserved covariant vector density $I\left(\xi \right),$ which is the divergence of a covariant antisymmetric tensor density, a “superpotential.” $I\left(\xi \right)$ is linear in the energy-momentum tensor perturbations of matter, which may be large; $I\left(\xi \right)$ does not contain the second order derivatives of the perturbed metric. The superpotential is identically zero when perturbations are absent. By integrating strongly conserved vectors over a part Σ of a hypersurface $S$ of the background, which spans a two-surface ∂Σ, we obtain integral relations between, on the one hand, initial data of the perturbed metric components and the energy-momentum perturbations on Σ and, on the other, the boundary values on ∂Σ. We show that there are as many such integral relations as there are different mappings, ξ’s, Σ’s, and ∂Σ’s. For given boundary values on ∂Σ, the integral relations may be interpreted as integral constraints on local initial data including the energy-momentum perturbations. Strong conservation laws expressed in terms of Killing fields $\overline{\xi }$ 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. Traschen’s “integral constraints” for linearized spatially localized perturbations of the energy-momentum tensor are examples of conservation laws with peculiar ξ vectors whose equations are rederived here. In Robertson-Walker spacetimes, the “integral constraint vectors” are the Killing vectors of a de Sitter background for a special mapping.