Spacetime and Geometry · Problem 7.10
Cosmology — Sean Carroll
Physics — General Relativity · gravitational-energy-hypersurface-independence
Problem
Show that the integral expression (7.173) for the total energy in gravitational perturbations is independent of the spatial hypersurface .Intuitive explanation
The key insight is that energy conservation in general relativity is encoded in the divergence of the stress-energy tensor: if energy truly is conserved, the total energy integrated over any spacelike hypersurface should remain constant. You'll use Gauss's theorem to relate integrals over different hypersurfaces to a volume integral of a divergence, then invoke the Einstein equations (or the constraint that perturbations satisfy the field equations) to show that divergence vanishes. This transforms a question about comparing two surfaces into a statement about the vanishing of a bulk quantity.
Intuitive explanation
The key insight is that energy conservation in general relativity is encoded in the divergence of the stress-energy tensor: if energy truly is conserved, the total energy integrated over any spacelike hypersurface should remain constant. You'll use Gauss's theorem to relate integrals over different hypersurfaces to a volume integral of a divergence, then invoke the Einstein equations (or the constraint that perturbations satisfy the field equations) to show that divergence vanishes. This transforms a question about comparing two surfaces into a statement about the vanishing of a bulk quantity.