Constraints on stress-energy perturbations in general relativity

Conditions are found for the existence of integral constraints on stress-energy perturbations in general relativity. The integral constraints can be thought of as a general-relativistic generalization of the conservation of energy and momentum of matter perturbations in special relativity. The constraints are stated in terms of a vector field V\ensuremath{\rightarrow}, and the Robertson-Walker spacetimes are shown to have such constraint vectors. Although in general V\ensuremath{\rightarrow} is not a Killing vector, in a vacuum spacetime the constraint vectors are precisely the Killing vectors.