Energy-momentum balance in general spacetimes, revisited

Abstract We revisit the balance equations for energy and momentum in general spacetimes. Specifically, we consider the action of gravity and matter fields as a theory in Minkowski spacetime, which is deformed through arbitrary given metric functions g μν ( x ). This differs from the standard approach, where the metric is considered as a dynamical field, in that the given metric coefficients introduce an explicit coordinate dependence into the theory which generically breaks all translational invariances. The advantage of our method is that it shows explicitly how the given spacetime metric breaks energy and momentum conservation. The results provide a complementary derivation, and a pedagogical illustration, that energy-momentum balance amounts to the covariant equation D μ T νμ = 0, even if T νμ refers to subdominant energy-momentum components, i.e., to matter fields which do not contribute to the Einstein equation because other matter fields dominate the evolution of spacetime.