The double role of Einstein's equations: as equations of motion and as vanishing energy-momentum tensor

Abstract

Diffeomorphism covariant theories with dynamical background metric, like gravity coupled to matter fields in the way expressed by Einstein-Hilbert's action or relativistic strings described by Polyakov's action, have `on-shell' vanishing energy-momentum tensor $t_{μν}$ because $t_{μν}$ is, essentially, the Eulerian derivative associated with the dynamical background metric and thus $t_{μν}$ vanishes `on-shell.' Therefore, the equations of motion for the dynamical background metric play a double role: as equations of motion themselves and as a reflection of the fact that $t_{μν}=0$. Alternatively, the vanishing property of $t_{μν}$ can be seen as a reflection of the so-called `problem of time' present in diffeomorphism covariant theories in the sense that $t_{μν}$ are written as linear combinations of first class constraints only.