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

TL;DR

This paper explores the dual role of Einstein's equations in diffeomorphism covariant theories, highlighting their function as equations of motion and as conditions for a vanishing energy-momentum tensor, related to the problem of time.

Contribution

It clarifies the dual role of Einstein's equations as both equations of motion and constraints reflecting a vanishing energy-momentum tensor in covariant theories.

Findings

Einstein's equations serve as equations of motion.

The vanishing energy-momentum tensor reflects gauge constraints.

This duality relates to the 'problem of time' in covariant theories.

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 $t_{\mu\nu}$ because $t_{\mu\nu}$ is, essentially, the Eulerian derivative associated with the dynamical background metric and thus $t_{\mu\nu}$ 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 $t_{\mu\nu}=0$. Alternatively, the vanishing property of $t_{\mu\nu}$ can be seen as a reflection of the so-called `problem of time' present in diffeomorphism covariant theories in the sense that $t_{\mu\nu}$ are written as linear combinations of first class constraints only.