Momentum density of spacetime and the gravitational dynamics

TL;DR

This paper introduces a covariant momentum density for spacetime geometry, linking it to gravitational field equations, energy conservation, and boundary thermodynamics, offering a new perspective on gravitational dynamics.

Contribution

It defines a covariant momentum density for spacetime that unifies gravitational field equations, energy conservation, and boundary thermodynamics in a novel framework.

Findings

Total momentum conservation leads to Einstein's equations.

On-shell total momentum vanishes for geodesic observers.

Boundary surface energy equals the total heat energy.

Abstract

I introduce a covariant four-vector $\mathcal{G}^a[v]$, which can be interpreted as the momentum density attributed to the spacetime geometry by an observer with velocity $v^a$, and describe its properties: (a) Demanding that the total momentum of matter plus geometry is conserved for all observers, leads to the gravitational field equations. Thus, how matter curves spacetime is entirely determined by this principle of momentum conservation. (b) The $\mathcal{G}^a[v]$ can be related to the gravitational Lagrangian in a manner similar to the usual definition of Hamiltonian in, say, classical mechanics. (c) Geodesic observers in a spacetime will find that the conserved total momentum vanishes on-shell. (d) The on-shell, conserved, total energy in a region of space, as measured by the comoving observers, will be equal to the total heat energy of the boundary surface. (e) The off-shell gravitational energy in a region will be the sum of the ADM energy in the bulk plus the thermal energy of the boundary. These results suggest that $\mathcal{G}^a[v]$ can be a useful physical quantity to probe the gravitational theories.