Gravitational energy as Noether charge

TL;DR

This paper proposes a new definition of gravitational energy based on Noether charge, providing a quasi-local energy concept applicable to various theories of gravity, and relates it to known energies in specific cases.

Contribution

It introduces a general, physically motivated gravitational energy definition using Noether charge, applicable to any diffeomorphism-invariant theory, and connects it to established energies in Einstein gravity.

Findings

Energy defined as Noether charge matches Bondi, Komar, and Misner-Sharp energies in specific cases.

The approach offers a quasi-local energy concept dependent on a chosen time flow vector.

Highlights the absence of a preferred energy in general relativity, reflecting quantum energy-time duality.

Abstract

A definition of gravitational energy is proposed for any theory described by a diffeomorphism-invariant Lagrangian. The mathematical structure is a Noether- current construction of Wald involving the boundary term in the action, but here it is argued that the physical interpretation of current conservation is conservation of energy. This leads to a quasi-local energy defined for compact spatial surfaces. The energy also depends on a vector generating a flow of time. Angular momentum may be similarly defined, depending on a choice of axial vector. For Einstein gravity: for the usual vector generating asymptotic time translations, the energy is the Bondi energy; for a stationary Killing vector, the energy is the Komar energy; in spherical symmetry, for the Kodama vector, the energy is the Misner-Sharp energy. In general, the lack of a preferred time indicates the lack of a preferred energy, reminiscent of the energy-time duality of quantum theory.