Gravitational energy is well defined

TL;DR

This paper demonstrates that using a 4D Minkowski reference frame, many quasi-local energy expressions in general relativity converge to a common, non-negative energy value, clarifying the longstanding issue of gravitational energy definition.

Contribution

It shows that a large class of quasi-local energy expressions agree when using a Minkowski reference, linking them to the Wang-Yau mass and resolving ambiguities.

Findings

Many energy expressions agree with the Einstein pseudotensor to linear order.

The Wang-Yau mass emerges as the common quasi-local energy value.

A Minkowski reference frame resolves key ambiguities in gravitational energy definitions.

Abstract

The energy of gravitating systems has been an issue since Einstein proposed general relativity: considered to be ill defined, having no proper local density. Energy-momentum is now regarded as \emph{quasi-local} (associated with a closed 2-surface). We consider the pseudotensor and quasi-local proposals in the Lagrangian-Noether-Hamiltonian formulations. There are two ambiguities: (i) many expressions, (ii) each depends on some non-dynamical structure, e.g., a reference frame. The Hamiltonian approach gives a handle on both problems. Our remarkable discovery is that with a 4D isometric Minkowski reference a large class of expressions---those that agree with the Einstein pseudotensor's Freud superpotential to linear order---give a common quasi-local energy value. With a best-matched reference on the boundary this value is the non-negative Wang-Yau mass.