Quasi-Local Gravitational Energy

TL;DR

This paper introduces a quasi-local gravitational energy definition based on the Hamiltonian in a 2+2 formulation, which is consistent with known mass measures and applicable to various spacetimes.

Contribution

It provides a new, well-defined quasi-local energy concept in general relativity that aligns with established mass measures and applies to diverse spacetime geometries.

Findings

Energy reduces to Hawking mass in shear-free, twist-free cases.

Energy approaches Bondi and ADM masses at null and spatial infinity.

Explicit calculations for Schwarzschild, Reissner-Nordström, and cosmological solutions.

Abstract

A dynamically preferred quasi-local definition of gravitational energy is given in terms of the Hamiltonian of a `2+2' formulation of general relativity. The energy is well-defined for any compact orientable spatial 2-surface, and depends on the fundamental forms only. The energy is zero for any surface in flat spacetime, and reduces to the Hawking mass in the absence of shear and twist. For asymptotically flat spacetimes, the energy tends to the Bondi mass at null infinity and the \ADM mass at spatial infinity, taking the limit along a foliation parametrised by area radius. The energy is calculated for the Schwarzschild, Reissner-Nordstr\"om and Robertson-Walker solutions, and for plane waves and colliding plane waves. Energy inequalities are discussed, and for static black holes the irreducible mass is obtained on the horizon. Criteria for an adequate definition of quasi-local energy are discussed.