Gravitational energy

TL;DR

This paper develops a method to calculate gravitational energy in static and stationary spacetimes, extending previous work and revealing that gravitational energy is generally negative under certain conditions.

Contribution

It provides a general approach to compute gravitational energy for any static or stationary spacetime with isolated sources, generalizing the results of MTW.

Findings

Gravitational energy can be calculated for any static or stationary spacetime.

Electromagnetic and gravitational energy densities have opposite signs in conformastatic spacetimes.

Gravitational energy tends to be negative in symmetric spacetimes or when energy conditions are met.

Abstract

Observers at rest in a stationary spacetime flat at infinity can measure small amounts of rest-mass+internal energies+kinetic energies+pressure energy in a small volume of fluid attached to a local inertial frame. The sum of these small amounts is the total "matter energy" for those observers. The total mass-energy minus the matter energy is the binding gravitational energy. Misner, Thorne and Wheeler evaluated the gravitational energy of a spherically symmetric static spacetime. Here we show how to calculate gravitational energy in any static and stationary spacetime for isolated sources with a set of observers at rest. The result of MTW is recovered and we find that electromagnetic and gravitational 3-covariant energy densities in conformastatic spacetimes are of opposite signs. Various examples suggest that gravitational energy is negative in spacetimes with special symmetries or when the energy-momentum tensor satisfies usual energy conditions.