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’, EM, for those observers. If Mc2 is the total mass energy, the difference Mc2 − EM is the binding gravitational energy. Misner, Thorne and Wheeler (MTW) evaluated the gravitational energy of a spherically symmetric static spacetime. Here we show how to calculate gravitational energy in any static and stationary spacetimes with 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.
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