Gravitational energy

The total energy of asymptotically flat, nonsingular gravitational fields is discussed in terms of the initial data on a spacelike hypersurface. The total energy is a surface integral which we relate to a volume integral over "sources," including the contributions of gravitational waves. This relationship follows from a recent formulation of the initial-value equations of general relativity and is free of coordinate conditions. We show that time-symmetric initial-data sets form minima of energy among all initial-data sets on maximal hypersurfaces. Combining this result with a result of Brill, it follows that every nonsingular, axisymmetric, asymptotically flat spacetime admitting at least one maximal slice has non-negative total energy. Negative "interaction energy" contributions are described and a discussion of nonmaximal initial data is given.