Equality of ADM mass and generalized Komar energy in asymptotically-flat dynamical spacetimes

We find a relation between the ADM mass and a generalized Komar energy in asymptotically-flat spacetime. We do not need to assume the existence of either a Killing or even asymptotically-Killing vector field. Instead, our generalized Komar energy is constructed from the normal evolution vector (the lapse function times the future-directed unit normal to the spacelike hypersurfaces on which the ADM mass is measured). We find equality between the ADM mass and this generalized Komar energy even for dynamical asymptotically-flat spacetimes provided the 3-dimensional Einstein tensor drops off quickly enough at spatial infinity, in particular, whenever ${}^{(3)}G_{ij}=o(r^{-3})$. No additional assumptions are required for equality. As this generalized energy is fully covariant, it may provide a powerful tool for analyzing energy content in dynamical spacetimes containing compact objects.