Coordinate Invariance and Energy Expressions in General Relativity

The invariance of various definitions proposed for the energy and momentum of the gravitational field is examined. We use the boundary conditions that $g_{\mu \nu }$ approaches the Lorentz metric as $\frac{1}{r}$, but allow $g_{\mu \nu ,\alpha }$ to vanish as slowly as $\frac{1}{r}$. This permits "coordinate waves." It is found that none of the expressions giving the energy as a two-dimensional surface integral are invariant within this class of frames. In a frame containing coordinate waves they are ambiguous, since their value depends on the location of the surface at infinity (unlike the case where $g_{\mu \nu ,\alpha }$ vanishes faster than $\frac{1}{r}$). If one introduces the prescription of space-time averaging of the integrals, one finds that the definitions of Landau-Lifshitz and Papapetrou-Gupta yield (equal) coordinate-invariant results. However, the definitions of Einstein, Møller, and Dirac become unambiguous, but not invariant.

The averaged Landau-Lifshitz and Papapetrou-Gupta expressions are then shown to give the correct physical energy-momentum as determined by the canonical formulations Hamiltonian involving only the two degrees of freedom of the field. It is shown that this latter definition yields that inertial energy for a gravitational system which would be measured by a nongravitational apparatus interacting with it. The canonical formalism's definition also agrees with measurements of gravitational mass by Newtonian means at spacial infinity. It is further shown that the energy-momentum may be invariantly calculated from the asymptotic form of the metric field at a fixed time.