Coordinate Invariance and Energy Expressions in General Relativity
Abstract
The invariance of various definitions proposed for the energy and momentum of the gravitational field is examined. We use the boundary conditions that g_{μν} approaches the Lorentz metric as 1r, but allow g_{μν,α} to vanish as slowly as 1r. This permits "coordinate waves." It is found that none of the expressions giving the energy as a twodimensional 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_{μν,α} vanishes faster than 1r). If one introduces the prescription of spacetime averaging of the integrals, one finds that the definitions of LandauLifshitz and PapapetrouGupta yield (equal) coordinateinvariant results. However, the definitions of Einstein, Møller, and Dirac become unambiguous, but not invariant. The averaged LandauLifshitz and PapapetrouGupta expressions are then shown to give the correct physical energymomentum 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 energymomentum may be invariantly calculated from the asymptotic form of the metric field at a fixed time.
 Publication:

Physical Review
 Pub Date:
 May 1961
 DOI:
 10.1103/PhysRev.122.997
 Bibcode:
 1961PhRv..122..997A