Asymptotic Covariant Conservation Laws for Gravitational Radiation

Abstract

The usual conserved quantities of Lorentz-covariant theories are associated with the descriptors of coordinate transformations in Killing directions. By suitably defining the concept of asymptotic-Killing vector fields, it is possible to extend the definition of the usual conserved quantities to situations in which, properly speaking there are no Killing vector fields, but in which, nevertheless, it is still meaningful to speak of the energy, momentum and perhaps angular momentum radiated by the gravitational field. Among the results obtained in the course of the investigation are: (a) an understanding of the circumstance which singles out the energy as a positive-definite quantity; (b) a possible global consequence that vanishing energy implies that the space is flat; (c) an understanding of the position of the Fock harmonic coordinate conditions in Trautman's treatment of gravitational radiation; (d) an extension of the validity of the Møller pseudotensor to the Trautman radiative solutions. Also a brief indication is given of how this work might be extended to give a proper treatment of energy, momentum, and angular momentum densities for general metrics.