Momentum in general relativity: local versus quasilocal conservation laws

Abstract

We construct a general relativistic conservation law for linear and angular momentum for matter and gravitational fields in a finite volume of space that does not rely on any spacetime symmetries. This work builds on our previous construction of a general relativistic energy conservation law with the same features (McGrath et al 2012 Class. Quantum Grav. 29 215012). Our approach uses the Brown and York (1993 Phys. Rev. D 47 1407–19) quasilocal stress–energy–momentum tensor for matter and gravitational fields, plus the concept of a rigid quasilocal frame (RQF) introduced in (Epp et al 2009 Class. Quantum Grav. 26 035015; 2012 Classical and Quantum Gravity: Theory, Analysis, and Applications (Nova Science)). The RQF approach allows us to construct, in a generic spacetime, frames of reference whose boundaries are rigid (their shape and size do not change with time), and that have precisely the same six arbitrary time-dependent degrees of freedom as the accelerating and tumbling rigid frames we are familiar with in Newtonian mechanics. These RQFs, in turn, give rise to a completely general conservation law for the six components of momentum (three linear and three angular) of a finite system of matter and gravitational fields. We compare in detail this quasilocal RQF approach to constructing conservation laws with the usual local one based on spacetime symmetries, and discuss the shortcomings of the latter. These RQF conservation laws lead to a deeper understanding of physics in the form of simple, exact, operational definitions of gravitational energy and momentum fluxes, which in turn reveal, for the first time, the exact, detailed mechanisms of gravitational energy and momentum transfer taking place in a wide variety of physical phenomena, including a simple falling apple. As a concrete example, we derive a general relativistic version of Archimedes’ law that we apply to understand electrostatic weight and buoyant force in the context of a Reissner–Nordström black hole.