Momentum in General Relativity: Local versus Quasilocal Conservation Laws

TL;DR

This paper develops a general relativistic conservation law for momentum using quasilocal frames, enabling precise definitions of gravitational energy and momentum fluxes without relying on spacetime symmetries.

Contribution

It introduces a novel quasilocal RQF approach to derive comprehensive momentum conservation laws in general relativity, overcoming limitations of traditional local methods.

Findings

Provides exact operational definitions of gravitational energy and momentum fluxes.

Derives a relativistic version of Archimedes' law for black holes.

Reveals detailed mechanisms of gravitational energy transfer in various phenomena.

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. Our approach uses the Brown and York quasilocal stress-energy-momentum tensor for matter and gravitational fields, plus the concept of a rigid quasilocal frame (RQF) introduced in previous work. 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-Nordstrom black hole.