Observer dependence of angular momentum in general relativity and its relationship to the gravitational-wave memory effect

TL;DR

This paper introduces a method for observers to measure angular momentum in curved spacetime, examines how these measurements depend on the observer's frame, and links this dependence to gravitational-wave memory effects.

Contribution

It provides a new procedure for measuring angular momentum locally in curved spacetime and relates observer dependence of angular momentum to gravitational-wave memory effects.

Findings

Observers' angular momentum measurements are observer dependent due to spacetime curvature.

Gravitational-wave bursts with memory induce nontrivial generalized holonomy.

The method recovers conventional results in stationary spacetimes near null infinity.

Abstract

We define a procedure by which observers can measure a type of special-relativistic linear and angular momentum $(P^a, J^{ab})$ at a point in a curved spacetime using only the spacetime geometry in a neighborhood of that point. The method is chosen to yield the conventional results in stationary spacetimes near future null infinity. We also explore the extent to which spatially separated observers can compare the values of angular momentum that they measure and find consistent results. We define a generalization of parallel transport along curves which gives a prescription for transporting values of angular momentum along curves that yields the correct result in special relativity. If observers use this prescription, then they will find that the angular momenta they measure are observer dependent, because of the effects of spacetime curvature. The observer dependence can be quantified by a kind of generalized holonomy. We show that bursts of gravitational waves with memory generically give rise to a nontrivial generalized holonomy: there is, in this context, a close relation between the observer dependence of angular momentum and the gravitational-wave memory effect.