Erratum: Observer dependence of angular momentum in general relativity and its relationship to the gravitational-wave memory effect [Phys. Rev. D92, 084057 (2015)]

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.

Gravitational wave memory in de Sitter spacetime

“…Gravitational wave memory, a permanent displacement of the gravitational wave detector after the wave has passed, has been known since the work of Zel'dovich and Polnarev [1], extended to the full nonlinear theory of general relativity by Christodoulou [2], and treated by several authors [3][4][5][6][7][8][9][10][11][12][13][14][15]. As is usual in the treatment of isolated systems, all these works consider asymptotically flat spacetimes.…”

Properties of an affine transport equation and its holonomy

“…Only after [5] was written did it come to the attention of the authors of [5] that the transport equations (1) and their solution (2) are related to other aspects of general relativity and differential geometry. The vector ∆ξ a also appears as the development of a curve on a manifold into the affine tangent space at the curve's starting point.…”

“…The aim of [5] was to define an operational method by which observers in asymptotically flat spacetimes could measure the linear and angular momentum of the spacetime geometry from the spacetime curvature and its derivatives in the viscinity of an observer's timelike worldline. The observers could then compare their measured values of linear and angular momentum by using a specific transport equation (which has the form of an affine map between the tangent spaces along a curve that connects two points along the two different worldlines).…”

“…The covariant method for transporting angular momentum and measuring gravitational-wave memory effects was based on a system of differential equations called "affine transport" in [5]. Given a vector ξ a along a curve x(λ), the affine transport of ξ a was defined bẏ…”

“…Similarly, for a spinning point particle, its 4-momentum and angular-momentum tensor are jointly transported through the coupled Mathisson-Papapetrou equations [2,3] (the dual of these equations, the Killing transport equations, also transport a vector and antisymmetric tensor in a related way-see, e.g., [4]). In [5], one of the authors and a collaborator introduced an affine transport equation for vectors that proved useful for measuring physical effects related to the gravitationalwave memory and for transporting a type of special-relativistic linear and angular momentum in general relativity. We review the definition of this transport equation and describe some of its properties in the next subsection.…”