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

TL;DR

The paper addresses the lack of a canonical angular momentum in general relativity by introducing a local, operational definition of (P^a, J^{ab}) at a point based solely on nearby geometry, and by developing a curve-dependent affine transport to compare angular momentum between observers. A central contribution is the generalized holonomy, which encodes observer dependence of angular momentum arising from spacetime curvature, and is closely tied to gravitational-wave memory. Through covariant analysis in linearized gravity, including plane waves and large-radius multipole waves, the authors show that memory effects generate nontrivial holonomies and therefore genuine observer-dependent ambiguities in angular momentum. The framework highlights a deep link between gravitational memory, BMS transformations, and the geometry of angular-momentum transport, with potential applications to post-Newtonian matching and the interpretation of quasi-local charges in dynamical spacetimes.

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.

Figures (2)

• Figure 1: Spacetime diagram of a burst of gravitational waves and the curve used to compute the generalized holonomy. The gray region represents the spacetime location of the gravitational waves, while the unshaded regions are Minkowski spacetimes before and after the burst. The curve bounded by $\mathcal{P}$ and $\mathcal{R}$ is the worldline of observer $A$, and that bordered by $\mathcal{Q}$ and $\mathcal{S}$ is that of $B$. The curves with endpoints $(\mathcal{P},\mathcal{Q})$ and $(\mathcal{R}, \mathcal{S})$ are spacelike geodesics before and after the burst, respectively, which are just straight lines in the flat spacetime regions.
• Figure 2: Curve used to compute the generalized holonomy in the Schwarzschild spacetime of mass $M$ at large radii. The lengths of the four segments that compose the curve, $\delta r_1$, $\delta r_2$, $\delta x_1$, and $\delta x_2$ are all of order $r$, where $r\gg M$ is the closest distance to the source along the segment labeled by $\delta x_1$. This curve has a nontrivial generalized holonomy as $r$ goes to spatial infinity, even though the Schwarzschild spacetime has a well-defined angular momentum.