Definitions of (super) angular momentum in asymptotically flat spacetimes: Properties and applications to compact-binary mergers

TL;DR

This paper investigates various definitions of angular momentum and super angular momentum in asymptotically flat spacetimes, analyzing their properties, differences, and implications for binary black hole mergers, with a focus on the effects of gravitational waves.

Contribution

It introduces a two-parameter family of angular momentum and super angular momentum definitions, clarifies their properties, and assesses their impact on binary merger observations.

Findings

Different angular momentum definitions differ at high post-Newtonian order during radiation.

Super angular momentum definitions differ at lower orders and show residual changes after gravitational waves.

The effects of these differences are small but potentially detectable with numerical relativity waveforms.

Abstract

The symmetries of asymptotically flat spacetimes in general relativity are given by the Bondi-Metzner-Sachs (BMS) group, though there are proposed generalizations of its symmetry algebra. Associated with each symmetry is a charge and a flux, and the values of these charges and their changes can characterize a spacetime. The charges of the BMS group are relativistic angular momentum and supermomentum (which includes 4-momentum); the extensions of the BMS algebra also include generalizations of angular momentum called "super angular momentum." Several different formalisms have been used to define angular momentum, and they produce nonequivalent expressions for the charge. It was shown recently that these definitions can be summarized in a two-parameter family of angular momenta, which we investigate in this paper. We find that requiring that the angular momentum vanishes in flat spacetime restricts the two parameters to be equal. If we do not require that the angular momentum agrees with a common Hamiltonian definition, then we are left with a one-parameter family of angular momenta that includes the definitions from the several different formalisms. We then also propose a similar two-parameter family of super angular momentum. We examine the effect of the free parameters on the values of the angular momentum and super angular momentum from nonprecessing binary-black-hole mergers. The definitions of angular momentum differ at a high post-Newtonian order for these systems, but only when the system is radiating gravitational waves. The different super-angular-momentum definitions occur at lower orders, and there is a difference in the change of super angular momentum even after the gravitational waves pass, which arises because of the gravitational-wave memory effect. We estimate the size of these effects using numerical-relativity surrogate waveforms and find they are small but resolvable.