Compact binary coalescences: The subtle issue of angular momentum

Abstract

In presence of gravitational radiation, the notion of angular momentum of an isolated system acquires an infinite dimensional supertranslation ambiguity. This fact has been emphasized in the mathematical general relativity literature over several decades. We analyze the issue in the restricted context of compact binary coalescence (CBC) where the initial total angular momentum of the binary and the final black hole spin generically refer to \emph{distinct} rotation subgroups of the Bondi-Metzner-Sachs group, related by \emph{supertranslations}. We show that this ambiguity can be quantified using gravitational memory and the `black hole kick'. Our results imply that, although the ambiguity is conceptually important, under assumptions normally made in the CBC literature, it can be ignored in practice for the current and foreseeable gravitational wave detectors.

Figures (1)

• Figure 1: A depiction of a compact binary space-time, together with an artist's impression of the wave form based on bohe2017improved. $\mathfrak{I}^{+}$ constitutes the future boundary of this space-time and has topology $\mathbb{S}^{2}\times \mathbb{R}$. The Bondi news --the time derivative of the waveform-- goes to zero in the distant past and distant future. Because of the black hole kick, the Bondi (conformal) frames $(\mathring{q}_{ab}, \mathring{n}^{a})$ and $(\mathring{q}_{ab}^{\,\prime}, \mathring{n}^{\prime\, a})$ adapted to rest-frames in the distant past and distant future are distinct. Cross-sections $u=u_{1}$ and $u=u_{2}$ belong to the center of mass family adapted to the distant past; the shear of this family of cross-sections vanishes as $u\to -\infty$. They define the past Poincaré group ${{\mathfrak{p}}}{(i^{\circ})}$ discussed in Section \ref{['s3.1']}. However, generically their shear fails to vanish in the distant future. There is a distinct family of cross-sections --such as $u^{\prime} = u_{2}^{\prime}$-- that are adapted to the rest frame in the future and become shear-free in the limit $u\to \infty$. This family defines the future Poincaré group ${\mathfrak{p}}{(i^{+})}$. Generically the two Poincaré groups are distinct subgroups of the BMS group $\mathfrak{B}$, related by a BMS supertranslation. The total angular momentum $\vec{J}_{i^{\circ}}$ in the distant past refers to ${{\mathfrak{p}}}{(i^{\circ})}$, while the final spin $\sf$ refers to ${\mathfrak{p}}{(i^{+})}$. Therefore, the difference $\vec{J}_{i^{\circ}} - \vec{S}_{i^{+}}$ involves flux of angular momentum as well as supermomentum.