Supertranslation invariance of angular momentum

TL;DR

The paper tackles the problem of defining angular momentum carried by gravitational radiation in general relativity in a way that is free from supertranslation ambiguities. It introduces the first supertranslation-invariant angular momentum at null infinity by taking the $u\to\infty$ limit of a refined quasilocal angular momentum, augmented by the correction term from the Wang-Yau optimal isometric embedding, and defines a corresponding center-of-mass quantity. It proves that the total fluxes of the resulting conserved quantities $(E, P^k, J^k, C^k)$ are invariant under supertranslations and that the full set forms a Poincaré-like charge algebra at null infinity, reducing to the known Kerr and Minkowski limits. This provides a rigorous, physically meaningful framework for analyzing gravitational radiation in isolated systems, with potential implications for interpreting gravitational-wave signals. The construction hinges on a Minkowski reference provided by the Wang-Yau embedding and yields a complete, translation-consistent set of charges with vanishing flux in flat spacetime.

Abstract

LIGO's successful detection of gravitational waves has revitalized the theoretical understanding of the angular momentum carried away by gravitational radiation. An infinite dimensional supertranslation ambiguity has presented an essential difficulty for decades of study. Recent advances were made to address and quantify the supertranslation ambiguity in the context of compact binary coalescence. Here we present the first definition of angular momentum in general relativity that is completely free from supertranslation ambiguity. The new definition was derived from the limit of the quasilocal angular momentum defined previously by the authors. A new definition of center of mass at null infinity is also proposed and shown to be supertranslation invariant. Together with the classical Bondi-Sachs energy-momentum, they form a complete set of conserved quantities at null infinity that transform according to basic physical laws.

Figures (1)

• Figure 1: A supertranslation that maps $\bar{u}$ cuts to $u=\bar{u}+f(x)$ cuts. $Y$ and $\bar{Y}$ are rotation BMS fields such that $Y$ is tangent to $u$ cuts and $\bar{Y}$ is tangent to $\bar{u}$ cuts.