Angular momentum and memory effect

TL;DR

Addresses the problem of supertranslation ambiguity in defining angular momentum for radiating gravitational systems. Proposes that gravitational memory is the essential obstruction and introduces a memory-subtraction procedure to obtain a supertranslation-invariant angular momentum at null infinity. The construction uses the Newman-Penrose formalism in the Newman-Unti gauge and a good-cut condition, solving $\eth^2 f = -\sigma_0$ to define a final-stage supertranslation that removes memory, resulting in the invariant current $J_G = -\frac{1}{P_s}[ Y^{\bar{z}} \Psi_{1}^{0\,G} + Y^{z} \Psi_{1}^{0\,G} ]$, with $\Psi_{1}^{0\,G} = \Psi_1^0 - 3 \eth f\, \Psi_2^0$. This recovers known invariant definitions in stationary cases and provides a practical route to compute radiated angular momentum in gravitational-wave contexts, with potential implications for post-Minkowskian analyses and black hole soft hair.

Abstract

It is a long-standing problem in general relativity that the notion of angular momentum of an isolated system has supertranslation ambiguity. In this paper, we argue that the ambiguity is essentially because of the gravitational wave memory. When properly subtracting the memory effect of the observer, one can introduce a supertranslation invariant definition of the angular momentum at null infinity.