Angular momentum balance in gravitational two-body scattering: Flux, memory, and supertranslation invariance

Two puzzles continue to plague our understanding of angular momentum balance in the context of gravitational two-body scattering. First, because the standard definition of the Bondi angular momentum $J$ is subject to a supertranslation ambiguity, it has been shown that when the corresponding flux $F_J$ is expanded in powers of Newton's constant $G$, it can start at either $O(G^2)$ or $O(G^3)$ depending on the choice of frame. This naturally raises the question as to whether the $O(G^2)$ part of the flux is physically meaningful. The second puzzle concerns a set of new methods for computing the flux that were recently developed using quantum field theory. Somewhat surprisingly, it was found that they generally do not agree with the standard formula for $F_J$, except in the binary's center-of-mass frame. In this paper, we show that the resolution to both of these puzzles lies in the careful interpretation of $J$: Generically, the Bondi angular momentum $J$ is \emph{not} equal to the mechanical angular momentum $\mathcal{J}$ of the binary, which is the actual quantity of interest. Rather, it is the sum of $\mathcal{J}$ and an extra piece involving the shear of the gravitational field. By separating these contributions, we obtain a new balance law, accurate to all orders in $G$, that equates the total loss in mechanical angular momentum $Δ_\mathcal{J}$ to the sum of a radiative term, which always starts at $O(G^3)$, and a static term, which always starts at $O(G^2)$. We show that each of these terms is invariant under supertranslations, and we find that $Δ_\mathcal{J}$ matches the result from quantum field theory at least up to $O(G^2)$ in all Bondi frames. The connection between our results and other proposals for supertranslation-invariant definitions of the angular momentum is also discussed.