The Wald-Zoupas prescription for asymptotic charges at null infinity in general relativity
Alexander M. Grant, Kartik Prabhu, Ibrahim Shehzad
TL;DR
This work derives covariant, conformally invariant Wald-Zoupas charges and fluxes for the BMS symmetries at null infinity in vacuum GR, avoiding reliance on Bondi-Sachs coordinates. It constructs a covariant framework using the universal structure at null infinity, the News tensor, and an auxiliary foliation to define charges on cross-sections that are independent of conformal frame, foliation, and extensions away from infinity. The resulting WZ charge and flux are shown to match known flux expressions (e.g., Ashtekar–Streubel) and yield consistent limits in Bondi-Sachs and conformal Gaussian null coordinates, while clarifying the relation to other formalisms (Komar/ linkage, twistor charges). The approach offers a robust, coordinate-free toolset for analyzing asymptotic symmetries and radiative data, with potential extensions to other gravity theories and horizons.
Abstract
We use the formalism developed by Wald and Zoupas to derive explicit covariant expressions for the charges and fluxes associated with the Bondi-Metzner-Sachs symmetries at null infinity in asymptotically flat spacetimes in vacuum general relativity. Our expressions hold in non-stationary regions of null infinity, are local and covariant, conformally-invariant, and are independent of the choice of foliation of null infinity and of the chosen extension of the symmetries away from null infinity. While similar expressions have appeared previously in the literature in Bondi-Sachs coordinates (to which we compare our own), such a choice of coordinates obscures these properties. Our covariant expressions can be used to obtain charge formulae in any choice of coordinates at null infinity. We also include detailed comparisons with other expressions for the charges and fluxes that have appeared in the literature: the Ashtekar-Streubel flux formula, the Komar formulae, and the linkage and twistor charge formulae. Such comparisons are easier to perform using our explicit expressions, instead of those which appear in the original work by Wald and Zoupas.