The partial Bondi gauge: Further enlarging the asymptotic structure of gravity

TL;DR

The paper expands the asymptotic analysis of gravity by employing a partial Bondi gauge, relaxing the determinant condition to admit a polyhomogeneous angular metric and time-dependent boundary data. Using both tensorial Einstein equations and the Newman–Penrose formalism, it derives a broadened solution space that accommodates a cosmological constant, logarithmic terms (when Λ=0), and an extra trace mode, while establishing a new AdS mass-loss-like structure through covariant functionals. It then presents a unified treatment of Bondi–Sachs and Newman–Unti gauges, showing how each gauge emerges from the partial gauge and highlighting a potential extra radial translation symmetry in NU (with a corresponding algebra extension). The work also performs holographic renormalization of the symplectic potential and develops the NP-based compact evolution equations, culminating in a Λ-dependent mass-loss formula and detailed transformation laws for asymptotic data. These results lay groundwork for future exploration of radial translation charges, higher-spin extensions, and sourced holography in (A)dS, as well as the role of logarithmic terms in memory and soft theorems.

Abstract

We present a detailed analysis of gravity in a partial Bondi gauge, where only the three conditions $g_{rr}=0=g_{rA}$ are fixed. We relax in particular the so-called determinant condition on the transverse metric, which is only assumed to admit a polyhomogeneous radial expansion. This is sufficient in order to build the solution space, which here includes a cosmological constant, time-dependent sources in the boundary metric, logarithmic branches, and an extra trace mode at subleading order in the transverse metric. The evolution equations are studied using the Newman-Penrose formalism in terms of covariant functionals identified from the Weyl scalars, and we build the explicit dictionary between this formalism and the tensorial Einstein equations. This provides in particular a new derivation of the (A)dS mass loss formula. We then study the holographic renormalisation of the symplectic potential, and the transformation laws under residual asymptotic symmetries. The advantage of the partial Bondi gauge is that it allows to contrast and treat in a unified manner the Bondi-Sachs and Newman-Unti gauges, which can each be reached upon imposing a further specific gauge condition. The differential determinant condition leads to the $Λ$-BMSW gauge, while a differential condition on $g_{ur}$ leads to a generalized Newman-Unti gauge. This latter gives access to a new asymptotic symmetry which acts on the asymptotic shear and further extends the $Λ$-BMSW group by an extra abelian radial translation. This generalizes results which we have recently obtained in three dimensions.