The partial Bondi gauge: Further enlarging the asymptotic structure of gravity
Marc Geiller, Céline Zwikel
Abstract
We present a detailed analysis of gravity in a partial Bondi gauge, where only the three conditions g_{rr}=0=g_{rA} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mi>r</mml:mi> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mi>r</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> 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 \Lambda <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Λ</mml:mi> </mml:math> -BMSW gauge, while a differential condition on g_{ur} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mi>u</mml:mi> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> </mml:math> 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 \Lambda <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Λ</mml:mi> </mml:math> -BMSW group by an extra abelian radial translation. This generalizes results which we have recently obtained in three dimensions.