(A)dS$\mathbf{_4}$ in Bondi gauge

TL;DR

This work derives the general asymptotic solutions of four-dimensional Einstein gravity in Bondi gauge for Λ=0, ±, including asymptotically locally AdS and dS spacetimes. It presents two integration schemes (boundary and hybrid) that either fix data at the conformal boundary or combine boundary and null-surface data to construct the full asymptotic solution, and it explicitly constructs the transformation to Fefferman-Graham gauge to extract holographic data. The authors show that for asymptotically AdS4 spacetimes the Bondi mass remains constant and establish a direct link between Bondi data and the holographic stress-energy tensor, including concrete examples such as global AdS4, AdS4 Schwarzschild, and AdS black branes. By providing a clear holographic dictionary within Bondi coordinates, this paper advances both the study of gravitational radiation in AdS/dS contexts and the potential holographic interpretation of asymptotically flat gravity through Bondi data.

Abstract

We obtain the general asymptotic solutions of Einstein gravity with or without cosmological constant in Bondi gauge. The Bondi gauge was originally introduced in the context of gravitational radiation in asymptotically flat gravity. In the original work, initial conditions were prescribed at a null hypersurface and the Einstein equations were shown to take a nested form, which may be used to explicitly integrate them asymptotically. We streamline the derivation of the general asymptotic solution in the asymptotically flat case, and derive the most general asymptotic solutions for the case of non-zero cosmological constant of either sign (asymptotically locally AdS and dS solutions). With non-zero cosmological constant, we present integration schemes which rely on either prescribing data on the conformal boundary or on a null hypersurface and part of the conformal boundary. We explicitly work out the transformation to Fefferman-Graham gauge and identity how to extract the holographic data directly in Bondi coordinates. We illustrate the discussion with a number of examples and show that for asymptotically AdS${}_4$ spacetimes the Bondi mass is constant.

Figures (11)

• Figure 1: Penrose diagram of null hypersurfaces, $\color{islamicgreen}{\mathcal{N}_{u_i}}$, foliating future null infinity, $\mathscr{I^+}$, of an asymptotically flat spacetime. As indicated by the solid red axis, the retarded time coordinate $\color{red}{u}$ ranges from $(- \infty, \infty)$ along $\mathscr{I^+}$ and thus the dashed green lines represent the $u=\text{constant}$ hypersurfaces. (The arrows show the direction of increasing radial coordinate $r$). The dotted blue curves represent timelike hypersurfaces of constant $\textcolor{blue}{r}$.
• Figure 2: Left panel. Penrose diagram of the asymptotic region of an asymptotically locally AdS spacetime, where the timelike boundary manifold $\partial \mathcal{M}$ is denoted $\mathscr{I}$. The dashed green curves represent null hypersurfaces $\color{islamicgreen}{\mathcal{N}_{u_i}} =\{ u= u_i \, | \, u_i = \text{constant} \}$ and the dotted blue curves timelike surfaces of constant $\color{blue}{r}.$Right panel. Penrose diagram of the asymptotic region of an asymptotically locally dS space-time, where we have chosen to foliate the future spacelike boundary $\partial \mathcal{M}=\mathscr{I}^+$. The dashed green curves represent null hypersurfaces $\color{islamicgreen}{\mathcal{N}_{u_i}} =\{ u= u_i \, | \, u_i = \text{constant} \}$ and the dotted blue curves spacelike surfaces of constant $\color{blue}{r}$. The difference in the properties of constant $r$ surfaces between AdS (timelike) and dS (spacelike) is due to the presence of a cosmological horizon in asymptotically locally $dS$ spacetimes.
• Figure 3: Causal diagram illustrating how one applies the BMS scheme when given suitable initial data on a null hypersurface $\mathcal{N}_{u_0}$
• Figure 4: Penrose diagram illustrating the integration scheme for asymptotically flat space time
• Figure 5: Penrose diagram illustrating the difference between $D^+(\mathcal{N}_{u_0})$ and $J^+(\mathcal{N}_{u_0})$ in asymptotically locally AdS spacetime.