(A)dS ₄ in Bondi gauge

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.

The Λ-BMS4 charge algebra

“…It is therefore necessary to translate all results in a language appropriate to take the flat limit. Bondi gauge (see appendix B.2) is a convenient gauge for that purpose since we have already showed that the solution space and the asymptotic symmetries can be mapped in a bijective correspondence between Bondi and Starobinsky/Fefferman-Graham gauges [49,100]. The asymptotic boundary I becomes the future null boundary I + in the flat limit.…”

“…A diffeomorphism between Starobinsky/Fefferman-Graham gauge [81][82][83][84][85] and Bondi gauge [13,14] has been explicitly constructed [49,100]. This allows to map each dynamical quantity between the two gauges.…”

Asymptotics with a cosmological constant: The solution space

“…The gauge choice in (6) is the so-called Newman-Unti gauge [40]. To compare to the results in Bondi gauge in [13,37], one just needs to apply a transformation in the radial direction [47]. A conformally flat boundary 2 metric explored in [9] corresponds to Ψ 0 4 = Ψ 0 3 =Im(Ψ 0 2 ) = 0 in the NP formalism.…”

“…The solution space of asymptotically dS space-times was recently derived in [9] using geometric tools à la Penrose [5,38] with special emphasis on the consequence of a conformally flat boundary 2 metric. In the metric formalism, the axi-symmetric solution space in Bondi gauge was obtained with a cosmological constant in [13,37]. The solution space of the NP formalism with a cosmological constant was derived in [17,29] with a special choice of the foliation of space-times geometry which is a family of null hyper-surfaces given by constant values of the u coordinate.…”

Holography for the very early universe and the classic puzzles of hot big bang cosmology

“…Note that the asymptotic structure near the timelike boundary of AdS [34,35] is mapped to the asymptotic structure near the spacelike boundary of de Sitter [36] via analytic continuation [37], see also [38] and [39]. The same analytic continuation also maps general perturbations (at least to quadratic order) around Domain-Walls/FRW cosmologies [33,[40][41][42], and this translates into specific analytic continuation on the QFT side, as discussed in [33,[40][41][42].…”