Fefferman--Graham and Bondi Gauges in the Fluid/Gravity Correspondence

Abstract

In three-dimensional gravity, we discuss the relation between the Fefferman-Graham gauge, the Bondi gauge and the Eddington-Finkelstein type of gauge, often referred to as the derivative expansion, involved in the fluid/gravity correspondence. Starting with a negative cosmological constant, for each gauge, we derive the solution space and the residual gauge diffeomorphisms. We construct explicitly the diffeomorphisms that relate the various gauges, and establish the precise matching of their boundary data. We show that Bondi and Fefferman-Graham gauges are equivalent, while the fluid/gravity derivative expansion, originating from a partial gauge fixing, exhibits an extra unspecified function that encodes the boundary fluid velocity. The Bondi gauge turns out to describe a subspace of the derivative expansion's solution space, featuring a fluid in a specific hydrodynamic frame. We pursue our analysis with the Ricci-flat limit of the Bondi gauge and of the fluid/gravity derivative expansion. The relations between them persist in this limit, which is well-defined and non-trivial. Moreover, the flat limit of the derivative expansion maps to the ultra-relativistic limit on the boundary. This procedure allows to unravel the holographic properties of the Bondi gauge for vanishing cosmological constant, in terms of its boundary Carrollian dual fluid.

Gauges in three-dimensional gravity and holographic fluids

“…An appendix supplements our exposition with some detailed expressions for the Bondi gauge (appendix A), and a note (appendix B) on the algebra of residual diffeomorphisms, as it emerges from our study in section 2.2, nicely fitting the results available in the current literature. An alternative and complementary presentation, including a useful Mathematica notebook and summarizing the current analysis is also available in the conference contribution [35].…”

“…Our conventions are: * uρ = u σ ησρ with ησρ = | det g| σρ and 01 = +1 (there is a minus sign with respect to[35]). Hence η µσ ησν = δ µ ν .…”

“…Our convention for the variation δ ξ is the opposite of that used in[35] 16. We also quote:δ ξ Θ = ψΘ + ω * Θ * + * u(ω * ) − u(ω) = SΘ − u(S) − f u(Θ) − Y * u(Θ) k + k (ZΘ * + * u(Z)) and δ ξ Θ * = ψ * Θ + ωΘ * − * u (ψ)) + u (ψ * ) = SΘ * − * u(S) − f u(Θ * ) − Y * u(Θ * ) k + k (ZΘ + u(Z)).…”

Conservation and integrability in lower-dimensional gravity

“…Indeed, it is particularly well-adapted to investigate the interplay between radiation and symmetries [17,19,28,[34][35][36][37][38][39]. Furthermore, it allows us to consider simultaneously asymptotically locally flat spacetimes exhibiting null boundaries and asymptotically locally AdS spacetimes with timelike boundaries [26,[40][41][42][43][44][45]. The analyses of these two types of asymptotics are related through a flat limit process [40,46,47].…”

“…To consider asymptotically locally AdS and flat spacetimes in Bondi gauge, one has to allow the boundary structure to fluctuate [19,26,36,[41][42][43]48]. These mild falloffs led to the proposal of an infinite-dimensional enhancement of the BMS group with smooth superrotations in the flat case [19,36,48] and the discovery of BMS-like symmetries in presence of non-vanishing cosmological constant [26,47].…”