Abstract
Solutions to Einstein’s vacuum equations in three dimensions are locally maximally symmetric. They are distinguished by their global properties and their investigation often requires a choice of gauge. Although analyses of this sort have been performed abundantly, several relevant questions remain. These questions include the interplay between the standard Bondi gauge and the Eddington-Finkelstein type of gauge used in the fluid/gravity holographic reconstruction of these spacetimes, as well as the Fefferman-Graham gauge, when available i.e. in anti de Sitter. The goal of the present work is to set up a thorough dictionary for the available descriptions with emphasis on the relativistic or Carrollian holographic fluids, which portray the bulk from the boundary in anti-de Sitter or flat instances. A complete presentation of residual diffeomorphisms with a preliminary study of their algebra accompanies the situations addressed here.
Highlights
Contrary to Fefferman-Graham, the Bondi gauge has not been significant in holography, it has common features with the derivative expansion of fluid/gravity correspondence: both are of the Eddington-Finkelstein type with one null radial coordinate and a retarded time [19,20,21,22,23,24,25,26,27]
The general solution space emerging from fluid/gravity correspondence is analyzed in section 2, grounded in two-dimensional hydrodynamics, which we recall for that purpose
What is the solution space of the fluid/gravity derivative expansion viewed as a gauge, and what are its residual diffeomorphisms? Where do the conventional Bondi and Fefferman-Graham gauges stand regarding fluid/gravity, and how is this web woven?
Summary
3.1 Gauge fixing, solution space and residual diffeomorphisms In Bondi gauge, the metric takes the form [1, 3, 5]. Irrespective of Λ, these were generated by four arbitrary functions f , Y , S and Z For these diffeomorphisms to respect the Bondi hydrodynamic frame, we must require that δξuφ = 0 or δξμφ = 0. The effect of these diffeomorphisms on the data defining the solutions (Γ, vφ, γ, ε, and χ versus ζ, or φ, U0, β0, M and N ) is inferred from the general expressions (2.43), (2.44), (2.45) and (2.46), or (2.64), (2.67) (2.68), by setting uφ = 0 or μφ = 0 and using Z as given in (3.17) or (3.18) We gather these formulas in appendix A and carry out the algebra of Bondi residual diffeomorphisms
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