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

TL;DR

The work analyzes how Fefferman–Graham, Bondi, and the fluid/gravity derivative expansion describe locally AdS$_3$ spacetimes and their flat limits, deriving explicit diffeomorphisms to relate their solution spaces and residual gauge diffeomorphisms. It shows FG and Bondi gauges are equivalent descriptions, while the derivative expansion carries an extra boundary-fluid velocity degree of freedom, corresponding to a hydrodynamic frame; Bondi emerges as a sub-gauge of the derivative expansion in a particular frame. In the Ricci-flat limit, Bondi remains compatible with a Carrollian boundary fluid, and the flat derivative expansion yields Carrollian hydrodynamics with a well-defined ultra-relativistic limit that preserves holographic structure. The results illuminate how holography in AdS$_3$ and flat space are connected through gauge choices, boundary data, and hydrodynamic frames, with implications for asymptotic symmetries and charges in both settings.

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.