The holographic fluid dual to vacuum Einstein gravity

TL;DR

This work constructs a direct holographic map between (d+2)-dimensional Ricci-flat spacetimes and (d+1)-dimensional incompressible Navier–Stokes fluids with higher-derivative corrections, extending the gravity/fluid correspondence to arbitrary order in a non-relativistic hydrodynamic expansion. Starting from equilibrium Rindler-like configurations with vanishing energy density, the authors develop a systematic all-orders algorithm to generate regular bulk solutions, and then extract the dual fluid's transport coefficients by matching to a relativistic hydrodynamic framework with vanishing equilibrium energy density. They provide explicit results up to fifth order, including the bulk metric components, the Brown–York stress tensor, and the relativistic hydrodynamic interpretation that yields precise coefficients and a quadratic stress-tensor constraint. A simple square-root scalar field model is offered to realize the non-dissipative part of the holographic fluid, highlighting a possible flat-space holographic description and opening avenues for extending holography beyond AdS. The work raises broad questions about the global extension, convergence, and potential string embeddings of flat-space holography, as well as the physical interpretation of the dual theory.

Abstract

We present an algorithm for systematically reconstructing a solution of the (d+2)-dimensional vacuum Einstein equations from a (d+1)-dimensional fluid, extending the non-relativistic hydrodynamic expansion of Bredberg et al in arXiv:1101.2451 to arbitrary order. The fluid satisfies equations of motion which are the incompressible Navier-Stokes equations, corrected by specific higher derivative terms. The uniqueness and regularity of this solution is established to all orders and explicit results are given for the bulk metric and the stress tensor of the dual fluid through fifth order in the hydrodynamic expansion. We establish the validity of a relativistic hydrodynamic description for the dual fluid, which has the unusual property of having a vanishing equilibrium energy density. The gravitational results are used to identify transport coefficients of the dual fluid, which also obeys an interesting and exact constraint on its stress tensor. We propose novel Lagrangian models which realise key properties of the holographic fluid.