The Robinson-Trautman spacetime and its holographic fluid

TL;DR

The paper develops a holographic reconstruction of four-dimensional AdS Robinson–Trautman spacetimes from boundary data using a resummed derivative expansion with a shearless velocity in the Eckart frame. The RT dynamics arise as a boundary heat equation controlled by the boundary curvature, while the dual fluid remains at rest with constant energy density and a conserved entropy current for zero chemical potential. The framework emphasizes the role of Weyl covariance, Cotton tensors, and Petrov type in linking boundary data to bulk algebraically special spacetimes. It also discusses the limitations and cautions of transforming to the Landau–Lifshitz frame and highlights a geometric, isentropic interpretation of the out-of-equilibrium evolution.

Abstract

We discuss the holographic reconstruction of four-dimensional asymptotically anti-de Sitter Robinson-Trautman spacetime from boundary data. We use for that a resummed version of the derivative expansion. The latter involves a vector field, which is interpreted as the dual-holographic-fluid velocity field and is naturally defined in the Eckart frame. In this frame the analysis of the non-perfect holographic energy-momentum tensor is considerably simplified. The Robinson-Trautman fluid is at rest and its time evolution is a heat-diffusion kind of phenomenon: the Robinson-Trautman equation plays the rôle of heat equation, and the heat current is identified with the gradient of the extrinsic curvature of the two-dimensional boundary spatial section hosting the conformal fluid, interpreted as an out-of-equilibrium kinematical temperature. The hydrodynamic-frame-independent entropy current is conserved for vanishing chemical potential, and the evolution of the fluid resembles a Moutier thermodynamic path. We finally comment on the general transformation rules for moving to the Landau-Lifshitz frame, and on possible drawbacks of this option.