The relativistic Rindler hydrodynamics

We consider a (d + 2)-dimensional class of Lorentzian geometries holographically dual to a relativistic fluid flow in (d + 1) dimensions. The fluid is defined on a (d + 1)-dimensional time-like surface which is embedded in the (d + 2)-dimensional bulk space-time and equipped with a flat intrinsic metric. We find two types of geometries that are solutions to the vacuum Einstein equations: the Rindler metric and the Taub plane symmetric vacuum. These correspond to dual perfect fluids with vanishing and negative energy densities respectively. While the Rindler geometry is characterized by a causal horizon, the Taub geometry has a timelike naked singularity, indicating pathological behavior. We construct the Rindler hydrodynamics up to second order in derivatives of the fluid variables and show the positivity of its entropy current divergence.