Local Fluid Dynamical Entropy from Gravity

TL;DR

This paper constructs and analyzes the event horizons of geometries dual to fluid flows in strongly coupled N=4 super Yang Mills theory, defining a local entropy current via the horizon's area form.

Contribution

It demonstrates the regularity of horizons in these geometries and establishes a local entropy current in the dual field theory using a boundary-to-horizon map.

Findings

Event horizons are regular in these geometries.

A local entropy current is defined from the horizon area.

The divergence of the entropy current is positive, consistent with the area theorem.

Abstract

Spacetime geometries dual to arbitrary fluid flows in strongly coupled N=4 super Yang Mills theory have recently been constructed perturbatively in the long wavelength limit. We demonstrate that these geometries all have regular event horizons, and determine the location of the horizon order by order in a boundary derivative expansion. Intriguingly, the derivative expansion allows us to determine the location of the event horizon in the bulk as a local function of the fluid dynamical variables. We define a natural map from the boundary to the horizon using ingoing null geodesics. The area-form on spatial sections of the horizon can then be pulled back to the boundary to define a local entropy current for the dual field theory in the hydrodynamic limit. The area theorem of general relativity guarantees the positivity of the divergence of the entropy current thus constructed.