A note on gravity and fluid dynamic correspondence on a null hypersurface

TL;DR

This paper extends the null foliation formalism for null hypersurfaces by deriving energy conservation and continuity equations, enriching the fluid-gravity correspondence framework beyond the previously established momentum conservation law.

Contribution

It introduces a comprehensive set of fluid dynamical equations within the null foliation formalism, including energy and momentum conservation, derived from covariant energy-momentum tensor conservation.

Findings

Derived energy conservation law for null hypersurfaces.

Established continuity equation in the null foliation framework.

Extended fluid-gravity correspondence to include energy dynamics.

Abstract

In the extensive literature on fluid-gravity correspondence formulated on null hypersurfaces, the Carrollian and membrane paradigm approaches have predominantly employed a timelike foliation. By contrast, within the null foliation formalism, only the momentum conservation law, expressed through the Damour-Navier-Stokes (DNS) equation, has been established. In this work, we revisit the null foliation formalism for a generic null hypersurface and extend it to include the energy conservation law, continuity equation, and related relations, all derived from the covariant conservation of an appropriately defined energy-momentum tensor. This development complements the existing literature on the fluid description of gravitational dynamics in the null foliation framework.