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

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.