Action principle for the Fluid-Gravity correspondence and emergent gravity

TL;DR

This paper develops an action principle based on null vectors that reproduces the fluid-like behavior of Einstein's equations on null surfaces, supporting the view of gravity as an emergent phenomenon.

Contribution

It introduces a null vector-based action principle in arbitrary spacetimes that links gravitational dynamics to fluid and entropy concepts, strengthening the emergent gravity paradigm.

Findings

Null surface degrees of freedom behave like a fluid with PA=TS.

The action principle reproduces Navier-Stokes form of Einstein's equations.

Null vectors, not the metric, are varied in the action.

Abstract

It has been known for a long time that Einstein's field equations when projected onto a black hole horizon looks very similar to a Navier-Stokes equation in suitable variables. More recently, it was shown that the projection of Einstein's equation on to any null surface in any spacetime reduces exactly to the Navier-Stokes form when viewed in the freely falling frame. We develop an action principle, the extremization of which leads to the above result, in an arbitrary spacetime. The degrees of freedom varied in the action principle are the null vectors in the spacetime and not the metric tensor. The same action principle was introduced earlier in the context of emergent gravity paradigm wherein it was shown that the corresponding Lagrangian can be interpreted as the entropy density of spacetime. The current analysis strengthens this interpretation and reinforces the idea that field equations in gravity can be thought of as emergent. We also find that the degrees of freedom on the null surface are equivalent to a fluid with equation of state PA = TS. We demonstrate that the same relation arises in the context of a spherical shell collapsing to form a horizon.