Gravitational field equations near an arbitrary null surface expressed as a thermodynamic identity

TL;DR

This paper generalizes the thermodynamic interpretation of gravitational field equations to arbitrary null surfaces in any spacetime, revealing a deep connection between gravity, thermodynamics, and fluid dynamics.

Contribution

It extends previous results by expressing Einstein's equations near any null surface as a thermodynamic identity and provides explicit formulas for associated energy, unifying different approaches.

Findings

Field equations near null surfaces can be written as a thermodynamic identity.

Explicit expression for thermodynamic energy associated with null surfaces.

Components of Einstein's equations relate to both thermodynamics and Navier-Stokes equations.

Abstract

Previous work has demonstrated that the gravitational field equations in all Lanczos-Lovelock models imply a thermodynamic identity TdS=dE+PdV (where the variations are interpreted as changes due to virtual displacement along the affine parameter) in the near-horizon limit in static spacetimes. Here we generalize this result to any arbitrary null surface in an arbitrary spacetime and show that certain components of the Einstein's equations can be expressed in the form of the above thermodynamic identity. We also obtain an explicit expression for the thermodynamic energy associated with the null surface. Under appropriate limits, our expressions reduce to those previously derived in the literature. The components of the field equations used in obtaining the current result are orthogonal to the components used previously to obtain another related result, viz. that some components of the field equations reduce to a Navier-Stokes equation on any null surface, in any spacetime. We also describe the structure of Einstein's equations near a null surface in terms of three well-defined projections and show how the different results complement each other.