Entropy density of spacetime and thermodynamic interpretation of field equations of gravity in any diffeomorphism invariant theory

TL;DR

This paper demonstrates that the field equations of any diffeomorphism invariant gravity theory can be expressed as a thermodynamic identity TdS=dE near any spacetime event, linking gravity and thermodynamics.

Contribution

It shows how to derive the thermodynamic form of gravitational field equations using Noether currents and local Rindler frames, clarifying conceptual subtleties involved.

Findings

Field equations can be written as TdS=dE locally in any diffeomorphism invariant gravity theory.

Using Noether current as local entropy density is key to this thermodynamic interpretation.

Inverting the derivation to obtain field equations from thermodynamics is generally incorrect.

Abstract

I argue that the field equations of any theory of gravity which is diffeomorphism invariant must be expressible as a thermodynamic identity, TdS=dE around any event in the spacetime. This fact can be demonstrated explicitly (and rather easily) if: (a) one accepts the Noether current of the theory as providing the definition for local entropy density and (b) one is allowed to introduce the local notions of a Rindler frame, acceleration horizon and a Killing vector (related to translation in Rindler time) around any event. It is conceptually incorrect - in general - to invert this argument and obtain the field equations of the theory from the thermodynamic identity. I discuss under what conditions this may be possible. Several subtleties related to these arguments are described.