Gauss-Codazzi thermodynamics on the timelike screen

TL;DR

This paper generalizes Jacobson's thermodynamic relation from null horizons to timelike screens with constant acceleration and specific geometric conditions, linking spacetime geometry with thermodynamics.

Contribution

It extends the thermodynamic interpretation of spacetime to timelike screens, relaxing some constraints and establishing conditions for the geometric setup.

Findings

Energy flux relates to the extrinsic curvature and expansion of the screen.

The extrinsic curvature must be flat or evolve proportionally to expansion.

The construction applies to observers with constant acceleration orthogonal to the screen.

Abstract

It is a known result by Jacobson that the flux of energy-matter through a local Rindler horizon is related with the expansion of the null generators in a way that mirrors the first law of thermodynamics. We extend such a result to a timelike screen of observers with finite acceleration. Since timelike curves have more freedom than null geodesics, the construction is more involved than Jacobson's and few geometrical constraints need to be imposed: the observers' acceleration has to be constant in time and everywhere orthogonal to the screen. Moreover, at any given time, the extrinsic curvature of the screen has to be flat. The latter requirement can be weakened by asking that the extrinsic curvature, if present at the beginning, evolves in time like on a cone and just rescales proportionally to the expansion.