Unified first law and some general prescription: a redefinition of surface gravity

TL;DR

This paper explores the unified first law in cosmological models, deriving key equations and redefining surface gravity to establish thermodynamic relations on horizons using the Kodama vector.

Contribution

It introduces a new approach to relate the unified first law with horizon thermodynamics by projecting along the Kodama vector, including a novel surface gravity definition.

Findings

Friedmann equations derived from the UFL via Kodama vector

Clausius relation obtained on horizons using the Kodama vector

Redefinition of surface gravity consistent with Unruh temperature

Abstract

The paper contains an extensive study of the unified first law (UFL) in the Friedmann-Robertson-Walker spacetime model. By projecting the UFL along the Kodama vector the second Friedmann equation can be obtained. Also studying the UFL on the event horizon it is found that Clausius relation cannot be obtained from the UFL by projecting it along the tangent to the event horizon as it can be for the trapping horizon. However, it is shown in the present work that Clausius relation can be obtained by projecting the UFL along the Kodama vector on the horizon and the result is found to be true for any horizon. Finally motivated by the Unruh temperature for the Rindler observer, surface gravity is redefined and a Clausius relation is obtained from the UFL by projecting it along a vector analogous to the Kodama vector.