Thermodynamic structure of a generic null surface and the zeroth law in scalar-tensor theory

TL;DR

This paper demonstrates that the equations of motion in scalar-tensor gravity can be expressed as thermodynamic identities on null surfaces, establishing a covariant framework and confirming the zeroth law in this context.

Contribution

It introduces a covariant approach to derive thermodynamic identities from scalar-tensor equations on null surfaces, applicable in both Einstein and Jordan frames, and proves the zeroth law for Killing horizons.

Findings

Equations of motion acquire thermodynamic form on null surfaces.

Thermodynamic quantities are covariant and frame-independent.

Zeroth law is validated for Killing horizons in scalar-tensor theory.

Abstract

We show that the equation of motion of scalar-tensor theory acquires thermodynamic identity when projected on a generic null surface. The relevant projection is given by $E_{ab}l^ak^b$, where $E_{ab} =8\pi T_{ab}^{(m)}$ represents the equation motion for gravitational field in presence of external matter, $l^a$ is the generator of the null surface and $k^a$ is the corresponding auxiliary null vector. Our analysis is done completely in a covariant way. Therefore all the thermodynamic quantities are in covariant form and hence can be used for any specific form of metric adapted to a null surface. We show this both in Einstein and Jordan frames and find that these two frames provide equivalent thermodynamic quantities. This is consistent with the previous findings for a Killing horizon. Also, a concrete proof of the zeroth law in scalar-tensor theory is provided when the null surface is defined by a Killing vector.