Thermodynamic structure of a generic null surface and the zeroth law in scalar-tensor theory
Sumit Dey, Krishnakanta Bhattacharya, Bibhas Ranjan Majhi
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 ${E}_{ab}{l}^{a}{k}^{b}$, where ${E}_{ab}=8\ensuremath{\pi}{T}_{ab}^{(m)}$ represents the equation of motion for a gravitational field in the 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.