Noether and Abbott-Deser-Tekin conserved quantities in scalar-tensor theory of gravity both in Jordan and Einstein frames

TL;DR

This paper investigates the thermodynamic properties of scalar-tensor gravity in both Jordan and Einstein frames, establishing conformally invariant conserved quantities via Noether and ADT formalisms that align with black hole thermodynamics.

Contribution

It demonstrates the conformal invariance of thermodynamic quantities and shows the equivalence of Noether and ADT conserved quantities in scalar-tensor gravity.

Findings

Conserved quantities are conformally invariant.

Noether and ADT formalisms yield equivalent thermodynamic descriptions.

The analysis is off-shell and independent of boundary conditions.

Abstract

We revisit the thermodynamic aspects of the scalar-tensor theory of gravity in the Jordan and in the Einstein frame. Examining the {\it missing links} of this theory carefully, we establish the thermodynamic descriptions from the conserved currents and potentials by following both the Noether and the Abbott-Deser-Tekin (ADT) formalism. With the help of conserved Noether current and potential, we define the thermodynamic quantities, which we show to be {\it conformally invariant}. Moreover, the defined quantities are shown to fit nicely in the laws of (the first and the second) black hole thermodynamics formulated by the Wald's method. We stretch the study of the conformal equivalence of the physical quantities in these two frames by following the ADT formalism. Our further study reveals that there is a connection between the ADT and the Noether conserved quantities, which signifies that the ADT approach provide the equivalent thermodynamic description in the two frames as obtained in Noether prescription. Our whole analysis is very general as the conserved Noether and ADT currents and potentials are formulated {\it off-shell} and the analysis is exempted from any prior assumption or boundary condition.