Thermodynamic equivalence of Jordan and Einstein frame in scalar-tensor theory

Among the various modified theories of gravity, the scalar-tensor theory is one of the most post popular one and one of the most studied as well. This theory is described in the two frames– the Jordan frame and the Einstein frame.

Although, the theory is successful in various avenues, there are two major controversies in the theory itself. These are– (1) Whether the conformal equivalence of the action in the two frames is merely a mathematical equivalence or whether these two frames are indeed equivalent. (2) What are the explicit covariant expressions of the physical quantities (energy, entropy, temperature) and how they are connected in the two frames. Although, the expression of the entropy and the temperature is widely accepted to some extent but, there is a controversy in the expression of the energy which can be used for the thermodynamic description in this theory.

We in Phys. Rev. D 95, no. 6, 064026 (2017) and Phys. Rev. D 97, no. 12, 124013 (2018) have explored these issues. We show that by properly defining the Lagrangians in the two frames, one can obtain the first law and the covariant expressions of the thermodynamic parameters (including energy) following the Iyer-Wald formalism. We also show that all the thermodynamic parameters are
conformally invariant.