# Fresh look at the scalar-tensor theory of gravity in Jordan and Einstein frames from undiscussed standpoints

## TL;DR

This paper critically examines the scalar-tensor theory of gravity in Jordan and Einstein frames, revealing previously unnoticed inequivalences at the action level while confirming thermodynamic equivalence and conformal invariance of key quantities.

## Contribution

It provides a detailed analysis showing inequivalences at the action level and confirms thermodynamic equivalence and invariance of entropy, energy, and temperature across frames.

## Key findings

- Thermodynamic quantities are conformally invariant.
- Inconsistencies in previous equivalence claims are identified.
- Thermodynamic and action-level analyses yield different conclusions.

## Abstract

We study the scalar-tensor theory of gravity profoundly in the action level as well as in the thermodynamic level. Contrary to the usual description in the literature about the equivalence in the two conformally connected frames, this paper addresses several incomplete inferences regarding it and mentions some inequivalences which were not pointed out earlier. In the thermodynamic level, our analysis shows the two frames are equivalent. In that process, we identify the entropy, the energy and the temperature for the thermodynamic description, and we find these quantities are conformally invariant even without any prior assumption. The same conclusion is reached from the gravitational action as well as from the Gibbons-Hawking-York boundary term, establishing the result in a more convincing manner.