Consistency of equations of motion in conformal frames

TL;DR

This paper demonstrates that equations of motion in scalar-tensor theories are consistent across conformal frames, ensuring solutions can be reliably translated between the Jordan and Einstein frames, confirming their mathematical equivalence.

Contribution

It establishes the conditions under which equations of motion in different conformal frames are equivalent, confirming the mathematical consistency of solutions across frames.

Findings

Equations of motion in Jordan and Einstein frames are equivalent under certain conditions.

Solutions in one frame can be translated to the other frame reliably.

Mathematical equivalence of conformal frames is confirmed.

Abstract

Four dimensional scalar-tensor theory is considered within two conformal frames, the Jordan frame (JF) and the Einstein frame (EF). The actions for the theory are equivalent and equations of motion can be obtained from each action. It is found that the JF equations of motion, expressed in terms of EF variables, translate directly into and agree with the EF equations of motion obtained from the EF action, provided that certain simple consistency conditions are satisfied, which is always the case. The implication is that a solution set obtained in one conformal frame can be reliably translated into a solution set for the other frame, and therefore the two frames are, at least, mathematically equivalent.