On the KG-constrained Bekenstein's disformal transformation of the Einstein-Hilbert action

TL;DR

This paper explores a simplified form of the Bekenstein disformal transformation of the Einstein-Hilbert action within scalar theories of gravity, revealing connections to Horndeski and beyond-Horndeski terms across different dimensions.

Contribution

It introduces a constrained disformal transformation that simplifies the transformed action, linking it to known scalar-tensor theories and extending understanding in lower dimensions.

Findings

Simplified action in four dimensions includes Horndeski and beyond-Horndeski terms.

Reduced complexity of the transformed action using Klein-Gordon conformal-disformal constraint.

Invariant topological action identified in two dimensions.

Abstract

Motivated by an inclination for symmetry and possible extension of the General Theory of Relativity within the framework of Scalar Theory, we investigate the Bekenstein's disformal transformation of the Einstein-Hilbert action. Owing to the complicated combinations of second order metric derivatives encoded in the Ricci scalar of the action, such a transformation yields an unwieldy expression. To ``tame'' the transformed action, we exploit the Klein-Gordon (KG) conformal-disformal constraint previously discovered in the study of the invariance of the massless Klein-Gordon equation under disformal transformation. The result upon its application is a surprisingly much more concise and simple action in four spacetime dimensions containing three out of four sub-Lagrangians in the Horndeski action, and three beyond-Horndeski terms. The latter group of terms may be attributed to the kinetic-{term} dependence of the conformal and disformal factors in the Bekenstein's disformal transformation. Going down to three dimensions, we find a relatively simpler resulting action but the signature of the three ``extraneous'' terms remains. Lastly, in two dimensions, we find an invariant action consistent with its topological nature in these dimensions.