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

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 KG 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.