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

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 threeextraneous'' terms remains. Lastly, in two dimensions, we find an invariant action consistent with its topological nature in these dimensions.