On the KG-constrained general disformal transformation of the Einstein–Hilbert action

Although with great successes in explaining phenomena and natural behavior involving the Universe or a part thereof, the General Theory of Relativity is far from a complete theory. Focusing on its extension within the framework of scalar tensor theory, we investigate the transformation of the Einstein–Hilbert action under the general disformal transformation coupled with a set of triple KG constraints. The idea is that the general disformal transformation leads to an extremely complicated action necessitating a set of constraints to tame the resulting form. Motivated by previous studies on the invertibility and invariance of the massless Klein–Gordon equation under the general disformal transformation, we identify three constraints that significantly simplifies the transformed Einstein–Hilbert action. In four spacetime dimonsions, we find that the transformed action is a sum of the original action and a disformal contribution involving a six-term Lagrangian that includes the Ricci tensor coupled to a sum of derivatives of scalar fields and kinetic terms. In three spacetime dimensions, the disformal contribution becomes a five-term Lagrangian. Last, in two spacetime dimensions, the Einstein–Hilbert action is invariant under the constrained disformal transformation.