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

Although with great successes in explaining phenomena and natural behaviour involving the Universe or a part thereof, the General Theory of Relativity is far from a complete theory. There are plenty of avenues to investigate its extensions or generalisations that, in the future, could pave the way for a much deeper understanding and possibly solutions of current problems involving the Universe; e.g., dark matter, dark energy, fine-tuning, etc. 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 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 dimensions, 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. Lastly, in two spacetime dimensions, the Einstein-Hilbert action is invariant under the constrained disformal transformation.