The Disformal Transformation of the Einstein-Hilbert Action

Abstract

A disformal transformation is a generalization of the well-known conformal transformation commonly elaborated in mainstream graduate texts on general relativity. This transformation is one of the most important operations in serious attempts in the literature to address pressing problems about the universe such as dark energy and dark matter. In this work, we derive the disformal transformation of the Einstein-Hilbert action following effectively the same logic as that for the conformal transformation. The resulting action, however, contains “anomalous” terms that could be construed as leading to equations of motion that could go beyond second order in spacetime derivatives, signaling instability of the transformed action. We demonstrate that these terms can be manipulated by way of decomposing the Riemann curvature tensor and shifting derivative indices through integration by parts, to end up with a manifestly stable action.