Gravitational wave energy momentum-tensor in reduced Horndeski theories

Abstract

We generalize, imposing the field equations only at dominant order, the Isaacson formula for the gravitational wave (GW) energy-momentum tensor (EMT) to the class of Horndeski theories in which the tensor modes travel at the speed of light (reduced Horndeski theories) and scalar waves are present. We discuss important particular cases such as: theories where scalar waves are also luminal and theories in which the transverse-traceless gauge can be achieved in an arbitrary open set. The vanishing of the trace of the gravitational wave energy-momentum tensor is obtained for theories in which all wave perturbations propagate at the speed of light. The trace is shown not to vanish trivially in other cases. We obtain, as a particular case of our general result, the GW EMTs, in a Brans-Dicke theory, both in the Einstein frame, recovering previous results in the literature, and in the Jordan frame, thereby showing the GW EMT is not conformally invariant. We further prove that there exists a subclass of reduced Horndeski theories where, in contrast to general relativity, the divergence of the GW EMT does not vanish even after the imposition of the full equations of motion, assuming an eikonal solution.

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