Slice energy in higher-order gravity theories and conformal transformations

We study the generic transport of slice energy between the scalar field generated by the conformal transformation of higher-order gravity theories and the matter component. We give precise relations for this exchange in the cases of dust and perfect fluids. We show that, unless we are in a stationary spacetime where slice energy is always conserved, in non-stationary situations, contributions to the total slice energy depend on whether or not test matter follows geodesics in both frame representations of the dynamics, that is, on whether or not the two conformally related frames are physically indistinguishable.