The first variation of the matter energy–momentum tensor with respect to the metric, and its implications on modified gravity theories

Abstract

The first order variation of the matter energy–momentum tensor Tμν with respect to the metric tensor gαβ plays an important role in modified gravity theories with geometry-matter coupling, and in particular in the f(R,T) modified gravity theory. We obtain the expression of the variation δTμν/δgαβ for the baryonic matter described by an equation given in a parametric form, with the basic thermodynamic variables represented by the particle number density, and by the specific entropy, respectively. The first variation of the matter energy–momentum tensor turns out to be independent on the matter Lagrangian, and can be expressed in terms of the pressure, the energy–momentum tensor itself, and the matter fluid four-velocity. We apply the obtained results for the case of the f(R,T) gravity theory, where R is the Ricci scalar, and T is the trace of the matter energy–momentum tensor, which thus becomes a unique theory, also independent on the choice of the matter Lagrangian. A simple cosmological model, in which the Hilbert–Einstein Lagrangian is generalized through the addition of a term proportional to Tn is considered in detail, and it is shown that it gives a very good description of the observational values of the Hubble parameter up to a redshift of z≈2.5.