Further matters in space-time geometry: f(R, T, RμνTμν) gravity

We consider a gravitational theory in which matter is nonminimally coupled to geometry, with the effective Lagrangian of the gravitational field being given by an arbitrary function of the Ricci scalar, the trace of the matter energy-momentum tensor, and the contraction of the Ricci tensor with the matter energy-momentum tensor. The field equations of the theory are obtained in the metric formalism, and the equation of motion of a massive test particle is derived. In this type of theory the matter energy-momentum tensor is generally not conserved, and this nonconservation determines the appearance of an extra force acting on the particles in motion in the gravitational field. It is interesting to note that in the present gravitational theory, the extra force explicitly depends on the Ricci tensor, which entails a relevant deviation from the geodesic motion, especially for strong gravitational fields, thus rendering the possibility of a space-time curvature enhancement by the R mu nu T mu nu coupling. The Newtonian limit of the theory is also considered, and an explicit expression for the extra acceleration that depends on the matter density is obtained in the small velocity limit for dust particles. We also analyze in detail the so-called Dolgov-Kawasaki instability and obtain the stability conditions of the theory with respect to local perturbations. A particular class of gravitational field equations can be obtained by imposing the conservation of the energy-momentum tensor. We derive the corresponding field equations for the conservative case by using a Lagrange multiplier method, from a gravitational action that explicitly contains an independent parameter multiplying the divergence of the energy-momentum tensor. The cosmological implications of the theory are investigated in detail for both the conservative and the nonconservative cases, and several classes of exact analytical and approximate solutions are obtained.