Fully conservative $f(R,T)$ gravity and Solar System constraints

The $f(R,T)$ gravity is a model whose action contains an arbitrary function of the Ricci scalar $R$ and the trace of the energy-momentum tensor $T$. We consider the separable model $f(R,T)=\chi (R)+\phi (T)$ and shown that, for perfect fluids, the dynamical equations are sufficient to determine how $\phi$ depends on $T$, independently of the matter state equation and the geometry of space-time. Imposing the energy-momentum tensor conservation we obtain that $\phi$ must be linear in $T$. However, the $T$ dependence is severely constrained using the full Will-Nordtvedt version of the parameterized post-Newtonian (PPN) formalism. The result of the PPN analysis is discussed and in addition it is shown that the diffeomorphism invariance of the matter action imposes strong constraints on conservative versions of $f(R,T)$ gravity.