In this work, we consider different forms of relativistic perfect fluid Lagrangian densities that yield the same gravitational field equations in general relativity (GR). A particularly intriguing example is the case with couplings of the form $[1+f_2(R)]{\mathcal{L}}_m$, where $R$ is the scalar curvature, which induces an extra force that depends on the form of the Lagrangian density. It has been found that, considering the Lagrangian density ${\mathcal{L}}_m=p$, where $p$ is the pressure, the extra-force vanishes. We argue that this is not the unique choice for the matter Lagrangian density, and that more natural forms for ${\mathcal{L}}_m$ do not imply the vanishing of the extra force. Particular attention is paid to the impact on the classical equivalence between different Lagrangian descriptions of a perfect fluid.

We study the vacuum solutions of a gravity model where Lorentz symmetry is spontaneously broken once a vector field acquires a vacuum expectation value. Results are presented for the purely radial Lorentz symmetry breaking (LSB), radial/temporal LSB and axial/temporal LSB. The purely radial LSB result corresponds to new black hole solutions. When possible, parametrized post-Newtonian parameters are computed and observational boundaries used to constrain the Lorentz symmetry breaking scale.

It is shown that a nonminimal coupling between the scalar curvature and the matter Lagrangian density may account for the accelerated expansion of the Universe and provide, through mimicking, for a viable unification of dark energy and dark matter. An analytical exploration is first performed, and a numerical study is then used to validate the obtained results. The encountered scenario allows for a better grasp of the proposed mechanism, and sets up the discussion for improvements that can lead to a complete agreement with the observational data.

We consider a modified action functional with a nonminimum coupling between the scalar curvature and the matter Lagrangian, and study its consequences on stellar equilibrium. Particular attention is paid to the validity of the Newtonian regime, and on the boundary and exterior matching conditions, as well as on the redefinition of the metric components. Comparison with solar observables is achieved through numerical analysis, and constraints on the nonminimum coupling are discussed.