We calculate static and spherically symmetric solutions for the Rastall modification of gravity to describe Neutron Stars (NS). The key feature of the Rastall gravity is the non-conservation of the energy-momentum tensor proportionally to the space-time curvature. Using realistic equations of state for the NS interior we place a conservative bound on the non-GR behaviour of the Rastall theory which should be 1% level. This work presents the more stringent constraints on the deviations of GR caused by the Rastall proposal.

We test a free ad hoc parametrization of the Tolman-Oppenheimer-Volkoff (TOV) equation. We do not have in mind any specific extended theory of gravity (ETG) but each new parameter introduced has a physical interpretation. Our aim is fully pedagogical rather than a proposal for a new ETG. Given a realistic neutron star equation of state we map the contributions of each new parameter into a shift in trajectories of the mass-radius diagram. This exercise allows us to make the correspondence between each TOV sector with some possible modifications of gravity and clarifies how neutron star observations are helpful for distinguishing theories.

We formulate a theory combining the principles of scalar-tensor gravity and Rastall's proposal of a violation of the usual conservation laws. We obtain a scalar-tensor theory with two parameters ω and λ, the latter quantifying the violation of the usual conservation laws (λ = 1 corresponding to the General Relativity limit). The only exact spherically symmetric solution is that of Robinson-Bertotti besides the Schwarzschild solution. A PPN analysis reveals that General Relativity results are reproduced when λ = 0. The cosmological case displays a possibility of deceleration/acceleration or acceleration/deceleration transitions during the matter dominated phase depending on the values of the free parameters.

Although general relativistic cosmological solutions, even in the presence of pressure, can be mimicked by using neo-Newtonian hydrodynamics, it is not clear whether there exists the same Newtonian correspondence for spherical static configurations. General relativity solutions for stars are known as the Tolman-Oppenheimer-Volkoff (TOV) equations. On the other hand, the Newtonian description does not take into account the total pressure effects and therefore can not be used in strong field regimes. We discuss how to incorporate pressure in the stellar equilibrium equations within the neo-Newtonian framework. We compare the Newtonian, neo-Newtonian and the full relativistic theory by solving the equilibrium equations for both three approaches and calculating the mass-radius diagrams for some simple neutron stars equation of state.

We analyse the vacuum static spherically symmetric space-time for a specific class of nonconservative theories of gravity based on the Rastall's theory. We obtain a new vacuum solution which has the same structure as the Schwarzschild-de Sitter solution in the General Relativity theory obtained with a cosmological constant playing the rôle of source. We further discuss the structure (in particular, the coupling to matter fields) and some cosmological aspects of the underline nonconservative theory.