We describe the first numerical Tolman-Oppenheimer-Volkoff solutions of compact objects in entangled relativity, which is an alternative to the framework of general relativity that does not have any additional free parameter. Assuming a simple polytropic equation of state and the conservation of the rest-mass density, we notably show that, for any given density, compact objects are always heavier (up to ∼ 8%) in entangled relativity than in general relativity-for any given central density within the usual range of neutron stars' central densities, or for a given radius of the resulting compact object.

Effective models of black holes interior have led to several proposals for regular black holes. In the so-called polymer models, based on effective deformations of the phase space of spherically symmetric general relativity in vacuum, one considers a deformed Hamiltonian constraint while keeping a non-deformed vectorial constraint, leading under some conditions to a notion of deformed covariance. In this article, we revisit and study further the question of covariance in these deformed gravity models. In particular, we propose a Lagrangian formulation for these deformed gravity models where polymer-like deformations are introduced at the level of the full theory prior to the symmetry reduction and prior to the Legendre transformation. This enables us to test whether the concept of deformed covariance found in spherically symmetric vacuum gravity can be extended to the full theory, and we show that, in the large class of models we are considering, the deformed covariance can not be realized beyond spherical symmetry in the sense that the only deformed theory which leads to a closed constraints algebra is general relativity. Hence, we focus on the spherically symmetric sector, where there exist non-trivial deformed but closed constraints algebras. We investigate the possibility to deform the vectorial constraint as well and we prove that non-trivial deformations of the vectorial constraint with the condition that the constraints algebra remains closed do not exist. Then, we compute the most general deformed Hamiltonian constraint which admits a closed constraints algebra and thus leads to a well-defined effective theory associated with a notion of deformed covariance. Finally, we study static solutions of these effective theories and, remarkably, we solve explicitly and in full generality the corresponding modified Einstein equations, even for the effective theories which do not satisfy the closeness condition. In particular, we give the expressions of the components of the effective metric (for static and spherically symmetric black holes interior) in terms of the functions that govern the deformations of the theory.

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In this manuscript, we present analytical external spherical solutions of entangled relativity, which we compare to numerical solutions obtained in a Tolman–Oppenheimer–Volkoff framework. Analytical and numerical solutions match perfectly well outside spherical compact objects, therefore validating both types of solutions at the same time. The analytical external (hairy) solutions – which depend on two parameters only – may be used in order to easily compute observables – such as X-ray pulse profiles – without having to rely on an unknown equation of state for matter inside the compact object.