In the context of Brans-Dicke scalar-tensor theories of gravity, it has recently been shown that the post-Newtonian parameters should be generalized in the context of strongly gravitating bodies, and that its generalization—the so-called exact parameters—actually depends on the pressure and energy density of the considered celestial body. Here we develop two new methods to numerically obtain the exact parameters by means of the usual Tolman-Oppenheimer-Volkoff computation, and find that the difference from the value of standard post-Newtonian parameters can be more than 80% in some situations. We also provide the connection with the Damour-Esposito-Farèse nonperturbative parameter ${\alpha }_{DEF}$. We then apply the methodology to the case of entangled relativity, and derive these exact parameters for the Sun and the Earth, as well as for neutron stars. We argue that current and foreseeable experiments are likely able to constrain the theory under the assumption that ${\mathcal{L}}_m=-\rho$, where $\rho$ is the total energy density. If ${\mathcal{L}}_m=T$ instead, as often advocated in the literature, then there is no deviation with respect to general relativity and the prospects of testing entangled relativity become much more remote in time, as only compact objects with extreme electric or magnetic fields could lead to some deviation from general relativity.

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• Alternative gravity theories
• Astrophysical studies of gravity
• Gravitation
• Gravitational waves