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In this paper, we generalize the Schwarzschild–Melvin solution within Einstein–Maxwell-dilaton theories to include non-null scalar charges, while remaining embedded in a magnetic or electric field à la Melvin . We then use this general solution to obtain the solution in the specific case of Entangled Relativity after a conformal transformation. This notably enables us to verify that the analytical solution used in Arruga and Minazzoli (Eur Phys J C 81(11):1027, 2021) in order to represent compact objects such as neutron stars in Entangled Relativity is indeed a good approximation of the exact solutions of Entangled Relativity when the background field goes to zero.
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Minazzoli et al. (2025) studied this question.
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