Static black holes in an external uniform electromagnetic field: Reissner-Nordstrom accelerating in Bertotti-Robinson

Abstract: We provide a detailed analysis of the non-twisting subcase of the large class of type D black holes with a non-aligned electromagnetic field, presented recently in [H. Ovcharenko and J. Podolsky, Phys. Rev. D 112 (2025) 064076]. We show that such exact solutions split into two main subclasses that (after a suitable re-parametrization) can be interpreted as either the uncharged Schwarzschild or C-metric in the external Bertotti-Robinson (BR) spacetime with geometry ${\mathrm{AdS}_2\times\mathrm{S}_2}$, or as the charged Reissner-Nordstrom black hole accelerating in the external BR electromagnetic field. The distinction between these two subclasses is determined by the parameter $r_0$ that encodes relations between the external Maxwell field (given by the non-aligned components of the Faraday tensor ${Φ_0=Φ_2}$) and the Maxwell field created by the charge of the black hole (given by the aligned component $Φ_1$). Namely, if ${r_0=0}$ then the electromagnetic field is fully determined by ${Φ_0=Φ_2}$, and one gets the C-metric in the BR universe (including also the non-accelerating Schwarzschild-BR black hole). But if ${r_0\neq 0}$ then the electromagnetic field is independently determined by both the external BR field and the field of a black hole itself, and this can be interpreted as the Reissner-Nordstrom black hole accelerating in the Bertotti-Robinson spacetime. Even though such an interpretation of the spacetime family is quite simple, it contains a lot of subtleties (e.g. the no-charge limit of the RN-BR spacetime, the non-trivial dependence on the signs of the mass and charge of a black hole, extreme black holes, and others) which we carefully investigate in this work. We also show the explicit relation to solutions previously found by Van den Bergh and Carminati, and we discuss the connection to the Alekseev-Garcia and Alexeev solutions.