Varying fine structure “constant” and charged black holes

Speculation that the fine-structure constant $\alpha$ varies in spacetime has a long history. We derive, in 4-D general relativity and in isotropic coordinates, the solution for a charged spherical black hole according to the framework for dynamical $\alpha$ J. D. Bekenstein, Phys. Rev. D 25, 1527 (1982).. This solution coincides with a previously known one-parameter extension of the dilatonic black hole family. Among the notable properties of varying-$\alpha$ charged black holes are adherence to a “no hair” principle, the absence of the inner (Cauchy) horizon of the Reissner-Nordström black holes, the nonexistence of precisely extremal black holes, and the appearance of naked singularities in an analytic extension of the relevant metric. The exteriors of almost extremal electrically (magnetically) charged black holes have simple structures which makes their influence on applied magnetic (electric) fields transparent. We rederive the thermodynamic functions of the modified black holes; the otherwise difficult calculation of the electric potential is done by a shortcut. We confirm that variability of $\alpha$ in the wake of expansion of the universe does not threaten the generalized second law.