Electrically charged regular black holes in higher dimensions: Maxwellian limit, branch point of the Lagrangian, and thermodynamics
Electrically charged regular black holes in general relativity coupled to nonlinear electrodynamics are known to require a double-valued Lagrangian density $L(F)$, with a branch point at the radius where the electromagnetic invariant attains its extremum. We extend the construction of such black holes to an arbitrary number of spacetime dimensions $D\geq4$ and obtain a new family of solutions that unifies the Bardeen and Hayward black holes in a single analytic form, with a single nonlinearity parameter $γ$. We show that the Maxwellian weak-field limit is recovered if and only if $γ=D-3$, so that the Bardeen and Hayward black holes, which lack this limit in four dimensions, become Maxwellian in five and six dimensions, respectively. The matter sector is written in the dual $P$ framework as a Hamiltonian density $H(P)$ that contains only fixed coupling constants, so that the mass and the charge of the black hole appear as integration constants of a fixed theory; the regular solutions form a one-parameter subfamily in which the mass is tied to the charge. We locate the branch point of $Ł(F)$ in closed form and find that it is always hidden inside the event horizon in four dimensions, whereas for $D\geq5$ it emerges from the horizon for near-extremal charges and becomes visible to a distant observer. The null and weak energy conditions are satisfied everywhere, the strong energy condition is violated only inside the horizon, and the dominant energy condition is violated at large distances only for $γ>D-3$. Finally, since the theory is fixed, the first law of black hole thermodynamics holds in its standard form; we derive the horizon potential in closed form, the extended first law with the couplings as thermodynamic variables, and the corresponding Smarr formula.