New class of rotating charged black holes with nonaligned electromagnetic field

TL;DR

This work constructs a new, large class of exact twisting solutions to the Einstein–Maxwell equations of algebraic type D in which the Faraday field is not aligned with the Weyl PNDs. The authors formulate a Plebański–Demiański–type metric with acceleration and twist, derive a Griffiths–Podolský–like parameterization, and reveal a nonaligned EM field characterized by three real control parameters and two duality-rotation angles. They systematically connect the general solution to known spacetimes via physically meaningful limits, including Kerr–Newman–NUT, Kerr–Bertotti–Robinson in a uniform external field, and the Van den Bergh–Carminati non-twisting case, while highlighting rich horizon and conical-structure properties. The results provide a versatile exact model for charged, accelerating black holes in external electromagnetic fields, with potential relevance to astrophysical magnetized black holes and supergravity contexts. The work also clarifies how duality rotations and external fields influence the nonalignment of EM and gravitational structures in type D spacetimes.

Abstract

We present a large family of twisting and expanding solutions to the Einstein-Maxwell equations of algebraic type D, for which the two double principal null directions (PNDs) of the Weyl tensor are not aligned with the null eigendirections of the Faraday tensor. In addition to systematically deriving this new class, we present its various metric forms and convenient parameterizations. We show that in Boyer-Lindquist-type coordinates these solutions depend on 7 parameters, namely the Kerr and NUT (Newman-Unti-Tamburino) twist parameters $a$ and $l$, mass parameter $m$, acceleration $α$, strength of the Maxwell field $|c|$, and angular parameters $β, γ$ that represent two duality rotations of the Faraday tensor, which include the rotation between the electric and magnetic charges generating the aligned part of the Maxwell field. This coordinate parameterization, analogous to the Griffiths-Podolský form of the Plebański-Demiański solutions, allows us to perform various limits, explicitly identify the subcases, and determine the physical interpretation of the new class. Interestingly, by considering the limit with no acceleration ($α\to 0$), one obtains either the famous Kerr-Newman-NUT black holes (if the parameter $|c|$ remains constant) or the novel Kerr-Bertotti-Robinson black holes, announced recently in our work [Kerr Black Hole in a Uniform Bertotti-Robinson Magnetic Field: An Exact Solution, Phys. Rev. Lett. {\bf 135} (2025) 18, 181401] (if $|c|\rightarrow \infty$ while $α|c|=\mathrm{const.}$). We may thus conclude that this new class of spacetimes represents twisting charged accelerating black holes, immersed in an external magnetic (or electric) field. In the non-twisting subcase, we obtain the previously known solution of Van den Bergh-Carminati.

Figures (5)

• Figure 1: Graphs showing the dependence of the Plebański-Demiański parameters $k$, $-n$, $\epsilon$ on $|c|$, calculated from \ref{['n_rel']}, \ref{['eps_rel']}, \ref{['b0tild']}, plotted in symmetric logarithmic scale. The Roman numbers I-V label different 5 possible roots. Colored solid lines represent the solutions with finite real limits as ${|c|\rightarrow 0}$ (these are the upper lines denoted as I), while the dashed lines represent the roots of \ref{['n_rel']}, \ref{['eps_rel']}, \ref{['b0tild']} which either diverge or are complex in the ${|c|\rightarrow 0}$ limit. Notice also that the roots IV and V occur only in a small restricted interval of ${|c|\ne0}$. The specific fixed values of the physical parameters are ${m=2.2}$, ${a=1.1}$, ${l=0.2}$, ${\omega=1}$, ${\alpha=0.14}$.
• Figure 2: Plots showing the positions of horizons depending on $|c|$ for the branch I of Fig \ref{['kne_plot']} (plotted in symmetric logarithmic scale). Different colors show different horizons, namely blue and red curves represent the outer and inner black hole horizons $r_b^{\pm}$, respectively, while the purple and orange curves represent the outer and inner acceleration horizons $r_a^{\pm}$ (notations are taken from Podolsk2021Podolsk2023). The parameters employed here are ${m=2.2},~{a=1.1},~{l=0.2},~{\alpha=0.14},~{\omega=1}$.
• Figure 3: Plots showing the positions of horizons depending on $|c|$ for the branches II and III of Fig. \ref{['kne_plot']}. Different colors show different horizons, namely blue and red curves represent the outer and inner black hole horizons $r_b^{\pm}$, while the purple and orange curves represent the outer and inner acceleration horizons $r_a^{\pm}$. Notice a nice and uniform ordering $r_a^+>r_b^+>r_b^->r_a^-$ for all values of $|c|$. The parameters are $m=2.2,~a=1.1,~{l=0.2},~\alpha=0.14,~\omega=1$.
• Figure 4: Values of the angles $\delta$ (left panel) and $\psi$ (right panel), defined by (\ref{['delt_psi_def']}), characterizing how the spacelike part of the eigendirections of the electromagnetic field is rotated relative to the spacelike part of the Weyl tensor PNDs for the branch I of Fig. \ref{['kne_plot']}. Black circles represent horizons, dashed curves represent ergoregions. The darkest blue means zero value, so that in such regions with ${\delta=0}$ the fields are aligned. The parameters are ${m=2.2,~a=1.1,~l=0.2,~|c|=0.2,~\alpha=0.14,~\omega=1}$.
• Figure 5: Values of the angles $\delta$ (left panel) and $\psi$ (right panel), defined by (\ref{['delt_psi_def']}), characterizing how the spacelike part of eigendirections of the electromagnetic field is rotated relative to the spacelike part of the Weyl tensor PNDs for the branch III of Fig. \ref{['kne_plot']}. Black circles represent horizons, dashed curves represent ergoregions. The parameters are ${m=2.2,~a=1.1,~l=0.2,~|c|=0.2,~\alpha=0.14,~\omega=1}$.