Kerr black hole in a uniform Bertotti-Robinson magnetic field: An exact solution

TL;DR

The paper presents Kerr–Bertotti–Robinson (Kerr-BR) black holes: exact, type D solutions of the Einstein–Maxwell equations describing rotating black holes immersed in an asymptotically uniform magnetic (or electric) field. The authors derive an explicit metric and Maxwell field depending on three parameters $m$, $a$, and $B$, with notable limits recovering Kerr, Bertotti–Robinson, and Schwarzschild–BR spacetimes. They perform a thorough physical analysis, determining curvature singularities, horizons, ergoregions, axis regularity, the non-aligned Maxwell field, geodesics (including ISCO in the Schwarzschild–BR limit), and thermodynamics, including a Smarr-type relation and a Meissner effect in the external field. The work provides a clean, analytically tractable model of a black hole in a uniform external field, offering insights for mathematical relativity and potential astrophysical applications where external fields influence black hole environments.

Abstract

A new class of exact spacetimes in Einstein's gravity, which are Kerr black holes immersed in an external magnetic (or electric) field that is asymptotically uniform and oriented along the rotational axis, is presented. These are axisymmetric stationary solutions to the Einstein-Maxwell equations such that (unlike in the Plebanski-Demianski spacetime) the null directions of the Faraday tensor are not aligned with neither of the two principal null directions of the Weyl tensor of algebraic type D (unlike the Kerr-Melvin spacetime). Three physical parameters are the black hole mass $m$, its rotation $a$, and the external field value $B$. For vanishing $B$ the metric directly reduces to standard Boyer-Lindquist form of the Kerr black hole, while for zero $m$ we recover conformally flat Bertotti-Robinson universe with a uniform Maxwell field. For zero $a$ the spacetime is contained in the Van den Bergh-Carminati solutions which can be understood as the Schwarzschild black hole in a magnetic field. Our family of black holes with non-aligned Maxwell hair - that can be called the Kerr-Bertotti-Robinson (Kerr-BR) black holes - may find application in various studies ranging from mathematical relativity to relativistic astrophysics.

Figures (2)

• Figure 1: Visualization of the magnetic field by the color plots (encoding the magnitude of the field) and the force lines (the arrows indicating the directions of the field) for ${B=0.2}$. The top left panel (${m=0}$, ${a=0}$) shows the uniform magnetic field of the Bertotti-Robinson (BR) background universe. Left column corresponds to the Schwarzschild-BR black holes without rotation (${a=0}$), while the right column depicts the general situation for Kerr-BR black holes (${a\not=0}$) with the rotation parameters ${a=0.35, 0.39, 0.399998 }$ (labeled as ${a=0.4}$) and ${m=0.4}$. The horizons are indicated by the black circles, the ergoregions are bounded by the dashed curves. In all the cases the external magnetic field is weakened in the equatorial plane ${\theta=\frac{\pi}{2}}$, and the force lines are "expelled away" --- exhibiting the Meissner effect. The blue discs and annuli indicate non-stationary region ${Q<0}$ inside the black hole where the ZAMO observer \ref{['ZAMO']} is not timelike and thus the field \ref{['EandB']} is not well defined.
• Figure 2: Effective potential $V(r)$ as a function of the radial coordinate $r$ for different values of the magnetic field $B$ (for ${m=1}$, ${L=4m_0\ne L_{\rm ISCO}}$). The points on the dashed curve show the positions of the corresponding stable circular orbits.