Black holes in the external Bertotti-Robinson-Bonnor-Melvin electromagnetic field

TL;DR

An exact analytic solution is constructed for a Schwarzschild-like black hole embedded in a general external electromagnetic field that combines the Levi-Civita-Bertotti-Robinson and Bonnor-Melvin backgrounds. The solution is generated via the Harrison transformation applied to a Schwarzschild seed in BR, yielding a BR-BM background with two physical intensities controlled by $b$ and $B$; limits recover Schwarzschild-Bertotti-Robinson, Schwarzschild-Bonnor-Melvin, and pure BR-BM backgrounds, and a swirling generalization is obtained by composing with Ehlers transformations. Conserved charges, thermodynamics, and horizon properties are discussed, including a dilatation factor required to satisfy the first law in non-flat backgrounds. The work provides a unified exact framework for axial-symmetric black holes in back-reacting external EM fields and outlines promising generalizations, such as complex Harrison parameters and cosmological-constant extensions.

Abstract

An exact and analytical solution, in four-dimensional general relativity coupled with Maxwell electromagnetism, is built by means of a Lie point symmetry of the Ernst equations, the Harrison transformation. The new spacetime describes a Schwarzschild-like black hole embedded into a general external back-reacting electromagnetic field, which is the superposition of the Levi-Civita-Bertotti-Robinson and the Bonnor-Melvin ones. The relation between the two homogeneous electromagnetic fields is clarified. Conserved charges and the first law of thermodynamics are analysed. Swirling generalisations are also considered. Limits to the known metrics such as Schwarzschild-Bertotti-Robinson, Schwarzschild-Bonnor-Melvin and Bertotti-Robinson-Bonnor-Melvin are discussed.

Figures (3)

• Figure 1: Embedding in Euclidean three-dimensional space $\mathbb{E}^3$ of the event horizon of the black hole distorted by the presence of both the external Bertotti-Robinson and Bonnor-Melvin magnetic backgrounds, for different values of the parameters $b$ and $B$, while $m=1$. When $b=0$ and $B=0$ we have the spherical Schwarzschild horizon. More pictures can be found in figure \ref{['fig:picture-horizons-more']}.
• Figure 2: Map of the solutions of the Einstein-Maxwell theory presented in this article. The novel spacetimes, not yet known in the literature, are emphasized in bold line rectangles. The new Schwarzschild-like solutions belong to the Petrov Type I. The more general black hole (\ref{['fbar-general']})-(\ref{['LWP-spherical-bar']}) carries, apart from the mass $m$ and the intensity of the swirling background $\jmath$, a couple of integrating constants related to the Bertotti-Robinson field ($B$ and $w$) and another one ($b$) for the Melvin-Bonnor field.
• Figure 3: More embeddings in Euclidean three-dimensional space $\mathbb{E}^3$ of the event horizon of the black hole described by metric (\ref{['LWP-spherical-new']})-(\ref{['A-new-magn']}), distorted by the presence of both the external Bertotti-Robinson and Bonnor Melvin magnetic backgrounds, for different values of the parameters $b$ and $B$, and $m=1$.