Black holes in rotating, electromagnetic backgrounds and topological Kerr-Newman-NUT spacetimes

Abstract: We notice that a large class of well behaved stationary and axisymmetric black hole solutions in general relativity and in the Einstein-Maxwell theory can be classified according to the properties of their background. Actually all these backgrounds belong to a unique family which includes simultaneously all the known axisymmetric and regular backgrounds: the swirling, the Bertotti-Robinson, the Bonnor-Melvin universe, the Witten's expanding bubble and also others novel, regular, rotating gravitational or electromagnetic environments. All these can be, fundamentally, re-conducted to the double Wick rotation of the topological generalisation of (accelerating) Kerr-Newman-NUT metric. We present a black hole embedded in an unexplored sector of the general background: Schwarzschild inside a generalised rotating (and possibly electromagnetic) universe. These results indicate that basically all the known analytical and exact single black hole solutions in the four-dimensional Einstein-Maxwell theory belong to the (accelerating) Kerr-Newman-NUT family embedded into backgrounds that are a subcase of the conjugated Kerr-Newman-NUT space-time with an angular manifold of arbitrary topology.

• The paper unifies diverse black hole backgrounds by demonstrating that all regular Einstein-Maxwell solutions can be viewed as deformations of the Kerr-Newman-NUT metric via double Wick rotation.
• It employs solution-generating techniques, such as the inverse scattering method, to reinterpret known spacetimes like Bertotti-Robinson and Bonnor-Melvin as special limits.
• The study introduces new avenues by embedding Schwarzschild and accelerating black holes into exotic, curling, and swirling electromagnetic and rotational environments.

Classification and Embedding of Black Holes in Rotating, Electromagnetic, and Topological Kerr-Newman-NUT Backgrounds

Overview and Motivation

The paper "Black holes in rotating, electromagnetic backgrounds and topological Kerr-Newman-NUT spacetimes" (2604.05017) provides a comprehensive analysis of stationary and axisymmetric black hole solutions in four-dimensional Einstein-Maxwell theory. The work systematically classifies all known exact, single black hole solutions (excluding those with matter sources beyond the Maxwell field) according to the structure of their external gravitational and electromagnetic backgrounds. A crucial assertion of the work is that essentially all these backgrounds are continuous deformations or analytical continuations (via double Wick rotation) of the topological generalisation of the (accelerating) Kerr-Newman-NUT metric. This framework unifies and extends the catalogue of admissible backgrounds, incorporating known solutions such as Kerr/Bonnor-Melvin/Bertotti-Robinson universes, their rotating and NUT-deformed analogues, as well as novel regular rotating and electromagnetic environments.

Backgrounds as Double Wick Rotations: The General Construction

The starting point is the observation that various well-known external backgrounds―Bertotti-Robinson, Bonnor-Melvin, and swirling universes―can be constructed as double Wick rotations of type-D black hole solutions of Einstein-Maxwell(-$\Lambda$) gravity. The generic axisymmetric, stationary background is identified as the double Wick rotation of the topological Kerr-Newman-NUT-(A)dS metric (with constant-curvature event horizon geometry indexed by $k$). This leads to a unified background metric with parameters capturing mass ($m$), rotation ($a$), NUT charge ($\ell$), electric ($e$) and magnetic ($p$) monopole charges, cosmological constant ($\Lambda$), and horizon topology/curvature ($k$).

After analytic continuation, the physical charges are rendered “imaginative” (e.g., $e \to i\hat{e}$), so the resulting backgrounds are physically inequivalent from their black hole progenitors but share algebraic properties such as Petrov type D. The new backgrounds incorporate the swirling and “curling” (a previously unexplored rotation-independent from the swirling parameter) effects, both gravitational and electromagnetic, and provide an arena for embedding black holes of arbitrary topology.

Recovery and Embedding of Known Solutions

By explicit parametrizations and coordinate transformations, the paper demonstrates that:

• Minkowski, Levi-Civita, Bonnor-Melvin, Bertotti-Robinson, Witten bubble, and swirling universes all arise as special limits of this general conjugated metric.
• The backgrounds admit several physical parameters (e.g., swirling $k$0, curling $k$1, external electromagnetic field strength $k$2, Bertotti-Robinson field $k$3), whose various settings and limits produce all previously catalogued regular, asymptotically non-flat backgrounds in which black holes may be embedded.
• All known legitimate, analytic, single black hole solutions (no conical or curvature singularities outside the horizon) in Einstein-Maxwell theory (without scalar or more general matter) can be interpreted as (possibly accelerating) Kerr-Newman-NUT black holes in these backgrounds.

The work goes further by generating new black hole solutions in backgrounds with independent swirling and curling rotational structure. For example, the Schwarzschild black hole is embedded in a "curling" Bonnor-Melvin electromagnetic background, and the resulting spacetime is shown to be regular (i.e., no curvature singularities outside the event horizon) for a nontrivial parameter domain. Limits are systematically analyzed to recover the classical Schwarzschild, Schwarzschild-Levi-Civita, and other black holes in less exotic backgrounds.

Theoretical Implications

A central claim is all regular axisymmetric and stationary black holes in the four-dimensional Einstein-Maxwell system can be constructed as Kerr-Newman-NUT black holes (possibly accelerating) embedded in double Wick rotated Kerr-Newman-NUT backgrounds. This strongly suggests a closure property of the solution space under this embedding. The extension to arbitrary topology in the angular sector further enlarges the class of admissible solutions and settles the background-independence of the uniqueness results in more general settings. The inverse scattering method plays a central role in the explicit realization of these solutions.

The treatment of acceleration—via inclusion of the C-metric and its double Wick rotated analogues—and the connection with the Plebanski-Demianski family further strengthens this unifying picture. In particular, the self-duality of acceleration under double Wick rotation indicates that the structure of the backgrounds is robust under this operation.

Practical Implications and Solution Generating Techniques

The paper highlights that all these backgrounds and embeddings can be generated systematically by exploiting axisymmetric and stationary solution-generating techniques (Ernst equations, Harrison, and Ehlers transformations, inverse scattering methods). Moreover, the analysis clarifies the role of continuous vs. discrete solution-generating transformations—showing, for instance, that the Levi-Civita background can be accessed as a singular limit of swirling or Bonnor-Melvin backgrounds, and thus does not yield genuinely new analytic families. This streamlines the generation of solutions best suited for modeling black holes in external fields that may be relevant for astrophysical or theoretical scenarios.

Future Directions

The formalism and classification appear to extend naturally to other metric theories sharing the underlying solution-generating structure, such as those with (conformally) coupled scalar fields, non-linear sigma models, and specific $k$4 modifications. The inverse scattering/soliton approach suggests further potential for constructing multi-horizon solutions or more general algebraic types, although the paper notes the need for caution in interpreting such solutions as genuine black holes (regularity and asymptotic conditions). The completeness or unicity conjecture (i.e., no further independent backgrounds beyond those classified) deserves further investigation, especially in the presence of additional matter couplings or higher-dimensional generalisations.

Conclusion

This work provides a unifying classification and constructive prescription for all known stationary, axisymmetric, and regular black hole solutions in four-dimensional Einstein-Maxwell theory. All such solutions correspond to (possibly accelerating) Kerr-Newman-NUT black holes embedded in a background obtainable via double Wick rotation of a generalized topological Kerr-Newman-NUT metric, supplemented by explicit treatment of rotation, NUT charge, and electromagnetic field parameters. The results clarify the relation between the various well-known (and lesser-known) spacetimes and regularize the landscape of legitimate black hole solutions. The methods and results are immediately relevant for further developments in gravitational solution theory, the application of external field backgrounds, and potentially in studies of gravitational memory, black hole uniqueness, and the formulation of new extensions in alternative theories.