Double Wick rotations between symmetries of Taub-NUT, near-horizon extreme Kerr, and swirling spacetimes

TL;DR

The article addresses how distinct spacetime symmetries in four-dimensional electro-vacuum geometries are connected through double Wick rotations in a theory-independent manner. By treating symmetries as infinitesimal group actions $\Gamma$ with 3D orbits and employing the Hicks classification, the authors map between symmetry-invariant metrics and electromagnetic fields across families such as Taub–NUT, Schwarzschild, NHEK, Melvin, and swirling spacetimes. The key contribution is a concrete, coordinate-only analytic continuation scheme that relates [4,3,{1--6}] to [4,3,{8--11}] and beyond, enabling automatic translation of vacuum solutions (and, with charge-continuations, electrovacuum solutions) from one symmetry class to another within GR and selected modified theories. This framework unifies disparate solutions under a single symmetry-structure paradigm and provides a practical tool for constructing new solutions in various gravity theories from known TNUT-type seeds. The results have potential implications for generating near-horizon and swirling geometries in alternative theories and for understanding the role of symmetry reductions in consistent solution-generation across gravitational models.

Abstract

We explicitly show that certain 4-dimensional infinitesimal group actions with 3-dimensional orbits are related by double Wick rotations. In particular, starting with the symmetries of the spherical/hyperbolic/planar Taub-NUT spacetimes, one can obtain symmetries of the near-horizon extreme Kerr (NHEK) geometry or swirling universe by complex analytic continuations of coordinates. Similarly, the static spherical/hyperbolic/planar symmetries (i.e., symmetries of the Schwarzschild spacetime and other A-metrics) are mapped to symmetries of the B-metrics (or Melvin spacetime). All these mappings are theory-independent -- they constitute relations among symmetries themselves, and, hence among the classes of symmetry-invariant metrics and electromagnetic field strengths, rather than among specific solutions. Consequently, finding, e.g., vacuum Taub-NUT-type solutions in a given gravitational theory automatically yields vacuum NHEK- or swirling-type solutions of that theory, with a possible extension to the electromagnetic case.

Figures (1)

• Figure 1: Diagram of possible double Wick rotations, indicated by double wiggly arrows ( ), among infinitesimal group actions denoted by ${[d,l,c]}$ according to the Hicks classification Hicks:thesisFrausto:2024egp (summarized in App. \ref{['sc:Hicks']}) and the corresponding classes of $\Gamma$-invariant tensors (metrics and field strengths). The signs $+$ and $-$ above these arrows indicate that the maps of the $\Gamma$-invariant metrics are restricted to $\mathrm{dS_2}$ and $\mathrm{AdS_2}$ structures (in metrics of [4,3,{8,9}]), respectively. The simple arrows () correspond to the limits from the actions with ${n\neq0}$ (equivalent for different non-zero values of $n$) to ${n=0}$. Description in sans-serif provides examples of spacetimes with such symmetries within electro-vacuum solutions of Einstein--Maxwell--$\Lambda$ theory. The shorthands stand for spherical (sph.), hyperbolic (hyp.), planar (pl.), charged (ch.), spherical horizon topology (sht.), and hyperbolic horizon topology (hht.).