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

This paper demonstrates that the symmetries of various 4-dimensional spacetimes, including Taub-NUT, near-horizon extreme Kerr, and swirling universes, are fundamentally interconnected through double Wick rotations, thereby establishing a theory-independent framework where finding solutions in one class automatically yields solutions in the others.

The Gist

Imagine the universe as a giant, infinite dance floor. In physics, we study the "symmetries" of this floor—essentially, the rules that tell us how the floor looks the same even if you spin it, slide it, or stretch it. These rules are like the choreography of the dance.

This paper is about discovering a secret "magic trick" that allows physicists to instantly swap one type of dance choreography for another, completely different one, without having to re-learn the whole dance from scratch.

Here is the breakdown of the paper's big idea, using simple analogies:

1. The "Double Wick Rotation" Magic Trick

In physics, there is a mathematical tool called a Wick rotation. Think of it like turning a real number (like 5) into an imaginary number (like $5i$). It's a bit like turning a clock that measures time into a clock that measures space.

Usually, physicists use this trick to solve specific problems for specific stars or black holes. But this paper says: "Wait a minute! This trick doesn't just change the numbers; it changes the dance moves themselves."

The authors found that if you apply this "double rotation" (doing it twice in a specific way) to the fundamental rules of how a Taub-NUT spacetime (a weird, twisted universe with a "magnetic" twist) moves, you instantly get the rules for a Near-Horizon Extreme Kerr (NHEK) spacetime (the area right next to a spinning black hole) or a Swirling Universe (a cosmos that spins like a tornado).

2. The "Universal Translator"

The most exciting part of this paper is that this trick is theory-independent.

• The Old Way: If you wanted to find a solution for a black hole in Einstein's gravity, you had to solve complex equations. If you then wanted to find a solution for a black hole in a different theory of gravity, you had to start over and solve the equations again.
• The New Way (This Paper): The authors show that the symmetries (the dance moves) are connected directly. It's like having a universal translator.
  • If you find a "vacuum solution" (a universe with no matter, just gravity) for the Taub-NUT dance, you automatically know the solution for the NHEK dance and the Swirling dance.
  • You don't need to know which theory of gravity you are using (Einstein's, or some new modified one). If the dance works for one, the "magic translation" guarantees it works for the others.

3. The "Lego Set" Analogy

Imagine you have a set of Lego instructions for building a Castle (Taub-NUT).

• Usually, if you want to build a Space Station (NHEK), you have to throw away the Castle instructions and start a new book from scratch.
• This paper says: "No! The Castle and the Space Station are built from the exact same Lego bricks, just arranged differently."
• By applying a simple "rotation" to the instructions (the double Wick rotation), the Castle instructions instantly transform into Space Station instructions.
• Even better: If you decide to build the Castle out of Wood instead of Plastic (changing the theory of gravity), the instructions for the Space Station automatically update to be made of Wood too.

4. What Did They Actually Do?

The authors looked at a specific classification of symmetries (called the Hicks classification). They took the "blueprints" for:

• Spherical/Hyperbolic/Planar Taub-NUT: Twisted universes.
• Schwarzschild: Standard, static black holes.

And they showed how to mathematically "twist" these blueprints to get:

• NHEK: The extreme environment around a spinning black hole.
• Swirling Universe: A universe that spins like a fluid.
• Melvin Space: A universe filled with a magnetic field.

They proved that these aren't just coincidental similarities between specific solutions; they are deep, structural connections between the rules of the game itself.

5. Why Does This Matter?

This is a huge time-saver for physicists.

• Discovery: If a physicist discovers a new, weird solution for a Taub-NUT universe in a new theory of gravity, they can immediately say, "Aha! I just found the solution for a Swirling Universe in that same theory!" without doing any new math.
• New Physics: It helps them find solutions they might have missed. For example, they suggest there might be a "near-horizon" version of a black hole with a hyperbolic (saddle-shaped) horizon that no one has written down explicitly yet, because they didn't know how to translate the "Taub-NUT" rules to that shape.

Summary

Think of the universe as a library of different "worlds." This paper discovered a secret elevator that connects the "Taub-NUT" floor to the "Spinning Black Hole" floor and the "Swirling Universe" floor.

You don't need to know the language of the building to use the elevator. If you have a key (a solution) for one floor, the elevator automatically gives you the key for the other floors, regardless of what the building is made of. It turns a massive, difficult search for new universes into a simple matter of pressing a button.

Technical Summary

1. Problem Statement

While specific exact solutions in General Relativity (GR) and other gravitational theories are known to be related via "double Wick rotations" (complex analytic continuations of coordinates and parameters), these relationships have historically been treated as isolated coincidences between specific solutions (e.g., mapping Schwarzschild to the "bubble of nothing" or Taub-NUT to Near-Horizon Extreme Kerr).

The authors identify a gap in the literature: the lack of a theory-independent analysis of the underlying spacetime symmetries. It has not been explicitly demonstrated that these mappings arise from relations between the infinitesimal group actions (Killing vector algebras) themselves, rather than just specific metric solutions. Consequently, it was unclear whether these transformations could systematically generate entire classes of solutions across different gravitational theories (including nonlinear electrodynamics) without re-deriving them from scratch.

2. Methodology

The authors employ a rigorous mathematical framework based on the Hicks classification of infinitesimal group actions on 4-dimensional manifolds.

• Symmetry Focus: They focus on Lie algebras of Killing vector fields that admit 4 independent generators with 3-dimensional orbits, denoted in the Hicks classification as the $[4,3,-]$ family.
• Unified Ansatz: They define a unified set of vector fields (Eq. 1) representing the symmetries of spherical ($k=1$), hyperbolic ($k=-1$), and planar ($k=0$) Taub-NUT (TNUT) spacetimes and Schwarzschild-like (A-metrics) spacetimes.
  • The vector fields depend on a parameter $n$ (NUT charge) and curvature $k$.
  • $n \neq 0$ corresponds to TNUT-type symmetries; $n=0$ corresponds to static Schwarzschild-type symmetries.
• General Invariant Tensors: Instead of solving field equations, they construct the most general $\Gamma$-invariant metric ($g$) and electromagnetic field strength ($F$) compatible with these symmetries. These are expressed in terms of arbitrary radial functions $a(r), b(r), c(r), d(r)$ and $f_1(r), f_2(r)$.
• Double Wick Rotations: The core methodology involves performing complex coordinate transformations (Wick rotations) on the vector fields and coordinates.
  • Examples include $t \to iq$, $\rho \to i\theta$, or mixing coordinates like $t \to iq - 2n\phi$.
  • Crucially, they track how these rotations transform the algebra of the Killing vectors and the form of the invariant tensors, ensuring the resulting vector fields remain real and the metric signature remains Lorentzian.

3. Key Contributions

A. Theory-Independent Symmetry Mapping

The paper establishes that the double Wick rotation is a map between symmetry classes, not just specific solutions.

• Taub-NUT $\leftrightarrow$ NHEK/Swirling: The symmetries of spherical/hyperbolic/planar TNUT spacetimes (with $n \neq 0$) map directly to the symmetries of the Near-Horizon Extreme Kerr (NHEK) geometry and the "swirling" universe.
• Schwarzschild (A-metrics) $\leftrightarrow$ B-metrics/Melvin: The static symmetries of Schwarzschild-type spacetimes ($n=0$) map to the symmetries of B-metrics (including the "bubble of nothing") and the Melvin universe.

B. Explicit Construction of Invariant Tensors

The authors provide explicit formulas for how the general $\Gamma$-invariant metrics and electromagnetic fields transform under these rotations.

• Metric Transformation: The general form of the metric is preserved, but the functions $a, b, c, d$ undergo specific redefinitions (e.g., $d \to i\tilde{d}$) to maintain reality. The 2-dimensional transverse metric $q_k$ transforms between Euclidean spheres/hyperboloids and Lorentzian $dS_2/AdS_2$ structures.
• Electromagnetic Transformation: The field strength tensor $F$ transforms, often requiring an analytic continuation of electromagnetic charges (e.g., $q_e \to i q_e$) to ensure the resulting field remains real.

C. The Hicks Classification Diagram

The authors map the specific $[4,3,c]$ classes involved in these rotations (Fig. 1 in the paper), showing a network of connections:

• $[4,3,3/1/6]$ (Static/Schwarzschild) $\leftrightarrow$ $[4,3,8/11]$ (BI/BII/BIII or Melvin).
• $[4,3,4/2/5]$ (TNUT) $\leftrightarrow$ $[4,3,9/10]$ (NHEK/Swirling).

4. Results and Applications

1. Vacuum Solutions

For any metric theory of gravity (assuming standard analyticity), a vacuum solution with a specific symmetry (e.g., a Taub-NUT solution) automatically generates a vacuum solution with the mapped symmetry (e.g., an NHEK solution) simply by applying the coordinate rotation. No new integration of field equations is required.

2. Electrovacuum Solutions (Einstein-Maxwell-$\Lambda$)

The authors demonstrate the mapping on known solutions:

• Charged RN $\to$ Charged BI/BII: Spherical/Hyperbolic Reissner-Nordström maps to charged BI/BII metrics.
• Charged Planar RN $\to$ Melvin: Maps to the Melvin spacetime.
• Massless Charged TNUT $\to$ NHEK: The massless charged hyperbolic TNUT maps to the NHEK geometry (specifically the Newman-NUT variant).
• Charged Planar TNUT $\to$ Swirling: Maps to the swirling universe solution.

3. Nonlinear Electrodynamics (Einstein-ModMax-$\Lambda$)

The methodology is extended to theories with nonlinear electrodynamics (ModMax).

• The authors show that the double Wick rotation successfully maps the charged TNUT solutions in ModMax theory to the corresponding swirling-Melvin solutions in the same theory.
• This confirms the universality of the symmetry mapping: it holds regardless of the specific form of the electromagnetic Lagrangian, provided the field equations respect the symmetry reduction.

4. New Potential Solutions

The mapping suggests the existence of previously unexplored solutions, such as the near-horizon extreme limits of rotating black holes with hyperbolic horizon topology, which are implied by the mapping of spherical TNUT to NHEK but have not been explicitly written in the literature.

5. Significance

• Unification of Spacetime Geometries: The paper provides a unifying mathematical framework explaining why seemingly distinct spacetimes (like the swirling universe and Taub-NUT) share structural similarities. They are different "projections" of the same underlying symmetry algebra.
• Solution Generation Tool: It offers a powerful, theory-independent algorithm for generating new exact solutions. Researchers can take a known solution in a complex theory (e.g., Einstein-Born-Infeld or ModMax) and immediately generate its "Wick-rotated" counterpart without solving the complex field equations anew.
• Clarification of Symmetry Reduction: By focusing on the infinitesimal group actions, the authors clarify the conditions under which symmetry reduction of Lagrangians is consistent (satisfying the principle of symmetric criticality), linking the Hicks classification directly to physical solution spaces.
• Bridging GR and Modified Gravity: The results demonstrate that these deep geometric connections are not artifacts of General Relativity but are fundamental properties of the spacetime symmetries themselves, applicable to a wide range of gravitational and electromagnetic theories.

In summary, the paper elevates the concept of double Wick rotation from a trick for finding specific solutions to a fundamental structural relationship between classes of spacetime symmetries, providing a robust toolkit for exploring the landscape of exact solutions in modern gravitational physics.