Rotating wormholes in five dimensions with equal angular momenta: large asymmetry regime

This paper investigates five-dimensional rotating wormholes with equal angular momenta and large asymmetry, revealing that the violation of the null energy condition is primarily governed by angular momentum rather than mass asymmetry, and that the spacetime geometry approaches the extremal Myers-Perry black hole in the limit of large angular momentum.

The Gist

Imagine the universe as a giant, stretchy sheet of fabric. In the world of physics, a wormhole is like a tunnel you could dig through that fabric to connect two distant points instantly, rather than traveling the long way around.

For a long time, scientists thought building these tunnels was impossible because they would collapse instantly. To keep them open, you'd need "exotic matter"—a weird substance that pushes outward instead of pulling inward, essentially breaking the standard rules of gravity (specifically, something called the Null Energy Condition or NEC).

This paper, written by a team of researchers from Japan, explores a new way to build these tunnels in a universe with five dimensions (our familiar four, plus one extra hidden dimension). They are asking: Can we make these wormholes more stable and require less "exotic matter" by spinning them really fast?

Here is the breakdown of their findings using simple analogies:

1. The Spinning Top Effect

Think of a spinning top. When it's still, it's wobbly and unstable. But when you spin it fast, it stands up straight and becomes very stable.

The researchers found that rotation acts like that spin.

• The Old Idea: Previous studies suggested that spinning a wormhole might reduce the amount of "exotic matter" needed to keep it open.
• The New Discovery: This paper confirms that yes, spinning the wormhole helps. The faster it spins, the less "exotic matter" you need. In fact, if you spin it fast enough, the violation of the physical rules becomes almost negligible.

2. The "Tug-of-War" (Asymmetry)

Imagine a wormhole connecting two planets. Usually, we imagine them as twins—identical in size and mass. But what if one planet is huge and the other is tiny? This is called asymmetry.

The researchers tested what happens when the two sides of the wormhole are very different (a massive "heavy" side and a light "light" side).

• The Surprise: They found that this "tug-of-war" between the two sides doesn't matter much for the stability of the tunnel. Whether the wormhole connects two identical worlds or two very different ones, the amount of exotic matter needed is determined almost entirely by how fast it is spinning, not by the difference in mass between the two ends.

3. The "Black Hole" Connection

There is a famous type of black hole called the Myers-Perry black hole. It's like a cosmic vacuum cleaner that spins. There are two types:

• Non-extremal: A normal spinning black hole.
• Extremal: A black hole spinning at the absolute maximum speed possible without falling apart.

The researchers discovered a fascinating limit:

• As they spun their wormhole faster and faster, it didn't turn into a normal black hole. Instead, it slowly morphed into the extremal (maximum speed) black hole.
• The Catch: You can never turn a wormhole into a normal (non-extremal) black hole just by changing the parameters. The wormhole and the normal black hole are like two different species that can't interbreed. The wormhole only becomes a black hole when it reaches the "speed limit" of the universe.

4. The "Stretched" Tunnel

When you spin a wormhole very fast, the "throat" (the narrowest part of the tunnel) gets stretched out.

• Analogy: Imagine pulling a piece of taffy. As you pull it, it gets longer and thinner.
• In the limit of maximum spin, the wormhole throat stretches out into an infinitely long cylinder. This shape looks exactly like the inside of an extremal black hole. This suggests that at the very edge of stability, a wormhole and a black hole might be two sides of the same coin.

The Bottom Line

This paper is a roadmap for understanding how rotation changes the rules of wormholes.

1. Spin is King: The speed of rotation is the main factor in making a wormhole stable and reducing the need for "magic" exotic matter.
2. Asymmetry is Irrelevant: Whether the two ends of the wormhole are different sizes doesn't really change the physics of the tunnel.
3. The Ultimate Limit: If you spin a wormhole as fast as physically possible, it stops being a wormhole and becomes an extremal black hole.

Why does this matter?
While we can't build wormholes today, understanding these rules helps physicists figure out if they are even possible in a realistic universe. It also helps us understand the behavior of black holes and the fundamental laws of gravity in higher dimensions. It's like learning the rules of a video game before you can actually play it.

Technical Summary

1. Problem Statement

The paper addresses the theoretical construction of traversable wormholes in five-dimensional spacetime coupled to a phantom scalar field. While static wormholes inevitably require exotic matter that violates the Null Energy Condition (NEC), previous studies suggested that rotation might mitigate this violation.

• Gap in Knowledge: Previous work by Dzhunushaliev et al. (Ref. [25]) investigated rotating wormholes with equal angular momenta in 5D but restricted the analysis to specific, discrete values of the asymmetry parameter (representing the mass difference between the two asymptotic regions). It remained unclear whether the reduction of NEC violation due to rotation holds generally across different asymmetry regimes and whether these wormhole solutions connect to non-extremal black hole geometries.
• Objective: This study aims to systematically explore the parameter space of 5D rotating wormholes with equal angular momenta, specifically focusing on the large asymmetry regime, to determine the relationship between rotation, asymmetry, NEC violation, and the connection to Myers–Perry black holes.

2. Methodology

The authors employ a numerical approach based on the Einstein equations coupled to a massless phantom scalar field ($\phi$) in 5D.

• Action and Field Equations:
  The system is governed by the action $S = \int d^5x \sqrt{-g} [\frac{1}{16\pi G}R + \frac{1}{2}\partial_\mu \phi \partial^\mu \phi]$. The phantom field implies a negative kinetic term, leading to NEC violation.
• Metric Ansatz:
  To handle the complexity of rotation, the authors utilize a metric ansatz with equal angular momenta in the two independent rotation planes. This symmetry reduces the problem from partial differential equations (PDEs) to a system of Ordinary Differential Equations (ODEs) depending only on the radial coordinate $l$. The metric is cohomogeneity-1, admitting $R_t \times SU(2) \times U(1)$ symmetry.
• Boundary Conditions and Asymptotics:
  • The spacetime is asymptotically flat on both sides ($l \to \pm \infty$).
  • The solutions are characterized by global charges: Mass ($M_+, M_-$), Angular Momentum ($J$), and Scalar Charge ($Q$).
  • The asymmetry parameter is defined via the difference in the time-scaling factors at infinity, $a_{-\infty}$, which relates to the mass difference.
• Numerical Procedure:
  • A shooting method is employed. Due to numerical instability when integrating from one side to the other, the authors integrate from both asymptotic ends ($l \to \pm \infty$) and match the solutions at an intermediate point ($l=0$).
  • The radial coordinate is mapped to a finite domain $x \in (-1, 1)$ to handle the infinite boundaries.
  • The free parameters are the angular momentum parameter ($c_\omega$) and the asymmetry parameter ($a_{-\infty}$).

3. Key Contributions

1. Extension of Parameter Space: The study significantly extends the previous work by exploring a continuous range of the asymmetry parameter $a_{-\infty}$ (from $-\infty$ to $+\infty$), rather than just discrete values.
2. Clarification of NEC Violation: It provides a definitive analysis of how the degree of NEC violation depends on both rotation and asymmetry.
3. Black Hole Limit Analysis: It rigorously investigates the limit where wormhole solutions might transition into black hole solutions, specifically comparing them to the Myers–Perry (MP) black hole family.

4. Key Results

A. Dependence of NEC Violation on Parameters

• Rotation Mitigates Violation: The degree of NEC violation (quantified by $\Xi = T_{\mu\nu}k^\mu k^\nu$) decreases as the magnitude of the angular momentum ($|J|$) increases. In the limit of rapid rotation, the violation becomes arbitrarily small.
• Independence from Asymmetry: Crucially, the study finds that the degree of NEC violation is essentially determined by the angular momentum and shows little to no dependence on the asymmetry parameter ($a_{-\infty}$). Increasing the mass difference between the two sides does not significantly alleviate the need for exotic matter; only rapid rotation does.

B. Phase Diagrams and Convergence

• Universal Upper Bound: For a fixed throat area, there appears to be a universal upper bound on the angular momentum. As $|J|$ increases, all solution curves (regardless of $a_{-\infty}$) converge toward a single point in the phase diagram.
• Connection to Extremal Black Holes: This convergence point corresponds exactly to the parameters of the extremal Myers–Perry black hole (where the horizon area replaces the throat area). The scalar charge $Q$ tends to zero in this limit, indicating a transition to a vacuum solution.
• Geometric Stretching: In the rapid rotation limit, the wormhole throat geometry stretches, resembling the near-horizon geometry of an extremal black hole (an infinitely long cylinder).

C. Relation to Myers–Perry Black Holes

• Extremal Limit: The wormhole solutions asymptotically approach the extremal Myers–Perry solution as $|J| \to \infty$. The authors analytically demonstrate that the metric ansatz can accommodate the extremal MP solution (where $\mu = 4\alpha^2$).
• Non-Extremal Gap: The study finds a "gap" in the phase diagrams. The wormhole solutions cannot reproduce the non-extremal Myers–Perry black hole geometry in any limit. Even as the asymmetry parameter $a_{-\infty} \to \pm \infty$, the solutions do not approach the non-extremal black hole branch; they remain distinct, separated by a region in the parameter space where no wormhole solutions exist.

5. Significance

• Theoretical Insight: The paper resolves the question of whether "large asymmetry" can reduce the exotic matter requirements for wormholes. The answer is negative; rotation is the dominant factor in reducing NEC violation, not asymmetry.
• Stability Implications: By establishing a clear link between rapidly rotating wormholes and extremal black holes, the results provide a foundation for future stability analyses. Since extremal black holes are often more stable or have unique stability properties, understanding this connection is vital for assessing the physical viability of such wormholes.
• Higher-Dimensional Gravity: The work advances the understanding of higher-dimensional gravity solutions, specifically how symmetry (equal angular momenta) simplifies the complex dynamics of rotating wormholes, allowing for a complete exploration of their phase space.

In summary, Uemichi et al. demonstrate that while rotation can make 5D wormholes "more physical" by minimizing NEC violation, this effect is driven solely by angular momentum, not by the mass asymmetry between the two universes. Furthermore, these wormholes represent a distinct class of solutions that only merge with the black hole family in the extremal limit, leaving non-extremal black holes inaccessible via this construction.