⚛️ general relativity

Geometry-Induced Termination of the Repetitive Penrose Process in Rotating Simpson-Visser Black Holes

This paper demonstrates that in rotating Simpson-Visser black holes, the repetitive Penrose process can be terminated by a geometry-induced condition where the regularization parameter forces the spacetime out of the two-horizon black hole regime before the conventional spin limit is reached, thereby reducing the cumulative energy extraction and efficiency compared to standard Kerr black holes.

Original authors: Mohammad Ali S. Afshar, Mohammad Reza Alipour, Saeed Noori Gashti, J. Sadeghi

Published 2026-07-23

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CC BY 4.0

Original authors: Mohammad Ali S. Afshar, Mohammad Reza Alipour, Saeed Noori Gashti, J. Sadeghi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a grand cosmic playground, where the most extreme playground equipment is a black hole. For decades, physicists have been fascinated by spinning black holes, not just because they are terrifyingly heavy, but because they are like cosmic batteries. In the 1960s, a physicist named Roger Penrose discovered a clever trick: if you throw a rock into the swirling "dance floor" just outside a spinning black hole (called the ergosphere), and that rock breaks apart, one piece can fall in with "negative energy" while the other flies away with more energy than the original rock had. It's like throwing a ball into a spinning carousel, having it shatter, and watching one piece fly off faster than you threw it, while the other gets sucked into the center, slowing the carousel down. This is the "Penrose Process," a way to steal rotational energy from a black hole.

But what happens if you keep doing this? What if you have an army of particles, one after another, breaking apart and stealing energy until the black hole stops spinning entirely? For a long time, scientists thought the process would just keep going until the spin ran out. However, real physics is rarely that simple. This new study asks a fresh question: What if the black hole isn't the standard, textbook kind? What if the very fabric of space around it is slightly "regularized"—meaning it's smoothed out so there's no crushing, infinite point at the center, but rather a gentle bounce? The researchers wanted to see if this smooth, weird shape of space would change the rules of the game, stopping the energy theft in a way standard black holes never would.

The Smooth Black Hole and the Energy Heist

In this study, the authors Mohammad Ali S. Afshar, Mohammad Reza Alipour, Saeed Noori Gashti, and J. Sadeghi decided to play a game of cosmic "heist" inside a specific, theoretical type of black hole called the Rotating Simpson-Visser (R-S-V) black hole. Think of a standard black hole (like the famous Kerr black hole) as a deep, infinite pit with a singularity at the bottom—a place where physics breaks down. The Simpson-Visser model is like a "black-bounce" or a wormhole. Instead of a bottomless pit, imagine the center is a smooth, curved tunnel. If you fall in, you don't get crushed; you hit a "bounce" and slide out the other side or into another region of space.

The researchers assumed that while the black hole loses energy and slows down (its mass and spin change), this "smoothness" parameter (called $\xi$ ) stays fixed. It's like a car that loses speed as you drive, but the shape of the road remains exactly the same. They simulated a repetitive version of the Penrose process: a particle falls in, splits, the "bad" piece gets eaten by the black hole (slowing it down), and the "good" piece escapes with extra energy. They asked: How many times can we do this before the process stops?

The Two Ways the Party Stops

The team discovered that the "heist" can stop for two very different reasons, and which one wins depends entirely on how "smooth" the black hole is.

1. The Usual Suspect: Running Out of Spin
In a normal black hole, the process stops when the black hole spins too slowly. To pull off the trick, the black hole needs to be spinning fast enough to create the conditions for "negative energy" particles. Once the black hole slows down past a certain point, the trick no longer works. This is the "dynamical termination." It's like a battery running out of charge; you just can't get any more energy out.

2. The New Twist: The Geometry Trap
Here is the paper's big surprise. In the smooth, bouncing black holes, the process can stop before the black hole runs out of spin. The authors found that as the black hole loses mass and spin, it might drift out of the "safe zone" where it is a two-horizon black hole.

Imagine the black hole's properties (mass and spin) as a car driving on a map. The "two-horizon black hole" is a specific neighborhood on that map. As the Penrose process repeats, the car (the black hole) drives through this neighborhood. If the "smoothness" of the road (the parameter $\xi$ ) is high enough, the car might drive right off the edge of the "black hole" neighborhood and into a "wormhole" neighborhood before the engine (the spin) runs out.

The authors call this geometry-induced termination. The process stops not because the black hole is too slow, but because the type of object it is has changed. The rules for the energy heist were written for a "two-horizon black hole," but now the object is no longer that; it's something else. The simulation has to stop because the original rules no longer apply.

The Results: Smoother Roads Mean Less Loot

The researchers ran the numbers to see how this plays out. They looked at different values for the smoothness parameter ( $\xi$ ):

Small Smoothness (Like a Normal Black Hole): If the smoothness is low (small $\xi$ ), the behavior looks just like the standard Kerr black hole. The black hole spins down, and the process stops when the spin gets too low. You can extract a decent amount of energy, and the efficiency is high.
Medium Smoothness: As the smoothness increases, the "safe zone" on the map gets smaller. The black hole drifts out of the "two-horizon" neighborhood faster. The process stops earlier, meaning you get less total energy out.
High Smoothness (The Big Trap): When the smoothness is very high (specifically, when the dimensionless parameter $\xi$ is around 0.88 or higher), the effect is dramatic. The simulation shows that the black hole leaves the "two-horizon" branch after just one or a very few Penrose events. Even though the black hole still has plenty of spin left, the geometry has changed so much that the process can't continue under the original assumptions.

The paper also looked at where the particles break apart (the decay radius). They found that breaking apart closer to the horizon is always better for getting energy, but as the smoothness increases, the total energy you can get drops significantly. The "Energy Return on Investment" (how much energy you get back compared to what you put in) drops steadily as the black hole gets smoother.

What This Means for the Universe

The authors are careful to point out that this doesn't mean energy extraction is impossible forever. It just means that in this specific model, the repetitive process as they defined it hits a wall. If the black hole transitions into a wormhole or a different type of object, you would need to write new rules to see if energy can still be extracted. But within the framework of the "two-horizon black hole," the geometry itself acts as a speed limit.

The study suggests that the shape of spacetime isn't just a passive background stage; it actively dictates how long a process can last. If the universe is filled with these "smooth" black holes, the amount of energy we could theoretically steal from them might be much less than we thought, and the process might end abruptly due to a change in the object's identity rather than a lack of fuel.

In short, the paper shows that in the world of regular black holes, the geometry of space is a strict bouncer. It doesn't just let you in; it checks your ID (the spin) and makes sure you haven't wandered out of the club (the parameter space) before you can finish your drink. If the club changes its name while you're inside, the party is over, even if you still have energy left in your tank.

Technical Summary: Geometry-Induced Termination of the Repetitive Penrose Process in Rotating Simpson–Visser Black Holes

Problem Statement
The paper investigates the long-term evolution of the Repetitive Penrose Process (RPP) within the Rotating Simpson–Visser (R-S-V) spacetime. While the classical RPP in Kerr spacetime is known to terminate when the black hole's dimensionless spin parameter drops below a kinematic minimum required to sustain negative-energy trajectories, this study asks whether the underlying spacetime geometry imposes additional constraints. Specifically, the authors examine whether the secular evolution of a black hole's mass and angular momentum, driven by repeated energy extraction, can force the system out of its initial geometric branch (the two-horizon regular black hole regime) before the conventional dynamical spin limit is reached. The R-S-V geometry is chosen for this analysis because its parameter space encompasses regular black holes, black-bounce geometries, and traversable wormholes, depending on the value of a fixed regularization parameter $\xi$ .

Methodology
The authors employ a nonlinear iterative framework to model the cumulative extraction of rotational energy. Unlike linear approximations that treat the background spacetime as static between events, this approach updates the macroscopic black-hole parameters ( $M$ and $L$ ) after each decay step while conserving global energy and angular momentum.

Kinematics: The process assumes an incident particle decays into two fragments within the ergoregion: one captured by the black hole (negative energy) and one escaping to infinity. The analysis utilizes the "triple turning-point" condition, where the radial momenta of all three particles vanish at the decay radius, maximizing energy extraction.
Iterative Evolution: The black hole's mass and angular momentum are updated according to the conservation laws following each capture. The dimensionless spin parameter $\hat{a}$ is recalculated at every step.
Termination Criteria: The sequence terminates when any of the following conditions are met:
- Kinematic Feasibility: The mass defect becomes non-positive.
- Negative Energy Infall: The captured fragment can no longer possess negative energy.
- Thermodynamic Consistency: The extractable energy becomes non-positive or the irreducible mass decreases.
- Marginal Stability: The effective potential constraints prevent the existence of required orbits.
- Geometric Branch Transition: The evolving parameters $(M, L)$ cross the boundary of the parameter domain defining the initial two-horizon regular black hole solution, even if the spin $\hat{a}$ remains above the kinematic minimum.

The study assumes the regularization parameter $\xi$ remains constant throughout the evolution, acting as a fixed scale for the geometry.

Key Contributions
The primary contribution of this work is the identification of a geometry-induced termination mechanism distinct from the conventional dynamical spin limit. The authors demonstrate that in the R-S-V spacetime, the RPP can be terminated not because the black hole has exhausted its extractable rotational energy, but because the evolution of its mass and spin parameters forces the solution out of the "two-horizon regular black hole" branch into a different geometric regime (e.g., a single-horizon black hole or a wormhole).

This introduces a competition between two termination regimes:

Dynamical Termination: Governed by the minimum spin threshold ( $\hat{a}_{min}$ ) required for negative-energy orbits.
Structural (Geometry-Induced) Termination: Governed by the boundary of the admissible parameter space for the initial geometric branch.

Results
Numerical simulations across various values of the dimensionless regularization parameter $\hat{\xi} = \xi/M$ reveal the following:

Small Deformations ( $\hat{\xi} \lesssim 0.44$ ): The evolution closely mirrors that of the Kerr spacetime. The process terminates due to the conventional minimum-spin condition, and the solution remains within the two-horizon branch.
Intermediate to Large Deformations ( $\hat{\xi} \gtrsim 0.44$ ): A transition occurs where the geometry-induced constraint becomes dominant. For sufficiently large $\hat{\xi}$ , the black hole parameters evolve such that the system crosses the boundary of the two-horizon regime before the spin reaches its kinematic minimum.
Impact on Efficiency: As $\hat{\xi}$ increases, the cumulative extracted energy, the number of admissible Penrose iterations, and the overall efficiency (measured by Energy Return on Investment and Energy Utilization Efficiency) are progressively reduced.
Decay Radius Dependence: The results show that near-horizon decay events generally yield higher efficiency. However, for large $\hat{\xi}$ , the geometry-induced termination severely restricts the number of iterations regardless of the decay radius, leading to a monotonic decrease in total extracted energy and efficiency measures.
Irreducible Mass: For large $\hat{\xi}$ , the growth of the irreducible mass is significantly suppressed compared to the Kerr case, not due to thermodynamic violations, but because the iterative sequence is cut short prematurely by the geometric boundary.

Significance and Claims
The paper claims that the underlying spacetime structure plays a decisive role in shaping the long-term evolution of idealized energy extraction processes. The findings highlight that in regular rotating black hole spacetimes with additional geometric parameters, the "endpoint" of the Penrose process is not solely a function of the black hole's spin but is also constrained by the global topology and parameter space of the solution.

The authors emphasize that this geometry-induced termination should not be interpreted as a fundamental prohibition of rotational energy extraction in general. Rather, it signifies that the specific iterative framework, constructed for the two-horizon branch, ceases to be valid once the system evolves into a different geometric regime (e.g., a wormhole). To continue the analysis beyond this point would require reformulating the RPP within the new geometric branch.

Ultimately, the study positions the RPP as a probe for understanding how global spacetime properties constrain the dynamical evolution of compact objects, suggesting that the interplay between secular evolution and fixed geometric parameters can fundamentally alter the viability and limits of energy extraction mechanisms.

The Smooth Black Hole and the Energy Heist

The Two Ways the Party Stops

The Results: Smoother Roads Mean Less Loot

What This Means for the Universe

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