⚛️ general relativity

The Penrose process in rotating regular black holes

This paper investigates energy extraction via the Penrose process in rotating regular black holes, demonstrating that while known solutions like the Neves-Saa class are bounded by the Kerr limit, a newly proposed regular black hole derived from a specific non-rotating seed can achieve significantly higher extraction efficiencies.

Original authors: Anjan Kar, Ayan Dey, Sayan Kar

Published 2026-09-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Anjan Kar, Ayan Dey, Sayan Kar

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

Black holes are often imagined as cosmic vacuum cleaners, devouring everything that crosses their threshold, but a rotating black hole is a far more dynamic object. When a massive star spins as it collapses, the resulting black hole drags the very fabric of space around with it, creating a swirling region just outside its point of no return. In this zone, known as the ergoregion, space itself moves so fast that nothing can stand still; even light is forced to move in the direction of the spin. Decades ago, physicists realized that this violent dragging of space offered a theoretical loophole. If a particle were to enter this region and split into two, one piece could fall into the black hole with negative energy, while the other piece escaped with more energy than the original particle possessed. This process, called the Penrose process, suggests that we could theoretically harvest rotational energy from a black hole, slowing its spin in the process. While this idea has been studied extensively for standard black holes, a new question has emerged: what happens if the black hole is not a standard, singular object, but a "regular" one?

In the standard view of general relativity, the center of a black hole is a singularity, a point where density becomes infinite and the laws of physics break down. However, many physicists suspect that a complete theory of gravity might smooth out this infinite point, replacing it with a dense but finite core. These hypothetical objects are called regular black holes. They look like normal black holes from a distance, with event horizons and ergoregions, but they lack the crushing singularity at their center. A team of researchers set out to investigate how the Penrose process would function in these regular black holes. They wanted to know if the absence of a singularity would change the amount of energy we could extract, and whether the specific way the black hole is "smoothed out" at the center would make a difference.

The researchers began by examining a well-known family of rotating regular black holes, which includes several famous models like the Bardeen and Hayward solutions. These models are constructed by taking a standard black hole and modifying its center to remove the singularity, controlled by a specific parameter that dictates how the smoothing occurs. When the team calculated the maximum energy that could be extracted from these objects, they found a consistent pattern: the presence of the regular, non-singular core actually reduced the efficiency of energy extraction. In every case they tested within this family, the maximum energy one could pull out was lower than what is possible with a standard, singular rotating black hole. The more the black hole deviated from the standard singular model, the less energy could be harvested. This suggests that for this entire class of regular black holes, the "regularization" that saves the center from becoming a singularity comes at the cost of reduced energy potential.

However, the story took a surprising turn when the team looked at a different, newly proposed type of regular black hole. This object was constructed from a seed solution that is fundamentally different from the standard models; it does not resemble a normal black hole when the rotation is turned off. Instead of fading into a standard non-rotating black hole, it approaches a strange state with zero mass and a unique charge-like property. When the researchers applied the Penrose process to this new geometry, the results were dramatically different. They found that for certain configurations of this new black hole, the energy extraction efficiency could be significantly higher than that of a standard singular black hole. In fact, under specific conditions, the efficiency could reach values far exceeding the known limits for standard black holes, with the maximum efficiency occurring when the smoothing parameter was zero, effectively making the object singular again.

The researchers discovered that the key to this difference lies in the "seed" from which the black hole is built. For the first family of black holes, the seed was a standard Schwarzschild black hole, and smoothing it out always lowered the energy yield. For the new geometry, the seed was a different, more exotic object, and in that context, the regular version allowed for much higher energy extraction than its singular counterpart. The study confirms that the ability to extract energy is not just about the spin of the black hole, but is deeply sensitive to the internal structure of the object and the nature of the solution it is based on. While the new model shows that higher efficiency is theoretically possible, the researchers note that the matter required to hold such a black hole together would violate standard energy conditions, meaning it likely cannot exist in our universe as we currently understand it.

Ultimately, this work clarifies that the potential to harvest energy from a black hole is not a universal constant but depends on the specific details of the spacetime geometry. The study demonstrates that while some regular black holes are less efficient energy sources than their singular cousins, others could theoretically be far more potent. This distinction highlights that the path to understanding black holes requires looking beyond the event horizon to the very core of the object. The findings suggest that if regular black holes exist, their behavior in energy extraction scenarios would be a powerful way to distinguish them from the singular black holes predicted by classical theory, provided we can one day observe such subtle differences in the extreme environment of a spinning black hole.

Technical Summary: The Penrose Process in Rotating Regular Black Holes

Problem Statement
The paper investigates the efficiency of rotational energy extraction via the Penrose process in the context of rotating regular black holes. While the Penrose process is well-understood for the singular Kerr spacetime (where the maximum extraction efficiency is approximately 20.7%), its behavior in regular black hole geometries—solutions designed to avoid central curvature singularities—remains less explored. The authors aim to determine how the "regularisation parameter" ( $g$ ) and the specific nature of the underlying non-rotating "seed" solution influence the maximum energy extraction efficiency ( $\delta E_{max}$ ) compared to their singular counterparts.

Methodology
The authors analyze two distinct classes of rotating regular black hole spacetimes using the Newman-Janis algorithm (specifically the Type-I prescription) to generate rotating metrics from static, spherically symmetric seeds.

Neves-Saa Class: The authors first examine the rotating versions of the Neves-Saa family of solutions, which include the Bardeen, Hayward, and Fan-Wang spacetimes as special cases. These metrics are defined by a mass function $m(r) = \frac{Mr^p}{(r^q + g^q)^{p/q}}$ . In the limit $g \to 0$ , these solutions reduce to the singular Kerr metric.
New Rotating Regular Black Hole: The authors construct a new rotating geometry based on a recently proposed static seed solution (from Gen. Rel. Grav. 56, 52 (2024)) that does not possess a Schwarzschild limit when $g=0$ . Instead, its $g=0$ limit corresponds to a singular Reissner-Nordström-type metric with zero mass and an "imaginary" charge.
Analytical Framework:
- Penrose Process: The analysis utilizes the Hamilton-Jacobi method to derive geodesic equations. The process involves a massive particle entering the ergo-region, decaying into two massless particles, where one falls into the black hole with negative energy ( $E < 0$ ) and the other escapes with energy greater than the incident particle.
- Efficiency Calculation: The maximum efficiency is calculated using the formula derived for the horizon radius ( $r_h$ ): $\delta E_{max} = \frac{1}{2} \left( \sqrt{1 + \frac{a^2}{r_h^2 E_{(0)}^2}} - 1 \right) E_{(0)}$ .
- Energy Conditions: For the new rotating metric, the authors evaluate the Null and Weak Energy Conditions (NEC and WEC) within a Locally Non-rotating Frame (LNRF) to characterize the matter required to support the geometry.
- Curvature Invariants: The regularity of the new rotating solution is verified by checking the finiteness of the Zakhary-McIntosh (ZM) curvature invariants.

Key Contributions and Results

Neves-Saa Class (Bardeen, Hayward, Fan-Wang):
- The study confirms that for the entire Neves-Saa class, the presence of a finite regularisation parameter ( $g \neq 0$ ) consistently reduces the maximum energy extraction efficiency compared to the singular Kerr limit ( $g=0$ ).
- The maximum efficiency is bounded above by the Kerr value (0.207 for $E_{(0)}=1$ ).
- An analytical argument is provided: since the radial derivative of the mass function $m'(r)$ is strictly positive for these regular solutions (whereas $m'=0$ for Kerr), the horizon condition leads to a lower maximum efficiency.
- Variations in parameters $p$ and $q$ affect the range of allowed spin parameters ( $a$ ) and regularisation parameters ( $g$ ) for which horizons exist, but the efficiency never exceeds the Kerr bound.
New Rotating Regular Black Hole:
- The authors construct the rotating counterpart of the new seed metric. This geometry is regular everywhere (finite ZM invariants) but violates the Weak and Null Energy Conditions, similar to its static counterpart.
- Significant Finding: Unlike the Neves-Saa class, this new rotating regular black hole allows for an extraction efficiency exceeding the Kerr limit for specific parameter choices.
- The maximum efficiency for this new geometry occurs in the singular limit ( $g=0$ ), reaching values as high as $\delta E_{max} \approx 3.04$ (for $a/b_0 = 0.99$ ).
- For regular cases ( $g > 0$ ), the efficiency decreases as $g$ increases, but it can still remain higher than the Kerr value for certain combinations of $a$ and $g$ .
- The authors address the apparent paradox of energy extraction from a geometry with zero ADM mass and angular momentum (in the $g=0$ limit). They demonstrate that the laws of black hole mechanics ( $\delta M - \Omega_H \delta J > 0$ ) still hold, where the parameter $b_0$ plays a role analogous to mass in the surface gravity and horizon area relations.
- Trajectory analysis shows that for $E_{(0)}=1$ , particles must originate from a finite distance (rather than infinity) to enter the ergo-region and facilitate the process, avoiding contradictions in angular momentum constraints.

Significance and Claims
The paper claims that the efficiency of the Penrose process is sensitive to both the regularisation parameter and the specific character of the seed non-rotating solution.

For the Neves-Saa class, regularisation acts as a suppressor of energy extraction efficiency relative to the singular case.
For the new class of regular black holes (derived from a non-Schwarzschild seed), the regularised spacetime can support energy extraction efficiencies significantly higher than the Kerr black hole, provided the seed geometry itself allows for such high efficiencies in its singular limit.
The authors conclude that different classes of regular black holes yield notably different outcomes for energy extraction, suggesting that the Penrose process could theoretically serve as a distinguishing feature between regular and singular black hole spacetimes, or between different types of regular black holes.

The work remains strictly theoretical, relying on general relativity and the specific metric constructions provided. The authors note that future work could explore the repetitive Penrose process in these spacetimes.

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