Penrose process in rotating black holes with quantum corrections: implications for energy extraction and irreducible mass

This paper analyzes the feedback of the rotational energy extraction from a Kerr black hole (BH) by the ‘ballistic method’, i.e. the test particle decay in the BH ergosphere pioneered by Roger Penrose. The focus is on the negative energy counterrotating particles (which can be massive or massless) going in towards the horizon, and the feedback on the BH irreducible mass is assessed. Generally, the change in irreducible mass is a function of the conserved quantities of the particle. For an extreme Kerr BH and in the limit μ1/M→0 , all the reduced transformable energy goes into the irreducible mass (i.e. ΔMirr/|E1|→∞ ), resulting in high irreversibility. The amount of extracted energy from the BH using test particles is much lower than the change of transformable energy. For non-extreme Kerr BHs, the effective potential of particle motion on the equatorial plane in Kerr spacetime is analyzed, and it is demonstrated that the Penrose process can only be undergone by BHs with a dimensionless spin a^>1/2 if the decay point coincides with the turning point. Based on that, the lower limit of the change in irreducible mass is provided as a function of the dimensionless spin of the BH. The significance of the increase in the irreducible mass of the BH during the energy extraction process is generally and concisely illustrated by introducing the concept of transformable energy of the BH. The feedback from the Penrose process on the irreducible mass demonstrates the irreversibility of energy extraction and highlights that the total amount of energy that can be extracted from a BH is less than previously anticipated.

The concept of the irreducible mass (Mirr) has led to the mass-energy (M) formula of a Kerr black hole (BH), in turn leading to its surface area S=16πMirr2. This also allowed the coeval identification of the reversible and irreversible transformations, soon followed by the concepts of and energy. This new conceptual framework avoids inconsistencies recently evidenced in a repetitive Penrose process. We consider repetitive decays in the ergosphere of an initially extreme Kerr BH and show the processes are highly irreversible. For each decay, the particle that the BH captures causes an increase of the irreducible mass (so the BH horizon), much larger than the extracted energy. The energy extraction process stops when the BH reaches a positive spin lower limit set by the process boundary conditions. Thus, the reaching of a final nonrotating Schwarzschild BH state through this accretion process is impossible. We have assessed such processes for selected decay radii and incoming particle with rest mass 1% of the BH initial mass M0. For r=1.2M and 1.9M, the sequence stops after 8 and 34 decays, respectively, at a spin 0.991 and 0.857, the energy extracted has been only 1.16%, and 0.42%, the extractable energy is reduced by 17% and 56%, and the irreducible mass increases by 5% and 22%, all values in units of M0. These results show the highly nonlinear change of the BH parameters, dictated by the BH mass-energy formula, and that the BH rotational energy is mainly converted into irreducible mass. Thus, evaluating the irreducible mass increase in any energy extraction processes in the Kerr BH ergosphere is mandatory. Published by the American Physical Society 2025

In this work we investigate the consequences of running gravitational coupling on the properties of rotating black holes. Apart from the changes induced in the space-time structure of such black holes, we also study the implications to Penrose process and geodetic precession. We are motivated by the functional form of gravitational coupling previously investigated in the context of infra-red limit of asymptotic safe gravity theory. In this approach, the involvement of a new parameter {tilde{xi }} in this solution makes it different from Schwarzschild black hole. The Killing horizon, event horizon and singularity of the computed metric is then discussed. It is noticed that the ergosphere is increased as {tilde{xi }} increases. Considering the black hole solution in equatorial plane, the geodesics of particles, both null and time like cases, are explored. The effective potential is computed and graphically analyzed for different values of parameter {tilde{xi }}. The energy extraction from black hole is investigated via Penrose process. For the same values of spin parameter, the numerical results suggest that the efficiency of Penrose process is greater in quantum corrected gravity than in Kerr Black Hole. At the end, a brief discussion on Lense–Thirring frequency is also done.

In this paper, we investigate the Penrose process in the purlieus of the axially symmetric magnetized Reissner-Nordstr\"om black hole for both neutral and charged particles. We start with the study of the geometry of the black hole and find the regions where the ${g}_{tt}$ component of the metric tensor is positive (i.e., ${g}_{tt}>0$). It is interestingly found that the condition ${g}_{tt}>0$ is fulfilled not only close to event horizon known as the ergosphere but also far away from the event horizon in the silhouette of potential wells. We also show that as the dimensionless magnetic field $B$ increases the silhouette of potential wells for which ${g}_{tt}>0$ grows correspondingly and eventually merges with the ergoregion when $B\ensuremath{\gtrsim}1.6$. Finally, we investigate the efficiency of the Penrose process for the axially symmetric magnetized black hole case and bring out the effect of magnetic field on it. Further, we also compare our results with the one for Kerr black hole. We show that when the charge $Q$ of the black hole is kept constant, the efficiency of the energy extraction process for the case of neutral particle (i.e., $q/m=0$) first increases and then begins to decrease with rise in the value of $B$ field, in contrast to Kerr black hole where it always increases as the rotation parameter grows. However, for the case of charged particle (i.e., $q\ensuremath{\ne}0$) the efficiency always increases with the rise in $B$ field and can go over 100%, when both $B$ and $q/m$ are large enough (say $B\ensuremath{\approx}1$ and $q/m>2.2$). It is worth noting that the existence of regions away from the horizon where ${g}_{tt}>0$ also favors the energy-extraction process away from effect of the black hole. However, the energy extraction from these regions is pure consequence of the magnetic field.

In the Penrose process and the Blandford-Znajek mechanism, the rotational energy of a black hole (BH) is extracted via particle fission and magnetic tension, respectively. Recently, inspired by a fundamental trait in plasma astrophysics known as magnetic reconnection (MR), a new energy extraction mechanism based on the fast reconnection of the magnetic field lines inside the ergosphere has been proposed by Comisso and Asenjo. In this paper, we investigate energy extraction caused by MR in the ergosphere of a rapidly spinning BH with broken Lorentz symmetry by a background bumblebee vector field. The desired rotating BH solution differentiates from the standard Kerr BH via the Lorentz symmetry breaking (LSB) parameter $l$, which comes from nonminimal coupling between the bumblebee field with nonzero vacuum expectation value and gravity. We find that incorporating $l<0$ in the background is in the interest of the energy extraction via MR for the fast-spinning BH surrounded by the plasma with weak magnetization, below what is expected from the scenario by Comisso and Asenjo. Our analysis robustly indicates that the power of energy extraction and efficiency of the plasma energization process through fast MR is more efficient than the Comisso-Asenjo solution, provided that the LSB parameter is negative, $l<0$. Compared to the Blandford-Znajek mechanism arising from the underlying background, we also show that MR is a more efficient energy extraction mechanism if $l<0$.

In the present work, we explore several gravitational aspects such as energy extraction (via the Penrose process and superradiance), particle collisions around a , U(1)2 rotating dyonic black hole (BH) in the gauged supergravity model. The influence of the rotation parameter (a) and the gauge coupling constant (g) on the behavior of the horizon and ergoregion of the BH is investigated. In comparison to the extremal Kerr BH, the gauge coupling constant, under certain constraints, can interestingly enhance the maximum efficiency of energy extraction through the Penrose process by almost twice. Under the same constraints, we can extract approximately 60.75% of the initial mass-energy from the BH which is noticeably higher and far different from that of the Kerr BH. The limit of energy extraction in terms of the local speeds of the fragments is also determined with the help of the Wald inequality. We discover an upper limit on the gauge coupling constant up to which superradiance is likely to occur. Finally, we estimate the center-of-mass energy (E CM) of two particles with the same rest mass moving in the equatorial plane of the BH. Our study also aims to sensitize E CM to the parameters a and g for both extremal and nonextremal spacetime. Especially, for the extremal case, an infinitely large amount of E CM can be achieved closer to the event horizon confirming a basic point of view that an extreme supergravity BH with dyons could serve as ultimate particle accelerators as compared to Kerr and any other generalized BHs in this family and other alternative theories of gravity. However, E CM for the nonextremal spacetime is shown to be finite and has an upper bound.

Extracting the rotational energy from a Kerr black hole (BH) is one of the crucial topics in relativistic astrophysics. Here, we give special attention to the Penrose ballistic process based on the fission of a massive particle μ_{0} into two particles μ_{1} and μ_{2}, occurring in the ergosphere of a Kerr BH. Bardeen etal. indicated that for the process to occur, some additional "hydrodynamical forces or superstrong radiation reactions" were needed. Wald and Chandrasekhar further expanded this idea. This animosity convinced Piran and collaborators to move from a simple three-body system characterizing the original Penrose process to a many-body system. This many-body approach was further largely expanded by others, some questionable in their validity. Here, we return to the simplest original Penrose process and show that the solution of the equations of motion, imposing the turning point condition on their trajectories, leads to the rotational energy extraction from the BH expected by Penrose. The efficiency of energy extraction by a single process is quantified for three different single decay processes occurring, respectively, at r=1.2M, r=1.5M, and r=1.9M. An interesting repetitive model has been proposed by Misner etal. [Gravitation (W. H. Freeman, San Francisco, 1973)]. Indeed, it would appear that a repetitive sequence of 246 decays of the above injection process at r=1.2M and the corresponding ones at r=1.5M and r=1.9M could extract 100% of the rotational energy of the BH, so violating energy conservation. The accompanying article, accounting for the existence of the BH irreducible mass, introduces a nonlinear approach that avoids violating energy conservation and leads to a new energy extraction process.

Black hole physics has been one of the most active areas of research in general relativity. A great deal of information has been gathered on the structure of black holes and physical phenomena that take place in their spacetimes. These spacetimes, such as those associated with the Schwarzschild and Kerr black holes, are time-independent and asymptotically flat. Time symmetry is equivalent to the requirement that the spacetime admit a global timelike Killing vector field. In a totally realistic model, however, the black hole should be imbedded in or associated with a cosmological background. In such a scenario, neither of the above two conditions would be valid. Being part of an expanding universe, the black hole would cease to be time independent, i.e., the spacetime will no longer admit a timelike Killing vector field. Furthermore, spacetime would become cosmological and non-flat at large distances from the black hole. Very little has been done in exploring such black holes. It is not at all unlikely that the structure and properties of these black holes may differ significantly, or even drastically, from the ones that have been studied. Even in the case of the latter it is well known that the introduction of rotation, i.e., the passage from the non-rotating, spherically symmetric Schwarzschild to the rotating Kerr black hole, brings about profound changes. For instance, in the case of the Schwarzschild black hole the timelike Killing vector becomes null (static limit) on the black hole which is itself a null surface (Killing event horizon) [1]. On the other hand, in the case of the Kerr black hole the stationary limit at which the timelike Killing vector becomes null does not coincide with the event horizon which is required to be a null surface. However, Kerr spacetime admits a globally hypersurface orthogonal, irrotational timelike vector field which does become null on the event horizon[2]. The separation of the stationary limit from the event horizon and the consequent existence of the ergosphere in between lead to several interesting phenomena such as the Penrose process and superradiance. Similarly, phenomena that occur in the Schwarzschild spacetime may not take place in the Kerr spacetime, for instance the generation of gravitational synchrotron radiation. In the same manner, the introduction of the cosmic background may radically transform the physics of black holes.

In the case involving particles the necessary and sufficient condition for the Penrose process to extract energy from a rotating black hole is absorption of particles with negative energies and angular momenta. No torque at the black-hole horizon occurs. In this article we consider the case of arbitrary fields or matter described by an unspecified, general energy-momentum tensor $T_{\mu \nu}$ and show that the necessary and sufficient condition for extraction of a black hole's rotational energy is analogous to that in the mechanical Penrose process: absorption of negative energy and negative angular momentum. We also show that a necessary condition for the Penrose process to occur is for the Noether current (the conserved energy-momentum density vector) to be spacelike or past directed (timelike or null) on some part of the horizon. In the particle case, our general criterion for the occurrence of a Penrose process reproduces the standard result. In the case of relativistic jet-producing "magnetically arrested disks" we show that the negative energy and angular-momentum absorption condition is obeyed when the Blandford-Znajek mechanism is at work, and hence the high energy extraction efficiency up to $\sim 300\%$ found in recent numerical simulations of such accretion flows results from tapping the black hole's rotational energy through the Penrose process. We show how black-hole rotational energy extraction works in this case by describing the Penrose process in terms of the Noether current.

Recently, references [1, 2] found that the repetitive Penrose process cannot extract all the extractable rotational energy of a Kerr black hole, and reference [3] found that the repetitive electric Penrose process cannot extract all the electrical energy of a Reissner-Nordström (RN) black hole. In this paper, we intend to study the repetitive Penrose process in the Kerr-de Sitter (Kerr-dS) black hole. We will explore influences of the cosmological parameter on the repetitive Penrose process. The results show that, apart from the comparable features documented earlier, the Kerr-dS black hole yields a higher energy return on investment (EROI) and single-extraction energy capability compared to the Kerr black hole. Specifically, the larger the cosmological parameter, the stronger the EROI and the single-extraction energy capability. Furthermore, we also find that at a lower decay radius, the Kerr black hole exhibits a higher energy utilization efficiency (EUE) and more extracted energy after the repetitive Penrose process is completed. However, at a higher decay radius, the situation is reversed, i.e., the Kerr-dS black hole exhibits a higher EUE and more extracted energy, which is due to the existence of stopping condition of the iteration.

Circular motion and energy extraction in a rotating black hole

High-energy astrophysical sources such as active galactic nuclei, quasars, X-ray binaries, and gamma-ray bursts are powered by mechanisms that convert gravitational or rotational energy into radiation, jets, and relativistic outflows. Understanding the physics of these processes remains a major challenge. Black holes have traditionally served as the central engines behind such phenomena, with well-established energy extraction mechanisms including the Penrose process, the Blandford–Znajek process, and the Banados–Silk–West mechanism. However, studies in general relativity indicate that, under certain conditions, gravitational collapse may lead to the formation of horizonless compact objects (HCOs), which could in principle allow more efficient energy extraction than classical black holes. This brief review summarizes recent progress on energy extraction mechanisms in HCO spacetimes. We examine the roles of rotation, electromagnetic fields, and particle interactions in shaping extraction efficiency and dynamics. Particular attention is given to negative energy orbits and ergoregion physics, which enable Penrose type and magnetic Penrose mechanisms without an event horizon. We also discuss collisional Penrose processes and particle acceleration near the horizonless singularity, emphasizing their potential astrophysical implications. By comparing extraction efficiencies and physical conditions in black holes and HCOs, we highlight how the absence of a horizon fundamentally alters the dynamics of energy release. These results suggest that HCOs may serve as natural laboratories for strong field gravity and as alternative engines for high-energy astrophysical phenomena in the era of multi-messenger observations.

In this paper, the Penrose process is used to extract rotational energy from regular black holes. Initially, we consider the rotating Simpson–Visser regular spacetime, which describes the class of geometries of Kerr black hole mimickers. The Penrose process is then studied through conformally transformed rotating singular and regular black hole solutions. Both the Simpson–Visser and conformally transformed geometries depend on mass, spin, and an additional regularisation parameter l. In both cases, we investigate how the spin and regularisation parameter l affect the configuration of an ergoregion and event horizons. Surprisingly, we find that the energy extraction efficiency from the event horizon surface is not dependent on the regularisation parameter l in the Simpson–Visser regular spacetimes, and hence, it does not vary from that of the Kerr black hole. Meanwhile, in conformally transformed singular and regular black holes, we obtain that the efficiency rate of extracted energies is extremely high compared to that of the Kerr black hole. This distinct signature of conformally transformed singular and regular black holes is useful to distinguish them from Kerr black holes in observation.

We analyze the extraction of the rotational energy of a Kerr black hole (BH) endowed with a test charge and surrounded by an external test magnetic field and ionized low-density matter. For a magnetic field parallel to the BH spin, electrons move outward(inward) and protons inward(outward) in a region around the BH poles(equator). For zero charge, the polar region comprises spherical polar angles -60∘≲θ≲60∘ and the equatorial region 60∘≲θ≲120∘. The polar region shrinks for positive charge, and the equatorial region enlarges. For an isotropic particle density, we argue the BH could experience a cyclic behavior: starting from a zero charge, it accretes more polar protons than equatorial electrons, gaining net positive charge, energy and angular momentum. Then, the shrinking(enlarging) of the polar(equatorial) region makes it accrete more equatorial electrons than polar protons, gaining net negative charge, energy, and angular momentum. In this phase, the BH rotational energy is extracted. The extraction process continues until the new enlargement of the polar region reverses the situation, and the cycle repeats. We show that this electrodynamical process produces a relatively limited increase of the BH irreducible mass compared to gravitational mechanisms like the Penrose process, hence being a more efficient and promising mechanism for extracting the BH rotational energy.

In this paper, we explore the geodesics motion of neutral test particles and the process of energy extraction from a regular rotating Hayward black hole (BH). We analyze the effect of spin as well as deviation parameter [Formula: see text], on ergoregion, event horizon and static limit of the said BH. By making use of geodesic equations on the equatorial plane, we determine the innermost stable circular and photon orbits. Moreover, we investigate the effective potentials and effective force to have information on motion and the stability of circular orbits. On studying the negative energy states, we figure out the energy limits of Penrose mechanism. Using Penrose mechanism, we found expression for the efficiency of energy extraction and observed that both spin and deviation parameters contribute to the efficiency of energy extraction. Finally, the obtained results are compared with that acquired from Kerr and braneworld Kerr BHs.