arXiv subjects

José P. S. Lemos

Publications and source records attributed to José P. S. Lemos.

At least 19 recordsLinked to original sources

arXiv2026

Quasinormal modes of Proca and Maxwell fields in $d$-dimensional Schwarzschild-AdS black holes

Proca and Maxwell fields in $d$-dimensional Schwarzschild black holes with anti-de Sitter (AdS) asymptotics are investigated through their linear perturbations and associated quasinormal modes (QNMs) with Dirichlet boundary conditions at infinity. The Proca field equations reduce to one decoupled and two coupled radial wave-like equations. We demonstrate how the Maxwell equations emerge from the zero-mass limit of the Proca system. Several analytical properties of the corresponding QNM spectrum are examined. To compute the QNM frequencies, we employ two complementary numerical methods particularly suited to asymptotically AdS spacetimes. Using these techniques, we determine the QNMs modes of Proca field perturbations in $4$, $5$, $6$, and $7$-dimensional Schwarzschild-AdS backgrounds. As a new result, we find numerically that scalar-type Maxwell perturbations in large $d\geq 5$ Schwarzschild-AdS black holes exhibit purely imaginary low-frequency modes, analogous to those found in vector-type gravitational perturbations. The presence of such modes is especially relevant within the AdS/CFT correspondence, as they correspond to the linearized hydrodynamic regime in the dual conformal field theory. We also analyze the influence of the Proca mass on the QNM spectrum, also emphasizing how Maxwell modes are recovered in the massless limit. The dependence of the spectrum on the black hole radius is explored. In addition, analytic expressions for the QNM frequencies of vector-type and monopole Proca perturbations, as well as Maxwell modes, are derived for small $d$-dimensional Schwarzschild-AdS black holes by matching asymptotic expansions using an intermediate region. These analytic results show good agreement with the numerical findings, confirming, in particular, the existence of purely imaginary low-frequency scalar-type Maxwell modes in large $d\geq 5$ Schwarzschild-AdS spacetimes.

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arXiv2026

Recursive Penrose processes in electrically charged black hole spacetimes: Backreaction and energy extraction

We study a recursive Penrose process and the energy extraction for the decay of electrically charged particles in a Reissner-Nordström black hole spacetime with anti-de Sitter (AdS) asymptotics, incorporating the backreaction on the black hole's mass and charge. A recursive process requires that the decay products are confined in a finite region so that the emitted particles bounce back for further decay. In AdS spacetimes, the confinement arises naturally. Outgoing particles encounter a turning point and are reflected. One may impose a mirror at finite radius, but in AdS, backreaction makes these two confinement methods equivalent. Let $Q_n$ be the black hole charge after $n$ decays, and define $n_{\rm c}$ as the index for which the black hole's charge is zero, $Q_{n_{\rm c}}=0$. For $n_{\rm c}$ integer the black hole's charge decreases and reaches exactly zero after a finite number of decays, terminating the process. However, the last particle turns back, and encountering zero charge, falls into the hole. The final state is a charged black hole whose charge equals the sum of the original black hole and the initial particle charges. For $n_{\rm c}$ noninteger, the black hole charge decreases and can be arbitrarily small, but is never zero. The last allowed decay occurs at $n=n_{c}^-$, where $n=n_{c}^-$ is the greatest integer less than $n_{\rm c}$. Any further decay invalidates the approximations, the particles would carry a charge comparable to the black hole mass, transforming the problem into a two-body problem. The would-be subsequent decay would violate cosmic censorship and the process terminates before any inconsistency arises. In the integer and noninteger cases, the system yields a finite energy gain. Backreaction ensures that the process extracts a finite amount of energy. No black hole bomb occurs, the system works at most as an energy factory.

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arXiv2026

The canonical ensemble of a self-gravitating matter thin shell in AdS

We build the canonical ensemble of a hot self-gravitating matter thin shell in anti-de Sitter (AdS) space by finding its partition function through the Euclidean path integral approach with fixed temperature at the conformal boundary. We obtain the reduced action of the system by restricting the path integral to spherically symmetric metrics with given boundary conditions and with the Hamiltonian constraint satisfied. The stationary conditions, i.e., the mechanical equilibrium and the thermodynamic equilibrium, are obtained from minimizing the reduced action. Evaluating the perturbed reduced action at the stationary points yields the mechanical stability condition and the thermodynamic stability condition. The reduced action calculated at the stationary points gives the partition function in the zero-loop approximation and from it the thermodynamic properties of the system are acquired. Within thermodynamics alone, the only stability condition that one can establish is thermodynamic stability, which follows from the computation of the heat capacity. For given specific pressure and temperature equations of state for the shell, we obtain the solutions of the ensemble. There are four different thin shell solutions, one of them is fully stable, i.e., is stable mechanically and thermodynamically. For the equations of state given, we find a first order phase transition from the matter thermodynamic phase to the Hawking-Page black hole phase. Moreover, there is a maximum temperature above which the shell ceases to exist, presumably at these high temperatures the shell inevitably collapses to a black hole.

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arXiv2025

When spacetime vibrates: An introduction to gravitational waves

This article presents a comprehensive analysis of the physics of gravitational waves, exploring both the theoretical foundations and the most recent experimental advances. After a general introduction to the theory of general relativity and its major implications, the article discusses the history of gravitational waves, from their prediction by Einstein to their actual detection. It then explains what gravitational waves are and how they interact with appropriate detectors. The main mechanisms of gravitational radiation emission are analyzed, with a focus on compact binary systems of compact objects, whose orbits typically evolve in three phases: inspiral, merger, and the final ringdown phase, each of these phases leaving distinct signatures in the emitted waves. The article highlights the fundamental role of the giant interferometers LIGO, Virgo, and KAGRA, true cathedrals of modern science, and revisits the historic event GW150914, the first direct detection of gravitational waves, which confirmed the predictions of general relativity and opened a new era for astronomy. This achievement was recognized with the 2017 Nobel Prize in Physics. Other observed events are also discussed, along with their astrophysical sources, and the possibility of detecting gravitational waves of cosmological origin, originating from the Big Bang itself. Finally, current and future projects are analyzed, including observatories based on increasingly sophisticated interferometers, as well as proposals for alternative detection methods, illustrating how gravitational-wave astronomy is shaping the present and future of our exploration of the universe. In concluding, the detection of gravitational waves is set in a broader context by examining the discoveries across the electromagnetic spectrum, thereby illustrating the complementary perspectives these different observational channels provide.

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arXiv2025

Gibbons-Hawking action for electrically charged black holes in the canonical ensemble and Davies' thermodynamic theory of black holes

We establish the connection between the Gibbons-Hawking Euclidean path integral approach applied to the canonical ensemble of a Reissner-Nordström black hole and the thermodynamic theory of black holes of Davies. We build the ensemble, characterized by a reservoir at infinity at temperature $T$ and electric charge $Q$, in $d$ dimensions. The Euclidean path integral yields the action and partition function. In zero loop, we uncover two solutions, one with horizon radius $r_{+1}$ the least massive, the other with $r_{+2}$, both meeting at a saddle point with radius $r_{+s}$ at temperature $T_s$. We derive the thermodynamics, finding that the heat capacity diverges at the turning point $T_s$ for each solution. The free energy of the stable solution is positive, so if the system is a black hole it makes a transition to hot flat space with charge at infinity. For a given $Q$ and $T>T_s$, there is only hot space. An interpretation of the results as energy wavelengths is attempted. For $d=4$, the thermodynamics from the path integral applied to the canonical ensemble is precisely the Davies thermodynamics theory of black holes, with $T_s$ being the Davies point. We sketch the case $d=5$.

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arXiv2025

Canonical ensemble of a $d$-dimensional Reissner-Nordström black hole in a cavity

We construct the canonical ensemble of a $d$-dimensional Reissner-Nordström black hole spacetime in a cavity surrounded by a heat reservoir through the Euclidean path integral formalism. The heat reservoir is described by the boundary of the cavity with fixed radius $R$, fixed temperature $T$, and fixed electric charge $Q$. We use York's approach to find the reduced action, and then perform a zero loop approximation. We find that the number of solutions for the black hole depends on the electric charge being smaller or larger than a critical $Q_s$, having two stable and one unstable solutions for the former case, and one stable solution for the latter. We obtain the system's thermodynamic properties from the partition function. We analyze thermodynamic stability, controlled by the positivity of the heat capacity at constant area and electric charge. We show that there is a discontinuity in the heat capacity, signaling a turning point. We investigate the favorable stable phases and the phase diagram of the system. We show that the two stable black hole solutions can suffer a first order phase transition from one to the other, and at the critical charge $Q_s$ this turns into a second order one. We introduce a model of charged hot flat space, i.e., hot flat space with charge near the boundary. We find that a first order phase transition between the large stable black hole and charged hot flat space occurs at a horizon radius larger than the Buchdahl bound, and comment on the physics. Finally, we recover the Davies thermodynamic solutions and a Rindler solution in the limit of infinite cavity, and the York solutions in the limit of zero charge. Hence, both York and Davies formalisms are unified and connected in our approach. In all instances we mention carefully the four-dimensional case, for which we accomplish new results, and study in detail all aspects of the five-dimensional case.

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arXiv2024

Penrose and super-Penrose energy extraction from a Reissner-Nordström black hole spacetime with a cosmological constant through the BSW mechanism: Full story

The Penrose process, a process that transfers energy from a black hole to infinity, together with the BSW mechanism, which uses collisions of ingoing particles at the event horizon of a black hole to locally produce large amounts of energy, is studied in a combined description for a $d$ dimensional extremal Reissner-Nordström black hole spacetime with negative, zero, or positive cosmological constant, i.e., for an asymptotically anti-de Sitter (AdS), flat, or de Sitter (dS) spacetime. In an extremal Reissner-Nordström black hole background, in the vicinity of the horizon, several types of radial collisions between electrically charged particles can be considered. The most interesting one is between a critical particle, with its electric charge adjusted in a specific way, and a usual particle, as it gives a divergent center of mass frame energy locally, this being a favorable but not sufficient condition to extract energy from the black hole. To understand whether energy can be extracted in such a collisional Penrose process, we investigate in detail a collision between ingoing particles 1 and 2, from which particles 3 and 4 emerge, with the possibility that particle 3 can carry energy far out from the black hole horizon. One finds that the mass, energy, electric charge, and initial direction of motion of particle 3 can have different values, depending on the collision internal process, but these values lie within some range. Moreover, the energy of particle 3 can be arbitrarily high but not infinite, characterizing a super-Penrose process. It is also shown that particle 4 has negative energy, living in its own electric ergosphere before being engulfed by the event horizon. For zero cosmological constant the results do not depend on the number of dimensions, but they do for nonzero cosmological constant, which also introduces differences in the lower bound for the energy extracted.

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arXiv2024

Normal modes of Proca fields in AdS$_d$ spacetime

The normal modes of Proca field perturbations in $d$-dimensional anti-de Sitter spacetime, AdS$_d$ for short, with reflective Dirichlet boundary conditions, are obtained exactly. Within the Ishibashi-Kodama framework, we decompose the Proca field in scalar-type and vector-type components, according to their tensorial behavior on the $(d-2)$-sphere $\mathcal{S}^{d-2}$. Two of the degrees of freedom of the Proca field are described by scalar-type components, which in general are coupled due to the mass of the field, but in AdS$_d$ we show that they can be decoupled. The other $d-3$ degrees of freedom of the field are described by a vector-type component that generically decouples completely. The normal modes and their frequencies for both the scalar-type and vector-type components of the Proca field are then obtained analytically. Additionally, we analyze the normal modes of the Maxwell field as the massless limit of the Proca field. We find that for scalar-type perturbations in $d=4$ there is a discontinuity in the massless limit, in $d=5$ the massless limit is well defined using Dirichlet-Neumann rather than Dirichlet boundary conditions, and in $d>5$ the massless limit is completely well defined, i.e., it is obtained smoothly from the massless limit of the scalar-type perturbations of the Proca field. For vector-type perturbations the Maxwell field limit is obtained smoothly for all $d$ from the massless limit of the vector-type perturbations of the Proca field.

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arXiv2024

Hot spaces with positive cosmological constant in the canonical ensemble: de Sitter solution, Schwarzschild-de Sitter black hole, and Nariai universe

In a space with positive cosmological constant $Λ$, we consider a black hole surrounded by a heat reservoir at radius $R$ and temperature $T$, i.e., we analyze the Schwarzschild-de Sitter black hole in a cavity. We use the Euclidean path integral approach to quantum gravity to study its canonical ensemble and thermodynamics. We give the action, energy, entropy, temperature, and heat capacity. $T$, $Λ$, the black hole radius $r_+$, and the cosmological horizon radius $r_{\rm c}$, are gauged in $R$ units to $RT$, $ΛR^2$, $\frac{r_+}{R}$, and $\frac{r_{\rm c}}{R}$. The whole extension of $ΛR^2$, $0\leqΛR^2\leq 3$, is divided into three ranges. The first, $0\leqΛR^2<1$, includes York's Schwarzschild black holes. The second range, $ΛR^2=1$, opens up a folder of Nariai universes. The third range, $1<ΛR^2\leq 3$, is unusual. One feature here is that it interchanges the cosmological horizon with the black hole horizon. The end point, $ΛR^2=3$, only existing for infinite $RT$, is a cavity filled with de Sitter space, except for a singularity, with the cosmological horizon coinciding with the reservoir. For the three ranges, for low temperatures, there are no black holes and no Nariai universes, the space is hot de Sitter. The value of $RT$ that divides the nonexistence from existence of black holes or Nariai universes, depends on $ΛR^2$. For each $ΛR^2\neq1$, for high temperatures, there is one small and thermodynamically unstable black hole, and one large and stable. For $ΛR^2=1$, for high temperatures, there is the unstable black hole, and the neutrally stable Nariai universe. Phase transitions can be analyzed. The transitions are between the black hole and hot de Sitter and between Nariai and hot de Sitter. The Buchdahl radius, the radius for collapse, plays an interesting role in the analysis.

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arXiv2024

Penrose process in Reissner-Nordström-AdS black hole spacetimes: Black hole energy factories and black hole bombs

The Penrose process for the decay of electrically charged particles in a Reissner-Nordström-anti-de Sitter black hole spacetime is studied. To extract large quantities of energy one needs to mount a recursive Penrose process where particles are confined and can bounce back to suffer ever again a decaying process in the black hole electric ergoregion. In an asymptotically anti-de Sitter (AdS) spacetime, two situations of confinement are possible. One situation uses a reflecting mirror at some radius, which obliges the energetic outgoing particles to return to the decaying point. The other situation uses the natural AdS property that sends back at some intrinsic returning radius those outgoing energetic particles. In addition, besides the conservation laws the decaying process must obey, one has to set conditions at the decaying point for the particles debris. These conditions restrain the possible scenarios, but there are still a great number of available scenarios for the decays. Within these, we choose two scenarios, scenario 1 and scenario 2, that pertain to the masses and electric charges of the final particles. Thus, in the mirror situation we find that scenario 1 leads to a black hole energy factory, and scenario 2 ends in a black hole bomb. In the no mirror situation, i.e., pure Reissner-Nordström-AdS, scenario 1 leads again to a black hole energy factory, but scenario 2 yields no bomb. This happens because the volume in which the particles are confined increases to infinity along the chain of decays, leading to a zero value of the extracted energy per unit volume and the bomb is demined. The whole treatment performed here involves no backreaction on the black hole mass and electric charge, nevertheless we speculate that the end state of the recursive process is a Reissner-Nordström-AdS black hole with very short hair, i.e., with one particle at rest at some definite radius.

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arXiv2023

Grand canonical ensemble of a $d$-dimensional Reissner-Nordström black hole in a cavity

The grand canonical ensemble of a $d$-dimensional Reissner-Nordström black hole space in a cavity is analyzed. The realization of this ensemble is made through the Euclidean path integral approach by giving the Euclidean action for the black hole with the correct topology, and boundary conditions corresponding to a cavity, where the fixed quantities are the temperature and the electric potential. One performs a zero loop approximation to find and analyze the stationary points of the reduced action. This yields two solutions for the electrically charged black hole, $r_{+1}$, which is the smaller and unstable, and $r_{+2}$, which is the larger and stable. One also analyzes the most probable configurations, which are either a stable charged black hole or hot flat space, mimicked by a nongravitating charged shell. Making the correspondence between the action and the grand potential, one can get the black hole thermodynamic quantities, such as the entropy, the mean charge, the mean energy, and the thermodynamic pressure, as well as the Smarr formula, shown to be valid only for the unstable black hole. We find that thermodynamic stability is related to the positivity of the heat capacity at constant electric potential and area of the cavity. We also comment on the most favorable thermodynamic phases and phase transitions. We then choose $d = 5$, which is singled out naturally from the other higher dimensions as it provides an exact solution for the problem, and apply all the results previously found. The case $d = 4$ is mentioned. We compare thermodynamic radii with the photonic orbit radius and the Buchdahl-Andréasson-Wright bound radius in $d$-dimensional Reissner-Nordström spacetimes and find they are unconnected, showing that the connections displayed in the Schwarzschild case are not generic, rather they are very restricted holding only in the pure gravitational situation.

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arXiv2023

Traversable wormholes with double layer thin shells in quadratic gravity

In quadratic gravity, the junction conditions are six and permit the appearance of double layer thin shells. Double layers arise typically in theories with dipoles, i.e., two opposite charges, such as electromagnetic theories, and appear exceptionally in gravitational theories, which are theories with a single charge. We explore this property of the existence of double layers in quadratic gravity to find and study traversable wormholes in which the two domains of the wormhole interior region, where the throat is located, are matched to two vacuum domains of the exterior region via the use of two double layer thin shells. The quadratic gravity we use is essentially given by a $R+αR^2$ Lagrangian, where $R$ is the Ricci scalar of the spacetime and $α$ is a coupling constant, plus a matter Lagrangian. The null energy condition, or NEC for short, is tested for the whole wormhole spacetime. The analysis shows that the NEC is satisfied for the stress-energy tensor of the matter in the whole wormhole interior region, notably at the throat, and is satisfied for some of the stress-energy tensor components of the matter at the double layer thin shell, but is not satisfied for some other components, namely, the double layer stress-energy distribution component, at the thin shell. This seems to mean that the NEC is basically impossible, or at least very hard, to be satisfied when double layer thin shells are present. Single layer thin shells are also admitted within the theory, and we present thin shell traversable wormholes, i.e., wormholes without interior, with a single layer thin shell at the throat for which the corresponding stress-energy tensor satisfies the NEC, that are asymmetric, i.e., with two different vacuum domains of the exterior region joined at the wormhole throat.

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arXiv2023

Black holes and hot shells in the Euclidean path integral approach to quantum gravity

We study a spherical black hole surrounded by a hot self-gravitating thin shell in the canonical ensemble, i.e., a black hole and a hot shell inside a heat reservoir acting as a boundary with its area and temperature fixed. To work out the quantum partition function, from which the thermodynamics of the system follows, we use the Euclidean path integral approach to quantum gravity that identifies the path integral of the gravitational system with the partition function. In a semiclassical approximation, one needs only to compute the classical action of the system. Then, one finds that the total entropy, i.e., the sum of black hole and matter entropies, is a function of the gravitational radius of the system alone. So, the black hole inside the shell has no direct influence on the entropy. One also finds the free energy, the thermodynamic energy, and the temperature stratification. The reservoir temperature is composed of a free function of the gravitational radius of the system divided by the redshift. Upon specification of the reduced temperature free function, the solutions for the gravitational radii compatible with the data are found. The black hole inside has two possible horizon radii. It is shown that there is a first law of thermodynamics for the system, another for the hot shell, and yet another for the black hole. A thermodynamic stability analysis is performed. By specifying for the free function the Hawking temperature for the gravitational radius of the system, which is not a black hole, one finds a remarkable exact thermodynamic solution. With it one establishes that pure black holes, hot shells with a black hole, pure hot shells, and hot flat spaces are phases that cohabit in the ensemble, with some acting as thermodynamic mimickers. This exact solution is a model to situations involving black holes and hot gravitons. The high temperature limits reveal important aspects.

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arXiv2023

Relativistic cosmology and intrinsic spin of matter: Results and theorems in Einstein-Cartan theory

We start by presenting the general set of structure equations for the 1+3 threading spacetime decomposition in 4 spacetime dimensions, valid for any theory of gravitation based on a metric compatible affine connection. We then apply these equations to the study of cosmological solutions of the Einstein-Cartan theory in which the matter is modeled by a perfect fluid with intrinsic spin. It is shown that the metric tensor can be described by a generic FLRW solution. However, due to the presence of torsion the Weyl tensors might not vanish. The coupling between the torsion and Weyl tensors leads to the conclusion that, in this cosmological model, the universe must either be flat or open, excluding definitely the possibility of a closed universe. In the open case, we derive a wave equation for the traceless part of the magnetic part of the Weyl tensor and show how the intrinsic spin of matter in a dynamic universe leads to the generation and emission of gravitational waves. Lastly, in this cosmological model, it is found that the torsion tensor, which has an intrinsic spin as its source, contributes to a positive accelerated expansion of the universe. Comparing the theoretical predictions of the model with the current experimental data, we conclude that torsion cannot completely replace the role of a cosmological constant.

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arXiv2023

Normal modes of Proca fields in AdS spacetime

A normal mode analysis for Proca fields in the anti-de Sitter (AdS) spacetime is given. It is found that the equations for the Proca field can be decoupled analytically. This is performed by changing the basis of the vector spherical harmonics (VSH) decomposition. The normal modes and the normal mode frequencies of the Proca equation in the AdS spacetime are then analytically determined. It is also shown that the Maxwell field can be recovered by taking the massless limit of the Proca field with care so that the nonphysical gauge modes are eliminated.

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arXiv2023

Stability of electrically charged stars, regular black holes, quasiblack holes, and quasinonblack holes

The stability of a class of electrically charged fluid spheres under radial perturbations is studied. Among these spheres there are regular stars, overcharged tension stars, regular black holes, quasiblack holes, and quasinonblack holes, all of which have a Reissner-Nordström exterior. We formulate the dynamical perturbed equations by following the Chandrasekhar approach and investigate the stability against radial perturbations through numerical methods. It is found that (i) under certain conditions that depend on the adiabatic index of the radial perturbation, there are stable charged stars and stable tension stars; (ii) also depending on the adiabatic index there are stable regular black holes; (iii) quasiblack hole configurations formed by, e.g., charging regular pressure stars or by discharging regular tension stars, can be stable against radial perturbations for reasonable values of the adiabatic index; (iv) quasinonblack holes are unstable against radial perturbations.

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arXiv2022

Electrically charged spherical matter shells in higher dimensions: Entropy, thermodynamic stability, and the black hole limit

We study the thermodynamic properties of a static electrically charged spherical thin shell in $d$ dimensions by imposing the first law of thermodynamics on the shell. The shell is at radius $R$, inside it the spacetime is Minkowski, and outside it the spacetime is Reissner-Nordström. We obtain that the shell thermodynamics is fully described by giving two additional reduced equations of state, one for the temperature and another for the electrostatic potential. We choose the equation of state for the temperature as a power law in the gravitational radius $r_+$ with exponent $a$, such that the $a=1$ case gives the temperature of a shell with black hole thermodynamic properties, and for the electrostatic potential we choose an equation of state characteristic of a Reissner-Nordström black hole spacetime. The entropy of the shell is found to be proportional to $A_+^a$, where $A_+$ is the gravitational area corresponding to $r_+$, with $a>0$. We are then able to perform the black hole limit $R=r_+$, find the Smarr formula, and recover the thermodynamics of a $d$-dimensional Reissner-Nordström black hole. We study the intrinsic thermodynamic stability of the shell with the chosen equations of state. We obtain that for $0 1$ all the configurations are unstable, except for the shell at its own gravitational radius, which is marginally stable. We rewrite the stability conditions in terms of laboratory variables. We find that the sufficient condition for the stability of these shells is when the isothermal electric susceptibility $χ_{p,T}$ is positive, marginal stability happening when this quantity is infinite, and instability arising for configurations with a negative electric susceptibility.

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arXiv2022

Extraction of energy from an extremal rotating electrovacuum black hole: Particle collisions in the equatorial plane

The collisional Penrose process received much attention when Banados, Silk and West (BSW) pointed out the possibility of test-particle collisions with arbitrarily high center-of-mass energy in the vicinity of the horizon of an extremally rotating black hole. However, the energy that can be extracted from the black hole in this promising, if simplified, scenario, called the BSW effect, turned out to be subject to unconditional upper bounds. And although such bounds were not found for the electrostatic variant of the process, this version is also astrophysically unfeasible, since it requires a maximally charged black hole. In order to deal with these deficiencies, we revisit the unified version of the BSW effect concerning collisions of charged particles in the equatorial plane of a rotating electrovacuum black hole spacetime. Performing a general analysis of energy extraction through this process, we explain in detail how the seemingly incompatible limiting cases arise. Furthermore, we demonstrate that the unconditional upper bounds on the extracted energy are absent for arbitrarily small values of the black hole electric charge. Therefore, our setup represents an intriguing simplified model for possible highly energetic processes happening around astrophysical black holes, which may spin fast but can have only a tiny electric charge induced via interaction with an external magnetic field.

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