Schwarzschild Black Hole Research Articles

Scrutinizing black hole stability in cubic vector Galileon theories

In a subclass of generalized Proca theories where a cubic vector Galileon term breaks the U(1) gauge invariance, it is known that there are static and spherically symmetric black hole (BH) solutions endowed with nonvanishing temporal and longitudinal vector components. Such hairy BHs are present for a vanishing vector-field mass (m=0) with a non-zero cubic Galileon coupling β 3. We study the linear stability of those hairy BHs by considering even-parity perturbations in the eikonal limit. In the angular direction, we show that one of the three dynamical perturbations has a nontrivial squared propagation speed c Ω,1 2, while the other two dynamical modes are luminal.We could detect two different unstable behaviors of perturbations in all the parameter spaces of hairy asymptotically flat BH solutions we searched for. In the first case, an angular Laplacian instability on the horizon is induced by negative c Ω,1 2. For the second case, it is possible to avoid this horizon instability, but in such cases, the positivity of c Ω,1 2 is violated at large distances. Hence these hairy BHs are generally prone to Laplacian instabilities along the angular direction in some regions outside the horizon. Moreover, we also encounter a pathological behavior of the radial propagation speeds c r possessing two different values of c r 2 for one of the dynamical perturbations. Introducing the vector-field mass m to cubic vector Galileons, however, we show that the resulting no-hair Schwarzschild BH solution satisfies all the linear stability conditions in the small-scale limit, with luminal propagation speeds of three dynamical even-parity perturbations.

Free-falling motion of an elastic rigid rod towards a Schwarzschild black hole

Abstract We study the motion of an elastic rigid rod which is radially free-falling towards a Schwarzschild black hole. This is accomplished by reducing the corresponding free-boundary partial differential equation problem to a sequence of ODEs, which we integrate numerically. Starting with a rod at rest, we show that it is possible to choose its initial compression profile so that its midpoint falls substantially faster, or slower, than a free-falling particle with the same initial conditions. This seems to be a purely kinematic effect, since on average there is no net transfer of elastic energy to mechanical energy.

Dual effect of string cloud and dark matter halos on particle motions, shadows and epicyclic oscillations around Schwarzschild black holes

In this work, we study the particle dynamics, shadows and quasi-periodic oscillations around the Schwarzschild black hole with three different dark matter halos and string cloud parameters. The motion of particles is considered with the string cloud parameter α and investigated for massive and massless particles. The photon orbits and inner stable circular orbits of these black holes are obtained from the effective potential. The black hole shadows are calculated using three different dark matter halos. The shadow radius of these black holes increases compared to a pure Schwarzschild black hole for different values of the string cloud parameter. Further, quasi-periodic oscillations are also discussed in the current analysis. Our study examines how the radial profiles of orbital and radial harmonic oscillation frequencies change with the string cloud parameter. This study examines the major properties of test particle's quasi-periodic oscillations near stable circular orbits in the black hole equatorial plane. Our findings indicate that observational tests for black holes influenced by dark matter halos and cloud strings are also feasible and viable.

The lensing effect of quantum-corrected black hole and parameter constraints from EHT observations

The quantum-corrected black hole model demonstrates significant potential in the study of gravitational lensing effects. By incorporating quantum effects, this model addresses the singularity problem in classical black holes. In this paper, we investigate the impact of the quantum correction parameter on the lensing effect based on the quantum-corrected black hole model. Using the black holes M87∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M87^*$$\\end{document} and SgrA∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Sgr A^*$$\\end{document} as our subjects, we explore the influence of the quantum correction parameter on angular position, Einstein ring, and time delay. Additionally, we use data from the Event Horizon Telescope observations of black hole shadows to constrain the quantum correction parameter. Our results indicate that the quantum correction parameter significantly affects the lensing coefficients a¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{a}$$\\end{document} and b¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{b}$$\\end{document}, as well as the Einstein ring. The position θ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta _{\\infty }$$\\end{document} and brightness ratio S of the relativistic image exhibit significant changes,with deviations on the order of magnitude of ∼1μas\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim 1\\,\\upmu \\! \ extrm{as}$$\\end{document} and ∼0.01μas\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim 0.01\\,\\upmu \\! \ extrm{as}$$\\end{document}, respectively. The impact of the quantum correction parameter on the time delay ΔT21\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta T_{21}$$\\end{document} is particularly significant in the M87∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M87^*$$\\end{document} black hole, with deviations reaching up to several tens of hours. Using observational data from the Event Horizon Telescope(EHT) of black hole shadows to constrain the quantum correction parameter, the constraint range under the M87∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M87^*$$\\end{document} black hole is 0≤αM2≤1.4087\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\le \\frac{\\alpha }{M^2}\\le 1.4087$$\\end{document} and the constraint range under the SgrA∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Sgr A^*$$\\end{document} black hole is 0.9713≤αM2≤1.6715\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.9713\\le \\frac{\\alpha }{M^2}\\le 1.6715$$\\end{document}. Although the current resolution of the EHT limits the observation of subtle differences, future high-resolution telescopes are expected to further distinguish between the quantum-corrected black hole and the Schwarzschild black hole, providing new avenues for exploring quantum gravitational effects.

A note on the area of event horizon of Kleinian black hole

We point out that the area of event horizon of Kleinian black hole is infinite due to the fact that its event horizon is not a sphere but a hyperboloid. Therefore, the usual interpretations of Schwarzschild black hole might not be applicable to the Kleinian black hole.

A New Gauge for Gravitational Perturbations of Kerr Spacetimes II: The Linear Stability of Schwarzschild Revisited

We present a new proof of linear stability of the Schwarzschild solution to gravitational perturbations. Our approach employs the system of linearised gravity in the new geometric gauge of Benomio (A new gauge for gravitational perturbations of Kerr spacetimes I: the linearised theory, 2022, https://arxiv.org/abs/2211.00602), specialised to the |a|=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|a|=0$$\\end{document} case. The proof fundamentally relies on the novel structure of the transport equations in the system. Indeed, while exploiting the well-known decoupling of two gauge invariant linearised quantities into spin ±2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pm 2$$\\end{document} Teukolsky equations, we make enhanced use of the red-shifted transport equations and their stabilising properties to control the gauge dependent part of the system. As a result, an initial-data gauge normalisation suffices to establish both orbital and asymptotic stability for all the linearised quantities in the system. The absence of future gauge normalisations is a novel element in the linear stability analysis of black hole spacetimes in geometric gauges governed by transport equations. In particular, our approach simplifies the proof of Dafermos et al. (Acta Math 222:1–214, 2019), which requires a future normalised (double-null) gauge to establish asymptotic stability for the full system.

Quasinormal modes and universality of the Penrose limit of black hole photon rings

We study the physics of photon rings in a wide range of axisymmetric black holes admitting a separable Hamilton-Jacobi equation for the geodesics. Utilizing the Killing-Yano tensor, we derive the Penrose limit of the black holes, which describes the physics near the photon ring. The obtained plane wave geometry is directly linked to the frequency matrix of the massless wave equation, as well as the instabilities and Lyapunov exponents of the null geodesics. Consequently, the Lyapunov exponents and frequencies of the photon geodesics, along with the quasinormal modes, can be all extracted from a Hamiltonian in the Penrose limit plane wave metric. Additionally, we explore potential bounds on the Lyapunov exponent, the orbital and precession frequencies, in connection with the corresponding inverted harmonic oscillators and we discuss the possibility of photon rings serving as effective holographic horizons in a holographic duality framework for astrophysical black holes. Our formalism is applicable to spacetimes encompassing various types of black holes, including stationary ones like Kerr, Kerr-Newman, as well as static black holes such as Schwarzschild, Reissner-Nordström, among others.

The shadow and gamma-ray bursts of a Schwarzschild black hole in asymptotic safety

Abstract The effects and rules of the dimensionless parameter $\xi$ on neutrino annihilation $\nu+\bar{\nu}\to e^{-}+e^{+}$ dominated gamma-ray bursts are analysed and investigated within the context of black holes in asymptotic safety. We also computationally model photon orbits around black holes, as photons and neutrinos have the same geodesic equations near black holes. We show that the black hole shadow radius decreases with increasing $\xi$. Calculations are made to determine the temperature of the accretion disk surrounding the black hole and the ratio $\dot{Q}/\dot{Q}_{Newt}$ of energy deposition per unit time to that of the Newtonian scenario. The accretion disk temperature peaks at a higher temperature due to quantum gravity corrections, which increases the probability of neutrino emission from the black hole. It is interesting to note that larger quantum gravity effects cause the ratio value to significantly decline. In the neutrino-antineutrino annihilation process, the energy deposition rate is sufficient even while the energy conversion is inhibited because of quantum corrections. Gamma-ray bursts might originate from the corrected annihilation process. Additionally, we examine the derivative $\mathrm{d}\dot{Q}/\mathrm{d}r$ about the star radius $r$. The findings demonstrate that the ratio is lowered by the black hole's quantum influence. The neutrino pair annihilation grows weaker the more prominent the influence of quantum gravity is.

The Asymptotic Expansion of the Spacetime Metric at the Event Horizon

AbstractHawking’s local rigidity theorem, proven in the smooth setting by Alexakis-Ionescu-Klainerman, says that the event horizon of any stationary non-extremal black hole is a non-degenerate Killing horizon. In this paper, we prove that the full asymptotic expansion of any smooth vacuum metric at a non-degenerate Killing horizon is determined by the geometry of the horizon. This gives a new perspective on the black hole uniqueness conjecture. In spacetime dimension 4, we also prove an existence theorem: Given any non-degenerate horizon geometry, Einstein’s vacuum equations can be solved to infinite order at the horizon in a unique way (up to isometry). The latter is a gauge invariant version of Moncrief’s classical existence result, without any restriction on the topology of the horizon. In the real analytic setting, the asymptotic expansion is shown to converge and we get well-posedness of this characteristic Cauchy problem.

Role of Rastall gravity in constructing new spherically symmetric stellar solutions

This paper formulates new stellar solutions corresponding to a static spherical spacetime through the minimal gravitational decoupling in the context of Rastall theory. For this, we initially adopt a seed source to be the isotropic and then add a new gravitating matter source whose impact is controlled by the decoupling parameter. The addition of the later source enhances the degrees of freedom in the corresponding equations of motion. To reduce them, we apply the geometric transformation only to a radial metric potential that divides the field equations into two sets, each of them govern the impact of only one source. Afterwards, we take into account the Schwarzschild solution to make the metric potentials known. Moreover, different constraints on the components of newly added source are employed to develop some suitable solutions of the second system. We construct three different solutions to the total fluid distribution and check their graphical behavior for certain parametric values. Finally, we conclude all our resulting models to be agree with the energy conditions in this extended theory.

Observable strong field effects of extra spacetime dimension in the braneworld black hole

Inspired by the string theory, the braneworld picture introduces extra dimensions beyond the four that may have observable non-trivial effects in short distance (strong field) gravity experiments. A case in point is the Randall–Sundrum braneworld picture that projects the 5d bulk Weyl tensor onto the 3d brane providing a stress tensor in the effective Einstein field equations on the brane. Dadhich, Maartens, Papadopoulos and Rezania (DMPR) derived an exact braneworld black hole solution of the brane vacuum field equations. The solution formally resembles that of Reissner–Nordström but is physically different from it since the ”tidal charge” Υ in the solution is not the electric charge but an imprint from the fifth dimension allowing both signs in the power law modification ±Υ2r2 to the Schwarzschild metric Υ=0. The corresponding black holes are designated as DMPR±. We study here the effect of Υ on strong field lensing observables and compare in the eikonal limit the ring down quasinormal mode (QNM) frequencies of DMPR− with those of DMPR＋ , the two variants of tidal charge modified Schwarzschild black hole (Υ=0). It turns out that the tidal charge can significantly modify the Schwarzschild lensing observables and QNM frequencies. In particular, we find that the Pretorius–Khurana critical exponent γ of circular null orbits in the DMPR− black hole has a lower value than that for the Schwarzschild black hole, which indicates a stronger Lyapunov instability suggesting that the accretion disks of DMPR− black holes would appear brighter. The case of the SgrA* black hole is considered for a possible constraint on Υ from the EHT observation of its shadow size.

Physically accessible and inaccessible quantum correlations of Dirac fields in Schwarzschild spacetime

In this study, we investigate the influence of Hawking decoherence on the quantum correlations of Dirac fields between Alice and Bob. Initially, they share a Gisin state near the Schwarzschild black hole (SBH) in an asymptotically flat region. Then, Alice remains stationary in this region, while Bob hovers near the event horizon (EH) of the SBH. We expect that Bob, using his excited detector, will detect a thermal Fermi-Dirac particle distribution. We assess the quantum correlations in the evolved Gisin state using quantum consonance and uncertainty-induced non-locality across physically accessible, physically inaccessible, and spacetime regions. Our investigation examines how these measures vary with Hawking temperature, Dirac particle frequency, and the parameters of the initial Gisin state. Additionally, we analyze the distribution of these quantum correlation measures across all possible regions, noting a redistribution towards the physically inaccessible region. Our findings demonstrate that Hawking decoherence reduces the quantum correlations of Dirac fields in the physically accessible region, with the extent of reduction depending on the initial state parameters. Moreover, as Hawking decoherence intensifies in the physically inaccessible and spacetime regions, the quantum correlations of Dirac fields reemerge and ultimately converge to specific values at infinite Hawking temperature. These results contribute to our understanding of quantum correlation dynamics within the framework of relativistic quantum information (RQI).

Physical characteristics of anisotropic solutions in f(Q,T) gravity under the vanishing complexity, embedding class one, conformally flat and conformally Killing conditions

The foundation of this paper is to present a comparative study of the properties of anisotropic solutions for compact stars in the framework of modified f(Q,T) gravity for the first time under the schemes of vanishing complexity, embedding class one, conformally flat and conformally Killing, where Q is the nonmetricity scalar and T is the trace of the energy-momentum tensor. The model f(Q,T)=ϕQ+αQn+1−γ(1−e−Q/γ)+βT is chosen in order to compare the results of the various forms f(Q,T) and f(Q) gravity theories. The f(Q,T) and f(Q) gravities in this model are then further simplified to depend on parameters ϕ,α,γ&β. Then, we discussed several approaches to determining the spherically symmetric spacetime components. Schwarzschild geometry represents the exterior spacetime, while a Tolman-Kuchowiz spacetime is used to examine the spherically symmetric spacetime inside the interior geometry. Many properties of compact stars are investigated, like energy density, energy conditions, pressure profiles, sound speeds, gradients, adiabatic index, TOV equation, mass function, compactness, and redshift function are all carefully examined in order to complete the analysis. We have compiled results for both stable and unstable scenarios, derived from gravity theories categorized into various cases, with solutions considered under different frameworks such as Tolman-Kuchowiz spacetime, embedding class 1 spacetime, the vanishing complexity condition, the conformally flat condition, and the conformal Killing vector condition.

Minimum measurable length inspired thermodynamic characteristics of the Schwarzschild black hole surrounded by the cloud of strings and quintessence

We explore the thermodynamic properties of the Schwarzschild black hole surrounded by quintessence matter and a cloud of strings within the framework of the generalized Heisenberg algebra. We provide precise computations of temperature, heat capacity, and entropy associated with this black hole and investigate the impacts of various deformations of the Heisenberg algebra on these thermodynamic parameters. The findings are compared with those for the usual Schwarzschild black hole. Additionally, the behavior of the isotherm related to this black hole is studied. It is anticipated that remnants will hold the key to solving the information loss problem. It is observed that different deformations of the Heisenberg algebra, which introduce minimum measurable length, maximum measurable momenta, or both simultaneously, could lead to the existence of a black hole remnant. Furthermore, it is noted that both exotic fields, cloud of strings and quintessence contribute to the quantum gravity-corrected entropy of the composite system.

Quantum Field Theory of Black Hole Perturbations with Backreaction: I General Framework

In a seminal work, Hawking showed that natural states for free quantum matter fields on classical spacetimes that solve the spherically symmetric vacuum Einstein equations are KMS states of non-vanishing temperature. Although Hawking’s calculation does not include the backreaction of matter on geometry, it is more than plausible that the corresponding Hawking radiation leads to black hole evaporation which is, in principle, observable. Obviously, an improvement of Hawking’s calculation including backreaction is a problem of quantum gravity. Since no commonly accepted quantum field theory of general relativity is available yet, it has been difficult to reliably derive the backreaction effect. An obvious approach is to use the black hole perturbation theory of a Schwarzschild black hole of fixed mass and to quantize those perturbations. However, it is not clear how to reconcile perturbation theory with gauge invariance beyond linear perturbations. In recent work, we proposed a new approach to this problem that applies when the physical situation has an approximate symmetry, such as homogeneity (cosmology), spherical symmetry (Schwarzschild), or axial symmetry (Kerr). The idea, which is surprisingly feasible, is to first construct the non-perturbative physical (reduced) Hamiltonian of the reduced phase space of fully gauge invariant observables and only then apply perturbation theory directly in terms of observables. The task to construct observables is then disentangled from perturbation theory, thus allowing to unambiguously develop perturbation theory to arbitrary orders. In this first paper of the series we outline and showcase this approach for spherical symmetry and second order in the perturbations for Einstein–Klein–Gordon–Maxwell theory. Details and generalizations to other matter and symmetry and higher orders will appear in subsequent companion papers.

Relaxation rate of ModMax–de Sitter black holes perturbed by massless neutral scalar fields

In this study, we investigate the behavior of ModMax-de Sitter black holes when perturbed by massless neutral scalar fields. Specifically, we analyze how the relaxation time, defined as the inverse of the fundamental imaginary frequency, varies with respect to two key parameters: the cosmological constant and the nonlinear parameter characterizing the ModMax theory. We explore scenarios both with and without a cosmological constant, focusing on the static charged ModMax black hole configuration. Our results reveal dependencies between the relaxation time and the nonlinear parameter, shedding light on the dynamical properties of these black hole systems. We also show the validity of WKB approximation under consideration.

Observational signature of QPOs with particle motion around non-commutative Schwarzschild black hole surrounded by perfect fluid dark matter

We analyze the orbital and oscillatory motion of test particles in the vicinity of a non-commutative black hole submerged in perfect fluid dark matter and derive analytical solutions for the specific angular momentum and radial profiles of energy. Using the effective potential approach, we discuss the stability of circular orbits. Furthermore, we calculate the innermost stable circular orbits. The effective force acting on particles has also been discussed. We find the expressions for frequencies of radial and latitudinal harmonic oscillations as a function of the black hole mass and the model’s parameters. The key features of quasi-periodic oscillations of test particles near the stable circular orbits in an equatorial plane of the black hole are discussed. Furthermore, Periastron precession has been discussed. We demonstrate that the parameters of the model have a strong influence on particle motion around black holes. By using the observational data of four different X-ray binary systems GRO J1655-40, XTE J1550-564, XTE J1859+226, and GRS 1915+105, within the scope of Monte Carlo Markov Chain, we constrain the involved parameters α and β. It is necessary to mention that our presented investigations through graphical behavior are viable with required physical behavior.

The shape of non-graviton operators for SU(2)

The BPS spectrum of AdS/CFT exhibits multi-gravitons at low energies, while having black hole states at higher energies. This can be studied concretely in AdS5/CFT4 in terms of classical cohomologies, even in the quantum regimes at finite 1/N. Recently, Chang and Lin found a threshold for non-graviton states in the SU(2) maximal super-Yang-Mills theory. We explicitly construct and present this threshold cohomology.

Stability Analysis of Stable Circular Orbit in Multi-Static Black Hole Spacetime

We herein study the circular orbit stability of a static black hole system composed of multiple Reissner–Nordstrom (RN) black holes. By comparing the circular orbits of two static black holes, three static black holes (TBHs), four static black holes and five static black holes at different spacetime, we find that the continuity of their stable circular orbits changes, i.e., the peaks of the effective potentials are transformed from single-peaked to bi-peaked, and that the distance a between the black holes is the main reason for this change. This characteristic is completely different from the continuity of the stable circular orbit interval of any kind of single black hole in the past. After calculation, we obtain several critical values that lead to the change in circular orbit stability. The three fundamental frequencies (orbital frequency, radial local frequency, and vertical local frequency) are derived and compared for two different spacetimes of double and three black holes. We also analyse the effect of the black hole distance a on the three fundamental frequencies of circular orbits.

Stabilizing effects of higher-order quantum corrections on charged BTZ black hole thermodynamics

In this paper, we study the thermodynamic properties and stability of static charged BTZ black holes with the inclusion of higher-order quantum corrections. The corrections to the entropy, mass, and Helmholtz free energy are derived, revealing the intricate interplay between quantum effects and classical gravitational forces in the context of black hole thermodynamics. The study of the specific heat capacity shows that higher-order corrections stabilize the system by removing the instabilities present at lower orders. The analysis of the van der Waals-like isotherms demonstrates the continuous transition from a highly compressible to an almost incompressible regime as the volume is decreased, akin to the behavior of supercritical fluids. Notably, the isotherms do not exhibit any regions of negative compressibility, indicating the absence of instabilities. Furthermore, the convexity of the Helmholtz free energy as a function of volume confirms the stability of the charged BTZ black hole system. These findings provide valuable insights into the complex thermodynamic landscape of three-dimensional black holes and the role of quantum corrections in shaping their behavior.