Schwarzschild Black Hole Research Articles

Scalar field perturbation of hairy black holes in EsGB theory

We investigate scalar field perturbations of the hairy black holes involved with spontaneous symmetry breaking of the global U(1) symmetry in Einstein-scalar-Gauss-Bonnet theory for asymptotically flat spacetimes. We consider the mechanism that black holes without hairs become unstable at the critical point of the coupling constant and undergo a phase transition to hairy black holes in the symmetry-broken phase driven by spontaneous symmetry breaking. This transition occurs near the black hole horizon due to the diminishing influence of the Gauss-Bonnet term at infinity. To examine such process, we introduce a scalar field perturbation on the newly formed background spacetime. We solve the linearized perturbation equation using Green’s function method. We begin by solving the Green’s function, incorporating the branch cut contribution. This allows us to analytically investigate the late-time behavior of the perturbation at both spatial and null infinity. We found that the late-time behavior only differs from the Schwarzschild black hole by a mass term. We then proceed to calculate the quasinormal modes (QNMs) numerically, which arise from the presence of poles in the Green’s function. Our primary interest lies in utilizing QNMs to investigate the stability of the black hole solutions both the symmetric and symmetry-broken phases. Consistent with the prior study, our analysis shows that hairy black holes in the symmetric phase become unstable when the quadratic coupling constant exceeds a critical value for a fixed value of the quartic coupling constant. In contrast, hairy black holes in the symmetry-broken phase are always stable at the critical value. These numerical results provide strong evidence for a dynamical process that unstable black holes without hairs transition into stable hairy black holes in the symmetry-broken phase through the spontaneous symmetry breaking.

Thermal fluctuations, deflection angle, and greybody factor of a high-dimensional Schwarzschild black hole in scalar–tensor–vector gravity

In this study, we examined the thermal fluctuations, deflection angle, and greybody factor of a high-dimensional Schwarzschild black hole in scalar–tensor–vector gravity (STVG). We calculated some thermodynamic quantities related to the correction of the black hole entropy caused by thermal fluctuations and discussed the effect of the correction parameters on these quantities. By analyzing the changes in the corrected specific heat, we found that thermal fluctuations made the small black hole more stable. It is worth noting that the STVG parameter did not affect the thermodynamic stability of this black hole. Additionally, by utilizing the Gauss–Bonnet theorem, the deflection angle was obtained in the weak field limit, and the effects of the two parameters on the results were visualized. Finally, we calculated the bounds on the greybody factor of a massless scalar field. We observed that as the STVG parameter around the black hole increased, the weak deflection angle became larger, and more scalar particles can reach infinity. However, the spacetime dimension has the opposite effect on the STVG parameter on the weak deflection angle and greybody factor.

Scalar field solutions and energy bounds for modeling spatial oscillations in Schwarzschild black holes based on the Regge–Wheeler equation

This text discusses the behavior of solutions and the energy stability within Schwarzschild spacetimes, with a particular emphasis on the behavior of massless scalar fields under the influence of a non-rotating and spherically symmetric black hole. The stability of solutions in the proximity of the event horizon of black holes in general relativity remains an open question, especially given the difficulties introduced by minor perturbations that may resemble Kerr solutions. To address this, this work explores a simplified model, including massless scalar fields, to better understand perturbation behaviors around black holes under the Schwarzschild approach. We depart from Richard Price’s work in connection with how scalar, electromagnetic, and gravitational fields behave. The tortoise coordinate transformation is considered to set the stage for numerical solutions to the wave equations. Afterward, we explore energy estimates, which are used to gauge stability and wave behavior over time. Our analysis reveals that the time evolution of the energy does not exceed twice its initial value. Further and under the assumption of initial conditions in L2−spaces, we obtain an exponential decreasing behavior in the energy time evolution. A question to continue exploring is how perturbations in L2 in the initial conditions that introduce Kerr solutions as a second-order effect in the linearized equations perturb this obtained exponential decay.

Constraining a disformal Schwarzschild black hole in DHOST theories with the orbit of the S2 star

With the observed data of the S2 orbit around the black hole Sgr A∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^*$$\\end{document} and the Markov Chain Monte Carlo method, we make a constraint on parameters of a disformal Schwarzschild black hole in quadratic degenerate higher-order scalar-tensor (DHOST) theories. This black hole belongs to a class of non-stealth solutions and owns an extra disformal parameter described the deviation from general relativity. Our results show that the best fit value of the disformal parameter is positive. However, in the range of 1σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\sigma $$\\end{document}, we also find that general relativity remains to be consistent with the observation of the S2 orbit.

Topology of black hole thermodynamics via Rényi statistics

In this paper, we investigate the topological numbers of the four-dimensional Schwarzschild black hole, d-dimensional Reissner–Nordström (RN) black hole, d-dimensional singly rotating Kerr black hole and five-dimensional Gauss–Bonnet black hole via the Rényi statistics. We find that the topological number calculated via the Rényi statistics is different from that obtained from the Gibbs–Boltzmann (GB) statistics. However, what is interesting is that the topological classifications of different black holes are consistent in both the Rényi and GB statistics: the four-dimensional RN black hole, four-dimensional and five-dimensional singly rotating Kerr black holes, five-dimensional charged and uncharged Gauss–Bonnet black holes belong to the same kind of topological class, and the four-dimensional Schwarzschild black hole and d(>5)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(>5)$$\\end{document}-dimensional singly rotating Kerr black holes belong to another kind of topological class. In addition, our results suggest that the topological numbers calculated via the Rényi statistics in asymptotically flat spacetime background are equal to those calculated from the standard GB statistics in asymptotically AdS spacetime background, which provides more evidence for the connection between the nonextensivity of the Rényi parameter λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document} and the cosmological constant Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}.

Embedding diagrams in stationary spacetimes

We find the spatial and dynamic embedding diagrams in stationary black hole spacetimes. The spatial embeddings include the NUT, pure NUT and Kerr spacetimes. In the case of pure NUT spacetime, the spatial embedding equations are solved in terms of the elliptic integrals. In other cases we obtain the spatial embedding diagrams by numerical integration of the corresponding embedding equations. These embedding diagrams are then compared through their Gaussian and mean curvatures. We also find the dynamic embedding diagrams of NUT and pure NUT spacetimes, and compare them with the dynamic embedding diagram of Schwarzschild spacetime.

Polarized image of a synchrotron emitting ring around a static hairy black hole in Horndeski theory

In this paper, we investigate the polarization images of a synchrotron-emitting fluid ring surrounding a static hairy black hole within the framework of Horndeski’s theory. Our findings indicate that the characteristics of these polarization images are predominantly influenced by the hairy parameter, alongside the magnetic field near the black hole, the fluid’s velocity, and the observer’s inclination angle. Specifically, the hairy parameter primarily affects the polarized intensity and the apparent radius of the ring in the images. Conversely, the impacts of the magnetic field, fluid velocity, and inclination angle on the polarization images are found to be independent of the hairy parameter and closely resemble those observed in the context of Schwarzschild black holes. Additionally, the polarization direction is significantly influenced by the magnetic field orientation, while the inclination angle crucially determines the apparent flatness of the images. Variations in the fluid velocity direction also markedly affect the trend in polarized intensity. Furthermore, we explore how these parameters influence the Stokes Q-U\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q-U$$\\end{document} loops, revealing distinct behaviors in response to changes in the aforementioned variables. This comprehensive analysis enhances our understanding of the intricate dynamics and observational signatures of black holes within alternative gravitational theories.

Quasinormal modes of the Schwarzschild black hole with a deficit solid angle and quintessence-like matter: Improved asymptotic iteration method

We study the quasinormal modes (QNM) for scalar, and electromagnetic perturbations in the Schwarzschild black hole with a deficit solid angle and quintessence-like matter. Using the sixth--order WKB approximation and the improved asymptotic iteration method (AIM) we can determine the dependence of the quasinormal modes on the parameters of the black hole and the parameters on the test fields. The values of the real part and imaginary parts of the quasi--normal modes increase with the decrease of the values of the deficit solid angle and density of quintessence-like matter. The quasinormal modes gotten by these two methods are in good agreement. Using the finite difference method, we obtain the time evolution profile of such perturbations in this Black Hole.

Relativistic periastron advance beyond Einstein theory: analytical solution with applications

We find a new solution to calculate the orbital periastron advance of a test body subject to a central gravitational force field for relativistic theories and models beyond Einstein. This analitycal formula has general validity that includes all the post-Newtonian (PN) contributions to the dynamics and is useful for high-precision gravitational tests. The solution is directly applicable to corrective potentials of various forms, without the need for numerical integration. Later, we apply it to the scalar tensor fourth order gravity (STFOG) and noncommutative geometry, providing corrections to the Newtonian potential of Yukawa-like form V(r)=αe-βrr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(r)=\\alpha \\frac{e^{-\\beta r}}{r}$$\\end{document}, and we conduct the first analysis involving all the PN terms for these theories. The same work is performed with a Schwarzschild geometry perturbed by a Quintessence Field, leading to a power-law potential V(r)=αqrq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(r)=\\alpha _q {r}^q$$\\end{document}. Finally, by using astrometric data of the Solar System planetary precessions and those of the S2 star around Sgr A*, we infer new theoretical constraints and improvements in the bounds for β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}. The resulting simulated orbits turn out to be compatible with general relativity.

Construction of fuzzy black holes and self-gravitational droplets in exponential f(R) gravity

This paper discusses the construction of non-commutative geometry based self-gravitational droplets or fuzzy black holes (BHs) within the context of exponential-f(R) gravity model 1βeβR−1 , where β is a parameter with units [length 2]. Considering the Einasto density profile, a generalization of the Gaussian profile, we extend the noncommutative-inspired mini BH metric to incorporate dark matter (DM) as a constituent of the BH. We investigate various Einasto DM fuzzy objects filled with anisotropic fluid content, considering different values of the shape parameter and rescaled mass. This study investigates the possibility of stable self-gravitational compact objects with fuzzy mass distributions inspired by a well-known DM profile, the Einasto parametrization. These intriguing astrophysical configurations fall into three categories: fuzzy self-gravitational droplets with no event horizon, and fuzzy black holes featuring either one or two horizons. It is important to highlight that all these astrophysical entities are solutions derived from the exponential-f(R) gravity model. The corresponding effective potentials exhibit a significant resemblance to the effective potential of the classical Schwarzschild Black Hole located at the center of the Galaxy. Specifically, we investigate the potential for these black holes to serve as central galactic entities.

Image of a black hole illuminated by a parabolic screen

A model of an illuminating black hole surface (“parabolic screen”) is proposed. This allows to avoid in a natural way the appearance of edge effects related with photons moving along the screen plane. The temperature distribution along the radius corresponds to the corresponding distribution for a relativistic disk (Novikov-Thorne disk). It is shown that the structure of the formed black hole shadow essentially differs from the case when the source of photons is a distant screen, since in the considered model the photons, subjected to strong gravitational lensing of the black hole, are emitted by the “backward” side of screen, which would not be visible in the absence of the black hole. In the thin screen approximation, shadow images are constructed for the Schwarzschild black hole in the cases when the angle between the symmetry axis of the illuminating screen and the direction to the observer is equal to 5, 30, 30 and 80 angular degrees, similar images for the Kerr black hole are constructed for the cases of 60 and 80 angular degrees.

Optical appearance of numerical black hole solutions in higher derivative gravity

The optical appearance of the numerically black hole solutions within the higher derivative gravity illuminated by an accretion disk context is discussed. We obtain solutions for non-Schwarzschild black holes with r0=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_0=1$$\\end{document}, r0=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_0=2$$\\end{document}, and r0=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_0=3$$\\end{document}. Further analysis of spacetime trajectories reveals properties similar to Schwarzschild black holes, while the r0=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_0=2$$\\end{document} black hole exhibits significant differences. The results reveal the presence of a repulsive potential barrier for the black hole, allowing only particles with energies exceeding a certain threshold to approach it, providing a unique gravitational scenario for non-Schwarzschild black holes. Additionally, the optical images are derived through numerical simulations by discussing the trajectories of photons in the black hole spacetime. The distribution of radiation flux and the effects of gravitational redshift and Doppler shift on the observed radiation flux are considered. Interestingly, previous analyses of the optical appearance of black holes were conducted within the framework of analytic solutions, whereas the analysis of numerical black hole solutions first appears in our analysis.

Black-Hole singularity and its possible mitigations: Reformulation of Penrose’s singularity theorem using null Raychaudhuri matrix

In this paper, we introduce a notion of null Raychaudhuri matrix motivated by the matrix representation of tensor fields. The evolution of this matrix gives the matrix form of null Raychaudhuri equation. Using this distinct geometric approach, we have reformulated the original Penrose’s singularity theorem on Black-Hole and have commented on the characteristic of the Raychaudhuri matrix at the singularity. The paper also suggests two possible mitigations for the physical singularity of Schwarzschild black hole via the Wheeler–DeWitt formalism and Bohmian formalism.

Massive Dirac equation in static Bumblebee black hole space-times, detailed derivations and novel exact solutions

In this work, we construct and investigate the relativistic spin-12 fermionic fields quantum dynamics in static spherically symmetric Bumblebee black hole background. The derivation of the Dirac equation in a general static spherically symmetric black hole space-time is carried out in detail via tetrad formalism. With the help of total angular momentum operator, the angular equation can be separated from the radial part where the solution is given in terms of the spinor harmonics. The radial Dirac equation has a well-known problem due to the presence of the square root terms appearing simultaneously with the rest mass that prevents us to find exact solutions. In this work, we present exact solutions of light mass fermion's wave function and energy levels bound in the static Bumblebee black hole. We discover the exact solutions of the massive Dirac's radial equation in terms of the Confluent Heun functions. Moreover, thanks to the well-known polynomial condition of the Confluent Heun functions, we also derive the energy quantization.

Entropy production and the generalised second law of black hole thermodynamics

The generalised second law of black hole thermodynamics states that the sum of a black hole’s entropy and the entropy of all matter outside the black hole cannot decrease with time. The violation of the generalised second law via the process in which a distant observer extracts work by lowering a box arbitrarily close to the event horizon of a black hole has two profound ramifications: (1) that the entropy of the Universe can be decreased arbitrarily via this process; and (2) that it is not appropriate to apply the laws of thermodynamics to systems containing black holes. In this paper, we argue that for the generalised second law to not be violated, entropy must be produced during the lowering process. To demonstrate this, we begin by deriving an equation for the locally measured temperature of the vacuum state of an observer that is a finite distance from the event horizon of a Schwarzschild black hole. Then, using this locally measured temperature and the Unruh effect, we derive an equation for the force required to hold this observer in a stationary position relative to a Schwarzschild black hole. These equations form a framework for calculating the change in black hole entropy as a result of the lowering process both in the case where the process is isentropic and in the case where entropy is produced during the lowering process. In the latter case, two requirements: (1) that the resultant change in black hole entropy is finite; and (2) that the resultant change in common entropy is finite, are used to identify two conditions that the form of an entropy production function must satisfy. These, in turn, are used to identify a set of possible functions describing the production of entropy. Using this set of functions, we demonstrate that the production of entropy limits the amount of work that the distant observer can extract from the lowering process. We find that this allows for the generalised second law to be preserved, provided that a coefficient in this set of functions satisfies a given bound. To conclude, we discuss two natural choices of this coefficient that allow for the generalised second law to be preserved in this lowering process. In addition to providing a resolution to this violation of the generalised second law, the framework presented in this paper can be applied to inform theories of gravity and quantum gravity on the form of their entropy relations, such that they do not violate the generalised second law.

On the wormhole-warp drive correspondence

We propose a correspondence between the Morris-Thorne wormhole metric and a warp drive metric, which generalizes an earlier result by H. Ellis regarding the Schwarzschild black hole metric and makes it possible to embed a warp drive in a wormhole background. We demonstrate that in order to do that, one needs to also generalize the Natario-Alcubierre definition of warp drive and introduce nonzero intrinsic curvature. Surprisingly, we find out that Alcubierre warp drives cannot traverse Morris-Thorne wormholes; nonetheless, this “no-go” theorem does not extend to spherically symmetric warp drives that are not localized in space.

Fermi model of a quantum black hole

We propose a quantum model of the Schwarzschild black hole as a quantum mechanics of a system of fermionic degrees of freedom. The system has a constant density of states and a Fermi energy that is inversely proportional to the size of the system. Assuming the equivalence principle, we show that the degeneracy pressure of the Fermi degrees of freedom is able to withstand the collapse of gravity if the radius of the system is given precisely by the horizon radius of the Schwarzschild black hole. In our model, the fermionic degrees of freedom at each energy level can be entangled in certain different ways, giving rise to a multitude of degenerate ground states of the system. The counting of these microstates reproduces precisely the Bekenstein-Hawking entropy. This simple Fermi model is universal and works also for the Reissner-Nordström charged black hole as well as black holes with a cosmological constant. From the properties of the Fermi variables, we propose that quantum gravity is characterized by a principle of where there can be no more than V/lP3 quantum states in any volume V. It implies a loss of spatial locality below the Planck length and suggests that any singularity predicted by general relativity is resolved and replaced by a quantum space in quantum gravity. In our model, a black hole spacetime is equipped with a uniform distribution of energy levels. This is another reason why black holes can be considered a simple harmonic oscillator of quantum gravity. Published by the American Physical Society 2024

Cosmology from Schwarzschild black hole revisited

Cosmology from Schwarzschild black hole revisited

Charged Particles Capture Cross-section by a Weakly Charged Schwarzschild Black Hole

We study the capture cross-section of charged particles by a weakly charged Schwarzschild black hole. The dependence of the maximum impact parameter for capture on the particle’s energy is investigated numerically for different values of the electromagnetic coupling strength between the particle and the black hole. The capture cross-section is then calculated. We show that the capture cross-section is independent of the electromagnetic coupling for ultra-relativistic particles. The astrophysical implications of our results are discussed.

Accretion mechanism for regular black holes with asymptotically Minkowski Cores and improved Schwarzschild black holes

In this paper, we investigate the fascinating implications of the accretion process for two different kinds of black holes. A detailed analysis of the moving fluid in this accretion process is provided. The radial velocity u(r), the energy density ρ(r), speed of sound a2s and the mass accretion rate Ṁ of the regular black holes and improved Schwarzschild black hole are explored. Interestingly, we consider the most generic black hole space–times with isotropic fluid for this accretion process. The required and necessary physical behaviors around the black holes, such as the radial velocity, the energy density of the moving fluid within the scope of dust, stiff fluid and quintessence, are achieved. Our findings suggest that critical parameters like l in regular black holes, Γ in improved Schwarzschild black holes and the state parameter k are responsible for the accretion mechanism.