Schwarzschild Case Research Articles

Scattering effects of bumblebee gravity in metric-affine formalism

In this work, we explore a Schwarzschild-like black hole within the framework of metric-affine bumblebee gravity. First, we investigate the behavior of the Kretschmann scalar and singularities in this modified gravity approach. Next, we introduce a newly defined time coordinate related to a stationary asymptotically flat spacetime. We also analyze the scattering effects and numerically calculate and comprehensively examine the partial and total absorption cross sections. At the high-frequency approximation, we find that the absorption cross section tends to the geodesic capture cross section. The continued fraction method is applied to investigate the quasinormal modes, and we explore the deviations of both the real and imaginary terms of the quasinormal modes from the Schwarzschild case in detail. We verify the relation between the shadow radius and the real part of the quasinormal frequencies at the eikonal limit within this modified gravity framework. Finally, we examine the energy emission rate.

Scalarization of Taub-NUT black holes in extended scalar-tensor-Gauss-Bonnet theory

Recently, the scalarization of the Schwarzschild black hole has been extensively studied. In this work, we explore the scalarization of the Taub-NUT black hole within the context of the extended scalar-tensor-Gauss-Bonnet theory, which admits a Ricci-flat Taub-NUT black hole as a solution. We carried out an analysis of the probe scalar field to identify the mass parameter and NUT parameter (m, n) where hairy black holes begin to emerge. Subsequently, we used the shooting method to construct the scalarized Taub-NUT black hole numerically. Unlike the Schwarzschild case, there are two branches of new hairy black holes that are smoothly connected. We calculated the entropy of the scalarized black holes and compared these entropies with those of scalar-free Taub-NUT black holes, finding that the entropies of the new hairy black holes are larger. A novel phenomenon emerges in this system: the entropy of the black holes at the bifurcation point is constant for a positive mass parameter. We then conjecture a maximal entropy bound for all scalarized black holes whose mass parameter at the bifurcation point is greater than zero.

Scalar Quasi-Normal Modes of a loop quantum black hole

We compute the Quasi-Normal Mode (QNM) frequencies for scalar perturbations for modified Schwarzschild black holes in Loop Quantum Gravity. We study the singularity-free polymerized metric characterized by two parameters encoding loop quantum effects: the minimal area gap a 0 and the polymeric deformation parameter P. We perform numerical computations using Leaver's continued fraction method and compare our results to other semi-analytical methods and existing literature. We study the effects on the QNM spectrum of variation of both deformation parameters and systematically compare to the standard Schwarzschild case. In particular we find that the scalar fundamental mode is modified from the third decimal for values of P in accordance with the most recent astrophysical constraints. We also show that qualitative differences arise for highly damped modes: on the one hand, a new crossing of the imaginary axis occurs for high values of a 0 and, on the other hand, increasing P produces a positive shift of the real part and an increase of the spacing in imaginary part between modes.

Expansions for semiclassical conformal blocks

We propose a relation the expansions of regular and irregular semiclassical conformal blocks at different branch points making use of the connection between the accessory parameters of the BPZ decoupling equations to the logarithm derivative of isomonodromic tau functions. We give support for these relations by considering two eigenvalue problems for the confluent Heun equations obtained from the linearized perturbation theory of black holes. We first derive the large frequency expansion of the spheroidal equations, and then compare numerically the excited quasi-normal mode spectrum for the Schwarzschild case obtained from the large frequency expansion to the one obtained from the low frequency expansion and with the literature, indicating that the relations hold generically in the complex modulus plane.

Thermodynamics of the quantum Schwarzschild black hole

We discuss some thermodynamic properties as well as the stability of a quantum Schwarzschild black hole, comparing the results with those obtained within a bumblebee gravity model. In particular, the Hawking temperature, TH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{{\ ext{H}}}$$\\end{document}, the entropy, S, the heat capacity, C, and the Gibbs free energy, G, are computed for both cases. In addition to that, we compute the Brown-York quasilocal energy and compare the solution with the Schwarzschild case. We find that in both cases (quantum Schwarzschild and bumblebee gravity model) the temperature, the entropy, and the heat capacity show the same functional form, under the replacement λ2→ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda ^2 \\rightarrow \\ell$$\\end{document} and vice versa. Specifically, the temperature is found to be lower compared to the classical (Schwarzschild) solution; whereas, the entropy is computed to be larger. Moreover, the heat capacity becomes more negative. Notably, a distinct contrast emerges in obtaining the Gibbs free energy between these two cases, and this distinction appears to stem from the ADM mass.

Constraints via the Event Horizon Telescope for Black Hole Solutions with Dark Matter under the Generalized Uncertainty Principle Minimal Length Scale Effect

AbstractFour spherically symmetric but non‐asymptotically flat black hole solutions surrounded with spherical dark matter distribution perceived under the minimal length scale effect is derived via the generalized uncertainty principle. Here, the effect of this quantum correction, described by the parameter , is considered on a toy model galaxy with dark matter and the three well‐known dark matter distributions: the cold dark matter, scalar field dark matter, and the universal rotation curve. The aim is to find constraints to by applying these solutions to the known supermassive black holes: Sagittarius A (Sgr. A*) and Messier 87* (M87*), in conjunction with the available Event Horizon telescope. The effect of is then examined on the event horizon, photonsphere, and shadow radii, where unique deviations from the Schwarzschild case are observed. As for the shadow radii, bounds are obtained for the values of on each black hole solution at confidence level. The results revealed that under minimal length scale effect, black holes can give positive (larger shadow) and negative values (smaller shadow) of , which are supported indirectly by laboratory experiments and astrophysical or cosmological observations, respectively.

Strong gravitational lensing of rotating regular black holes in non-minimally coupled Einstein-Yang-Mills theory* *Support of the National Natural Science Foundation of China (11865018); the Natural Science Foundation of Henan Province, China (232300421351); the Talent Introduction Fund at Henan University of Technology, China(2018BS042, 2020BS035)

The strong gravitational lensing of a regular and rotating magnetic black hole in non-minimally coupled Einstein-Yang-Mills theory is studied. We find that, with the increase of any characteristic parameters of this black hole, such as the rotating parameter a, magnetic charge q and EYM parameter λ, the angular image position and relative magnification decrease while deflection angle and image separation s increase. The results will degenerate to that of the Kerr case, RN case with magnetic charge and Schwarzschild case when we take some specific values for the black hole parameters. The results also show that, due to the small influence of magnetic charge and EYM parameters, it is difficult for current astronomical instruments to tell this black hole apart from a General Relativity one.

The shadows of accelerating Kerr-Newman black hole and constraints from M87*

In this paper, we study the influence of the parameters for the accelerating Kerr-Newman black hole on the shadows and the constraints, extensively. We find that the rotating parameter a, the charge parameter e, and the inclination angle θ0 affect the shadow qualitatively similar to that of Kerr-Newman black holes. The result shows that the size of the shadow will scale down with the accelerating factor A. Besides, the factor A also can affect the best viewing angles, which make the observations maximum deviate from θ0=π2, and the degree of the deviations are less than 1%. Then, we assume the M87* as an accelerating Kerr-Newman black hole with the mass M=6.5×109M⊙ and the distance r0=16.8Mpc. Combining the EHT observations, we find that neither the observations, circularity deviation ΔC or axial ratio Dx can distinguish the accelerating black hole or not. However, the characteristic areal-radius of the shadow curve Ra can give corresponding constraints on the parameters of the accelerating Kerr-Newman black hole. The result shows that the bigger accelerating factor A is, the stronger constraints on the rotating parameter a and charged parameter e. The maximum range of the accelerating factor is Ar0≤0.558 for a accelerating Schwarzschild case with (a/M=e/M=0), and for an extremely slow accelerating case (Ar0≤0.01), the ranges of rotating parameter a and charged parameter e are a/M∈(0,1) and e/M∈(0,0.9).

Quasiperiodic oscillations around hairy black holes in Horndeski gravity

Testing gravity theories and their parameters using observations is an important issue in relativistic astrophysics. In this context, we investigate the motion of test particles and their harmonic oscillations in the spacetime of non-rotating hairy black holes (BHs) in Hordeski gravity, together with astrophysical applications of quasiperiodic oscillations (QPOs). We show possible values of upper and lower frequencies of twin-peak QPOs which may occur in the orbits from innermost stable circular orbits to infinity for various values of the Horndeski parameter q in relativistic precession, warped disk models, and three different sub-models of the epicyclic resonant model. We also study the behaviour of the QPO orbits and their position relative to innermost stable circular orbits (ISCOs) with respect to different values of the parameter q. It is obtained that at a critical value of the Horndeski parameter ISCO radius takes 6M which has been in the pure Schwarzschild case. Finally, we obtain mass constraints of the central BH of microquasars GRS 1915+105 and XTE 1550-564 at the GR limit and the possible value of the Horndeski parameter in the frame of the above-mentioned QPO models. The analysis of orbits of twin peak QPOs with the ratio of upper and lower frequencies 3:2, around the BHs in the frame of relativistic precession (RP) and epicyclic resonance (ER4) QPO models have shown that the orbits locate close to the ISCO. It is obtained that the distance between QPO orbits and ISCO is less than the error of the observations.

Spectral method for the gravitational perturbations of black holes: Schwarzschild background case

We develop a novel technique through spectral decompositions to study the gravitational perturbations of a black hole, without needing to decouple the linearized field equations into master equations and separate their radial and angular dependence. We first spectrally decompose the metric perturbation in a Legendre and Chebyshev basis for the angular and radial sectors respectively, using input from the asymptotic behavior of the perturbation at spatial infinity and at the black hole event horizon. This spectral decomposition allows us to then transform the linearized Einstein equations (a coupled set of partial differential equations) into a linear matrix equation. By solving the linear matrix equation for its generalized eigenvalues, we can estimate the complex quasinormal frequencies of the fundamental mode and various overtones of the gravitational perturbations simultaneously and to high accuracy. We apply this technique to perturbations of a nonspinning, Schwarzschild black hole in general relativity and find the complex quasinormal frequencies of two fundamental modes and their first two overtones. We demonstrate that the technique is robust and accurate, in the Schwarzschild case leading to relative fractional errors of $\leq 10^{-10} - 10^{-8}$ for the fundamental modes, $\leq 10^{-7} - 10^{-6}$ for their first overtones, $\leq 10^{-7} - 10^{-4}$ for their second overtones. This method can be applied to any black hole spacetime, irrespective of its Petrov type, making the numerical technique extremely powerful in the study of black hole ringdown in and outside general relativity.

Quasinormal modes of a holonomy corrected Schwarzschild black hole

We analyze the quasinormal modes (QNMs) of a recently obtained solution of a Schwarzschild black hole (BH) with corrections motivated by Loop Quantum Gravity (LQG). This spacetime is regular everywhere and presents the global structure of a wormhole, with a minimal surface whose radius depends on a LQG parameter. We focus on the investigation of massless scalar field perturbations over the spacetime. We compute the QNMs with the WKB approximation, as well as the continued fraction method. The QNM frequency orbits, for $l=0$ and $n>0$, where $l$ and $n$ are the multipole and overtone numbers, respectively, are self-intersecting, spiraling curves in the complex plane. These orbits accumulate to a fixed complex value corresponding to the QNMs of the extremal case. We obtain that, for small values of the LQG parameter, the overall damping decreases as we increase the LQG parameter. Moreover the spectrum of the quantum corrected black hole exhibits an oscillatory pattern, which might imply in the existence of QNMs with vanishing real part. This pattern suggests that the limit $n\rightarrow \infty$ for the real part of the QNMs is not well-defined, what differs from Schwarzschild's case. We also analyze the time-domain profiles for the scalar perturbations, showing that the LQG correction does not alter the Schwarzschild power-law tail. We compute the fundamental mode from the time profile by means of the Prony method, obtaining excellent agreement with the two previously mentioned methods.

Weak Gravitational Lensing around Bardeen Black Hole with a String Cloud in the Presence of Plasma

The effect of spacetime curvature on optical properties may provide an opportunity to suggest new tests for gravity theories. In this paper, we investigated gravitational weak lensing around a Bardeen black hole with the string clouds parameter. First, we examined the horizon structure in the presence of string clouds around the gravitational compact object defined by Bardeen spacetime. The effect of gravitational weak lensing in a plasma medium is also discussed. According to the findings, the influence of the string cloud parameter on the circular orbits of a light ray around the black hole is greater than that in the Schwarzschild case, while the influence of the charge is reversed. The deflection angle of light rays in weak lensing is also used to study how much the image is magnified.

Gravitational Lensing of Acoustic Charged Black Holes

We study the gravitational lensing of acoustic charged black holes in strong and weak field limit approximations. For this purpose, we first numerically obtain the deflection limit coefficients and deflection angle in the strong field limit. We observe that the strong deflection angle α D increases with increasing magnitude of the charged parameter Q and that the strong deflection angle α D of an acoustic charged black hole with tuning parameter ξ = 4 is greater than that of a standard Reissner–Nordström black hole (ξ = 0). We also study the astrophysical consequences via strong gravitational lensing by taking the example of various supermassive black holes in the center of several galaxies and observe that the acoustic charged black hole could be quantitatively distinguished from standard Reissner–Nordström (ξ = 0) and standard Schwarzschild (ξ = 0, Q = 0) black holes. Furthermore, by using the Gauss–Bonnet theorem, we derive the weak deflection angle in the background of an acoustic charged black hole in the curved spacetime. We find that, for fixed values of the charged parameter Q and the tuning parameter (ξ = 0 or 4), the weak deflection angle σ D decreases with the impact parameter b. We also observe that the weak deflection angle σ D decreases with increasing magnitude of the charged parameter Q for a fixed value of the tuning parameter (ξ = 0 or 4). Our results suggest that the observational test for an acoustic charged black hole is indeed feasible, and it is generalized to the cases of acoustic Schwarzschild (Q = 0), standard Reissner–Nordström (ξ = 0), and standard Schwarzschild (ξ = 0, Q = 0) black holes.

Are nonsingular black holes with super-Planckian hair ruled out by S2 star data?

We investigate a nonsingular black hole spacetime representing a strong deformation of the Schwarzschild solution with mass $M$ by an additional hair $\ensuremath{\ell}$, which may be hierarchically larger than the Planck scale. The spacetime is an exact solution of Einstein's equations sourced by an anisotropic fluid. The model presents a de Sitter core and $\mathcal{O}({\ensuremath{\ell}}^{2}/{r}^{2})$ slow-decaying corrections to the Schwarzschild solution. These solutions are thermodynamically preferred when $0.2\ensuremath{\lesssim}\ensuremath{\ell}/GM\ensuremath{\lesssim}0.3$ and are characterized by strong deviations in the orbits of test particles from the Schwarzschild case. In particular, we find corrections to the perihelion precession angle scaling linearly with $\ensuremath{\ell}$. We test our model using the available data for the orbits of the S2 star around ${\text{SgrA}}^{*}$. These data strongly constrain the value of the hair $\ensuremath{\ell}$, casting an upper bound on it of $\ensuremath{\sim}0.47GM$ but do not rule out the possible existence of regular black holes with super-Planckian hair.

Calculating quasinormal modes of Schwarzschild anti–de Sitter black holes using the continued fraction method

We investigate the scalar, gravitational, and electromagnetic quasinormal mode spectra of Schwarzschild anti-de Sitter black holes using the numerical continued fraction method. The spectra have similar, almost linear structures. With a few exceptions, the low overtone quasinormal modes are consistent with previously obtained results in the literature that use other numerical techniques. The intermediate and high overtone quasinormal modes, in comparison to the Schwarzschild case, converge very quickly to the asymptotic formulas previously obtained by analytic monodromy techniques. In addition, we find a connection between the analytic asymptotic formulas and the purely imaginary modes. In particular, these formulas can be used to predict the bifurcation of the lowest damped electromagnetic modes. Finally, we find no high overtone quasinormal modes with high oscillation frequency and low damping, which had been previously predicted.

Testing Born–Infeld f(T) teleparallel gravity through Sgr hbox {A}^star observations

We use observational data from the S2 star orbiting around the Galactic Center to constrain a black hole solution of extended teleparallel gravity models. Subsequently, we construct the shadow images of Sgr hbox {A}^{star } black hole. In particular, we constrain the parameter alpha =1/lambda which appears in the Born–Infeld f(T) model. In the strong gravity regime we find that the shadow radius increases with the increase of the parameter alpha . Specifically, from the S2 star observations, we find within 1sigma that the parameter must lie between 0 le alpha /M^2 le 6 times 10^{-4}. Consequently, we used the best fit parameters to model the shadow images of Sgr hbox {A}^{star } black hole and then using the Gauss-Bonnet theorem we analysed the deflection angle for leading order expansions of the parameter alpha . It is found that within the parameter range, these observables are very close to the Schwarzschild case. Furthermore, using the best fit parameters for the Born–Infeld f(T) model we show that the angular diameter is consistent with recent observations for the Sgr hbox {A}^{star } with angular diameter (51.8 pm 2.3) mu arcsec and difficult to be distinguished from the GR. For the deflection angle of light, in leading order terms, we find that the deflection angle expressed in the ADM mass coincides with the GR, but the ADM mass in the Born–Infeld f(T) gravity increases with the increase of alpha and the overall deflection angle is expected to me greater in f(T) gravity. As a consequence of this fact, we have shown that the electromagnetic intensity observed in shadow images is smaller compared to GR.

Generalized Extended Uncertainty Principle Black Holes: Shadow and Lensing in the Macro- and Microscopic Realms

Motivated by the recent study about the extended uncertainty principle (EUP) black holes, we present in this study its extension called the generalized extended uncertainty principle (GEUP) black holes. In particular, we investigated the GEUP effects on astrophysical and quantum black holes. First, we derive the expression for the shadow radius to investigate its behavior as perceived by a static observer located near and far from the black hole. Constraints to the large fundamental length scale, L*, up to two standard deviations level were also found using the Event Horizont Telescope (EHT) data: for black hole Sgr. A*, L*=5.716×1010 m, while for M87* black hole, L*=3.264×1013 m. Under the GEUP effect, the value of the shadow radius behaves the same way as in the Schwarzschild case due to a static observer, and the effect only emerges if the mass, M, of the black hole is around the order of magnitude of L* (or the Planck length, lPl). In addition, the GEUP effect increases the shadow radius for astrophysical black holes, but the reverse happens for quantum black holes. We also explored GEUP effects to the weak and strong deflection angles as an alternative analysis. For both realms, a time-like particle gives a higher value for the weak deflection angle. Similar to the shadow, the deviation is seen when the values of L* and M are close. The strong deflection angle gives more sensitivity to GEUP deviation at smaller masses in the astrophysical scenario. However, the weak deflection angle is a better probe in the micro world.

Polarization distribution in the image of a synchrotron emitting ring around a regular black hole

The polarized images of a synchrotron emitting ring are studied in the regular Hayward and Bardeen black hole spacetimes. These regular black holes carry a magnetic field in terms of gravity coupled to nonlinear electrodynamics. Results show that the main features of the polarization images of the emitting rings are similar in these two regular black hole spacetimes. As the magnetic charge parameter increase, the polarization intensity and the electric vector position angle in the image plane increase in Hayward and Bardeen black hole spacetimes. Moreover, the polarization intensity and electric vector position angle in the image of the emitting ring in the Hayward black hole spacetime are closer to those in the Schwarzschild case. The effects of the magnetic charge parameter on the Strokes $Q-U$ loops are also slightly smaller in the Hayward black hole spacetime. This information stored in the polarization images around Hayward and Bardeen black holes could help understand regular black holes and the gravity coupled to nonlinear electrodynamics.

Instability of the charged massive scalar field on the Kerr–Newman black hole spacetime

We investigate the quasibound states of charged massive scalar fields in the Kerr–Newman black hole spacetime by using a new approach recently developed, which uses the polynomial conditions of the Heun functions. We calculate the resonant frequencies related to the spectrum of quasibound states, as well as its corresponding angular and radial wave eigenfunctions. We also analyze the instability of the system. These results are particularized to the cases of Schwarzschild and Kerr black holes. Additionally, we compare our analytical results with the numerical ones known in the literature. Finally, we apply the obtained results to compute the characteristic times of growth and decay of bosonic particles around a supermassive black hole situated at the center of the M87 galaxy.

A study of McVittie Spacetime in Stationary and Dynamical Regime

AbstractOne of the solutions of the Einstein equations, called McVittie solution, signifying a black hole embedded by the dynamic spacetime is studied. In the stationary spacetime the Mcvittie metric becomes the Schwarzschild‐de Sitter metric (SdS). The geodesic of a freely falling test particle towards the black hole is examined in the SdS spacetime. It is found that unlike the Schwarzschild case the potential of such particle becomes maximum at a point where it eventually stops for a while and then resumes its motion towards the center of the black hole. It is shown that an observer or system of particles is spaghettified near the black hole singularity in the SdS spacetime. The dynamics of the universe in the framework of McVittie metric, being a generalized time dependent SdS solution, is represented in terms of that point, called stationary or turning point. The motion of the stationary point is studied in various regimes of the expanding universe and the possible outcomes are discussed in brief.