Schwarzschild Case Research Articles

Near-horizon geometry and the entropy of a minimally coupled scalar field in the Kerr black hole

In this article we will discuss a Lorentzian sector calculation of the entropy of a minimally coupled scalar field in the Kerr black hole background. We will use the brick wall model of t' Hooft. In the Kerr black hole, complications arise due to the absence of a global timelike Killing field and the presence of the ergosphere. Nevertheless, it is possible to calculate the entropy of a thin shell of matter field in the near-horizon region using the brick wall model. The corresponding leading order entropy of the nonsuperradiant modes is found to be proportional to the area of the horizon and is logarithmically divergent. Thus, the entropy of a three dimensional system in the near-horizon region is proportional to the boundary surface. This is similar to that of the black hole entropy itself. The corresponding internal energy remains finite if the entropy is chosen to be of the order of the black hole entropy itself. For a fixed value of the brick wall cut-off, the leading order entropy in the Kerr black hole is found to be half of the corresponding term in the Schwarzschild black hole. This is consistent with the preferential emission of particles in the Kerr black hole with azimuthal angular momentum in the same direction as that of the black hole itself. However, we can obtain the Schwarzschild case expression by including a subleading term and taking the appropriate limit. The results obtained in this article may be relevant to entropy bound and holographic principle.

The Hawking cascade from a black hole is extremely sparse

The average time between emission of subsequent quanta in the Hawking process is extremely large. While this sparsity result has been known for a long time, it is neither well-known, nor do (semi-)analytic results currently exist, the prior focus being placed on numerical results. We define several ways of quantifying this sparsity, and starting from a black body approximation for the Schwarzschild case, we derive analytic expressions for a lower bound on this average time. We also check the validity of the results in presence of greybody factors by numerical analysis. Furthermore, we show how to separate the super-radiance in the low-frequency regime from the genuine Hawking effect itself. This enables us to extend the previous lower bounds to Reissner–Nordström, Kerr and dirty black holes in addition to different particle species. Lastly, we want to draw attention to some of the physical consequences of this under-appreciated fact of the Hawking process.

Parameters of innermost stable circular orbits of spinning test particles: Numerical and analytical calculations

The motion of classical spinning test particles in the equatorial plane of a Kerr black hole is considered for the case where the particle spin is perpendicular to the equatorial plane. We review some results of our recent research of the innermost stable circular orbits (ISCO) [P.I. Jefremov, O.Yu. Tsupko and G.S. Bisnovatyi-Kogan, Phys.Rev. D 91 124030 (2015)] and present some new calculations. The ISCO radius, total angular momentum, energy, and orbital angular frequency are considered. We calculate the ISCO parameters numerically for different values of the Kerr parameter $a$ and investigate their dependence on both black hole and test particle spins. Then we describe in details how to calculate analytically small-spin corrections to the ISCO parameters for an arbitrary values of $a$. The cases of Schwarzschild, slowly rotating Kerr and extreme Kerr black hole are considered. The use of the orbital angular momentum is discussed. We also consider the ISCO binding energy. It is shown that the efficiency of accretion onto an extreme Kerr black hole can be larger than the maximum known efficiency (42 %) if the test body has a spin.

Tidal forces in Reissner–Nordström spacetimes

We analyze the tidal forces produced in the spacetime of Reissner–Nordstrom black holes. We point out that the radial component of the tidal force changes sign just outside the event horizon if the charge-to-mass ratio is close to 1, unlike in Schwarzschild spacetime of uncharged black holes, and that the angular component changes sign between the outer and inner horizons. We solve the geodesic deviation equations for radially falling bodies toward the charged black hole. We find, for example, that the radial component of the geodesic deviation vector starts decreasing inside the event horizon unlike in the Schwarzschild case.

Quasinormal modes of Gauss-Bonnet black holes at large D

Einstein’s General Relativity theory simplifies dramatically in the limit that the spacetime dimension D is very large. This could still be true in the gravity theory with higher derivative terms. In this paper, as the first step to study the gravity with a Gauss-Bonnet(GB) term, we compute the quasi-normal modes of the spherically symmetric GB black hole in the large D limit. When the GB parameter is small, we find that the non-decoupling modes are the same as the Schwarzschild case and the decoupled modes are slightly modified by the GB term. However, when the GB parameter is large, we find some novel features. We notice that there are another set of non-decoupling modes due to the appearance of a new plateau in the effective radial potential. Moreover, the effective radial potential for the decoupled vector-type and scalar-type modes becomes more complicated. Nevertheless we manage to compute the frequencies of the these decoupled modes analytically. When the GB parameter is neither very large nor very small, though analytic computation is not possible, the problem is much simplified in the large D expansion and could be numerically treated. We study numerically the vector-type quasinormal modes in this case.

Horizon Wavefunction of Generalized Uncertainty Principle Black Holes

We study the Horizon Wavefunction (HWF) description of a Generalized Uncertainty Principle inspired metric that admits sub-Planckian black holes, where the black hole mass m is replaced by M=m1+β/2MPl2/m2. Considering the case of a wave-packet shaped by a Gaussian distribution, we compute the HWF and the probability PBH that the source is a (quantum) black hole, that is, that it lies within its horizon radius. The case β<0 is qualitatively similar to the standard Schwarzschild case, and the general shape of PBH is maintained when decreasing the free parameter but shifted to reduce the probability for the particle to be a black hole accordingly. The probability grows with increasing mass slowly for more negative β and drops to 0 for a minimum mass value. The scenario differs significantly for increasing β>0, where a minimum in PBH is encountered, thus meaning that every particle has some probability of decaying to a black hole. Furthermore, for sufficiently large β we find that every particle is a quantum black hole, in agreement with the intuitive effect of increasing β, which creates larger M and RH terms. This is likely due to a “dimensional reduction” feature of the model, where the black hole characteristics for sub-Planckian black holes mimic those in (1+1) dimensions and the horizon size grows as RH~M-1.

Classical and quantum Reissner-Nordström black hole thermodynamics and first order phase transition

First we consider CRNBH metric which is obtained by solving Einstein-Maxwell metric equation for a point electric charge $e$ inside of a spherical static body with mass $M$. It has 2 interior and exterior horizons. Using Bekenestein-Hawking entropy theorem we calculate interior and exterior entropy, temperature, Gibbs free energy and heat capacity at constant electric charge. We calculate first derivative of the Gibbs free energy with respect to temperature which become a singular function having a singularity at critical point with corresponding temperature $T_c=\frac{1}{24\pi \sqrt{3}|e|}.$ Hence we clime first order phase transition is happened there. Temperature same as Gibbs free energy takes absolutely positive (negative) values on the exterior (interior) horizon. The Gibbs free energy takes two different positive values synchronously for $0<T<T_c$ but not for negative values which means the system is made from two subsystem. For negative temperatures entropy reaches to zero value at $T\to-\infty$ and so takes Bose-Einstein condensation single state. Entropy increases monotonically in case $0<T<T_c$. Regarding results of the work presented at Ref. \citep{Bob01} we calculate again the mentioned thermodynamical variables for remnant stable final state of evaporating QRNBH and obtained results same as one in case of the CRNBH. Finally, we solve mass loss equation of QRNBH against advance Eddington-Finkelstein time coordinate and derive luminosity function. We obtain switching off of QRNBH evaporation before than the mass completely vanishes. It reaches to a could Lukewarm type of RN black hole which its final remnant mass is $m_{final}=|e|$ in geometrical units. Its temperature and luminosity vanish but not in Schwarzschild case of evaporation. Our calculations can be takes some acceptable statements about information loss paradox .

Nakedly singular non-vacuum gravitating equilibrium states

Non-vacuum static spherically symmetric spacetimes with central point-like repulsive gravity sources are investigated. Both the symmetries of spacetime and the degree of irregularity of curvature invariants, are the same as for the Schwarzschild case. The equilibrium configurations are modelled using the neutron star polytrope equation of state.

Hypersurface foliation approach to renormalization of ADM formulation of gravity

We carry out ADM splitting in the Lagrangian formulation and establish a procedure in which (almost) all of the unphysical components of the metric are removed by using the 4D diffeomorphism and the measure-zero 3D symmetry. The procedure introduces a constraint that corresponds to the Hamiltonian constraint of the Hamiltonian formulation, and its solution implies that the 4D dynamics admits an effective description through 3D hypersurface physics. As far as we can see, our procedure implies potential renormalizability of {the ADM formulation of} 4D Einstein gravity for which a complete gauge-fixing in the ADM formulation and hypersurface foliation of geometry are the key elements. If true, this implies that the alleged unrenormalizability of 4D Einstein gravity may be due to the presence of the unphysical fields. The procedure can straightforwardly be applied to quantization around a flat background; the Schwarzschild case seems more subtle. We discuss a potential limitation of the procedure when applying it to explicit time-dependent backgrounds.

Time-Like Geodesic Motion in Schwarzschild Spacetime with Weak-Field Limit

We analyze the geodesic motion in Schwarzschild spacetime with the weak-field limit. We investigate all geodesic types of the test particle by solving the geodesic equation and analyzing the behavior of effective potential. At the same time, all kinds of orbits, which are allowed according to the energy level corresponding to the effective potential, are numerically simulated in detail. Then we discuss the effect of different parameters on the effective potential energy. We also find that the test particle falls rapidly along the fall-into orbit and the radius of stable (unstable) circular orbits become larger in the Schwarzschild spacetime with the weak-field limit than those in the Schwarzschild case.

Innermost stable circular orbits of spinning test particles in Schwarzschild and Kerr space-times

We consider the motion of classical spinning test particles in Schwarzschild and Kerr metrics and investigate innermost stable circular orbits (ISCO). The main goal of this work is to find analytically the small-spin corrections for the parameters of ISCO (radius, total angular momentum, energy, orbital angular frequency) of spinning test particles in the case of vectors of black hole spin, particle spin and orbital angular momentum being collinear to each other. We analytically derive the small-spin linear corrections for arbitrary Kerr parameter $a$. The cases of Schwarzschild, slowly rotating and extreme Kerr black hole are considered in detail. For a slowly rotating black hole, the ISCO parameters are obtained up to quadratic in $a$ and particle's spin $s$ terms. From the formulas obtained it is seen that the spin-orbital coupling has attractive character when spin and angular momentum are parallel and repulsive when they are antiparallel. For the case of the extreme Kerr black hole with co-rotating particle we succeed to find the exact analytical solution for the limiting ISCO parameters for arbitrary spin. It has been shown that the limiting values of ISCO radius and frequency do not depend on the particle's spin while values of energy and total angular momentum depend on it. We have also considered circular orbits of arbitrary radius and have found small-spin linear corrections for the total angular momentum, energy and frequency at given radius. System of equations for numerical calculation of ISCO parameters for arbitrary $a$ and $s$ is also explicitly written.

Critical escape velocity for a charged particle in Ernst spacetime

Motion of a charged particle in Ernst spacetime is discussed. We study the conditions that a charged particle, originally revolving around this black hole in the innermost stable circular orbit (ISCO), will escape to infinity after being kicked by another particle or photon. The motion of the kicked particle is chaotic. The critical escape energy and velocity of the charged particle are obtained in the present paper. Comparing to the Schwarzschild case, the kicked charged particle without the radial velocity needs more energy to escape in Ernst case for l &gt; 0 and less energy to escape for l &lt; 0.

Accretion Dynamics in SdS Space

the role of the gravitational powerhouse of gigantic scale in AGN. The accretion process for such a black hole may be studied using general relativistic as well as post Newtonian scheme for various possible flow geometries with different equation of states of the accreted matter. For the sake of simplicity, hydro-dynamical flow models are generally adopted to get an analytical estimate for those accretion process. In our work, accretion process over a region far beyond the dimension of a single galaxy by a much more massive object (» 10 18 Mfl, a cluster of galaxies or a possible super-giant black hole) has been studied where the large scale feature of the space-time structure of the Universe may be relevant. In the non-rotating limit of the accreting black-hole with accelerated expansion of the Universe, Schwarzschild-de Sitter (SdS) space is chosen to be a suitable one. Pseudo-Newtonian potential describing the gravitational field of static and spherically symmetric black holes in the Universe with a repulsive cosmological constant, i.e. for SdS space, has been recently introduced by Stuchlik and Kovar. Using this potential, here we have studied the phase topology of the flow and also the possibility of formation of stationary shocks both in adiabatic and isothermal limits in accretion discs. In AGN, the existence of stationary shock is believed to be a plausible explanation of the extremely hot annular region there. In our study, in adiabatic (and in isothermal) cases, a significant region of parameter space allow shock formation and those regions shift in comparison with the same for Schwarzschild case due to the effect of cosmological constant in SdS space.

Tidal invariants for compact binaries on quasicircular orbits

We extend the gravitational self-force approach to encompass ``self-interaction'' tidal effects for a compact body of mass $\ensuremath{\mu}$ on a quasicircular orbit around a black hole of mass $M\ensuremath{\gg}\ensuremath{\mu}$. Specifically, we define and calculate at $O(\ensuremath{\mu})$ (conservative) shifts in the eigenvalues of the electric- and magnetic-type tidal tensors, and a (dissipative) shift in a scalar product between their eigenbases. This approach yields four gauge-invariant functions, from which one may construct other tidal quantities such as the curvature scalars and the speciality index. First, we analyze the general case of a geodesic in a regular perturbed vacuum spacetime admitting a helical Killing vector and a reflection symmetry. Next, we specialize to focus on circular orbits in the equatorial plane of Kerr spacetime at $O(\ensuremath{\mu})$. We present accurate numerical results for the Schwarzschild case for orbital radii up to the light ring, calculated via independent implementations in Lorenz and Regge-Wheeler gauges. We show that our results are consistent with leading-order post-Newtonian expansions, and demonstrate the existence of additional structure in the strong-field regime. We anticipate that our strong-field results will inform (e.g.) effective one-body models for the gravitational two-body problem that are invaluable in the ongoing search for gravitational waves.

Localized Energy Estimates for Wave Equations on (1+4)-dimensional Myers--Perry Space-times

Localized energy estimates for the wave equation have been increasingly used to prove various other dispersive estimates. This article focuses on proving such localized energy estimates on (1+4)-dimensional Myers--Perry black hole backgrounds with small angular momenta. The Myers--Perry space-times are generalizations of higher-dimensional Kerr backgrounds where additional planes of rotation are available while still maintaining axial symmetry. Once it is determined that all trapped geodesics have constant $r$, the method developed by Tataru and the fourth author, which perturbs off of the Schwarzschild case by using a pseudodifferential multiplier, can be adapted.

Stationary spherically symmetric one-kink model in Saez-Ballester theory of gravitation

In this paper we consider stationary Spherically symmetric kink space–time in the scalar–tensor theory of gravitation proposed by Saez and Ballester (Phys. Lett. A 113:467, 1986) in the presence of perfect fluid distribution. It is shown that spherically symmetric kink space-time does not accommodate perfect fluid distribution in this theory. Hence a vacuum model is obtained which is asymptotically flat. This model corresponds to a one kink metric in this theory. This can be considered as an analogue of usual spherically symmetric Schwarzschild case in this theory.

Late-time evolution of Dirac field around Schwarzschild-quintessence black hole

The late-time evolution of Dirac field around spherically symmetric black hole surrounded by quintessece is studied numerically. Our results show that for lower values of the quintessence state parameter ϵ, Dirac field decays as power-law tail but with a slower decay rate than the corresponding Schwarzschild case. But for ϵ&lt;-1/3, all the ℓ-poles of the Dirac field give up the power-law decay form and relax to a constant residual field at asymptotically late-times. The value of this residual field for which the field settles down varies on different surfaces. It has the lowest value on the black hole event horizon, increases as the radial distance increases and maximizes on the cosmological horizon.

Regular rotating black holes and the weak energy condition

We revisit here a recent work on regular rotating black holes. We introduce a new mass function generalizing the commonly used Bardeen and Hayward mass functions and extend the recently proposed solutions in order to accommodate a cosmological constant Λ. We discuss some aspects of the causal structure (horizons) and the ergospheres of the new proposed solutions. We also show that, in contrast with the spherically symmetrical case, the black hole rotation will unavoidably lead to the violation of the weak energy condition for any physically reasonable choice of the mass function, reinforcing the idea that the description of the interior region of a Kerr black hole is much more challenging than in the Schwarzschild case.

Absorption and Scattering Cross Section of Regular Black Holes

By using the partial wave method, we investigate the absorption of massless scalar wave from regular black hole. We numerically carry out the absorption cross section and find that the larger angular momentum quantum number l is, the smaller the corresponding maximum value of partial absorption cross section is. Comparing with Schwarzschild case, the absorption cross section of regular black holes is strengthened in both low and high frequency regions, and the absorption cross section oscillates around the geometric optical value in the high frequency region. Generally speaking, the scattering flux is strengthened and its scattering width becomes narrower in the forward direction. There are obvious contrast of scattering properties of different type of regular black hole.

Harrison Metrics for the Schwarzschild Black Hole

Based on the hidden conformal symmetry, some authors have proposed a Harrison metric for the Schwarzschild black hole. We give a procedure which can generate a family of Harrison metrics starting from a general set of SL(2, R) vector fields. By analogy with the subtracted geometry of the Kerr black hole, we find a new Harrison metric for the Schwarzschild case. Its conformal generators are also investigated using the Killing equations in the near-horizon limit.