Second Order Perturbations of a Schwarzschild Black Hole: Inclusion of Odd Parity Perturbations

We consider perturbations of a Schwarzschild black hole that can be of both even and odd parity, keeping terms up to second order in perturbation theory, for the l = 2 axisymmetric case. We develop explicit formulae for the evolution equations and radiated energies and waveforms using the Regge–Wheeler–Zerilli approach. This formulation is useful, for instance, for the treatment in the "close limit approximation" of the collision of counterrotating black holes.     

Entropy Quantization of Schwarzschild Black Hole

The surface gravity of Schwarzschild black hole can be quantized from the test particle moving around different energy states analog to the Bohr's atomic model. We have quantized the Hawking temperature and entropy of Schwarzschild black hole from quantization of surface gravity. We also have shown that the change of entropy reduces to zero when the boundary shrinks to very small size.     

The region interior to the event horizon of the regular Hayward black hole

The Painlevé–Gullstrand coordinates allow us to explore the interior of the regular Hayward black hole. The behavior of an infalling particle in traversing the Hayward black hole is compared with the one inside the Schwarzschild and Reissner–Nordstrom singular black holes. When approaching the origin the test particle trajectories present differences depending if the center is regular or singular. The velocities of the infalling test particle into the modified Hayward black hole are analyzed as well. As compared with the normal Hayward, in the modified Hayward black hole the particle moves faster and the surface gravity is smaller.     

Falling into a Schwarzschild black hole

Consider a radially freely falling observer who plunges into a Schwarzschild black hole. In contrast to a static observer, he will have a different view of the black hole and of the outer sky. Furthermore, the relationship between the proper time of the falling observer and the proper time of a distant static observer differs from the relationship between the proper times of two static observers or two freely falling observers.     

Black hole entropy and the modified uncertainty principle: A heuristic analysis

Recently Ali et al. (2009) proposed a Generalized Uncertainty Principle (or GUP) with a linear term in momentum (accompanied by Plank length). Inspired by this idea here we calculate the quantum corrected value of a Schwarzschild black hole entropy and a Reissner-Nordström black hole with double horizon by utilizing the proposed generalized uncertainty principle. We find that the leading order correction goes with the square root of the horizon area contributing positively. We also find that the prefactor of the logarithmic contribution is negative and the value exactly matches with some earlier existing calculations. With the Reissner-Nordström black hole we see that this model-independent procedure is not only valid for single horizon spacetime but also valid for spacetimes with inner and outer horizons.     

Exact Harmonic Metric for a Uniformly Moving Schwarzschild Black Hole

The harmonic metric for Schwarzschild black hole with a uniform velocity is presented. In the limit of weak field and low velocity, this metric reduces to the post-Newtonian approximation for one moving point mass. As an application, we derive the dynamics of particle and photon in the weak-field limit for the moving Schwarzschild black hole with an arbitrary velocity. It is found that the relativistic motion of gravitational source can induce an additional centripetal force on the test particle, which may be comparable to or even larger than the conventional Newtonian gravitational force.     

Thermodynamics of the black holes under the extended generalized uncertainty principle with linear terms

In this paper, we employ the extended generalized uncertainty principle with linear terms (LEGUP) to investigate the thermodynamics properties of the Schwarzschild and Reissner–Nordström (RN) black holes. Firstly, by constructing the theoretical framework of LEGUP, the minimal temperature of the Schwarzschild black hole and the modified mass–temperature function for the black hole are calculated. Furthermore, the heat capacity function for the Schwarzschild black hole is obtained. After that, we compare LEGUP black hole thermodynamics with EGUP black hole and with the usual forms. Besides, the modification of black hole entropy is discussed, which involves a heuristic analysis of particles absorbed by the black hole. Finally, we derive the LEGUP-corrected temperature, heat capacity and entropy functions of the RN black hole.     

Analytic observation of a star orbiting a Schwarzschild black hole

A star orbiting a Schwarzschild black hole can be used as a toy model for an educational study of the relativistic effects like bending of light, geodesic precession, and frequency shift. Additionally, the finiteness of the speed of light plays a crucial role for the visual appearance of the star. We will develop an analytic method to show the difference between the actual and the apparent position of the star depending on the observation time and the observer’s inclination to the orbital plane.     

Negative mode of Schwarzschild black hole from the thermodynamic instability

The thermodynamic instability, for example the negative heat capacity, of a black hole implies the existence of off-shell negative mode(s) (unstable perturbation(s)) around the black hole geometry in the Euclidean path integral formalism of quantum gravity. We explicitly construct an off-shell unstable perturbation inspired from the negative heat capacity in the case of Schwarzschild black hole with/without a cosmological constant. We carefully check the boundary conditions, i.e. the regularity at the horizon, the traceless condition, and the normalizability.     

A simpler method for researching fermions tunneling from black holes

A new simpler mathematic method is proposed to study fermions tunneling from black holes. According to this method, by using semiclassical approximation theory, it simplifies the Dirac equation of curved spacetime and then the relationship of the gamma matrix and the component of contravariant metric is considered in order to transform the set of difficult quantum equations into a simple equation. Finally, the fermion tunneling and Hawking radiation of black holes are obtained. The method is very effective and simple, and we will take the Schwarzschild black hole with global monopole and the higher-dimensional Reissner-Nordstrom de Sitter black hole as two examples to show the fact.     

On the Ricci Tensor of Non-Stationary Axisymmetric Space-Times

We recalculate the Ricci tensors of non-stationary axisymmetric space-times originally calculated by Chandrasekhar, and we find that in the linear regime there are some common factors that did not appear in the original results. We also find some discrepancies in the non-linear terms. However, these discrepancies do not affect the well-known results concerning linear perturbations of a Schwarzschild black hole.     

Quantum black hole and the modified uncertainty principle

Recently Ali et al. (2009) [13] proposed a Generalized Uncertainty Principle (or GUP) with a linear term in momentum (accompanied by Planck length). Inspired by this idea we examine the Wheeler-DeWitt equation for a Schwarzschild black hole with a modified Heisenberg algebra which has a linear term in momentum. We found that the leading contribution to mass comes from the square root of the quantum number n which coincides with Bekenstein?s proposal. We also found that the mass of the black hole is directly proportional to the quantum number n when quantum gravity effects are taken into consideration via the modified uncertainty relation but it reduces the value of mass for a particular value of the quantum number.     

Letter: A Moving Medium Simulation of Schwarzschild Black Hole Optics

An explicit fluid flow simulation of electromagnetic wave propagation in the gravitational field of a Schwarzschild black hole is given. The fluid has a constant refractive index and a spherically symmetric inward directed flow. The resulting form of the metric leads to a new coordinate system in which the Schwarzschild vacuum is written in Gordon's form. It is shown that a closely related coordinate system interpolates between the Kerr-Schild and Painlevé-Gullstrand coordinates.     

Electromagnetic quasinormal modes of D-dimensional black holes II

By using the sixth order WKB approximation we calculate for an electromagnetic field propagating in D-dimensional Schwarzschild and Schwarzschild de Sitter (SdS) black holes its quasinormal (QN) frequencies for the fundamental mode and first overtones. We study the dependence of these QN frequencies on the value of the cosmological constant and the spacetime dimension. We also compare with the results for the gravitational perturbations propagating in the same background. Moreover we compute exactly the QN frequencies of the electromagnetic field propagating in D-dimensional massless topological black hole and for the charged D-dimensional Nariai spacetime we calculate exactly the QN frequencies of the coupled electromagnetic and gravitational perturbations.     

对自由落体的诺维科夫时间与牛顿绝对时间关系讨论

文章首先将史瓦西黑洞场中自由下落质点的固有时(诺维科夫坐标时)公式,由自然单位制化成了国际单位制中的形式.然后,根据牛顿第二定律和万有引力定律,推导出了自由下落质点经历的绝对时间公式,进而证明了广义相对论中自由落体经历的固有时,恰好等于牛顿力学给出的绝对时间.最后,对自由下落质点在黑洞内外经历的时间进行了特例计算.     

Quasinormal Modes of the Schwarzschild Black Hole Surrounded by the Quintessence Field in Rastall Gravity

The quasinormal modes of the Schwarzschild black hole surrounded by the quintessence in Rastall gravity are studied using the sixth-order Wentzel-Kramers-Brillouin approximative approach. The effect of the Rastall parameter on the quasinormal modes of gravitational, electromagnetic and massless scalar perturbations is explored. Compared to the case of Einstein gravity, it is found that, when η < 0, the gravitational field, electromagnetic field as well as massless scalar field damp more rapidly and have larger real frequency of oscillation in Rastall gravity, while when η > 0, the gravitational field, electromagnetic field as well as massless scalar field damp more slowly and have smaller real frequency of oscillation in Rastall gravity. It is also found that the gravitational field, electromagnetic field as well as massless scalar field damp more and more slowly and the real frequency of oscillation for the gravitational perturbation, electromagnetic perturbation as well as massless scalar perturbation becomes smaller and smaller as the Rastall parameter η increases. Compared among the quasinormal frequencies of gravitational, electromagnetic and massless scalar perturbations, I find that, for fixed η, (l, n), ∈ and N_q, the oscillation damps most slowly for the gravitational perturbation, mediate for the electromagnetic perturbation and most rapidly for the massless scalar perturbation, and the real frequency of oscillation is the smallest for the gravitational perturbation, mediate for the electromagnetic perturbation and the largest for the massless scalar perturbation in Rastall gravity.     

The quantum tunnelling radiation of Schwarzschild de Sitter black hole with a global monopole

Applying Parikh's quantum tunnelling method, this paper has studied the quantum tunnelling radiation of Schwarzschild de Sitter black hole with a global monopole. The result shows that the tunnelling rates at the event horizon and the cosmological horizon are related to Bekenstein--Hawking entropy and the derived radiation spectrum is not precisely thermal when considering energy conservation and self-gravitation interaction.     

Ideal hydrodynamics outside and inside a black hole: Hamiltonian description in Painlevé-Gullstrand coordinates

We show that when the Painlevé-Gullstrand coordinates are used in their Cartesian version, the Hamiltonian of relativistic ideal hydrodynamics in the vicinity of a nonrotating black hole differs by only one simple term from the corresponding Hamiltonian in a flat spacetime. The interior region of the black hole is also described in a unified way, because there is no singularity on the event horizon in Painlevé-Gullstrand coordinates. We present the exact solution describing the steady accretion of extremely hard matter (? ∝ n ²) onto a moving black hole up to the central singularity. In the local induction approximation, we derive the equation of motion for a thin vortex filament against the background of such an accretion flow. We explicitly calculate the Hamiltonian for a fluid with an ultrarelativistic equation of state, ? ∝ n ^4/3, and solve the problem of a centrally symmetric steady flow of such matter.     

Scattering and absorption of particles by a black hole involving a global monopole

Under the conditions that the wavelength of a particle is much larger than its radius of central mass, and the Schwarzschild field is weak, the scattering of a particle has been studied by many researchers. They obtained that scalar and vector particles abide by Rutherford’s angle distribution by using the low level perturbation method and the scattered field’s approximation in a weak field. The scattering cross section of a photon coincides with the section in Newton’s field of point mass. We can obtain the photon’s polarization effect by calculating the second-order perturbation in the linear Schwarzschild field. This article discusses the scattering and absorption of a particle by a black hole involving a global monopole by using the aforesaid method.     

Schwarzschild and Synge once again

We complete the historical overview about the geometry of a Schwarzschild black hole at its horizon by emphasizing the contribution made by Synge in [6] to its clarification.