Schwarzschild Case Research Articles

Hawking radiation with angular momentum and the entropy variation in a Kerr black hole

Considering Hawking radiation with angular momentum, we propose a modified Stefan–Boltzmann law about the particles with both energy and angular momentum radiated from a Kerr black hole. Based on the modified Stefan–Boltzmann law, we calculate the entropy in the interior volume of the Kerr black hole and investigate the proportional relation between the variation of this entropy and the variation of Bekenstein–Hawking entropy under Hawking radiation. Comparing to Hawking radiation without angular momentum, we find that whether the particles radiated from the black hole take angular momentum cannot affect the proportional relation between the two types of entropy. Furthermore, the proportionality coefficient of the two types of entropy is investigated. It is found that this coefficient is approximately a constant except the late stage of Hawking radiation. Moreover, the proportional relation can degenerate to the Schwarzschild case when the angular momentum of the black hole completely disappears. For a Kerr black hole, the extremal black hole can technically not prevent and stop the evaporation anymore under Hawking radiation with angular momentum.

Information paradox in a Kerr-Newman black hole under generalized Hawking radiation

Analogous to the calculation method of the interior volume inside a spherically symmetric black hole, we calculate the interior volume of a Kerr-Newman black hole. After that, we propose the generalized Stefan-Boltzmann law which can be used to investigate Hawking radiation with energy, charge and angular momentum. Based on it, the proportional relation between the entropy of the scalar field in the interior volume of the Kerr-Newman black hole and Bekenstein-Hawking entropy under generalized Hawking radiation has been investigated. Comparing to Hawking radiation without charge and angular momentum, we find that no matter the particles radiated from the black hole take charge and angular momentum or not, the two types of entropy remain the same proportional relation expression. Furthermore, the proportionality coefficient of the two types of entropy has been analyzed and discussed. It is found that the two types of entropy are approximately proportional to each other in an infinitesimal process except the late stage of Hawking radiation. Moreover, the proportional relation can degenerate to the Schwarzschild case when the charge and angular momentum of the black hole completely disappear. It illustrates that, for a Kerr-Newman black hole, different from Hawking radiation carrying only energy, the extremal black hole will technically not prevent and stop the evaporation anymore under generalized Hawking radiation, and the proportional relation obtained is applicable to investigate the evolution relation between the two types of entropy under an arbitrary Hawking radiation. Finally, based on the evolution of the proportional relation under generalized Hawking radiation, the black hole information paradox is brought up again and discussed more deeply.

Dynamical evolution of non-minimally coupled scalar field in spherically symmetric de Sitter spacetimes

We investigate the dynamical behavior of a scalar field non-minimally coupled to Einstein’s tensor and Ricci scalar in geometries of asymptotically de Sitter spacetimes. We show that the quasinormal modes remain unaffected if the scalar field is massless and the black hole is electrically chargeless. In the massive case, the coupling of both parameters produces a region of instability in the spacetime determined by the geometry and field parameters. In the Schwarzschild case, every solution for the equations of motion with ell >0 has a range of values of the coupling constant that produces unstable modes. The case ell =0 is the most unstable one, with a threshold value for stability in the coupling. For the charged black hole, the existence of a range of instability in eta is strongly related to geometry parameters presenting a region of stability independent of the chosen parameter.

Seeing relativity-II: Revisiting and visualizing the Reissner–Nordström metric

In this paper we study some features of the Reissner–Nordström metric both from an analytic and a visual point of view. We perform an accurate ray tracing and study of null geodesics in various situations. Among the issues we focus on are (i) the comparison with the Schwarzschild case, (ii) the naked singularity case, where, if the electric charge is not too large, some dark shell appears on images despite there is no horizon in the metric, and (iii) the wormhole crossing case, i.e. a visual exploration of the maximal analytic extension of the metric.

Multi-wavelength torus–jet model for Sagittarius A*

Context. The properties of the accretion/ejection flow surrounding the supermassive central black hole of the Galaxy Sgr A* will be scrutinized by the new-generation instrument GRAVITY and the Event Horizon Telescope (EHT). Developing fast, robust, and simple models of such flows is therefore important and very timely. Aims. We want to model the quiescent emission of Sgr A* from radio to mid-infrared wavelengths, using thermal and nonthermal synchrotron. The radiation is emitted by the overlay of a magnetized compact torus close to the black hole, and a large-scale magnetized jet. We compare model spectra and images to the multi-wavelength observable constraints available to date. We simulate EHT observations at 1.3 mm of the best-fit model for different inclinations. methods. We use a simple analytic description for the geometry of the torus and jet. We model their emission by thermal synchrotron and κ-distribution synchrotron, respectively. We use relativistic ray tracing to compute simulated spectra and images, restricting our analysis to the Schwarzschild (zero spin) case. A best-fit is found by adjusting the simulated spectra to the latest observed data, and we check the consistency of our spectral best fits with the radio-image sizes and infrared spectral index constraints. We use the open-source eht-imaging library to generate EHT-reconstructed images. Results. We find perfect spectral fit ( χred2 ≈ 1) both for nearly face-on and nearly edge-on views. These best fits give parameter values very close to those found by the most recent numerical simulations, which are much more complex than our model. The intrinsic radio size of Sgr A* is found to be in reasonable agreement with the centimetric observed constraints. Our best-fit infrared spectral index is in perfect agreement with the latest constraints. Our emission region at 1.3 mm, although larger than the early-EHT Gaussian best fit, does contain bright features at the ≲40 μas scale. EHT-reconstructed images show that torus/jet-specific features persist after the reconstruction procedure, and that these features are sensitive to inclination. Conclusions. The main interest of our model is to provide a simple and fast model of the quiescent state of Sgr A*, which gives extremely similar results to those of state-of-the-art numerical simulations. Our model is easy to use and we publish all the material necessary to reproduce our spectra and images, meaning that anyone interested can use our results relatively straightforwardly. We hope that such a public tool will be useful in the context of the recent and near-future GRAVITY and EHT results.

Boundedness and Decay for the Teukolsky Equation on Kerr Spacetimes I: The Case |a|ll M

We prove boundedness and polynomial decay statements for solutions of the spin pm ,2 Teukolsky equation on a Kerr exterior background with parameters satisfying |a|ll M. The bounds are obtained by introducing generalisations of the higher order quantities P and {underline{P}} used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters |a|<M. As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.

Charged Hawking radiation and the entropy variation in a Reissner–Nordström black hole

Under charged Hawking radiation, we calculate the entropy of the scalar field in the interior volume of a Reissner–Nordström (RN) black hole and investigate the proportional relation between this entropy and the Bekenstein–Hawking entropy. Comparing with uncharged Hawking radiation, we find that whether the particles radiated from the black hole are charged cannot affect the proportional relation between the two types of entropy. Furthermore, the proportionality coefficient of the two types of entropy has been analyzed and discussed. The result shows that the variation of scalar field entropy inside the black hole is approximately proportional to the variation of Bekenstein–Hawking entropy in an infinitesimal evaporation process. Moreover, the proportional relation can degenerate to the Schwarzschild case when the charge of the black hole completely disappears. For a charged black hole, the extremal black hole will technically not prevent and stop the evaporation anymore under charged Hawking radiation.

Entropy in the interior of a Kerr black hole

Christodoulou et al have shown that the interior volume of a Schwarzschild black hole grows linearly with time. Subsequently, their conclusion has been extended to the Reissner–Nordström (RN) and Kerr black holes. Meanwhile, the entropy of the scalar field inside a Schwarzschild black hole has also been calculated. In this paper, a general method calculating the number of quantum states of the scalar field inside the black hole is given, which can be used in an arbitrary black hole. After introducing the two important assumptions as the black-body radiation assumption and the quasi-static process assumption, the entropy of the scalar field inside a Kerr black hole is calculated using the differential form, and we find that the variation of the entropy is proportional to the variation of the Bekenstein–Hawking entropy except the ending of the black hole evaporation. Similarly, we recalculate the entropy of the scalar field inside a Schwarzschild black hole and demonstrate that the entropy inside a Kerr black hole can exactly degenerate to the Schwarzschild black hole. As well as, we find that the proportionality coefficient between the entropy of the scalar field and the Bekenstein–Hawking entropy in Schwarzschild case, which is obtained using the differential form, is half of that given in the previous literature. Furthermore, we investigate the total entropy of Kerr and Schwarzschild black holes and find that they all increase with time. It means that the black holes, evolution with Hawking radiation satisfies the second law of thermodynamics. Finally, the black hole information paradox is brought up again and discussed.

Gravitational Perturbations in Einstein Aether Black Hole Spacetime**Supported by the National Natural Science Foundation of China under Grant No 11247013, and the Fund under Grant No QSQC1708.

We study the gravitational perturbations in Einstein aether black hole spacetime and find that the quasinormal modes (QNMs) of the first kind of aether black hole are similar to that of a Lorentz violation (LV) model, the quantum electrodynamics (QED) extension limit of standard model extension. These similarities between completely different backgrounds may imply that LV in the gravity sector and LV in the matter sector have some connections: damping QNMs more rapidly and prolonging its oscillation period. Compared to the Schwarzschild case, the first kind of black holes have larger damping rates and the second ones have lower damping rates, and they all have smaller real oscillation frequency. These differences could be detected by the new generation of gravitational antennas.

Light deflection and Gauss–Bonnet theorem: definition of total deflection angle and its applications

In this paper, we re-examine the light deflection in the Schwarzschild and the Schwarzschild-de Sitter spacetime. First, supposing a static and spherically symmetric spacetime, we propose the definition of the total deflection angle $\alpha$ of the light ray by constructing a quadrilateral $\Sigma^4$ on the optical reference geometry ${\cal M}^{\rm opt}$ determined by the optical metric $\bar{g}_{ij}$. On the basis of the definition of the total deflection angle $\alpha$ and the Gauss-Bonnet theorem, we derive two formulas to calculate the total deflection angle $\alpha$; (i) the angular formula that uses four angles determined on the optical reference geometry ${\cal M}^{\rm opt}$ or the curved $(r, \phi)$ subspace ${\cal M}^{\rm sub}$ being a slice of constant time $t$ and (ii) the integral formula on the optical reference geometry ${\cal M}^{\rm opt}$ which is the areal integral of the Gaussian curvature $K$ in the area of a quadrilateral $\Sigma^4$ and the line integral of the geodesic curvature $\kappa_g$ along the curve $C_{\Gamma}$. As the curve $C_{\Gamma}$, we introduce the unperturbed reference line that is the null geodesic $\Gamma$ on the background spacetime such as the Minkowski or the de Sitter spacetime, and is obtained by projecting $\Gamma$ vertically onto the curved $(r, \phi)$ subspace ${\cal M}^{\rm sub}$. We demonstrate that the two formulas give the same total deflection angle $\alpha$ for the Schwarzschild and the Schwarzschild--de Sitter spacetime. In particular, in the Schwarzschild case, the result coincides with Epstein--Shapiro's formula when the source $S$ and the receiver $R$ of the light ray are located at infinity. In addition, in the Schwarzschild--de Sitter case, there appear order ${\cal O}(\Lambda m)$ terms in addition to the Schwarzschild-like part, while order ${\cal O}(\Lambda)$ terms disappear.

Tidal Forces in Dyonic Reissner-Nördstrom Black Hole

This paper investigates the tidal as well as magnetic charge effects produced in dyonic Reissner-Nordström black hole. We evaluate Newtonian radial acceleration using radial geodesics for freely falling test particles. We establish system of equations governing radial and angular tidal forces using geodesic deviation equation and discuss their solutions for bodies falling freely towards this black hole. The radial tidal force turns out to be compressing outside the event horizon whereas the angular tidal force changes sign between event and Cauchy horizons unlike Schwarzschild black hole. The radial geodesic component starts decreasing in dyonic Reissner-Nordström black hole unlike Schwarzschild case. We conclude that magnetic charge strongly affects the radial as well as angular components of tidal force.

The geodesics structure of Schwarzschild black hole immersed in an electromagnetic universe

The geodesic equations are considered in a spacetime that represents a Schwarzschild metric coupled to a uniform external electromagnetic (em) field. Due to the em field horizon shrinks and geodesics are modified. By analyzing the behavior of the effective potentials for the massless and massive particle we study the radial and circular trajectories. Radial geodesics for both photons and particles are solved exactly. It is shown that a particle that falls toward the horizon in a finite proper time slows down so that the particle reaches the singularity slower than Schwarzschild case. Timelike and null circular geodesics are investigated. We have shown that, there are no stable circular orbits for photons, however stable and unstable second-kind orbits exist for the massive particle. An exact analytical solution for the innermost stable circular orbits (ISCO) has been obtained. It has been shown that the radius of the ISCO shrinks due to the presence of the em field.

Observational signatures of spherically-symmetric black hole spacetimes

A binary system composed of a supermassive black hole and a pulsar orbiting around it is studied. The motivation for this study arises from the fact that pulsar timing observations have proven to be a powerful tool in identifying physical features of the orbiting companion. In this study, taking into account a general spherically-symmetric metric, we present analytic calculations of the geodesic motion, and the possible deviations with respect to the standard Schwarzschild case of General Relativity. In particular, the advance at periastron is studied with the aim of identifying corrections to General Relativity. A discussion of the motion of a pulsar very close the supermassive central black hole in our Galaxy (Sgr A*) is reported.

Effect of the Kerr Metric on Photosperic Radius Expansion in X Ray Burst

The main objective of this paper is to study general relativistic effects on the photospheric radius expansion during an X-ray burst. We examine how the Kerr metric causes a shift in the effective temperature and radiation flux with respect to the Schwarzschild values during mass accretion onto a neutron star or a black hole resulting in the X-ray burst. The spin of the compact object is used up to the maximal Kerr limit χ = 0.99 with different latitudes of accretion emission. The amplitude of temperature shift relative to the Schwarzschild case is found to be δT/T ≈ − (10−3 − 10−4) for the range χ = 0.1 − 0.99 at latitudes θ = 0o , 30o, 45o and 88o. The ratio of emission flux in the Kerr metric to that in the Schwarzschild metric, F(K)/F(S), is found to be less than unity. It goes up to a maximum of 0.9 for the lowest nonzero value of the spin parameter (i.e., 0.1). For the maximal Kerr limit, χ = 0.99 , it saturates near 0.8. This effect is more prominent towards the pole. This reduction in temperature and flux is found to be consistent with the absence of photospheric radius expansion in the X Ray burst LMXB 4U 1608-52, observed by NuSTAR. Although this is not uniquely ascribed to the metric, it is believed that the spacetime metric effect in the burst phenomena can be used as a probe for testing general relativity. Also, the shift in temperature or the radiation flux might have an observable signature in the element synthesis processes in such environments.

Universal black hole stability in four dimensions

We show that four-dimensional black holes become stable below certain mass when the Einstein-Hilbert action is supplemented with higher-curvature terms. We prove this to be the case for an infinite family of ghost-free theories involving terms of arbitrarily high order in curvature. The new black holes, which are non-hairy generalizations of Schwarzschild's solution, present a universal thermodynamic behavior for general values of the higher-order couplings. In particular, small black holes have infinite lifetimes. When the evaporation process makes the semiclassical approximation break down (something that occurs after a time which is usually infinite for all practical purposes), the resulting object retains a huge entropy, in stark contrast with Schwarzschild's case.

Kerr/CFT correspondence in a 4D extremal rotating regular black hole with a non-linear magnetic monopole

We carry out the Kerr/CFT correspondence in a four-dimensional extremal rotating regular black hole with a non-linear magnetic monopole (NLMM). One problem in this study would be whether our geometry can be a solution or not. We search for the way making our rotating geometry into a solution based on the fact that the Schwarzschild regular geometry can be a solution. However, in the attempt to extend the Schwarzschild case that we can naturally consider, it turns out that it is impossible to construct a model in which our geometry can be a exact solution. We manage this problem by making use of the fact that our geometry can be a solution approximately in the whole space-time except for the black hole's core region. As a next problem, it turns out that the equation to obtain the horizon radii is given by a fifth-order equation due to the regularization effect. We overcome this problem by treating the regularization effect perturbatively. As a result, we can obtain the near-horizon extremal Kerr (NHEK) geometry with the correction of the regularization effect. Once obtaining the NHEK geometry, we can obtain the central charge and the Frolov–Thorne temperature in the dual CFT. Using these, we compute its entropy through the Cardy formula, which agrees with the one computed from the Bekenstein–Hawking entropy.

Null and timelike geodesics of the Schwarzschild black hole with string cloud background

The trajectories of the time-like and null geodesics for radial and circular motion of the Schwarzschild black hole with string cloud background are investigated and compared with the Schwarzschild case without string clouds. It is found that in the presence of the string cloud parameter, the radius of the orbits is larger than the radius of the orbits in the case of the Schwarzschild black hole without string cloud parameter. Effective potential is calculated and it is observed that as the value of string cloud parameter increases the particle can more easily escape to infinity. Stability of the circular orbits is also discussed.

A Zeroth Law Compatible Model to Kerr Black Hole Thermodynamics

We consider the thermodynamic and stability problem of Kerr black holes arising from the nonextensive/nonadditive nature of the Bekenstein–Hawking entropy formula. Nonadditive thermodynamics is often criticized by asserting that the zeroth law cannot be compatible with nonadditive composition rules, so in this work we follow the so-called formal logarithm method to derive an additive entropy function for Kerr black holes also satisfying the zeroth law’s requirement. Starting from the most general, equilibrium compatible, nonadditive entropy composition rule of Abe, we consider the simplest non-parametric approach that is generated by the explicit nonadditive form of the Bekenstein–Hawking formula. This analysis extends our previous results on the Schwarzschild case, and shows that the zeroth law-compatible temperature function in the model is independent of the mass–energy parameter of the black hole. By applying the Poincaré turning point method, we also study the thermodynamic stability problem in the system.

Scalar field self-force effects on a particle orbiting a Reissner-Nordström black hole

Scalar field self-force effects on a scalar charge orbiting a Reissner-Nordstr\"om black hole are investigated. The scalar wave equation is solved analytically in a post-Newtonian framework, and the solution is used to compute the self-field as well as the components of the self-force at the particle's location up to 7.5 post-Newtonian order. The energy fluxes radiated to infinity and down the hole are also evaluated. Comparison with previous numerical results in the Schwarzschild case shows a good agreement in both strong-field and weak-field regimes.

Versatile Method for Renormalized Stress-Energy Computation in Black-Hole Spacetimes.

We report here on a new method for calculating the renormalized stress-energy tensor (RSET) in black-hole (BH) spacetimes, which should also be applicable to dynamical BHs and to spinning BHs. This new method only requires the spacetime to admit a single symmetry. Thus far, we have developed three variants of the method, aimed for stationary, spherically symmetric, or axially symmetric BHs. We used this method to calculate the RSET of a minimally coupled massless scalar field in Schwarzschild and Reissner-Nordström backgrounds for several quantum states. We present here the results for the RSET in the Schwarzschild case in the Unruh state (the state describing BH evaporation). The RSET is type I at weak field, and becomes type IV at r≲2.78M. Then we use the RSET results to explore violation of the weak and null energy conditions. We find that both conditions are violated all the way from r≃4.9M to the horizon. We also find that the averaged weak energy condition is violated by a class of (unstable) circular timelike geodesics. Most remarkably, the circular null geodesic at r=3M violates the averaged null energy condition.