We study the thermodynamics of a moving Schwarzschild black hole, identifying the temperature and entropy in a relativistic scenario. Furthermore, we set arguments in a framework relating invariant geometrical quantities under global spacetime transformations and the dispersion relation of the system. We then extended these arguments in order to consider more general dispersion relations, and identify criteria to rule them out.
In this manuscript, we investigate the effects of a modified dispersion relation on the thermodynamics of Schwarzschild black hole surrounded by the quintessence matter. We find that the MDR-correction states the same lower bound on the horizon, while the quintessence matter specifies the upper bound to the horizon depending on the state parameter. Due to MDR-correction and quintessence matter presence, we observe modifications in equation of state and specific heat functions of black hole. We show that a remnant can occur according to quintessence matter, and the black hole's stability depend…
We expand upon our previous analysis of numerical moving-puncture simulations of the Schwarzschild spacetime. We present a derivation of the family of analytic stationary 1+log foliations of the Schwarzschild solution, and outline a transformation to isotropic-like coordinates. We discuss in detail the numerical evolution of standard Schwarzschild puncture data, and the new time-independent 1+log data. Finally, we demonstrate that the moving-puncture method can locate the appropriate stationary geometry in a robust manner when a numerical code alternates between two forms of 1+log slicing duri…
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 Painleve-Gullstrand coordinates.
New developments related to our recent study of the accretion of the Vlasov gas onto a moving Schwarzschild black hole are presented. We discuss the low-temperature limit of the mas accretion rate and a simple Monte Carlo simulation used to check the results obtained in this limit. We also comment on several numerical aspects related with momentum integrals expressing the particle density current and the particle density.
In this paper we investigate the impact of modified dispersion relations (MDR) on (Anti)de Sitter-Schwarzschild black holes. In this context we find the temperature of black holes can be derived with important corrections. In particular given a specific MDR the temperature has a maximal value such that it can prevent black holes from total evaporation. The entropy of the (A)dS black holes is also obtained with a logarithmic correction.
We investigate accretion of dark matter onto a moving Schwarzschild black hole. The dark matter is modeled by the collisionless Vlasov gas, assumed to be in thermal equilibrium at infinity. We derive an exact stationary solution and provide a compact formula for the mass accretion rate. In general, the mass accretion rate is a nonmonotonic function of the black hole velocity. A monotonic relation (the accretion rate proportional to the Lorentz factor associated with the velocity of the black hole) is obtained for high asymptotic temperatures of the gas. The derived accretion rates are relevant…
The motion of matter immersed in a radiation field is affected by radiation drag, as a result of scattering or absorption and re-emission. The resulting friction-like drag, also known as Poynting-Robertson effect, has been recently studied in the general relativistic background of the Schwarzschild and Kerr metric, under the assumption that all photons in the radiation field possess the same angular momentum. We calculate here the signal produced by an emitting point-like specific source moving in a Schwarzschild spacetime under the influence of such a radiation field. We derive the flux, reds…
The success of the moving puncture method for the numerical simulation of black hole systems can be partially explained by the properties of stationary solutions of the 1+log coordinate condition. We compute stationary 1+log slices of the Schwarzschild spacetime in isotropic coordinates in order to investigate the coordinate singularity that the numerical methods have to handle at the puncture. We present an alternative integration method to obtain isotropic coordinates that simplifies numerical integration and that gives direct access to a local expansion in the isotropic radius near the punc…
We derive an exact, axially symmetric solution representing stationary accretion of the relativistic, collisionless Vlasov gas onto a moving Schwarzschild black hole. The gas is assumed to be in thermal equilibrium at infinity, where it obeys the Maxwell-Jüttner distribution. The Vlasov equation is solved analytically in terms of suitable action-angle variables. We provide explicit expressions for the particle current density and accretion rates. In the limit of infinite asymptotic temperature of the gas, we recover the qualitative picture known form the relativistic Bondi-Hoyle-Lyttleton accr…
We discuss charged particles motion in a spacetime of a weakly magnetized static non-rotating black hole. We study under which conditions a charged particle originally revolving around the black hole at a circular orbit after being kicked by another particle or photon can escape to infinity. We determine the escape velocity for particles at the innermost stable circular orbits and discuss the properties of particles moving with near-critical velocity. We show that in a general case such a motion is chaotic.
The behavior of isothermal plasma waves has been analyzed near the Schwarzschild horizon. We consider a non-rotating background with non-magnetized and magnetized plasmas. The general relativistic magnetohydrodynamical equations for the Schwarzschild planar analogue spacetime with an isothermal state of the plasma are formulated. The perturbed form of these equations is linearized and Fourier analyzed by introducing simple harmonic waves. The determinant of these equations in each case leads to a complex dispersion relation, which gives complex values of the wave number. This has been used to …
We provide simple and accurate analytical expressions for the Regge poles of the Schwarzschild black hole. This is achieved by using third-order WKB approximations to solve the radial wave equations for spins 0, 1 and 2. These results permit us to obtain analytically the dispersion relation and the damping of the "surface waves" lying on the photon sphere of the Schwarzschild black hole and which generate the weakly damped quasinormal modes of its spectrum. Our results could be helpful in order to simplify considerably the description of wave scattering from the Schwarzschild black hole as wel…
We analyze the dispersion modes of hot plasma around Schwarzschild de-Sitter event horizon in the presence of Veselago medium. For this purpose, we apply ADM 3+1 formalism to develop GRMHD equations for the Schwarzschild de-Sitter spacetime. Implementation of linear perturbation yields perturbed GRMHD equations that are used for the Fourier analysis of rotating (non-magnetized and magnetized) plasma. Wave analysis is described by the graphical representation of the wave vector, refractive index, change in refractive index, phase and group velocities. The outcome of this work supports the valid…
We apply the ADM 3+1 formalism to derive the general relativistic magnetohydrodynamic equations for cold plasma in spatially flat Schwarzschild metric. Respective perturbed equations are linearized for non-magnetized and magnetized plasmas both in non-rotating and rotating backgrounds. These are then Fourier analyzed and the corresponding dispersion relations are obtained. These relations are discussed for the existence of waves with positive angular frequency in the region near the horizon. Our results support the fact that no information can be extracted from the Schwarzschild black hole. It…
A new method is applied for deriving the redshift and shadow of a Schwarzschild black hole moving freely in the de Sitter expanding universe as recorded by a co-moving remote observer. This method is manly algebraic focusing on the transformation of the conserved quantities under the de Sitter isometry relating the black hole proper co-moving frame to observer's one. Hereby one extracts the general expressions of the redshifts and shadows of the black holes having peculiar velocities but their expressions are too extended to be written down here. Therefore, only some particular cases and intui…
The kinematical effect induced by the transversal motion of a gravitational lens on the frequency shift of light has been investigated in detail, while the effect of the radial motion is thought to be much smaller than the transversal one and thus has usually been neglected. In this work, we find that the radial velocity effect on the frequency shift has the same order of magnitude as that of the transversal velocity effect, when the light emitter (or the receiver) is close to the gravitational lens with the distance between them being an impact parameter scale. The significant velocity effect…
We model a radiating, moving black hole in terms of a worldtube-nullcone boundary value problem. We evolve this data in the region interior to the worldtube but exterior to a trapped surface by means of a characteristic evolution based upon a family of ingoing null hypersurfaces. Data on the worldtube is induced from a Schwarzschild spacetime but the worldtube is allowed to move relative to the static Schwarzschild trajectories. When the worldtube is stationary (static or rotating in place), a distorted black hole inside it evolves to equilibrium with the Schwarzschild boundary. A boost of the…
A particle in the vicinity of a Schwarzschild black hole is known to trace a geodesic of the Schwarzschild background, to a first approximation. If the interaction of the particle with its own field (scalar, electromagnetic or gravitational) is taken into account, the path is no longer a background geodesic and the self-force that the particle experiences needs to be taken into consideration. In this dissertation, a recently proposed method for the calculation of the self-force is implemented. According to this method the self-force comes from the interaction of the particle with the Regular-R…
Currently the most popular method to evolve black-hole binaries is the ``moving puncture'' method. It has recently been shown that when puncture initial data for a Schwarzschild black hole are evolved using this method, the numerical slices quickly lose contact with the second asymptotically flat end, and end instead on a cylinder of finite Schwarzschild coordinate radius. These slices are stationary, meaning that their geometry does not evolve further. We will describe these results in the context of maximal slices, and present time-independent puncture-like data for the Schwarzschild spaceti…
The 3+1 GRMHD equations for Schwarzschild spacetime in Rindler coordinates with isothermal state of plasma are formulated. We consider the cases of non-rotating and rotating backgrounds with non-magnetized and magnetized plasmas. For these cases, the perturbed form of these equations are linearized and Fourier analyzed by introducing plane wave type solutions. The determinant of these equations in each case leads to two dispersion relations which give value of the wave number $k$. Using the wave number, we obtain information like phase and group velocities etc. which help to discuss the nature…
It is well-known that the $(1+1)$ dimensional Schwarzschild and spatially flat FLRW spacetimes are conformally flat. This work examines entanglement harvesting from the conformal field vacuums in these spacetimes between two Unruh-DeWitt detectors, moving along outgoing null trajectories. In $(1+1)$ dimensional Schwarzschild spacetime, we considered the Boulware and Unruh vacuums for our investigations. In this analysis, one observes that while entanglement harvesting is possible in $(1+1)$ dimensional Schwarzschild and $(1+3)$ dimensional de Sitter spacetimes, it is not possible in the $(1+1)…
Particles moving on a radial ray with respect to a Schwarzschild mass are shown to have SU(1,1)/U(1) dynamical symmetry. This symmetry is used to identify a global time variable shared by all test particles moving on a radial ray. With this time variable, Hamilton's equations for particles on a radial ray have the form of the harmonic oscillator. The SU(1,1)/U(1) symmetry, in conjunction with the well defined time variable and observer, assists in determining the quantization of the motion of the test particles.
This paper is devoted to investigate the cold plasma wave properties outside the event horizon of the Schwarzschild planar analogue. The dispersion relations are obtained from the corresponding Fourier analyzed equations for non-rotating and rotating, non-magnetized and magnetized backgrounds. These dispersion relations provide complex wave numbers. The wave numbers are shown in graphs to discuss the nature and behavior of waves and the properties of plasma lying in the vicinity of the Schwarzschild event horizon.
We consider a synchrotron radiation from a charged particle moving in a bound orbit around a weakly magnetized Schwarzschild black hole (a static black hole immersed into a constant uniform magnetic field) in its equatorial plane, perpendicular to the magnetic field. In particular, we study the case when the Lorentz force acting on the charged particle is directed outward from the black hole. The particle is initially moving in a nongeodesic bound orbit which due to synchrotron radiation decays to a nongeodesic circular orbit. We study this transition and calculate the radiated power and energ…
Point charge, radially moving in the vicinity of a black hole is considered. Electromagnetic field in wave zone and in the small neighbourhood of the charge is calculated. Numerical results of the calculation of the spectrum of electromagnetic radiation of the point charge are presented. Covariant approach for the calculation of electromagnetic self-force is used for the case of the slowly moving charge. Numerical results for the self-force in the case of slow motion of the particle are obtained and compared to the results in literature.
In the present work, our main objective is to investigate the orbits of spinning test particles around a Schwarzschild black hole under the influence of a quintessence matter field (SQBH). We begin with the dynamics of the spinning test particles around SQBH which is governed by the Mathisson-Papapetrou-Dixon (MPD) equations under the pole-dipole approximation, where the gravitational field and the higher multipoles of the particle are neglected. Depending on the types of saddle points,the effective potential are classified and the possibility of chaotic orbits is discussed. The inner most sta…
We calculate the self-force acting on a particle with scalar charge moving on a generic geodesic around a Schwarzschild black hole. This calculation requires an accurate computation of the retarded scalar field produced by the moving charge; this is done numerically with the help of a fourth-order convergent finite-difference scheme formulated in the time domain. The calculation also requires a regularization procedure, because the retarded field is singular on the particle's world line; this is handled mode-by-mode via the mode-sum regularization scheme first introduced by Barack and Ori. Thi…
In this paper, we have extended the previous study of the thermodynamics and phase transition of the Schwarzschild black hole in the rainbow gravity to the Schwarzschild-AdS black hole where metric depends on the energy of a probe. Making use of the Heisenberg uncertainty principle and the modified dispersion relation, we have obtained the modified local Hawking temperature and thermodynamic quantities in an isothermal cavity. Moreover, we carry out the analysis of constant temperature slices of a black hole. As a result, we have shown that there also exists another Hawking-Page-like phase tra…
The modified dispersion relation as a common feature of all quantum gravity scenarios provides a perturbation framework upon which the black hole thermodynamics can be corrected. In this letter, we obtain the corrections to the $d$-dimensional Schwarzschild black hole thermodynamics by utilizing the extra dimensional form of the modified dispersion relation, leading to the modification of the Cardy-Verlinde formula. Furthermore, we show that the modified dispersion relation corrections to the Cardy-Verlinde formula can be taken into account by redefining the Virasoro operator and the central c…
We study a quantum-corrected Schwarzschild black hole proposed recently in Loop Quantum Gravity. Prompted by the fact that corrections to the innermost stable circular orbit of Schwarzschild diverge, we investigate timelike and null radial geodesics. Massive particles moving radially outwards are confined, while photons make it to infinity with infinite redshift. This unsettling physics, which deviates radically from both Schwarzschild (near the horizon) and Minkowski (at infinity) is due to repulsion by the negative quantum energy density that makes the quasilocal mass vanish as one approache…
We study the thermodynamic phase transition in the rainbow Schwarzschild black hole where the metric depends on the energy of the test particle. Identifying the black hole temperature with the energy from the modified dispersion relation, we obtain the modified entropy and thermodynamic energy along with the modified local temperature in the cavity to provide well defined black hole states. It is found that apart from the conventional critical temperature related to Hawking-Page phase transition there appears an additional critical temperature which is of relevance to the existence of a locall…
The moving puncture method is analyzed for a single, non-spinning black hole. It is shown that the puncture region is not resolved by current numerical codes. As a result, the geometry near the puncture appears to evolve to an infinitely long cylinder of finite areal radius. The puncture itself actually remains at spacelike infinity throughout the evolution. In the limit of infinite resolution the data never become stationary. However, at any reasonable finite resolution the grid points closest to the puncture are rapidly drawn into the black hole interior by the Gamma-driver shift condition. …
We study tidal effects induced by a particle moving along a slightly eccentric equatorial orbits in a Schwarzschild spacetime within the gravitational self-force framework. We compute the first order (conservative) corrections in the mass-ratio to the eigenvalues of the electric-type and magnetic-type tidal tensors up to the second order in eccentricity and through the 9.5 post-Newtonian order. Previous results on circular orbits are thus generalized and recovered in a proper limit.
The Hamiltonian formulation of modified dispersion relations (MDRs) allows for their implementation on generic curved spacetimes. In turn it is possible to derive phenomenological effects. I will present how to construct the kappa-Poincare dispersion relation on curved spacetimes, its spherically symmetric realizations, among them the kappa deformation of Schwarzschild spacetime, and its implementation on Friedmann-Lemaitre-Robertson-Walker spacetimes with arbitrary scale factor. In addition we will construct the general first order modifications of the general relativistic dispersion relation…
Stochastic motion of massless particles near event horizons in Schwarzschild and Reissner-Nordström spacetimes
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；School of Mathematics, Guangxi University（广西大学数学学院）；Guangxi Center for Mathematical Research, Guangxi University（广西大学广西数学研究中心）；Ningde No.5 Middle School of Fujian（福建省宁德第五中学）
Examination of the free-fall motion of particles of extremely high-speed in the Schwarzschild geometry reveals that the gravitational acceleration of such particles is reversed when measured in Schwarzschild coordinates. High-speed particles decelerate when moving radially downward, and they accelerate when moving upward. The onset of this abnormal behavior occurs at a speed of 1/Sqrt(3) times the local value of the speed of light. However, the gravitational force always remains attractive. PACS numbers: 04.20.-q, 04.20.Cv, 01.65+g
The motion of a charged particle is influenced by the self-force arising from the particle's interaction with its own field. In a curved spacetime, this self-force depends on the entire past history of the particle and is difficult to evaluate. As a result, all existing self-force evaluations in curved spacetime are for particles moving along a fixed trajectory. Here, for the first time, we overcome this long-standing limitation and present fully self-consistent orbits and waveforms of a scalar charged particle around a Schwarzschild black hole.
A modified (Anti-)de Sitter Schwarzschild black hole solution is presented in the framework of rainbow gravity with a cosmological constant. Its thermodynamical properties are investigated. In general the temperature of modified black holes is dependent on the energy of probes which take the measurement. However, a notion of intrinsic temperature can be introduced by identifying these probes with radiation particles emitted from black holes. It is interesting to find that the Hawking temperature of this sort of black holes can be reproduced by employing the extended uncertainty principle and m…
In an accompanying paper, we have formulated two types of regulariz_ation methods to calculate the scalar self-force on a particle of charge $q$ moving around a black hole of mass $M$, one of which is called the ``power expansion regularization''. In this paper, we analytically evaluate the self-force (which we also call the reaction force) to the third post-Newtonian (3PN) order on the scalar particle in circular orbit around a Schwarzschild black hole by using the power expansion regularization. It is found that the $r$-component of the self-force arises at the 3PN order, whereas the $t$- an…
We discuss the problem of canonical quantization of a free real massive scalar field in the Schwarzschild spacetime. It is shown that a consistent procedure of canonical quantization of the field can be carried out without taking into account the black hole interior, so that in the resulting theory the canonical commutation relations are satisfied exactly, and the Hamiltonian has the standard form. However, unlike some papers, in which the expansion of the quantum field in spherical harmonics is used, here we use an expansion in scatteringlike states for energies larger than the mass of the fi…
In this work the dynamics of a spinning particle moving in the Schwarzschild background is studied. In particular, the methods of Poincaré section and recurrence analysis are employed to discern chaos from order. It is shown that the chaotic or regular nature of the orbital motion is reflected on the gravitational waves.
We investigate a Boltzmann gas at equilibrium with its center of mass moving on a circular geodesics in the Schwarzschild field. As a consequence of Tolman's law we find that a central comoving observer measures oscillations of the temperature and of other thermodynamic quantities with twice the frequencies that are known from test-particle motion. We apply this scheme to the gas dynamics in the gravitational fields of the planets of the Solar System as well as to strong-field configurations of neutron stars and black holes.
Using the monodromy method we calculate the asymptotic quasinormal (QN) frequencies of an electromagnetic field moving in D-dimensional Schwarzschild and Schwarzschild de Sitter (SdS) black holes ($D\geq 4$). For the D-dimensional Schwarzschild anti-de Sitter (SadS) black hole we also compute these frequencies with a similar method. Moreover, we calculate the electromagnetic normal modes of the D-dimensional anti-de Sitter (AdS) spacetime.
We revisit the problem of computing the self-force on a scalar charge moving along an eccentric geodesic orbit around a Schwarzschild black hole. This work extends previous scalar self-force calculations for circular orbits, which were based on a regular "effective" point-particle source and a full 3D evolution code. We find good agreement between our results and previous calculations based on a (1+1) time-domain code. Finally, our data visualization is unconventional: we plot the self-force through full radial cycles to create "self-force loops", which reveal many interesting features that ar…
Waves propagating in the relativistic electron-positron or ions plasma are investigated in a frame of two-fluid equations using the 3+1 formalism of general relativity developed by Thorne, Price and Macdonald (TPM). The plasma is assumed to be freefalling in the radial direction toward the event horizon due to the strong gravitational field of a Schwarzschild black hole. The local dispersion relations for transverse and longitudinal waves have been derived, in analogy with the special relativistic formulation as explained in an earlier paper, to take account of relativistic effects due to the …
We use our previously developed identification of dispersion relations with Hamilton functions on phase space to locally implement the $κ$-Poincaré dispersion relation in the momentum spaces at each point of a generic curved spacetime. We use this general construction to build the most general Hamiltonian compatible with spherical symmetry and the Plank-scale-deformed one such that in the local frame it reproduces the $κ$-Poincaré dispersion relation. Specializing to Planck-scale-deformed Schwarzschild geometry, we find that the photon sphere around a black hole becomes a thick shell since pho…
We consider a spherically symmetric black hole metric in (3+1)-dimensions in presence of a positive cosmological constant $Λ$. We use a general approach as proposed in \cite{1} to transform the metric in co-moving coordinates. Then for a general metric we have evaluated the angular shadow size both in absence and presence of plasma. By using the aberration relation, we then find the corresponding shadow radius as viewed by a co-moving observer for Schwarzschild de-Sitter or Kottler spacetime. The effect of cosmological constant $Λ$ and plasma parameter $k$ on the angular size of black hole sha…
We study two different ways to analyze the Hawking evaporation of a Schwarzschild-de Sitter black hole. The first one uses the standard approach of surface gravity evaluated at the possible horizons. The second method derives its results via the Generalized Uncertainty Principle (GUP) which offers a yet different method to look at the problem. In the case of a Schwarzschild black hole it is known that this methods affirms the existence of a black hole remnant (minimal mass $M_{\rm min}$) of the order of Planck mass $m_{\rm pl}$ and a corresponding maximal temperature $T_{\rm max}$ also of the …
The 3+1 formalism of Thorne, Price and Macdonald has been used to derive the linear two-fluid equations describing transverse and longitudinal waves propagating in the two-fluid ideal collisionless plasmas surrounding a Schwarzschild black hole. The plasma is assumed to be falling in radial direction toward the event horizon. The relativistic two-fluid equations have been reformulate, in analogy with the special relativistic formulation as explained in an earlier paper, to take account of relativistic effects due to the event horizon. Here a WKB approximation is used to derive the local disper…
We compute the first-order self-force contribution to Detweiler's redshift invariant for extended bodies endowed with both dipolar and quadrupolar structure (with spin-induced quadrupole moment) moving along circular orbits on a Schwarzschild background. Our analysis includes effects which are second order in spin, generalizing previous results for purely spinning particles. The perturbing body is assumed to move on the equatorial plane, the associated spin vector being orthogonal to it. The metric perturbations are obtained by using a standard gravitational self-force approach in a radiation …
We investigate transverse electromagnetic waves propagating in a plasma influenced by the gravitational field of the Schwarzschild-de Sitter black hole. Applying 3+1 spacetime split we derive the relativistic two-fluid equations to take account of gravitational effects due to the event horizon and describe the set of simultaneous linear equations for the perturbations. We use a local approximation to investigate the one-dimensional radial propagation of Alfvén and high frequency electromagnetic waves. We derive the dispersion relation for these waves and solve it for the wave number $k$ numeri…
