Pseudo-Newtonian gravitational potential describing the gravitational field of static and spherically symmetric black holes in the universe with a repulsive cosmological constant is introduced. In order to demonstrate the accuracy of the pseudo-Newtonian approach, the related effective potential for test-particle motion is constructed and compared with its general relativistic counterpart given by the Schwarzschild-de Sitter geometry. The results indicate that such an approach could be useful in applications of developed Newtonian theories of accretion discs in astrophysically interesting situations in large galactic structures for the Schwarzschild-de Sitter spacetimes with the cosmological parameter y=(1/3)ΛM^2<10^{-6}.
When de Sitter first introduced his celebrated spacetime, he claimed, following Schwarzschild, that its spatial sections have the topology of the real projective space RP^3 (that is, the topology of the group manifold SO(3)) rather than, as is almost universally assumed today, that of the sphere S^3. (In modern language, Schwarzschild was disturbed by the non-local correlations enforced by S^3 geometry.) Thus, what we today call "de Sitter space" would not have been accepted as such by de Sitter. There is no real basis within classical cosmology for preferring S^3 to RP^3, but the general feel…
The propagation of electromagnetic plane waves with negative phase velocity (NPV) is considered in Schwarzschild-(anti-)de Sitter spacetime. It is demonstrated that NPV propagation occurs in Schwarzschild-de Sitter spacetime at lower values of the cosmological constant than is the case for de Sitter spacetime. Furthermore, we report that neither is NPV propagation observed in Schwarzschild-anti-de Sitter spacetime, nor is it possible outside the event horizon of a Schwarzschild blackhole.
In a recent paper we investigated stationary, relativistic Bondi-type accretion in Schwarzschild-(anti-)de Sitter spacetimes. Here we study their stability, using the method developed by Moncrief. The analysis applies to perturbations satisfying the potential flow condition. We prove that global isothermal flows in Schwarzschild-anti-de Sitter spacetimes are stable, assuming the test-fluid approximation. Isothermal flows in Schwarzschild-de Sitter geometries and polytropic flows in Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter spacetimes can be stable, under suitable boundary condit…
We review several aspects of anti-De Sitter (AdS) spaces in different dimensions, and of four dimensional Schwarzschild anti-De Sitter (SAdS) black hole.
We revisit the Parikh-Wilczek tunneling through the de Sitter horizon and obtain the tunneling rate in Schwarzschild-de Sitter space, which is non-thermal and closely related to the change of entropy. We discuss the thermodynamics of Schwarzschild-de Sitter space and show existence of correlation which ensured the conservation of the total entropy in the transition process of Schwarzschild-de Sitter space to de Sitter space. The correlation and the conserved entropy enlighten a way to explain the entropy in empty de Sitter space.
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…
With the post-Newtonian approxination, the energy of Schwarzschild spacetime in the Weinberg prescription is obtained. The energy for the first post-Newtonian approximation $E^{(1)} = m$ gives the Newtonian treatment of Schwarzschild spacetime. However, for the second post-Newtonian approximation, the erergy is shown that $E^{(1)}$ adds extra terms $E^{(2)}$ which consist of the energy stored in the configuration $E_{\rm config}$, in the gravitational field $E_{\rm field}$. and a term of surface integral. These extra terms gives post-Newtonian corrections to the Newtonian treatment.
The conformal structure of the Schwarzschild-de Sitter spacetime is analysed using the extended conformal Einstein field equations. To this end, initial data for an asymptotic initial value problem for the Schwarzschild-de Sitter spacetime is obtained. This initial data allows to understand the singular behaviour of the conformal structure at the asymptotic points where the horizons of the Schwarzschild-de Sitter spacetime meet the conformal boundary. Using the insights gained from the analysis of the Schwarzschild-de Sitter spacetime in a conformal Gaussian gauge, we consider nonlinear pertur…
The deep reason why the equations describing axial and polar perturbations of Schwarzschild black holes have the same spectrum is far from trivial. In this article, we revisit the original proof and try to make it clearer. Still focusing on uncharged and non-rotating black holes, we extend the results to spacetimes including a cosmological constant, which have so far mostly been investigated numerically from this perspective.
We study the thermodynamic properties of high dimensional Schwarzschild de Sitter spacetimes with the consideration of quantum effects. It is shown that by considering the cosmological constant as a variable state parameter and adding an extra term which denotes the vacuum energy, both the differential and integral mass formulas of the first law of Schwarzschild de Sitter spacetimes can be directly derived from the general Schwarzschild de Sitter metrics in a simple and natural way. Furthermore, after taking quantum effects into account, we can see that the cosmological constant must decrease …
We obtain an analytical expression for the electromagnetic quasinormal spectrum of the higher-dimensional nearly-extremal Schwarzschild-de Sitter black hole. The WKB method is used to verify the results, and a comparison with known results from previous works is briefly made as well.
The metric of arbitrary dimensional Schwarzschild black hole in the background of Friedman-Robertson-Walker universe is presented in the cosmic coordinates system. In particular, the arbitrary dimensional Schwarzschild-de Sitter metric is rewritten in the Schwarzschild coordinates system and basing on which the even more generalized higher dimensional Schwarzschild-de Sitter metric with another extra dimensions is found. The generalized solution shows that the cosmological constant may roots in the extra dimensions of space.
We analyze the scalar radiation emitted by a source in geodesic circular orbits around a Schwarzschild-de Sitter black hole. We obtain the emitted power using quantum field theory in curved spacetimes framework at tree level. We compare our results with the scalar synchrotron radiation in Schwarzschild spacetime.
The Lenz-Sommerfeld argument allows an ingenious and simple derivation of the Schwarzschild solution of Einstein equations of general relativity. In this paper, we use the same reasoning to construct the de Sitter line element.
We investigate the motion of a test particle in a d-dimensional, spherically symmetric and static space-time supported by a mass $M$ plus a $Λ$-term. The motion is strongly dependent on the sign of $Λ$. In Schwarzschild-de Sitter (SdS) space-time ($Λ> 0$), besides the physical singularity at $r=0$ there are cases with two horizons and two turning points, one horizon and one turning point and the complete absence of horizon and turning points. For Schwarzschild-Anti de Sitter (SAdS) space-time ($Λ< 0$) the horizon coordinate is associated to a unique turning point.
We introduce a calculation, based on purely topological reasoning, of the global equilibrium Hawking temperature of the Schwarzschild-de Sitter spacetime, where a Schwarzschild black hole horizon coexists with a de Sitter cosmological horizon. Our method is based on the careful calculation of the Euler characteristic of the total system, showing that this quantity completely determines the thermodynamical features of the system. The method is universal and can be applied to any structure possessing multiple horizons in general relativity.
A quantum Schwarzschild spacetime and a quantum Schwarzschild-de Sitter spacetime with cosmological constant $Λ$ are constructed within the framework of a noncommutative Riemannian geometry developed in an earlier publication. The metrics and curvatures of the quantum Schwarzschild spacetime and the quantum Schwarzschild-de Sitter spacetime are computed. It is shown that up to the second order in the deformation parameter, the quantum spacetimes are solutions of a noncommutative Einstein equation which is proposed in this paper.
As it is well known, the $0-0$ component of the Schwarzschild space can be obtained by the requirement that the geodesic of slowly moving particles match the Newtonian equation. Given this result, we show here that the remaining components can be obtained by requiring that the inside of a Newtonian ball of dust matched at a free falling radius with the external space determines that space to be Schwarzschild, if no pathologies exist. Also we are able to determine that the constant of integration that appears in the Newtonian Cosmology coincides with the spacial curvature of the FLRW metric.
Within the general class of Asymptotically Anti-de Sitter spacetimes that are asymptotic to the A-de-S Schwarzschild metric, we give a simple positive mass theorem based on arguments from causal structure. A general result for all asymptotically A-de-S spacetimes will be given elsewhere.
We study the scalar radiation emitted by a source orbiting a Schwarzschild--anti--de Sitter black hole in circular geodesics. We obtain the emitted power using quantum field theory in curved spacetimes at tree level with reflective boundary conditions at infinity. We compare our results with the ones obtained in Schwarzschild and Schwarzschild--de Sitter ($Λ> 0$) spacetimes. In the Schwarzschild-de Sitter spacetime we observe an enhancement of the emitted power associated to the lower multipole modes, when compared with the Schwarzschild case. In contrast, for the Schwarzschild-anti-de Sitter …
We provide an infinity of spacetimes which contain part of both the Schwarzschild vacuum and de Sitter space. The transition, which occurs below the Schwarzschild event horizon, involves only boundary surfaces (no surface layers). An explicit example is given in which the weak and strong energy conditions are satisfied everywhere (except in the de Sitter section) and the dominant energy condition is violated only in the vicinity of the boundary to the Schwarzschild section. The singularity is avoided by way of a change in topology in accord with a theorem due to Borde..
The weak field approximation in a model of de Sitter gravity is investigated in the static and spherically symmetric case, under the assumption that the vacuum spacetime without perturbations from matter fields is a torsion-free de Sitter spacetime. It is shown on one hand that any solution should be singular at the center of the matter field, if the exterior is described by a Schwarzschild-de Sitter spacetime and is smoothly connected to the interior. On the other, all the regular solutions are obtained, which might be used to explain the galactic rotation curves without involving dark matter…
We study the quantum evolution of black holes immersed in a de Sitter background space. For black holes whose size is comparable to that of the cosmological horizon, this process differs significantly from the evaporation of asymptotically flat black holes. Our model includes the one-loop effective action in the s-wave and large N approximation. Black holes of the maximal mass are in equilibrium. Unexpectedly, we find that nearly maximal quantum Schwarzschild-de Sitter black holes anti-evaporate. However, there is a different perturbative mode that leads to evaporation. We show that this mode …
A classical model for the interior structure of a Schwarzshild black hole which consists in creating multiple de Sitter universes with lightlike boundaries is proposed.The interaction of the boundaries is studied and a scenario leading to disconnected de Sitter universes is described.
It is widely believed that in the presence of a positive cosmological constant, heavy black holes can exhibit non-standard behaviour, namely there is a possibility that such objects would grow instead of evaporating. We point out that all those results (obtained in different frameworks) rely heavily upon the identification of the Nariai spacetime with the Schwarzschild--de Sitter (Kottler) black hole. In this note we argue that it is an incorrect assumption. As a result, previous treatments need revisiting. In particular, we show that within effective action approach, there is no solution corr…
The binary potential technique of interpolation (by M. Riesz, Acta Math. 81, 1 (1949)) is applied to some well-known metrics of general relativity. These include Schwarzschild, de Sitter and 2+1-dimensional BTZ spacetimes. In particular, the Schwarzschild-de Sitter solution is analyzed in some detail with a finite range parameter. Reasoning by the high level of non-linearity and absence of a superposition law necessitates search for alternative approaches. We propose the method of interpolation between different spacetimes as one such possibility paving the way toward controlling the two-metri…
Motivated by the existence of black holes with various topologies in four-dimensional spacetimes with a negative cosmological constant, we study axisymmetric static solutions describing any large distortions of Schwarzschild-anti-de Sitter black holes parametrized by the mass $m$. Under the approximation such that $m$ is much larger than the anti-de Sitter radius, it is found that a cylindrically symmetric black string is obtained as a special limit of distorted spherical black holes. Such a prolonged distortion of the event horizon connecting a Schwarzschild-anti-de Sitter black hole to a bla…
Using Einstein's and Weinberg's energy complex, we evaluate the energy distribution of the vaccum nonsingularity black hole solution. The energy distribution is positive everywhere and be equal to zero at origin.
Upper limits for the mass-radius ratio are derived for arbitrary general relativistic matter distributions in the presence of a cosmological constant. General restrictions for the red shift and total energy (including the gravitational contribution) for compact objects in the Schwarzschild-de Sitter geometry are also obtained in terms of the cosmological constant and of the mean density of the star.
We employ the minimal geometric deformation approach to gravitational decoupling (MGD- decoupling) in order to build an exact anisotropic version of the Schwarzschild interior solution in a space-time with cosmological constant. Contrary to the well-known Schwarzschild interior, the matter density in the new solution is not uniform and possesses subluminal sound speed. It therefore satisfies all standard physical requirements for a candidate astrophysical object.
We deal with phantom energy accretion onto the Schwarzschild de-Sitter black hole. The energy flux conservation, relativistic Bernoulli equation and mass flux conservation equation are formulated to discuss the phantom accretion. We discuss the conditions for critical accretion. It is found that mass of the black hole decreases due to phantom accretion. There exist two critical points which lie in the exterior of horizons (black hole and cosmological horizons). The results for the phantom energy accretion onto the Schwarzschild black hole can be recovered by taking $Λ\rightarrow0$.
We study thermodynamics of the Schwarzschild-de Sitter black hole in five dimensions by introducing two temperatures based on the standard and Bousso-Hawking normalizations. We use the first-law of thermodynamics to derive thermodynamic quantities. The two temperatures indicate that the Nariai black hole is thermodynamically unstable. However, it seems that black hole thermodynamics favors the standard normalization, and does not favor the Bousso-Hawking normalization.
We briefly revisit the Schwarzschild-de Sitter solution in the context of five-dimensional general relativity. We obtain a class of five-dimensional solutions of Einstein vacuum field equations into which the four-dimensional Schwarzschild-de Sitter space can be locally and isometrically embedded. We show that this class of solutions is well-behaved in the limit of lambda approaching zero. Applying the same procedure to the de Sitter cosmological model in five dimensions we obtain a class of embedding spaces which are similarly well-behaved in this limit. These examples demonstrate that the pr…
We show in this note that the Padmanabhan expression $E = a/2$ for the energy of the Schwarzschild and de Sitter spacetimes (a - the horizon radius) is also valid for the Rindler horizon, using an analogy with the electrostatic case.
In this paper we establish energy decay for solutions to the Klein-Gordon equation on the positive mass hyperboloidal anti-de Sitter Schwarzschild black hole, subject to Dirichlet, Neumann and Robin boundary conditions at infinity, for a range of the (negative) mass squared parameter. To do so we use vector field methods with a renormalised energy to avoid divergences that would otherwise appear in the energy integrals. For another region of the parameter space, we use the existence of negative energy solutions to demonstrate linear instability.
The quasinormal modes of the electromagnetic and gravitational perturbation on Schwarzschild-AdS black hole calculated by Cardoso and Lemos has been revisited. Although the equations of motion are correct some frequencies calculated previously by the authors are not. We present the new values of quasinormal modes and discuss the possible sources of problems and implications on the conclusions presented.
After considering the reference case of the motion of spinning test bodies in the equatorial plane of the Schwarzschild space-time, we generalize the results to the case of the motion of a spinning particle in the equatorial plane of the Schwarzschild-de Sitter space-time. Specifically, we obtain the loci of turning points of the particle in this plane. We show that the cosmological constant affect the particle motion when the particle distance from the black hole is of the order of the inverse square root of the cosmological constant.
The complete set of analytic solutions of the geodesic equation in a Schwarzschild--(anti) de Sitter space--time is presented. The solutions are derived from the Jacobi inversion problem restricted to the theta--divisor. In its final form the solutions can be expressed in terms of derivatives of Kleinian sigma functions. The solutions are completely classified by the structure of the zeros of the characteristic polynomial which depends on the energy, angular momentum, and the cosmological constant.
We combine notions of a maximal curvature scale in nature with that of a minimal curvature scale to construct a non-singular Schwarzschild-de Sitter black hole. We present an exact solution within the context of two-dimensional dilaton gravity. For a range of parameters the solution approaches Schwarzschild-de Sitter at large values of the radial coordinate, asymptotically approaching a de Sitter metric with constant minimal curvature, while approaching a maximal constant curvature smooth spacetime as the radial coordinate approaches zero. The spacetime is geodesically complete and generically…
We show asymptotic completeness for linear massive Dirac fields on the Schwarzschild-Anti-de Sitter spacetime. The proof is based on a Mourre estimate. We also construct an asymptotic velocity for this field.
We construct a mass or energy function for the non-Nariai class Schwarzschild-de Sitter black hole spacetime in the region between the black hole and the cosmological event horizons. The mass function is local, positive definite, continuous and increases monotonically with the radial distance from the black hole event horizon. We derive the Smarr formula using this mass function, and demonstrate that the mass function reproduces the two-temperature Schwarzschild-de Sitter black hole thermodynamics, along with a term corresponding to the negative pressure due to positive cosmological constant. …
Earlier constructed a simple nonlocal de Sitter gravity model has a cosmological solution in a very good agreement with astronomical observations. In this paper, we continue the investigation of the nonlocal de Sitter model of gravity, focusing on finding an appropriate solution for the Schwarzschild-de Sitter metric. We succeeded to solve the equations of motion in a certain approximation. The obtained approximate solution is of particular interest for examining the possible role of non-local de Sitter gravity in describing the effects in galactic dynamics that are usually attributed to dark …
We examine the orbital dynamics of the planar pseudo-Newtonian Copenhagen problem, in the case of a binary system of Schwarzschild-like primaries, such as super-massive black holes. In particular, we investigate how the Jacobi constant (which is directly connected with the energy of the orbits) influences several aspects of the orbital dynamics, such as the final state of the orbits. We also determine how the relativistic effects (i.e., the Schwarzschild radius) affect the character of the orbits, by comparing our results with the classical Newtonian problem. Basin diagrams are deployed for pr…
Perturbations of the Dirac field in the Schwarzschild spacetime are characterized by two wave equations (for two chiralities) with effective potentials which are iso-spectral and one of which is positive definite. Therefore, the stability of a Dirac field in the Schwarzschild background is straightforward. This is not so for the Schwarzschild-de Sitter case, because potentials for both chiralities have negative gaps and the fact that the stability is not automatically guaranteed in the asymptotically de Sitter case was apparently omitted in the literature. Performing the time-domain integratio…
Following the monodromy technique performed by Motl and Neitzke, we consider the analytic determination of the highly damped (asymptotic) quasi-normal modes of small Schwarzschild-de Sitter (SdS) black holes. We comment the result as compared to the recent numerical data of Konoplya and Zhidenko.
There is constructed, for each member of a one-parameter family of cosmological models, which is obtained from the Kottler-Schwarzschild-de Sitter spacetime by identification under discrete isometries, a slicing by spherically symmetric Cauchy hypersurfaces of constant mean curvature. These slicings are unique up to the action of the static Killing vector. Analytical and numerical results are found as to when different leaves of these slicings do not intersect, i.e. when the slicings form foliations.
We present a gluing construction which adds, via a localized deformation, exactly Delaunay ends to generic metrics with constant positive scalar curvature. This provides time-symmetric initial data sets for the vacuum Einstein equations with positive cosmological constant with exactly Kottler-Schwarzschild-de Sitter ends, extending the results in arXiv:0710.3365 [gr-qc].
Based on the perspective that continuum gravitational physics is an emergent quantum gravitational phenomenon, and that spacetime thermodynamic is the natural langauge in which it can be described, we derive a modified Schwarzschild-de Sitter geometry in static coordinates from quantum gravity inspired logarithmic corrections to the entropy-area law. The resulting quantum corrections to classical geometry can be encoded in explicitly radius dependent black hole and cosmological constant parameters. By considering the pure black hole case we study linear perturbations on this modified backgroun…
Recent studies suggest that the cosmological constant affects the gravitational bending of photons although the orbital equation for light in Schwarzschild-de Sitter space time is free from cosmological constant. Here we argued that the very notion of cosmological constant independency of photon orbit in the Schwarzschild-de Sitter space time is not proper. Consequently the cosmological constant has some clear contribution to the deflection angle of light rays. We stressed the importance of the study of photon trajectories from the reference objects in bending calculation, particularly for asy…
Trumpet geometries play an important role in numerical simulations of black hole spacetimes, which are usually performed under the assumption of asymptotic flatness. Our Universe is not asymptotically flat, however, which has motivated numerical studies of black holes in asymptotically de Sitter spacetimes. We derive analytical expressions for trumpet geometries in Schwarzschild-de Sitter spacetimes by first generalizing the static maximal trumpet slicing of the Schwarzschild spacetime to static constant mean curvature trumpet slicings of Schwarzschild-de Sitter spacetimes. We then switch to a…
By considering particles as smeared objects, we investigate the effects of space noncommutativity on the geodesic structure in Schwarzschild-AdS spacetime. By means of a detailed analysis of the corresponding effective potentials for particles, we find the possible motions which are allowed by the energy levels. Radial and non-radial trajectories are treated and the effects of space noncommutativity on the value of the precession of the perihelion are estimated. We show that the geodesic structure of this black hole presents new types of motion not allowed by the Schwarzschild spacetime.
Recently it has been shown that despite previous claims the cosmological constant affects light bending. In the present article we study light bending and the advance of Mercury's perihelion in the context of the Newtonian limit of Schwarzschild-de Sitter spacetime employing the special relativistic equivalence of mass and energy. In both cases, up to a constant factor, we find the same results as in the full general relativistic treatment of the same phenomena. These approximate and intuitive arguments demonstrate clearly what effects should have been expected from the presence of $Λ$ in the …
