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 corresponding to the Schwarzschild--de Sitter black hole.
The Lemaitre and Schwarzschild analytical solutions for a relativistic spherical body of constant density are linked together through the use of the Weyl quadratic invariant. The critical radius for gravitational collapse of an incompressible fluid is shown to vary continuously from 9/8 of the Schwarzschild radius to the Schwarzschild radius itself while the internal pressures become locally anisotropic.
All possible orbital trajectories and their analytical expressions in the Schwarzschild metric are presented in a single complete map characterized by two dimensionless parameters. While three possible pairs of parameters with different advantages are described, the parameter space that gives the most convenient reduction to the Newtonian case is singled out and used which leads to a new insight on Newtonian limits among other results. Numerous analytic relations are presented. A comparison is made with the widely used formulation and presentation given by S. Chandrasekhar.
We study the problem of a four-dimensional brane lying in the five-dimensional degenerate Schwarzschild-de Sitter (Nariai) black hole, in five-dimensional $F(R)$-gravity. We show that there cannot exist the brane in the Nariai bulk space except the case that the brane tension vanishes. We demonstrate that the five-dimensional Nariai bulk is unstable in a large region of the parameter space. In particular, the Nariai bulk can have classical (anti-)evaporation instabilities. The bulk instability back-reacts on the four-dimensional brane, in case that the brane tension vanishes, and the unstable …
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…
For the radial motion of massive particles with large angular momenta in Schwarzschild geometry and that of massive charged particles with large angular momenta or energy in a particular range in Reissner--Nordström geometry, there exist classically forbidden regions on the outside of the respective event horizons which scatter certain infalling geodesics or screen some of the black holes' evaporation by reflecting the emitted particles back into the black holes. Quantum tunneling across this forbidden regions is studied.
We studied the anti-evaporation of degenerate Schwarzschild-de Sitter black hole (so-called Nariai space-time) in modified $F(R)$ gravity. The analysis of perturbations of the Nariai black hole is done with the conclusion that anti-evaporation may occur in such a theory already on classical level. For several power-law $F(R)$ gravities which may describe the inflation and/or dark energy eras we presented the theory parameters bounds for occurrence of anti-evaporation and conjectured creation of infinite number of horizons.
We study the stability under the perturbation and the related anti-evaporation of the Nariai space-time in bigravity. If we impose specific condition for the solutions and parameters, we obtain asymptotically de Sitter space-time, and show the existence of the Nariai space-time as a background solution. Considering the perturbation around the Nariai space-time up to first order, we investigate the behavior of black hole horizon. We show that the anti-evaporation does not occur on the classical level in the bigravity.
We revisit the classical stability of Schwarzschild black hole in fourth-order theories of gravity. On the contrary of the stability of this black hole, it turns out that the linearized perturbations exhibit unstable modes featuring the Gregory-Laflamme instability of five-dimensional black string. This shows clearly an instability of black hole in fourth-order gravity with $α=-β/3$.
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 …
We construct a one-loop effective metric describing the evaporation phase of a Schwarzschild black hole in a spherically symmetric null-dust model. This is achieved by quantising the Vaidya solution and by chosing a time dependent quantum state. This state describes a black hole which is initially in thermal equilibrium and then the equilibrium is switched off, so that the black hole starts to evaporate, shrinking to a zero radius in a finite proper time. The naked singularity appears, and the Hawking flux diverges at the end-point. However, a static metric can be imposed in the future of the …
The question of the physical reality of the black hole interior is a recurrent one. An objection to its existence is the well known fact that the velocity of a material particle, refered to the stationary frame, tends to the velocity of light as it approaches the horizon. It is shown, using Kruskal coordinates, that a timelike radial geodesic does not become null at the event horizon.
The Schwarzschild solution has played a fundamental conceptual role in general relativity, and beyond, for instance, regarding event horizons, spacetime singularities and aspects of quantum field theory in curved spacetimes. However, one still encounters the existence of misconceptions and a certain ambiguity inherent in the Schwarzschild solution in the literature. By taking into account the point of view of an observer in the interior of the event horizon, one verifies that new conceptual difficulties arise. In this work, besides providing a very brief pedagogical review, we further analyze …
We propose that the late-time tail problem in the Reissner-Nordström (RN) spacetime is dual to a tail problem in the Schwarzschild spacetime with a different initial data set: at a fixed observation point the asymptotic decay rate of the fields are equal. This duality is used to find the decay rate for tails in RN. This decay rate is exactly as in Schwarzschild, including the case of the extremely-charged RN spacetime (ERN). The only case where any deviation from the Schwarzschild decay rate is found is the case of the tails along the event horizon of an ERN spacetime, where the decay rate is …
A model is proposed to describe a transition from a Schwarzschild black hole of mass $M_{0}$ to a Schwarzschild black hole of mass $M_{1}$ $\leq M_{0}$. The basic equations are derived from the non-vacuum Einstein field equations taking a source representing a null scalar field with a nonvanishing trace anomaly. It is shown that the nonvanishing trace anomaly of the scalar field prevents a complete evaporation.
Both the canonical and microcanonical ensembles are utilized to study the thermodynamic and evaporation properties of a closed black cosmic string whose spacetime is asymptotically anti deSitter. There are similarities and differences to the Schwarzschild-anti deSitter and 2+1 BTZ black hole solutions. It is found that there exist regimes of black string/thermal radiation equilibrium as well as stable remnant regimes. The relevance to black hole evaporation is discussed.
We investigate the future evolution of Nariai black hole which is extremal limit of Schwarzschild-de Sitter one in modified gravity. The perturbations equations around Nariai black hole are derived in static and cosmological patches for general $F(R)$-gravity. The analytical and numerical study of several realistic $F(R)$-models shows the occurence of rich variety of scenarios: instabilities, celebrated Hawking evaporation and anti-evaporation of black hole. The realization of specific scenario depends on the model under consideration. It is remarkable that the presence of such primordial blac…
In this work, we study the thermodynamic properties on a non--commutative background via gravitational gauge field potentials. This procedure is accomplished after contracting de Sitter (dS) group, $\mathrm{SO}(4,1)$, with the Poincarè group, $\mathrm{ISO}(3,1)$. Particularly, we focus on a static spherically symmetric black hole. In this manner, we calculate the modified Hawking temperature and the other deformed thermal state quantities, namely, entropy, heat capacity, Helmholtz free energy and pressure. Finally, we also investigate the black hole evaporation process in such a context.
We consider the gravastar model where the vacuum phase transition between the de Sitter interior and the Schwarzschild or Schwarzschild-de Sitter exterior geometries takes place at a single spherical delta-shell. We derive sharp analytic bounds on the surface compactness (2m/r) that follow from the requirement that the dominant energy condition (DEC) holds at the shell. In the case of Schwarzschild exterior, the highest surface compactness is achieved with the stiff shell in the limit of vanishing (dark) energy density in the interior. In the case of Schwarzschild-de Sitter exterior, in additi…
We present a modified Schwarzschild solution for a model of evaporation of a black hole with information preservation. By drawing a direct analogy to the quantum pure accelerating mirror (dynamical Casimir effect of a 1D horizon), we derive a Schwarzschild metric with not only the usual Schwarzschild radius but an additional length scale related to the Planck length. The black hole has thermal particle production that leads to complete evaporation of the black hole, resulting in non-divergent entanglement entropy, Page curve turn-over, and an asymptotic quantum pure state with no information l…
We investigate the thermodynamics of the noncommutative black hole whose static picture is similar to that of the nonsingular black hole known as the de Sitter-Schwarzschild black hole. It turns out that the final remnant of extremal black hole is a thermodynamically stable object. We describe the evaporation process of this black hole by using the noncommutativity-corrected Vaidya metric. It is found that there exists a close relationship between thermodynamic approach and evaporation process.
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.
In the classical theory of general relativity black holes can only absorb and not emit particles. When quantum mechanical effects are taken into account, then the black holes emit particles as hot bodies with temperature proportional to $κ$, its surface gravity. This thermal emission can lead to a slow decrease in the mass of the black hole, and eventually to its disappearance, also called black hole evaporation. This characteristic allows us to analyze what happens with the mass of the black hole when its temperature is increased or decreased, and how the energy is exchanged with the external…
The Unruh temperature calculated from the global embedding of the Schwarzschild AdS spacetime into Minkowski spacetime was identified with the local temperature measured by a free-fall observer; however, it would be imaginary in a certain region outside the event horizon. So, the temperature was assumed to be zero of no thermal radiation for that region. In this paper, we revisit this issue in the exactly soluble two-dimensional Schwarzschild AdS black hole and present an alternative resolution to this problem by using the Tolman's procedure. However, the process is not straightforward in the …
We calculate the one-loop effective action for conformal matter (scalars, spinors and vectors) on spherically symmetric background. Such effective action (in large $N$ approximation and expansion on curvature) is used to study quantum aspects of Schwarzschild-de Sitter black holes (SdS BHs) in nearly degenerated limit (Nariai BH). We show that for all types of above matter SdS BHs may evaporate or anti-evaporate in accordance with recent observation by Bousso and Hawking for minimal scalars. Some remarks about energy flow for SdS BHs in regime of evaporation or anti-evaporation are also done. …
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.
We investigate the behavior of a noncommutative radiating Schwarzschild black hole. It is shown that coordinate noncommutativity cures usual problems encountered in the description of the terminal phase of black hole evaporation. More in detail, we find that: the evaporation end-point is a zero temperature extremal black hole even in the case of electrically neutral, non-rotating, objects; there exists a finite maximum temperature that the black hole can reach before cooling down to absolute zero; there is no curvature singularity at the origin, rather we obtain a regular DeSitter core at shor…
The large dimension ($D$) limit of general relativity has been used in problems involving black holes as an analytical approximation tool. Further it has been proposed that both linear and nonlinear problems involving black holes can be systematically studied in a $1/D$ expansion. Certain quasinormal modes of higher-dimensional Schwarzschild black holes with $ω\sim \mathcal{O}(1)$ were studied in the large $D$ limit using a $1/D$ expansion for the mode function. In this paper, we revisit this linear perturbation problem and obtain an analytical equation for the vector quasinormal modes $ω\sim …
The theory of Schwarzschild geodesics is revisited. Basing on a result by Weierstrass and Biermann, we derive a formula describing all non radial, timelike and null trajectories in terms of Weierstrass elliptic functions. Quite remarkably, a single formula works for an entire geodesic trajectory, even if it passes through turning points. Using this formula, we derive expressions for the proper and coordinate time along the geodesic.
We investigate the possibility that quantum effects responsible for black hole radiation do not allow for hori- zon crossing of gravitationally collapsing matter in a finite time as seen by distant observers. We consider this in the context of the collapse of evaporating massive thin dust shells using two families of metrics to describe the exterior geometry: the outgoing Vaidya metric and the retarded Schwarzschild metric. We describe how this hypothesis results in a modified equation of motion for the shell. In each case the collapse is accelerated due to evaporation, but the Schwarzschild r…
We study tunneling process through quantum horizon of a Schwarzschild black hole in noncommutative spacetime. This is done by considering the effect of smearing of the particle mass as a Gaussian profile in flat spacetime. We show that even in this noncommutative setup there will be no correlation between the different modes of radiation which reflects the fact that information doesn't come out continuously during the evaporation process at least at late-time. However, due to spacetime noncommutativity, information might be preserved by a stable black hole remnant.
If the situation of quantum gravity nowadays is nearly the same as that of the quantum mechanics in it's early time of Bohr and Sommerfeld, then a first step study of the quantum gravity from Sommerfeld's quantum condition of action might be helpful. In this short paper the spectra of Schwarzschild black hole(SBH) in quasi-classical approach of quantum mechanics is given. We find the quantum of area, the quantum of entropy and the Hawking evaporation will cease as the black hole reaches its ground state.
The interior of Schwarzschild black hole is quantized by the method of loop quantum gravity. The Hamiltonian constraint is solved and the physical Hilbert space is obtained in the model. The properties of a Dirac observable corresponding to the ADM mass of the Schwarzschild black hole are studied by both analytical and numerical techniques. It turns out that zero is not in the spectrum of this Dirac observable. This supports the existence of a stable remnant after the evaporation of a black hole. Our conclusion is valid for a general class of schemes adopted for loop quantization of the model.
We present the public code \texttt{BlackHawk}, a powerful tool to compute the evaporation of any distribution of Schwarzschild and Kerr Black Holes and the emission spectra of Hawking radiation.
Inhomogeneity of the actual value of the vacuum energy density is considered in a black hole background. We examine the back-reaction of a Schwarzschild black hole to the highly inhomogeneous vacuum density, and argue the fluctuations lead to deviations from general relativity in the near-horizon region. In particular, we found that vacuum fluctuations onto the horizon trigger adiabatic release of quantum information, while vacuum fluctuations in the vicinity of the horizon produce potentially observable metric fluctuations of order the Schwarzschild radius. Consequently, we propose a form of …
We present a new proof of linear stability of the Schwarzschild solution to gravitational perturbations. Our approach employs the system of linearised gravity in the new geometric gauge of \cite{benomio_kerr}, specialised to the $|a|=0$ case. The proof fundamentally relies on the novel structure of the transport equations in the system. Indeed, while exploiting the well-known decoupling of two gauge invariant linearised quantities into spin $\pm 2$ Teukolsky equations, we make enhanced use of the red-shifted transport equations and their stabilising properties to control the gauge dependent pa…
The black-hole evaporation implies that the quantum-field propagators in a local Minkowski frame acquire a correction, which gives rise to this process. The modification of the propagators causes, in turn, non-trivial local effects due to the radiative/loop diagrams in non-linear QFTs. In particular, there should be imprints of the evaporation in QED, if one goes beyond the tree-level approximation. Of special interest in this respect is the region near the black-hole horizon, which, already at tree level, appears to show highly non-classical features, e.g., negative energy density and energy …
In this essay we argue that once quantum gravitational effects change the classical geometry of a black hole and remove the curvature singularity, the black hole would not evaporate entirely but approach a remnant. In a modified Schwarzschild spacetime characterized by a finite Kretschmann scalar, a minimal mass of the black hole is naturally bounded by the existence of the horizon rather than introduced by hand. A thermodynamical analysis discloses that the temperature, heat capacity and the luminosity are vanishing naturally when the black hole mass approaches the minimal value. This phenome…
Primordial black holes (PBHs) in the asteroid-mass range $(10^{-17}$--$10^{-12})~M_{\odot}$ almost entirely escape constraints as dark matter candidates. In the early universe, however, the Schwarzschild metric is no longer applicable and we must carefully consider PBH metrics with a proper cosmological embedding. In particular, the Thakurta solution stands out as perhaps the most adequate such solution. Notably, the Thakurta metric contains a time-dependent apparent horizon, proportional to the cosmological scale-factor. We show that this implies the PBHs are hotter than in the standard Schwa…
The accretion of relativistic and non-relativistic fluids into a Reissner-Nordström black hole is revisited. The position of the critical point, the flow velocity at this point and the accretion rate are only slightly affected with respect to the Schwarzschild case when the fluid is non-relativistic. On the contrary, relativistic fluids cross the critical point always subsonically. In this case, the sonic point is located near the event horizon, which is crossed by the fluid with a velocity less than the light speed. The accretion rate of relativistic fluids by a Reissner-Nordström black hole …
It is shown that the internal stationary state of the Schwarzschild black hole can be represented by a maximally entangled two-mode squeezed state of collapsing matter and infalling Hawking radiation. The final boundary condition at the singularity is then described by the random unitary transformation acting on the collapsing matter field. The outgoing Hawking radiation is obtained by the final state projection on the total wave function, which looks like a quantum teleportation process without the classical information transmitted. The black hole evaporation process as seen by the observer o…
It has recently been shown that the Hawking evaporation process of $(3+1)$-dimensional Schwarzschild black holes is characterized by the dimensionless ratio $η\equivτ_{\text{gap}}/τ_{\text{emission}}\gg1$, where $τ_{\text{gap}}$ is the characteristic time gap between the emissions of successive Hawking quanta and $τ_{\text{emission}}$ is the characteristic timescale required for an individual Hawking quantum to be emitted from the Schwarzschild black hole. This strong inequality implies that the Hawking cascade of gravitons from a $(3+1)$-dimensional Schwarzschild black hole is extremely {\it …
We discuss how in certain theories of spacetime admitting a maximal proper acceleration Hawking radiation does not completely evaporate the black hole. The black hole remnant's mass depends on the inverse of the maximal acceleration. Furthermore, as a consequence of a duality between the minimum mass that a Schwarzschild black hole can have in such spacetimes and the maximal acceleration, we show that in certain theories with a maximal acceleration there must be an upper uniform bounds on the value of the force that can be exerted on test particles.
In this work, we follow the recently revisited f(R) theory of gravity for studying the interaction between quantum scalar particles and the gravitational field of a generalized black hole with an f(R) global monopole. This background has a term playing the role of an effective cosmological constant, which permits us to call it as Schwarzschild-Anti-de Sitter (SAdS) black hole with an f(R) global monopole. We examine the separability of the Klein-Gordon equation with a non-minimal coupling and then we discuss both the massless and massive cases for a conformal coupling. We investigate some phys…
In this article is shown that the thermodynamical evolution of a Schwarzschild de Sitter space is the evaporation of its black hole. The result is extended in higher dimensions to Lovelock theories of gravity with a single positive cosmological constant.
