We study the sparsity of Hawking radiation from nonrotating singularity-free black holes in conformal gravity. We give a rigorous bound on the greybody factor for massless scalar field and calculate the sparsity of Hawking radiation from the black hole. Besides, we investigate the dependence of the greybody factor and the sparsity of Hawking radiation on the conformal parameters. Our study shows that the Hawking radiation from the black hole is extremely sparse. When the conformal parameters are large, the increase of conformal parameters will lead to an even more sparse Hawking radiation, while to a less sparse Hawking radiation if the conformal parameters are small.
We show that Raychaudhuri's recently proposed theorem on nonrotating universes cannot be used to rule out realistic singularity-free descriptions of the universe, as suggested by him in PRL 80, 654 (1998).
In this paper, we extend the calculation of the entropy of the nonrotating isolated horizons in 4 dimensional spacetime to that in a higher dimensional spacetime. We show that the boundary degrees of freedom on an isolated horizon can be described effectively by a punctured $SO(1,1)$ BF theory. Then the entropy of the nonrotating isolated horizon can be calculated out by counting the microstates. It satisfies the Bekenstein-Hawking law.
In the presence of a Killing symmetry, various self-gravitating field theories with massless scalars (moduli) and vector fields reduce to sigma-models, effectively coupled to 3-dimensional gravity. We argue that this particular structure of the Einstein-matter equations gives rise to quadratic relations between the asymptotic flux integrals and the area and surface gravity (Hawking temperature) of the horizon. The method is first illustrated for the Einstein-Maxwell system. A derivation of the quadratic formula is then also presented for the Einstein-Maxwell-axion-dilaton model, which is relev…
Hawking's singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a non-negative effective energy density (EED). The SEC ensures the timelike convergence property. However, for both classical and quantum fields, violations of the SEC can be observed even in the simplest of cases, like the Klein-Gordon field. Therefore there is a need to develop theorems with weaker restrictions, namely energy conditions averaged over an entire geodesic and weighted local averages of energy densities such as quantum energy inequalities (QEIs). We …
We apply the technique of complex paths to obtain Hawking radiation in different coordinate representations of the Schwarzschild space-time. The coordinate representations we consider do not possess a singularity at the horizon unlike the standard Schwarzschild coordinate. However, the event horizon manifests itself as a singularity in the expression for the semiclassical action. This singularity is regularized by using the method of complex paths and we find that Hawking radiation is recovered in these coordinates indicating the covariance of Hawking radiation as far as these coordinates are …
We examine Hawking radiation for a (2+1)-dimensional spinning black hole and study the interesting possibility of tunneling through the event horizon which acts as a classically forbidden barrier. Our finding shows it to be much lower than its nonrotating counterpart. We further explore the associated thermodynamics in terms of Hawking temperature and give estimates of black hole parameters like the surface gravity and entropy.
We re-examine Hawking radiation for a nonrotating (2+1)-dimensional BTZ black hole and evaluate the transmission probability of tunneling through the barrier of the event horizon employing the standard method of WKB approximation. Our results are presented for both uncharged and charged cases. We also explore the associated thermodynamics in terms of Hawking temperature and provide estimates of black hole parameters like the surface gravity and entropy.
We point out that the singular instanton of Hawking-Turok type, in which the singularity occurs due to the divergence of a massless scalar field, can be generated by Euclideanized regular $p$-brane solutions in string (or M-) theory upon compactification to four dimensions.
Spacetime singularity has always been of interest since the proof of the Penrose-Hawking singularity theorem. Naked singularity naturally emerges from reasonable initial conditions in the collapsing process. A recent interesting approach in black hole information problem implies that we need a firewall to break the surplus entanglements among the Hawking photons. Classically, the firewall becomes a naked singularity. We find some vacuum analytical solutions in $R^n$-gravity of the firewall-type and use these solutions as concrete models to study the naked singularities. By using standard quant…
It has recently been pointed out that, under certain conditions, the energy of particles accelerated by black holes in the center-of-mass frame can become arbitrarily high. In this paper, we study the collision of two particles in the case of four-dimensional charged nonrotating, extremal charged rotating and near-extremal charged rotating Kaluza-Klein black holes as well as the naked singularity case in Einstein-Maxwell-dilaton theory. We find that the center-of-mass energy for a pair of colliding particles is unlimited at the horizon of charged nonrotating Kaluza-Klein black holes, extremal …
We study the natural extension of spacetime across Schwarzschild's central singularity and the behavior of the geodesics crossing it. Locality implies that this extension is independent from the future fate of black holes. We argue that this extension is the natural $\hbar\!\to\!0$ limit of the effective quantum geometry inside a black hole, and show that the central region contains causal diamonds with area satisfying Bousso's bound for an entropy that can be as large as Hawking's radiation entropy. This result sheds light on the possibility that Hawking radiation is purified by information c…
The internal structure of a charged spherical black hole is still a topic of debate. In a nonrotating but aspherical gravitational collapse to form a spherical charged black hole, the backscattered gravitational wave tails enter the black hole and are blueshifted at the Cauchy horizon. This has a catastrophic effect if combined with an outflux crossing the Cauchy horizon: a singularity develops at the Cauchy horizon and the effective mass inflates. Recently a numerical study of a massless scalar field in the \RN background suggested that a spacelike singularity may form before the Cauchy horiz…
We provide a detailed proof of Hawking's singularity theorem in the regularity class $C^{1,1}$, i.e., for spacetime metrics possessing locally Lipschitz continuous first derivatives. The proof uses recent results in $C^{1,1}$-causality theory and is based on regularisation techniques adapted to the causal structure.
The notion of the black hole singularity and the proof of the singularity theorem were considered great successes in classical general relativity. Singularities had presented deep puzzles to physicists. Conceptual challenges were set up by the intractability of the singularity. The existence of black hole horizons which cover up the interior, including the singularity of the black hole from outside observers, builds an information curtain, further hindering physicists from understanding the nature of the singularity and the interior structure of black holes. The regular black hole is a concept…
In this work we construct several black hole metrics which are consistent with the generalized uncertainty principle logarithmic correction to the Bekenstein-Hawking entropy formula. After preserving the event horizon at the usual position, a singularity at the Planck scale is found. Finally, these geometries are shown to be realized by certain model of nonlinear electrodynamics, which resembles previously studied regular black hole solutions.
The boundary of any observer's spacetime is the boundary that divides what the observer can see from what they cannot see. The boundary of an observer's spacetime in the presence of a black hole is not the true (future event) horizon of the black hole, but rather the illusory horizon, the dimming, redshifting surface of the star that collapsed to the black hole long ago. The illusory horizon is the source of Hawking radiation seen by observers both outside and inside the true horizon. The perceived acceleration (gravity) on the illusory horizon sets the characteristic frequency scale of Hawkin…
We apply the technique of complex paths to obtain Hawking radiation in different coordinate representations of the Schwarzschild space-time. The coordinate representations we consider do not possess a singularity at the horizon unlike the standard Schwarzschild coordinate. However, the event horizon manifests itself as a singularity in the expression for the semi-classical action. This singularity is regularized by using the method of complex paths and we find that Hawking radiation is recovered in these coordinates indicating the covariance of Hawking radiation. This also shows that there is …
The recent renaissance of wormhole physics has led to a very disturbing observation: If traversable wormholes exist then it appears to be rather easy to to transform such wormholes into time machines. This extremely disturbing state of affairs has lead Hawking to promulgate his chronology protection conjecture. This paper continues a program begun in an earlier paper [Physical Review {\bf D47}, 554--565 (1993), hepth@xxx/9202090]. An explicit calculation of the vacuum expectation value of the renormalized stress--energy tensor in wormhole spacetimes is presented. Point--splitting techniques ar…
Hawking radiation is computed in different coordinate systems using the method of complex paths. In this procedure the event horizon of the 2D Schwarzschild stringy black hole is treated as a singularity for the semiclassical action functional. After the regularization of the event horizon's singularity the emission/absorption probabilities and the Hawking temperature in the different coordinate representations are derived. The identical results obtained indicate the covariance of the Hawking radiation.
Starting from the Hawking--Page solutions, we consider the corresponding Lorentzian cone metrics. These represent cone interior scale-invariant vacuum solutions, defined in the chronological past of the scaling origin. We extend the Lorentzian Hawking--Page solutions to the cone exterior region in the class of $(4+1)$-dimensional scale-invariant vacuum solutions with an $SO(3)\times U(1)$ isometry, using the Kaluza--Klein reduction and the methods of Christodoulou. We prove that each Lorentzian Hawking--Page solution has extensions with a null curvature singularity, extensions with a spacelike…
The spacetime inside the white hole is like an anisotropic cosmological background with the past singularity playing the role of a big bang singularity. The scale factor along the extended spatial direction is contracting while the scale factor along the two-sphere is expanding. We consider an eternal Schwarzschild manifold and study quantum cosmological perturbations generated near the white hole singularity which propagate towards the past event horizon and to exterior of the black hole. It is shown that an observer deep inside the white hole and an observer far outside the black hole both s…
The behaviour of geometric quantities close to geometric pathologies of a spacetime is relevant to deduce the physical behaviour of the system. In this work, we compute the quasi-local mass quantities - the Hawking mass, the Brown-York mass and the Liu-Yau mass in the maximal extensions of the spherically symmetric solutions of the Einstein equations inside the black hole region, at the singularity, the event horizon, and the null infinity, in the limiting sense of a geometric flow.
The Hawking and Penrose singularity theorems for low regularity Finsler spacetimes
Dipartimento di Matematica, Università degli Studi di Pisa（比萨大学数学系）；Department of Mathematics, University of Toronto（多伦多大学数学系）；Department of Mathematics, University of Osaka（大阪大学数学系）；RIKEN Center for Advanced Intelligence Project (AIP)（理化学研究所先进智能研究中心）；Department of Mathematics, University of Vienna（维也纳大学数学系）
We consider the late-time accelerated universe in the Friedmann-Robertson-Walker (FRW) spacetime with a nonzero curvature, and investigate cosmological models when the cosmic fluid is taken to be inhomogeneous and viscous (bulk viscous), coupled to dark matter. We consider the influence from thermal effects caused by Hawking radiation on the formation of singularities of various classified types, within a finite time. It is shown that under the influence of Hawking radiation the time of formulation of a singularity, and the nature of the singularity itself, can change. It is also shown that by…
The interior of the Kerr solution is singular and achronological. The classic singularity theorem by Hawking and Penrose relies on chronology, and thus does not apply to the Kerr solution. An improvement of their theorem by Kriele partially removes the requirement of chronology. However, both of these singularity theorems fail to give any information on the type or location of the incomplete geodesics. Here, using recent results of Minguzzi, we prove a new singularity theorem, specifically designed to apply to black holes, which enables locating null incomplete geodesics within the black hole …
It is shown that static solutions with a finite curvature at the horizon may exist in dilaton gravity at temperatures $T\neq T_{H}$ (including T=0) where $T_{H} $is the Hawking one. Hawking radiation is absent and the state of a system represents thermal excitation over the Boulware vacuum. The horizon remains unattainable for a observer because of thermal divergences in the stress-energy of quantum fields there. However, the curvature at the horizon is finite, when measured from outside, since these divergences are compensated by those in gradients of a dilaton field. Spacetimes under conside…
We extend both the Hawking-Penrose Theorem and its generalisation due to Galloway and Senovilla to Lorentzian metrics of regularity $C^1$. For metrics of such low regularity, two main obstacles have to be addressed. On the one hand, the Ricci tensor now is distributional, and on the other hand, unique solvability of the geodesic equation is lost. To deal with the first issue in a consistent way, we develop a theory of tensor distributions of finite order, which also provides a framework for the recent proofs of the theorems of Hawking and of Penrose for $C^1$-metrics [7]. For the second issue,…
In this review article we compare the recent work of Peter Lynds, "On a finite universe with no beginning or end", with that of Stephen Hawking, primarily "Quantum Cosmology, M-Theory, and the Anthropic Principle", and two foundational works by Sean M. Carroll and Jennifer Chen, "Does Inflation Provide Natural Conditions for the Universe" and "Spontaneous Inflation and the Origin of the Arrow of Time", in order to evaluate their comparative treatments of the nature and role of causality, time ordering, thermodynamic reversibility, singularities and boundary conditions in the formation of the e…
Acoustic analogues of Kerr black hole in plasmas are considered, by taking for granted the existence of acoustic ion waves in plasmas. An effective black holes (BH) in curved Riemannian spacetime in a random walk plasmas is endowed with a naked singularity, when plasmas are in the lowest diffusion mode. The plasma particle diffusion is encoded in the effective metric. The diffusive solution has a horizon when the plasma flow reaches the sound velocity in the medium and a shock wave is obtained inside the slab. The sonic black hole curved Riemannian metric is also found in terms of particle num…
By allowing the lightcones to tip over according to the conservation laws of an one-kink in static, Schwarzschild metric, we show that there also exists an instanton which represents production of pairs of chargeless, nonrotating black holes with mass $M$, joined on an interior surface beyond the horizon ar $r=M$. Evaluation of the thermal properties of each of the black hole in a pair leads one to check that each black hole is exactly the antiblack hole to the other black hole in the pair. The instantonic action has been calculated and seen to be smaller than that corresponding to pair produc…
We study an internal structure of (2+1)-dimensional black hole with the neutral scalar matter in the spherically symmetric geometry by using a quantum theory of gravity which holds in the both vicinities of the singularity and the apparent horizon. A special attention is paid to the quantum-mechanical behavior of the singularity in the black hole. We solve analytically the Wheeler-DeWitt equation of a minisuperspace model where the ingoing Vaidya metric is used as a simple model representing a dynamical black hole. The wave function obtained in this way leads to interesting physical phenomena …
A sufficiently massive collapsing star will end its life as a spacetime singularity. The nature of the Hawking radiation emitted during collapse depends critically on whether the star's boundary conditions are such as would lead to the eventual formation of a black hole or, alternatively, to the formation of a naked singularity. This latter possibility is not excluded by the singularity theorems. We discuss the nature of the Hawking radiation emitted in each case. We justify the use of Bogoliubov transforms in the presence of a Cauchy horizon and show that if spacetime is assumed to terminate …
We study the Hawking radiation in two dimensional dilaton black hole by means of quantum gravity holding near the apparent horizon. First of all, we construct the canonical formalism of the dilaton gravity in two dimensions. Then the Vaidya metric corresponding to the dilaton black hole is established where it is shown that the dilaton field takes a form of the linear dilaton. Based on the canonical formalism and the Vaidya metric, we proceed to analyze quantum properties of a dynamical black hole. It is found that the mass loss rate of the Hawking radiation is independent of the black hole ma…
Hawking's singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a nonnegative effective energy density (EED), thereby guaranteeing the timelike convergence property. However, there are models that do not satisfy the SEC and therefore lie outside the scope of Hawking's hypotheses, an important example being the massive Klein-Gordon field. Here we derive lower bounds on local averages of the EED for solutions to the Klein-Gordon equation, allowing nonzero mass and nonminimal coupling to the scalar curvature. The averages are take…
We model the gravitational collapse of heavy massive shells including its main quantum corrections. Among these corrections, quantum improvements coming from Quantum Einstein Gravity are taken into account, which provides us with an effective quantum spacetime. Likewise, we consider dynamical Hawking radiation by modeling its back-reaction once the horizons have been generated. Our results point towards a picture of gravitational collapse in which the collapsing shell reaches a minimum non-zero radius (whose value depends on the shell initial conditions) with its mass only slightly reduced. Th…
If a black hole starts in a pure quantum state and evaporates completely by a unitary process, the von Neumann entropy of the Hawking radiation initially increases and then decreases back to zero when the black hole has disappeared. Here numerical results are given for an approximation to the time dependence of the radiation entropy under an assumption of fast scrambling, for large nonrotating black holes that emit essentially only photons and gravitons. The maximum of the von Neumann entropy then occurs after about 53.81% of the evaporation time, when the black hole has lost about 40.25% of i…
It is shown in a variant of two dimensional dilaton gravity theories that an arbitrary, localized massive source put in an initially regular spacetime gives rise to formation of the wormhole classically, without accompanying the curvature singularity. The semiclassical quantum correction under this wormhole spacetime yields Hawking radiation. It is expected, with the quantum back reaction added to the classical equation, that the information loss paradox may be resolved in this model.
We present numerical evidence for the existence of new black hole solutions in $d\geq 6$ spacetime dimensions. They approach asymptotically the Minkowski background and have an event horizon topology $S^2\times S^{d-4}$. These static solutions share the basic properties of the nonrotating black rings in five dimensions, in particular the presence of a conical singularity.
The Raychaudhuri equation and its consequences for chronality are studied in the context of Finsler spacetimes. It is proved that the notable singularity theorems of Lorentzian geometry extend to the Finslerian domain. Indeed, so do the theorems by Hawking, Penrose, Hawking and Penrose, Geroch, Gannon, Tipler, or Kriele, but also the Topological Censorship theorem and so on. It is argued that the notable results in causality theory connected to achronal sets, future sets, domains of dependence, limit curve theorems, length functional, Lorentzian distance, geodesic connectedness, extend to the …
We describe a classical configuration of conformal matter forming a naked singularity and discuss its subsequent Hawking evaporation within the context of two dimensional dilaton gravity. The one loop analysis is credible for a large mass naked singularity and suggests the existence of a weak cosmological censorship that would cause it to explode into radiation upon forming. (Hardcopies of figures available on request)
An event horizon for "relativistic" fermionic quasiparticles can be constructed in a thin film of superfluid 3He-A. The quasiparticles see an effective "gravitational" field which is induced by a topological soliton of the order parameter. Within the soliton the "speed of light" crosses zero and changes sign. When the soliton moves, two planar event horizons (black hole and white hole) appear, with a curvature singularity between them. Aside from the singularity, the effective spacetime is incomplete at future and past boundaries, but the quasiparticles cannot escape there because the nonrelat…
Considering the role of black hole singularity in quantum evolution, a resolution to the firewall paradox is presented. It is emphasized that if an observer has the singularity as a part of his spacetime, then the semi-classical evolution would be non-unitary as viewed by him. Specifically, a free-falling observer inside the black hole would have a Hilbert space with non-unitary evolution; a quantum jump for particles encountering the singularity to outside of the horizon as late Hawking radiations. The non-unitariness in the jump resembles the one in collapse of wave function, but preserves e…
Assuming that time exists, a new, effective formulation of gravity is introduced, which lies in between the Wheeler-DeWitt approach and ordinary QFT. Remarkably, the Penrose-Hawking singularity of usual Friedman-Robertson-Walker cosmologies is naturally avoided there. The theory is made explicit via specific examples, and compared with loop quantum cosmology. It is argued that it is the regularization of the classical Hamiltonian performed in this last theory what avoid the singularity, rather than quantum effects as in our case.
The singularity in Hawking and Turok's model (hep-th/9802030) of open inflation has some appealing properties. We suggest that this singularity should be regularized with matter. The singular instanton can then be obtained as the limit of a family of ``no-boundary'' solutions where both the geometry and the scalar field are regular. Using this procedure, the contribution of the singularity to the Euclidean action is just 1/3 of the Gibbons-Hawking boundary term. Unrelated to this question, we also point out that gravitational backreaction improves the behaviour of scalar perturbations near the…
We study the causal structure of dynamical charged black holes, with a sufficient number of massless fields, using numerical simulations. Neglecting Hawking radiation, the inner horizon is a null Cauchy horizon and a curvature singularity due to mass inflation. When we include Hawking radiation, the inner horizon becomes space-like and is separated from the Cauchy horizon, which is parallel to the out-going null direction. Since a charged black hole must eventually transit to a neutral black hole, we studied the neutralization of the black hole and observed that the inner horizon evolves into …
The radiation rate of an evaporating black hole is calculated in a toy model in which the geometry outside the collapsing matter is described by a Vaidya metric. When back reaction consistency is imposed, the singularity in the blueshift factor near the horizon is softened, suppressing the evaporation rate in the Schwarzschild case by the fourth power of the external time, thus rendering the hole eternal.
A model is proposed in which the Hawking particles emitted by a black hole are treated as an envelope of matter that obeys an equation of state, and acts as a source in Einstein's equations. This is a crude but interesting way to accommodate for the back reaction. For large black holes, the solution can be given analytically, if the equation of state is $p=κρ$, with $0<κ<1$. The solution exhibits a singularity at the origin. If we assume $N$ free particle types, we can use a Hartree-Fock procedure to compute the contribution of one such field to the entropy, and the result scales as expected a…
Recently, the possibility was pointed out by one of the present authors and his collaborators that an effective naked singularity referred to as "a visible border of spacetime" is generated by high-energy particle collision in the context of large extra dimensions or TeV-scale gravity. In this paper, we investigate the particle creation by a naked singularity in general dimensions, while adopting a model in which a marginally naked singularity forms in the collapse of a homothetic lightlike pressureless fluid. We find that the spectrum deviates from that of Hawking radiation due to scattering …
The black hole information paradox is the incompatibility of quantum mechanics with the semi-classical picture of Hawking radiation. Hawking radiation appears thermal and eventually leads to the complete disappearance of a black hole. However, black holes could be formed from a pure quantum state. The transition from such an initial state to the final state of pure Hawking radiation cannot be described by unitary time evolution. In this paper, we present an analysis in quantum gravity that shows how boundary conditions in the future prevent a loss of quantum mechanical information from the spa…
A star of mass greater than a critical mass is believed to undergo gravitational collapse to form a singularity, owing to Hawking-Penrose singularity theorem which is based on the Raychaudhuri equation in the absence of torsion. We argue that the spin-aspect of matter can lead to the evasion of singularity, caused by its mass-aspect, via torsion in asymptotically safe gravity.
