24 pages, 2 figures. arXiv admin note: text overlap with arXiv:2011.11815, v2: analysis extended, a new prescription to deal with fiducial elements introduced, figures are slightly modified and are now in color
We consider the classical Hamiltonian of the interior of the Schwarzschild black hole in Ashtekar-Barbero connection formalism. Then, inspired by generalized uncertainty principle models, we deform the classical canonical algebra and derive the effective dynamics of the model under this modification. We show that such a deformation leads to the resolution of the singularity of the black hole and a minimum nonzero radius for the infalling 2-spheres, provided that the deformation parameters are chosen to be negative.
Effective models of black holes interior have led to several proposals for regular black holes. In the so-called polymer models, based on effective deformations of the phase space of spherically symmetric general relativity in vacuum, one considers a deformed Hamiltonian constraint while keeping a non-deformed vectorial constraint. In this article, we revisit and study further the question of covariance in these deformed gravity models. In particular, we propose a Lagrangian formulation for these deformed gravity models where polymer-like deformations are introduced at the level of the full th…
In this paper we study the effects of noncommutative phase space deformations on the Schwarzschild black hole. This idea has been previously studied in {Friedmann-Robertson-Walker (FRW)} cosmology, where this "noncommutativity" provides a simple mechanism that can explain the origin of the cosmological constant. In this paper we obtain the same relationship between the cosmological constant and the deformation parameter that appears in deformed phase space cosmology, but in the context of the deformed phase space black holes. This was achieved by comparing the entropy of the deformed Schwarzsc…
We construct the first examples of deformed non-abelian black strings in a 5-dimensional Einstein-Yang-Mills model. Assuming all fields to be independent of the extra coordinate, we construct deformed black strings, which in the 4-dimensional picture correspond to axially symmetric non-abelian black holes in gravity-dilaton theory. These solutions thus have deformed S^2 x R horizon topology. We study fundamental properties of the black strings and find that for all choices of the gravitational coupling two branches of solutions exist. The limiting behaviour of the second branch of solutions ho…
A family of deformed black branes is employed to examine the confinement/deconfinement phase transition in AdS/QCD. The holographic entanglement entropy (HEE) plays the role of the order parameter driving the confinement/deconfinement phase transition. The binding energy of quark-antiquark bound states and the critical length that makes a null variation of the HEE, together with the critical temperature separating the deconfined quark-gluon plasma to the confined hadronic phase, are discussed in the deformed black brane background, matching current experimental hadronic data.
Inspired by the Einstein-Born-Infeld black hole, we introduce the isolated horizon to study the Kehagias-Sfetsos (KS) black hole in the deformed Hořava-Lifshitz gravity. This is because the KS black hole is more close to the Einstein-Born-Infeld black hole than the Reissner-Nordström black hole. We find the horizon and ADM masses by using the first law of thermodynamics and the area-law entropy. The mass parameter $m$ is identified with the quasilocal energy at infinity. Accordingly, we discuss the phase transition between the KS and Schwarzschild black holes by considering the heat capacity a…
We study the quasinormal modes of the massless scalar perturbation in the background of a deformed black hole in the Hořava-Lifshitz gravity with coupling constant $λ=1$. Our results show that the quasinormal frequencies depend on the parameter in the Hořava-Lifshitz gravity and the behavior of the quasinormal modes is different from those in the Reissner-Norstr\"{om} and Einstein-Born-Infeld black hole spacetimes. The absolute value of imaginary parts is smaller and the scalar perturbations decay more slowly in the deformed Hořava-Lifshitz black hole spacetime. This information can help us un…
We find generators of the conformal symmetry for the class of Kerr-Newman-NUT-AdS black holes from the deformed scalar probe equation. We find two classes of solutions for the generators (also known as conformal J and Q pictures). The two classes of deformed generators are the extension of similar generators for the regular conformal symmetry. Moreover, we find that the two pictures can be generalized and extended into a general picture. In each picture, the generators produce an extended local family of SL(2,R)_L \times SL(2,R)_R hidden conformal symmetries for the Kerr-Newman-NUT-AdS black h…
A deformation of the wave equation on a two-dimensional black hole is considered as a toy-model for possible gravitational or stringy nonlocal effects. The deformed wave-equation allows for an initial-value problem despite being nonlocal. The classical singularity present in the classical geometry is resolved by the deformation, so that propagation of a wave-packet can be continued through the classically singular region, ultimately reaching another asymptotically ``flat'' region.
We study the interior of distorted static axisymmetric black holes. We obtain a general interior solution and study its asymptotics both near the horizon and singularity. As a special example, we apply the obtained results to the case of the so-called `caged' black holes.
Using perturbative techniques, we investigate the existence and properties of a new static solution for the Einstein equation with a negative cosmological constant, which we call the deformed black hole. We derive a solution for a static and axisymmetric perturbation of the Schwarzschild-anti-de Sitter black hole that is regular in the range from the horizon to spacelike infinity. The key result is that this perturbation simultaneously deforms the two boundary surfaces--i.e., both the horizon and spacelike two-surface at infinity. Then we discuss the Abbott-Deser mass and the Ashtekar-Magnon o…
We calculate the interior volume for the three-dimensional Banados-Teitelboim-Zanelli black hole in this paper. To that end, we obtain the volume functional and numerically find a path to extremize it, before choosing a spacelike hypersurface to get an asymptotic analytical expression for the interior volume based on the characteristics of the path.
A family of deformed AdS4-Reissner-Nordström black branes, governed by a free parameter, is derived using the ADM formalism, in the context of the membrane paradigm. Their new event horizons, the Hawking temperature and other aspects are scrutinized. AdS/CFT near-horizon methods are then implemented to compute the shear viscosity-to-entropy ratio for the deformed AdS4-Reissner-Nordström metric. The Killing equation is shown to yield new values for the free parameter and the shear viscosity-to-entropy ratio is used to derive a reliable range for tidal charge.
The motion of a massive test particle around a Kehagias-Sfetsos black hole in deformed Hourava-Lifshitz gravity is studied. Employing the effective potential, the marginally bound orbits and the innermost stable circular orbits are analyzed. For the marginally bound orbits, their radius and angular momentum decrease with the parameter $ω$ of the gravity. For the innermost stable circular orbits, the energy and angular momentum also decrease with $ω$. Based on these results, we investigate the periodic orbits in the Kehagias-Sfetsos black holes. It is found that the apsidal angle parameter incr…
In this paper, a four dimensional rotating Kaluza Klien (K-K) black hole was deformed using rainbow functions derived from loop quantum gravity and non-commutative geometry. We studied the thermodynamic properties and critical phenomena of this deformed black hole. The deformed temperature and entropy showed the existence of a Planckian remnant. The calculation of Gibbs free energy $ G$ for the ordinary and deformed black holes showed that both share a similar critical behaviour.
A family of deformed AdS$_5$-Schwarzschild black branes is here derived, employing the membrane paradigm of AdS/CFT. The solution of the Einstein-Hilbert action, with the Gibbons-Hawking term and a counter-term that eliminates eventual divergences, yields a partition function associated with the dual theory which allows the computation of the entropy, pressure and free energy, as state functions, in the canonical ensemble. AdS/CFT near-horizon methods are then implemented to compute the shear viscosity-to-entropy ratio, then restricting the range of the parameter that defines a family of defor…
We discuss the "quantum deformed Schwarzschild spacetime" as originally introduced by Kazakov and Solodukhin in 1993, and investigate the precise sense in which it does and does not satisfy the desiderata for being a "regular black hole". We shall carefully distinguish (i) regularity of the metric components, (ii) regularity of the Christoffel components, and (iii) regularity of the curvature. We shall then embed the Kazakov-Solodukhin spacetime in a more general framework where these notions are clearly and cleanly separated. Finally we analyze aspects of the classical physics of these "quant…
We consider nonuniform black strings inside their event horizon. We present numerical evidence, that the singularity touches the horizon as the horizon topology changing transition is reached.
A nonstatic Schwarzschild black hole interior solution in Painleve-Gullstrand coordinates is proposed in this paper, by means of a coordinate transformation that changes the spatial coordinate but the timelike one is preserved. The timelike and null geodesic equations are obtained exactly, taking advantage that the spatial z-coordinate is cyclic.
We discuss the interior of a black hole in quantum gravity, in which black holes form and evaporate unitarily. The interior spacetime appears in the sense of complementarity because of special features revealed by the microscopic degrees of freedom when viewed from a semiclassical standpoint. The relation between quantum mechanics and the equivalence principle is subtle, but they are still consistent.
It was argued in a number of papers that the gravitational potential calculated by using the modified QFT that follows from the Planck-length deformed uncertainty relation implies the existence of black-hole remnants of the order of the Planck-mass. Usually this sort of QFTs are endowed with two specific features, the modified dispersion relation, which is universal, and the concept of minimum length, which, however, is not universal. While the emergence of the minimum-length most readily leads to the idea of the black hole remnants, here we examine the behaviour of the potential that follows …
We investigate the thermodynamic geometry and phase transition of Kehagias-Sfetsos black hole in the deformed Horava-Lifshitz gravity with coupling constant $λ=1$. The phase transition in black hole thermodynamics is thought to be associated with the divergence of the capacities. And the structures of these divergent points are studied. We also find that the thermodynamic curvature produced by the Ruppeiner metric is positive definite for all $r_+ > r_-$ and is divergence at $η_2=0$ corresponded to the divergent points of $C_Φ$ and $C_T$. These results suggest that the microstructure of the bl…
We study thermodynamical properties of static dyonic AdS black holes in four-dimensional $ω$-deformed Kaluza-Klein gauged supergravity theory, and find that the differential first law requires a modification via introducing a new pair of thermodynamical conjugate variables (X, Y). To ensure such a modification, we then apply the quasi-local ADT formalism developed in Ref. [20] to calculate the quasi-local conserved charge and identify that the new pair is precisely the one previously introduced to modify the differential form of first law.
We construct a new class of vacuum black hole solutions whose geometry is deformed and twisted by the presence of NUT charges. The solutions are obtained by `unspinning' the general Kerr-NUT-(A)dS spacetimes, effectively switching off some of their rotation parameters. The resulting geometry has a structure of warped space with the Kerr-like Lorentzian part warped to a Euclidean metric of deformed and/or twisted sphere, with the deformation and twist characterized by the `Euclidean NUT' parameters. In the absence of NUTs, the solution reduces to a well known Kerr-(A)dS black hole with several …
In this paper we discuss to what extent one can infer details of the interior structure of a black hole based on its horizon. Recalling that black hole thermal properties are connected to the non-classical nature of gravity, we circumvent the restrictions of the no hair theorem by postulating that the black hole interior is singularity free due to violations of the usual energy conditions. Further these conditions allow one to establish a one-to-one, holographic projection between Planckian areal "bits" on the horizon and "voxels", representing the gravitational degrees of freedom in the black…
The aim of this paper is to obtain the solution of the Einstein equation in the interior of the black holes by using arbitrary distribution functions; corresponding to Gaussian, Rayleigh, Maxwell-Boltzmann and non-Gaussian distributions. Also we calculate the Hawking temperature, the mass and heat capacity for cosmological horizon and the black hole horizon.
The information loss paradox is often discussed from the perspective of the observers who stay outside of a black hole. However, the interior spacetime of a black hole can be rather nontrivial. We discuss the open problems regarding the volume of a black hole, and whether it plays any role in information storage. We also emphasize the importance of resolving the black hole singularity, if one were to resolve the information loss paradox.
Analytical studies on 3D hairy rotating black hole interiors
Center for Gravitational Physics, Department of Space Science, Beihang University（北京航空航天大学空间科学学院引力物理中心）；Peng Huanwu Collaborative Center for Research and Education, Beihang University（北京航空航天大学彭桓武教研中心）
We study a Hamiltonian quantum formalism of a spherically symmetric space-time which can be identified with the interior of a Schwarzschild black hole. The phase space of this model is spanned by two dynamical variables and their conjugate momenta. It is shown that the classical Lagrangian of the model gives rise the interior metric of a Schwarzschild black hole. We also show that the the mass of such a system is a Dirac observable and then by quantization of the model by Wheeler-DeWitt approach and constructing suitable wave packets we get the mass spectrum of the black hole.
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…
Interior volume within the horizon of a black hole is a non-trivial concept which turns out to be very important to explain several issues in the context of quantum nature of black hole. Here we show that the entropy, contained by the {\it maximum} interior volume for massless modes, is proportional to the Bekenstein-Hawking expression. The proportionality constant is less than unity implying the horizon bears maximum entropy than that by the interior. The derivation is very systematic and free of any ambiguity. To do so the precise value of the energy of the modes, living in the interior, is …
The investigation about the volume of a black hole is closely related to the quantum nature of the black hole. The entropy is a significant concept for this. A recent work by Majhi and Samanta [Phys. Lett. B 770 (2017) 314] after us presented a similar conclusion that the entropy associated with the volume is proportional to the surface area of the black hole, but the proportionality coefficient is different from our earlier result. In this paper, we clarify the difference and show that their calculation is unrelated to the interior of the black hole.
Recently it was shown that folded strings are spontaneously created behind the horizon of the $SL(2,\mathbb{R})_k/U(1)$ black hole. Here we show that these folded strings violate the averaged null energy condition {\it macroscopically}. We discuss possible consequences of this observation on black hole physics in general, and on the information puzzle in particular.
We point out that scenarios in which the universe is born from the interior of a black hole may not posses many of the problems of the Standard Big-Bang (SBB) model. In particular we demonstrate that the horizon problem, flatness, and the structure formation problem might be solved naturally, not necessarily requiring a long period of cosmological inflation. The black hole information loss problem is also discussed. Our conclusions are completely independent of the details of general models.
A new class of black hole solutions of the five dimensional minimal gauged supergravity is presented. They are characterized by the mass, the electric charge, two equal magnitude angular momenta and the magnitude of the magnetic potential at infinity. These black holes possess a horizon of spherical topology; however, both the horizon and the sphere at infinity can be arbitrarily squashed, with nonextremal solutions interpolating between black strings and black branes. A particular set of extremal configurations corresponds to a new one-parameter family of supersymmetric black holes. While the…
Causal Space-Time Structure and non-Hausdorff Extension of Schwarzschild Black Hole Interior
Landau Phystech-School, Moscow Institute of Physics and Technology（兰道物理技术学院，莫斯科物理技术学院）；Institute for High Energy Physics of National Research Centre "Kurchatov Institute"（库尔恰托夫国家研究中心高能物理研究所）
Recently, two kinds of deformed schwarzschild spacetime have been proposed, which are the black-bounces metric (\cite{2019JCAP...02..042S,2021PhRvD.103h4052L}) and quantum deformed black hole (BH) (\cite{2021arXiv210202471B}). In present work, we investigate the rotating spacetime of these deformed Schwarzschild metric. They are exact solutions to the Einstein$'$s field equation. We analyzed the properties of these rotating spacetimes, such as event horizon (EH), stationary limit surface (SIS), structure of singularity ring, energy condition (EC), etc., and found that these rotating spacetime …
We present approximate exterior and interior solutions of Einstein's equations which describe the gravitational field of a static deformed mass distribution. The deformation of the source is taken into account up to the first order in the quadrupole.
We construct numerically static non-uniform black string solutions in six dimensions by using pseudo-spectral methods. An appropriately designed adaptation of the methods in regard of the specific behaviour of the field quantities in the vicinity of our numerical boundaries provides us with extremely accurate results, that allows us to get solutions with an unprecedented deformation of the black string horizon. Consequently, we are able to investigate in detail a critical regime within a suitable parameter diagram. In particular, we observe a clearly pronounced maximum in the mass curve, which…
We study the interior structure of five dimensional neutral helical black holes in Einstein gravity and charged helical black holes in Einstein-Maxwell gravity. Inside the neutral helical black holes, the systems evolve to a stable spacelike Kasner singularity. The metric field related to the helical deformation strength exhibits oscillation behavior close to the horizon at low temperature and small helical deformation strength. Inside the charged helical black holes, we show that the inner Cauchy horizon can not exist. The systems also evolve from the horizon to a stable Kasner singularity. W…
We reveal an $\mathfrak{iso}(2,1)$ Poincaré algebra of conserved charges associated with the dynamics of the interior of black holes. The action of these Noether charges integrates to a symmetry of the gravitational system under the Poincaré group ISO$(2,1)$, which allows to describe the evolution of the geometry inside the black hole in terms of geodesics and horocycles of AdS${}_2$. At the Lagrangian level, this symmetry corresponds to Möbius transformations of the proper time together with translations. Remarkably, this is a physical symmetry changing the state of the system, which also nat…
The phenomenon of mass inflation is shown to occur for a rotating black hole. We demonstrate this feature in $(2+1)$ dimensions by extending the charged spinning BTZ black hole to Vaidya form. We find that the mass function diverges in a manner quantitatively similar to its static counterparts in $(3+1)$, $(2+1)$ and $(1+1)$ dimensions.
We study a higher order conformally coupled scalar tensor theory endowed with a covariant geometric constraint relating the scalar curvature with the Gauss-Bonnet scalar. It is a particular Horndeski theory including a canonical kinetic term but without shift or parity symmetry for the scalar. The theory also stems from a Kaluza-Klein reduction of a well defined higher dimensional metric theory. Properties of an asymptotically flat spherically symmetric black hole are analyzed, and new slowly rotating and radiating extensions are found. Through disformal transformations of the static configura…
We consider a charged Lifshitz black hole in the large transverse dimension limit. In this setup, the dynamics near the black hole horizon are shown to be effectively governed by a family of two-dimensional models of dilaton gravity depending on the ratio of the dynamical parameter characterizing the black hole and the dimension of spacetime. This family includes the Callan-Giddings-Harvey-Strominger (CGHS) and Jackiw-Teitelboim (JT) models and their charged equivalents. This family also contains classes of asymptotically anti-de Sitter models beyond JT, characterized by a running Ricci scalar…
The possibility that a black hole could tunnel into a white hole has recently received attention. Here we present a metric that improves the 'firework' metric: it describes the entire process and solves the Einstein equations everywhere except on a small transition surface that corresponds to the quantum tunnelling. We compute the corresponding ray-tracing map from past infinity to future infinity explicitly.
Different approaches to quantum gravity conclude that black holes may possess an inner horizon, in addition to the (quantum corrected) outer `Schwarzschild' horizon. In this paper we assume the existence of this inner horizon and explain the physical process that might lead to the tunneling of particles through it. It is shown that the tunneling would produce a flux of particles with a spectrum that deviates from the pure thermal one. Under the appropriate approximation the extremely high temperature of this horizon is calculated for an improved quantum black hole. It is argued that the flux o…
We study the entropy of black holes in the deformed Horava-Lifshitz gravity with coupling constant lambda. For lambda=1, the black hole resembles the Reissner-Norstrom black hole with a geometric parameter acting like the electric charge. Therefore, we obtain some differences in the entropy when comparing with the Schwarzschild black hole. Finally, we study the heat capacity and the thermodynamical stability of this solution.
