47 pages, 5 figures, A short addendum is added after the last section to address misleading remarks in a recent pre-print [50] that may confuse the reader
A new description of macroscopic Kruskal black holes that incorporates the quantum geometry corrections of loop quantum gravity is presented. It encompasses both the `interior' region that contains classical singularities and the `exterior' asymptotic region. Singularities are naturally resolved by the quantum geometry effects of loop quantum gravity. The resulting quantum extension of space-time has the following features: (i) It admits an infinite number of trapped, anti-trapped and asymptotic regions; (ii) All curvature scalars have uniform (i.e., mass independent) upper bounds; (iii) In the large mass limit, all asymptotic regions of the extension have the same ADM mass; (iv) In the low curvature region (e.g., near horizons) quantum effects are negligible, as one would physically expect; and (v) Final results are insensitive to the fiducial structures that have to be introduced to construct the classical phase space description (as they must be). Previous effective theories [1-10] shared some but not all of these features. We compare and contrast our results with those of these effective theories and also with expectations based on the AdS/CFT conjecture [11]. We conclude with a discussion of limitations of our framework, especially for the analysis of evaporating black holes.
By extending the exterior Schwarzschild spacetime in two opposite directions with the Kruskal method, we get an extension which has the same T-X spacetime diagram as has the conventional Kruskal extension, while allowing its regions I and IV to correspond to different directions of the original spacetime. We further extend the exterior Schwarzschild spacetime in all directions and get a 4-dimensional form of the Kruskal extension. The new form of extension includes the conventional one as a part of itself. From the point of view of the 4-dimensional form, region IV of the conventional extensio…
We find a specific coordinate system that goes from the Painlevé-Gullstrand partial extension to the Kruskal-Szekeres maximal extension and thus exhibit the maximal extension of the Schwarzschild metric in a unified picture. We do this by adopting two time coordinates, one being the proper time of a congruence of outgoing timelike geodesics, the other being the proper time of a congruence of ingoing timelike geodesics, both parameterized by the same energy per unit mass $E$. $E$ is in the range $1\leq E<\infty$ with the limit $E=\infty$ yielding the Kruskal-Szekeres maximal extension. So, thro…
An explicit calculation shows that the model constructed in arXiv:1806.00648 violates general covariance. It is therefore unphysical and does not determine properties of space-time or black holes.
We present a new effective description of macroscopic Kruskal black holes that incorporates corrections due to quantum geometry effects of loop quantum gravity. It encompasses both the `interior' region that contains classical singularities and the `exterior' asymptotic region. Singularities are naturally resolved by the quantum geometry effects of loop quantum gravity, and the resulting quantum extension of the full Kruskal space-time is free of all the known limitations of previous investigations [1-11] of the Schwarzschild interior. We compare and contrast our results with these investigati…
Recently it was shown that, in an effective description motivated by loop quantum gravity, singularities of the Kruskal space-time are naturally resolved [1,2]. In this note we explore a few properties of this quantum corrected effective metric. In particular, we (i) calculate the Hawking temperature associated with the horizon of the effective geometry and show that the quantum correction to the temperature is completely negligible for macroscopic black holes, just as one would hope; (ii) discuss the subtleties associated with the asymptotic properties of the space-time metric, and show that …
We apply the conical singularity method to the two dimensional version of the Schwarzschild metric to obtain the Kruskal coordinates of the black hole in a very simple and direct way. Then we make use of this metric in an approximated version and calculate the expected value of energy-momentum tensor of a massless quantum field near the horizon, resulting in regular expressions for its components.
Banerjee and Majhi's recent work shows that black hole's emission spectrum could be fully reproduced in the tunneling picture, where, as an intriguing technique, the Kruskal extension was introduced to connect the left and right modes inside and outside the horizon. Some attempt, as an extension, was focused on producing the Hawking emission spectrum of the (charged) Reissner-Nordström black hole in the Banerjee-Majhi's treatment. Unfortunately, the Kruskal extension in their observation was so badly defined that the ingoing mode was classically forbidden traveling towards the center of black …
We perform the shadow calculation of the loop quantum gravity motivated regular black hole recently proposed by Ashtekar, Olmedo and Singh (will be termed AOS black hole hereafter). In the process, we also construct the rotating loop quantum gravity inspired solution of the originally proposed static spherically symmetric AOS black hole by applying the modified Newman-Janis algorithm. We study the quantum effects on the shadows of both the non-rotating and rotating loop quantum black hole solutions. It is observed that the general shape of the shadow for non-rotating AOS black hole is circular…
We study the Hamiltonian formulation of the Ashtekar-Olmedo-Singh model for the description of the interior geometry of non-rotating, uncharged black holes. This model incorporates loop quantum effects through the introduction of two regularization parameters. We consider an extended phase space formalism proposed by the creators of the model that includes such parameters as configuration variables, constrained to be functions of the black hole mass. We generalize this restriction, allowing for an off-shell phase space dependence. We then introduce a gauge fixing procedure and reduce the syste…
We reconsider space-time singularities in classical Einsteinian general relativity: with the help of several new co-ordinate systems we show that the Schwarzschild solution can be extended beyond the curvature singularity at r=0. The extension appears as an infinite covering of standard Kruskal space-time. While the two-dimensional reduction of this infinite sequence of Kruskal-Szekeres domains obtained by suppressing the angular degrees of freedom is still a topological manifold - albeit one for which the metric structure is singular on one-dimensional submanifolds - we obtain for the full fo…
We construct a Kruskal-Szekeres-type analytic extension of the Emparan-Reall black ring, and investigate its geometry. We prove that the extension is maximal, globally hyperbolic, and unique within a natural class of extensions. The key to those results is the proof that causal geodesics are either complete, or approach a singular boundary in finite affine time. Alternative maximal analytic extensions are also constructed.
Kuchař has recently given a detailed analysis of the classical and quantum geometrodynamics of the Kruskal extension of the Schwarzschild black hole. In this paper we adapt Kuchař's analysis to the exterior region of a Schwarzschild black hole with a timelike boundary. The reduced Lorentzian Hamiltonian is shown to contain two independent terms, one from the timelike boundary and the other from the bifurcation two-sphere. After quantizing the theory, a thermodynamical partition function is obtained by analytically continuing the Lorentzian time evolution operator to imaginary time and taking t…
We investigate the massless scalar field perturbations of a new loop quantum gravity motivated regular black hole proposed by Ashtekar {\it et al.} in [Phys.Rev.Lett. 121, 241301 (2018), Phys.Rev.D 98, 126003 (2018)]. The spacetime of this black hole is distinguished by its asymptotic properties: in Schwarzschild coordinates one of the metric functions diverges as $r\to \infty$ even though the spacetime is asymptotically flat. We show that despite this unusual asymptotic behavior, the quasinormal mode potential is well defined everywhere when Schwarzschild coordinates are used. We propose a us…
Boundary conditions and the corresponding states of a quantum field theory depend on how the horizons are taken into account. There is ambiguity as to which method is appropriate because different ways of incorporating the horizons lead to different results. We propose that a natural way of including the horizons is to first consider the Kruskal extension and then define the quantum field theory on the Euclidean section. Boundary conditions emerge naturally as consistency conditions of the Kruskal extension. We carry out the proposal for the explicit case of the Schwarzschild-de Sitter manifol…
Our starting point is an iterative construction suited to combinatorics in arbitarary dimensions d, of totally anisymmetrised p-Riemann 2p-forms (2p\le d) generalising the (1-)Riemann curvature 2-forms. Superposition of p-Ricci scalars obtained from the p-Riemann forms defines the maximally Gauss--Bonnet extended gravitational Lagrangian. Metrics, spherically symmetric in the (d-1) space dimensions are constructed for the general case. The problem is directly reduced to solving polynomial equations. For some black hole type metrics the horizons are obtained by solving polynomial equations. Cor…
Recently, there are comparable revised interests in bubble nucleation seeded by black holes. However, it is debated in the literature that whether one shall interpret a static bounce solution in the Euclidean Schwarzschild spacetime (with periodic Euclidean Schwarzschild time) as describing a false vacuum decay at zero temperature or at finite temperature. In this paper, we show a correspondence that the static bounce solution describes either a thermal transition of vacuum in the static region outside of a Schwarzschild black hole or a quantum transition in a maximally extended Kruskal-Szeker…
We present a class of spherically symmetric hypersurfaces in the Kruskal extension of the Schwarzschild space-time. The hypersurfaces have constant negative scalar curvature, so they are hyperboloidal in the regions of space-time which are asymptotically flat.
We present a smooth extension of the Schwarzschild exterior geometry, where the singular interior is superceded by a vacuum phase with vanishing metric determinant. Unlike the Kruskal-Szekeres continuation, this solution to the first-order field equations in vacuum has no singularity in the curvature two-form fields, no horizon and no global time. The underlying non-analytic structure provides a distinct geometric realization of `mass' in classical gravity. We also find that the negative mass Schwarzschild solution does not admit a similar extension within the first-order theory. This is consi…
We conjecture that (when the notion of Hadamard state is suitably adapted) there is no isometry-invariant Hadamard state for the massive or massless covariant Klein-Gordon equation defined on the region of the Kruskal spacetime to the left of a surface of constant Schwarzschild radius in the right Schwarzschild wedge when Dirichlet boundary conditions are put on that surface. We also prove that, with a suitable definition for 'boost-invariant Hadamard state' (which we call 'strongly boost-invariant globally-Hadamard') which takes into account both the existence of the timelike boundary and the…
We present the complete set of analytical solutions of the geodesic equation in Taub-NUT space-times in terms of the Weierstrass elliptic function. We systematically study the underlying polynomials and characterize the motion of test particles by its zeros. Since the presence of the "Misner string" in the Taub-NUT metric has led to different interpretations, we consider these in terms of the geodesics of the space-time. In particular, we address the geodesic incompleteness at the horizons discussed by Misner and Taub, and the analytic extension of Miller, Kruskal and Godfrey, and compare with…
The well-known Schwarzschild black hole was first obtained as a stationary, spherically symmetric solution of the Einstein's vacuum field equations. But until thirty years later, efforts were made for the analytic extension from the exterior area $(r>2GM)$ to the interior one $(r<2GM)$. As a contrast to its maximally extension in the Kruskal coordinates, we provide a comoving coordinate system from the view of the observers freely falling into the black hole in the radial direction. We find an interesting fact that the spatial part in this coordinate system is maximally symmetric $(E_3)$, i.e.…
The tunnel process of the quantum wave from the light cone is carefully discussed. They are applied in the massive quantum particles from the Schwarzschild black hole in the Kruskal metric. The tortoise coordinates prevent one from understanding the tunnel process, and are investigated with care. Furthermore, the massive particles could come out of the black hole either by the Hawking radiation or by the tunnel effect; the tunnel effect might give more information about the black hole.
We study a manifestly unitary formulation of 2d dilaton quantum gravity based on the reduced phase space quantization. The spacetime metric can be expanded in a formal power series of the matter energy-momentum tensor operator. This expansion can be used for calculating the quantum corrections to the classical black hole metric by evaluating the expectation value of the metric operator in an appropriate class of the physical states. When the normal ordering in the metric operator is chosen to be with respect to Kruskal vacuum, the lowest order semiclassical metric is exactly the one-loop effec…
We present the Hamiltonian, quasilocal energy, and angular momentum for a spacetime region spatially bounded by two timelike surfaces. The results are applied to the particular case of a spacetime representing an eternal black hole. It is shown that in the case when the boundaries are located in two different wedges of the Kruskal diagram, the Hamiltonian is of the form $H = H_+ - H_-$, where $H_+$ and $H_-$ are the Hamiltonian functions for the right and left wedges respectively. The application of the obtained results to the thermofield dynamics description of quantum effects in black holes …
The historical postulates for the point mass are shown to be satisfied by an infinity of space-times, differing as to the limiting acceleration of a radially approaching test particle. Taking this limit to be infinite gives Schwarzschild's result, which for a point mass at x = y = z = 0 has C(0+) = a^2, where a = 2m and C(r) denotes the coefficient of the angular terms in the polar form of the metric. Hilbert's derivation used the variable r* =[C(0+)]^1/2. For Hilbert, however, C was unknown, and thus could not be used to determine r*(0). Instead he asserted, in effect, that r* = (x^2 + y^2 + …
We use a microcanonical path integral closely related to that introduced by Brown and York in 1992 to add new entries to the AdS/CFT dictionary concerning the interiors of small black holes. Stationary points of such path integrals are also stationary points of more standard canonical-type path integrals with fixed boundary metric, but the condition for dominance is now maximizing Hubeny-Rangamani-Takayanagi entropy at fixed energy. As a result, such path integrals can bring to the fore saddles that fail to dominate in more familiar contexts. We use this feature to argue that the standard Krus…
We quantize the Reissner-Nordström black hole using an adaptation of Kuchař's canonical decomposition of the Kruskal extension of the Schwarzschild black hole. The Wheeler-DeWitt equation turns into a functional Schroedinger equation in Gaussian time by coupling the gravitational field to a reference fluid or dust. The physical phase space of the theory is spanned by the mass, $M$, the charge, $Q$, the physical radius, $R$, the dust proper time, $τ$, and their canonical momenta. The exact solutions of the functional Schroedinger equation imply that the difference in the areas of the outer and …
The BTZ stationary black hole solution is considered and its mass and angular momentum are calculated by means of Noether theorem. In particular, relative conserved quantities with respect to a suitably fixed background are discussed. Entropy is then computed in a geometric and macroscopic framework, so that it satisfies the first principle of thermodynamics. In order to compare this more general framework to the prescription by Wald et al. we construct the maximal extension of the BTZ horizon by means of Kruskal-like coordinates. A discussion about the different features of the two methods fo…
We investigate here the behavior of a few spherically symmetric static acclaimed black hole solutions in respect of tidal forces in the geodesic frame. It turns out that the forces diverge on the horizon of cold black holes (CBH) while for ordinary ones, they do not. It is pointed out that Kruskal-like extensions do not render the CBH metrics nonsingular. We present a CBH that is available in the Brans-Dicke theory for which the tidal forces do not diverge on the horizon and in that sense it is a better one.
We discuss quantum properties of the single-exterior, geon-type black (and white) holes that are obtained from the Kruskal spacetime and the spinless Banados-Teitelboim-Zanelli hole via a quotient construction that identifies the two exterior regions. For the four-dimensional geon, the Hartle-Hawking type state of a massless scalar field is thermal in a limited sense, but there is a discrepancy between Lorentzian and Riemannian derivations of the geon entropy. For the three-dimensional geon, the state induced for a free conformal scalar field on the conformal boundary is similarly thermal in a…
Enforcing the spacetime of supermassive cosmic strings to satisfy the symmetry of a gravitational topological defect, that is a spacetime kink, it is shown that the energy of these strings becomes quantized so that only defects whose internal geometry is that of a hemisphere (critical string) and that of a sphere (extreme string) are allowed. In the latter case, the exterior conical singularity becomes an event-horizon singularity which can be removed by using the Kruskal technique in the kink metric. It is also shown that the geodesically complete metric of the extreme string can accommodate …
We study the renormalized stress-energy tensor (RSET) for static quantum states on (n+1)-dimensional, static, spherically symmetric black holes. By solving the conservation equations, we are able to write the stress-energy tensor in terms of a single unknown function of the radial co-ordinate, plus two arbitrary constants. Conditions for the stress-energy tensor to be regular at event horizons (including the extremal and ``ultra-extremal'' cases) are then derived using generalized Kruskal-like co-ordinates. These results should be useful for future calculations of the RSET for static quantum s…
We study the behaviour of scalar fields on background geometries which undergo quantum tunneling. The two examples considered are a moving mirror in flat space which tunnels through a potential barrier, and a false vacuum bubble which tunnels to form a black hole. WKB approximations to the Schrodinger and Wheeler-DeWitt equations are made, leading one to solve field equations on the Euclidean metric solution interpolating between the classically allowed geometries. The state of the field after tunneling can then be determined using the method of non-unitary Bogolubov transformations developed …
We examine the Wheeler-DeWitt equaton for a static, eternal Schwarzschild black hole in Kucha\v r-Brown variables and obtain its energy eigenstates. Consistent solutions vanish in the exterior of the Kruskal manifold and are non-vanishing only in the interior. The system is reminiscent of a particle in a box. States of definite parity avoid the singular geometry by vanishing at the origin. These definite parity states admit a discrete energy spectrum, depending on one quantum number which determines the Arnowitt-Deser-Misner (ADM) mass of the black hole according to a relation conjectured long…
Quantum field effects on a classical background spacetime may be obtained from the semiclassical equations of General Relativity with the expectation value of the stress-energy tensor of the quantum field as a source. This expectation value can be calculated from Hadamard's elementary two-point function, which in practice is given in terms of sums of products of field modes evaluated at two spacetime points. We derive expressions for the two-point function for a massless scalar field in the Unruh state on a Kerr black hole spacetime. Our main result in this paper is a novel expression valid wh…
To apply the laws of General Relativity to quantum black holes, one first needs to remove the horizon singularity by means of Kruskal-Szekeres coordinates. This however doubles spacetime, which thereby is equipped with an exact binary symmetry. All particles near a black hole share the same symmetry, and conservation of this symmetry may completely remove the information paradox: the quantum black hole has no interior, or equivalently, the black hole interior is a quantum clone of the exterior region. These observations, totally overlooked in most of the literature on quantum black holes, reso…
We consider a Hamiltonian theory of spherically symmetric vacuum Einstein gravity under Kruskal-like boundary conditions in variables associated with the Einstein-Rosen wormhole throat. The configuration variable in the reduced classical theory is the radius of the throat, in a foliation that is frozen at the left hand side infinity but asymptotically Minkowski at the right hand side infinity, and such that the proper time at the throat agrees with the right hand side Minkowski time. The classical Hamiltonian is numerically equal to the Schwarzschild mass. Within a class of Hamiltonian quantiz…
The Hawking temperature for Schwarzschild black hole $T_H=1/8πM$ is singular in the limit of vanishing mass $M\to 0$. However, the Schwarzschild metric itself is regular in this limit, it is reduced to the Minkowski metric and there are no reasons to believe that the temperature becomes infinite. This discrepancy may be due to the singularity of the Kruskal coordinates in this limit. To improve the situation, new coordinates for the Schwarzschild metric are introduced, called thermal coordinates, which depend on the black hole mass $ M $ and the parameter $b$. The thermal coordinates are regul…
We compute the spectra and total fluxes of quantum mechanically produced particles crossing the black hole and cosmological horizons in Schwarzschild de Sitter (SdS). Particle states are defined with respect to well-behaved, Kruskal coordinates near the horizons, and as a consequence we find that these spectra are generally non-thermal. The non-thermal Bogoliubov coefficient for a vacuum fluctuation near the black hole horizon to produce a particle that crosses the cosmological horizon is shown to equal to the convolution of two thermal coefficients, one at the cosmological temperature and one…
These lecture notes are intended as a guide to Graduate level readers that are already familiar with basic General Relativity. They present in a concise way some advanced concepts and problems encountered in the study of gravitation. In these notes are covered: Alternates forms of the Schwarzschild Black Hole solution, including the classic Kruskal extension; An account of the building of Conformal, Carter-Penrose, diagrams; A discussion of Birkhoff Theorem; A discussion of tools for Geodesics and congruences, including Energy Conditions; A discussion of Horizons and an approach to some of the…
The basic properties of the C-metric are well known. It describes a pair of causally separated black holes which accelerate in opposite directions under the action of forces represented by conical singularities. However, these properties can be demonstrated much more transparently by making use of recently developed coordinate systems for which the metric functions have a simple factor structure. These enable us to obtain explicit Kruskal-Szekeres-type extensions through the horizons and construct two-dimensional conformal Penrose diagrams. We then combine these into a three-dimensional pictur…
It is shown how the technique of restricted path integrals (RPI) or quantum corridors (QC) may be applied for the analysis of relativistic measurements. Then this technique is used to clarify the physical nature of thermal effects as seen by an accelerated observer in Minkowski space-time (Unruh effect) and by a far observer in the field of a black hole (Hawking effect). The physical nature of the "thermal atmosphere" around the observer is analysed in three cases: a) the Unruh effect, b) an eternal (Kruskal) black hole and c) a black hole forming in the process of collapse. It is shown that t…
We extend the work of Oppenheimer & Synder to model the gravitational collapse of a star to a black hole by including quantum mechanical effects. We first derive closed-form solutions for classical paths followed by a particle on the surface of the collapsing star in Schwarzschild and Kruskal coordinates for space-like, time-like and light-like geodesics. We next present an application of these paths to model the collapse of ultra-light dark matter particles, which necessitates incorporating quantum effects. To do so we treat a particle on the surface of the star as a wavepacket and integrate …
There is rich literature on regular black holes from loop quantum gravity (LQG), where quantum geometry effects resolve the singularity, leading to a quantum extension of the classical space-time. As we will see, the mechanism that resolves the singularity can also trigger conceptually undesirable features that can be subtle and are often uncovered only after a detailed examination. Therefore, the quantization scheme has to be chosen rather astutely. We illustrate the new physics that emerges first in the context of the eternal black hole represented by the Kruskal space-time in classical gene…
We investigate the Hawking effect on entangled fields. By considering a scalar field which is in a two-mode squeezed state from the point of view of freely falling (Kruskal) observers crossing the horizon of a Schwarzschild black hole, we study the degradation of quantum and classical correlations in the state from the perspective of Schwarzschild observers confined outside the horizon. Due to monogamy constraints on the entanglement distribution, we show that the lost bipartite entanglement is recovered as multipartite entanglement among modes inside and outside the horizon. In the limit of a…
Schwarzschild's actual exterior solution (Gs) is resurrected and together with the manifold M is shown to constitute a space-time possessing all the properties historically thought to be required of a point mass. On the other hand, the metric that today is ascribed to Schwarzschild, but which was in fact first obtained by Droste and Weyl, is shown to give rise to a space-time that is neither equivalent to Schwarzschild's nor derivable from the "historical" properties of a point mass. Consequently, the point-mass interpretation of the Kruskal-Fronsdal space-time (Mw, Gkf) can no longer be justi…
