In this article, we introduce a new effective model for the Kantowski-Sachs spacetime in the context of loop quantum gravity, and we use it to evaluate departures from general relativity in the case of Schwarzschild black hole interior. The model is based on an effective Hamiltonian constructed via the regularized Thiemann identities in the $\bar μ$-scheme. We show that, in contrast with the $μ_o$-scheme studied in [1], the classical limit imposes certain alterations of Thiemann identities as well as restrictions on the choice of regulators. Once we define the Hamiltonian, we derive the equations of motion for the relevant variables and proceed with the solving using numerical methods, focusing on a specific choice of $\bar μ$. We establish that for a Schwarzschild black hole interior, the effective dynamics leads to a resolution of the classical singularity and the emergence of an anti-trapped region bounded by a second Killing horizon. We then perform a comparison of the dynamical trajectories and their properties obtained in the new model and some models present in the literature. We finally conclude with few comments on other choices of the regulators and their consequences.
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.
We calculate the maximum interior volume, enclosed by the event horizon, of a ($1+D$)-dimensional Schwarzschild black hole. Taking into account the mass change due to Hawking radiation, we show that the volume increases towards the end of the evaporation. This fact is not new as it has been observed earlier for four dimensional case. The interesting point we observe is that this increase rate decreases towards the higher value of space dimensions $D$; i.e. it is a decelerated expansion of volume with the increase of spacial dimensions. This implies that for a sufficiently large $D$, the maximu…
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 …
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）
In recent work, the Schwarzschild interior of a black hole was investigated, incorporating quantum gravitational modifications due to loop quantum gravity. The central Schwarzschild singularity was shown to be replaced by a Nariai type universe. In this brief report we show that this interior solution is stable with respect to small perturbations, in contrast to the classical Nariai universe.
We employ the minimal geometric deformation approach to gravitational decoupling (MGD- decoupling) in order to build an exact anisotropic version of the Schwarzschild interior solution in a space-time with cosmological constant. Contrary to the well-known Schwarzschild interior, the matter density in the new solution is not uniform and possesses subluminal sound speed. It therefore satisfies all standard physical requirements for a candidate astrophysical object.
We present a time-dependent uniform-density interior Schwarzschild solution, an exact solution to the Einstein field equations. Our solution describes the collapse (or the time-reversed expansion) of an object from an infinite radius to an intermediate radius of 9/8 of the Schwarzschild radius, at which time a curvature singularity appears at the origin, and then continues beyond the singularity to a gravastar with radius equal to the Schwarzschild radius.
We solve exactly the Regge-Wheeler equation for axial perturbations of the Schwarzschild metric in the black hole interior in terms of Heun functions and give a description of the spectrum and the eigenfunctions of the interior problem. The phenomenon of attraction and repulsion of the discrete eigenvalues of gravitational waves is discovered.
We embed the Schwarzschild interior solution in a five-dimensional flat space and show that the systems of the interior and the exterior solution are based on the same geometrical principles. It turns out that the energy tensor of the matter has its origin in the five-dimensional structure of the geometry and is built up by the generalized second fundamental forms. Thus, the matter is already geometrized. The Codazzi equations are the field equations for the matter field.
We investigate inner structure of Schwarzschild black hole on a five-dimensional spacetime S^3xR^2. To do this, we exploit a fünfbein scheme. In particular, we construct an equation of state of hydrostatic equilibrium for the five-dimensional Schwarzschild black hole, which is a five-dimensional version of the Tolman-Oppenheimer-Volkoff equation on four-dimensional manifold. We also investigate uniform density interior configuration of the five-dimensional black hole which consists of incompressible fluid of density, to find a general relativistic expression for pressure.
The analytic expression obtained in the preceding project for the massless conformal scalar propagator in the Hartle-Hawking vacuum state for small values of the Schwarzschild radial coordinate above $ r = 2M$ is analytically extended into the interior of the Schwarzschild black hole. The result of the analytical extension coincides with the exact propagator for a small range of values of the Schwarzschild radial coordinate below $ r = 2M$ and is an analytic expression which manifestly features its dependence on the background space-time geometry. This feature as well as the absence of any ass…
In a recent paper Liu considered the complete Schwarzschild interior and exterior solution in harmonic coordinates. There he argued about the necessity to keep an integration constant, in contrast with previous treatments. The purpose of this comment is to show that the above conclusion cannot be traced from the matching conditions between the vacuum exterior and the uniform density interior perfect fluid, as claimed.
In the context of the Randall-Sundrum braneworld, the minimal geometric deformation approach, which has been successfully used to generate exact interior solutions to Einstein's field equations for static braneworld stars with local and non-local bulk terms, is used to obtain the braneworld version of the Schwarzschild's interior solution. Using this new solution, the behaviour of the Weyl functions is elucidated in terms of the compactness for different stellar distributions.
Interior evolution of regular Schwarzschild black holes
Research Centre for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava（奥帕瓦西里西亚大学物理研究所理论与天体物理研究中心）；Universidad de Tarapacá（塔拉帕卡大学）
We consider the modified Einstein equations obtained in the framework of effective spherically symmetric polymer models inspired by Loop Quantum Gravity. When one takes into account the anomaly free point-wise holonomy quantum corrections, the modification of Einstein equations is parametrized by a function $f(x)$ of one phase space variable. We solve explicitly these equations for a static interior black hole geometry and find the effective metric describing the trapped region, inside the black hole, for any $f(x)$. This general resolution allows to take into account a standard ambiguity inhe…
We wish to construct a model for charged star as a generalization of the uniform density Schwarzschild interior solution. We employ the Vaidya and Tikekar ansatz [{\it Astrophys. Astron.} {\bf 3} (1982) 325] for one of the metric potentials and electric field is chosen in such a way that when it is switched off the metric reduces to the Schwarzschild. This relates charge distribution to the Vaidya-Tikekar parameter, $k$, indicating deviation form sphericity of three dimensional space when embedded into four dimensional Euclidean space. The model is examined against all the physical conditions …
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 …
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 a simple approach for obtaining Kerr interior solutions with the help of the Newman-Janis algorithm (NJA) starting with static space-times describing physically sensible interior Schwarzschild solutions. In this context, the Darmois-Israel (DI) junction conditions are analyzed. Starting from the incompressible Schwarzschild solution, a class of Kerr interior solutions is presented, together with a discussion of the slowly rotating limit. The energy conditions are discussed for the solutions so obtained. Finally, the NJA algorithm is applied to the static, anisotropic, conformally fl…
We discuss a sufficiently large 4-dimensional Schwarzschild black hole which is in equilibrium with a heat bath. In other words, we consider a black hole which has grown up from a small one in the heat bath adiabatically. We express the metric of the interior of the black hole in terms of two functions: One is the intensity of the Hawking radiation, and the other is the ratio between the radiation energy and the pressure in the radial direction. Especially in the case of conformal matters we check that it is a self-consistent solution of the semi-classical Einstein equation, $G_{μν}=8πG \langl…
We examine the Schwarzschild interior of a black hole, incorporating quantum gravitational modifications due to loop quantum gravity. We consider an improved loop quantization using techniques that have proven successful in loop quantum cosmology. The central Schwarzschild singularity is resolved and the implications for the fate of an in-falling test particle in the interior region is discussed. The singularity is replaced by a Nariai type Universe. We discuss the resulting conformal diagram, providing a clear geometrical interpretation of the quantum effects.
We present the detailed analyses of a model of loop quantum Schwarzschild interior coupled to a massless scalar field and extend the results in our previous rapid communication arXiv:2006.08313 to more general schemes. It is shown that the spectrum of the black hole mass is discrete and does not contain zero. This indicates the existence of a black hole remnant after Hawking evaporation due to loop quantum gravity effects. Besides to show the existence of a stable black hole remnant in the vacuum case, the quantum dynamics for the non-vacuum case is also solved and compared with the effective …
We discuss the properties of the previously constructed model of a Schwarzschild black hole interior where the singularity is replaced by a regular bounce, ultimately leading to a white hole. The model is semiclassical in nature and uses as a source of gravity the effective stress-energy tensor (SET) corresponding to vacuum polarization of quantum fields, and a minimum spherical radius is a few orders of magnitude larger than the Planck length, so that the effects of quantum gravity should be still negligible. We estimate the other quantum contributions to the effective SET, caused by a nontri…
We study the interior of a Schwarzschild Black-Hole (B-H) using Relativistic Quantum Geometry described in \cite{rb} and \cite{rb1}. We found discrete energy levels for a scalar field from a polynomial condition for Heun Confluent functions expanded around the Schwarzschild radius. From the solutions it is obtained that the uncertainty principle is valid for each energy level of space-time, in the form: $E_n\, r_{sh,n}=\hbar/2$. Temperature, entropy and the B-H mass are dependent on the number of states in the B-H, such that the Bekenstein-Hawking (BH) results are obtained in a limit case.
In this paper the interior of the Schwarzschild black hole, which is presented as a vacuum homogeneous and anisotropic Kantowski-Sachs minisuperspace cosmological model, is considered. Lagrangian of the model is reduced by a suitable coordinate transformation to Lagrangian of two decoupled oscillators with the same frequencies and with zero energy in total (an oscillator-ghost-oscillator system). The model will be presented in a classical, a p-adic and a noncommutative case. Then, within the standard quantum approach Wheeler-DeWitt equation and its general solutions, i.e. a wave function of th…
In this paper we calculate modifications to the Schwarzschild solution by using a semiclassical analysis of loop quantum black hole. We obtain a metric inside the event horizon that coincides with the Schwarzschild solution near the horizon but that is substantially different at the Planck scale. In particular we obtain a bounce of the two-sphere for a minimum value of the radius and that it is possible to have another event horizon close to the r=0 point.
The relativistic wave equation for spin-1/2 particles in the interior Schwarzschild solution in the presence of a uniform magnetic field is obtained. The fully relativistic regime is considered, and the energy levels occupied by the particles are derived as functions of the magnetic field, the radius of the massive sphere and the total mass of the latter. As no assumption is made on the relative strengths of the particles' interaction with the gravitational and magnetic fields, the relevance of our results to the physics of the interior of neutron stars, where both the gravitational and the ma…
We review, as well as provide some new results regarding the study of the structure of spacetime and the singularity in the interior of the Schwarzschild black hole in both loop quantum gravity and generalized uncertainty principle approaches, using congruences and their associated expansion scalar and the Raychaudhuri equation. We reaffirm previous results that in loop quantum gravity, in all three major schemes of polymer quantization, the expansion scalar, Raychaudhuri equation and the Kretschmann scalar remain finite everywhere in the interior. In the context of the eneralized uncertainty …
We study the behaviour of smooth solutions to the wave equation, $\square_gψ=0$, in the interior of a fixed Schwarzschild black hole. In particular, we obtain a full asymptotic expansion for all solutions towards $r=0$ and show that it is characterised by its first two leading terms, the principal logarithmic term and a bounded second order term. Moreover, we characterise an open set of initial data for which the corresponding solutions blow up logarithmically on the entirety of the singular hypersurface $\{r=0\}$. Our method is based on deriving weighted energy estimates in physical space and…
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.
Revisiting Schwarzschild black hole singularity through string theory
College of Physics, Sichuan University（四川大学物理学院）；DAMTP, Centre for Mathematical Sciences, University of Cambridge（剑桥大学数学科学中心应用数学与理论物理系）；Department of Physics, Chongqing University（重庆大学物理学院）
We show that loop quantum gravity effects leads to the finiteness of expansion and its rate of change in the effective regime in the interior of the Schwarzschild black hole. As a consequence the singularity is resolved. We find this in line with previous results about curvature scalar and strong curvature singularities in Kantowski-Sachs model which is isometric to Schwarzschild interior.
The constant density interior Schwarzschild solution for a static, spherically symmetric collapsed star has a divergent pressure when its radius $R\le\frac{9}{8}R_s=\frac{9}{4}GM$. We show that this divergence is integrable, and induces a non-isotropic transverse stress with a finite redshifted surface tension on a spherical surface of radius $R_0=3R\sqrt{1-\frac{8}{9}\frac{R}{R_s}}$. For $r < R_0$ the interior Schwarzschild solution exhibits negative pressure. When $R=R_s$, the surface is localized at the Schwarzschild radius itself, $R_0=R_s$, and the solution has constant negative pressure …
We show that the Kantowski--Sachs model of a Schwarzschild black hole interior can be slightly generalized in order to accommodate spatial metrics of different orientations, and in this formulation the equations of motion admit a variable redefinition that makes the system regular at the singularity. This system will then traverse the singularity in a deterministic way (information will be conserved through it), and evolve into a time-reversed and orientation-flipped Schwarzschild white hole interior.
The loop quantization of the Schwarzschild interior region, as described by a homogeneous anisotropic Kantowski-Sachs model, is re-examined. As several studies of different -inequivalent- loop quantizations have shown, to date there exists no fully satisfactory quantum theory for this model. This fact poses challenges to the validity of some scenarios to address the black hole information problem. Here we put forward a novel viewpoint to construct the quantum theory that builds from some of the models available in the literature. The final picture is a quantum theory that is both independent o…
We reconsider the study of the interior of the Schwarzschild black hole now including inverse triad quantum corrections within loop quantization. We derive these corrections and show that they are are related to two parameters $δ_b, δ_c$ associated to the minimum length in the radial and angular directions, that enter Thiemann's trick for quantum inverse triads. Introduction of such corrections may lead to non-invariance of physical results under rescaling of the fiducial volume needed to compute the dynamics, due to noncompact topology of the model. So, we put forward two prescriptions to res…
Matching Radial Geodesics in Two Schwarzschild Spacetimes (e.g. Black-to-White Hole Transition) or Schwarzschild and de Sitter Spacetimes (e.g. Interior of a Non-singular Black Hole)
A reformulation of the Schwarzschild solution of the linearised Einstein field equations in post-Riemannian Finsler spacetime is derived. The solution is constructed in three stages: the exterior solution, the event-horizon solution and the interior solution. It is shown that the exterior solution is asymptotically similar to Newtonian gravity at large distances implying that Newtonian gravity is a low energy approximation of the solution. Application of Eddington-Finklestein coordinates is shown to reproduce the results obtained from standard general relativity at the event horizon. Further a…
Our goal in this paper is to explain the internal properties of the black hole by considering its density as a function of the reciprocal of its radius and the temperature as a function of the reciprocal of the density. We then set the temperature to the Hawking temperature. This gives all the macroscopic quantities of the black hole, such as heat capacity, pressure, surface gravity, and equation of state. In this work, we only consider a black hole with mass M , radius r+ and vacuum. Two internal forces and the corresponding potential are obtained. In addition, the potential of the vacuum, th…
An anisotropic fluid with a negative radial pressure $p = - ρ$ is supposed to exist near the horizon of a Schwarzschild black hole. The constant energy density $ρ$ depends only on the black hole mass. The radial acceleration of the static observers is constant but its modulus has exactly the Rindler form, as was recently suggested by Dahia and Felix da Silva. The structure of the stress tensor is similar with that obtained previously for the interior region of a black hole, excepting the time dependence.
We investigate the properties of relativistic star spheres made of an electrically charged incompressible fluid, generalizing, thus, the Schwarzschild interior solution. The investigation is carried by integrating numerically the hydrostatic equilibrium equation, i.e., the Tolman-Oppenheimer-Volkoff (TOV) equation, with the hypothesis that the charge distribution is proportional to the energy density. We match the interior to a Reissner-Nordström exterior, and study some features of these star spheres such as the total mass $M$, the radius $R$, and the total charge $Q$. We also display the pre…
Though loop quantization of several spacetimes has exhibited existence of a bounce via an explicit evolution of states using numerical simulations, the question about the way central singularity is resolved in the black hole interior has remained open. The quantum Hamiltonian constraint in loop quantization turns out to be a finite difference equation whose stability is important to understand to gain insights on the viability of the underlying quantization and resulting physical implications. We take first steps towards addressing these issues for a loop quantization of the Schwarzschild inte…
The interior of a Schwarzschild black hole is investigated at the level of phenomenological dynamics with the discreteness corrections of loop quantum geometry implemented in two different improved quantization schemes. In one scheme, the classical black hole singularity is resolved by the quantum bounce, which bridges the black hole interior with a white hole interior. In the other scheme, the classical singularity is resolved and the event horizon is also diffused by the quantum bounce. Jumping over the quantum bounce, the black hole gives birth to a baby black hole with a much smaller mass.…
In this paper, we explore the interior dynamics of neutral and charged black holes. Scalar collapses in flat, Schwarzschild, and Reissner-Nordstrom geometries are simulated. We examine the dynamics in the vicinities of the central singularity of a Schwarzschild black hole and of the inner horizon of a Reissner-Nordstrom black hole. In simulating scalar collapses in Schwarzschild and Reissner-Nordstrom geometries, Kruskal and Kruskal-like coordinates are used, respectively, with the presence of a scalar field being taken into account. It is found that, besides near the inner horizons of Reissne…
The exterior and interior Schwarzschild solutions are rewritten replacing the usual radial variable with an angular one. This allows to obtain some results otherwise less apparent or even hidden in other coordinate systems.
In this contribution we present a noncanonical phase-space noncommutative (NC) extension of a Kantowski Sachs (KS) cosmological model to describe the interior of a Schwarzschild black hole (BH). We evaluate the thermodynamical quantities inside this NC Schwarzschild BH and compare with the well known quantities. We find that for a NCBH the temperature and entropy have the same mass dependence as the Hawking quantities for a Schwarzschild BH.
We propose a scheme for quantizing a scalar field over the Schwarzschild manifold including the interior of the horizon. On the exterior, the timelike Killing vector and on the horizon the isometry corresponding to restricted Lorentz boosts can be used to enforce the spectral condition. For the interior we appeal to the need for CPT invariance to construct an explicitly positive definite operator which allows identification of positive and negative frequencies. This operator is the translation operator corresponding to the inexorable propagation to smaller radii as expected from the classical …
Schwarzschild black-hole interiors border on space-like singularities representing classical information leaks. We show that local quantum physics is decoupled from these leaks due to dynamically generated boundaries, called Zeno borders. Beyond Zeno borders black-hole interiors become asymptotically silent, and quantum fields evolve freely towards the geodesic singularity with vanishing probability measure for populating the geodesic boundary. Thus Zeno borders represent a probabilistic completion of Schwarzschild black holes within the semiclassical framework.
Christodoulou and Rovelli have shown that the maximal interior volume of a Schwarzschild black hole linearly grows with time. Recently, their conclusion has been extended to the Reissner{-}Nordstr$\ddot{\text{o}}$m and Kerr black holes. Meanwhile, the entropy of interior volume in a Schwarzschild black hole has also been calculated. Here, a new method calculating the entropy of interior volume of the black hole is given and it can be used in more general cases. Using this method, the entropy associated with the volume of a Kerr black hole is calculated and it is found that the entropy is propo…
Using the Hamiltonian constraint derived by Ashtekar and Bojowald, we look for pre-classical wave functions in the Schwarzschild interior. In particular, when solving this difference equation by separation of variables, an inequality is obtained relating the Immirzi parameter $γ$ to the quantum ambiguity $δ$ appearing in the model. This bound is violated when we use a natural value for $δ$ based on loop quantum gravity together with a recent proposal for $γ$. We also present numerical solutions of the constraint.
The success of loop quantum cosmology to resolve classical singularities of homogeneous models has led to its application to the classical Schwarszchild black hole interior, which takes the form of a homogeneous Kantowski-Sachs model. The first steps of this were done in pure quantum mechanical terms, hinting at the traversable character of the would-be classical singularity, and then others were performed using effective heuristic models capturing quantum effects that allowed a geometrical description closer to the classical one but avoided its singularity. However, the problem of establishin…
Within the Relativistic Theory of Gravitation it is shown that the equation of state $p=ρ$ holds near the center of a black hole. For the stiff equation of state $p=ρ-ρ_c$ the interior metric is solved exactly. It is matched with the Schwarzschild metric, which is deformed in a narrow range beyond the horizon. The solution is regular everywhere, with a specific shape at the origin. The gravitational redshift at the horizon remains finite but is large, $z\sim 10^{23}$$M_\odot/M$. Time keeps its standard role also in the interior. The energy of the Schwarzschild metric, shown to be minus infinit…
