An Exploration of the Physics of Spherically Symmetric Dynamic Horizons
James Lindesay
备注：
Invited talk presented at the 2008 Annual Conference of the National Society of Black Physicists and National Society of Hispanic Physicists, Feb. 20-23, Washington, D.C., 37 pages, 12 figures, symbol fonts in figures corrected
Geometries with horizons offer insights into relationships between general relativity and quantum physics. For static spherically symmetric space-times, the event horizon is coincident with a coordinate anomaly that introduces complications in descriptions of near horizon physics. Naive introduction of dynamics using coordinates with anomalous behavior coincident with the horizon also introduces invariant singular physical content at that horizon. However, the introduction of a temporal coordinate that is non-orthogonal to spatial coordinates near the horizon, but asymptotically orthogonal, provides a dynamic description without singular physical content at the horizon itself. Penrose diagrams will be presented exhibiting temporal dependencies for accreting and evaporating black holes, and near horizon light-like trajectories will be examined. In addition, the quantum mechanics of simple quantum fields will be explored. Finally, a two-fluid cosmology will be suggested to describe dynamic coherent aspects of the universe as a whole.
We determine sufficient and necessary conditions for a spherically symmetric initial data set to satisfy the dynamical horizon conditions in the spacetime development. The constraint equations reduce to a single second order linear master equation, which leads to a systematic construction of all spherically symmetric dynamical horizons (SSDH) satisfying certain boundedness conditions. We also find necessary and sufficient conditions for a given spherically symmetric spacetime to contain a SSDH.
Using tunnelling approach, Hawking radiation is derived for a general class of non-static spherically symmetric space time. The standard tunnelling rate formula is obtained using the unified first law of thermodynamics on the trapping horizon and Hawking temperature is measured by an Kodama observer inside the horizon.
We apply the Dirac-Bergmann algorithm for the analysis of constraints to gauge theories defined on spherically symmetric black hole backgrounds. We find that the constraints for a given theory are modified on such spacetimes through the presence of additional contributions from the horizon. As a concrete example, we consider the Maxwell field on a black hole background, and determine the role of the horizon contributions on the dynamics of the theory.
It is well known that quasi-local black hole horizons depend on the choice of a time coordinate in a spacetime. This has implications for notions such as the surface of the black hole and also on quasi-local physical quantities such as horizon measures of mass and angular momentum. In this paper, we compare different horizons on non-spherically symmetric slicings of Vaidya spacetimes. The spacetimes we investigate include both accreting and evaporating black holes. For some simple choices of the Vaidya mass function function corresponding to collapse of a hollow shell, we compare the area for …
Isolated horizon conditions specialized to spherical symmetry can be imposed directly at the quantum level. This answers several questions concerning horizon degrees of freedom, which are seen to be related to orientation, and its fluctuations at the kinematical as well as dynamical level. In particular, in the absence of scalar or fermionic matter the horizon area is an approximate quantum observable. Including different kinds of matter fields allows to probe several aspects of the Hamiltonian constraint of quantum geometry that are important in inhomogeneous situations.
We consider a quantum field which is in a Hartle-Hawking state propagating in a general spherically symmetric black hole space-time. We make use of uniform approximations to the radial equation to calculate the components of the stress tensor, renormalized using the Hadamard form of the Green's function, on the horizons of this space-time. We then specialize these results to the case of the `lukewarm' Reissner-Nordstrom-de Sitter black hole and derive some conditions on the stress tensor for the regularity of the Hartle-Hawking state.
The exact static and spherically symmetric solutions of the vacuum field equations for a Higgs Scalar-Tensor theory (HSTT) are derived in Schwarzschild coordinates. It is shown that in general there exists no Schwarzschild horizon and that the fields are only singular (as naked singularity) at the center (i.e. for the case of a pont-particle). However, the Schwarzschild solution as in usual general relativity (GR) is obtained for the vanishing limit of Higgs field excitations.
We examine potential deformations of inner black hole and cosmological horizons in Reissner-Nordström de-Sitter spacetimes. While the rigidity of the outer black hole horizon is guaranteed by theorem, that theorem applies to neither the inner black hole nor past cosmological horizon. Further for pure deSitter spacetime it is clear that the cosmological horizon can be deformed (by translation). For specific parameter choices, it is shown that both inner black hole and cosmological horizons can be infinitesimally deformed. However these do not extend to finite deformations. The corresponding res…
We consider a spherically symmetric line element which admits either a black hole geometry or a wormhole geometry and show that in both cases the apparent horizon or the wormhole throat is partially characterized by the zero-set of a single curvature invariant. The detection of the apparent horizon by this invariant is consistent with the geometric horizon detection conjectures and implies that it is a geometric horizon of the black hole, while the detection of the wormhole throat presents a conceptual problem for the conjectures. To distinguish between these surfaces, we determine a set of cu…
We establish sufficient conditions for the appearance of apparent horizons in spherically symmetric initial data when spacetime is foliated extrinsically. Let $M$ and $P$ be respectively the total material energy and the total material current contained in some ball of radius $\ell$. Suppose that the dominant energy condition is satisfied. We show that if $M- P \ge \ell$ then the region must possess a future apparent horizon for some non -trivial closed subset of such gauges. The same inequality holds on a larger subset of gauges but with a larger constant of proportionality which depends weak…
We establish necessary conditions for the appearance of both apparent horizons and singularities in the initial data of spherically symmetric general relativity when spacetime is foliated extrinsically. When the dominant energy condition is satisfied these conditions assume a particularly simple form. Let $ρ_{Max}$ be the maximum value of the energy density and $\ell$ the radial measure of its support. If $ρ_{Max}\ell^2$ is bounded from above by some numerical constant, the initial data cannot possess an apparent horizon. This constant does not depend sensitively on the gauge. An analogous ine…
Horizon solutions for the axial perturbations of the spherically symmetric metric are analyzed in the framework of the relativistic theory of gravitation. The gravitational perturbations can not be absorbed by the horizon that results in the excitation of the new type of the normal modes trapped by the Regge-Wheeler potential. The obtained results demonstrate testable differences between the collapsar and the black hole near-horizon physics.
Understood in terms of pure states evolving into mixed states, the possibility of information loss in black holes is closely related to the global causal structure of spacetime, as is the existence of event horizons. However, black holes need not be defined by event horizons, and in fact we argue that in order to have a fully unitary evolution for black holes, they should be defined in terms of something else, such as a trapping horizon. The Misner-Sharp mass in spherical symmetry shows very simply how trapping horizons can give rise to black hole thermodynamics, Hawking radiation and singular…
By using of the Euler-Lagrange equations, we find a static spherically symmetric solution in the Einstein-aether theory with the coupling constants restricted. The solution is similar to the Reissner-Nordstrom solution in that it has an inner Cauchy horizon and an outer black hole event horizon. But a remarkable difference from the Reissner-Nordstrom solution is that it is not asymptotically flat but approaches a two dimensional sphere. The resulting electric potential is regular in the whole spacetime except for the curvature singularity. On the other hand, the magnetic potential is divergent…
In this work, we consider a semiclassical description of the spherically symmetric gravitational collapse with a massless scalar field. In particular, we employ an effective scenario provided by holonomy corrections from loop quantum gravity, to the homogeneous interior spacetime. The singularity that would arise at the final stage of the corresponding classical collapse, is resolved in this context and is replaced by a bounce. Our main purpose is to investigate the evolution of trapped surfaces during this semiclassical collapse. Within this setting, we obtain a threshold radius for the colla…
It has been shown by Bañados, Silk and West (BSW) that the center of mass energy (E_cm) of test particles starting from rest at infinity and colliding near the horizon of a Schwarzschild black hole is always finite. In this communication, we extent the BSW scenario and study two particles with different energies colliding near the horizon of a static spherically symmetric black hole. Surprisingly, we find that even for the static spherically symmetric (i.e., Schwarzschild like) black holes it is possible to obtain an arbitrarily high E cm from the two test particles colliding near the horizon …
We aim to study the thermodynamic properties of the spherically symmetric reference frames with uniform acceleration, including the spherically symmetric generalization of Rindler reference frame and the new kind of uniformly accelerated reference frame. We find that, unlike the general studies about the horizon thermodynamics, one cannot obtain the laws of thermodynamics for their horizons in the usual approaches, despite that one can formally define an area entropy (Bekenstein-Hawking entropy). In fact, the common horizon for a set of uniformly accelerated observers is not always exist, even…
We consider spherically-symmetric black holes in semiclassical gravity. For a collapsing radiating thin shell we derive a sufficient condition on the exterior geometry that ensures that a black hole is not formed. This is also a sufficient condition for an infalling test particle to avoid the apparent horizon of an existing black hole and approach it only within a certain minimal distance. Taking the presence of a trapped region and its outer boundary --- the apparent horizon--- as the defining feature of black holes, we explore the consequences of their finite time of formation according to a…
Some dynamical aspects of gravitational collapse are explored in this paper. A time-dependent spherically symmetric metric is proposed and the corresponding Einstein field equations are derived. An ultrarelativistic dust-like stress-momentum tensor is considered to obtain analytical solutions of these equations, with a perfect fluid consisting of two purely radial fluxes -the inwards flux of collapsing matter and the outwards flux of thermally emitted radiation. Thermal emission is calculated by means of a simplistic but illustrative model of unintercting collapsing shells. Our results show an…
We introduce the concept of a geometric horizon, which is a surface distinguished by the vanishing of certain curvature invariants which characterize its special algebraic character. We motivate its use for the detection of the event horizon of a stationary black hole by providing a set of appropriate scalar polynomial curvature invariants that vanish on this surface. We extend this result by proving that a non-expanding horizon, which generalizes a Killing horizon, coincides with the geometric horizon. Finally, we consider the imploding spherically symmetric metrics and show that the geometri…
The Einstein-Yang-Mills equations are the source of many interesting solutions within general relativity, including families of particle-like and black hole solutions, and critical phenomena of more than one type. These solutions, discovered in the last thirty years, all assume a restricted form for the Yang-Mills gauge potential known as the "magnetic" ansatz. In this thesis we relax that assumption and investigate the most general solutions of the Einstein-Yang-Mills system assuming spherically symmetry, a Yang-Mills gauge group of SU(2), and zero cosmological constant. We proceed primarily …
We consider the spherically symmetric, asymptotically flat, non-vacuum Einstein equations, using as matter model a collisionless gas as described by the Vlasov equation. We find explicit conditions on the initial data which guarantee the formation of a trapped surface in the evolution which in particular implies that weak cosmic censorship holds for these data. We also analyze the evolution of solutions after a trapped surface has formed and we show that the event horizon is future complete. Furthermore we find that the apparent horizon and the event horizon do not coincide. This behavior is a…
It is shown analytically that every static, spherically symmetric solution to the Einstein Yang Mills equations with SU(2) gauge group that is defined in the far field has finite ADM mass. Moreover, there can be at most two horizons for such solutions. The three types of solutions possible, Bartnik-McKinnon particle-like solutions, Reissner-Nordstrom-like solutions, and black hole solutions having only one horizon are distinguished by the behavior of the metric coefficients at the origin.
A spherical dust cloud which is initially at rest and which has a monotonously decaying density profile collapses and forms a shell-focussing singularity. Provided the density profile is not too flat, meaning that its second radial derivative is negative at the center, this singularity is visible to local, and sometimes even to global observers. According to the strong cosmic censorship conjecture, such naked singularities should be unstable under generic, nonspherical perturbations of the initial data or when more realistic matter models are considered. In an attempt to gain some understandin…
The choice of the area operator in loop quantum gravity is by no means unique. In addition to the area operator commonly used in loop quantum gravity there is also an area operator introduced by Krasnov in 1998, which gives uniformly spaced area spectra for the horizons of spacetime. Using Krasnov's area operator we consider the thermodynamics of spherically symmetric spacetimes equipped with horizons in loop quantum gravity. Among other things, our approach implies, in a pretty simple manner, that every horizon of spacetime emits thermal radiation and possesses emtropy which, in the natural u…
In this paper we discuss some mathematical aspects of the horizon wave-function formalism, also known in the literature as horizon quantum mechanics. In particular, first we review the structure of both the global and local formalism for static spherically symmetric sources. Then, we present an extension of the global analysis for rotating black holes and we also point out some technical difficulties that arise while attempting the local analysis for non-spherically symmetric sources.
There is an intriguing analogy between the gravitational dynamics of the horizons and thermodynamics. In case of general relativity, as well as for a wider class of Lanczos-Lovelock theories of gravity, it is possible to interpret the field equations near any spherically symmetric horizon as a thermodynamic identity TdS = dE + PdV. We study this approach further and generalize the results to two more generic cases within the context of general relativity: (i) stationary axis-symmetric horizons and (ii) time dependent evolving horizons. In both the cases, the near horizon structure of Einstein …
We study collapse of evaporating spherically-symmetric thin dust shells and dust balls assuming that quantum effects are encapsulated in a spherically-symmetric metric that satisfied mild regularity conditions. The evaporation may accelerate collapse, but for a generic metric the Schwarzschild radius is not crossed. Instead the shell (or the layer in the ball of dust) is always at a certain sub-Planckian distance from it.
We consider here the existence and structure of trapped surfaces, horizons and singularities in spherically symmetric static massless scalar field spacetimes. Earlier studies have shown that there exists no event horizon in such spacetimes if the scalar field is asymptotically flat. We extend this result here to show that this is true in general for spherically symmetric static massless scalar field spacetimes, whether the scalar field is asymptotically flat or not. Other general properties and certain important features of these models are also discussed.
Universal Bounds on Horizons, Photon Spheres, and Shadows: The Role of Energy Conditions in Spherically Symmetric Black Holes
Herzen state Pedagogical University of Russia（俄罗斯赫尔岑国立师范大学）；Center for Theoretical Physics, Khazar University（哈扎尔大学理论物理中心）；SPB branch of SAO RAS（俄罗斯科学院特别天体物理台圣彼得堡分部）
A short review on spherically symmetric static regular black holes and spherically symmetric non singular cosmological space-time is presented. Several models of regular black holes, including new ones, are considered. First, a large class of regular black holes having an inner de Sitter core with the related issue of Cauchy horizon is investigated. Then, black bounce space-times, where the Cauchy horizon and therefore the related instabilities are absent, are discussed as valid alternatives of regular black holes with inner de Sitter core. Friedman-Lemaitre-Robertson-Walker space-times admitt…
We present a family of spherically symmetric spacetimes, specified by the density profile of a vacuum dark energy, which have the same global structure as the de Sitter spacetime but the reduced symmetry which leads to a time-evolving and spatially inhomogeneous cosmological term. It connects smoothly two de Sitter vacua with different values of cosmological constant and corresponds to anisotropic vacuum dark fluid defined by symmetry of its stress-energy tensor which is invariant under the radial boosts. This family contains a special class distinguished by dynamics of evaporation of a cosmol…
Given a static Schwarzschild spacetime of ADM mass M, it is well-known that no ingoing causal geodesic starting in the outer domain r>2M will cross the event horizon r=2M in finite Schwarzschild time. In the present paper we show that in gravitational collapse of Vlasov matter this behaviour can be very different. We construct initial data for which a black hole forms and all matter crosses the event horizon as Schwarzschild time goes to infinity, and we show that this is a necessary condition for geodesic completeness of the event horizon. In addition to a careful analysis of the asymptotic b…
We show that it is possible to embed the 1+1 dimensional reduction of certain spherically symmetric black hole spacetimes into 2+1 Minkowski space. The spacetimes of interest (Schwarzschild de-Sitter, Schwarzschild anti de-Sitter, and Reissner-Nordstrom near the outer horizon) represent a class of metrics whose geometries allow for such embeddings. The embedding diagrams have a dynamic character which allows one to represent the motion of test particles. We also analyze various features of the embedding construction, deriving the general conditions under which our procedure provides a smooth e…
Analytic approximations for the stress-energy of quantized fields in the Hartle-Hawking state in static black hole spacetimes predict divergences on the event horizon of the black hole for a number of important cases. Such divergences, if real, could substantially alter the spacetime geometry near the event horizon, possibly preventing the black hole from existing. The results of three investigations of these types of effects are presented. The first involves a new analytic approximation for conformally invariant fields in Reissner-Nordstrom spacetimes which is finite on the horizon. The secon…
The Horizon Quantum Mechanics is an approach that allows one to analyse the gravitational radius of spherically symmetric systems and compute the probability that a given quantum state is a black hole. We first review the (global) formalism and show how it reproduces a gravitationally inspired GUP relation. This results leads to unacceptably large fluctuations in the horizon size of astrophysical black holes if one insists in describing them as (smeared) central singularities. On the other hand, if they are extended systems, like in the corpuscular models, no such issue arises and one can in f…
In this methodological paper we consider geodesic motion of particles in a spherically symmetric black hole space-times. We develop an approach based on splitting the velocity of a freely falling particle to the flow velocity, which depends only on a metric, and deviation from it (a peculiar velocity). It applies to a wide class of spherically symmetric metrics and is exploited under the horizon of the Schwarzschild black hole. The present work generalizes previous results obtained for pure radial motion. Now, the motion is, in general, nonradial, so that an observer can have a nonzero angular…
We present a simple spherically symmetric and regular solution of Einstein's equations with two parameters $k$ and $M$, which matches to Schwarzschild's solution, satisfies the weak energy conditions in the interior region and for small $r$ behaves like the de Sitter solution. Its energy density $ρ$ and its radial pressure $p_r$ satisfy the relation $ρ+p_r=0$. For some values of $k/M$ the solution does not have an event horizon and the event horizon of Schwarzschild's solution is inside the matching surface. Therefore it describes the formation of a gravitational soliton, which is shown to be …
A static spherically symmetric metric in Einstein-scalar-tensor gravity theory with a scalar field potential $V[ϕ]$ is non-singular for all real values of the coordinates. It does not have a black hole event horizon and there is no essential singularity at the origin of coordinates. The weak energy condition $ρ_ϕ> 0$ fails to be satisfied for $r\lesssim 1.3r_S$ (where $r_S$ is the Schwarzschild radius) but the strong energy condition $ρ_ϕ+3p_ϕ> 0$ is satisfied. The classical Einstein-scalar-tensor solution is regular everywhere in spacetime without a black hole event horizon. However, the viol…
The thermodynamic properties of a shell of bosons with the inner surface locating at Planck length away from the horizon of Schwarzschild black holes by using statistical mechanics are studied. The covariant partition function of bosons is obtained, from which the Bose-Einstein condensation of bosons is found at a non-zero temperature in the curved spacetimes. As a special case of bosons, we analyze the entropy of photon gas near the horizon of the Schwarzschild black hole, which shows an area dependence similar to the Bekenstein-Hawking entropy. The results may offer new perspectives on the s…
We consider a space-time of a spherically symmetric black hole with one simple horizon. As a standard coordinate frame fails in its vicinity, this requires continuation across the horizon and constructing frames which are regular there. Up to now, several standard frames of such a kind are known. It was shown in literature before, how some of them can be united in one picture as different limits of a general scheme. However, some types of frames (the Kruskal-Szekeres and Lema\^ıtre ones) and transformations to them from the original one remained completely disjoint. We show that the Kruskal-Sz…
We consider the problem of determination of a volume of some bounded space-like hypersurfaces in the case of spherically symmetric spacetimes. In the case when the hypersurfaces is cut or bounded by a light-like hypersurface the problem may not be well defined. In order to define properly the volume we introduce the volume forms related to the given foliation (observer) of the considered spacetime. In the case of the constant curvature slice the volume of the hypersurface cut by the horizon (light-like surface) becomes composed of the two parts, outer and inner, treated differently. We compute…
We firstly present a new asymptotical flat and spherically symmetric solution in the generalized Einstein-Cartan-Kibble-Sciama (ECKS) theory of gravity and then investigate the propagation of photon in this background. This solution possesses three independent parameters which affect sharply photon sphere, deflection angle of light ray and gravitational lensing. Since the condition of existence of horizons is not inconsistent with that of photon sphere, there exists a special case where there is horizon but no photon sphere in this spacetime. Especially, we find that in this special case, the …
We consider the interesting "near-horizon test" reported in (PRD84, 084006(2011), arXiv:1107.5727) for any static, spherically symmetric (SSS) black hole solution admitted in f(R) gravity. Before adopting the necessary conditions for the test, however, revisions are needed as we point out in this Comment.
Numerical studies of gravitational collapse to black holes make use of apparent horizons, which are intrinsically foliation-dependent. We expose the problem and discuss possible solutions using the Hawking quasilocal mass. In spherical symmetry, we present a physically sensible approach to the problem by restricting to spherically symmetric spacetime slicings. In spherical symmetry the apparent horizons are gauge-independent in any spherically symmetric foliation but physical quantities associated with them, such as surface gravity and temperature, are not. The widely used comoving and Kodama …
The Jang equation in the spherically symmetric case reduces to a first order equation. This permits an easy analysis of the role apparent horizons play in the (non)existence of solutions. We demonstrate that the proposed derivation of the Penrose inequality based on the Jang equation cannot work in the spherically symmetric case. Thus it is fruitless to apply this method, as it stands, to the general case. We show also that those analytic criteria for the formation of horizons that are based on the use of the Jang equation are of limited validity for the proof of the trapped surface conjecture…
We discuss different kinds of Killing horizons possible in static, spherically symmetric configurations and recently classified as "usual", "naked" and "truly naked" ones depending on the near-horizon behavior of transverse tidal forces acting on an extended body. We obtain necessary conditions for the metric to be extensible beyond a horizon in terms of an arbitrary radial coordinate and show that all truly naked horizons, as well as many of those previously characterized as naked and even usual ones, do not admit an extension and therefore must be considered as singularities. Some examples a…
We investigate the formation of acoustic horizons for an inviscid fluid moving in a pipe in the case of stationary and axi-symmetric flow. We show that, differently from what is generally believed, the acoustic horizon forms in correspondence of either a local minimum or maximum of the flux tube cross-section. Similarly, the external potential is required to have either a maximum or a minimum at the horizon, so that the external force has to vanish there. Choosing a power-law equation of state for the fluid, $P\propto ρ^{n}$, we solve the equations of the fluid dynamics and show that the two p…
In this paper we study properties that the vacuum must possess in the minimal extension to the teleparallel equivalent of general relativity (TEGR) where the action is supplemented with a quadratic torsion term. No assumption is made about the weakness of the quadratic term although in the weak-field regime the validity of our previously derived perturbative solution is confirmed. Regarding the exact nature of the vacuum, it is found that if the center of symmetry is to be regular, the mathematical conditions on the tetrad at the isotropy point mimic those of general relativity. With respect t…
