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 the numerically found axisymmetric trapping horizons with the area of the spherically symmetric trapping horizon and event horizon. We find that as expected, both the location and area are dependent on the choice of foliation. However, the area variation is not large, of order $0.035\%$ for a slowly evolving horizon with $\dot{m}=0.02$. We also calculate analytically the difference in area between the spherically symmetric quasi-local horizon and event horizon for a slowly accreting black hole. We find that the difference can be many orders of magnitude larger than the Planck area for sufficiently large black holes.
Marginally trapped surfaces (MTSs) are commonly used in numerical relativity to locate black holes. For dynamical black holes, it is not known generally if this procedure is sufficiently reliable. Even for Schwarzschild black holes, Wald and Iyer constructed foliations which come arbitrarily close to the singularity but do not contain any MTSs. In this paper, we review the Wald-Iyer construction, discuss some implications for numerical relativity, and generalize to the well known Vaidya spacetime describing spherically symmetric collapse of null dust. In the Vaidya spacetime, we numerically lo…
A large family of solutions, representing, in general, spherically symmetric Type II fluid, is presented, which includes most of the known solutions to the Einstein field equations, such as, the monopole-de Sitter-charged Vaidya ones.
In this talk we show that any spherically symmetric spacetime admits locally a maximal spacelike slicing. The above condition is reduced to solve a decoupled system of first order quasi-linear partial differential equations. The solution may be accomplished analytical or numerically. We provide a general procedure to construct such maximal slicings.
Kerr-Vaidya metrics are the simplest nonstationary extensions of the Kerr metric. We explore their properties and compare them with the near-horizon limits of the spherically symmetric self-consistent solutions (the ingoing Vaidya metric with decreasing mass and the outgoing Vaidya metric with increasing mass) for the evaporating and accreting physical black holes. The Newman-Janis transformation relates the corresponding Vaidya and Kerr-Vaidya metrics. For nonzero angular momentum, the energy-momentum tensor violates the null energy condition (NEC). However, we show that its structure differs…
We study the occurrence of naked singularities in the spherically symmetric collapse of a charged null fluid in an expanding deSitter background - a piece of charged Vaidya-deSitter spacetime. The necessary conditions for the formation of a naked singularity are found. The results for the uncharged solutions can be recovered from our analysis.
We study the matching of a general spherically symmetric spacetime with a Vaidya-Reissner-Nordstrom solution. To that end, we study the properties of spherically symmetric electromagnetic fields and develop the proper gravitational and electromagnetic junction conditions. We prove that generic spacetimes can be matched to a Vaidya-Reissner-Nordstrom solution or one of its specializations, and that these matchings have clear physical interpretations. Furthermore, the non-spacelike nature of the matching hypersurface is proved under very general hypotheses. We obtain the fundamental result that …
Motion of massive test particles in the nonvacuum spherically symmetric radiating Vaidya spacetime is investigated, allowing for physical interaction of the particles with the radiation field in terms of which the source energy-momentum tensor is interpreted. This "Poynting-Robertson-like effect" is modeled by the usual effective term describing a Thomson-type radiation drag force. The equations of motion are studied for simple types of motion including free motion (without interaction), purely radial and purely azimuthal (circular) motion, and for the particular case of "static" equilibrium; …
We analyze a class of exact type II solutions of the Robinson-Trautman family which contain pure radiation and (possibly) a cosmological constant. It is shown that these spacetimes exist for any sufficiently smooth initial data, and that they approach the spherically symmetric Vaidya-(anti-)de Sitter metric. We also investigate extensions of the metric, and we demonstrate that their order of smoothness is in general only finite. Some applications of the results are outlined.
We study the foliation of a $D$-dimensional spherically symmetric black-hole spacetime with $D\ge 5$ by two kinds of one-parameter family of maximal hypersurfaces: a reflection-symmetric foliation with respect to the wormhole slot and a stationary foliation that has an infinitely long trumpet-like shape. As in the four-dimensional case, the foliations by the maximal hypersurfaces have the singularity avoidance nature irrespective of dimensionality. This indicates that the maximal slicing condition will be useful for simulating higher-dimensional black-hole spacetimes in numerical relativity. F…
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 present here the field equations describing a non-stationary spherically symmetric n-dimensional charged black hole with varying mass m(v) and/or electric charge q(v), described by a generic charged Vaidya metric with cosmological constant Lambda in double null coordinates. This formulation of the metric has been shown to be particularly useful for perturbative studies and it was used in some recent works. Here we also discuss some issues related to the apparent and event horizons of the black hole.
We present a kind of generalized Vaidya solutions of a new cubic gravity in five dimensions whose field equations in spherically spacetime are always second order like the Lovelock gravity. We also study the thermodynamics of its apparent horizon and get its entropy expression and generalized Misner-Sharp energy. Finally we present the first law and second law hold in this gravity. Although all the results are analogue to those in Lovelock gravity, we in fact introduce the contribution of new cubic term in five dimensions where cubic Lovelock term is just zero.
In this paper, we have studied nonstationary radiative spherically symmetric spacetime, in general covariant theory ($U(1)$ extension) of Hořava-Lifshitz gravity without the projectability condition and in the infrared limit. The Newtonian prepotential $φ$ was assumed null. We have shown that there is not the analogue of the Vaidya's solution in the Hořava-Lifshitz Theory (HLT), as we know in the General Relativity Theory (GRT). Therefore, we conclude that the gauge field $A$ should interact with the null radiation field of the Vaidya's spacetime in the HLT.
We express Einstein's field equations for a spherically symmetric ball of general fluid such that they are conducive to an initial value problem. We show how the equations reduce to the Vaidya spacetime in a non-null coordinate frame, simply by designating specific equations of state. Furthermore, this reduces to the Schwarzschild spacetime when all matter variables vanish. We then describe the formulation of an initial value problem, whereby a general fluid ball with vacuum exterior is established on an initial spacelike slice. As the system evolves, the fluid ball collapses and emanates null…
Hawking evaporation of Dirac particles and scalar fields in a Vaidya-type black hole is investigated by the method of generalized tortoise coordinate transformation. It is shown that Hawking radiation of Dirac particles does not exist for $P_1, Q_2$ components but for $P_2, Q_1$ components in any Vaidya-type black holes. Both the location and the temperature of the event horizon change with time. The thermal radiation spectrum of Dirac particles is the same as that of Klein-Gordon particles. We demonstrates that there is no new quantum ergosphere effect in the thermal radiation of Dirac partic…
We establish a formal connection between the {\bf K}-essence emergent gravity scenario and generalizations of Vaidya spacetime. Choosing the {\bf K}-essence action to be of the Dirac-Born-Infeld variety, the physical spacetime to be a general static spherically symmetric black hole and restricting the {\bf K}-essence scalar field to be a function solely of the advanced or the retarded time, we show that the emergent gravity metric resembles closely the generalized Vaidya metrics for null fluid collapse proposed by Husain. Imposing null energy conditions on the emergent energy-momentum tensor d…
A simple observation about the action for geodesics in a stationary spacetime with separable geodesic equations leads to a natural class of slicings of that spacetime whose orthogonal geodesic trajectories represent freely falling observers. The time coordinate function can then be taken to be the observer proper time, leading to a unit lapse function. This explains some of the properties of the original Painlevé-Gullstrand coordinates on the Schwarzschild spacetime and their generalization to the Kerr-Newman family of spacetimes, reproducible also locally for the Gödel spacetime. For the stat…
Using the formalism of the interaction picture we calculate an expression for the Wightman function for only the spherically symmetric modes of a quantum Klein-Gordon scalar field in a general Vaidya spacetime with ingoing null dust. It is demonstrated that particle detectors following time-like trajectories that are in the ground state at some time outside of the collapsing shell will respond independently from the configuration of the ingoing null dust if the response is taken at any time outside of the collapsing shell. For detectors that are taken to be in the ground state at a time interi…
In this paper we derive exact analytical formulas for the evolution of the photon sphere and for the angular radius of the shadow in a special Vaidya spacetime. The Vaidya metric describes a spherically symmetric object that gains or loses mass, depending on a mass function $m(v)$ that can be freely chosen. Here we consider the case that $m(v)$ is a linearly increasing or decreasing function. The first case can serve as a simple model for an accreting black hole, the second case for a (Hawking) radiating black hole. With a linear mass function the Vaidya metric admits a conformal Killing vecto…
Using the 3+1 formalism of general relativity we obtain the equations governing the dynamics of spherically symmetric spacetimes with arbitrary sources. We then specialize for the case of perfect fluids accompanied by a flow of interacting massless or massive particles (e.g. neutrinos) which are described in terms of relativistic transport theory. We focus in three types of coordinates: 1) isotropic gauge and maximal slicing, 2) radial gauge and polar slicing, and 3) isotropic gauge and polar slicing.
The aim of this paper is to study the black hole evaporation and Hawking radiation for a noncommutative charged Vaidya black hole. For this purpose, we determine spherically symmetric charged Vaidya model and then formulate a noncommutative Reissner-Nordstr$\ddot{o}$m-like solution of this model which leads to an exact $(t-r)$ dependent metric. The behavior of temporal component of this metric and the corresponding Hawking temperature is investigated. The results are shown in the form of graphs. Further, we examine the tunneling process of the charged massive particles through the quantum hori…
A large family of inhomogeneous non-static spherically symmetric solutions of the Einstein equation for null fluid in higher dimensions has been obtained. It encompasses higher dimensional versions of many previously known solutions such as Vaidya, charged Vaidya and Husain solutions and also some new solutions representing global monopole or string dust. It turns out that physical properties of the solutions carry over to higher dimensions.
The Einstein equations are integrated in the presence of two (incoming and outgoing) streams of null dust, under the assumptions of spherical symmetry and staticity. The solution is also written in double null and radiation coordinates and it is reinterpreted as an anisotropic fluid. Interior matching with a static fluid and exterior matching with the Vaidya solution along null hypersurfaces is discussed. The connection with two-dimensional dilaton gravity is established.
We complete the intrinsic characterization of spherically symmetric solutions partially accomplished in a previous paper [Class.Quant.Grav. (2010) 27 205024]. In this approach we consider every compatible algebraic type of the Ricci tensor, and we analyze specifically the conformally flat case for perfect fluid and Einstein-Maxwell solutions. As a direct application we obtain the {\em ideal} labeling (exclusively involving explicit concomitants of the metric tensor) of the Schwarzschild interior metric and the Vaidya solution. The Stephani universes and some significative subfamilies are also …
It is known that spherically symmetric spacetimes admit flat spacelike foliations. We point out a simple method of seeing this result via the Hamiltonian constraints of general relativity. The method yields explicit formulas for the extrinsic curvatures of the slicings.
The spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates (i.e. polar slicing and areal radial coordinate) is considered. An improved continuation criterion for global existence of classical solutions is given. Two other types of criteria which prevent finite time blow-up are also given.
In general relativity, the endpoint of spherically symmetric gravitational collapse is a Schwarzschild--[(A)dS] black hole. In bimetric gravity, it has been speculated that a static end state must also be Schwarzschild--[(A)dS]. To this end, we present a set of exact solutions, including collapsing massless dust particles. For these, the speculation is confirmed.
It is shown that a scalar field, minimally coupled to gravity may have collapsing modes even when the energy condition is violated, that is, for $(ρ+3p)<0$. This result may be useful in the investigation of the possible clustering of dark energy. All the examples dealt with have apparent horizons which form before the formation of the singularity. The singularities formed are shell focusing in nature. The density of the scalar field distribution is seen to diverge at singularity. The Ricci scalar also diverges at the singularity. The interior spherically symmetric metric is matched with exteri…
Ringdown in Vaidya spacetimes: time-dependent frequencies, Penrose limit and time-domain analyses
Graduate School of Science, Nagoya University（名古屋大学理学研究科）；Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI)（小林-益川粒子宇宙起源研究所）；Department of Information, Artificial Intelligence and Data Science, Daiichi Institute of Technology（第一工业大学信息、人工智能与数据科学系）；Department of Physics, Rikkyo University（立教大学物理学系）
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…
In this article, we shed some light on the field theoretic aspect of the spherically symmetric trapping horizon in Vaidya spacetime. The effective field equations are that of a Chern-Simons theory coupled to bulk sources through two different couplings, one is purely geometric and the other is matter dependent. This is an effective generalization of the equilibrium horizon scenario where the Chern-Simons theory is coupled to the bulk geometry through a constant matter-independent coupling. Further, we note that the field equations pulled-back to a cross-section of the horizon is inadequate to …
We show that any spherically symmetric spacetime locally admits a maximal spacelike slicing and we give a procedure allowing its construction. The construction procedure that we have designed is based on purely geometrical arguments and, in practice, leads to solve a decoupled system of first order quasi-linear partial differential equations. We have explicitly built up maximal foliations in Minkowski and Friedmann spacetimes. Our approach admits further generalizations and efficient computational implementation. As by product, we suggest some applications of our work in the task of calibratin…
The covariance of loop quantum gravity studies of spherically symmetric space-times has recently been questioned. This is a reasonable worry, given that they are formulated in terms of slicing-dependent variables. We show explicitly that the resulting space-times, obtained from Dirac observables of the quantum theory, are covariant in the usual sense of the way -- they preserve the quantum line element -- for any gauge that is stationary (in the exterior, if there is a horizon). The construction depends crucially on the details of the Abelianized quantization considered, the satisfaction of th…
The Riemann, Ricci and Einstein tensors for N-dimensional spherically symmetric spacetimes in various systems of coordinates are studied, and the general metric for conformally flat spacetimes is given. As an application, all the Friedmann-Robertson-Walker-like solutions for a perfect fluid with an equation of state $p = k ρ$ are found. Then, these solutions are used to model the gravitational collapse of a compact ball, by first cutting them along a timelike hypersurface and then joining them with asymptotically flat Vaidya solutions. It is found that when the collapse has continuous self-sim…
We consider non-stationary spherically symmetric $n$-dimensional charged black holes with varying mass $m(v)$ and/or electric charge $q(v)$, described by generic charged Vaidya metrics with cosmological constant $Λ$ in double null coordinates, and perform a comprehensive numerical analysis of the fundamental quasinormal modes (QNM) for minimally coupled scalar fields. We show that the "instantaneous" quasinormal frequencies exhibit the same sort of non-stationary behavior reported previously for the four-dimensional uncharged case with $Λ=0$. Such property seems to be very robust, independent …
Spherically symmetric Black Holes of the Vaidya type are examined in an asymptotically de Sitter, higher dimensional spacetime. The various horizons are located. The structure and dynamics of such horizons are studied.
We consider several families of functions $f(α)$ that appear in the Bona-Masso slicing condition for the lapse function $α$. Focusing on spherically symmetric and time-independent slices we apply these conditions to the Schwarzschild spacetime in order to construct analytical expressions for the lapse $α$ in terms of the areal radius $R$. We then transform to isotropic coordinates and determine the dependence of $α$ on the isotropic radius $r$ in the vicinity of the black-hole puncture. We propose generalizations of previously considered functions $f(α)$ for which, to leading order, the lapse …
Based on the observation that the exterior space-times of Schwarzschild-type solutions allow two symmetric slicings, a static spherically symmetric one and a timelike homogeneous one, modifications of gravitational dynamics suggested by symmetry-reduced models of quantum cosmology can be used to derive corresponding modified spherically symmetric equations. Generally covariant theories are much more restricted in spherical symmetry compared with homogeneous slicings, given by $1+1$-dimensional dilaton models if they are local. As shown here, modifications used in loop quantum cosmology do not …
Recently we have studied, using a boundary-value approach, quantum amplitudes resulting from gravitational collapse to a black hole. Suitable boundary data for all fields present are posed on initial and final space-like asymptotically flat hypersurfaces $Σ_{I,F}$. The Lorentzian proper-time separation between the surfaces, as measured at spatial infinity, is denoted by $T$. Following Feynman's $+iε$ approach, we rotate $T$ into the complex: $T\to {\mid}T{\mid} \exp(-iθ)$, where $0<θ\leqπ/2$. The corresponding {\it classical} complex boundary-value problem is expected to be well-posed for $θ> …
We prove a theorem that characterizes a large family of non-static solutions to Einstein equations in $N$-dimensional space-time, representing, in general, spherically symmetric Type II fluid. It is shown that the best known Vaidya-based (radiating) black hole solutions to Einstein equations, in both four dimensions (4D) and higher dimensions (HD), are particular cases from this family. The spherically symmetric static black hole solutions for Type I fluid can also be retrieved. A brief discussion on the energy conditions, singularities and horizons is provided.
We utilize a recent formulation of a spherically symmetric spacetime endowed with a general decomposition of the energy momentum tensor [Phys. Rev. D, 75, 024031 (2007)] to derive equations governing spherically symmetric distributions of electromagnetic matter. We show the system reduces to the Reissner-Nordstrom spacetime in general, spherically symmetric coordinates in the vacuum limit. Furthermore, we show reduction to the charged Vaidya spacetime in non-null coordinates when certain equations of states are chosen. A model of gravitational collapse is discussed whereby a charged fluid resi…
I consider the appearance of shocks in hyperbolic formalisms of General Relativity. I study the particular case of the Bona-Masso formalism with zero shift vector and show how shocks associated with two families of characteristic fields can develop. These shocks do not represent discontinuities in the geometry of spacetime, but rather regions where the coordinate system becomes pathological. For this reason I call them coordinate shocks. I show how one family of shocks can be eliminated by restricting the Bona-Masso slicing condition to a special case. The other family of shocks, however, can …
We present two classes of inhomogeneous, spherically symmetric solutions of the Einstein-Maxwell-Perfect Fluid field equations with cosmological constant generalizing the Vaidya-Shah solution. Some special limits of our solution reduce to the known inhomogeneous charged perfect fluid solutions of the Einstein field equations and under some other limits we obtain new charged and uncharged solutions with cosmological constant. Uncharged solutions in particular represent cosmological models where the universe may undergo a topology change and in between is a mixture of two different Friedmann-Rob…
We study some quantum mechanical aspects of dynamical black holes where the Vaidya metric is used as a model representing evaporating black holes. It is shown that in this model the Wheeler-DeWitt equation is solvable in whole region of spacetime, provided that one considers the ingoing (or outgoing) Vaidya metric and selects a suitable coordinate frame. This wave function has curious features in that near the curvature singularity it oscillates violently owing to large quantum effects while in the other regions the wave function exhibits a rather benign and completely regular behavior. The ge…
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 study quasi-normal modes of black holes, with a focus on resonant (or quasi-normal mode) expansions, in a geometric frame based on the use of conformal compactifications together with hyperboloidal foliations of spacetime. Specifically, this work extends the previous study of Schwarzschild in this geometric approach to spherically symmetric asymptotically flat black hole spacetimes, in particular Reissner-Nordström. The discussion involves, first, the non-trivial technical developments needed to address the choice of appropriate hyperboloidal slices in the extended setting as well as the ge…
We derive spherically symmetric solutions of the classical λ-R model, a minimal, anisotropic modification of general relativity with a preferred foliation and two local degrees of freedom. Starting from a 3 + 1 decomposition of the four-metric in a general spherically symmetric ansatz, we perform a phase space analysis of the reduced model. We show that its constraint algebra is consistent with that of the full λ-R model, and also yields a constant mean curvature or maximal slicing condition as a tertiary constraint. Although the solutions contain the standard Schwarzschild geometry for the ge…
Motivated by the recent work of Robinson and Wilczek, we evaluate the gravitational anomaly of a chiral scalar field in a Vaidya spacetime of arbitrary mass function, and thus the outgoing flux from the time-dependent horizon in that spacetime. We show that this flux differs from that of a perfect blackbody at a fixed temperature. When this flux is taken into account, general covariance in that spacetime is restored. We also generalize their results to the most general static, and spherically symmetric spacetime.
We study gravitational collapse of the general spherically symmetric null strange quark fluid having the equation of state, $p = (ρ- 4B)/n$, where $B$ is the bag constant. An interesting feature that emerges is that the initial data set giving rise to naked singularity in the Vaidya collapse of null fluid gets covered due to the presence of strange quark matter component. Its implication to the Cosmic Censorship Conjecture is discussed.
The general structure of the spherically symmetric solutions in the Weyl conformal gravity is described. The corresponding Bach equations are derived for the special type of metrics, which can be considered as the representative of the general class. The complete set of the pure vacuum solutions is found. It consists of two classes. The first one contains the solutions with constant two-dimensional curvature scalar of our specific metrics, and the representatives are the famous Robertson--Walker metrics. One of them we called the "gravitational bubbles", which is compact and with zero Weyl ten…
