Vaidya Metric Research Articles

Quest for localized 4D black holes in brane worlds. II. Removing the bulk singularities

We analyze further the possibility of obtaining localized black hole solutions in the framework of Randall-Sundrum-type brane-world models. We consider black hole line elements analytic at the horizon, namely, generalizations of the Painlev\'e and Vaidya metrics, which are taken to have a decaying dependence of the horizon on the extra dimension. These backgrounds have no other singularities apart from the standard black hole singularity which is localized in the direction of the fifth dimension. Both line elements can be sustained by a regular, shell-like distribution of bulk matter of a non-standard form. Of the two, the Vaidya line element is shown to provide the most attractive, natural choice: despite the scaling of the horizon, the five-dimensional spacetime has the same topological structure as the one of a Randall-Sundrum--Schwarzschild spacetime and demands a minimal bulk energy-momentum tensor.

SPHERICALLY SYMMETRIC GRAVITATIONAL FIELDS: BLACK HOLES AND MIDISUPERSPACE QUANTIZATION NEAR THE APPARENT HORIZON

The purpose of this paper is to investigate the quantum vacua directly implied by the wave function of a gravitational configuration characterized by the presence of an apparent horizon, namely the Vaidya space–time solution. Spherical symmetry is a main feature of this configuration, with a scalar field constituting a source [a Klein–Gordon geon or Berger–Chitre–Moncrief–Nutku (BCMN) type model]. The subsequent analysis requires solving a Wheeler–DeWitt equation near the apparent horizon (following the guidelinesintroduced by A. Tomimatsu,18; M. Pollock,19 and developed by A. Hosoya and I. Oda20,21) with the scalar field herein expanded in terms of S2 spherical harmonics: midisuperspace quantization. The main results present in this paper are as follows. It is found that the mass function characteristic of the Vaidya metric is positive definite within this quantum approach. Furthermore, the inhomogeneous matter sector determines a descrip-tion in terms of open quantum (sub)systems, namely in the form of an harmonic oscillator whose frequency depends on the mass function. For this open (sub)system, a twofold approach is employed. On the one hand, an exact invariant observable is obtained from the effective Hamiltonian for the inhomogeneous matter modes. It is shown that this invariant admits a set of discrete eigenvalues which depend on the mass function. The corresponding set of eigenstates is constructed from a particular vacuum state. On the other hand, exact solutions are found for the Schrädinger equation associated with the inhomogeneous matter modes. This paper is concluded with a discussion, where two other issues are raised: (i) the possible application to realistic black hole dynamics of the results obtained for a simplified (BCMN) model and (ii) whether such vacuum states could be related with others defined instead within scalar field theories constructed in classical backgrounds.

Some radiation universes which generalize Vaidya

A generalization of the Vaidya metric is presented, describing a spherically-symmetric radiation spacetime with anisotropic pressure. The constituent radiation moves radially on spherical shells, as in Vaidya, but also propagates tangentially within the shells, which produces an effective tangential pressure. The solution is derived by taking the null limit of an equivalent dust spacetime. The solution contains three arbitrary functions of the shell label, R—the mass M(R), the ‘impact parameter’ D(R) and the initial distribution of radiation r(0, R), but one of these functions is gauge and represents the freedom of choice for the radial coordinate. The types of solution that exist and possible applications are described, singularity formation is discussed and conditions for matching onto exterior solutions are given. Some examples are presented, including separable solutions.

Black holes in nonflat backgrounds: The Schwarzschild black hole in the Einstein universe

As an example of a black hole in a non-flat background a composite static spacetime is constructed. It comprises a vacuum Schwarzschild spacetime for the interior of the black hole across whose horizon it is matched onto the spacetime of Vaidya representing a black hole in the background of the Einstein universe. The scale length of the exterior sets a maximum to the black hole mass. To obtain a non-singular exterior, the Vaidya metric is matched to an Einstein universe. The behavior of scalar waves is studied in this composite model.

DILATION DARK MATTER IN VAIDYA–de SITTER SPACETIME

The exterior of a relativistic star can be modeled with the Vaidya radiating metric. It is started from the generalized Vaidya metric that allows a type II fluid and studied the conditions of generating new analytical solutions of the Einstein's field equations. It is shown that the mass parameter solution gives the classical de Sitter universe in the static case and the extended de Sitter metric coupled with a dilation scalar field in the time-dependent case. It is concluded that in the time-dependent case the atmosphere of a relativistic star consists on an anisotropic string fluid coupled with a dark matter null fluid and interpreted the scalar field as the particle that produces the dark matter.

Collapsing void in a spatially flat Robertson-Walker universe

We consider here a model of the spherical void (or its precursor) containing low density conducting fluid surrounded by a thick spherical shell of radiation embedded in a RobertsonWalker (RW) universe with flat space sections. The underdense region has a metric which is the special case of a solution given by Maiti [1] surrounded by Vaidya metric. We also assume the RW universe to be filled with a perfect fluid with a linear equation of state. The matching conditions indicate that if the time coordinate in each region is future directed then the underdense region appears to go on contracting to a comoving observer in the universe as the latter expands until it disappears. However, if the pressure in the RW universe vanishes, (approximately the present day condition), the underdense region remains static. We have also extended the space-time coordinates of Vaidya metric to the interior of the underdense region as well as the RW universe. It remains to be seen if the region having Vaidya metric disappears earlier than the interior or vice versa.

ON THE QUANTIZATION OF CHARGED BLACK HOLES

The Wheeler–DeWitt equation for the wave function Ψ of the Schwarzschild black hole has been derived by Tomimatsu in the form of a Schrödinger equation, valid on the apparent horizon, using the two-dimensional Hamiltonian formalism of Hajicek and the radiating Vaidya metric. Here, the analysis is generalized to the Reissner–Nordström black hole. At constant charge Q, the evaporation rate is calculated from the solution for Ψ to be [Formula: see text], where k is a constant and [Formula: see text] are the radii of the outer event horizon and inner Cauchy horizon. In the extremal limit M → Q, however, the Hawking temperature [Formula: see text] tends to zero, suggesting, when the back reaction is taken into account, that the evaporation cannot occur this way and in agreement with the known discharging process of the hole via the Schwinger electron–positron pair-production mechanism. The more general charged dilaton black holes obtained from the theory L4 = [R4 - 2 (∇ Φ)2 - e-2aΦF2 ]/16π are also discussed, and it is explained why this quantization procedure cannot be applied when a is non-zero.

Black Hole Radiation inside Apparent Horizon in Quantum Gravity

We study a black hole radiation inside the apparent horizon in quantum gravity. First we perform a canonical quantization for spherically symmetric geometry where one of the spatial coordinates is dealt as the time variable since we would like to consider the interior region of a black hole. Next this rather general formalism is applied for a specific model where the ingoing Vaidya metric is used as a simple model of an evaporating black hole. Following Tomimatsu's idea, we will solve analytically the Wheeler-DeWitt equation in the vicinity of the apparent horizon and see that mass-loss rate of a black hole by thermal radiation is equal to the result obtained by Hawking in his semiclassical treatment. The present formalism may have a wide application in quantum gravity inside the horizon of a black hole such as mass inflation and strong cosmic censorship etc.

ON THE WHEELER-DEWITT EQUATION FOR BLACK HOLES

Integration over the angular coordinates of the evaporating, four-dimensional Schwarzschild black hole leads to a two-dimensional action, for which the Wheeler-DeWitt equation has been found by Tomimatsu, on the apparent horizon, where the Vaidya metric is valid, using the Hamiltonian formalism of Hajicek. For the Einstein theory of gravity coupled to a massless scalar field ζ, the wave function Ψ obeys the Schrödinger equation [Formula: see text], where M is the mass of the hole. The solution is [Formula: see text], where k2 is the separation constant, and for k2>0 the hole evaporates at the rate Ṁ=−k2/4M2, in agreement with the result of Hawking. Here, this analysis is generalized to the two-dimensional theory [Formula: see text], which subsumes the spherical black holes formulated in D≥4 dimensions, when A = ½ (D - 2) (D - 3)ϕ2 (D - 4)/(D - 2), B=2(D−3)/(D−2), C=1, and also the twodimensional black hole identified by Witten and by Gautam et al., when A=4/α′, B=2, C=1/8π, c=+8/α′ being (minus) the central charge. In all cases an analogous Schrödinger equation is obtained. The evaporation rate is [Formula: see text] when D≥4 and [Formula: see text] when D=2. Since Ψ evolves without violation of unitarity, there is no loss of information during the evaporation process, in accord with the principle of black-hole complementarity introduced by Susskind et al. Finally, comparison with the four-dimensional, cosmological Schrödinger equation, obtained by reduction of the ten-dimensional heterotic superstring theory including terms [Formula: see text], shows in both cases that there is a positive semi-definite potential which evolves to zero, this corresponding to the ground state, which is Minkowski space.

On the Vaidya limit of the Tolman model.

We show that the only Tolman models which permit a Vaidya limit are those having a dust distribution that is hollow, such as the self-similar case. Thus the naked shell-focusing singularities found in Tolman models that are dense through the origin have no Vaidya equivalent. This also casts light on the nature of the Vaidya metric. We point out a hidden assumption in Lemos' demonstration that the Vaidya metric is a null limit of the Tolman metric, and in generalizing his result, we find that a different transformation of coordinates is required.

Radiating spherical collapse with shear and heat flow

We consider a spherical body consisting of a fluid with heat flow which radiates in its exterior a null fluid described by the outgoing Vaidya's metric, we prove that this solution matched with the outgoing Vaidya's metric represents a physically reasonable collapsing model. Our model has the remarkable property: it is shear-free and the motion of the fluid is geodesic.

Evaporating black hole in a Vaidya space-time.

The model of an evaporating black hole proposed by Hiscock was based on using a two-dimensional Vaidya metric of mass {ital M}({ital v}) decreasing to zero in some moment {ital v}={ital v}{sub 0}. One of his main results says that the total Hawking energy emitted to {ital J}{sup +} from this system is finite if {ital dM}({ital v}{sub 0})/{ital dv}=0. An error in Hiscock's calculations is pointed out. The corrected result says that the Hawking energy obtained in this Vaidya model is always divergent. A simple, more general, proof of this statement is also given.

Black hole radiation in the Vaidya metric

An evaporating black hole model is considered for the Vaidya metric by using Teukolsky's perturbation method. The asymptotic solution in the far region, the solution at the final time of evaporation, and the solution near the event horizon are examined. It is also shown that the black hole temperature is (1−4M) 8πM up to M from the solution near the horizon without calculating the stress-energy tensor.

Conformally flat solution with heat flux.

It is shown that the spherically symmetric solution previously given by Maiti is not the most general conformally flat solution for a shear-free and rotation-free fluid with heat flux. We have presented a more general solution for such a distribution and have considered the conditions of fit at the boundary of a simple spherically symmetric model with heat flux across the boundary with the exterior Vaidya metric.

Friedmann-like collapsing model of a radiating sphere with heat flow

This paper considers a spherical body consisting of a fluid with heat flow which radiates in its exterior a null fluid described by the outgoing Vaidya's metric. A Friedmann-like exact solution of the interior Einstein field equations is given. It is proved that this solution, matched with the outgoing Vaidya matric, represents a physically reasonble collapsing model which, when the heat flow is switched off, reduces to the well-known collapsing model with dust. The proposed model has the remarkable property that even if the heat flow is small, the horizon will never be formed because, before this happens, the collapsing body will be destroyed by opposite gradients of pressure. 6 references.

Backscattered radiation in the Vaidya metric near zero mass

We examine the importance of backscattered radiation (via a radial null test field) in the Vaidya metric near zero mass and show that the backscattered radiation for an outgoing Vaidya metric becomes blueshifted without bound. The Vaidya metric cannot, therefore, physically model the late stages of radiating gravitational collapse to zero mass.

Evaporating black hole in Vaidya metric

The energy momentum tensor for an evaporating black hole modelled with Vaidya metric is computed. The result indicates that the evaporation does not create a naked singularity.

Models of evaporating black holes. II. Effects of the outgoing created radiation

The two-dimensional stress-energy tensor of a quantized masssless scalar field is studied on fixed background evaporating-black-hole spacetimes (EBH's). The spherically symmetric EBH backgrounds are constructed by patching together ingoing- and outgoing-null-fluid Vaidya metrics along a timelike hypersurface which is near (but outside) the apparent horizon of the EBH. The timelike hypersurface on which the metrics are joined may be viewed as a shell of pair-creation events. It is found that a necessary (but not sufficient) condition for the quantized field's stress-energy to remain finite on the Cauchy horizon of the EBH is that $\mathrm{lim}(\frac{\mathrm{dM}}{\mathrm{du}})=0$ as $M\ensuremath{\rightarrow}0$, where $u$ is the usual outgoing null coordinate defined on ${\mathcal{I}}^{+}$. This is particularly interesting because $\frac{\mathrm{dM}}{\mathrm{du}}$ is in principle measurable by a distant observer. If $\mathrm{lim}(\frac{\mathrm{dM}}{\mathrm{du}})\ensuremath{\ne}0$ as $M\ensuremath{\rightarrow}0$, then the quantized field's outgoing energy flux (${T}_{\mathrm{uu}}$) and its integral (the total energy radiated) both diverge as the Cauchy horizon is approached.

Models of evaporating black holes. I

Classical spacetimes which contain evaporating black holes (EBH's) are constructed using the Vaidya metric. These model EBH spacetimes are spherically symmetric, asymptotically flat, and contain event horizons which terminate after finite duration. These model EBH spacetimes are then used as fixed backgrounds for particle-creation calculations, in an attempt to learn something of the dynamics of real, semiclassical EBH spacetimes while bypassing the difficulties associated with the back-reaction problem. The stress-energy tensor of a quantized massless scalar field is studied on the two-dimensional EBH spacetimes obtained by setting $d\ensuremath{\theta}=d\ensuremath{\varphi}=0$. If $\mathrm{lim}(\frac{\mathrm{dM}}{\mathrm{dv}})\ensuremath{\ne}0$ as $M\ensuremath{\rightarrow}0$ ($v$ the usual ingoing null coordinate), then an infinite flux of outgoing radiation is produced along the Cauchy horizon of the EBH. This behavior suggests that a correct, self-consistent semiclassical EBH spacetime must have $\mathrm{lim}(\frac{\mathrm{dM}}{\mathrm{dv}})=0$ as $M\ensuremath{\rightarrow}0$, contrary to the behavior deduced by naively extrapolating Hawking's result all the way down to $M=0$, which gives $\frac{\mathrm{dM}}{\mathrm{dv}}\ensuremath{\sim}{M}^{\ensuremath{-}2}$. It also shows that not all zero-mass naked singularities left at the end point of an evaporating black hole are benign; they may produce a divergent flux of created particles.

Complexified Vaidya's metric

It is shown that the complexification of Vaidya's metric results in a metric that bears the same relation to Vaidya's metric as does the Kerr metric to the Schwarzschild metric. However, the energy-momentum tensor consists of two parts: (1) radiative and (2) nonradiative. The nonradiative part also corresponds to a trace-free field.