Vaidya Spacetime Research Articles

Generalized Vaidya solutions and Misner-Sharp mass for n -dimensional massive gravity

Dynamical solutions are always of interest to people in gravity theories. We derive a series of generalized Vaidya solutions in the $n$-dimensional de Rham-Gabadadze-Tolley (dRGT) massive gravity with a singular reference metric. Similar to the case of the Einstein gravity, the generalized Vaidya solution can describe shining/absorbing stars. Moreover, we also find a more general Vaidya-like solution by introducing a more generic matter field than the pure radiation in the original Vaidya spacetime. As a result, the above generalized Vaidya solution is naturally included in this Vaidya-like solution as a special case. We investigate the thermodynamics for this Vaidya-like spacetime by using the unified first law, and present the generalized Misner-Sharp mass. Our results show that the generalized Minser-Sharp mass does exist in this spacetime. In addition, the usual Clausius relation $\delta Q= TdS$ holds on the apparent horizon, which implicates that the massive gravity is in a thermodynamic equilibrium state. We find that the work density vanishes for the generalized Vaidya solution, while it appears in the more general Vaidya-like solution. Furthermore, the covariant generalized Minser-Sharp mass in the $n$-dimensional de Rham-Gabadadze-Tolley massive gravity is also derived by taking a general metric ansatz into account.

Vaidya spacetime in the diagonal coordinates

We have analyzed the transformation from initial coordinates $(v,r)$ of the Vaidya metric with light coordinate $v$ to the most physical diagonal coordinates $(t,r)$. An exact solution has been obtained for the corresponding metric tensor in the case of a linear dependence of the mass function of the Vaidya metric on light coordinate $v$. In the diagonal coordinates, a narrow region (with a width proportional to the mass growth rate of a black hole) has been detected near the visibility horizon of the Vaidya accreting black hole, in which the metric differs qualitatively from the Schwarzschild metric and cannot be represented as a small perturbation. It has been shown that, in this case, a single set of diagonal coordinates $(t,r)$ is insufficient to cover the entire range of initial coordinates $(v,r)$ outside the visibility horizon; at least three sets of diagonal coordinates are required, the domains of which are separated by singular surfaces on which the metric components have singularities (either $g_{00}=0$ or $g_{00}=\infty$.). The energy-momentum tensor diverges on these surfaces; however, the tidal forces turn out to be finite, which follows from an analysis of the deviation equations for geodesics. Therefore, these singular surfaces are exclusively coordinate singularities that can be referred to as false firewalls because there are no physical singularities on them. We have also considered the transformation from the initial coordinates to other diagonal coordinates $(\eta,y)$, in which the solution is obtained in explicit form, and there is no energy-momentum tensor divergence.

Conditions for the naked singularity formation in generalized Vaidya spacetime

The gravitational collapse of generalised Vaidya spacetime is considered. It is known that the endstate of gravitational collapse, as to whether a black hole or a naked singularity is formed, depends on the mass function M(v,r). Here we give conditions for the mass function which corresponds to the equation of the state P = αρ where α ∈ (0, ⅓] and according to these conditions we obtain either a black hole or a naked singularity at the endstate of gravitational collapse. Also we give conditions for the mass function when the singularity is gravitationally strong. Also we provide the metric which is the analogue of Vaidya metric in case of rotation.

Quasilocal energy exchange and the null cone

Energy is at best defined quasilocally in general relativity. Quasilocal energy definitions depend on the conditions one imposes on the boundary Hamiltonian, i.e., how a finite region of spacetime is "isolated". Here, we propose a method to define and investigate systems in terms of their matter plus gravitational energy content. We adopt a generic construction, that involves embedding of an arbitrary dimensional world sheet into an arbitrary dimensional spacetime, to a 2 + 2 picture. In our case, the closed 2-dimensional spacelike surface $\mathbb{S}$, that is orthogonal to the 2-dimensional timelike world sheet $\mathbb{T}$ at every point, encloses the system in question. The integrability conditions of $\mathbb{T}$ and $\mathbb{S}$ correspond to three null tetrad gauge conditions once we transform our notation to the one of the null cone observables. We interpret the Raychaudhuri equation of $\mathbb{T}$ as a work-energy relation for systems that are not in equilibrium with their surroundings. We achieve this by identifying the quasilocal charge densities corresponding to rotational and nonrotational degrees of freedom, in addition to a relative work density associated with tidal fields. We define the corresponding quasilocal charges that appear in our work-energy relation and which can potentially be exchanged with the surroundings. These charges and our tetrad conditions are invariant under type-III Lorentz transformations, i.e., the boosting of the observers in the directions orthogonal to $\mathbb{S}$. We apply our construction to a radiating Vaidya spacetime, a $C$-metric and the interior of a Lanczos-van Stockum dust metric. The delicate issues related to the axially symmetric stationary spacetimes and possible extensions to the Kerr geometry are also discussed.

Lanczos potential for some non-vacuum spacetimes

The Lanczos potential is a general relativity analogue of potential in electrodynamics. This potential can be obtained by solving Weyl-Lanczos relations. This computation is easy in the case of vacuum spacetimes and/or spacetimes of particular Petrov type, while obtaining the Lanczos potential for arbitrary spacetime is very difficult. In this paper, we have obtained the Lanczos potential for non-vacuum spacetimes: Vaidya spacetime and Van Stockum spacetime (which is not of any Petrov type).

Thermalization of Wightman functions in AdS/CFT and quasinormal modes

We study the time evolution of Wightman two-point functions of scalar fields in AdS$_3$-Vaidya, a spacetime undergoing gravitational collapse. In the boundary field theory, the collapse corresponds to a quench process where the dual 1+1 dimensional CFT is taken out of equilibrium and subsequently thermalizes. From the two-point function, we extract an effective occupation number in the boundary theory and study how it approaches the thermal Bose-Einstein distribution. We find that the Wightman functions, as well as the effective occupation numbers, thermalize with a rate set by the lowest quasinormal mode of the scalar field in the BTZ black hole background. We give a heuristic argument for the quasinormal decay, which is expected to apply to more general Vaidya spacetimes also in higher dimensions. This suggests a unified picture in which thermalization times of one- and two-point functions are determined by the lowest quasinormal mode. Finally, we study how these results compare to previous calculations of two-point functions based on the geodesic approximation.

Vaidya spacetime for Galileon gravity's rainbow

In this paper, we analyze Vaidya spacetime with an energy dependent metric in Galileon gravity's rainbow. This will be done using the rainbow functions which are motivated from the results obtained in loop quantum gravity approach and noncommutative geometry. We will investigate the Gravitational collapse in this Galileon gravity's rainbow. We will discuss the behavior of singularities formed from the gravitational collapse in this rainbow deformed Galileon gravity.

Naked singularity formation in generalized Vaidya space-time

The gravitational collapse in generalized Vaidya space-time is considered. It is known that the end state of gravitational collapse, as to whether a black hole or a naked singularity is formed, depends on the mass function M(v, r). Here we give conditions for the mass function which correspond to the equation of the state P = αρ. where α ∈ (0, 1/3], and according to these conditions we obtain either a black hole or a naked singularity as the end state of gravitational collapse. We also give the conditions for the mass function under which the singularity is gravitationally strong. We present simple examples showing when the result of gravitational collapse is a naked singularity and when this singularity is strong.

Generalized Vaidya spacetime for cubic gravity

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.

Gravitational collapse of Vaidya spacetime

The gravitational collapse of generalized Vaidya spacetime is considered. It is known that the endstate of gravitational collapse, as to whether a black hole or a naked singularity is formed, depends on the mass function [Formula: see text]. Here we give conditions for the mass function which corresponds to the equation of the state [Formula: see text] where [Formula: see text] and according to these conditions we obtain either a black hole or a naked singularity at the endstate of gravitational collapse. Also we give conditions for the mass function when the singularity is gravitationally strong.

Is cosmic censorship restored in higher dimensions?

In this paper we extend the analysis of gravitational collapse of spherically symmetric generalized Vaidya spacetimes to higher dimensions, in the context of the cosmic censorship conjecture. We present the sufficient conditions on the generalized Vaidya mass function that will generate a locally naked singular end state. Our analysis here generalizes all the earlier works on collapsing higher dimensional generalized Vaidya spacetimes. With specific examples, we show the existence of classes of mass functions that lead to a naked singularity in four dimensions, which gets covered on transition to higher dimensions. Hence for these classes of mass function cosmic censorship gets restored in higher dimensions and the transition to higher dimensions restricts the set of initial data that results in a naked singularity.

Thermalization of Green functions and quasinormal modes

We develop a new method to study the thermalization of time dependent retarded Green function in conformal field theories holographically dual to thin shell AdS Vaidya space times. The method relies on using the information of all time derivatives of the Green function at the shell and then evolving it for later times. The time derivatives of the Green function at the shell is given in terms of a recursion formula. Using this method we obtain analytic results for short time thermalization of the Green function. We show that the late time behaviour of the Green function is determined by the first quasinormal mode. We then implement the method numerically. As applications of this method we study the thermalization of the retarded time dependent Green function corresponding to a minimally coupled scalar in the AdS3 and AdS5 thin Vaidya shells. We see that as expected the late time behaviour is determined by the first quasinormal mode. We apply the method to study the late time behaviour of the shear vector mode in AdS5 Vaidya shell. At small momentum the corresponding time dependent Green function is expected to relax to equilibrium by the shear hydrodynamic mode. Using this we obtain the universal ratio of the shear viscosity to entropy density from a time dependent process.

Non-equilibrium scalar two point functions in AdS/CFT

In the first part of the paper, we discuss different versions of the AdS/CFT dictionary out of equilibrium. We show that the Skenderis - van Rees prescription and the "extrapolate" dictionary are equivalent at the level of "in-in" two point functions of free scalar fields in arbitrary asymptotically AdS spacetimes. In the second part of the paper, we calculate two point correlation functions in dynamical spacetimes using the "extrapolate" dictionary. These calculations are performed for conformally coupled scalar fields in examples of spacetimes undergoing gravitational collapse, the AdS$_2$-Vaidya spacetime and the AdS$_3$-Vaidya spacetime, which allow us to address the problem of thermalization following a quench in the boundary field theory. The computation of the correlators is formulated as an initial value problem in the bulk spacetime. Finally, we compare our results for AdS$_3$-Vaidya to results in the previous literature obtained using the geodesic approximation and we find qualitative agreement.

Gravitational collapse in Vaidya space–time for Galileon gravity theory

The motive of this work is to study gravitational collapse in Vaidya space–time embedded in Galileon gravity theory. Galileon gravity is in fact an infrared modification of Einstein gravity, which was proposed as a generalization of the four-dimensional effective theory in the DGP brane model. Vaidya’s metric is used all over to follow the nature of future outgoing radial null geodesics. Detecting whether the central singularity is naked or wrapped by an event horizon, by the existence of future directed radial null geodesic emitted in the past from the singularity is the basic objective. To point out the existence of a positive trajectory tangent solution, both particular parametric cases (through tabular forms) and wide range contouring process have been applied. Precisely, the equation of state (EoS) in a perfect fluid satisfies a wide range of phenomena: from dust to exotic fluids like dark energy. We have used the EoS parameter, k, to determine the end collapse state in different cosmological eras.

Gravitational collapse of generalized Vaidya spacetime

We study the gravitational collapse of a generalized Vaidya spacetime in the context of the cosmic censorship hypothesis. We develop a general mathematical framework to study the conditions on the mass function so that future directed nonspacelike geodesics can terminate at the singularity in the past. Thus our result generalizes earlier works on gravitational collapse of the combinations of Type-I and Type-II matter fields. Our analysis shows transparently that there exist classes of generalized Vaidya mass functions for which the collapse terminates with a locally naked central singularity. We calculate the strength of these singularities to show that they are strong curvature singularities and there can be no extension of spacetime through them.

Effects of some physical factors on the stability of collapsing stars

This paper formulates the instability regions of a general charged spherically symmetric matter distribution with anisotropic pressure, shear viscosity, radiation density and radial heat flux. Matching conditions are found by taking Reissner–Nordström–Vaidya spacetime as an exterior geometry. The perturbed forms of Einstein–Maxwell field equations and conservation laws are constructed to formulate the collapse equation. The instability regions are identified under N and pN limits for the stability analysis. We conclude that the adiabatic index identifies the instability range which depends upon physical parameters like anisotropic pressure, energy density, shear viscosity and electromagnetic field and heat flux but radiation density has no role.

Vaidya solutions in general covariant Hořava–Lifshitz gravity without projectability: Infrared limit

In this paper, we have studied nonstationary radiative spherically symmetric spacetime, in general covariant theory (U(1) extension) of Hořava–Lifshitz (HL) gravity without the projectability condition and in the infrared (IR) 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.

Gravitating dyons in Vaidya geometry

Gravitating monopoles and dyons in Einstein–Yang–Mills (EYM) or Einstein–Yang–Mills–Higgs (EYMH) systems have been extensively studied for various curved space–times, including those of black holes. We construct dyonic solutions of the EYMH theory in Vaidya space–time using a set of generalized Julia–Zee ansatz for the fields. It is found that the dyonic charge is static in nature and it does not contribute to the energy of the null dust.

Revisiting Vaidya Horizons

In this study, we located and compared different types of horizons in the spherically symmetric Vaidya solution. The horizons we found were trapping horizons, which can be null, timelike, or spacelike, null surfaces with constant area change and also conformal Killing horizons. The conformal Killing horizons only exist for certain choices of the mass function. Under a conformal transformation, the conformal Killing horizons can be mapped into true Killing horizons. This allows conclusions drawn in the dynamical Vaidya spacetime to be related to known properties of static spacetimes. We found the conformal factor that performs this transformation and wrote the new metric in explicitly static coordinates. Using this construction we found that the tunneling argument for Hawking radiation does not umabiguously support Hawking radiation being associated with the trapping horizon. We also used this transformation to derive the form of the surface gravity for a class of physical observers in Vaidya spacetimes.

Thermodynamics in Higher Dimensional Vaidya Space-Time

In this work, we have considered the Vaidya spacetime in null radiating fluid with perfect fluid in higher dimension and have found the solution for barotropic fluid. We have shown that the Einstein’s field equations can be obtained from Unified first law i.e., field equations and unified first law are equivalent. The first law of thermodynamics has also been constructed by Unified first law. From this, the variation of entropy function has been derived on the horizon. The variation of entropy function inside the horizon has been derived using Gibb’s law of thermodynamics. So the total variation of entropy function has been constructed at apparent and event horizons both. If we do not assume the first law, then the entropy on the both horizons can be considered by area law and the variation of total entropy has been found at both the horizons. Also the validity of generalized second law (GSL) of thermodynamics has been examined at both apparent and event horizons by using the first law and the area law separately. When we use first law of thermodynamics and Bekenstein-Hawking area law of thermodynamics, the GSL for apparent horizon in any dimensions are satisfied, but the GSL for event horizon can not be satisfied in any dimensions.