Vaidya Solution in Non-Stationary de Sitter Background: Hawking’s Temperature

In this paper we propose a class of non-stationary solutions of Einstein’s field equations describing an embedded Vaidya-de Sitter solution with a cosmological variable function Λ(u). Vaidya-de Sitter solution is interpreted as the radiating Vaidya black hole which is embedded into the non-stationary de Sitter space with variable Λ(u). The energymomentum tensor of the Vaidya-de Sitter black hole may be expressed as the sum of the energy-momentum tensor of the Vaidya null fluid and that of the non-stationary de Sitter field, and satisfies the energy conservation law. We also find that the equation of state parameter w= p/ρ = −1 of the non-stationary de Sitter solution holds true in the embedded Vaidya-de Sitter solution. It is also found that the space-time geometry of non-stationary Vaidya-de Sitter solution with variable Λ(u) is type D in the Petrov classification of space-times. The surface gravity, temperature and entropy of the space-time on the cosmological black hole horizon are discussed.


Introduction
The Vaidya solution having a variable mass m(u) with retarded time u is a non-stationary generalization of Schwarzschild black hole of constant mass m [1]. The Schwarzschild solution is regarded as a black hole in an asymptotically flat space. The Schwarzschild-de Sitter solution is interpreted as a black hole in an asymptotically de Sitter space with non-zero cosmological constant Λ [2]. The Schwarzschild-de Sitter solution is also considered as an embedded black hole that the Schwarzschild solution is embedded into the de Sitter space with cosmological constant Λ to produce the Schwarzschildde Sitter black hole [3]. Mallett [4] has introduced Vaidya-de Sitter solution with constant Λ by making the Schwarzschild mass m variable with respect to the retarded time u as m(u) and studied the nature of the spacetime [5].
Here the aim of this paper is to propose an exact solution of Einstein's field equations describing Vaidya black hole embedded into the non-stationary de Sitter space to obtain Vaidya-de Sitter black hole with variable Λ(u). This Vaidya-de Sitter solution with variable Λ(u) will have the limit m(u) = ± (1/3) Λ(u) (−1/2) of the Vaidya mass m(u) in the extreme black hole case 9Λ(u)m 2 (u) = 1, which could not be explained with the constant Λ in [4]. This situation can be seen in the next section of this paper. Now we introduce the non-stationary de Sitter solution with variable Λ(u) briefly. The line-element describing a non-stationary de Sitter solution of Einstein's field equations with cosmological function Λ(u) in the null coordinates (u, r, θ, ϕ) is given in [6] as where dΩ 2 = dθ 2 + sin 2 θdϕ 2 . Here Λ(u) is an arbitrary non-increasing function of the retarded time coordinate u = t − r. The line-element (1.1) possesses an energy-momentum tensor as is a null vector and the universal constant K = 8πG/c 4 . The trace of the tensor (1.2) is given by KT = 4Λ(u). Here it is worth mentioning that the energymomentum tensor (1.2) involves a Vaidya-like null term   u a b , which arises from the non-stationary state of motion of an observer traveling in the non-N. ISHWARCHANDRA, K. Y. SINGH 495 stationary de Sitter universe (1.1). This non-stationary part of the energy-momentum tensor (1.2) contributes the nature of null matter field present in (1.1) whose energy- has zero trace and will vanish when r → 0. However, it still maintains the non-stationary status that and the space-time of the observer are naturally time-dependent even at the origin r → 0. Here "NS" in stands for non-stationary as it arises from the non-stationary state of the universe.
Using the energy-momentum tenor (1.2) we could write Einstein's field equations as follows arises from the non-stationary state of motion depending on the derivative of coordinate u and is given by It is noted that the right side of the Equation (1.3) does not involve the universal constant K. The derivation of this non-stationary de Sitter cosmological universe (1.1) is in agreement with the original stationary de Sitter model [7] when Λ(u) takes a constant value Λ.
Now expressing the metric tensor in terms of null tet- where ρ and p denote the density and the pressure of the non-stationary de Sitter model respectively and are obtained as: The trace of energy-momentum tensor T ab (1.5) is given as Here it is observed that ρ − p > 0 for the non-stationary de Sitter model. From (1.6) we find the equation of state as with the negative pressure p in (1.6). This shows the fact that the non-stationary de Sitter solution (1.1) is in agreement with the cosmological constant (Λ) de Sitter model possessing the equation of state w = −1 in the dark energy scenario [9][10][11], when Λ(u) takes a constant value Λ.
The black hole embedded into de Sitter space plays an important role in classical general relativity that the cosmological constant is found present in the inflationary scenario of the early universe in a stage where the uni-verse is geometrically similar to the original de Sitter space [7]. Also embedded black holes can avoid the direct formation of negative mass naked singularities during Hawking's black hole evaporation process [12].

Vaidya in Non-Stationary de Sitter Space
In this section we propose an exact solution describing the Vaidya-de Sitter solution with a non-stationary variable Λ(u), which may be treated as the non-stationary Vaidya-de Sitter black hole or the Vaidya black hole on the non-stationary de Sitter background with variable Λ(u).
Wang and Wu [13] have expressed the mass function as Here the u-coordinate is related to the retarded time in flat space-time. The u-constant surfaces are null cones open to the future. The r-constant is null coordinate. The retarded time coordinate is used to evaluate the radiating (or outgoing) energy momentum tensor around the astronomical body [14]. For deriving the Vaidya-de Sitter solution with Λ(u) we consider the Wang-Wu functions q(u) in (2.1) as follows: such that the mass function (2.1) takes the form Then using this mass function in the Schwarzschildlike metric we obtain a non-stationary metric, describing the Vaidya metric embedded into the non-stationary de Sitter model to produce Vaidya-de Sitter solution with variable cosmological function Λ(u) as we find the energy-momentum tensor describing the matter field for the non stationary space-time (2.5) as where the coefficients ρ, p and μ are the density, the pressure and the null density, respectively and are obtained as: The trace of energy-momentum tensor (2.7) is given by Here it is observed that ρ − p > 0 for the non-stationary de Sitter model and the trace T does not involve the Vaidya mass m(u), showing the null fluid distribution of the space-time (2.5). We also find the ratio of the pressure p to the energy density ρ as the equation of state for the solution w = p/ρ = −1, which is one of the important properties of de Sitter space to be regarded as a dark energy candle [9][10][11]  As in general relativity the physical properties of a space-time geometry are determined by the nature of the matter distribution in the space, we must express the energy-momentum tensor (2.7) in such a way that one should be able to understand it easily. Thus, the total energy-momentum tensor (EMT) for the solution (2.5)  is a time-like vector field for a a U Vaidy n observer in the a-de Sitter metric. This violation of the strong energy condition is due to the negative pressure in (2.8), and may lead to a repulsive gravitational force of the matter field in the space-time (2.5). The violation of the strong energy condition indicates various properties of the energy-momentum tensor (2.7) or (2.11) which are different from those of the ordinary matter fields, like perfect fluid, electromagnetic field etc., having positive pressures. In particular, it is worth to note that this strong energy condition is satisfied by the energy-momentum tensor of electromagnetic field with   Here we shall justify the nature of the embedded solution in the form of Kerrounds. The Vaidya-de Sitter metric can be expressed in Kerr-Schild ansatz is the nonand  is geo-stationary ven in de Sitter metric gi hear free, expanding a [6] ist n a desic, s nd zero tw ull vector for both   ce-time (2.5) with variable Λ(u) is a solution of Einstein's field equations. They establish the structure of embedded black hole that either "the null radiating Vaidya black hole is embedded into the non-stationary de Sitter cosmological universe to produce Vaidya-de Sitter black hole" or the non-stationary de Sitter universe is embedded into the Vaidya black hole to obtain the de Sitter-Vaidya black holeboth nomenclature possess the same geometrical meaning. That is, we cannot physically predict which space started first to embed into another. When the cosmological function Λ(u) takes a constant value Λ, the line element (2.5) will reduce to the Vaidya-de Sitter black hole [4]. If one sets M(u) and Λ(u) both constant, the metric (2.5) becomes the stationary Schwarzschild-de Sitter black hole.
Surface gravity: The metric (2.5) will describe a cosmological bl r which the polynomial equation vector field . From these we obse he mass he variable Λ(u) do a all the three equations. When m(u) = 0, the rem ation onary de Sitte these will is a space-like rve that both t ppear in aining equ 1 a a v v   m(u) and t s will be for non-stati r space, whereas, if Λ(u) = 0, be for Vaidya radiating black hole. This means that the time-like observer in the Vaidya space will have a four velocity vector field which is expanding, accelerating, shearing but zero-twist. This is also true for the observer in non-stationary de Sitter space, when m(u) = 0.
The Vaidya-de Sitter metric (2.5) describes a non-stationary embedded spherically symmetric solution whose Weyl curvature tensor is type D   3  or ℓ a given in = ψ 1 = ψ non-rotating Vaidya-de Sit- which does not involve any derivative term of m(u) and Λ(u), and will not change its form when the mass m(u) and the variable Λ(u) take the constant va these constant values the Kretschmann scalar will reduce bedded in the nonkground with variable Λ(u) [6]. e regarded as a generalization of hole in asymptotically de Sitter space [3]. Thus, in this lues. With to that of Schwarzschild-de Sitter solution. This invariant does not diverge at the origin r → 0.