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Vaidya Solution in Non-Stationary De Sitter Background: Hawking's

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Vaidya Solution in Non-Stationary De Sitter Background: Hawking's International Journal of Astronomy and Astrophysics, 2013, 3, 494-499 Published Online December 2013 (http://www.scirp.org/journal/ijaa) http://dx.doi.org/10.4236/ijaa.2013.34057 Vaidya Solution in Non-Stationary de Sitter Background: Hawking’s Temperature Ngangbam Ishwarchandra, Kangujam Yugindro Singh Department of Physics, Manipur University, Imphal, India Email: [email protected], [email protected] Received September 30, 2013; revised October 25, 2013; accepted November 3, 2013 Copyright © 2013 Ngangbam Ishwarchandra, Kangujam Yugindro Singh. This is an open access article distributed under the Crea- tive Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT 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 ra- diating Vaidya black hole which is embedded into the non-stationary de Sitter space with variable Λ(u). The energy- momentum 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. Keywords: Vaidya Solution; Non-Stationary de Sitter; Einstein’s Equations; Energy-Momentum Tensor 1. Introduction mass m(u) in the extreme black hole case 9Λ(u)m2(u) = 1, which could not be explained with the constant Λ in [4]. The Vaidya solution having a variable mass m(u) with This situation can be seen in the next section of this paper. retarded time u is a non-stationary generalization of Now we introduce the non-stationary de Sitter solution Schwarzschild black hole of constant mass m [1]. The with variable Λ(u) briefly. The line-element describing a Schwarzschild solution is regarded as a black hole in an non-stationary de Sitter solution of Einstein’s field equa- asymptotically flat space. The Schwarzschild-de Sitter tions with cosmological function Λ(u) in the null coordi- solution is interpreted as a black hole in an asymptoti- nates (u, r, θ, ϕ) is given in [6] as cally de Sitter space with non-zero cosmological constant Λ [2]. The Schwarzschild-de Sitter solution is also con- 2221 22 d1sruuurr d2ddd (1.1) sidered as an embedded black hole that the Schwarzs- 3 child solution is embedded into the de Sitter space with where dΩ2 = dθ2 + sin2θdϕ2. Here Λ(u) is an arbitrary cosmological constant Λ to produce the Schwarzschild- non-increasing function of the retarded time coordinate u de Sitter black hole [3]. Mallett [4] has introduced = t − r. The line-element (1.1) possesses an energy-mo- Vaidya-de Sitter solution with constant Λ by making the mentum tensor as Schwarzschild mass m variable with respect to the re- 1 tarded time u as m(u) and studied the nature of the space- KTruab , u a b ugab (1.2) time [5]. 3 1 Here the aim of this paper is to propose an exact solu- where a a is a null vector and the universal con- tion of Einstein’s field equations describing Vaidya black stant K = 8πG/c4. The trace of the tensor (1.2) is given by hole embedded into the non-stationary de Sitter space to KT = 4Λ(u). Here it is worth mentioning that the energy- obtain Vaidya-de Sitter black hole with variable Λ(u). momentum tensor (1.2) involves a Vaidya-like null term This Vaidya-de Sitter solution with variable Λ(u) will 13ru ,uab , which arises from the non-statio- have the limit m(u) = ± (1/3) Λ(u)(−1/2) of the Vaidya nary state of motion of an observer traveling in the non- Open Access IJAA N. ISHWARCHANDRA, K. Y. SINGH 495 stationary de Sitter universe (1.1). This non-stationary verse is geometrically similar to the original de Sitter part of the energy-momentum tensor (1.2) contributes the space [7]. Also embedded black holes can avoid the di- nature of null matter field present in (1.1) whose energy- rect formation of negative mass naked singularities dur- NS momentum tensor Tab has zero trace and will vanish ing Hawking’s black hole evaporation process [12]. when r → 0. However, it still maintains the non-station- ary status that u constant and the space-time of 2. Vaidya in Non-Stationary de Sitter Space the observer are naturally time-dependent even at the ori- NS In this section we propose an exact solution describing gin r → 0. Here “NS” in T stands for non-stationary ab the Vaidya-de Sitter solution with a non-stationary vari- as it arises from the non-stationary state of the universe. able Λ(u), which may be treated as the non-stationary Using the energy-momentum tenor (1.2) we could Vaidya-de Sitter black hole or the Vaidya black hole on write Einstein’s field equations as follows the non-stationary de Sitter background with variable 1 RRgugTNS (1.3) Λ(u). ab 2 ab ab ab Wang and Wu [13] have expressed the mass function NS as where Tab arises from the non-stationary state of motion depending on the derivative of coordinate u and n is given by M ur,. qn u r (2.1) n NS 1 Here the u-coordinate is related to the retarded time in Truab ,u a b . (1.4) 3 flat space-time. The u-constant surfaces are null cones It is noted that the right side of the Equation (1.3) does open to the future. The r-constant is null coordinate. The not involve the universal constant K. The derivation of retarded time coordinate is used to evaluate the radiating this non-stationary de Sitter cosmological universe (1.1) (or outgoing) energy momentum tensor around the as- is in agreement with the original stationary de Sitter tronomical body [14]. For deriving the Vaidya-de Sitter model [7] when Λ(u) takes a constant value Λ. solution with Λ(u) we consider the Wang-Wu functions Now expressing the metric tensor in terms of null tet- q(u) in (2.1) as follows: rad vectors ,,nmm , [8] as aa a b mu ,when0 n gab22 () anmm b ( a b ), the energy momentum tensor qun u6, when n 3 (2.2) (1.2) takes the form 0, whenn 0,3 Tnab a b 22 ()a b pm (am b), (1.5) such that the mass function (2.1) takes the form where ρ and p denote the density and the pressure of the 1 non-stationary de Sitter model respectively and are ob- M ur,. mu r3 u (2.3) tained as: 6 u r Then using this mass function in the Schwarzschild- p ， u, . (1.6) KK3 u like metric The trace of energy-momentum tensor T (1.5) is 2,Mur ab d1suur22 d2dddr22, (2.4) given as TgTab 24 p K u. Here it ab r is observed that ρ − p > 0 for the non-stationary de Sitter 2222 model. From (1.6) we find the equation of state as with ddsind , we obtain a non-statio- nary metric, describing the Vaidya metric embedded into p w 1 (1.7) the non-stationary de Sitter model to produce Vaidya-de Sitter solution with variable cosmological function Λ(u) with the negative pressure p in (1.6). This shows the fact as that the non-stationary de Sitter solution (1.1) is in 222mu 1 222 agreement with the cosmological constant (Λ) de Sitter d1sruuur d2dddr,(2.5) r 3 model possessing the equation of state w = −1 in the dark energy scenario [9-11], when Λ(u) takes a constant value where M(u) is the mass of the Vaidya black hole and Λ(u) Λ. denotes the de Sitter cosmological function of retarded The black hole embedded into de Sitter space plays an time coordinate u. When we set the function Λ(u) to be a important role in classical general relativity that the constant Λ, the line element (2.5) will become the Vaid- cosmological constant is found present in the inflationary ya de Sitter space-time with cosmological constant. scenario of the early universe in a stage where the uni- When the Vaidya mass sets to zero, and function Λ(u) to Open Access IJAA 496 N. ISHWARCHANDRA, K. Y. SINGH be a constant Λ, the line element will recover the original TTVNSdS T T , (2.11) de Sitter solution. The solution (2.5) may have a singu- ab ab ab ab 2 V larity when 12mu r 13 r u 0. where the Tab for the Vaidya null radiating fluid, NS The complex null vectors for the Vaidya-de Sitter so- Tab the non-stationary contribution of de Sitter field lution can be chosen as follows: Λ(u) associated with the derivative term u,u and T dS the cosmological de Sitter matter are given , re- 1 ab 11,,n 2 spectively aaa2r 2 aa (2.6) T VV , (2.12) r 34 ab a b miaa sin a 2 NS NS Tab a b , (2.13) 24 where rr 2 muur 3.
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