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 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 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- 2221 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 13ru ,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 RRgugTNS (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  gab22 () 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  d2dddr22, (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 222mu  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 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 TTVNSdS T    T  , (2.11) de Sitter solution. The solution (2.5) may have a singu- ab ab ab ab 2 V larity when 12mu 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. Here  a , na are TnpdS22  dS  dS mm, (2.14) real null vectors and ma is complex with the normaliza- ab() a b (a b ) tion conditions  naa1 mm and the other inner aa with the coefficients products of the null vectors are zero. From Einstein’s field equations RRgK12 T, we find the V1 NSr ab ab ab  mu,, u,, Kr 2 uu3K energy-momentum tensor describing the matter field for (2.15) the non stationary space-time (2.5) as  u  dS  pdS . Tnab a b 22 ()a b pm (am b) (2.7) K Here  V is the null density for the Vaidya null fluid where the coefficients ρ, p and μ are the density, the V NS Tab ,  the non-stationary null density associated pressure and the null density, respectively and are ob- dS dS tained as: with the derivative of Λ(u),  and p are the density and the pressure of de Sitte r matter. W hen the u  p  , function Λ(u) becomes constant, it will provide K NS . (2.8) Tab  0, then the space-time will be that of the Vai- 1 r dya-de Sitter with constant Λ. If m(u) = 0,  2 mu,,uu   u V Kr 3K (u ) constant we have Tab  0, then the remaining space-time (2.5) will be the non-stat ionary de Sitter mo- The trace of energy-momentum tensor Tab (2.7) is NS del with variable Λ(u) [6]. At that time the EMTs Tab given by dS and Tab will exist indicating the non-stationary de dS ab 4  Sitter matt er distribution; and for constant Λ, Tab  0, TgTab 2 p u. (2.9) K then the total energy-momentum tensor (2.11) will re- dS Here it is observed that ρ − p > 0 for the non-station- duce to TTab ab g ab for the well-known de Sitter ary de Sitter model and the trace T does not involve the model with constant Λ. Vaidya mass m(u), showing the null fluid distribution of Here, it is interesting to mention that the energy-mo- the space-time (2.5). We also find the ratio of the pres- mentum tensor (2.7) does not satisfy the strong energy a b sure p to the energy density ρ as the equation of state for condition TabU U ≥ (1/2)T, which is equivalent to the solution w = p/ρ = −1, which is one of the important  0,pp 0, 0. (2.16) properties of de Sitter space to be regarded as a dark en- ergy candle [9-11] and references there in. The Ricci Ua is a time-like vector field for an observer in the Vaidya-de Sitter metric. This violation of the strong en- scalar * 1 g ab R , describing matter field takes  24 ab  ergy condition is due to the negative pressure in (2.8), 1 and may lead to a repulsive gravitational force of the the form * u . The energy-momentum tensor 6 matter field in the space-time (2.5). The violation of the (2.7) satisfies the energy conservation equation strong indicates various properties of the energy-momentum tensor (2.7) or (2.11) which are ab T ;b  0 (2.10) different from those of the ordinary matter fields, like As in general relativity the physical properties of a perfect fluid, electromagnetic field etc., having positive space-time geometry are determined by the nature of the pressures. In particular, it is worth to note that this strong matter distribution in the space, we must express the en- energy condition is satisfied by the energy-momentum 24 ergy-momentum tensor (2.7) in such a way that one tensor of electromagnetic field with  pe Kr should be able to understand it easily. Thus, the total of Reissner-Nordstrom space-time. It is a lso to note that energy-momentum tensor (EMT) for the solution (2.5) the energy-momentum tensor (2.11) may be written in may be written in the following decomposition form as: the form of Guth’s modification of TTab ab g ab

Open Access IJAA N. ISHWARCHANDRA, K. Y. SINGH 497

[15] for early inflation of the universe as These roots satisfy the following relation VNS    u TTab ab T ab  ug ab , (2.17) 3   3 rr12 rr rr3  r 2 mu  r  u 3  where gab is the Vaidya-de Sitter metric. This equation shows the possibility of the inflation of radiating Vaidya (2.22) space in th e non-stationary de Sitter space with cosmo- Here we are interested only the real root r1 which may logical variable Λ(u). describe the horizon of the Vaidya-de Sitter cosmological Here we shall justify the nature of the embedded solu- black hole, as the complex roots have less physical inter- tion in the form of Kerr-Schild ansatze in different back- pretation. grounds. The Vaidya-de Sitter metric can be expressed in The surface gravity  of a horizon is defined by the ba a a Kerr-Schild ansatz Relation nnb  n, where the null vector n is pa- b VdS  dS rameterized by the coordinate u, such that ddun b ggQurab ab 2,a b (2.18) [16,17]. b is the covariant derivative. The surface 1 dS where Qur,  M  ur . Here, gab is the non- gravities at rr i , (i = 1, 2, 3) are found as stationary de Sitter metric given in [6] and  is geo- 3 a 1  ur desic, shear free, expanding and zero twist n ull vector for  rmu   , (2.23) dS VdS 2  both g as well as g . The above Kerr -Schild r 6 ab ab rr i form can also be recast on the Vaidya background as and the entropy of the horizon is found as, ggQurVdS  V 2, (2.19) ab ab a b Sr 2 (2.24) where Qur,  ur2 6. These two Kerr-Schild rr i forms (2.18) and (2.19) support the fact that the non- Here we consider a case of extreme Vaidya-de Sitter stationary Vaidya-de Sitter space-time (2.5) with variable black hole having the mass function 12 Λ(u) is a solution of Einstein’s field equations. They es- mu13  u , if u 0 . This implies 12 tablish the structure of embedded black hole that either that the real root r1 take the values ru1 2,  “the null radiating Vaidya black hole is embedded into and the two complex roots are coincided: the non-stationary de Sitter cosmological universe to 12 rr23 u . The surface gravity on the cosmo- produce Vaidya-de Sitter black hole” or the non-statio- logical black hole horizon rr takes the form nary de Sitter universe is embedded into the Vaidya 1 black hole to obtain the de Sitter-Vaidya black hole—  34  u 12 . (2.25) both nomenclature possess the same geometrical mean- ing. That is, we cannot physically predict which space However, it is vanished at rr23 r. Then we obtain started first to embed into another. When the cosmologi- the Hawking’s temperature of the cosmological black cal function Λ(u) takes a constant value Λ, the line ele- hole horizon at rr 1 from the relation T = κ/2π as ment (2.5) will reduce to the Vaidya-de Sitter black hole 3  1 Tu 2 . (2.26) [4]. If one sets M(u) and Λ(u) both constant, the metric 8 (2.5) becomes the stationary Schwarzschild-de Sitter black hole. The temperature associated with the real root r1 in Surface gravity: The metric (2.5) will describe a cos- (2.20) will never vanish as long as the de Sitter Λ(u) ex- mological black hole with the horizons at the values of r ists in the space-time geometry of the Vaidya-de Sitter black hole. The condition for which the polynomial equation 12 12 2 13 umuu  13 of the Vai- 12mu r 13 r  u 0 has three roots r1, r2 and rr . The explicit roots are given as dya mass m(u) is the non-stationary generalization of the 32 condition 13 12 m 13 12 of the Sch- 1   11 3 rQ1  1  3 warzschild-de Sitter black hole with constant mass and Λ 3Q 3  u [2].

1  11Properties of the time-like vector: The physical pro- 33 riuQiQ2 13 3   133 perties of the space-time geometry is determined by the 2 u  motion of a matter field distribution (2.7) whose the time (2.20) like vector field is unaa12 a. The four ve- where locity vector ua measures the kinematical properties of 1 a b 22 a fluid expansion u , acceleration uuu  , Q umuu 13 2 9  umu  1  ;a  aab; shear  ab and twist wab  . It is found that the fluid (2.21) flow of the Vaidya-de Sitter model having variable Λ(u)

Open Access IJAA 498 N. ISHWARCHANDRA, K. Y. SINGH

a  is expanding u ;a 0, accelerating ua  0, regard, our Vaidya-de Sitter solution with variable Λ(u) shearing  ab  0 and zero twist wab  0. (2.5) may be interpreted as a black hole in asymptotically non-stationary de Sitter background. We also observe 123 rmu   r  u (2.27) that in the extreme Vaidya-de Sitter black hole, the Vai- 2 3 2r dya mass m(u) has the limit

113  12  12 umauruva (2.28) 13 umuu  13 2r 2 3  for Λ(u) > 0 as mentioned above. This range of the mass 113 limit is a straightforward generalization of Schwarzschild  ab  rmur4   uvva b  mm()a b  32r 2 3 de Sitter case 13 12 m 13 12 associ- (2.29) ated with constant mass m and Λ(> 0). We have also observed that the variable Λ(u) does not where vnaa12 a is a space-like vector field a involve the Weyl scalar ψ2, which shows the conformally vva 1. From these we observe that both the mass m(u) and the variable Λ(u) do appear in all the three flat property of de Sitter space with variable Λ(u) even in equations. When m(u) = 0, the remaining equ ations will the embedded Vaidya-de Sitter solution (2.5). Also there be for non-stationary de Sitter space, whereas, if Λ(u) = 0, is no involvement of Vaidya mass m(u) in the expres- these will be for Vaidya radiating black hole. This means sions of the pressure p and the density ρ (2.8) keeping the that the time-like observer in the Vaidya space will have equation of state parameter w = p/ρ = −1 unaffected for a four velocity vector field which is expanding, acceler- the embedded solution. We have also seen that the ating, shearing but zero-twist. This is also true for the 4-velocity vector ua of the fluid flow of the Vaidya-de observer in non-stationary de Sitter space, when m(u) = Sitter solution having variable Λ(u) is expanding a  b 0. u ;a 0, accelerating uuuaab; 0 , shearing The Vaidya-de Sitter metric (2.5) describes a non-sta-  ab  0 and zero twist wab  0 . It is emphasized that tionary embedded spherically symmetric solution whose the embedded Vaidya-de Sitter solution (2.5) discussed Weyl curvature tensor is type D here includes the following solutions: 1) non-stationary de Sitter solution u 0 , pqrs 3   2 Cmmnmurpqrs    (2.30) NS  dS V TTab ab  0 when m(u) = 0, Tab  0 [6]; in Petrov classification possessing a geodesic, shear free, V 2) Vaidya solution mu   0 , Tab  0 when Λ(u) = expanding and zero-twist null vector ℓ given in (2.6), as NS  dS a 0, TTab ab  0 [1]; other Weyl scalars are vanished ψ0 = ψ1 = ψ3 = ψ4 = 0. Vd S  3) Vaidya-de Sitter mu   0, Tab Tab 0 The Kretschmann scalar for non-rotating Vaidya-de Sit- NS ter model (2.5) takes the form when Λ(u) becomes constant and Tab  0 [4]; dS  4) Schwarzschild-de Sitter Tab  0 when m(u) and abcd 4822 8 RRabcd 6 mu  u, (2.31) VN S  r 3 Λ(u) are constant with Tab Tab 0 [2]; dS  which does not involve any derivative term of m(u) and 5) de Sitter Tab  0 when m(u) = 0 and Λ(u) is a Λ(u), and will not change its form when the mass m(u) constant with TTVNS   0 [7]. and the variable Λ(u) take the constant values. With ab ab It is noted that all known results with constant Λ may these constant values the Kretschmann scalar will reduce be extended with this variable Λ(u) of the Vaidya-de to that of Schwarzschild-de Sitter solution. This invariant Sitter space-time. does not diverge at the origin r → 0. 4. Acknowledgements 3. Conclusions The work of Ishwarchandra is supported by the Univer- We present an exact solution of sity Grants Commission (UGC), New Delhi, in the form for the radiating Vaidya solution embedded in the non- of Rajiv Gandhi National Fellowship for SC/ST. stationary de Sitter background with variable Λ(u) [6]. This solution may be regarded as a generalization of Vaidya-de Sitter solution with constant Λ [4]. The REFERENCES Vaidya-de Sitter solution with mass m(u) and constant Λ [1] P. C. Vaidya, “The External Field of a Radiating Star in is also a generalized form of the Schwarzschild-de Sitter General Relativity,” Current Science, Vol. 12, 1943, pp. model with constant mass m and constant Λ. The Sch- 183-184; reprinted General Relativity and Gravitation, warzschild-de Sitter solution is interpreted as a black Vol. 31, No. 1, 1999, pp. 119-120. hole in asymptotically de Sitter space [3]. Thus, in this http://dx.doi.org/10.1023/A:1018871522880

Open Access IJAA N. ISHWARCHANDRA, K. Y. SINGH 499

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