Radiating black hole solutions in arbitrary dimensions
aa r X i v : . [ g r- q c ] J a n Radiating black hole solutions in arbitrary dimensions
S. G. Ghosh ∗ BITS, Pilani DUBAI Campus, P.B. 500022,Knowledge Village, DUBAI, UAE andInter-University Center for Astronomy and Astrophysics, Post Bag 4Ganeshkhind, Pune - 411 007, INDIA
A. K. Dawood
Inter-University Center for Astronomy and Astrophysics, Post Bag 4Ganeshkhind, Pune - 411 007, INDIA (Dated: October 24, 2018)We prove a theorem that characterizes a large family of non-static solutions to Einstein equationsin N -dimensional space-time, representing, in general, spherically symmetric Type II fluid. It isshown that the best known Vaidya-based (radiating) black hole solutions to Einstein equations, inboth four dimensions (4D) and higher dimensions (HD), are particular cases from this family. Thespherically symmetric static black hole solutions for Type I fluid can also be retrieved. A briefdiscussion on the energy conditions, singularities and horizons is provided. PACS numbers: 04.20.Jb, 04.70.Bw, 04.40.Nr
I. INTRODUCTION
In recent time, it was demonstrated that the string theory requires higher dimensions for its consistency. Modelswith the space-time with large extra dimensions were recently proposed in order to solve the hierarchy problem, thatis to explain why the gravitational coupling constant is much smaller than the coupling constants of other physicalinteractions. These new concepts of higher dimensional physics have a number of interesting applications in moderncosmology and theory of gravity. This has triggered, during the past decade, a significant increase in interest in blackholes in higher dimensions (HD) (see, for example, the review articles of Horowitz [1] and Peet [2]). There is nowan extensive literature of solutions in string theory with horizons, which represent black holes, and related objectsin arbitrary dimensions. The physical properties of these solutions have been widely studied. Interest in black holesin HD has been further intensified in recent years, due, for example, to the role they have played in the conjecturedcorrespondence between string theory (or supergravity) on asymptotically locally anti-de Sitter backgrounds and thelarge-N limit of certain conformal field theories defined on the boundary-at-infinity of these backgrounds [3, 4, 5].Static and spherically symmetric space-times are one of the simplest kinds of space-times that one can imagine ingeneral relativity. Yet, even in this simple situation, solving the Einstein field equations may be far from trivial. Infact, it turns out that such a problem is intractable due to complexity of the Einstein field equations. Hence, thereare very few inhomogeneous and nonstatic solutions known, one of them is the Vaidya solution. The Vaidya solution[6] is a solution of Einstein’s equations with spherical symmetry for a null fluid (radiation) source (a Type II fluid)described by energy momentum tensor T ab = ψl a l b , l a being a null vector field. The Vaidya’s radiating star metricis today commonly used for two purposes: (i) As a testing ground for various formulations of the Cosmic CensorshipConjecture (CCC). (Actually CCC is a famous conjecture, first formulated by Penrose [7]. The conjecture, in it’sweak version, essentially state that any naked singularity which is created by evolution of regular initial data will beshielded from the external view by an event horizon. According to the strong version of the CCC, naked singularitiesare never produced, which in the precise mathematical terms demands that space-time should be globally hyperbolic.)(ii) As an exterior solution for models of objects consisting of heat-conducting matter. Recently, it has also proved tobe useful in the study of Hawking radiation, the process of black-hole evaporation [8], and in the stochastic gravityprogram [9]. It has also advantage of allowing a study of the dynamical evolution of horizon associated with a radiatingblack hole.Also, several solutions in which the source is a mixture of a perfect fluid and null radiation have been obtained inlater years [10]. This includes the Bonnor-Vaidya solution [11] for the charge case, the Husain solution [12] with anequation of state P = kρ . Glass and Krisch [13] further generalized the Vaidya solution to include a string fluid, while ∗ E-mail: [email protected] charged strange quark fluid (SQM) together with the Vaidya null radiation has been obtained by Harko and Cheng[14] (see also [15]). Wang and Wu [16] further extrapolated the Vaidya solution to more general case, which includea large family of known solutions.Motivated by this and by a recent work Salgado [17], we [18] have proved a theorem characterizing a three parameterfamily of solutions, representing, in general, spherically symmetric Type II fluid that includes most of the knownsolutions to Einstein field equations. This is done by imposing certain conditions on the energy momentum tensor(EMT) (see also [19, 20, 21, 22]).In this paper, we consider an extension of our work [18], so that a large family of exact spherically symmetric TypeII fluid solutions, in arbitrary dimensions, are possible, including it’s generalization to asymptotically de Sitter/anti-deSitter.
II. THE RADIATING BLACK-HOLE SOLUTIONS
Theorem - I : Let (
M, g ab ) be an N-dimensional space-time [sign ( g ab ) = +( N − ] such that (i) It is non-static andspherically symmetric, (ii) it satisfies Einstein field equations, (iii) in the Eddington-Bondi coordinates where ds = − A ( v, r ) f ( v, r ) dv + 2 ǫA ( v, r ) dv dr + r ( d Ω N − ) , where, ( d Ω N − ) = dθ + sin ( θ ) dθ + sin ( θ ) sin ( θ ) dθ + . . . + h(cid:16)Q N − j =1 sin ( θ j ) (cid:17) dθ N − i , the energy-momentum tensor T ab satisfies the conditions T vr = 0 and T θ θ = kT rr , ( k = const. ∈ R ) (iv) it possesses a regular Killing horizon or a regular origin. Then the metric ofthe space-time is given by ds = − (cid:20) − m ( v, r )( N − r N − (cid:21) dv + 2 ǫdvdr + r ( d Ω N − ) , ( ǫ = ±
1) (1)where m ( v, r ) = M ( v ) if C ( v ) = 0 ,M ( v ) − πC ( v ) (cid:16) N − N − (cid:17) N − k +1 r ( N − k +1 if C ( v ) = 0 and k = − / ( N − ,M ( v ) − πC ( v ) (cid:16) N − N − (cid:17) ln r if C ( v ) = 0 and k = − / ( N − . (2) T ab = C ( v ) r ( N − − k ) diag[1 , , k, . . . , k ] . (3)and T rv = πr N − (cid:16) N − N − (cid:17) ∂M∂v − N − k +1 ∂C∂v r ( N − k − if k = − / ( N − , πr N − (cid:16) N − N − (cid:17) ∂M∂v − r N − ∂C∂v ln r if k = − / ( N − . (4) Here, M(v) and C(v) are the arbitrary functions whose values depend on the boundary conditions and the fundamentalconstants of the underlying matter.
Proof : Expressed in terms of Eddington coordinate, the metric of general spherically symmetric space-time in N -dimensional space-times [23, 24, 25, 26] is, ds = − A ( v, r ) f ( v, r ) dv + 2 ǫA ( v, r ) dv dr + r ( d Ω N − ) . (5)Here A ( v, r ) is an arbitrary function. It is useful to introduce a local mass function m ( v, r ) defined by f ( v, r ) =1 − m ( v, r ) / ( N − r ( N − . For m ( v, r ) = m ( v ) and A = 1, the metric reduces to the N -dimensional Vaidya metric[23].In the static limit, this metric can be obtained from the metric in the usual, spherically symmetric form, ds = − f ( r ) dt + dr f ( r ) + r ( d Ω N − ) . (6)by the coordinate transformation dv = A ( r ) − ( dt + ǫ drf ( r ) ) (7)In case of spherical symmetry, even when f ( r ) is replaced by f ( t, r ), one can cast the metric in the form (5) [27].The non-vanishing components of the Einstein tensor [24] are G vr = ( N − rA ∂A∂r , (8a) G rv = − ( N − r ∂f∂v , (8b) G vv = ( N − r (cid:20) r ∂f∂r − ( N − − f ) (cid:21) , (8c) G rr = ( N − r (cid:20) r ∂f∂r − ( N − − f ) (cid:21) + ( N − rA f ∂A∂r , (8d)2 r G θ i θ i = r ∂ f∂r + ( N − (cid:18) r ∂f∂r − ( N − − f ) (cid:19) + 2( N − rfA ∂A∂r + 2 r A ∂ A∂v∂r (8e)+ 3 r A ∂f∂r ∂A∂r + 2 r fA ∂ A∂r − r A ∂A∂r ∂A∂v , where { x a } = { v, r, θ , . . . θ N − } . We shall consider the special case T vr = 0 (hypothesis), which means fromEq. (8a), A ( v, r ) = g ( v ). This also implies that G vv = G rr (Eq. (8d)). However, by introducing another null coordinate v = R g ( v ) dv , we can always set without the loss of generality, A ( v, r ) = 1. Hence, the metric takes the form, ds = − (cid:20) − m ( v, r )( N − r ( N − (cid:21) dv + 2 ǫdvdr + r ( d Ω N − ) . (9)Therefore the entire family of solutions we are searching for is determined by a single function m ( v, r ). Henceforth,we adopt here a method similar to Salgado [17] which we modify here to accommodate the non static case. In whatfollows, we shall consider ǫ = 1. The Einstein field equations are R ab − Rg ab = 8 πT ab , (10)and combining Eqs. (8) and (10), we have if a = b , T ab = 0 except for a non-zero off-diagonal components T rv . Inaddition, we observe that T vv = T rr . Thus the EMT can be written as : T ab = T vv T vr . .T rv T rr . . T θ θ . .. . . . . .. . . . . T θ N − θ N − . which in general belongs to a Type II fluid with T θ θ = T θ θ = . . . = T θ N − θ N − It may be recalled that EMT of a Type IIfluid has a double null eigen vector, whereas an EMT of a Type I fluid has only one time-like eigen vector [28]. Onthe other hand, from the Einstein equations, it follows that ∇ a T ab = 0 . (11)Enforcing the conservation laws ∇ a T ab = 0, yields the following non-trivial differential equations: ∂T rr ∂r = − ( N − r ( T rr − T θ θ ) , (12)and, using, T rr = T vv , ∂T vv ∂v = − ∂T rv ∂r − ( N − r T rv . (13)Using the assumption made above that T θ θ = kT rr , we obtain the following linear differential equation ∂T rr ∂r = − ( N − r (1 − k ) T rr , (14)which can be easily integrated to give T rr = C ( v ) r ( N − − k ) , (15)where C ( v ) is an arbitrary function of v , arising as an integration constant. Then, using hypothesis (iii), we concludethat T ab = C ( v ) r ( N − − k ) diag[1 , , k, . . . , k ] . (16)Now using Eqs. (8) (cid:2) with f ( v, r ) = 1 − m ( v, r ) / ( N − r ( N − (cid:3) , (10) and (15), we get ∂m∂r = − π (cid:18) N − N − (cid:19) C ( v ) r − ( N − k , (17)which trivially integrates to m ( v, r ) = M ( v ) if C ( v ) = 0 ,M ( v ) − πC ( v ) (cid:16) N − N − (cid:17) N − k +1 r ( N − k +1 if C ( v ) = 0 and k = − / ( N − ,M ( v ) − πC ( v ) (cid:16) N − N − (cid:17) ln r if C ( v ) = 0 and k = − / ( N − . (18)Here the function M ( v ) arises as a result of integration. What remains to be calculated is the only non-zero off-diagonalcomponent T rv of the EMT. From Eqs. (8) and (10), one gets T rv = 18 πr N − (cid:18) N − N − (cid:19) ∂m∂v , (19)which, on using Eq. (18), gives T rv = πr N − (cid:16) N − N − (cid:17) ∂M∂v − N − k +1 ∂C∂v r ( N − k − if k = − / ( N − , πr N − (cid:16) N − N − (cid:17) ∂M∂v − r N − ∂C∂v ln r if k = − / ( N − . (20)It is seen that Eq. (13) is identically satisfied. Hence the theorem is proved. The theorem proved above represents ageneral class of non-static, N -dimensional spherically symmetric solutions to Einstein’s equations describing radiatingblack-holes with the EMT which satisfies the conditions in accordance with hypothesis (iii). The solutions generatedhere highly rely on the assumption ( iii ). On the other hand, although hypothesis (iv) is not used a priori for provingthe result, but it is indeed suggested by regularity of the solution at the origin, from which, T vv = T rr | r =0 (see [17] forfurther details).The family of the N -dimensional solutions outlined here contains N -dimensional version of, for instance, Vaidya[23, 29] Bonnor-Vaidya [24, 30], dS/AdS [24], global monopole [24, 25, 31], Husain [12, 24, 25], and Harko-ChengSQM solution [14, 23]. Obviously, by proper choice of the functions M ( v ) and C ( v ), and k − index, one can generateas many solutions as required. The above solutions include most of the known Vadya-based spherically symmetricsolutions of the Einstein field equations. When N = 4, the 4 D solutions derived in [16, 18] can be recovered. Thestatic black holes solutions, in both HD [32] and in 4D [17], can be recovered by setting M ( v ) = M, C ( v ) = C , withM and C as constants, in which case matter is Type I.In summary, we have shown that the metric ds = − (cid:20) − M ( v )( N − r N − + 16 πC ( v )( N − N − k + 1] r ( N − k − (cid:21) dv + 2 dvdr + r ( d Ω N − ) , (21)is a solution of the Einstein equations for the stress energy tensor Eqs. (3) and (4). A metric is considered to beasymptotically flat if in the vicinity of a spacelike hypersurface its components behave as g ab → η ab + α ab ( x c /r, t ) r + O (cid:18) r ǫ (cid:19) , (22)as r → ∞ . ( ǫ > η ab is the Minkowski metric, α ab is an arbitrary symmetric tensor, and x c is a flat coordinate systemat spacelike infinity). According to this definition, our metrics Eq. (1) are asymptotically flat for k > − / ( N −
2) andare cosmological for k < − / ( N − M ( v ) = M and 2 C ( v ) = Q , the metric is just higherdimensional Reissner-Nordstr¨ o m. The detailed of the asymptotic structure of spatial infinity in higher-dimensionalspace-times can be found in Ref. [33] and the different conformal diagrams for maximal extension of 4D Vaidya isdiscussed by Fayos et al. [34].The theorem proved shows that rather than a mathematical coincidence, the above form of the metric is a conse-quence of the features of the energy-momentum tensor considered. In the above exact solutions, the associated energymomentum tensors share some properties that are taken into account in the theorem in a general fashion withoutspecifying the nature of the matter. Therefore, the theorem helps to characterize a whole two-parameter family ofsolutions to the Einstein field equations.The solutions discussed in the section are characterized by two arbitrary functions M ( v ) and C ( v ), and thecosmological constant Λ. Thus one would like to generalize the above theorem to include Λ. We can show thatthe energy momentum tensor components, in general, can be written as, T ab = T a ( f ) b − Λ8 π δ ab [17, 22], where Λ isthe cosmological constant and T a ( f ) b is energy momentum tensor of the matter fields that satisfy T θ ( f ) θ = kT r ( f ) r .A trivial extension of the theorem allows one to cover a three-parameter family of solutions, with one of the pa-rameters being a cosmological constant Λ. Next, we just state (proof being similar) the generalization of the Theorem I. Theorem - II : Let (
M, g ab ) be an N-dimensional space-time [sign ( g ab ) = +( N − ] such that (i) It is non-staticand spherically symmetric, (ii) it satisfies Einstein field equations, (iii) the total energy-momentum tensor is givenby T ab = T a ( f ) b − Λ8 π δ ab , where Λ is the cosmological constant and T a ( f ) b is energy momentum tensor of the matter fields,(iv) in the Eddington coordinates where ds = − A ( v, r ) f ( v, r ) dv + 2 A ( v, r ) dv dr + r ( d Ω N − ) , the EMT T a ( f ) b satisfies the conditions T v ( f ) r = 0 , T θ ( f ) θ = kT r ( f ) r , ( k = const. ∈ R ), (v) it possesses a regular Killing horizon or aregular origin. Then the metric of the space-time is given by metric (1) , where m ( v, r ) = M ( v ) + ( N − N − N − Λ r N − if C ( v ) = 0 ,M ( v ) − πC ( v ) (cid:16) N − N − (cid:17) N − k +1 r ( N − k +1 + ( N − N − N − Λ r N − if C ( v ) = 0 and k = − / ( N − ,M ( v ) − πC ( v ) (cid:16) N − N − (cid:17) ln r + ( N − N − N − Λ r N − if C ( v ) = 0 and k = − / ( N − . (23) T ab = C ( v ) r ( N − − k ) diag[1 , , k, . . . , k ] − Λ8 π diag[1 , , , . . . ,
1] (24)and T rv = πr N − (cid:16) N − N − (cid:17) ∂M∂v − N − k +1 ∂C∂v r ( N − k − if k = − / ( N − , πr N − (cid:16) N − N − (cid:17) ∂M∂v − r N − ∂C∂v ln r if k = − / ( N − . (25) Here, M(v) and C(v) are arbitrary functions of v , arising as integration constants, whose values depend on theboundary conditions and the fundamental constants of the underlying matter. III. ENERGY CONDITIONS
The family of solutions discussed here, in general, belongs to Type II fluid defined in [28]. When m = m ( r ), wehave µ =0, and the matter field degenerates to type I fluid [16]. In the rest frame associated with the observer, theenergy-density of the matter will be given by (assuming Λ = 0), µ = T rv , ρ = − T tt = − T rr = − C ( v ) r ( N − − k ) , (26)and the principal pressures are P i = T ii (no sum convention). Therefore P r = T rr = − ρ and P θ = kP r = − kρ (hypothesis ( iii )). a) The weak energy conditions (WEC): The energy momentum tensor obeys inequality T ab w a w b ≥ µ ≥ , ρ ≥ , P θ = P θ = . . . = P θ ( N − ≥ . (27)We say that strong energy condition (SEC), holds for Type II fluid if, Eq. (27) is true., i.e., both WEC and SEC, fora Type II fluid, are identical. b) The dominant energy conditions : For any timelike vector w a , T ab w a w b ≥
0, and T ab w a is non-spacelike vector,i.e., µ ≥ , ρ ≥ P θ , P θ = . . . = P θ ( N − ≥ . (28)Clearly, ( a ) is satisfied if C ( v ) ≤ , k ≤
0. However, µ > M ( v )and C ( v ). From Eq. (4), ( k = − / ( n − , we observe µ > πr N − (cid:18) N − N − (cid:19) ∂M∂v − N − k + 1 ∂C∂v r ( N − k − > . (29)This, in general, is satisfied, if (cid:18) N − N − (cid:19) ∂M∂v > , and, either ∂C∂v > k < − / ( N − , or ∂C∂v < k > − / ( N − . (30)On the other hand, for k = − / ( N − µ ≥ ∂M /∂C ≥ π (cid:16) N − N − (cid:17) ln r . The DEC holds if C ( v ) ≤ − ≤ k ≤
0, and the function M is subject to the condition (30). Clearly, 0 ≤ − k ≤ IV. SINGULARITY AND HORIZONS
The invariants are regular everywhere except at the origin r = 0, where they diverge. Hence, the space-time hasthe scalar polynomial singularity [28] at r = 0. The nature (a naked singularity or a black hole) of the singularitycan be characterized by the existence of radial null geodesics emerging from the singularity. The singularity is atleast locally naked if there exist such geodesics, and if no such geodesics exist, it is a black hole. The study of causalstructure of the space-time is beyond the scope of this paper and will be discussed elsewhere.In order to further discuss the physical nature of our solutions, we introduce their kinematical parameters. FollowingYork [35] a null-vector decomposition of the metric (1) is made of the form g ab = − n a l b − l a n b + γ ab , (31)where, n a = δ va , l a = 12 (cid:20) − m ( v, r )( N − r N − (cid:21) δ va + δ ra , (32a) γ ab = r δ θ a δ θ b + r i − Y j =1 sin ( θ j ) δ θ i a δ θ i b , (32b) l a l a = n a n a = 0 l a n a = − ,l a γ ab = 0; γ ab n b = 0 , (32c)with m ( v, r ) given by Eq. (2). The optical behavior of null geodesics congruences is governed by the Carter [36] formof the Raychaudhuri equation d Θ dv = K Θ − R ab l a l b − ( γ cc ) − Θ − σ ab σ ab + ω ab ω ab , (33)with expansion Θ, twist ω , shear σ , and surface gravity K . Here R ab is the N -dimensional Ricci tensor, γ cc is the traceof the projection tensor for null geodesics. The expansion of the null rays parameterized by v is given byΘ = ∇ a l a − K , (34)where the ∇ is the covariant derivative. In the present case, σ = ω = 0 [35], and the surface gravity is, K = − n a l b ∇ b l a . (35)Spherically symmetric irrotational space-times, such as under consideration, are vorticity and shear free. The structureand dynamics of the horizons are then only dependent on the expansion, Θ. As demonstrated by York [35], horizonscan be obtained by noting that (i) apparent horizons are defined as surface such that Θ ≃ d Θ /dv ≃
0. Substituting Eqs. (23), (32) and (35) into Eq. (34), we get, ( k = − / r (cid:20) − M ( v ) r N − + Q ( v ) r ( N − k − ( N − − χ r (cid:21) , (36)where, M ( v ) = M ( v )( N − , χ = 2Λ( N − N − , Q ( v ) = 16 πC ( v )( N − N − k + 1) . (37)Since the York conditions require that at apparent horizons Θ vanish, it follows form the Eq. (36) that apparenthorizons will satisfy χ r N − − Q r ( N − k +1 − r N − + 2 M ( v ) , = 0 , (38)which in general has two positive solutions. For χ = Q = 0, we have Schwarzschild horizon r = (2 M ) N − , andfor M = Q = 0 we have de Sitter horizon r = 1 /χ . As mentioned above, for k = −
1, one gets Bonnor-Vaidyasolution, in which case the various horizons are identified and analyzed by Mallett [37] and hence, to conserve space,we shall avoid the repetition of same. For general k , as it stands, Eq. (38) will not admit simple closed form solutions.However, for Q = Q c = − ( N − N − k + 1) (cid:20) M (( N − k + 1)( N − N − k − ( N − (cid:21) ( N − − ( N − k ( N − , (39)with χ = 0, the two roots of the Eq. (38) coincide and there is only one horizon r = (cid:20) M (( N − k + 1)( N − N − k − ( N − (cid:21) N − (40)For Q ≤ Q c there are two horizons, namely a cosmological horizon and a black hole horizon. On the other hand if,the inequality is reversed, Q > Q c no horizon would form. V. CONCLUDING REMARKS
In the study of the Einstein equations in the 4D space-time several powerful mathematical tools were developed,based on the space-time symmetry, algebraical structure of space-time, internal symmetry and solution generationtechnique, global analysis, and so on. It would be interesting how to develop some of these methods to higherdimensional space-time. With this as motivation, plus the fact that exact solutions are always desirable and valuable,we have extended to higher dimensional space-time, a recent theorem [18] and it’s trivial extension (that includescosmological term Λ), which, with certain restrictions on the EMT, characterizes a large family of radiating black holesolutions in N-dimensions, representing, in general, spherically symmetric Type II fluid. In particular, the monopole-de Sitter-charged Vaidya and Husain solutions can be generated form our analysis and when n = 4, one recovers the4D black solutions. If M = C = constant, we have µ =0, and the matter field degenerates to type I fluid and we cangenerate static black hole solutions by a proper choice of these constants. Since many known solutions are identifiedas particular case of this family and hence it would be interesting to ask whether there exist realistic matter thatfollows the restrictions of the theorem that would generate a new black hole solution.The solutions depend on one parameter k , and two arbitrary functions M ( v ) and C ( v ) (modulo energy conditions).It is possible to generate various solutions by proper choice of these functions and parameter k . Further, most of theknown static spherically symmetric black hole solutions, in 4D and HD, can be recovered from our analysis.It should be interesting to apply these metrics to study the gravitation collapse and naked singularities formation.Finally, the result obtained would also be relevant in the context of string theory which is often said to be next ”theoryof everything” and in the study of gravitational collapse. Acknowledgments
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