Radiating fluid sphere immersed in an anisotropic atmosphere

Transcription

1 Geneal Relativity and Gavitation manuscipt No. (will be inseted by the edito) Radiating fluid sphee immesed in an anisotopic atmosphee N. F. Naidu, M. Govende, S. Thiukkanesh, S. D. Mahaaj the date of eceipt and acceptance should be inseted late axiv: v1 [g-qc] 7 Sep 2017 Abstact We model a adiating sta undegoing dissipative gavitational collapse in the fom of adial heat flux. The exteio of the collapsing sta is descibed by the genealised Vaidya solution epesenting a mixtue of null adiation and stings. Ou model genealises peviously known esults of constant sting density atmosphee to include inhomogeneities in the exteio spacetime. By utilising a causal heat tanspot equation of the Maxwell-Cattaneo fom we show that elaxational effects ae enhanced in the pesence of inhomogeneities due to the sting density. 1 Intoduction Oppenheime and Snyde [31] pioneeed eseach and inteest in gavitational collapse with the examination of a dust sphee undegoing collapse. Spacetime singulaities ae egions in spacetime whee densities and spacetime cuvatues divege. They ae contained within geneal elativity, as a consequence of the mathematics involved in the theoy. The Cosmic Censoship Conjectue [32], N. F. Naidu, S. D. Mahaaj Astophysics and Cosmology Reseach Unit, School of Mathematics, Statistics and Compute Science, Univesity of KwaZulu-Natal, Pivate Bag X54001, Duban 4000, South Afica. nolene.naidu@physics.og mahaaj@ukzn.ac.za M. Govende Depatment of Mathematics, Faculty of Applied Sciences, Duban Univesity of Technology, Duban, 4000, South Afica. megandheng@dut.ac.za S. Thiukkanesh Depatment of Mathematics, Easten Univesity, Chenkalady, Si Lanka. thiukkanesh@yahoo.co.uk

2 2 N. F. Naidu, M. Govende, S. Thiukkanesh, S. D. Mahaaj states that any easonable matte distibution that undegoes continued gavitational collapse will always fom a black hole. Howeve, thee ae seveal factos that can influence the outcome of such collapse. Exceptions have been poposed by eseaches since inteest in the study of late stage collapse has peaked [17,20,18,19]. In 1951, Vaidya discoveed a nonstatic solution of the Einstein field equations which descibes the atmosphee of a adiating sta [36]. Late, Santos deived the junction conditions to match any type of stella inteio to a adiating Vaidya exteio [34]. To date, thee have been many models of gavitational collapse unde seveal sets of initial conditions and physical factos, deived fom exact solutions to the Einstein field equations and appopiate vesions of Santos matching conditions. The Santos junction conditions wee genealised to include the effects of an electomagnetic field and sheaing anisotopic stesses duing dissipative stella collapse by de Oliveia and Santos [6], and Mahaaj and Govende [22]. Pinheio and Chan [33] examined shea-fee nonadiabatic collapse in the pesence of electic chage. The influence of pessue anisotopy, shea and bulk viscosity wee also studied by Chan [3,4]. Abebe et al. [1,2] employed Lie symmeties to investigate the behaviou of adiating stas in confomally flat spacetime manifolds. Mahaaj and Govende [23] studied gavitational collapse with isotopic pessue and vanishing Weyl stesses and showed that the stella coe was moe unstable than the oute egions. Matinez [26], Heea and Santos [13], Naidu et al. [29], and Naidu and Govende [30], focussed on the themodynamics of adiating stas and the impotance of both the elaxation and mean collision time. Mahaaj et al. [24] showed the impact of the genealised Vaidya adiating metic on the junction conditions on the bounday of a adiating sta. Thei esults descibe a moe geneal atmosphee suounding the sta, which is a supeposition of the pessueless null dust and a sting fluid. The sting density was shown to affect the fluid pessue at the suface of the sta. It was demonstated that the sting density educes the pessue on the stella bounday. The usual junction conditions fo the Vaidya spacetime ae egained in the absence of the sting fluid. Although seveal applications of ievesible themodynamics in geneal elativity to date have employed Eckat theoy, thee ae seveal shotcomings to this appoach. In this theoy, if a themodynamic foce is suddenly set equal to zeo, then the coesponding themodynamic flux vanishes instantaneously. This is a violation of elativistic causality, since it implies that the signal would popagate though the fluid at an infinite speed. This led to the development of causal theoies of dissipative fluids (both elativistic and non-elativistic). Thee ae seveal advantages to taking a causal appoach: (i) Thee ae causal popagations of dissipative signals fo stable fluid configuations, (ii) Thee is no geneic shot-wavelength secula instability in causal theoies, and (iii) The petubations have a easonably posed initial value poblem, including the

3 Radiating fluid sphee immesed in an anisotopic atmosphee 3 case of otating fluids. Causal theoies extend the space of vaiables of conventional theoies by incopoating the dissipative quantities concened (such as heat flux, paticle cuents, shea and bulk stesses). These physical quantities ae teated simila to the conseved vaiables (such as enegy density, paticle numbes, etc.). This leads to a moe compehensive theoy with a lage numbe of vaiables and paametes [21, 11, 12]. The layout of this pape is as follows: In 2, we pesent the inteio spacetime of a adiating sta. In 3, we focus on the exteio atmosphee. In 4, we list the junction conditions fo the smooth matching of the exteio and inteio spacetimes. In 5, we solve the junction condition and povide seveal classes of new exact solutions. Finally, ou esults ae discussed in 6. 2 Confomally flat inteio The inteio spacetime of the fluid sphee is epesented by a spheically symmetic, shea-fee, confomally flat line element ( ds 2 C1 (t) 2 ) 2 +1 = C 2 (t) 2 dt 2 +C 3 (t) ( ) C 2 (t) 2 [d dω 2 ], (1) +C 3 (t) wheedω 2 = dθ 2 +sin 2 θdφ 2 andc 1 (t), C 2 (t) andc 3 (t) aetempoalfunctions yet to be detemined [14]. The matte content of the inteio stella egion is descibed by an enegy momentum tenso of a pefect fluid with heat flux T ab = (µ+p)u a u b +pg ab +q a u b +q b u a, (2) whee the enegy density is epesented by µ, p epesents the pessue and the magnitude of the heat flux is q = (q a q a ) 2. 1 The comoving fluid fou velocity u has the fom u a = C 2(t) 2 +C 3 (t) C 1 (t) 2 δ a (3) With the assumption that heat flow is in the adial diection and q a u a = 0, the heat flow vecto is given by q a = (0,q,0,0). (4) The fluid collapse ate is given by Θ = u a ;a. We calculate this fo ou fluid sphee as C Θ = C 3 C 1 (t) (5) The line element (1) has been extensively utilised by vaious authos to study dissipative collapse in which the Weyl stesses vanish within the stella inteio

4 4 N. F. Naidu, M. Govende, S. Thiukkanesh, S. D. Mahaaj [14,23,15,27]. In these investigations the line element is matched to Vaidya s outgoing solution which descibes an isotopic and homogeneous atmosphee. The Weyl-fee collapse model was subsequently utilised to model a adiating sta with a twofluid atmosphee [10]. The atmosphee consisted of a mixtue of stings and null adiation. The density of the stings was assumed to be constant making the junction conditions mathematically tactable. When the sting density vanishes the two-fluid atmosphee model is ecoveed [23]. Although the assumption of constant sting density is highly simplified, it was shown that the pesence of stings in the extenal spacetime leads to highe coe tempeatues. In this pape we elax the assumption of constant sting density to incopoate inhomogeneity and anisotopy in the sting density. The bounday condition equied fo the smooth matching of the inteio spacetime to the extenal genealised Vaidya solution is solved to poduce an exact model of a adiating sta with an inhomogeneous atmosphee. In the case of constant sting density, ou model educes to the solution studied by Govende [10]. Fo the line element (1) the Einstein field equations give µ = 3 p = (Ċ2 2 +Ċ3 C ) 2 +12C 2 C 3, (6) 1 [ (C ) 2 2( C C 3 )(C 2 2 +C 3 ) Ċ 1 3(Ċ2 2 +Ċ3) 2 2 C (Ċ2 2 +Ċ3) (C 2 2 +C 3 ) 2] + 4 C [C 2 (C 2 2C 1 C 3 ) 2 +C 3 (C 1 C 3 2C 2 )], (7) ( C2 2 ) 2 +C 3 q = 4(Ċ3C 1 Ċ2) C 1 2. (8) +1 The tempoal behaviou of the metic functions will be taken up in the next section. 3 Exteio atmosphee The exteio spacetime is taken to be the genealised Vaidya solution which epesents a two-fluid atmosphee made up of null adiation and a sting fluid [7] ( ds 2 = 1 2 m(v,) ) dv 2 2dvd + 2 (dθ 2 +sin 2 θdφ 2 ), (9)

5 Radiating fluid sphee immesed in an anisotopic atmosphee 5 whee m(v, ) is the mass function which epesents the gavitational enegy within a sphee of adius. The genealised Vaidya solution has been extensively studied within the context of gavitational collapse and Cosmic Censoship. Dawood and Ghosh [5] found a class of nonstatic black hole solutions fo a type-two fluid using the line element (9). They futhe showed that the end state of continued gavitational collapse is sensitive to the initial matte configuation. Recently, Mkenyeleye et al. [28] studied the end states esulting fom the gavitational collapse of a genealised Vaidya cloud. They showed that naked singulaities ae stable end states esulting fom the collapse of a two-fluid composite such as null adiation and stings. The enegy momentum tenso consistent with the line element (9) is T + ab = µl al b +(ρ+p)(l a n b +l b n a )+Pg ab, (10) whee we have intoduced two null vectos l a = δ 0 a, (11) n a = 1 2 [ 1 2 m(v,) ] δa 0 +δ 1 a, (12) and l a l a = n a n a = 0 and l a n a = 1. The above enegy momentum tenso descibes a composite atmosphee made up of a supeposition of pessueless null dust and a null sting fluid [16,37]. In (10) we intepet µ as the enegy density of the null dust adiation, ρ as the null sting enegy density, and P as the null sting pessue. It is assumed that the stings diffuse, and that sting diffusion is simila to point paticle diffusion, i.e. that the numbe density diffuses fom highe to lowe numbes. Within this context, Glass and Kisch [8] povided the following density pofiles ρ = ρ 0 +k 1 /, (13) ρ = ρ 0 +k 3 v 3/2 exp[ 2 /(4Dv)], (14) ρ = ρ 0 +(k 4 /)(1+( π/2)ef[(4dv) 1/2 ]), (15) whee ρ 0 is a constant. The fist type epesents fee steaming adiation, the second epesents diffusion and the thid epesents adiation. The adial pessue within the Vaidya atmosphee can be attibuted to the sting tension. In the standad Vaidya envelope with photons caying enegy away fom the stella coe, Santos [34] was able to show that the adial pessue is popotional to the heat flux at the bounday of the adiating sta. This bounday condition detemines the tempoal evolution of the model. With the genealised Vaidya atmosphee it is the photons that cay enegy to the exteio spacetime with the stings diffusing inwads. Ou intention is to detemine

6 6 N. F. Naidu, M. Govende, S. Thiukkanesh, S. D. Mahaaj the effect of the sting flux on physical paametes such as tempeatue and luminosity. 4 Matching conditions Fo a complete model of a adiating sta the line element (1) is matched smoothly to the exteio spacetime (9) acoss a time-like hypesuface. The eade is efeed to Mahaaj et al. [25] fo moe insight into the junction conditions. The mass pofile of the collapsing sphee is given by [ ] m(v,) = 3 4(1+C 1 2 ) 2 C 2 C 3 +(Ċ2 2 +Ċ3) 2 2(1+C 1 2 )(C 2 2 +C 3 ) 3 Σ, (16) which is the total gavitational enegy contained within the stella suface Σ. The continuity of the momentum flux acoss the bounday of the sta yields [ ] q p = C 2 (t) 2 +C 3 (t) ρ, (17) Σ which genealises the esults obtained by Santos [34]. It is clea fom (17) that the pessue at the bounday of the collapsing sta depends on the magnitude of the heat flux q and the exteio sting density ρ. Fo ou line element (1) and the assumption of vanishing Weyl stesses, (17) educes to the nonlinea equation C 2 b 2 + C 3 3 ( C 2 b 2 + C 3 ) 2 C 2 C 2 b 2 1 b 2 ( C 2 b 2 + C 3 ) +C 3 C 1 b (C 1 C 3 C 2 )b+2 (C 1b 2 +1) C 2 b 2 [C 2 (C 2 2C 1 C 3 )b 2 +C 3 +C 3 (C 1 C 3 2C 2 )] + 1 (C 1 b 2 +1) 2 ( C 2 b 2 ) +C 3 2 C 2 b 2 ρ 0 +k 1 = 0, +C 3 b (18) whee = b detemines the bounday of the sta. In the above we have utilised the sting density given in (13). The sting density chosen genealises pevious attempts at modeling adiating stas with a genealised Vaidya atmosphee. Ealie woks adopted a constant sting density pofile which made the junction condition (18) mathematically tactable. By setting k 1 = 0 we obtain peviously known esults fo constant density sting atmosphee [25]. We should also point out a subtle featue of the junction condition which is highlighted fo the fist time hee. Recall that (17) is evaluated at the bounday ( = Σ ).

7 Radiating fluid sphee immesed in an anisotopic atmosphee 7 In addition, the continuity of the metic functions connects the inteio adial coodinate to the exteio adial coodinate by ( ) C 2 (t) 2 = Σ. (19) +C 3 (t) Hence ou choice of the sting density pofile ρ = ρ 0 +k 1 / implies that Σ ρ = ρ 0 + k 1(C 2 (t) 2 +C 3 (t), (20) at the bounday. This means that the sting density pofile (20) genealises the constant sting density pofile to include anisotopy and inhomogeneity in the exteio since ρ = ρ(,t) at the bounday. This shows a diect link between the inteio and exteio matte vaiables which is absent fo the usual Vaidya metic with m = m(v). 5 Exact solution In ode to geneate exact solutions of (18) we adopt the appoach taken by Thiukkanesh et al. [35]. If we take U(t) = C 1 (t)b 2 +1, (21) then the govening equation can be witten as [ ] ( C 2 b 2 + C 3 ) U C 3 +2 b (C2 2 b4 C3 2) b 2 (C 2 b 2 U 2 +C 3 ) [ 3( C 2 b 2 + ] C 3 ) 2 2 C 2 b 2 2 +C 3 b ( C 2 b 2 + C 3 ) ( C 2 b 2 + C 3 ) U [ 1 2(C 2 b 2 4(C 3 2C 2 b 2 ) C 3 +C 3 ) b 2 +ρ 0 + k ] 1 b (C 2b 2 +C 3 ) U 3 = 0. (22) This equation has the geneic stuctue whee we have set A U +BU +CU 2 +DU 3 = 0, (23) A = C 2 b 2 + C 3, B = 3 ( C 2 b 2 + C 3 ) 2 2 C 2 b 2 2 +C 3 b ( C 2 b 2 + C 3 ) ( C 2 b 2 + C 3 ),

8 8 N. F. Naidu, M. Govende, S. Thiukkanesh, S. D. Mahaaj [ ] C 3 C = 2 b (C2 2b 4 C3) 2 b 2 (C 2 b 2, +C 3 ) [ 1 D = 2(C 2 b 2 4(C 3 2C 2 b 2 ) C 3 +C 3 ) b 2 +ρ 0 + k 1 b (C 2b 2 +C 3 ) ]. Equation (23) is an Abelian equation of the fist kind in the vaiable U. We have the following class of solution ( ) C 1 = 1 e 2t/b ż b 2 z 3/2[ K λ ] 1, 1/2 whee (24) 2 C 3 b± C 3 b2 4C 3 (C 3 + C 3 b) C 2 = 2b 2, (25) C 3 = abitay function, (26) λ = e4t/b ż ( 4(C 3 2C 2 b 2 ) C3 b 2 +ρ 0 + k1 b (z)) z 4, z = C 2 b 2 +C 3, (27) and K is a constant. It is inteesting to note that this class of solution educes to a paticula categoy of the Misthy et al. [27] solution in the limit of vanishing sting density (i.e. ρ 0 = k 1 = 0). 6 Physical analysis Ou aim is to detemine the influence of an inhomogeneous and anisotopic atmosphee on the tempeatue distibution of the collapsing body. By utilizing a causal heat tanspot equation, it was shown that tempeatue is inceased at each inteio point of the collapsing sta when the atmosphee is composed of a mixtue of pue adiation and stings with unifom density [10]. The tuncated causal tanspot equation fo the line element (1) is given by [10] ( q ) β C 2 2 +C T σ + q(c ) 3 (C 2 2 +C 3 ) 2 ( )( α (C1 2 ) +1)T = C 2 2 +C 3 C 2 2 T 3 σ, (28) +C 3 fo the physically motivated choices of the themal conductivity κ, the mean collision time τ c, and the elaxation time τ: ( ) ( ) α βγ κ = γt 3 τ c, τ c = T σ, τ = τ c. (29) γ α

9 Radiating fluid sphee immesed in an anisotopic atmosphee 9 The quantities α 0, β 0, γ 0 and σ 0 ae constants. Fo the special case of constant collision time which coesponds to σ = 0, it is possible to integate (28) [9]. The noncasual heat tanspot equation is ecoveed when we set β = 0 in (28). In ode to investigate the behaviou of the tempeatue pofiles in the pesence of a cloud of anisotopic and inhomogeneous stings and null adiation we need to specify the abitay tempoal functions. To this end we choose C 3 (t) = ae bt in (24) (26) which ensues that the function C 3 (t) is continuous and fee of singulaities. With this choice, elation (25) implies that C 2 (t) = C 3 (t) and since C 1 (t) is defined in tems of C 2 (t), C 3 (t) and thei deivatives, the tempoal behaviou of ou model is now fully specified. We ae now in a position to obtain the causal tempeatue pofile fo the collapsing body. Fig. 1 shows the Eckat tempeatue distibution as a function of the adial coodinate fo both the isotopic sting density (ρ = constant) and the nonunifom, inhomogeneous sting atmosphee. Fig. 1 indicates that the tempeatue pofiles ae almost identical duing the ealy stages of collapse. This is expected as the sta is in quasi-static equilibium and the anisotopy and inhomogeneity in the young atmosphee ae indistinguishable fom an isotopic and homogeneous atmosphee. As the collapse poceeds, anisotopic effects and density inhomogeneities become moe pominent in the atmosphee. The esult of this is a eduction of heat tansfe to the exteio spacetime which esults in a highe coe tempeatue. We expect the cental tempeatue to be significantly diffeent to the suface tempeatue as the suface layes of the sta ae much coole than the cental egion. In Fig. 2 we have plotted the causal tempeatue pofile anging fom the cente of the sta to the bounday. We obseve that the causal tempeatue pofile fo the nonconstant density pofile is highe at each inteio point when compaed to the unifom density pofile. This diffeence is much highe at the coe than at the suface layes. In addition, the diffeence in the causal tempeatue pofiles is moe accentuated than thei noncausal countepats close to the cente of the sta. In obseving Fig. 1 and Fig. 2, we must bea in mind that that noncausal tempeatue is calculated fo β = 0, when the sta is close to equilibium (ealy times). The causal tempeatue is obtained fo lage values of β which coespond to late times of the collapse pocess. The causality index β is intimately linked to the tempoal evolution of the collapsing body. In ode to detemine the impact of the inhomogeneity we have plotted the atio of the tempeatue (T) aising fom nonunifom density to the tempeatue (T ) esulting fom a unifom sting density. The esults ae plotted in Fig. 3 (noncausal tempeatue) and Fig. 4 (causal tempeatue). We obseve that inhomogeneity effects on the tempeatue pofiles ae much moe maked at the coe compaed to the coole suface layes. Fig. 3 shows that the atio of T/T is close to unity thoughout the collapsing body. We expect this since

10 10 N. F. Naidu, M. Govende, S. Thiukkanesh, S. D. Mahaaj 2.25 Tempeatue Fig. 1 Non-causal tempeatue pofile fo constant density (solid line) and non-constant density (dashed line), with β = 0, A 0 = 10, 0 = 1, b = 10, K = , ρ 0 = 1, t = 10, and k 1 = 0 fo non-constant density. duing this epoch inhomogeneity effects in the exteio ae small. As the collapse poceeds, thee will be a steady flux of adiation fom the coe to the exteio thus enhancing inhomogeneity in the atmosphee. Fig. 4 shows the deviation of T/T fom unity. In this wok we have modeled dissipative collapse of a adiating sta in which the atmosphee is composed of null adiation and anisotopic sting distibution. We wee able to integate the bounday condition fo a nonunifom sting density thus genealising pevious investigations involving constant sting densities. We showed that the nonunifom sting density pesent in the atmosphee leads to highe coe tempeatues with the enhancement being moe ponounced close to the cental egions of the collapsing body. A natual extension of this investigation is to include sheaing effects within the collapsing coe and to detemine the subsequent evolution of the tempeatue pofiles of the adiating sta. Acknowledgements The authos ae gateful to the National Reseach Foundation (NRF) and both the Duban Institute of Technology and Univesity of KwaZulu-Natal. SDM acknowledges that this eseach is suppoted by the South Afican Reseach Chai Initiative of the Depatment of Science and Technology and the NRF. Refeences 1. Abebe, G.Z., Mahaaj, S.D., Govinde K.S.: Gen. Relativ. Gav. 46, 1650 (2014) 2. Abebe, G.Z., Mahaaj, S.D., Govinde, K.S.: Gen. Relativ. Gav. 46, 1733 (2014) 3. Chan, R.: Mon. Not. R. Aston. Soc. 288, 589 (1997) 4. Chan, R.: Aston. Astophys. 368, 325 (2001) 5. Dawood, A.K., Ghosh, S.G.: Phys. Rev. D 70, (2004) 6. De Oliveia, A.K.G., Santos, N.O.: Astophys. J. 312, 640 (1987)

11 Radiating fluid sphee immesed in an anisotopic atmosphee Tempeatue Fig. 2 Causal tempeatue pofile fo constant density (solid line) and non-constant density (dashed line), with β = 10 6, A 0 = 10, 0 = 1, b = 10, K = , ρ 0 = 1, t = 10 5, and k 1 = fo non-constant density NoncausalT/T * Fig. 3 Ratio of Non-causal tempeatue pofile: non-constant density ove constant density with t = CausalT/T * Fig. 4 Ratio of Causal tempeatue pofile: non-constant density ove constant density with t = 10.

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