Hindawi Publishing Corporation Journal of Gravity Volume 2013, Article ID 659605, 8 pages http://dx.doi.org/10.1155/2013/659605

Research Article Collapse of a Relativistic Self-Gravitating Star with Radial Heat Flux: Impact of Anisotropic Stresses

Ranjan Sharma and Shyam Das

Department of Physics, P.D. Women’s College, Jalpaiguri, West Bengal 735101, India

Correspondence should be addressed to Ranjan Sharma; [email protected]

Received 28 February 2013; Accepted 22 April 2013

Academic Editor: Sergei Odintsov

Copyright © 2013 R. Sharma and S. Das. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We develop a simple model for a self-gravitating spherically symmetric relativistic star which begins to collapse from an initially static configuration by dissipating energy in the form of radial heat flow. We utilize the model to show how local anisotropy affects the collapse rate and thermal behavior of gravitationally evolving systems.

1. Introduction and Santos [5] formulated the junction conditions joining the interior space time of the collapsing object to the Vaidya In cosmology and astrophysics, there exist many outstanding exterior metric [4]. These developments have enabled many issues relating to a dynamical system collapsing under the investigators to construct realistic models of gravitationally influence of its own gravity. In view of Cosmic Censorship evolving systems and also to analyze critically relevance of Conjecture, the general relativistic prediction is that such a various factors such as shear, density inhomogeneity, local collapse must terminate into a space-time singularity covered anisotropy, electromagnetic field, viscosity, and so forth, on under its event horizon though there are several counter the physical behaviour of collapsing bodies [6–52]. In the examples where it has been shown that a naked singularity absence of any established theory governing gravitational is more likely to be formed (see [1] and references therein). collapse, such investigations have been found to be very use- In astrophysics, the end stage of a massive collapsing star ful to get a proper understanding about systems undergoing has long been very much speculative in nature [1, 2]. From gravitational collapse. classical gravity perspective, to get a proper understanding of The aim of the present work is to develop a simple the nature of collapse and physical behavior of a collapsing model of a collapsing star and investigate the impact of system, construction of a realistic model of the collapsing pressure anisotropy on the overall behaviour of the collapsing system is necessary. This, however, turns out to be a difficult body. Anisotropic stresses may occur in astrophysical objects task because of the highly nonlinear nature of the governing for various reasons which include phase transition, density field equations. To reduce the complexity, various simplifying inhomogeneity, shear, and electromagnetic field10 [ , 53, 54]. methods are often adopted and the pioneering work of In [53],ithasbeenshownthatinfluencesofshear,elec- Oppenheimer and Snyder [3] was a first step in this direction tromagnetic field, and so forth on self-bound systems can when collapse of a highly idealized spherically symmetric be absorbed if the system is considered to be anisotropic, dust cloud was studied. Since then, various attempts have in general. Local anisotropy has been a well-motivated been made to develop realistic models of gravitationally factor in the studies of astrophysical objects and its role on collapsing systems to understand the nature and properties the gross features of static stellar configurations have been of collapsing objects. It got a tremendous impetus when investigated by many authors (see, e.g., [10, 11, 55–59]and Vaidya [4] presented a solution describing the exterior references therein). For dynamical systems, though pressure gravitational field of a stellar body with outgoing radiation anisotropy is, in general, incorporated in the construction, 2 Journal of Gravity

0 1 2 very few works have been reported till date where impacts of self-gravity (in standard coordinates 𝑥 =𝑡, 𝑥 =𝑟, 𝑥 =𝜃, 3 of anisotropy have been discussed explicitly [12–14, 26, 60]. and 𝑥 =𝜙)as Appropriate junction conditions for an anisotropic fluid collapsing on the background space time described by the 2 2 2 𝑑𝑠− =−𝐴0 (𝑟) 𝑑𝑡 Vaidya metric have been obtained in [27]. Considering a (1) spheroidal geometry of Vaidya and Tikekar [61], Sarwe and 2 2 2 2 2 2 2 +𝑓 (𝑡) [𝐵0 (𝑟) 𝑑𝑟 +𝑟 (𝑑𝜃 + sin 𝜃𝑑𝜙 )] , Tikekar [50] have analyzed the impact of geometry vis-a-vis matter composition on the collapse of stellar bodies which 𝐴 (𝑟) 𝐵 (𝑟) 𝑓(𝑡) begin to collapse from initial static configurations possess- where, 0 , 0 ,and are yet to be determined. Note 𝑓(𝑡) =1 ing equal compactness. Sharma and Tikekar [51, 52]have that in (1), if we set , the metric corresponds to a static investigated the evolution of nonadiabatic collapse of a shear- spherically symmetric configuration. free spherically symmetric object with anisotropic stresses We assume that the matter distribution of the collaps- on the background of space-time obtained by introducing ing object is an imperfect fluid described by an energy- an inhomogeneous perturbation in the Robertson-Walker momentum tensor of the form space-time. 𝑇 =(𝜌+𝑝)𝑢 𝑢 +𝑝𝑔 +(𝑝 −𝑝)𝜒 𝜒 Inthepresentwork,wehavedevelopedamodeldescrib- 𝛼𝛽 𝑡 𝛼 𝛽 𝑡 𝛼𝛽 𝑟 𝑡 𝛼 𝛽 (2) ing a shear-free spherically symmetric fluid distribution radi- +𝑞 𝑢 +𝑞 𝑢 , ating away its energy in the form of radial heat flux. The 𝛼 𝛽 𝛽 𝛼 starbeginsitscollapsefromaninitiallystaticconfiguration whose energy-momentum tensor describing the material where 𝜌 represents the energy density 𝑝𝑟 and 𝑝𝑡,respectively, denote fluid pressures along the radial and transverse direc- composition has been assumed to be anisotropic, in general. 𝛼 𝛼 To develop the model of the initial static star, we have utilized tions; 𝑢 is the 4-velocity of the fluid; 𝜒 is a unit space-like 𝛼 1 theFinchandSkea[62]ansatzwhichhasearlierbeenfound 4-vector along the radial direction; and 𝑞 =(0,𝑞,0,0)is the to be useful to develop physically acceptable models capable heat flux vector which is orthogonal to the velocity vector so 𝑢𝛼𝑢 =−1 𝑢𝛼𝑞 =0 of describing realistic stars in equilibrium [63–66]. The back that 𝛼 and 𝛼 . ground space-time for a static configuration for the given The Einstein’s field equations governing the evolution of ansatz has a clear geometrical interpretation as may be found the system are then obtained as (we set 𝐺=𝑐=1) in [65]. By assuming a particular form of the anisotropic parameter, we have solved the relevant field equations and 1 1 1 2𝐵󸀠 3𝑓2̇ 8𝜋𝜌 = [ − + 0 ]+ , constructedamodelfortheinitialstaticstellarconfiguration 𝑓2 𝑟2 𝑟2𝐵2 𝑟𝐵3 𝐴2 𝑓2 (3) which could either be isotropic or anisotropic in nature. The 0 0 0 solution provided by Finch and Skea [62] is a subclass of 1 1 1 2𝐴󸀠 1 𝑓̈ 𝑓2̇ 8𝜋𝑝 = [− + + 0 ]− [2 + ], the solution provided here. Since, the solution presented here 𝑟 2 2 2 2 2 2 2 𝑓 𝑟 𝐵 𝑟 𝑟𝐴 0𝐵 𝐴 𝑓 𝑓 provides a wider range of values of the anisotropic parameter, 0 0 0 (4) it enables us to examine the impact of anisotropic stresses on the evolution of a large class of initial static configurations. 1 𝐴󸀠󸀠 𝐴󸀠 𝐵󸀠 𝐴󸀠 𝐵󸀠 8𝜋𝑝 = [ 0 + 0 − 0 − 0 0 ] Our paper has been organized as follows. In Section 2, 𝑡 2 2 2 3 3 𝑓 𝐴0𝐵0 𝑟𝐴 0𝐵0 𝑟𝐵0 𝐴0𝐵0 we have presented the basic equations governing the system (5) undergoing non-adiabatic radiative collapse. In Section 3,by 1 𝑓̈ 𝑓2̇ − [2 + ], assuming a particular anisotropic profile, we have solved the 2 2 𝐴0 𝑓 𝑓 relevant field equations to develop a model for the initial static star. In Section 4, by stipulating the boundary conditions 2𝐴󸀠 𝑓̇ 8𝜋𝑞1 =− 0 . across the surface separating the stellar configuration from 2 2 3 (6) 𝐴0𝐵0𝑓 the Vaidya [4] space-time, we have solved the surface equa- tion which governs the evolution of the initial static star that In (3)–(6), a “prime” denotes differentiation with respect to 𝑟 begins to collapse when the equilibrium is lost. In Section 5, 𝑡 wehaveanalyzedtheimpactofanisotropyongravitationally and a “dot” denotes differentiation with respect to .Making collapsing systems by considering evolution of initial static use of (4)and(5), we define the anisotropic parameter of the stars which could either be isotropic or anisotropic. Impact of collapsing object as anisotropy on the evolution of temperature has been analyzed in Section 6. Finally, some concluding remarks have been Δ (𝑟,) 𝑡 =8𝜋(𝑝𝑟 −𝑝𝑡) made in Section 7. 1 𝐴󸀠󸀠 𝐴󸀠 𝐵󸀠 𝐴󸀠 𝐵󸀠 = [− 0 + 0 + 0 + 0 0 𝑓2 𝐴 𝐵2 𝑟𝐴 𝐵2 𝑟𝐵3 𝐴 𝐵3 2. Equations Governing the Collapsing System 0 0 0 0 0 0 0 (7) 1 1 We write the line element describing the interior space-time + − ]. 2 2 2 of a spherically symmetric star collapsing under the influence 𝑟 𝐵0 𝑟 Journal of Gravity 3

We rewrite (3)–(5)as where 𝛼 is the measure of anisotropy. The motivation for choosing the particular form of the anisotropy parameter is 8𝜋𝜌 3𝑓2̇ 8𝜋𝜌 = 𝑠 + , the following: (1) it is physically reasonable as the anisotropy 2 2 2 (8) 𝑓 𝐴0𝑓 vanishes at the center (𝑟=0, i.e., 𝑥=1) as expected, and ̈ 2̇ (2) it provides a solution of (13)inclosedform.Notethat 8𝜋(𝑝𝑟) 1 𝑓 𝑓 𝛼=0 8𝜋𝑝 = 𝑠 − [2 + ], corresponds to an initial static star which is isotropic 𝑟 2 2 2 (9) 𝑓 𝐴0 𝑓 𝑓 in nature. Substituting (15)in(13), we get 2 8𝜋(𝑝 ) 1 𝑓̈ 𝑓2̇ 𝑑2𝐴 2 𝑑𝐴 𝛼(𝑥 −2) 8𝜋𝑝 = 𝑡 𝑠 − [2 + ], 0 − 0 +[1− ]𝐴 =0, 𝑡 2 2 2 (10) 2 2 0 (16) 𝑓 𝐴0 𝑓 𝑓 𝑑𝑥 𝑥 𝑑𝑥 𝑥 where 𝜌𝑠, (𝑝𝑟)𝑠 and (𝑝𝑡)𝑠 denote the energy-density, radial whose solution is obtained as pressure, and tangential pressure, respectively, of the initial 𝐴 (𝑥) =𝑃𝑥3/2𝐽 (−𝑖𝑥√−1 + 𝛼) static star. 0 (1/2)√9−8𝛼 (17) 3/2 √ +𝑄𝑥 𝑌 √ (−𝑖𝑥 −1 + 𝛼) , 3. Interior Space-Time of the Initially (1/2) 9−8𝛼 Static Configuration where 𝑃 and 𝑄 are integration constants, √ 𝐽(1/2)√9−8𝛼(−𝑖𝑥 −1 + 𝛼) is the Bessel function of first In our construction, we assume that an initially static star (1/2)√9−8𝛼 𝑌 (−𝑖𝑥√−1 + 𝛼) 𝑓(𝑡) =1 𝐴 (𝑟) kind of order ,and (1/2)√9−8𝛼 is (with in (1)), described by metric potentials 0 , √ 𝐵 (𝑟) Δ (𝑟) the Bessel function of second kind of order (1/2) 9−8𝛼.It 0 ,andanisotropy 𝑠 , starts collapsing if, for some 𝛼<1 𝛼=1 reasons, it loses its equilibrium. To develop a model of the is obvious that the solution is valid for .At ,the Bessel function encounters a singularity and, therefore, we initially static configuration, we first assume that the aniso- 𝛼=1 tropic parameter is separable in its variables so that Δ(𝑟, 𝑡) = shall deal with the case separately. 2 Δ 𝑠(𝑟)/𝑓 (𝑡).Equation(7)thenreducesto Special Cases 󸀠󸀠 󸀠 󸀠 󸀠 󸀠 𝐴0 𝐴0 𝐵0 𝐴0𝐵0 1 1 Δ (𝑟) =[− + + + + − ], Case 1(𝛼=0(Δ 𝑠(𝑟) =). 0 That is, initial static configuration 𝑠 𝐴 𝐵2 𝑟𝐴 𝐵2 𝑟𝐵3 𝐴 𝐵3 𝑟2𝐵2 𝑟2 0 0 0 0 0 0 0 0 is isotropic in nature). Equation (13)reducesto (11) 𝑑2𝐴 2 𝑑𝐴 0 − 0 +𝐴 =0, (18) which is independent of 𝑡.Equation(11) can only be solved if 𝑑𝑥2 𝑥 𝑑𝑥 0 any two of the unknown functions (𝐴0, 𝐵0,andΔ 𝑠(𝑟))are specified. To develop a physically reasonable model of the whose solution is found to be initial static configuration, we first utilize the Finch and Skea 𝐴0 (𝑥) = [(𝐶−𝐷𝑥) cos 𝑥+(𝐶𝑥 +𝐷) sin 𝑥] , (19) [62]ansatzforthemetricpotential𝐵0 given by where, 𝐶 and 𝐷 are integration constants. This particular 𝑟2 2 solution has been found previously in [62]. However, the 𝐵0 (𝑟) =(1+ ), (12) 𝑅2 solution (19) can be obtained directly from the solution (17) 𝛼=0 𝐶=−√2/𝜋𝑄 𝐷= 𝑅 by substituting and setting and where is the curvature parameter describing the geometry √2/𝜋𝑃 of the configuration. In the static case, the ansatz12 ( )has . a clear geometric characterization and has been found to Case 2(𝛼=1(Δ 𝑠(𝑟) =0̸ )). Equation (16)reducesto yield realistic models for compact stellar objects [64–66]. Substituting (12)in(11), we obtain 𝑑2𝐴 2 𝑑𝐴 2 0 − 0 + 𝐴 =0, 2 2 0 (20) 1−𝑥2 𝑑2𝐴 1−𝑥2 𝑑𝐴 𝑑𝑥 𝑥 𝑑𝑥 𝑥 ( ) 0 −2( ) 0 𝑅2𝑥4 𝑑𝑥2 𝑅2𝑥5 𝑑𝑥 whose solution is given by (13) 2 1−𝑥2 𝐴0 (𝑥) =𝐴𝑥+𝐵𝑥 , (21) +[( )−Δ ]𝐴 =0, 2 4 𝑠 0 𝑅 𝑥 where 𝐴 and 𝐵 are integration constants [64]. Unfortunately, the above solution can not be regained from the general where we have used the following transformation: solution (17) due to the properties of Bessel functions and 𝑟2 should be treated separately. 𝑥2 =(1+ ). 𝑅2 (14) We thus have a model for an initially static stellar configu- ration which could either be isotropic or anisotropic in To solve (13), we assume a particular anisotropic profile nature. After loss of equilibrium, the initially static star starts collapsing, and to generate a solution of the subsequent evol- 2 2 𝛼(𝑥 −1)(2−𝑥 ) ving system, we need to determine 𝑓(𝑡).Thiswillbetakenup Δ (𝑥) = , (15) 𝑠 𝑅2𝑥6 in the following section. 4 Journal of Gravity

4. Exterior Space-Time and which admits a solution Junction Conditions 1 𝑓 𝑡= [ + √𝑓+ (1 − √𝑓)] . 𝑛 2 ln (32) In our construction, evolution of the collapsing object is governed by the function 𝑓(𝑡) which can be determined from Note that at 𝑡→−∞, that is, at the onset of collapse, 𝑓=1 the boundary conditions across the boundary surface joining and 𝑓(𝑡) →0 as 𝑡→0. the interior space-time and the exterior space-time described We, therefore, have a complete description of the interior by the Vaidya [4]metric and exterior space-times of the collapsing body. In the 2𝑚 (V) following section, we shall analyze the impact of anisotropy 𝑑𝑠2 =−(1− )𝑑V2 −2𝑑V𝑑𝑟+𝑟2𝑑(𝑑𝜃2 + 2𝜃𝑑𝜙2). by making use of the solutions thus obtained. + 𝑟 sin (22) 5. Physical Analysis In (22), V denotes the retarded time and 𝑚(V) represents the total mass of the collapsing star. The junction conditions are To understand the nature of collapse, we assume that a star 3 Σ starts its collapse at 𝑡=−∞(i.e., 𝑓(𝑡))andtheinitial =1 determined by assuming a time-like -surface which sep- 𝐴 𝐵 Δ arates the interior and the exterior manifolds [5]. Continuity static star is described by the parameters 0, 0,and 𝑠. 2 2 2 We assume that the collapsing object has an initial radius of the metric space times ((𝑑𝑠−)Σ =(𝑑𝑠+)Σ =𝑑𝑠Σ)andthe − + 𝑟Σ(𝑡 → −∞) =𝑠 𝑟 and mass 𝑚(𝑟𝑠,−∞)= 𝑚𝑠 satisfying the extrinsic curvatures (𝐾𝑖𝑗 =𝐾𝑖𝑗 ) across the surface Σ then condition 2𝑚𝑠/𝑟𝑠 <1. If for some reason, instability develops yield the following matching conditions linking smoothly the inside the star, it begins to collapse. The comoving boundary interior (𝑟≤𝑟Σ) and the exterior (𝑟≥𝑟Σ)space-timesacross surface (𝑟𝑓)Σ =𝑟𝑠𝑓(𝑡) then starts shrinking until it reaches its the boundary: [𝑟𝑓(𝑡 )] =𝑟𝑓(𝑡 ) = 2𝑚(V) Schwarzschild horizon value bh Σ 𝑠 bh , 𝑡=𝑡 1/2 where bh denotes the time of formation of the black hole 2𝑚 𝑑𝑟 𝑓(𝑡) =𝑓 𝐴 (𝑟 )𝑑𝑡=(1− +2 ) 𝑑V, (23) corresponding to the value of bh. 0 Σ 𝑟 𝑑V Σ From (28), the mass of the evolving star at any instant 𝑡 within the boundary radius 𝑟Σ may be written as 𝑟Σ𝑓 (𝑡) = 𝑟Σ (V) , (24) 2𝑛2𝑟3 2 𝑟𝑓 (𝑡) 𝑟2 𝑟𝑅̇ 2 𝑚 (𝑟 ,𝑡) = [𝑚 𝑓+ (1−√𝑓) ] , 𝑚 (V) = ( [ + ( ) ]) , Σ 𝑠 𝐴2 (33) 2 𝑅2 +𝑟2 𝐴 (25) 0 Σ 0 Σ where mass of the initial static star has the form [𝑝𝑟]Σ =[𝑞𝑓(𝑡) 𝐵0]Σ, (26) 3 𝑟𝑠 𝑟 𝑚 = ∫ 4𝜋𝑟2𝜌 𝑑𝑟 = 𝑠 . Σ 𝑠 𝑠 2 2 (34) 𝑚 (𝑟,) 𝑡 =𝑚(V) . (27) 0 2(𝑟𝑠 +𝑅 )

Consequently, the condition (𝑟𝑓 )Σ = 2𝑚(V) yields From (25)and(27), the mass enclosed within the boundary bh 2 surface at any instant 𝑡 within a boundary surface 𝑟≤𝑟Σ can [ 2𝑛𝑟/𝐴 ] be written as 𝑓 = [ 0 ] , bh (35) 2 2 ̇ (2𝑛𝑟/𝐴0)+√1−(2𝑚𝑠/𝑟) 𝑟𝑓 (𝑡) 𝑟 𝑟𝑓 [ ]Σ 𝑚 (𝑟,) 𝑡 = [ + ( ) ] . (28) 2 𝑅2 +𝑟2 𝐴 0 and the time of black hole formation is obtained as 1 𝑓 Combining (6), (9), and (26), together within the condition 𝑡 = [ bh + √𝑓 + (1 − √𝑓 )] . bh bh ln bh (36) (𝑝𝑟)𝑠(𝑟 =Σ 𝑟 )=0,wededucethesurfaceequationgoverning 𝑛 2 thecollapsingmatterintheform The model parameters (namely, 𝑅, 𝑃,and𝑄)involvingthe 2 initial static star are determined using the following boundary 2𝑓𝑓̈ + 𝑓̇ −2𝑛𝑓=0,̇ (29) conditions: 1/2 where we have defined 2𝑚𝑠 𝐴0 (𝑟𝑠)=(1− ) , (37) 󸀠 𝑟𝑠 𝐴0 𝑛=[ ] . (30) 𝐵 −1/2 0 Σ 2𝑚 𝐵 (𝑟 )=(1− 𝑠 ) , (38) 0 𝑠 𝑟 Note that for a given initial static configuration 𝑛 appears as 𝑠 aconstantin(29). Following Bonnor et al. [6], we write (29) (𝑝𝑟)𝑠 (𝑟𝑠)=0, (39) as a first order differential equation wherewehavematchedthestaticinteriorspace-timetothe ̇ 2𝑛 Schwarzschild exterior and imposed the condition that the 𝑓=− (1 − √𝑓) , (31) √𝑓 radial pressure ((𝑝𝑟)𝑠)mustvanishattheboundary. Journal of Gravity 5

Table 1: Data for collapsing systems having different values of the anisotropic parameter 𝛼. All the configurations start collapsing with initial masses 𝑚𝑠 = 3.25 M⊙ and radii 𝑟𝑠 =20km.

𝛼𝑃 𝑄𝑅 𝑛𝑓 𝑡 𝑟 𝑚 (km) bh bh bh (km) bh (M⊙) 0.9 0.2848 −0.6341 20.8427 0.0244 0.2298 −2.3914 4.596 2.2980 0.6 0.3800 −0.4994 20.8427 0.0163 0.2298 −3.5816 4.596 2.2980 0.3 0.4065 −0.4619 20.8427 0.0142 0.2298 −4.1074 4.596 2.2980 0 0.42067 −0.4418 20.8427 0.0131 0.2298 −4.4578 4.596 2.2980 −0.3 0.4300 −0.4287 20.8427 0.0124 0.2298 −4.7239 4.596 2.2980 −0.6 0.4367 −0.4192 20.8427 0.0118 0.2298 −4.9401 4.596 2.2980 −0.9 0.4418 −0.4120 20.8427 0.0114 0.2298 −5.1230 4.596 2.2980 −1.5 0.4493 −0.4012 20.8427 0.0108 0.2298 −5.4231 4.596 2.2980

0 construction, turns out to be negative. However, from the absolute values of Θ, we note that for a positive anisotropy −50 (𝑝𝑟 >𝑝𝑡) the collapse rate increases as compared to an iso- tropic star while for a negative value of 𝛼 (𝑝𝑡 >𝑝𝑟), the rate decreases. From Table 1,wenotethefollowing. −100 𝑡 Case 1. When 𝛼>0(implying 𝑝𝑟 >𝑝𝑡), the horizon is formed −150 at a faster rate as compared to 𝛼=0,thatis,isotropiccase. 𝛼<0 𝑝 >𝑝 −200 Case 2. When (implying 𝑡 𝑟), the horizon is formedataslowerrateascomparedto 𝛼=0,thatis,isotropic 0.0 0.2 0.4 0.6 0.8 1.0 case. 𝑓 Similar observations may be found in [12]. However, we note that mass and radius of the collapsed configuration do 𝛼 = 0.9 not depend on anisotropy in our formulation. 𝛼=0 𝛼 = −0.9 6. Thermal Behaviour Figure 1: Variation of 𝑓(𝑡) for different anisotropic parameters. To analyze the impact of anisotropy on the evolution of temperature of the collapsing system, we use the relativistic Now, to get an insight about the effects of anisotropy, we Maxwell-Cattaneo relation for temperature governing the consider different initial static configurations, both isotropic heat transport [60, 67, 68]givenby (𝛼 = 0) and anisotropic (𝛼 =0̸ ). We consider different initially static stellar configurations of identical initial masses and 𝜏(𝑔𝛼𝛽 +𝑢𝛼𝑢𝛽)𝑢𝛿𝑞 +𝑞𝛼 radii (we have assumed 𝑚𝑠 = 3.25𝑀⊙ and radius 𝑟𝑠 =20km) 𝛽;𝛿 with different values of the anisotropic parameter 𝛼.Using (41) =−𝜅(𝑔𝛼𝛽 +𝑢𝛼𝑢𝛽)[𝑇 +𝑇𝑢̇ ], numerical procedures, we have calculated the corresponding ,𝛽 𝛽 model parameters and also evaluated the time of formation 𝑡 𝑟 =𝑟𝑓 𝑚 of black holes bh and radius ( bh 𝑠 bh)andmass( bh) where 𝜅(≥ 0) is the thermal conductivity and 𝜏(≥ 0) is the oftheblackholeformed.Ourresultshavebeencompiled relaxation time. For the line element (1), (41)reducesto in Table 1.Variationsof𝑓(𝑡) for different choices of the anisotropic parameter 𝛼 have been shown in Figure 1.We 𝑑 1 1 𝑑 have also calculated the rate of collapse in our model for 𝜏 (𝑞𝑓𝐵0)+𝑞 𝑓𝐴 0𝐵0 =−𝜅 (𝐴0𝑇) . (42) different anisotropic parameters. The collapse rate, inour 𝑑𝑡 𝑓𝐵0 𝑑𝑟 model, is obtained as Following an earlier work [51], we write the relativistic 3𝑓̇ 6𝑛 (√𝑓−1) 𝛽 Fourier heat transport equation by setting 𝜏=0in (42). For Θ=𝑢;𝛽 = = . (40) 𝐴0𝑓 𝐴0𝑓√𝑓 𝜏=0,combining(6)and(42), we get

The collapse rate turns out to be Θ = −0.101464, −0.102096, 󸀠 ̇ 󸀠 2𝐴 𝑓 −0.100749 for 𝛼 = 0, 0.9, −0.9,respectively.Since,the 8𝜋𝜅(𝐴 𝑇) = 0 . (43) 0 𝐴 𝑓 collapsing object contracts in size as time evolves Θ,inour 0 6 Journal of Gravity

0.0030 capture the impact of anisotropy on gravitational collapse successfully. Though, as pointed out in [53], effects of factors 0.0025 like shear, charge, and so forth on self-bound systems can 0.0020 be absorbed by considering the system to be anisotropic, in general, we intend to formulate a more general framework

∑ 0.0015 so as to examine the combined impacts of various factor 𝑇 relevant to gravitationally collapsing systems. These issues 0.0010 will be taken up elsewhere.

0.0005 Acknowledgment 0.0000 0.0 0.2 0.4 0.6 0.8 1.0 R. Sharma is thankful to the Inter University Center for 𝑓 Astronomy and Astrophysics (IUCAA), Pune, India, where a part of this work was carried out under its Visiting Research 𝛼=0 𝛼 = −0.6 Associateship Program. 𝛼 = −0.9 References Figure 2: Evolution of surface temperature for different anisotropic parameters. [1] P. S. Joshi and D. Malafarina, “Recent developments in grav- itational collapse and spacetime singularities,” International Journal of Modern Physics D,vol.20,ArticleID2641,2011. Let us now assume that the thermal conductivity varies as 𝜅= [2] S. Chandrasekhar, “Stellar configurations with degenerate 𝜔 𝛾𝑇 ,where𝛾 and 𝜔 are constants. Equation (43) then yields cores,” Observatory,vol.57,pp.373–377,1934. [3] J. R. Oppenheimer and H. Snyder, “On continued gravitational 󸀠 −𝜔 󸀠 2𝐴 𝑇 2𝑛√ ( 𝑓−1) contraction,” Physical Review,vol.56,no.5,pp.455–459,1939. 8𝜋𝛾(𝐴 𝑇) = 0 [ ], (44) 0 𝐴 𝑓√𝑓 [4] P. C. Vaidya, “The gravitational field of a radiating star,” Pro- 0 ceedings of the Indian Academy of Science A,vol.33,no.5,pp. 264–276, 1951. wherewehaveused(31). Integrating the above equation, we [5] N. O. Santos, “Non-adiabatic radiating collapse,” Monthly get Notices of the Royal Astronomical Society,vol.216,pp.403–410, 1985. 𝑛(√𝑓−1) 𝜔+1 ln 𝐴0 [6] W.B. Bonnor, A. K. G. de Oliveira, and N. O. Santos, “Radiating 𝑇 = ( )+𝑇0 (𝑡) , (45) 2𝜋𝛾𝑓√𝑓 𝐴0 spherical collapse,” Physics Reports,vol.181,no.5,pp.269–326, 1989. To get a simple estimate of temperature evolution, we set 𝜔= [7]A.K.G.deOliveiraandN.O.Santos,“Nonadiabaticgravita- 0, 𝛾=1,and𝑇0(𝑡) = 0 andevaluatethesurfacetemperature tional collapse,” Astrophysical Journal,vol.312,pp.640–645, at any instant as 1987. [8]A.K.G.deOliveira,N.O.Santos,andC.A.Kolassis,“Collapse 𝑛(√𝑓−1) of a radiating star,” Monthly Notices of the Royal Astronomical ln 𝐴0 𝑇(𝑟 ,𝑡)= ( ) . (46) Society,vol.216,pp.1001–1011,1985. Σ 𝐴 2𝜋𝑓√𝑓 0 Σ [9]A.K.G.deOliveira,C.A.Kolassis,andN.O.Santos,“Collapse of a radiating star revisited,” Monthly Notices of the Royal Astro- Time evolution of the surface temperature for different nomical Society, vol. 231, pp. 1011–1018, 1988. anisotropic parameter 𝛼 has been shown in Figure 2,where [10] L. Herrera, J. Ospino, and A. di Prisco, “All static spherically we have used the data from Table 1. symmetric anisotropic solutions of Einstein’s equations,” Physi- cal Review D,vol.77,no.2,ArticleID027502,2008. [11] L. Herrera, N. O. Santos, and A. Wang, “Shearing expansion- 7. Discussions free spherical anisotropic fluid evolution,” Physical Review D, vol. 78, no. 8, Article ID 084026, 10 pages, 2008. We have developed a minimalistic and basic framework of [12] L. Herrera, A. di Prisco, J. L. Hernandez-Pastora, and N. a gravitationally collapsing system which has allowed us to O. Santos, “On the role of density inhomogeneity and local examine the impact of pressure anisotropy explicitly. In our anisotropyinthefateofsphericalcollapse,”Physics Letters A, construction, we have ignored the effects of various other vol.237,no.3,pp.113–118,1998. factors relevant to collapsing systems such as shear, viscosity, [13]L.HerreraandN.O.Santos,“Thermalevolutionofcompact and charge. Moreover, the line element describing the interior objects and relaxation time,” Monthly Notices of the Royal space-timehasbeenassumedtobesphericallysymmetric Astronomical Society,vol.287,no.1,pp.161–164,1997. wherethemetricfunctionshavebeenchosentobeseparable [14] L.Herrera,A.diPrisco,J.Martin,J.Ospino,N.O.Santos,andO. in its variables. Though formulation of a more general Troconis, “Spherically symmetric dissipative anisotropic fluids: frameworkisalwayspreferredtodescribearealisticsituation, a general study,” Physical Review D,vol.69,no.8,ArticleID the simple model developed here provides a mechanism to 084026, 2004. Journal of Gravity 7

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