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

In cosmology and astrophysics, there exist many outstanding issues relating to a dynamical system collapsing under the influence of its own gravity. In view of Cosmic Censorship Conjecture, the general relativistic prediction is that such a collapse must terminate into a space-time singularity covered under its event horizon though there are several counter examples where it has been shown that a naked singularity is more likely to be formed (see [1] and references therein). In astrophysics, the end stage of a massive collapsing star has long been very much speculative in nature [1, 2]. From classical gravity perspective, to get a proper understanding of the nature of collapse and physical behavior of a collapsing system, construction of a realistic model of the collapsing system is necessary. This, however, turns out to be a difficult task because of the highly nonlinear nature of the governing field equations. To reduce the complexity, various simplifying methods are often adopted and the pioneering work of Oppenheimer and Snyder [3] was a first step in this direction when collapse of a highly idealized spherically symmetric dust cloud was studied. Since then, various attempts have been made to develop realistic models of gravitationally collapsing systems to understand the nature and properties of collapsing objects. It got a tremendous impetus when Vaidya [4] presented a solution describing the exterior gravitational field of a stellar body with outgoing radiation and Santos [5] formulated the junction conditions joining the interior space time of the collapsing object to the Vaidya exterior metric [4]. These developments have enabled many investigators to construct realistic models of gravitationally evolving systems and also to analyze critically relevance of various factors such as shear, density inhomogeneity, local anisotropy, electromagnetic field, viscosity, and so forth, on the physical behaviour of collapsing bodies [6–52]. In the absence of any established theory governing gravitational collapse, such investigations have been found to be very useful to get a proper understanding about systems undergoing gravitational collapse.

The aim of the present work is to develop a simple model of a collapsing star and investigate the impact of pressure anisotropy on the overall behaviour of the collapsing body. Anisotropic stresses may occur in astrophysical objects for various reasons which include phase transition, density inhomogeneity, shear, and electromagnetic field [10, 53, 54]. In [53], it has been shown that influences of shear, electromagnetic field, and so forth on self-bound systems can be absorbed if the system is considered to be anisotropic, in general. Local anisotropy has been a well-motivated factor in the studies of astrophysical objects and its role on the gross features of static stellar configurations have been investigated by many authors (see, e.g., [10, 11, 55–59] and references therein). For dynamical systems, though pressure anisotropy is, in general, incorporated in the construction, very few works have been reported till date where impacts of anisotropy have been discussed explicitly [12–14, 26, 60]. Appropriate junction conditions for an anisotropic fluid collapsing on the background space time described by the Vaidya metric have been obtained in [27]. Considering a spheroidal geometry of Vaidya and Tikekar [61], Sarwe and Tikekar [50] have analyzed the impact of geometry vis-a-vis matter composition on the collapse of stellar bodies which begin to collapse from initial static configurations possessing equal compactness. Sharma and Tikekar [51, 52] have investigated the evolution of nonadiabatic collapse of a shear-free spherically symmetric object with anisotropic stresses on the background of space-time obtained by introducing an inhomogeneous perturbation in the Robertson-Walker space-time.

In the present work, we have developed a model describing a shear-free spherically symmetric fluid distribution radiating away its energy in the form of radial heat flux. The star begins its collapse from an initially static configuration whose energy-momentum tensor describing the material composition has been assumed to be anisotropic, in general. To develop the model of the initial static star, we have utilized the Finch and Skea [62] ansatz which has earlier been found to be useful to develop physically acceptable models capable of describing realistic stars in equilibrium [63–66]. The back ground space-time for a static configuration for the given ansatz has a clear geometrical interpretation as may be found in [65]. By assuming a particular form of the anisotropic parameter, we have solved the relevant field equations and constructed a model for the initial static stellar configuration which could either be isotropic or anisotropic in nature. The solution provided by Finch and Skea [62] is a subclass of the solution provided here. Since, the solution presented here provides a wider range of values of the anisotropic parameter, it enables us to examine the impact of anisotropic stresses on the evolution of a large class of initial static configurations.

Our paper has been organized as follows. In Section 2, we have presented the basic equations governing the system undergoing non-adiabatic radiative collapse. In Section 3, by assuming a particular anisotropic profile, we have solved the relevant field equations to develop a model for the initial static star. In Section 4, by stipulating the boundary conditions across the surface separating the stellar configuration from the Vaidya [4] space-time, we have solved the surface equation which governs the evolution of the initial static star that begins to collapse when the equilibrium is lost. In Section 5, we have analyzed the impact of anisotropy on gravitationally collapsing systems by considering evolution of initial static stars which could either be isotropic or anisotropic. Impact of anisotropy on the evolution of temperature has been analyzed in Section 6. Finally, some concluding remarks have been made in Section 7.

2. Equations Governing the Collapsing System

We write the line element describing the interior space-time of a spherically symmetric star collapsing under the influence of self-gravity (in standard coordinates x⁰ = t, x¹ = r, x² = θ, and x³ = ϕ) as

(1)

where, A₀(r), B₀(r), and f(t) are yet to be determined. Note that in (1), if we set f(t) = 1, the metric corresponds to a static spherically symmetric configuration.

We assume that the matter distribution of the collapsing object is an imperfect fluid described by an energy-momentum tensor of the form

(2)

where ρ represents the energy density p_r and p_t, respectively, denote fluid pressures along the radial and transverse directions; u^α is the 4-velocity of the fluid; χ^α is a unit space-like 4-vector along the radial direction; and q^α = (0, q¹, 0,0) is the heat flux vector which is orthogonal to the velocity vector so that u^αu_α = −1 and u^αq_α = 0.

The Einstein’s field equations governing the evolution of the system are then obtained as (we set G = c = 1)

(3)

(4)

(5)

(6)

In (3)–(6), a “prime” denotes differentiation with respect to r and a “dot” denotes differentiation with respect to t. Making use of (4) and (5), we define the anisotropic parameter of the collapsing object as

(7)

We rewrite (3)–(5) as

(8)

(9)

(10)

where ρ_s, (p_r) _s and (p_t) _s denote the energy-density, radial pressure, and tangential pressure, respectively, of the initial static star.

3. Interior Space-Time of the Initially Static Configuration

In our construction, we assume that an initially static star (with f(t) = 1 in (1)), described by metric potentials A₀(r), B₀(r), and anisotropy Δ_s(r), starts collapsing if, for some reasons, it loses its equilibrium. To develop a model of the initially static configuration, we first assume that the anisotropic parameter is separable in its variables so that Δ(r, t) = Δ_s(r)/f²(t). Equation (7) then reduces to

(11)

which is independent of t. Equation (11) can only be solved if any two of the unknown functions (A₀, B₀, and Δ_s(r)) are specified. To develop a physically reasonable model of the initial static configuration, we first utilize the Finch and Skea [62] ansatz for the metric potential B₀ given by

(12)

where R is the curvature parameter describing the geometry of the configuration. In the static case, the ansatz (12) has a clear geometric characterization and has been found to yield realistic models for compact stellar objects [64–66]. Substituting (12) in (11), we obtain

(13)

where we have used the following transformation:

(14)

To solve (13), we assume a particular anisotropic profile

(15)

where α is the measure of anisotropy. The motivation for choosing the particular form of the anisotropy parameter is the following: (1) it is physically reasonable as the anisotropy vanishes at the center (r = 0, i.e., x = 1) as expected, and (2) it provides a solution of (13) in closed form. Note that α = 0 corresponds to an initial static star which is isotropic in nature. Substituting (15) in (13), we get

(16)

whose solution is obtained as

(17)

where P and Q are integration constants,

is the Bessel function of first kind of order

, and

is the Bessel function of second kind of order

. It is obvious that the solution is valid for α < 1. At α = 1, the Bessel function encounters a singularity and, therefore, we shall deal with the α = 1 case separately.

Special Cases

Case 1 (α = 0 (Δ_s(r) = 0). That is, initial static configuration is isotropic in nature). Equation (13) reduces to

(18)

whose solution is found to be

(19)

where, C and D are integration constants. This particular solution has been found previously in [62]. However, the solution (19) can be obtained directly from the solution (17) by substituting α = 0 and setting

and

Case 2 (α = 1 (Δ_s(r) ≠ 0)). Equation (16) reduces to

(20)

whose solution is given by

(21)

where A and B are integration constants [64]. Unfortunately, the above solution can not be regained from the general solution (17) due to the properties of Bessel functions and should be treated separately.

We thus have a model for an initially static stellar configuration which could either be isotropic or anisotropic in nature. After loss of equilibrium, the initially static star starts collapsing, and to generate a solution of the subsequent evolving system, we need to determine f(t). This will be taken up in the following section.

4. Exterior Space-Time and Junction Conditions

In our construction, evolution of the collapsing object is governed by the function f(t) which can be determined from the boundary conditions across the boundary surface joining the interior space-time and the exterior space-time described by the Vaidya [4] metric

(22)

In (22), v denotes the retarded time and m(v) represents the total mass of the collapsing star. The junction conditions are determined by assuming a time-like 3-surface Σ which separates the interior and the exterior manifolds [5]. Continuity of the metric space times (

) and the extrinsic curvatures (

) across the surface Σ then yield the following matching conditions linking smoothly the interior (r ≤ r_Σ) and the exterior (r ≥ r_Σ) space-times across the boundary:

(23)

(24)

(25)

(26)

(27)

From (25) and (27), the mass enclosed within the boundary surface at any instant t within a boundary surface r ≤ r_Σ can be written as

(28)

Combining (6), (9), and (26), together within the condition (p_r) _s(r = r_Σ) = 0, we deduce the surface equation governing the collapsing matter in the form

(29)

where we have defined

(30)

Note that for a given initial static configuration n appears as a constant in (29). Following Bonnor et al. [6], we write (29) as a first order differential equation

(31)

which admits a solution

(32)

Note that at t → −∞, that is, at the onset of collapse, f = 1 and f(t) → 0 as t → 0.

We, therefore, have a complete description of the interior and exterior space-times of the collapsing body. In the following section, we shall analyze the impact of anisotropy by making use of the solutions thus obtained.

5. Physical Analysis

To understand the nature of collapse, we assume that a star starts its collapse at t = −∞ (i.e., f(t) = 1) and the initial static star is described by the parameters A₀, B₀, and Δ_s. We assume that the collapsing object has an initial radius r_Σ(t → −∞) = r_s and mass m(r_s, −∞) = m_s satisfying the condition 2m_s/r_s < 1. If for some reason, instability develops inside the star, it begins to collapse. The comoving boundary surface (rf) _Σ = r_sf(t) then starts shrinking until it reaches its Schwarzschild horizon value , where t = t_bh denotes the time of formation of the black hole corresponding to the value of f(t) = f_bh.

From (28), the mass of the evolving star at any instant t within the boundary radius r_Σ may be written as

(33)

where mass of the initial static star has the form

(34)

Consequently, the condition (rf_bh) _Σ = 2m(v) yields

(35)

and the time of black hole formation is obtained as

(36)

The model parameters (namely, R, P, and Q) involving the initial static star are determined using the following boundary conditions:

(37)

(38)

(39)

where we have matched the static interior space-time to the Schwarzschild exterior and imposed the condition that the radial pressure ((p_r) _s) must vanish at the boundary.

Now, to get an insight about the effects of anisotropy, we consider different initial static configurations, both isotropic (α = 0) and anisotropic (α ≠ 0). We consider different initially static stellar configurations of identical initial masses and radii (we have assumed m_s = 3.25M_⊙ and radius r_s = 20 km) with different values of the anisotropic parameter α. Using numerical procedures, we have calculated the corresponding model parameters and also evaluated the time of formation of black holes t_bh and radius (r_bh = r_sf_bh) and mass (m_bh) of the black hole formed. Our results have been compiled in Table 1. Variations of f(t) for different choices of the anisotropic parameter α have been shown in Figure 1. We have also calculated the rate of collapse in our model for different anisotropic parameters. The collapse rate, in our model, is obtained as

(40)

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

α	P	Q	R (km)	n	f_bh	t_bh	r_bh (km)	m_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

Details are in the caption following the image — **Figure 1**
Open in figure viewer PowerPoint

Variation of f(t) for different anisotropic parameters.

The collapse rate turns out to be Θ = −0.101464, −0.102096, −0.100749 for α = 0,0.9, −0.9, respectively. Since, the collapsing object contracts in size as time evolves Θ, in our construction, turns out to be negative. However, from the absolute values of Θ, we note that for a positive anisotropy (p_r > p_t) the collapse rate increases as compared to an isotropic star while for a negative value of α (p_t > p_r), the rate decreases. From Table 1, we note the following.

Case 1. When α > 0 (implying p_r > p_t), the horizon is formed at a faster rate as compared to α = 0, that is, isotropic case.

Case 2. When α < 0 (implying p_t > p_r), the horizon is formed at a slower rate as compared to α = 0, that is, isotropic case.

Similar observations may be found in [12]. However, we note that mass and radius of the collapsed configuration do not depend on anisotropy in our formulation.

6. Thermal Behaviour

To analyze the impact of anisotropy on the evolution of temperature of the collapsing system, we use the relativistic Maxwell-Cattaneo relation for temperature governing the heat transport [60, 67, 68] given by

(41)

where κ(≥0) is the thermal conductivity and τ(≥0) is the relaxation time. For the line element (1), (41) reduces to

(42)

Following an earlier work [51], we write the relativistic Fourier heat transport equation by setting τ = 0 in (42). For τ = 0, combining (6) and (42), we get

(43)

Let us now assume that the thermal conductivity varies as κ = γT^ω, where γ and ω are constants. Equation (43) then yields

(44)

where we have used (31). Integrating the above equation, we get

(45)

To get a simple estimate of temperature evolution, we set ω = 0, γ = 1, and T₀(t) = 0 and evaluate the surface temperature at any instant as

(46)

Time evolution of the surface temperature for different anisotropic parameter α has been shown in Figure 2, where we have used the data from Table 1.

7. Discussions

We have developed a minimalistic and basic framework of a gravitationally collapsing system which has allowed us to examine the impact of pressure anisotropy explicitly. In our construction, we have ignored the effects of various other factors relevant to collapsing systems such as shear, viscosity, and charge. Moreover, the line element describing the interior space-time has been assumed to be spherically symmetric where the metric functions have been chosen to be separable in its variables. Though formulation of a more general framework is always preferred to describe a realistic situation, the simple model developed here provides a mechanism to capture the impact of anisotropy on gravitational collapse successfully. Though, as pointed out in [53], effects of factors like shear, charge, and so forth on self-bound systems can be absorbed by considering the system to be anisotropic, in general, we intend to formulate a more general framework so as to examine the combined impacts of various factor relevant to gravitationally collapsing systems. These issues will be taken up elsewhere.

Acknowledgment

R. Sharma is thankful to the Inter University Center for Astronomy and Astrophysics (IUCAA), Pune, India, where a part of this work was carried out under its Visiting Research Associateship Program.

References

1 Joshi P. S. and Malafarina D., Recent developments in gravitational collapse and spacetime singularities, International Journal of Modern Physics D. (2011) 20, 2641, https://doi.org/10.1142/S0218271811020792.
Google Scholar
2 Chandrasekhar S., Stellar configurations with degenerate cores, Observatory. (1934) 57, 373–377.
Google Scholar
3 Oppenheimer J. R. and Snyder H., On continued gravitational contraction, Physical Review. (1939) 56, no. 5, 455–459, 2-s2.0-36149009919, https://doi.org/10.1103/PhysRev.56.455.
Google Scholar
4 Vaidya P. C., The gravitational field of a radiating star, Proceedings of the Indian Academy of Science A. (1951) 33, no. 5, 264–276.
Google Scholar
5 Santos N. O., Non-adiabatic radiating collapse, Monthly Notices of the Royal Astronomical Society. (1985) 216, 403–410.
Google Scholar
6 Bonnor W. B., de Oliveira A. K. G., and Santos N. O., Radiating spherical collapse, Physics Reports. (1989) 181, no. 5, 269–326, 2-s2.0-0141803643.
Google Scholar
7 de Oliveira A. K. G. and Santos N. O., Nonadiabatic gravitational collapse, Astrophysical Journal. (1987) 312, 640–645.
Google Scholar
8 de Oliveira A. K. G., Santos N. O., and Kolassis C. A., Collapse of a radiating star, Monthly Notices of the Royal Astronomical Society. (1985) 216, 1001–1011.
Google Scholar
9 de Oliveira A. K. G., Kolassis C. A., and Santos N. O., Collapse of a radiating star revisited, Monthly Notices of the Royal Astronomical Society. (1988) 231, 1011–1018.
Google Scholar
10 Herrera L., Ospino J., and di Prisco A., All static spherically symmetric anisotropic solutions of Einstein’s equations, Physical Review D. (2008) 77, no. 2, 027502.
Google Scholar
11 Herrera L., Santos N. O., and Wang A., Shearing expansion-free spherical anisotropic fluid evolution, Physical Review D. (2008) 78, no. 8, 10, 084026.
Google Scholar
12 Herrera L., di Prisco A., Hernandez-Pastora J. L., and Santos N. O., On the role of density inhomogeneity and local anisotropy in the fate of spherical collapse, Physics Letters A. (1998) 237, no. 3, 113–118.
Google Scholar
13 Herrera L. and Santos N. O., Thermal evolution of compact objects and relaxation time, Monthly Notices of the Royal Astronomical Society. (1997) 287, no. 1, 161–164, https://doi.org/10.1093/mnras/287.1.161.
Google Scholar
14 Herrera L., di Prisco A., Martin J., Ospino J., Santos N. O., and Troconis O., Spherically symmetric dissipative anisotropic fluids: a general study, Physical Review D. (2004) 69, no. 8, 084026.
Google Scholar
15 Herrera L. and Barreto W., Relativistic gravitational collapse in comoving coordinates: the post-quasistatic approximation, International Journal of Modern Physics D. (2011) 20, no. 07, 1265–1288.
Google Scholar
16 Herrera L. and Santos N. O., Local anisotropy in self-gravitating systems, Physics Reports. (1997) 286, no. 2, 53–130.
Google Scholar
17 Herrera L., Martin J., and Ospino J., Anisotropic geodesic fluid spheres in general relativity, Journal of Mathematical Physics. (2002) 43, no. 10, 4889–4897.
Google Scholar
18 Herrera L., Physical causes of energy density inhomogenization and stability of energy density homogeneity in relativistic self-gravitating fluids, International Journal of Modern Physics D. (2011) 20, no. 9, 1689.
Google Scholar
19 di Prisco A., Herrera L., Denmat G. L., MacCallum M. A. H., and Santos N. O., Nonadiabatic charged spherical gravitational collapse, Physical Review D. (2007) 76, no. 6, 064017.
Google Scholar
20 DiPrisco A., Herrera L., and Varela V., Cracking of homogeneous self-gravitating compact objects induced by uctuations of local anisotropy, General Relativity and Gravitation. (1997) 29, no. 10, 1239–1256.
Google Scholar
21 Pinheiro G. and Chan R., Radiating gravitational collapse with shear viscosity revisited, General Relativity and Gravitation. (2008) 40, no. 10, 2149–2175.
Google Scholar
22 Pinheiro G. and Chan R., Radiating gravitational collapse with an initial inhomogeneous energy density distribution, General Relativity and Gravitation. (2011) 43, no. 5, 1451–1467.
Google Scholar
23 Pinheiro G. and Chan R., Radiating shear-free gravitational collapse with charge, General Relativity and Gravitation. (2013) 45, no. 1, 243–261.
Google Scholar
24 Chan R., Radiating gravitational collapse with shear viscosity, Monthly Notices of the Royal Astronomical Society. (2000) 316, no. 3, 588–604, https://doi.org/10.1046/j.1365-8711.
Google Scholar
25 Chan R., Radiating gravitational collapse with shearing motion and bulk viscosity, Astronomy and Astrophysics. (2001) 368, no. 1, https://doi.org/10.1051/00046361:20000525.
Google Scholar
26 Chan R., da Silva M. F. A., and da Rocha J. F. V., Gravitational collapse of self-similar and shear-free fluid with heat flow, International Journal of Modern Physics D. (2003) 12, article 347, no. 3, https://doi.org/10.1142/S021827180300327X.
Google Scholar
27 Chan R., Herrera L., and Santos N. O., Dynamical instability for shearing viscous collapse, Monthly Notices of the Royal Astronomical Society. (1994) 267.
Google Scholar
28 Tikekar R. and Patel L. K., Non-adiabatic gravitational collapse of charged radiating fluid spheres, Pramana. (1992) 39, no. 1, 17–25.
Google Scholar
29 Maharaj S. D. and Govender M., Collapse of a charged radiating star with shear, Pramana. (2000) 54, no. 5, 715–727, 2-s2.0-0010008329.
Google Scholar
30 Sharif M. and Iqbal K., Spherically symmetric gravitational collapse, Modern Physics Letters A. (2009) 24, no. 19, 1533.
Google Scholar
31 Sharif M. and Siddiqe A., Singularity in gravitational collapse of plane symmetric charged vaidya spacetime, Modern Physics Letters A. (2010) 25, no. 33, 2831.
Google Scholar
32 Sharif M. and Siddiqe A., Dynamics of charged plane symmetric gravitational collapse, General Relativity and Gravitation. (2011) 43, no. 1, 73–91.
Google Scholar
33 Sharif M. and Abbas G., Charged perfect fluid cylindrical gravitational collapse, Journal of the Physical Society of Japan. (2011) 80, 104002.
Google Scholar
34 Sharif M. and Abbas G., Dynamics of non-adiabatic charged cylindrical gravitational collapse, Astrophysics and Space Science. (2011) 335, no. 2, 515–521.
Google Scholar
35 Sharif M. and Fatima S., Charged cylindrical collapse of anisotropic fluid, General Relativity and Gravitation. (2011) 43, no. 1, 127–142.
10.1007/s10714-010-1076-3
Web of Science® Google Scholar
36 Barreto W., Rodríguez B., Rosales L., and Serrano O., Self-similar and charged radiating spheres: an anisotropic approach, General Relativity and Gravitation. (2007) 39, no. 1, 23–39.
Google Scholar
37 Ghezzi C., Relativistic structure, stability, and gravitational collapse of charged neutron stars, Physical Review D. (2005) 72, 104017.
Google Scholar
38 Goswami R. and Joshi P. S., Gravitational collapse of an isentropic perfect fluid with a linear equation of state, Classical and Quantum Gravity. (2004) 21, no. 15, 3645.
Google Scholar
39 Goswami R. and Joshi P. S., What role do pressures play in determining the final end state of gravitational collapse?, Classical and Quantum Gravity. (2002) 19, article 129, no. 20.
Google Scholar
40 Goswami R. and Joshi P. S., Black hole formation in perfect fluid collapse, Physical Review D. (2004) 69, 027502.
Google Scholar
41 Chakraborty S. and Bandyopadhyay T., Collapsing inhomogeneous dust in the background of perfect (or anisotropic) fluid, International Journal of Modern Physics D. (2008) 17, no. 08, 1271–1281.
Google Scholar
42 Eardley D. M. and Smarr L., Time functions in numerical relativity: marginally bound dust collapse, Physical Review D. (1979) 19, no. 8, 2239–2259, 2-s2.0-35949034546, https://doi.org/10.1103/PhysRevD.19.2239.
Google Scholar
43 Mena F. C., Nolan B. C., and Tavakol R., Role of anisotropy and inhomogeneity in Lemaitre-Tolman-Bondi collapse, Physical Review D. (2004) 70, no. 8, 084030.
Google Scholar
44 Thirukkanesh S. and Maharaj S. D., Charged relativistic spheres with generalized potentials, Mathematical Methods in the Applied Sciences. (2009) 32, no. 6, 684–701.
Google Scholar
45 Govinder K. S., Govender M., and Maartens R., On radiating stellar collapse with shear, Monthly Notices of the Royal Astronomical Society. (1998) 299, no. 3, 809–810.
Google Scholar
46 Schäfer D. and Goenner H. F., The gravitational field of a radiating and contracting spherically-symmetric body with heat flow, General Relativity and Gravitation. (2000) 42, no. 11, 2119–2130.
Google Scholar
47 Ivanov B. V., Collapsing shear-free perfect fluid spheres with heat flow, General Relativity and Gravitation. (2012) 44, no. 7, 1835–1855.
Google Scholar
48 Ivanov B. V., Self-gravitating spheres of anisotropic fluid in geodesic flow, International Journal of Modern Physics D. (2011) 20, no. 3, article 319.
Google Scholar
49 Ivanov B. V., Evolving spheres of shear-free anisotropic fluid, International Journal of Modern Physics A. (2010) 25, no. 20, 3975–3991.
Google Scholar
50 Sarwe S. and Tikekar R., Non-adiabatic gravitational collapse of a superdense star, International Journal of Modern Physics D. (2010) 19, no. 12, 1889–1904, 2-s2.0-77957578702, https://doi.org/10.1142/S0218271810018098.
Google Scholar
51 Sharma R. and Tikekar R., Space-time inhomogeneity, anisotropy and gravitational collapse, General Relativity and Gravitation. (2012) 44, no. 10, 2503–2520.
Google Scholar
52 Sharma R. and Tikekar R., Non-adiabatic radiative collapse of a relativistic star under different initial conditions, Pramana. (2012) 79, no. 3, 501–509.
Google Scholar
53 Ivanov B. V., The importance of anisotropy for relativistic fluids with spherical symmetry, International Journal of Theoretical Physics. (2010) 49, no. 6, 1236–1243.
Google Scholar
54 Bowers R. L. and Liang E. P. T., Anisotropic spheres in general relativity, Astrophysical Journal. (1974) 188, 1262–1274.
Google Scholar
55 Ivanov B. V., Maximum bounds on the surface redshift of anisotropic stars, Physical Review D. (2002) 65, no. 10, 104011.
Google Scholar
56 Mak M. K. and Harko T., Anisotropic stars in general relativity, Proceedings of the Royal Society A. (2003) 459, no. 2030, 393–408.
Google Scholar
57 Harko T. and Mak M. K., Anisotropic relativistic stellar models, Annalen Der Physik. (2002) 11, no. 1, 3–13.
Google Scholar
58 Dev K. and Gleiser M., Dark energy and the statistical study of the observed image separations of the multiply-imaged systems in the class statistical sample, International Journal of Modern Physics D. (2004) 13, no. 06.
Google Scholar
59 Karmakar S., Mukherjee S., Sharma R., and Maharaj S. D., The role of pressure anisotropy on the maximum mass of cold compact stars, Pramana. (2007) 68, no. 6, 881–889, 2-s2.0-34250646615, https://doi.org/10.1007/s12043-007-0088-3.
Google Scholar
60 Martinez J., Transport processes in the gravitational collapse of an anisotropic fluid, Physical Review D. (1996) 53, no. 12, 6921–6940.
Google Scholar
61 Vaidya P. C. and Tikekar R., Exact relativistic model for a superdense star, Journal of Astrophysics and Astronomy. (1982) 3, no. 3, 325–334.
Google Scholar
62 Finch M. R. and Skea J. E. F., A realistic stellar model based on an ansatz of Duorah and Ray, Classical and Quantum Gravity. (1989) 6, no. 4, article 467.
Google Scholar
63 Hansraj S. and Maharaj S. D., Charged analogue of finch-skea stars, International Journal of Modern Physics D. (2006) 15, no. 08, 1311.
Google Scholar
64 Tikekar R. and Jotania K., A relativistic two-parameter core-envelope model of compact stars, Gravitation and Cosmology. (2009) 15, no. 2, 129–133, 2-s2.0-67149137842, https://doi.org/10.1134/S0202289309020042.
Google Scholar
65 Tikekar R. and Jotania K., On relativistic models of strange stars, Pramana. (2007) 68, no. 3, 397–406, 2-s2.0-33947146568, https://doi.org/10.1007/s12043-007-0043-3.
Google Scholar
66 Jotania K. and Tikekar R., Paraboloidal space-times and relativistic models of strange stars, International Journal of Modern Physics D. (2006) 15, no. 8, 1175–1182, 2-s2.0-33750199183, https://doi.org/10.1142/S021827180600884X.
Google Scholar
67 Israel W. and Stewart J., Transient relativistic thermodynamics and kinetic theory, Annals of Physics. (1979) 118, no. 2, 341–372.
Google Scholar
68 Maartens R., Dissipative cosmology, Classical and Quantum Gravity. (1995) 12, no. 6, 1455.
Google Scholar

All articles

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

Abstract

1. Introduction

2. Equations Governing the Collapsing System

3. Interior Space-Time of the Initially Static Configuration

4. Exterior Space-Time and Junction Conditions

5. Physical Analysis

6. Thermal Behaviour

7. Discussions

Acknowledgment

References

Figures

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

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

Abstract

1. Introduction

2. Equations Governing the Collapsing System

3. Interior Space-Time of the Initially Static Configuration

4. Exterior Space-Time and Junction Conditions

5. Physical Analysis

6. Thermal Behaviour

7. Discussions

Acknowledgment

References

Figures

References

Related

Information