In this study, we create matching conditions for a spherically symmetric radiating star in the presence of shear. We study the role of anisotropy in the thermal evolution of a radiating star undergoing continuous dissipative gravitational collapse in the presence of shear.
Introduction
Field equations
The metric relation Γ is defined in terms of the metric tensor field and its derivatives by. Rab = Γdab,d−Γdad,b+ ΓeabΓded−ΓeadΓdeb (2.2.3) A further contraction of the Ricci tensor (2.2.3) yields the Ricci scalar.
Spherical spacetimes
Shearing spacetimes
The field equations (2.3.6) describe the interaction of a distribution of shear matter that expands and accelerates with the heat flow.
Shear-free spacetimes
Vaidya spacetime
In this chapter, we review the contact conditions corresponding to two spherically symmetric space-times over the time-like hypersurface Σ (Bonnoret al1989, Santos 1985). The results obtained in this chapter apply to the Einstein field equations (2.3.6) with a vanishing cosmological constant.
Matching hypersurfaces
The second connection condition is generated by requiring the outer curvature of Σ to be continuous over the boundary. Different forms of the first and second connection terms have been obtained by other researchers. We should point out that the crossing conditions (3.2.3) and (3.2.4) given above are equivalent to the crossing conditions generated earlier by Lichnerowicz (1955) and O'.
Lake (1987) provides a comprehensive review of junction conditions for boundary surfaces and surface layers with several applications in general relativity and cosmology.
Shearing junction conditions
Thus, the necessary and sufficient conditions for the spacetimes which ensure the validity of the first crossing condition (3.2.3) are that. This is an important result which relates the radial pressure pR to the magnitude of the heat flow Q=qB. Relation (3.3.11b) implies that the isotropic pressure pR is proportional to the magnitude of the heat flow q, which generally does not disappear.
The radial momentum flux of the radiation on both sides of Σ is given by F± =e±a0 n±bTab±. Therefore, the result (3.3.11b) corresponds to the continuity of the radiation radial momentum flux over the surface Σ, i.e. it expresses the local conservation of momentum.
Shear-free junction conditions
The contact conditions have also been extended to include non-spherical collapse and rotation in the slow approximation (Nath et al 2008, Herrera et al 1998b). Glass (1989) is the first attempt to generalize the above coupling conditions to include shear forces for neutral matter. In order to obtain a complete solution for radiative gravitational collapse, we need to solve for the pressure isotropy condition (∆ = 0) together with the contact condition (3.4.8b).
Junction conditions for shearing spacetimes for the special case with geodesic motion were created by Tomimura and Nunes (1993). Several solutions were found for the contact conditions when the shear vanishes; recent models generated include research by Herrera et al (2004b), Maharaj and Govender (2005) and Misthryet al (2008) where the Weyl tensor components vanish.
Aspects of irreversible thermodynamics
The rate at which entropy is produced is given by the divergence of the entropy four current. The assumption is that nonequilibrium states can be fully specified by the hydrodynamic tensors nα, Tαβ alone. In irreversible thermodynamics, the shape of Rα in the standard theory is different from that in the extended theory.
Several applications of irreversible thermodynamics in general relativity have so far used Eckart's theory as presented above. If the thermodynamic force in this theory is suddenly zero, then the corresponding thermodynamic current instantly disappears.
Causal thermodynamics
4.3.3) Following a similar approach as in the standard theory, we impose linear relationships between the thermodynamic fluxes and forces (extended). This is consistent with an assumption that these terms are negligible compared to the other terms in the equations. In the analysis of the gravitational collapse of liquid spheres, the thermal relaxation time τ is frequently ignored, as it is usually very small compared to the typical time scales of collapse for gravitational systems (for phonon-electron interaction τ ~ 10−10− 11 sec , and for phonon-phonon and free electron interactionτ ~10−13 sec at room temperature).
Furthermore, events prior to relaxation can significantly affect the subsequent evolution of the system (ie for longer than τ). It has been shown for spherically symmetric stars with radial heat flow, the temperature gradient arising as a result of the perturbation, and thus the luminosity, is highly dependent on the product of the relaxation time in the period of the stellar oscillation.
Thermodynamics of radiating stars
We consider the situation when energy is carried away from the stellar core by massless particles that are thermally generated with energies of the order of kT. This is determined by the expression for the temperature of the star at its surface Σ. The above expressions for temperature T were first created by Govender and Govinder (2001b) who extended Govenderet al's (1998) model of causal radiative collapse.
To investigate the relaxation effects due to shear stress, we use (4.3.10) as the definition for the shear stress relaxation time. This allows for both cases where the relaxation time scale is larger than the collision time (β > 1, Maartens and Triginer 1998), as well as the case of pertubative deviations from the quasi-stationary case (β1 < 1).
Introduction
To fully describe the evolution of such a system, one must satisfy the junction conditions for the smooth matching of the inner and outer spacetimes. In the case of a shear-free, spherically symmetric star undergoing dissipative gravitational collapse, these conditions were first derived by Santos (1985) . It is in this spirit that we seek an exact solution of the Einstein field equations representing a spherically symmetric radiating star undergoing dissipative gravitational collapse with non-zero shear.
In this chapter, we investigate the role of anisotropy in the thermal evolution of a spherically symmetric star undergoing dissipative gravitational collapse in the presence of shear. We are also in a position to integrate the causal heat transport equation and obtain the corresponding causal temperature profile in the interior of the star.
Radiating anisotropic collapse
We show that previous assumptions that the relaxation time is proportional to the corresponding collision time only apply to a limited regime of stellar evolution. In §5.2 we give the Einstein field equations for the most general, non-rotating, spherically symmetric line element. The energy momentum tensor for the internal spacetime is that of an imperfect fluid with heat conduction and pressure anisotropy.
The general solution for the special case of constant mean collision time in both causal and noncausal theory is presented. Given this, we expect that the luminosity radius of the star in both the radiative and the non-radiative case should have the same time dependence.
Physical considerations
The redshift for an observer at infinity diverges at the time of formation of the horizon determined from. In particular, we observe that the causal temperature is larger than the non-causal temperature at every interior point of the star. For small values of β, the temperature profile is similar to that of the non-causal theory; but asβis increased, i.e.
From the plots in Figure 5.2, the relaxation time of the shear stress also exhibits significantly different behavior when the fluid is close to hydrostatic equilibrium as opposed to collapse at a later time. We further note that while the relaxation time of the heat flux is assumed to be constant, the relaxation time of the shear stress increases as the collapse progresses.
Discussion
For a more realistic comparison of relaxation times, an analytical solution of the causal temperature equation for non-constant relaxation times is required. The final outcome of a star's gravitational collapse is still very much open to debate with the discovery of models that allow bare singularities (Haradaet al 1998, Kudohet al 2000). Shear effects have been shown to slow the formation of the apparent horizon so that it becomes finite.
We find that relaxation effects increase the temperature at each interior point of the stellar configuration. Our studies show that the BCD model shows physically reasonable temperature profiles throughout the star's evolution.
The BCD radiating model revisited
The internal matter distribution is that of an imperfect fluid with heat flux and the external spacetime is described by the Vaidya radiative metric (Vaidya 1951). The crossing conditions required for the smooth matching of inner and outer space in a time-like four-dimensional hypersurface are solved exactly. To generate an accurate model of radiative gravitational collapse, the following ansatz was adopted for the metric functions in (6.2.1).
The crossing conditions required for the smooth matching of the inner metric (Banerjee et al 2002) and the outer Vaidya metric (6.2.7), along a time hypersurface Σ, are. 6.2.8c) where mΣ represents the total mass of the stellar configuration of radius r within Σ. It is interesting to note that the parameters in (6.2.11) can be chosen so that 2mΣ/¯rΣ < 1 to avoid the appearance of the horizon at the boundary.
Causal temperature profiles
6.3.9), which diverges for an observer to infinity at the time of horizon appearance. The brightness of the star as perceived by an observer at infinity is given by We now turn our attention to the development of the temperature profiles for the BCD model.
To this end, we use the causal transport equation for the heat flux, which in the absence of rotation and viscous stress is given by. In the case of variable collision time, Figure 6.2, we see that the causal temperature everywhere is greater than the corresponding non-causal temperature in the interior of the star.
Concluding remarks
In this study, involving the dynamics of dissipative gravity collapse, an analytical model for the anisotropic shear case was presented. It was shown that the relaxation time for heat flow differs from that for shear flow. An analysis of the thermodynamics of a radiating star undergoing gravitational collapse without forming a horizon was presented in Chapter 6.
Radiative stellar models studied here could also act as precursor models for the study of the nature of the singularities found during the collapse. Our models narrow the window on the initial conditions for the study of ongoing gravitational collapse.
