Abstract
The junction conditions between static and nonstatic space–times are studied for analysing gravitational collapse in the presence of a cosmological constant. We have discussed about the apparent horizon and their physical significance. We also show the effect of cosmological constant in the collapse and it has been shown that cosmological constant slows down the collapse of matter.
1 INTRODUCTION
Gravitational collapse is one of the most important problems in classical general relativity. Usually, the formation of compact stellar objects such as white dwarf and neutron star are preceded by a period of collapse. Hence for astrophysical collapse, it is necessary to describe the appropriate geometry of interior and exterior regions and to determine proper junction conditions which allow the matching of these regions.
The study of gravitational collapse was started by Oppenhiemer & Snyder (1939). They studied collapse of dust with a static Schwarzschild exterior while interior space–time is represented by Friedmanlike solution. Since then several authors have extended the above study of collapse of which important and realistic generalizations are the following: (i) the static exterior was studied by Misner & Sharp (1964) for a perfect fluid in the interior; (ii) using the idea of outgoing radiation of the collapsing body by Vaidya (1951), Santos and collaborations (Santos 1984; de Oliveira, Santos & Kolassis 1985; Santos 1985; de Oliveira, de F. Pacheco & Santos 1986; de Oliveira & Santos 1987; de Oliveira, Kolassis & Santos 1988) included the dissipation in the source by allowing radial heat flow (while the body undergoes radiating collapse). Ghosh & Deshkar (2003) have considered collapse of a radiating star with a plane symmetric boundary (which has a close resemblance with spherical symmetry Ghosh & Deshkar 2000) and have concluded with some general remarks. On the other hand, Cissoko et al. (2001) and Goncalves (2001) have studied junction conditions between static and nonstatic space–times for analysing gravitational collapse in the presence of dark energy has been investigated by Mota & Van de Bruc (2004) and Cai & Wang (2005). The effect of cosmological constant (a source of dark energy) in cosmology has been shown by Lahav et al. (1991) and BalagueraAntolinez et al. (2006).
So far most of the studies have considered in a star whose interior geometry is spherical. But in the real astrophysical situation the geometry of the interior of a star may not be exactly spherical, rather quasispherical in form. Recently, solutions for arbitrary dimensional Szekeres' model with perfect fluid (or dust) (Chakraborty & Debnath 2004) has been found for quasispherical or quasicylindrical symmetry of the space–time. Also a detailed analysis of the gravitational collapse (Szekeres 1975a; Debnath, Chakraborty & Barrow 2004) has been done for quasispherical symmetry of the Szekeres' model. The junction conditions between quasispherical interior geometry of radiating star and exterior Vaidya metric (Debnath, Nath & Chakraborty 2005) have also been studied. In this paper, we have considered the interior space–time V^{−} by Szekeres' model (Szekeres 1975a,b; Chakraborty & Debnath 2004; Debnath et al. 2004, 2005) while for exterior geometry V^{+} we have considered Schwarzschild–de Sitter space–time. The plan of the paper is as follows. The junction conditions has been presented in Section 2. The apparent horizons and their physical interpretations are shown in Section 3. The paper ends with a short conclusion in Section 4.
2 JUNCTION CONDITIONS
Let us consider a timelike 3D hypersurface Σ, which divides 4D space–time into two distinct 4D manifolds V^{−} and V^{+}. For junction conditions we follow the modified version of Israel (1966) by Santos (1984, 1985). Now the geometry of the space–time V^{−} inside the boundary Σ is given by the Szekeres space–time
where α and β are functions of all space–time variables.
The metric coefficients α and β have the explicit form for dust matter with cosmological constant Λ (Mota & Van de Bruc 2004; BalagueraAntolinez, Boehmer & Nowakowski 2006):
The evolution equation for R is
where F(r) (>0) and f(r) are arbitrary functions of r and A(r), B_{1}(r), B_{2}(r) and C(r) are arbitrary functions of r along with the restriction
Assuming , equation (1) becomes
For exterior space–time V^{+} to Σ, we have considered the Schwarzschild–de Sitter space–time
where N(z) = 1 − 2M/z− (Λ/3) z^{2}, M is a constant.
The intrinsic metric on the boundary Σ of the hypersurface r=r_{Σ} is given by
Now Israel's junction conditions (as described by Santos 1984, 1985) are:

The continuity of the line element, i.e.
where ( )_{Σ} means the value of () on Σ.

The continuity of extrinsic curvature over Σ gives
where due to Eisenhart (1949) the extrinsic curvature has the expression
Here, ξ^{i}= (τ, x, y) are the intrinsic coordinates to Σ, χ^{σ}_{±}, σ= 0, 1, 2, 3 are the coordinates in V^{±} and n^{±}_{α} are the components of the normal vector to Σ in the coordinates χ^{σ}_{±}.
Now for the interior space–time described by the metric (1) the boundary of the interior matter distribution (i.e. the surface Σ) is characterized by
where r_{Σ} is a constant. As the vector with components ∂f/∂χ^{σ}_{−} is orthogonal to Σ so we take
So comparing the metric ansatzs given by equations (1) and (9) for d r= 0 we have from the continuity relation (10)
Also the components of the extrinsic curvature for the interior space–time are
On the other hand, for the exterior Schwarzschild–de Sitter metric described by the equation (8) with its interior boundary, given by
the unit normal vector to Σ is given by
and the components of the extrinsic curvature are
Hence, the continuity of the extrinsic curvature due to junction condition (equation 11) gives
Now using the junction condition (20) with the help of equations (2)–(4), we have (on the boundary) (Szekeres 1975a)
which can be interpreted as the energy conservation equation on the boundary. It is to be noted that the cosmological term leads to a repulsive term to the Newtonian potential (BalagueraAntolinez et al. 2006), i.e.
3 TRAPPED SURFACES: COSMOLOGICAL AND BLACK HOLE HORIZONS
As the present space–time geometry is complicated, so it is difficult to find the formation of event horizon. However, trapped surfaces which are spacelike 2surfaces with normals on both sides are future pointing converging null geodesic families, may be considered here. In fact, if the 2surface S_{r,t}(r= constant, t= constant) is a trapped surface then it and its entire future development lie behind the event horizon unless the density falls off fast enough at infinity. So if K^{μ} is the tangent vector field to the null geodesics orthogonal to the trapped surface then K^{μ} should satisfy (i) K_{μ}K^{μ}= 0, (ii) K^{μ}_{;ν}K^{ν}= 0.
Also the convergence (or divergence) of the null geodesics on the trapped surface is characterized by the sign of the scalar K^{μ}_{;μ} (K^{μ}_{;μ} < 0 for convergence, K^{μ}_{;μ} > 0 for divergence). It is to be noted that the inward geodesics converges initially and throughout the collapsing process but the outward geodesics diverges initially but becomes convergent after a time t_{ah}(r) (the time of formation of apparent horizon) given by
Then from the evolution equation (4), we have
The possible solutions of equation (25) for different choices of Λ and F(r) are shown in Table 1.
Restrictions on Λ, F(r)  Solutions of equation (25): different horizons 

(i) Λ= 0  R=F(r), Schwarzschild horizon 
(ii) F(r) = 0  R= 0 (black hole) 
(de Sitter horizon)  
(iii)  Two horizons: 
(iv)  R= 0 
(v)  No horizon 
Restrictions on Λ, F(r)  Solutions of equation (25): different horizons 

(i) Λ= 0  R=F(r), Schwarzschild horizon 
(ii) F(r) = 0  R= 0 (black hole) 
(de Sitter horizon)  
(iii)  Two horizons: 
(iv)  R= 0 
(v)  No horizon 
For marginally bound case [i.e. f(r) = 0] the evolution equation (4) can be solved as
where t=t_{c}(r) is the time of collapse of a shell of radius R[i.e. R= 0 at t=t_{c}(r)].
Hence, the time of formation of apparent horizon t_{ah}(r) is given by
where R_{H} is a root of the equation (25).
Thus from Table 1, we see that in the fourth case [i.e. ] we have two horizons namely cosmological and black hole horizons (R_{1}≥R_{2}) and let t_{1} and t_{2} be their time of formation then from equation (27), t_{1}≤t_{2}, i.e. cosmological horizon forms earlier than the formation of black hole horizon.
Further, if T_{1} and T_{2} be the time differences between the formation of cosmological horizon and singularity and the formation of black hole horizon and singularity, respectively, then
A straightforward calculation shows
Thus, the time difference between the formation of singularity and cosmological horizon decreases with F increases while the time difference between the formation of singularity and black hole horizon increases with F. As F is related to the mass of the collapsing system so for more massive quasispherical model, the time of formation of singularity and cosmological horizon become close to each other while the time difference between the formation of black hole horizon and that of cosmological horizon becomes smaller.
4 CONCLUSION
In this paper, the collapse of a quasispherical star is considered where the exterior geometry corresponds to Schwarzschild–de Sitter space–time. The junction conditions on the boundary show a energy conservation equation on it.
Due to the presence of the cosmological constant Λ, the Newtonian force is given by (see equation 23) (Engineer, Kanekar & Padmanabhan 2000)
For collapsing process the force should be attractive in nature and as a result R should always be less than (3M/Λ)^{1/3}. Further, the rate of collapse has the expression
which shows that the presence of Λ term slows down the collapsing process and hence influences the time difference between the formation of the apparent horizon and the singularity.
As the presence of a cosmological constant (dark energy) induces a potential barrier to the equation of motion so particles with a small velocity are unable to reach the central object. This idea can be used astrophysically for a particle orbiting a black hole, which contains dark energy and an estimation of minimum velocity can be done for which the particle enters inside the black hole. Consequently, the amount of dark energy in the black hole can be calculated.
Lastly, due to the presence of the cosmological constant, there are two physical horizons – the black hole horizon and the cosmological horizon. Further, for more massive collapsing system, the time of formation of the two horizons become very close to each other. Moreover, asymptotic flatness of the space–time is violated due to the presence of the cosmological constant.
One of the authors (SC) is thankful to CSIR, Government of India for providing a research project no. 25(0141)/05/EMRII. Also the authors are thankful to IUCAA for warm hospitality where the major part of the work was done.