1. QUASI-SPHERICAL GRAVITATIONAL COLLAPSE Ujjal Debnath Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah , India. ITP, CAS, Beijing, China 13th June, 2012

2. Szekeres Metric

3. Szekeres Metric Quasi-Spherical Solutions

4. Szekeres Metric Quasi-Spherical Solutions Dust Collapse

5. Szekeres Metric Quasi-Spherical Solutions Dust Collapse Collapse with Pressure

6. Szekeres Metric
The Cosmic Censorship Conjecture is yet one of the unresolved problems in General Relativity. It states that the space-time singularity produced by Gravitational Collapse must be covered by the horizon. In fact the conjecture has not yet any precise mathematical proof. Most of the counter examples of Cosmic Censorship Conjecture present in the literature either belong to Vaidya space-time or to the Tolman-Bondi space-time. Here the objective is to fully investigate the situation in the background of higher dimensional Szekeres space-time.

7. An extension of this metric in (n+2)D :
Szekeres [1975] gave a class of inhomogeneous solutions representing irrotational dust for the metric of the form (known as Szekeres metric) :
[P. Szekeres, Comm. Math. Phys. 41 , 55 (1975)]
An extension of this metric in (n+2)D :
[S. Chakraborty, U. Debnath, IJMPD, 13 , 1085 (2004)]
where

8. Both radial and transverse stresses, the energy- momentum tensor has the form:
The Einsteins field equations are:

9. Last two equations give:
This implies:

10. This type of solution is called Quasi-Spherical solution
Dust Solutions
Choose
The solutions are ( ) :
This type of solution is called Quasi-Spherical solution

11. The expression for energy density for dust model:
Assume:
with transformation:

12. The Szekeres metric reduces to (n+2)D spherically symmetric metric:
This is known as Tolman-Bondi metric.
It is to be noted that if the arbitrary functions depends on r, then we can not get the above spherical form of the space-time. So these arbitrary functions play an important role to identify the nature of the space-time.
But if we get quasi-cylindrical solution.

13. Dust Collapse
Initial choice of R is r i.e., [P. Szekeres, PRD, 12, 2941 (1976)]
The curve defines the shell-focusing singularity and is characterized by
The curve is the equation of apparent horizon which marks the boundary of the trapped region and is characterized by
The necessary condition for possibility of naked singularity :

14. The coefficients of and are related as :
For the collapse from regular initial hypersurface we can choose [U. Debnath, S. Chakraborty and J. D. Barrow, Gen. Rel. Grav., 36, 231 (2004)]
The coefficients of and are related as :

15. OR

16. Formation of Trapped Surfaces
Marginally bound case :
The time difference between the formation of apparent horizon and central singularity is
For , if initial density gradient falls off rapidly near the centre (i.e., ) then one can not get any restriction on the dimension of the space-time for existence of NS if

17. For , we have the following results :
However for and initial density gradient falls off rapidly near the centre ( ) then NS is possible if
For , we have the following results :
Possibility of NS:

18. Non-Marginally bound case : For consistency of the field equations,
Here NS is possible in any dimension for

19. Radial Null Geodesics
For marginally bound case, it is possible to examine whether one or more outgoing radial null geodesics which terminate in the past at the central singularity in the form (near r = 0) [U. Debnath and S. Chakraborty, JCAP , 0405 , 001 (2004)]
Also for central singularity near r = 0 ,
Comparing these two equations,

20. Then starting from the metric, an outgoing radial null geodesic must satisfy
which implies
For , result is same as TBL model.
For one gets,
Thus,
so, : NS is possible in all dimension.
: NS is possible only in 4 dimension.

21. For the other case, i.e., , a comparative study shows which implies the possible choice for n is 2 or 3. Thus as before, NS is possible only for 4D and 5D and for higher dimensions ( ) all singularities are covered by trapped surfaces leading to black hole.
Similarly, for non-marginally bound case, one can shows that outgoing radial null geodesic terminated in the past at the singularity in any dimensions.
Note: If the assumption on the density gradient is removed then NS is possible in any dimension.

22. Geodesics and Nature of Singularity
Assume
[P.S.Joshi and I.H.Dwivedi, PRD, 47, 5357 (1993) ; U.Debnath, S.Chakraborty and J.D.Barrow, GRG, 36, 231 (2004)]
Determine the nature of singularity:
Radial null geodesic :
Write :
Now study limiting behaviour of X as we approach singularity at R=0, u=0 along radial null geodesic.

23. The polynomial equation of :
Decrease : other parameters fixed --- Formation of NS is more
probable in higher dimensions.
Decrease : other parameters fixed --- do
Decrease : Result is opposite.

24. Table Positive roots ( X0 )
4D D D D D

25. Collapse with Pressure
[S. Chakraborty, S. Chakraborty and U. Debnath, IJMPD, 14, 1707 (2005) ; gr-qc/ ]
In this case:
Since is regular initially at the center and blows up at the singularity, choose

26. This implies : where, Assume: So on apparent horizon,

27. which implies : ( for )
NS cannot possible for
For NS is possible upto 5D.

28. In dust collapse, we have BH or NS if initial density gradient at the centre is +ve or –ve definite. In definiteness in the sign of one may note that if initial density and radial pressure has identical behaviour (increase or decrease) then even with –ve initial density gradient it is possible to have BH while if initial density and pressure have opposite nature (one increase when other decrease and vice versa) then the behaviour is identical to dust collapse. Therefore, pressure tries to resist the formation of NS.

29. Effect of Equation of State
Assume,
In this case,
For simplicity,
We get,

30. For , possibility of NS :
Thus validity of CCC for higher dimension strongly depends on equation of state.

31. Geodesic Study for Imperfect Fluid Collapse
Define,
We have the polynomial equation of :

32. If is invariant then the final fate of the singularity remains same i
If is invariant then the final fate of the singularity remains same i.e., we have space-time dimension and equation of state and we have another space-time dimension and the equation of state
Thus, the state of collapse (BH or NS) remain invariant for imperfect fluid with equation of state
in n dimension, dust in (n+1) dimension and exotic matter with in (n+2) dimension are equivalent from the point of view of final state of collapse. This is a interesting feature for nature of the singularity.

33. Summary 1. Consider (n+2)D Szekeres metric.
2. Get quasi-spherical solution.
3. For dust collapse NS is possible upto 5D for
and any dimension for if initial density gradient is zero near the centre.
4. For radial pressure with equation of state
, the state of collapse remains invariant as long as the sum is fixed.

34. Thanking You