Layout 1.Junction conditions 2.Solution of Einstein Mexwell field

Gravitational Charged Perfect FluidGravitational Charged Perfect FluidCollapse in the Friedman Universe Collapse in the Friedman Universe

Models Models

Ghulam Abbas Supervised by

Prof. Dr. Muhammad SharifDepartment of the Mathematics

University of the Punjab

Basic Definitions

Literature Review

Motivation

Research Work

Conclusion

References

LayoutLayout

Department of MathematicsGhulam Abbas University of the Punjab

Gravitational CollapseGravitational Collapse


Basic DefinitionsBasic Definitions

http://www.sonic.net/~nbs/projects/astro305-3/collapse/gravity_vs_pressure.gif

Curvature, Energy densityand pressure diverges.


Basic DefinitionsBasic DefinitionsSpacetime SingularitySpacetime Singularity

• Curvature or essential singularity• Coordinate Singularity

ExampleExample

)sin()21()21( 22222122 φθθ ddrdrrmdt

rmds +−−−−= −

r=0, r=2m



Kinds of SingularitiesKinds of Singularities

A spacetime singularity is said to be naked when it is observable to local or distant observer.

A spacetime singularity that can not be observed is called covered singularity or black hole.

According to this hypothesis, the singularities that appear in the gravitational collapse are always covered by an event horizon.

Naked Singularity

Black Hole

Cosmic Censorship Hypothesis



Oppenheimer and Snyder [1] Schwarzschild solution in exterior and Friedman like solution in interior- black hole

Misner and Sharp [2] For perfect fluid in interior.


Literature ReviewLiterature Review

Markovic and Shapiro [3] Generalized the work of Oppenheimer and Snyder with positive cosmological constant.

Lake [4] Both positive and negative cosmological constant.

Rocha et al. [5] Self-similar gravitational collapse of perfect fluid using Israel's method.

Herrera and Santos [6] Investigated the matching conditions for the collapse of perfect fluid.



Ghosh and Deshkar [7-8]Collapse of radiating star with plane and spherical symmetric boundaries. Also they discuss the higher dimensional dust collapse.

Debnath et al. [9]Non-adiabatic collapse of a quasi-spherical radiating star and discussed some thermo-dynamical relations.

Sharif and his Collaborators [10]Darmois and Israel Junction conditions, High speed approximation scheme.




MotivationMotivation

The cosmic censorship conjecture is a major motivation to study gravitational collapse. For this purpose, we study the gravitational collapse in the presence of electromagnetic field and cosmological constant. Main objectives of this work are•To check the validity of CCH.• To see the effects of electromagnetic field on

the rate of collapse.

Junction Conditions

)sin()()( 222222222 φθθχ ddftadtadtds k +−−=−

1sinh0,1,sin)(

−====+==

kkkfk

χχχχ

The interior metric is given by

)1(

)2(

where

)(ta is a scale factor.

Ghulam Abbas Department of Mathematics University of the Punjab

)sin(1 2222222 φθθ ddRdRF

FdTds +−−=+

.3

21)(2

2

2 RRQ

RMRF Λ

−+−=

The exterior metric is taken as

)3(

where

Research WorkResearch Work

( ) ( )Σ+Σ− = 22 dsds (4)

• The continuity of line elements



• The continuity of extrinsic curvature over ∑gives

0][ =−= −+ababab kkk

where the extrinsic curvature abk

(5)

)3,2,0,()(2

=∂∂

±∂∂Γ+

∂∂±∂

= ±±± baxxxnk babaab

νμσμν

σ

σξξξξ

(6)

(7)0),( =−= Σ− χχχ th


The defining equations for hyper- surface in terms of interior and exterior co-ordinates

0)(),( =−= Σ+ TRRTRh

Using Eq.(7) in Eq.(1)

(9)

(8)


)sin)(()()( 2222222 φθθχ ddftadtds k +−= ΣΣ−

)sin(])()(

1)([)( 222222 ϕθθ ddRdTdTdR

RFRFds +−−= Σ

Σ

ΣΣΣ+

0)()(

1)( 2 >− Σ

ΣΣ dT

dRRF

RF

(10)


Similarly using Eqs.(2) and (8), we get

From Eqs. (4), (9) and (10), it follows that

dtdTdTdR

RFRF =⎥

⎦

⎤⎢⎣

⎡− Σ

ΣΣ

21

2)()(

1)(

(11)

(12)


)()( ΣΣ = χkftaR

Now the outward normals

)0,0,,(

)0,0),(,0(

dtdT

dtdRn

tan

Σ+

−

−=

=

μ

μ (13)

(14)



000 =−k

( )Σ−− ′== ffakk 332

22 csc θ

Σ

+ ⎟⎠⎞

⎜⎝⎛ +−−= 23

00 23

2RT

dRdF

FT

dRdFFTRTRk &&&&&&&&&

(15)

(16)

(17)

( )Σ++ == TFRkk &33

222 csc θ (18)

Department of Mathematics Abbas University of the PunjabGhulam


The continuity of extrinsic curvature gives

+−

+

=

=

2222

00 0

kk

k

Using equations (16)-(20) along with Eqs.(3), (12) and (13) . The junction conditions turns out to be

(21)0)( =′ Σf&

(19)

(20)

(22)



Equations (11), (12), (21) and (22) are the necessary and sufficient conditions for the smooth matching of the interior and exterior regions of a star.

Σ′−++Λ

−= )22262

( 2322

33 ffafaaafQfaafM &

μνμν

μνμννμμν ρρ

gTg

TgpuupR

em

em

Λ−−

+−++=

)(

)(

21

})(21){(

(23)

Solution of Einstein Field EquationsSolution of Einstein Field Equations



)41(

41)( δω

δωμννωμδδω

μν πFFgFFgT em +−= (24)

where

(26)

νμμνμν φφ ,, −=F

)0,0,0),,(( tχφφμ = (27)

(28)



(25)

μμνν π JF 4; =

μμ σ uJ =

χφ∂∂

−=−= 1001 FF (29)

22

2

42 aff σπ

χφ

=′

+∂∂

(30)

Putting in (28) and using (29),

0)3()1( 32 =∂∂

−∂∂

∂∂

χφ

χφ

aa

at&

(31)

0=μ

1=μ



For in Eq. (27) and using Eq.(29), we get

Integrating Eq. (30), we get

2

1)(af

q χχφ=

∂∂

(32)

where χσπχ

χdfaq 2

0

34)( ∫= (33)

Also

EEY

qE ~4,)(2 πχ

== (34)



aEFF −=∂∂

−=−=χφ

1001 (35)

22)(11

2)(00 8

1,81 aETET emem

ππ−== (36)

)(22

2)(33

22)(22 sin,)(

81 ememem TTafET θπ

==

(37) 0)( =emT



(38)

(39)

The Einstein Field Equations are

Λ+−−=−′

+′′

+−−

Λ++−=′′

+−−

Λ−++=−

222

2

22

2

222

2

2

)(4]1[1

)(422

)3(43

Epff

fff

aaa

aa

Epfa

faa

aa

Epaa

ρπ

ρπ

ρπ

&&&

&&&

&&

(40)

The presence of electromagnetic field in the Freidmann model causes the distortion of its generic properties. To overcome this problem, we follow Tsagas [11] and assume that electromagnetic field is weak relative to matter , i.e., if is the electromagnetic field contribution in the system then


2E

ρ<<2E

(41)

Using Eqs.(41)and (22), we get

( )af

mpEafWfa 23

8)(1)( 022

22 +−+Λ

+−=π

&



where is arbitrary function.)(χW

6)()]8[()(

2

3

02

2 afpErmaf

QM π−+Λ++= (42)

Integrating Eq.(22), we get)()( χwf =′

6)(]8[)(

2),(~ 3

02

2 afpErmafqrtM π−+Λ++= (43)

Also

and the condition0)8( 0

2 >−+Λ pE π

),(sinh8

6 323

1

02 t

pmaf

Eχα

π ⎟⎟⎠

⎞⎜⎜⎝

⎛

−+Λ=

Using Eqs.(41) and (44)

(45)

])([2

)8(3),( 0

2

ttpE

t s −−+Λ

= χπ

χα

where(46)


Research WorkResearch WorkHere, we assume

1)( =χW (44)

For line element (1), the equation for apparent horizon is given by

(47)

Apparent HorizonsApparent Horizons



0)()()( 22,, =′−= ffaafafg &βα

αβ

063))(8( 30

2 =+−−+Λ maffapE π (48)

The positive real roots of Eq.(48) will give apparent horizons

Case (1) For0

2 813

pEm

π−+Λ<

3cos

82)(

021

ψπ pE

af−+Λ

= (49)



)3

sin3(cos8

1)(0

22ψψ

π−

−+Λ

−=

pEaf (50)

where0

2 83cos pEm πψ −+Λ−= (51)

For , it follows from Eqs.(49-51) that 0=m

0)(,)8(

3)( 20

21 =−+Λ

= afpE

afπ (52)

1)(af 2)(afis called cosmological horizon and

is called black hole horizon.Case(2) For

02 813

pEm

π−+Λ=

)(8

1)()(0

221 afpE

afaf =−+Λ

==π

(53)

unique horizon.



)8(3)(

)8(1)(0

021

022 pE

afpE

afππ −+Λ

≤≤−+Λ

≤≤

The range for cosmological horizon and black hole horizon turns out to be

The black hole horizon has area

02

2

84)(4

pEaf

πππ−+Λ

=

and the cosmological horizon has area between

(54)

)8(4

02 pE ππ−+Λ



)8(12

02 pE ππ−+Λand

(55)

(56)

)2,1()12

)((sinh)8(3

2 1

02

=−−+Λ

−= − nm

afpE

tt nsn

π(57)

Eqs.(45) and (48) , we get

)8(13

02 pE

mπ−+Λ

>Case(3) For

we have no positive real roots, hence no horizon.





From Eq.(54) it is clear that 21 )()( afaf ≥

Above inequality yields 21 tt ≤

Using Eqs.(12) and (44) in Eq. (58) for exterior metric the Newtonian potential takes the form

6]8[)(

2

02 RpE

RmR πφ −+Λ+= (59)

)1(21

00g−=φ (58)

The Newtonian potential is



Formulation of Newtonian ModelFormulation of Newtonian Model

3)8( 0

22

RpERmF π−+Λ+−=

The acceleration of the collapsing system from Eq. (41)

The Newtonian force is

(60)

3)8(

)()( 0

22

afpEafmaf π−+Λ+−=&& (61)

Raf =)(From Junction conditions

)( &&afF =





)1

02 pE

R−+Λ

=0

2 831

pEm

π−+Λ=

For

F vanishs.

* Generalization.

* Generic properties of the model require weak electromagnetic field slow rate of collapse as compared to any other spacetime.

* Horizons form earlier than singularity, CCH is valid.

* Space-like singularity.

* Two horizons, instead of one.

208 Ep −>Λ π


Conclusion Conclusion

[1] Oppenheimer, J.R. and Snyder, H.: Phys. Rev.56 (1939)455.

[2] Misner, C.W. and Sharp, D.: Phys. Rev. 136(1964)B571.

[3] Markovic, D. and Shapiro, S.L.: Phys. Rev.D 61 (2000)084029.

[4] Lake, K.: Phys. Rev. D62 (2000)027301.


ReferencesReferences

[5] Sharif, M. and Ahmad, Z.: J. Korean Physical Socity. 52 (2008) 980. Mod. Phys. Lett. A22 (2007) 1493. Int. J. Mod. Phys. A23 (2008)181. Gen. Relativ. Grav . 39(2007)1331 .

[6] Sharif, M. and Iqbal, K.: Mod. Phys. Lett. A24 (2009)1533.

[7] Sharif, M. and Rehmat, Z.: Gen. Relativ. Grav .(to appear, 2010).



[8] Debnath, U., Nath, S. and Chakraborty, S.: Gen.Relativ. Grav.37(2005)215.

[9] Sharif, M. and Abbas, G.: Mod. Phys. Lett. A24(2009)2551. J. Korean PhysicalSoc. (to appear , 2010). Astrophys. SpaceSc. (to appear , 2010).

[10] Tsagas, C.G.: Class. Quantum Grav.22(2005)393.



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Layout 1.Junction conditions 2.Solution of Einstein Mexwell field

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