Charged Perfect Fluid Cylindrical Gravitational Collapse

Muhammad SHARIF� and Ghulam ABBASy

Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan

(Received April 15, 2011; accepted July 26, 2011; published online September 15, 2011)

This paper is devoted to study the charged perfect fluid cylindrical gravitational collapse. For this purpose, we find anew analytical solution of the field equations for non-static cylindrically symmetric spacetime. We discuss physicalproperties of the solution which predict gravitational collapse. It is concluded that in the presence of electromagneticfield the outgoing gravitational waves are absent. Further, it turns out that when longitudinal length reduces to zero dueto resultant action of gravity and electromagnetic field, then the end state of the gravitational collapse is a conicalsingularity. We also explore the smooth matching of the collapsing cylindrical solution to a static cylindricallysymmetric solution. In this matching, we take a special choice of constant radius of the boundary surface. We concludethat the gravitational and Coulomb forces of the system balance each other.

KEYWORDS: gravitational collapse, junction conditions, cylindrical symmetry, electromagnetic field

1. Introduction

Gravitational collapse of a massive star occurs when allthe thermonuclear reactions in the interior of a star could notfavor the pressure against gravity. Gravitational collapse isone of the most important problems in general relativity.The singularity theorems1) state that there exist spacetimesingularities in the realistic gravitational collapse. Toinvestigate the nature of spacetime singularity, Penrose2)

suggested a hypothesis known as cosmic censorshiphypothesis (CCH). It states that the final fate of gravitationalcollapse of a massive astrophysical object is always a blackhole. This is equivalent to saying that the singularitiesappearing in gravitational collapse are always clothed by anevent horizon.

Many attempts predicted that final fate of gravitationalcollapse of the massive star might be a black hole or nakedsingularity depending upon the choice of initial data. In thischain, Virbhadra et al.3) introduced a new theoretical toolusing the gravitational lensing phenomena. In a recentpaper,4) Virbhadra used the gravitational lensing phenomenato find an improved form of the CCH. The classical paper ofOppenheimer and Snyder5) is devoted to study dust collapseaccording to which singularity is neither locally or globallynaked. In other words, the final fate of the dust collapse is ablack hole. Many people6) extended this work for physicallyexisting form of fluid with cosmological constant inspherically symmetric background.

In order to generalize the geometry of the star, peopleworked on gravitational collapse using cylindrical symme-try. The existence of cylindrical gravitational wavesprovides a strong motivation in this regard. Bronnikov andKovalchuk7) were the poineers to the work on gravita-tional collapse with non-spherical symmetry. Later on, thesame authors8) extended it for some non-spherical exactmodel. Nolan9) investigated the naked singularities in thecylinderical gravitational collapse of counter rotating dustshell.

Hayward10) studied gravitational waves, black holes andcosmic strings in cylindrical symmetry. Sharif and Ahmad11)

analyzed cylindrically symmetric gravitational collapse oftwo perfect fluids using the high speed approximationscheme. They investigated the emission of gravitationalradiation from cylindrically symmetric gravitational col-lapse. Di Prisco et al.12) discussed the shear free cylindricalgravitational collapse using junction conditions. Nakaoet al.13) studied gravitational collapse of a hollow cylindercomposed of dust.

Gravity is the weakest interaction among all the naturalforces. The behavior of electromagnetic field in gravitationalfield has been the subject of interest for many years.Thorne14) developed the concept of cylindrical energy andinvestigated that a strong magnetic field along the symmetryaxis may halt the cylindrical collapse of a finite cylinderbefore it reached to singularity. Oron15) studied the rela-tivistic magnetized star with the poloidal and toroidalfields. Thirukkanesh and Maharaj16) found that the inclusionof electromagnetic field in gravitational collapse wouldcounterbalance the gravitational attraction by the Coulombrepulsive force along with pressure gradient. In recentpapers,17–19) we have investigated the effects of electro-magnetic field on the perfect fluid collapse by using junctionconditions in spherically symmetric background with posi-tive cosmological constant. It has been found that electro-magnetic field reduces the pressure and favors the nakedsingularity formation but cannot play a dominant role. Thusblack hole was formed as a final state of the gravitationalcollapse.

In this paper, we study the cylindrically symmetriccharged perfect fluid gravitational collapse. The mainobjective of this work is to study the final fate of chargedperfect fluid gravitational collapse in the cylindricallysymmetric background. The plan of the paper is as follows:In the next section, we discuss the solution of the Einstein–Maxwell field equations. The physical properties of thesolution are discussed in x3. Section 4 gives the derivationof the matching conditions. We summarize the results in thelast section.

Geometrized units (i.e., the gravitational constant G ¼ 1

and speed of light in vacuum c ¼ 1) are used. All the Latinand Greek indices run from 0 to 3, otherwise, it will bementioned.�E-mail: msharif.math@pu.edu.pk

yE-mail: abbasg91@yahoo.com

Journal of the Physical Society of Japan 80 (2011) 104002

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#2011 The Physical Society of Japan

DOI: 10.1143/JPSJ.80.104002

2. Solution of the Einstein Field Equations

This section is devoted to the solution of the Einstein fieldequations coupled with the charged perfect fluid as thesource of gravitation distributed per unit length of thecylinder. The general cylindrically symmetric spacetime isgiven by the following line element12)

ds2� ¼ A2ðdt2 � dr2Þ � B2 d�2 � C2 dz2; ð2:1Þwhere A, B, and C are functions of t and r. Here we take thefollowing restrictions on the coordinates in order to preservethe cylindrical symmetry of the spacetime

�1 � t � 1; r � 0; 0 � � � 2�; �1 < z < 1:

ð2:2ÞThe proper unit length of the cylinder for the line element(2.1) is defined by

l ¼ 2�BC: ð2:3ÞThe Einstein field equations are given by

G�� ¼ �ðT�

� þ T��ðemÞÞ: ð2:4Þ

The energy–momentum tensor for perfect fluid is

T�� ¼ ð�þ pÞu�u� � p���; ð2:5Þ

where � is the energy density, p is the pressure andu� ¼ A�0� is the four-vector velocity in co-moving coordi-nates. The energy–momentum tensor for the electromagneticfield is given by

T ��ðemÞ ¼ 1

4�ð�F�F� þ 1

4���F�F

� Þ; ð2:6Þ

where F�� is the Maxwell field tensor. Now we solve theMaxwell’s field equations

F�� ¼ �;� � �;�; F��;� ¼ 4�J�; ð2:7Þ

where � is the four potential and J� is the four current. Inco-moving coordinate system, the charge per unit length ofthe cylinder is assumed to be at rest so that the magneticfield will be zero. Thus we can choose the four potential andfour current as follows

� ¼ ððt; rÞ; 0; 0; 0Þ; J� ¼ �u�; ð2:8Þwhere � is charge density. The only non-zero component ofthe field tensor is

F01 ¼ �F10 ¼ � @

@r: ð2:9Þ

Thus the Maxwell field equations take the following form

@2

@r2þ @

@r

B0

Bþ C0

C� 2

A0

A

� �¼ 4��A3; ð2:10Þ

@

@t

1

A4

@

@r

� �þ 1

A4

@

@r

� �_B

Bþ

_C

Cþ 2

_A

A

� �¼ 0; ð2:11Þ

where dot and prime indicate derivatives with respect totime t and radial coordinate r, respectively. Integration ofeq. (2.10) implies that

@

@r¼ 2qA2

BC; ð2:12Þ

where

qðrÞ ¼ 2�

Z r

0

�ðABCÞ dr;

being the consequence of conservation law of charge, i.e.,J�;� ¼ 0 is known as the total amount of charge per unit

length of the cylinder. It is mentioned here that eq. (2.11) isidentically satisfied by the solution of eq. (2.10). We canwrite eq. (2.9) as follows

F01 ¼ �F10 ¼ � 2qA2

BC: ð2:13Þ

The non-zero components of T��ðemÞ turn out to be

T 00

ðemÞ ¼ T 11

ðemÞ ¼ �T 22

ðemÞ ¼ �T 33

ðemÞ ¼ 1

2�

q2

ðBCÞ2 :

The electric field intensity is defined by

Eðr; tÞ ¼ q

2�ðBCÞ : ð2:14Þ

We assume that the charged perfect fluid distributed per unitlength of the cylinder follows along the geodesics in theinterior of the cylindrical symmetry. This requires thatvelocity should be uniform and acceleration must be zerowhich is only possible if A is constant and in particular, wetake A ¼ 1 (for simplicity). Thus the field equation (2.4)takes the following form

�B00

B� C00

Cþ

_C

C� B0C0

BCþ

_B _C

BC¼ 8�ð�þ 2�E2Þ; ð2:15Þ

_B0

Bþ

_C0

C¼ 0; ð2:16Þ

B0C0

BC�

_B _C

BC�

€B

B�

€C

C¼ 8�ð p� 2�E2Þ; ð2:17Þ

�€C

Cþ C00

C¼ 8�ð pþ 2�E2Þ; ð2:18Þ

�€B

Bþ B00

B¼ 8�ð pþ 2�E2Þ: ð2:19Þ

We note that there are five equations and five unknowns B,C, p, �, and E, thus we can find a unique solution.

For this purpose, we adopt the method of separationof variables. The comparison of eqs. (2.18) and (2.19)give

�€C

Cþ C00

C¼ �

€B

Bþ B00

B; ð2:20Þ

which yields the necessary condition for pressure to beisotropic. We take

B ¼ f ðrÞgðtÞ; C ¼ hðrÞkðtÞ: ð2:21ÞUsing eq. (2.21) in eq. (2.16), we get

f ¼ �hL; k ¼ �g�L; ð2:22Þwhere L ( 6¼ 0, for non-trivial solution) is a separationconstant while � and � are integration constants. Usingeq. (2.22) in eq. (2.20), it follows that

€g

g�

€k

k¼ f 00

f� h00

h: ð2:23Þ

Since both sides are functionally independent, we put themequal to constant say M (6¼ 0)

M. SHARIF and G. ABBASJ. Phys. Soc. Jpn. 80 (2011) 104002 FULL PAPERS

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€g

g�

€k

k¼ M ¼ f 00

f� h00

h: ð2:24Þ

Application of eq. (2.22) to eq. (2.24) leads to

€g

g� _g2

g2¼ M

Lþ 1;

h00

hþ h02

h2¼ M

L� 1: ð2:25Þ

The solution to these equations is

gðtÞ ¼ 0 cos1

1�LðWt þ t0Þ;hðrÞ ¼ 1 cosh

11þLðSr þ r0Þ; ð2:26Þ

where 0, 1, t0, and r0 are constants of integration. Further,W and S are given by the following relations

W ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMðL� 1ÞLþ 1

r; S ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMðLþ 1ÞL� 1

r: ð2:27Þ

Using eq. (2.26) in eq. (2.22), it follows that

kðtÞ ¼ 2 cosL

L�1ðWt þ t0Þ;f ðrÞ ¼ 3 cosh

L1þLðSr þ r0Þ; ð2:28Þ

where 2 and 3 are constants of integration. Thus the metriccoefficients, given by eq. (2.21), turn out to be

B ¼ �coshL

1þLðSr þ r0Þ cos 11�LðWt þ t0Þ; ð2:29Þ

C ¼ �cosh1

1þLðSr þ r0Þ cos LL�1ðWt þ t0Þ; ð2:30Þ

where � ¼ 0 3, � ¼ 1 2. Consequently, the spacetime(2.1) takes the form

ds2� ¼ dt2 � dr2 ��2

� cosh2L1þLðSr þ r0Þ cos 2

1�LðWt þ t0Þ d�2

��2 cosh2

1þLðSr þ r0Þ� cos

2LL�1ðWt þ t0Þ dz2: ð2:31Þ

Using the following transformations

Sr0 ¼ Sr þ r0;

Wt0 ¼ Wt þ t0;

�0 ¼ ��;

z0 ¼ �z;

this metric reduces to

ds2� ¼ dt02 � dr02 � cosh2L1þLðSr0Þ

� cos2

1�LðWt0Þ d�02� cosh

21þLðSr0Þ cos 2L

L�1ðWt0Þ dz02: ð2:32ÞBy assuming � ¼ 1, it is clear that the above metricpreserves cylindrical symmetry with the restriction oncoordinates given by eq. (2.2). Here we take

~B ¼ coshL

1þLðSr0Þ cos 11�LðWt0Þ;

~C ¼ cosh1

1þLðSr0Þ cos LL�1ðWt0Þ:

3. Physical Properties of the Solution

Here, we discuss some physical and geometrical proper-ties of the solution. The physical parameters, i.e., pressure p,density �, and the electric field intensity E for the metric(2.32) are given by

p ¼ 1

16�

S2

1þ L� 4 tan2ðWt0ÞW2L

ð1� LÞ2 þ W2

ðL� 1Þð2L� 1Þ� �

; ð3:1Þ

E ¼�

1

32�2

�2Lð1þ LÞ2W2 sec2ðWt0Þ � ð1þ LÞ3W2 � ðL� 1Þ3S2

ð1� L2Þ2 þ 2LS2 sec2 hðSr0ÞðLþ 1Þ2

��1=2; ð3:2Þ

� ¼ 1

8�

�S2ð1þ Lþ L2 þ L sec2 hðSr0ÞÞðLþ 1Þ2 þ LW2 tan2ðWt0Þ

ðL� 1Þ2� �

: ð3:3Þ

We would like to mention here that eqs. (2.29), (2.30), and(3.1)–(3.3) satisfy all the field equations with the restrictionon constants given by eq. (2.27). The proper unit length ofthe cylinder for the new metric is given by

l ¼ 2� ~B ~C � 2� coshðSr0Þ cosðWt0Þ ð3:4Þand the longitudinal length in this case is

~l ¼ ~B ~C � coshðSr0Þ cosðWt0Þ: ð3:5ÞThe rate of change of longitudinal length is

_~l ¼ �W coshðSr0Þ sinðWt0Þ; ð3:6Þwhere negative sign shows that motion is directed inward.20)

Thus such motion represents gravitational collapse of thecharged perfect fluid distributed per unit length of thecylinder.

In order to analyze the nature of singularity of thesolution, we use the curvature invariants. Many scalars canbe constructed from the Riemann tensor but symmetryassumption can be used to find only a finite number ofindependent scalars. Some of these are

R1 ¼ R ¼ gabRab;

R2 ¼ RabRab;

R3 ¼ RabcdRabcd;

R4 ¼ RabcdR

cdab:

Here, we give the analysis for the first invariant commonlyknown as the Ricci scalar. For the metric (2.32), it is given by

R ¼ 2

~lð €~B ~C� ~B €~C� ~B00 ~C� ~C00 ~Bþ _~B _~C� ~B0 ~C0Þ; ð3:7Þ

where ~l is given by eq. (3.5). We see that Ricci scalar as wellas all the other curvature invariants and physical parametersof the solution are finite for r0 ! 0. Thus r0 ¼ 0 is theconical singularity of the metric (2.32).

Now we analyze the values of the constants for which thesolution is physical. In this solution, L and M are non-zeroseparation constants for the non-trivial solution while therest are integration constants that are removed by applyingthe transformations to eq. (2.31) and by evaluating thephysical parameters from the field equations. Fromeq. (2.27), it is clear that the constants W and S cannot bechosen arbitrarily. These are non-zero because M 6¼ 0 fornon-trivial solution. Further, forW and S to be real, there arefollowing four possible solutions:

M. SHARIF and G. ABBASJ. Phys. Soc. Jpn. 80 (2011) 104002 FULL PAPERS

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1: L < �1; M > 0; 2: L > �1; M < 0;

3: L > 1; M > 0; 4: L < 1; M < 0:

Keeping in mind these restrictions on the constants, we findthat the cases 1 and 2 lead to non-physical solutions (i.e.,negative energy density for the arbitrary choice ofcoordinates). In the case 3, for 0 < M � 0:5 and 1 < L �1:9, there exists a physical solution which represents thegravitational collapse. The graphs 1–4 in this case indicatethat all the physical quantities become homogeneous. Thusthe geodesic model with charged perfect fluid distributed perunit length of the cylinder is free of initial inhomogeneities.

It is interesting to mention here that pressure remainsfunction of time only for the geodesic model that isanalogous to the spherical case.21) In the case 4 for 0:63 �L � 0:95 and �1 < M � �0:10, all quantities exceptpressure behave like the case 3, in this case pressure isnegative indicating a dark energy solution. As long as therealistic energy condition �þ 3p > 0 holds, the gravityremains attractive.22) However, the violation of this condi-tion i.e., �þ 3p < 0 due to negative pressure, leads tothe repulsive gravitational effects. Thus in the relativisticphysics, negative pressure acting as a repulsive gravityplays the role of preventing the gravitational collapse. Weare interested to study the gravitational collapse which isthe consequence of attractive gravity, so the case 4 isnot interesting here. Thus the only interesting case is thecase 3.

Now we proceed to discuss the energy conditions forcase 3 which are given by the following relations1)

1. Weak energy condition:

� � 0; �þ p � 0; ð3:8Þ2. Dominant energy condition:

�þ p � 0; �� p � 0; ð3:9Þ3. Strong energy condition:

�þ 3p � 0: ð3:10ÞFor the pressure and energy density given by eqs. (3.1) and(3.3) respectively, it follows that

�þ p ¼ 1

16�

�S2

1þ L1� 2

1þ Lþ L2 þ L sec2 hðSr0Þ1þ L

� �

þ W2

L� 1

1

2L� 1� 2L

tan2 Wt0

L� 1

� ��; ð3:11Þ

�� p ¼ 1

16�

� �S2

1þ L1þ 2

1þ Lþ L2 þ L sec2 hðSr0Þ1þ L

� �

þ W2

L� 16L

tan2 Wt0

L� 1� 1

2L� 1

� ��; ð3:12Þ

�þ 3p ¼ 1

16�

�S2

1þ L3� 2

1þ Lþ L2 þ L sec2 hðSr0Þ1þ L

� �

þ W2

L� 1

1

2L� 1� 10L

tan2 Wt0

L� 1

� ��: ð3:13Þ

Notice that all these equations satisfy eqs. (3.8)–(3.10) for0 < M � 0:5, 1 < L � 1:9, 0 � t0 � 1, and 0 < r0.

The rate of change of longitudinal length in Fig. 1 showsthat the longitudinal length is a decreasing function oftime, thus the resulting solution represents the gravitational

collapse.23) The collapse starts at some finite time and endsat t0 ¼ 1, where longitudinal length of the cylinder reducesto zero. Further, energy density is an increasing function oftime shown in Fig. 2. This is the strong argument for amodel to collapse. The pressure in the interior of cylinderstarts decreasing as shown in Fig. 3. This causes to initiatethe gravitational collapse, more matter is concentrated in thesmall volume, hence density goes on increasing.

It is to be noted that decrease in the proper unit length ofthe cylinder, increases the interaction between the electriccharges and a strong electromagnetic tension inside thecylinder is created. This is an increasing function of time asshown in Fig. 4. The coupled action of electromagnetic andgravitational forces play a dominant role to reduce longi-tudinal length of cylinder to zero.

The nature of the collapse can be seen as follows: Whenlatitudinal and vertical lengths of the cylinder reduce to zero,there is a complete collapse. From the metric (2.32), we haveg�� ¼ ~B2, gzz ¼ ~C2. Since singularity analysis implies thatthe Ricci scalar diverges at a point where the longitudinallengths ~l ¼ ~B ~C ¼ 0. Thus when the longitudinal length aswell as the latitudinal and vertical lengths reduce to zero, weobtain a conical singularity at r0 ¼ 0.

The conical singularity has effects on the gravitationalcollapse of non-zero mass objects. The conical singularitybelongs to the class of line singularities which aregravitationally weaker than point singularities and strongerthan the plane singularities. The tidal forces for point andline singularities are so strong that these can crush an object

Velocity Graph

00.2

0.40.6

0.81

t ′0

0.2

0.4

0.6

0.81

r ′–0.3–0.2–0.1

0

Fig. 1. (Color online) Decrease in longitudinal length with the passage of

time for 0 < M � 0:5 and 1 < L � 1:9.

Matter Density Graph

1t ′

0

0.2

0.4

0.6

0.81

r ′

00.1

0.2

00.2

0.40.6

0.8

Fig. 2. (Color online) Increase of density with the passage of time for

0 < M � 0:5 and 1 < L � 1:9.

M. SHARIF and G. ABBASJ. Phys. Soc. Jpn. 80 (2011) 104002 FULL PAPERS

104002-4 #2011 The Physical Society of Japan

of finite size and non-zero mass to zero volume. However,it was pointed out by Nakao et al.24) that only conicalsingularity is the exceptional line singularity which does notcrush an object to zero volume that collapses onto it. Hence,it is concluded that massive objects of finite size arecollapsed on it without crushing to zero volume. The reasonof non-crushing to zero volume does not imply that tidalforces are weak but it may be due to the geometric structureof the conical singularity.

4. Matching Conditions

Following,21,25) we proceed to cut the spacetime andmatch it with another spacetime, which represents theexterior region of the collapsing cylinder. We match thecharged perfect fluid solution with the electro-vacuumsolution. For this purpose, we consider the Darmois junctionconditions,26) which require that the first and secondfundamental forms (that are the line elements and theextrinsic curvature respectively) must be continuous over theboundary surface �.

We assume that the 3D timelike boundary surface � splitsthe two 4D cylindrically symmetric spacetimes Vþ and V�.The metric which describes the internal region V� is thecharged perfect fluid solution given by eq. (2.32). For therepresentation of the exterior region Vþ, a charged cylin-drically symmetric electro-vacuum solution is taken as27)

ds2þ ¼ H dT 2 � 1

HdR2 � R2ðd�2 þ dz2Þ; ð4:1Þ

where

HðRÞ ¼ 2Q2

R2� 4M

R;

M and Q are the mass and charge per unit length of thecylinder, respectively. This choice of the exterior solutionin Vþ region is compatible with the charged perfect fluidsolution in the interior region V� for their smooth matchingover the boundary surface �.

Now the boundary surface � in terms of interior andexterior coordinates can be described by the followingequations

f�ðr0; t0Þ ¼ r0 � r0� ¼ 0; ð4:2ÞfþðR; T Þ ¼ R� R�ðT Þ ¼ 0; ð4:3Þ

where r0� is a constant. Using these equations, the interiorand exterior metrics on � take the following form

ðds2�Þ� ¼ dt02 � ~B2 d�02 � ~C2 dz02; ð4:4Þ

ðds2þÞ� ¼ HðR�Þ � 1

HðR�ÞdR�

dT

� �2" #

dT 2

� R2�ðd�2 þ dz2Þ: ð4:5Þ

We assume g00 > 0 in eq. (4.5) so that T is a timelikecoordinate.

The continuity of the first fundamental form gives

ð ~BÞ� ¼ R�; ð ~CÞ� ¼ R�; ð4:6Þ

HðR�Þ � 1

HðR�ÞdR�

dT

� �2" #1=2

dT ¼ ðdt0Þ�: ð4:7Þ

The components of extrinsic curvature K�i j in terms of

interior and exterior coordinates are

K�00 ¼ 0; K�

22 ¼ K�33 ¼ ð ~B �~BÞ�;

Kþ00 ¼ ðRyT yy � T yRyy � H

2

dZ

dRT yy3 þ 3

2H

dH

dRT yRy2 Þ�;

Kþ22 ¼ Kþ

33 ¼ ðHRT yÞ�; ð4:8Þwhere dagger and bar represent differentiation with respectto the new coordinates t0 and r0 respectively. The continuityof the extrinsic curvature components with eqs. (4.6) and(4.7) leads to

ð �~ByÞ� ¼ 0; ð4:9Þ

M ¼ Q2

2 ~Bþ

~B

4ð ~By2 � �~B

2Þ� �

�

: ð4:10Þ

Here eq. (4.9) implies that the boundary surface � representsa cylinder with constant proper unit length which behaves asboundary of the interior charged perfect fluid distributed perunit length of the cylinder. Thus it connects the interiorcharged perfect fluid solution to the exterior electro-vacuumsolution. Using this equation, eq. (4.10) reduces toM ¼ ðQ2=2 ~BÞ�. The parametric representation of this equation impliesthat the gravitational and Coulomb forces of the systembalance each other on the boundary surface �. This conse-quence in the absence of pressure on the boundary, is equiva-lent to the result found by Thirukkanesh and Maharaj.16)

5. Discussion

In this paper, we find an analytical solution to the Einsteinfield equations coupled with the charged perfect fluiddistributed per unit length of the cylinder. It has been found

Electric Field Intensity Graph

00.2

0.40.6

0.81

t ′0

1

2

3

r ′0

0.050.1

0.15

Fig. 4. (Color online) Increase of electric intensity with the passage of

time for 0 < M � 0:5 and 1 < L � 1:9.

0.2 0.4 0.6 0.8 1 t ′

0.01765

0.0177

0.01775

0.0178

0.01785

0.0179

p Pressure–Time Graph

Fig. 3. Decrease in pressure with the passage of time for 0 < M � 0:5

and 1 < L � 1:9.

M. SHARIF and G. ABBASJ. Phys. Soc. Jpn. 80 (2011) 104002 FULL PAPERS

104002-5 #2011 The Physical Society of Japan

that all the physical parameters become homogenous butisotropic pressure is function of time only. This property(i.e., pressure being only time dependent) of cylindricallysymmetric geodesic model is similar to the spherical case.Further, all the energy conditions are satisfied for a range ofseparation parameters and coordinates for which the solutionis physical. All sorts of singularities of the solutions arediscussed in detail. It is found that physical singularity of thesolution occurs at a point where proper areal radius of thecylinder reduces to zero.

The physical and geometrical properties of the solutionsuch as increase of density and decrease in proper unitlength of the cylinder with respect to time represents thegravitational collapse. Also, the time interval for the collapsehas been investigated. It is found that the electric fieldintensity of the system increases with time. This implies thatas longitudinal length of the cylinder decreases, chargescome close to each other and the interaction between thecharges is increased.

In general, it is known19,28) that the presence of anelectromagnetic field in a geometry causes to disturb itsgeneric properties which may result in the form ofoscillation of spacetime. Here, we discuss the absence ofoscillation in geometry by showing the absence of outgoinggravitational waves.

Following Pereira andWang,29) the component of theWeyltensor for our solution, is �0 ¼ �C���L

�M�LM� ¼ 0,where L� and M� are null vectors. It means that there doesnot exist outgoing gravitational waves implying that oscilla-tions are absent in the geometry of the spacetime. Thus thereis no energy loss and hence no bouncing. Further, the absenceof fluctuations in the energy density graph 2 indicates theabsence of bounce, oscillation and gravitational waves. Theprediction that the fluctuation in energy density representsgravitational waves is recently made by Hussain et al.30)

We would like to remark here that in this caseelectromagnetic field is weak as compared to matter fieldwhich is obvious from Figs. 2 and 4. This condition (E2 < �)for the weak electromagnetic field was stated by Tasagas28)

and used by us19) to preserve the generic properties of theFriedmann universe models. In a paper,29) �0 6¼ 0 impliesthat there exist outgoing gravitational waves from thecylindrical symmetric radiating fluid gravitational collapse.It would be interesting to find solution of the field equationwith radiating fluid and electromagnetic field in order topredict collective role of the electromagnetic field andradiation flux for the existence of the gravitational waves.

The Darmois criteria for the smooth matching of thecylindrically symmetric charged perfect fluid solution to anelectro-vacuum charged static cylindrically symmetric solu-tion leads to the following consequences: (a) boundarysurface represents a cylinder with constant proper unit lengthand behaves as the boundary of the charged perfect fluid

distributed per unit length of the cylinder; (b) the gravita-tional and Coulomb forces of the system balance each otheron the boundary surface �.

Acknowledgment

We would like to thank the Higher Education Commis-sion, Islamabad, Pakistan for its financial support throughthe Indigenous Ph. D. 5000 Fellowship Program Batch-IV.

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