for review only - university of toronto t-space · 2017-01-06 · for review only collapse and...

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Collapse and Expansion of Plane Symmetric Charged

Anisotropic Source

Journal: Canadian Journal of Physics

Manuscript ID cjp-2016-0741

Manuscript Type: Article

Date Submitted by the Author: 11-Oct-2016

Complete List of Authors: Abbas, G.; The Islamia University Bahawalpur Shah, S.; The Islamia University of Bahawalpur , Mathematics Zubair, M.; COMSATS Institute of Information Technology, Mathematics

Keyword: Anisotropic Fluids, Electromagnetic Field, Gravitational Collapse, Expanding solutions, Plane symmetry

https://mc06.manuscriptcentral.com/cjp-pubs

Canadian Journal of Physics

For Review O

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Source

G. Abbas1 ∗, S. M. Shah1 †and M. Zubair2 ‡

1 Department of Mathematics, The IslamiaUniversity of Bahawalpur, Bahawalpur-63100, Pakistan.

2 Department of Mathematics, COMSATSInstitute of Information Technology, Lahore-54000, Pakistan.

Abstract

In this paper, we have investigated the final evolutionary stages ofcharged non-static plane symmetric anisotropic source. To this end,we have solved the Einstein-Maxwell field equations with the chargedplane symmetric source. We have found that vanishing of radial heatflux in the gravitating source provides the parametric form of the met-ric functions. The new form of the metric functions can generate aclass of physically acceptable solutions depending on the choice pa-rameter. These solutions may be classified as expanding or collapsingsolutions with the particular values of generating parameter. Thegravitational collapse in this case end with the formation of single ap-parent horizon while there exists two such horizon in case of chargedspherical anisotropic source.

Keywords: Anisotropic Fluids; Electromagnetic Field; Expansion; Gravi-tational Collapse.PACS: 04.20.Cv; 04.20.Dw

∗[email protected]†[email protected]‡[email protected], [email protected]

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1 Introduction

In 1939, Oppenheimer and Snyder [1] were the pioneer, who studied thespherically symmetric gravitational collapse of a massive star consisting ofdust matter. Initially, it was a particular and simple statement of the gravi-tational collapse of the star because dust could not be considered as a generalrealistic matter and the role of matter stress on the formation of singularitycannot be neglected. The general treatment for the problem of gravitationalcollapse was carried out by Misner and Sharp [2]. They consider the isotropicperfect fluid in the interior region of a star and constructed the equationsof motions which governs the perfect fluid gravitational collapse. Further,Misner and Sharp [3] examined the gravitational collapse for anisotropic fluidtaking a more realistic form of star geometry. After this there has been anextensive interest among the physicist to discuss the relativistic gravitationalcollapse of anisotropic gravitating source [4, 5].

The analytic solutions of anisotropic sources has achieved much attentionin General Relativity due to their implication in astronomy as astrophysics[6]-[10]. Barcelo et al.[11] have investigated the evolution of anisotropicsources in quantum gravity and proposed a new model of gravitational col-lapse without horizons and trapped surfaces. Many researchers [11]-[15] haveinvestigated the anisotropic behavior of dark energy fluid source in modifiedf(R) gravity. Further, Herrera and Santos [16] have studied the models ofgravitational collapse of massive stars and pointed out the factors affectingstable state of stars under the linear perturbation. Herrera et al. [4] pro-posed a single generating function in the modeling of anisotropic source, wehave used such generating function in the present paper for the modeling ofcharged anisotropic plane symmetric source.

The generating solution approach for the non-adiabatic gravitating sourcecollapse was proposed for the first time by Glass [17]. For this purpose, he uti-lized adiabatic fluid static/nonstatic solutions of shearfree collapsing modelto a shear-free collapsing source which has dissipation in the form of outwardradial flow. Recently, Glass [18] has determined the anisotropic perfect fluidspherically symmetric gravitating model. Abbas and his collaborators [19]-[22] have generalized this work for the plane symmetric anisotropic sourceand charged anisotropic spherical and cylindrical sources. In this paper, weexamined the effects of electric charge parameter on the dynamics of planesymmetric anisotropic source. The geometries with plane shapes would beinteresting models for numerical relativity, quantum gravity, cosmic censor-

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ship hypothesis and hoop conjuncture. Sharif and Zaeem [23] have proposedthat plane symmetric gravitating sources are viable as compared to spheri-cal models for determining the different evolutionary stages of our universe.The present paper is a systematic study of Glass [18] work to discuss theevolution of the charged plane symmetric source.

This paper has been arranged as follows: In the next section, we drive theequations of motions of charged anisotropic plane symmetric source. Section3 deals with the parametric solutions of the field equations which exhibit theevolution of the source as expansion or collapse depending on the choice ofparameters of the proposed model. The last section contains the summaryof results.

2 Gravitating Source and Einstein-Maxwell

Field Equations

The plane symmetric anisotropic charged fluid source is defined by the fol-lowing line element

ds2− = −X2(t, z)dt2 + Y 2(t, z)(dx2 + dy2) + Z2(t, z)dz2. (1)

The charged anisotropic source has following form of stress energy tensor

Tab = (µ+ P⊥)UaUb + P⊥gab + (Pz − P⊥)ψaψb + prgαβ (2)

+1

4π

(F caFbc −

1

4F cdFcdgab

), (3)

here Pz and P⊥ are pressures along z-direction and perpendicular direction,ψa is a four vector along z-direction and Ua is a co-moving four velocity.Further, Fab = −ϕa,b + ϕb,a is electromagnetic field tensor with four vectorpotential ϕa.

The Maxwell field equations are

Fαβ;β = 4πJα, F[αβ;γ] = 0,

here, we can define the four-potential and four-current as follows:

ϕα = ϕδ0α, Jα = ϱUα,

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where ϱ and ϕ are electric charge density and electric scalar potential, re-spectively.

The expansion scalar is

Θ =1

X(2Y

Y+Z

Z). (4)

Here dot is partial derivative with to time coordinate t. The fractionalanisotropy of the fluid is defined as

∆a = 1− P⊥

Pz

. (5)

The set of Einstein field equations is given by

8πµX2 +16π2Q2X2

Y 4=Y

Y

(2Z

Z+Y

Y

)+

(X

Z

)2(−2Y ′′

Y+

(2Z ′

Z− X ′

X

)X ′

X

),

(6)

− Y′

Y+X ′Y

XY+ZY ′

Y Z= 0, (7)

8πP⊥Y2 +

16π2Q2

Y 2= −

(Y

X

)2[Y

Y+Z

Z− X

X

(Y

Y+Z

Z

)+Y Z

Y Z

]

+Y 2

Z2

[X ′′

X+Y ′′

Y− X ′

X

(Z ′

Z− Y ′

Y

)− Y ′Z ′

Y Z

], (8)

8πPzZ2 − 16π2Q2Z2

Y 4= −

(Z

X

)22YY

+

(Y

Y

)2

− 2XY

XY

+

(Y ′

Y

)2

+2X ′Y ′

XY. (9)

By the direct calculation, we note that if X and Z have following func-tional form

X =Y

Y αZ = Y α, (10)

then Eq.(4), can be satisfied easily. By Eq.(4), expansion scalar takes thefollowing form

Θ = (2 + α)Y (1−α). (11)

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It is interesting to mention here that if α > −2 and α < −2, then theevolution of source may results as expansion and collapse.

Hence, by Eq.(10), we get the following form of Einstein field equations

8πµ+Q2Y 2

Y 2α+4= (1 + 2α)Y 2α−2 − 1

Y 2α

[2Y ′′

Y+ (1− 2α)

(Y ′

Y

)2]+

1

Y 2, (12)

8πPz −16π2Q2Y 2α

Y 4= (1 + 2α)Y 2α−2 +

1

Y 2+

1

Y 2α

[(2α− 1)

(Y ′

Y

)2

− 2Y ′

Y

Y ′

Y

], (13)

8πP⊥ +16π2Q2

Y 4= −α(1 + 2α)Y 2α−2

+1

Y 2α

[(1− α)

Y ′′

Y− (3α− 1)

Y ′

Y

(Y ′

Y

)+Y ′′

Y+ α(2α− 1)

(Y ′

Y

)2].

(14)

The mass function defined by Taub [24] for plane symmetry with the contri-bution of electromagnetic field is

m(z, t) =(g11)

3/2

2R12

12 +Q2

2Y.

Plugging the values of g11 and R1212 from given spacetime along with X =

YY α , Z = Y α, the Taub’s mass takes the following form

2m(t, z)

Y− Q2

Y 2=

(Y 2α − Y ′2

Y 2α

). (15)

For Y ′ = Y 2α, one gets Y = Q2

2m, which implies that in this case a trapping

surface corresponding to single horizon exist at Y = Q2

2m. The trapping

condition Y ′ = Y 2α, has the integral

Y(1−2α)trap = z(1− 2α) + g(t), (16)

where g(t) is the function of integration.

3 Parametric Solutions

In this section, we determine the values of parameter α, for solutions revealthe expansion and collapse of the source.

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3.1 Gravitational Collapse for α = −52

We know that for a collapsing object, the expansion scalar should be negative.Therefore, we investigate that Θ < 0 ( see Eq.(4)), if α < −2.Here, withoutthe loss of generality, we take α = −5

2and trapping condition Y ′ = Y 2α,

becomes Y ′ = Y −5, which further leads to

Ytrap = (6z + g1(t))16 , (17)

where g1(t) is function of integration. For α = −52, Eqs.(6), (7) and (8) give

8πµ+ 16π2Q2Y 2Y = −4Y −7 − 2Y 5

[2Y ′′

Y+ 3

(Y ′

Y

)2]+

1

Y 2, (18)

8πPz −16π2Q2

Y 9= −4Y −7 +

1

Y 2− 2Y 5

[3

(Y ′

Y

)2

− 2Y ′

Y

Y ′

Y

], (19)

8πP⊥ +16π2Q2

Y 4= −10Y −7 + Y 5

[7

2

Y ′′

Y+

17

2

Y ′

Y

(Y ′

Y

)+Y ′′

Y+ 15

(Y ′

Y

)2].

(20)

In order to get a class of solutions, we take Ytrap = k(6z + g1(t))16 , the above

equations in this case reduces to

8πµ = (6z + h1)−1/3 − 8π2Q2

3h1

2(6z + h1)

3/2, (21)

8πPz = (6z + h1)−1/3 +Q2(6z + h1)

−3/2, (22)

8πP⊥ = Q2(6z + h1)−1/3. (23)

The dimensionless measure of anisotropy defined by Eq.(5) is

△a = 1 +Q2(6z + h1)

5/6

Q2(6z + h1)−1/3 + (6z + h1)5/6. (24)

It is noted that for α = −52, we get Θ < 0 and energy density remains

positive and goes on increasing to attains a finite value for the specific valueof charge parameter Q and time dependent function g1(t). As g1 = 1 + t, sog1 = 1, and the electric charge term in case contributes (being the multipleof g1) to effect the energy density of the fluid. In this way energy density of

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0

5

10

z 0

5

10

t

2.1

2.2

2.3

2.4

2.5

Μ

0

5

10

z

0

5

10

t

40

50

60

70

80

pz

Figure 1: Both graphs have been plotted for Q = 2, g1 = 1 + t.

0

5

10

z

0

5

10

t

50

100

0

5

10

z2

4

6

8

10

t

-0.1

0.0

0.1

Da


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the gravitating source becomes homogeneous and it increases with the timeas shown in left graph of Fig.1. Similarly, for the above choice of temporalprofile, the pressure Pz is decreasing function of time and homogeneous everywhere inside the gravitating source as shown in right graph of Fig.1. Thetransverse pressure P⊥ and anisotropic parameter △a (with the above choiceof Q and g1(t)), become inhomogeneous and time independent as shown inFig.2. The anisotropy increases from negative value to finite positive value.

3.2 Expansion for α = 32

Here, we are interested to find such values of α for which expansion scalarbecomes positive. It is noted that for α ≥ 0, we get Θ > 0 (from Eq.(4)). Inthis case, we take α = 3

2, and also assume that

Y = (z2 + z02)−1 + g2(t), (25)

where g2(t) is an arbitrary function of time and z0 an arbitrary constant.When α = 3/2, the field equations reduce to

8πµ = 4Y − 2Y −4

(Y ′′ − Y ′2

Y

)+ Y −2 −Q2Y 2Y −7, (26)

8πPz = 4Y + Y −2 + 2Y −5

((Y ′)

2 − Y ′

YY ′Y

)+Q2Y −1, (27)

8πP⊥ = −6Y + Y −5

(−1

2Y ′′Y − 7

2

Y ′Y ′

Y+ Y ′′Y + 3 (Y ′)

2

)−Q2Y −4.

(28)

With F (t, z) = 1+g2(t)(z2+z0

2) and Y = F(z2+z02)

, the density and pressures

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5

10

z

0

5

10

t

0.45

0.50

0.55

0.60

Μ

0

5

10

z

0

5

10

t

10

20

30

40

pz


can be written as

8πµ =4F

(z2 + z02)+

(z2 + z02)2

F 2+

(z2 + z02)(z0

2 − 3z2)

F 4+

8z2(z2 + z02)

F 5

− Q2h1(z2 + z0

2)7

F 7, (29)

(30)

8πPz =4F

(z2 + z02)+

(z2 + z02)2

F 2+

8z2(z2 + z02)

F 5+Q2(z2 + z0

2)

F, (31)

8πP⊥ =−6F

(z2 + z02)+

(z2 + z02)(z0

2 − 3z2)

F 4+

12z2(z2 + z02)

F 5

− Q2(z2 + z02)4

F 4. (32)

The anisotropy (Eq.(5)) can be written as

△a = 1+

6F 6

(z2+z02)2− 12z2 − F 3(z2 + z0

2) + F [z6Q2 − z02 + 3z4Q2z0

2 + 3z2(1 +Q2z02)](

4F 6

(z2+z02)2+ F 4Q2 + 8z02 + F 3(z2 + z02)

) .

(33)All these quantities are shown graphically in figures 3 and 4.

It is interesting for α = 32, we get Θ > 0 and energy density remains pos-

itive and decreases to a finite value for the specific value of charge parameterQ and time dependent function g2(t). As g2 = 1 + t, so g2 = 1, and theelectric charge term in case contributes (being the multiple of g1) to effectthe energy density of the fluid. In this way energy density of the gravitating

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5

10

z

0

5

10

t

-10

-8

-6

0

5

10

z

0

5

10

t

12

13

14

15

Da


source becomes homogeneous and it decreases with the time as shown in leftgraph of Fig.3. The decrease in energy density confirms the expansion ofgravitating source. Similarly, for the above choice of temporal profile, thepressure Pz is increasing function of time and homogeneous as shown in rightgraph of Fig.3. The transverse pressure P⊥ and anisotropic parameter △a,become homogeneous and time dependent as shown in Fig.2. The anisotropyincreases as the gravitating source goes on expanding.

4 Conclusion

This paper aims to discuss the evolutionary stages of the charged non-staticplane symmetric sources. The source fluid is non-radiating with plane sym-metry and have electrical conduction property. The off diagonal componentof the field equations provides the metric functions in terms of parametricform of another metric function. The Taub’s mass in the presence of electriccharge has been calculated which helps to determine the trapped surfaces.The trapping condition Y ′ = Y 2α, leads to the existence of single horizonat Y = Q2

2m. Thus the non-vanishing charges in the plane gravitating source

plays an important rule for the formation of horizon. It is interesting to notethat no such horizon exist in non-charged plane gravitating source dynamics.

During the trapping situation, the relation Y ′ = Y 2α must holds and withthis relation, we get Θ = (α+2)Y (1−α). The behavior of Θ depends on valuesof α, it can be summarized as follows:

• For α = −2, we get Θ = 0, which corresponds to bouncing of thegravitating source

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• For α < −2, we get Θ < 0, which corresponds to the collapse of thegravitating source

• For α ≥ 0, we get Θ > 0, which corresponds to expansion of thegravitating source.

We would like to mention here that in case of gravitational collapse,for the particular choice of the electric charge parameter Q and time profileg1(t), the energy density and Pz pressure along z-direction are increasing anddecreasing function of time respectively. Both these quantities also, becomehomogenous. The P⊥ transverse pressure and △a anisotropic parameter areincreasing function of z, so both are non-homogenous and time independent.Since in this case Pz decreases and P⊥ increases, therefore, we get non-zero value of the anisotropic parameter △a. On the other hand in caseof expansion, for the particular choice of the electric charge parameter Qand time profile g2(t), the energy density and pressure imply the reversebehavior as compared to gravitational collapse case. In this case, energydensity decreases and pressure along z-direction increases with time. Here,both P⊥ and △a, are homogenous and increasing function of time. Further,the anisotropy of the source becomes zero for the particular choice of theparameters.

References

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[9] M. Sharif and G. Abbas : Astrophys. Space Sci. 335(2011)515

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[14] S. Nojiri, S.D. Odintov and O.G. Gorbunova: J. Phys. A39(2006)6627

[15] M. Gasperini and G. Veneziano: Astropart. Phys. 1(1993)317

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[17] E.N. Glass: Phys. Lett. A86(1981)351

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[24] T. Zannias: Phys. Rev. D41(1990)3252.

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