Int J Theor PhysDOI 10.1007/s10773-013-1662-8

Gravitational Collapse with Heat Flux and GravitationalWaves

Zahid Ahmad · Qazi Zahoor Ahmed · Abdul Sami Awan

Received: 19 March 2013 / Accepted: 24 May 2013© Springer Science+Business Media New York 2013

Abstract In this paper, we investigated the cylindrical gravitational collapse with heat fluxby considering the appropriate geometry of the interior and exterior spacetimes. For this pur-pose, we matched collapsing fluid to an exterior containing gravitational waves.The effectsof heat flux on gravitational collapse are investigated and matched with the results obtainedby Herrera and Santos (Class. Quantum Gravity 22:2407, 2005).

Keywords Gravitational collapse · Junction conditions · Heat flux

1 Introduction

Einstein was the first who put forward the concept of gravity in terms of spacetime curva-ture. According to him, mass causes spacetime to bend and this bending of spacetime is anagency that produces gravity. He formulated the field equations called Einstein field equa-tions that have non-linear nature. The first exact solution of these equations was given bySchwarzschild. The Friedmann cosmological models are also the one of the initial exact so-lutions of these equations. These solutions and many others contain spacetime singularities.

The outcome of gravitational collapse is a spacetime singularity, where physical quan-tities such as energy density and spacetime curvature diverge [2]. Gravitational collapseoccurs when a massive star contracts due to excess of gravity and its entire mass is confinedto a very minute area reducing its size to a black hole. Black holes are bounded by a surfaceknown as event horizon. Its gravitational field is stronger to such an extent that a particleor light can not escape it once entered. The spacetime singularity without event horizon iscalled naked singularity.

Using cylindrical symmetry many people studied the properties of a star. The thing thatgives much inspiration to consider cylindrical symmetry is the existence of cylindrical grav-itational waves. Initially Bronnikov and Kovalchuk [3] worked on cylindrical symmetry

Z. Ahmad (�) · Q.Z. Ahmed · A.S. AwanDepartment of Mathematics, COMSATS Institute of Information Technology, University Road,Post Code 22060, Abbottabad, KPK, Pakistane-mail: zahidahmad@ciit.net.pk

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and found some exact solutions. Using cylindrical spacetime Shapiro and Teukolsky [4]investigated the collapse of dust spheroid numerically. They showed that the end state ofcollapse was either a black hole or a naked singularity depending on the compactness ofthe spheroid. The exact solution for collapsing convex shell was studied by Barrabes et al.[5]. They showed that apparent horizons were not formed in some cases. High-speed cylin-drical collapse of two perfect fluids was investigated by Sharif and Ahmad [6]. Some morereferences on cylindrical collapse can be seen in [6].

Applying matching conditions the gravitational collapse of an infinite cylindrical distri-bution of time-like dust was examined by Gutti et al. [7]. Sharif and Abbas [8] studied thedynamics of non-adiabatic charged cylindrical gravitational collapse and examined the roleof anisotropy, electric charge and radial heat flux over the dynamics of the collapse withthe help of coupled equation. Using matching conditions, Charged cylindrical collapse ofanisotropic fluid was discussed by Sharif and Fatima [9]. They showed that homogeneityin energy density and conformal flatness of spacetime are necessary and sufficient for eachother. A lot of wok on gravitational collapse has been done by using matching conditions(references can be seen in [1]). Prisco et al. [10] discussed Shearfree cylindrical gravitationalcollapse.

Different researchers [11–13] did work on cylindrical gravitational waves. It is shownthat they carry energy and reduces the mass of the source when emerging out from it. Onthe other hand, they do not give support to fluid pressure of the source. Herrera and Santos[1] discussed the cylindrical collapse and gravitational waves by considering non-dissipativefluid and matched it to an exterior containing gravitational waves. They showed that radialpressure on the surface is non zero. In this paper, we extended the work done by Herrera andSantos [1] by considering the energy momentum tensor with heat flux. We also investigatedthe effects of heat flux on gravitational collapse. The results obtained are compared with byHerrera and Santos [1].

The paper is organized as follows. In Sect. 2, we formulated field equations for the prob-lem considered. Matching conditions are derived in Sect. 3. The results are discussed inSect. 4. Finally the summary of the work done is given in last section.

2 Field Equations

Here we divide the given four dimensional cylindrical symmetric spacetime into interiorand an exterior regions by a timelike three dimensional hypersurface

∑. These regions are

denoted by V −and V + respectively. We take general cylindrical symmetric spacetime ininterior region given by

ds2− = −A2

(dt2 − dr2

) + B2dz2 + C2dφ2, (1)

where A, B and C are functions of t and r . The cylindrical symmetric condition restrictsthe coordinates as

−∞ ≤ t ≤ ∞, 0 ≤ r ≤ ∞, −∞ < z < ∞, 0 ≤ φ ≤ 2π. (2)

The energy momentum tensor in interior region is taken in the following form

T −αβ = (μ + P⊥)VαVβ + P⊥gαβ + (Pr − P⊥)XαXβ + Vαqβ + qαVβ, (3)

where μ is the amount of energy per unit volume, Pr the radial pressure, P⊥ the tangentialpressure, qα the thermal flux, V α the four velocity of fluid and Xα a unit four vector alongthe radial direction respectively. These quantities satisfy the following relations

V αVα = −1, V αqα = 0, XαXα = 1, XαVα = 0. (4)

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For interior metric (1) in comoving coordinates, we have

V α = A−1δα0 , qα = qA−1δα

1 , Xα = A−1δα1 . (5)

The non-zero components of Einstein field equation for metric (1) are five but we needonly the following two components

G−01 = −B,rt

B+ A,rB,t

AB+ A,tB,r

AB− C,rt

C+ A,rC,t

AC+ A,tC,r

AC= −κqA2, (6)

G−11 = −B,tt

B− C,tt

C+ A,tB,t

AB

A,rB,r

AB+ A,tC,t

AC+ A,rC,r

AC− C,tB,t

BC

+ C,rB,r

BC= κPrA

2. (7)

For the exterior vacuum region, we take the metric

ds2+ = −e2(γ−Ψ )dT 2 + e2(γ−Ψ )dR2 + e2Ψ dz2 + e−2Ψ R2dφ2, (8)

where γ and Ψ are functions of T and R. For the exterior vacuum spacetime (8), the Einsteinfield Equations are given by

Ψ,T T − Ψ,RR − Ψ,R

R= 0, (9)

γ,R = R(Ψ 2

,T + Ψ 2,R

), (10)

γ,T = 2RΨ,T Ψ,R. (11)

3 Junction Conditions

For the interior and exterior regions, the equations of hypersurface are given by

f − = r − rΣ = 0, (12)

f + = R − RΣ(T ) = 0, (13)

here rΣ is constant. Using Eq. (12) in Eq. (1) the interior metric on the hypersurface Σ takesthe following form

ds2− = −dτ 2 + B2dz2 + C2dφ2, (14)

where on Σ

dτ = Adt. (15)

From Eqs. (13) and (8), the exterior metric on Σ becomes

ds2+ = −e2(γ−Ψ )

(

1 −(

dR

dT

)2)

dT 2 + e2Ψ dz2 + e−2Ψ R2dφ2. (16)

Comparing Eqs. (14) and (16) on Σ , we get

e(γ−Ψ )

(

1 −(

dR

dT

)2) 12

dT = dτ, (17)

eΨ = B, (18)

e−Ψ R = C. (19)

Int J Theor Phys

For T a timelike coordinate, we assume

1 −(

dR

dT

)2

> 0. (20)

From Eqs. (12) and (13), the outward unit normals to Σ are given by

n+α = (−R, T ,0,0), (21)

n−α = (0,A,0,0), (22)

where dot denotes differentiation with respect to τ .The extrinsic curvature Kab is defined by

Kab = −n±α

(∂2xα

∂ξa∂ξb+ Γ α

βγ

∂xβ∂xγ

∂ξa∂ξb

)

, (23)

where a, b = (0,2,3) and the Christoffel symbols are determined from interior and exteriormetrics. Non-zero components of Kab for interior and exterior spacetimes on Σ are givenby

K−00 = −A,r

A2, (24)

K−22 = BB,r

A, (25)

K−33 = CC,r

A, (26)

K+00 = e2(γ−Ψ )

{

RT − RT −[

(γ,R − Ψ,R)T

+(γ,T − Ψ,T )R

](T 2 − R2

)}

, (27)

K+22 = e2Ψ (RΨ,T + T Ψ,R), (28)

K+33 = −e−2Ψ R2

(

RΨ,T + T Ψ,R − T

R

)

. (29)

The Eqs. (15) and (17)–(19) together with the continuity of Kab across Σ are completejunction conditions.

4 Results

Here we derive some basic formulae to calculate the results. Eq. (17) can be written as

e2(γ−Ψ )(T 2 − R2

) = 1. (30)

From Eqs. (18) and (19), we get

R = BC. (31)

Differentiating above equation with respect to τ and using Eq. (15), it yields

R = (BC),t

A. (32)

From the continuity of K22 and K33 on Σ , using Eqs. (18) and (19), we obtain the followingrelation

T = (BC),r

A. (33)

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From the continuity of K22, we have

eΨ (RΨ,T + T Ψ,R) = B,r

A. (34)

Differentiation of Eq.(17) with respect to τ on Σ yields

eΨ (Ψ,T T + Ψ,RR) = B,t

A. (35)

From Eqs. (34) and (35) with Eqs. (33) and (32), the values of Ψ,T and Ψ,R are given by

Ψ,R = B,r(BC),r − B,t (BC),t

B{(BC)2,r − (BC)2

,t }, (36)

Ψ,T = B,t (BC),r − B,r(BC),t

B{(BC)2,r − (BC)2

,t }. (37)

Differentiating Eqs. (33) and (32) with respect to τ with Eqs. (6), (7) and (15), we getthe following expression

T R − T R = 1

A4

[(BC),t {B,t (AC),r + (AB),rC,t + κqA3BC}

+(BC),r{κPrCA3B − AC,tB,t − A,r(BC),r − C,rB,rA}

]

. (38)

From continuity of K00 and K22 on Σ , with Eqs. (10), (11), (17), (18), (31), (32), (33),(36), (37) and (38), following relation is obtained

(T 2 − R2

)2Ψ 2

,T = 1

A4

{(B,tC,r − B,rC,t )

2

+κqA2(BC),t (BC),r + κPrA2(BC)2

,r

}

. (39)

Differentiating Eq. (18) with respect to τ and using Eq. (17), we get

e2Ψ −γ

(

1 −(

dR

dT

)2)− 12

Ψ,T = B,t

A. (40)

Similarly differentiating Eq. (19) with respect to τ and using Eq. (17), we get

e−γ

(

1 −(

R

T

)2)− 12{

R

T− RΨ,T

}

= C,t

A. (41)

The continuity of K22 and K33 along with Eqs. (40)and (41), yields

B,tC,r − B,rC,t

A2= {(

T 2 − R2)Ψ,T − RT Ψ,R

}. (42)

Using Eq. (42) in Eq. (39) with Eqs. (10), (11) and (32), it follows

κq = e2(Ψ −γ )Ψ 2,R

{

2Ψ,T

Ψ,R

− v

[1 − v2]}

− κPr

v, (43)

where v = dRdT

, is the radial velocity on the surface. For sufficient large value of v, we have

κq = 2e2(Ψ −γ )Ψ,RΨ,T . (44)

This equation indicates that for large values of T heat flux is positive because the quantityΨ,RΨ,T on the right hand side is positive. It means that flux is outward. From Eq. (43), wecan write

κPr = e2(Ψ −γ )Ψ 2,R

{

2vΨ,T

Ψ,R

− v2

[1 − v2]}

− κqv. (45)

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The above equation indicates that the radial pressure Pr on the surface∑

of the collapsingperfect fluid is non zero. This is because of the flux of momentum of the gravitational waveemerging from the cylinder. Here it is also mentioned that for q = 0, all the results reduceto the results obtained by Herrera and Santos [1].

5 Conclusion

In this paper, we investigated the cylindrical gravitational collapse with heat flux and us-ing junction conditions. For this purpose, we matched the collapsing fluid to an exteriorcontaining gravitational waves. We have calculated the radial pressure and heat flux on thehypersurface

∑. It is concluded that if cylindrical source is continuously emerging grav-

itational waves then the radial pressure of the collapsing cylinder is non vanishing on thehypersurface

∑. While for the static source, the pressure on the surface is zero because the

source is not emitting gravitational waves, that is similar to the result investigated by Bondi[14] in the slowly evolving case. These results agree with [1]. It is also concluded that forthe static case heat flux does not contribute to the radial pressure.It is also mentioned thatfor q = 0, all the results reduce to the results obtained by Herrera and Santos [1].

It is of much importance that for large values of T , the expression Ψ,RΨ,T is positive onthe hypersurface

∑. This quantity corresponds to the derivative of light energy and shows

that heat flux is positive on the hypersurface. It means that flux is outward and cylindricalsource emits gravitational waves.

In the end, we can summarize it as if a cylinder with sources is collapsing, it will contin-uously emerge gravitational waves outside the source. This grantees the non-zero pressureon the surface of the source.

References

1. Herrera, L., Santos, N.O.: Class. Quantum Gravity 22, 2407 (2005)2. Joshi, P.S.: Global Aspects in Gravitation and Cosmology. Oxford University Press, London (1993)3. Bronnikov, K.A., Kovalchuk, M.A.: Gen. Relativ. Gravit. 15, 809 (1983)4. Shapiro, S.L., Teukolsky, S.A.: Phys. Rev. Lett. 66, 994 (1991)5. Barrabes, C., Israel, W., Letelier, P.S.: Phys. Lett. A 160, 41 (1991)6. Sharif, M., Ahmad, Z.: Gen. Relativ. Gravit. 39, 1331 (2007)7. Gutti, S., Singh, T.P., Sundararajan, P.A., Vaz, C.: arXiv:gr-qc/02120898. Sharif, M., Abbas, G.: Astrophys. Space Sci. 335, 515 (2011)9. Sharif, M., Fatima, S.: Gen. Relativ. Gravit. 43, 127 (2011)

10. Prisco, A.Di., Herrera, L., MacCallum, M.A.H., Santos, N.O.: Phys. Rev. D 80, 064031 (2009)11. Marder, L.: Proc. R. Soc. Lond. A 244, 524 (1958)12. Bondi, H., van der Burg, M.G.J., Metzner, A.W.K.: Proc. R. Soc. Lond. A 269, 21 (1962)13. Apostolatos, T.A., Throne, K.S.: Phys. Rev. D 46, 2435 (1992)14. Bondi, H.: Proc. R. Soc. Lond. A 42, 7259 (1990)