Dynamics of Charged Plane Symmetric Gravitational Collapse
aa r X i v : . [ g r- q c ] A ug Dynamics of Charged PlaneSymmetric Gravitational Collapse
M. Sharif ∗ and Aisha Siddiqa † Department of Mathematics, University of the Punjab,Quaid-e-Azam Campus, Lahore-54590, Pakistan.
Abstract
In this paper, we study dynamics of the charged plane symmetricgravitational collapse. For this purpose, we discuss non-adiabatic flowof a viscous fluid and deduce the results for adiabatic case. The Ein-stein and Maxwell field equations are formulated for general planesymmetric spacetime in the interior. Junction conditions betweenthe interior and exterior regions are derived. For the non-adiabaticcase, the exterior is taken as plane symmetric charged Vaidya space-time while for the adiabatic case, it is described by plane Reissner-Nordstr¨ o m spacetime. Using Misner and Sharp formalism, we obtaindynamical equations to investigate the effects of different forces overthe rate of collapse. In non-adiabatic case, a dynamical equation isjoined with transport equation of heat flux. Finally, a relation betweenthe Weyl tensor and energy density is found. Keywords : Gravitational collapse; Junction conditions; Dynamical equa-tions; Transport equations.
Gravitational collapse is defined as the implosion of a celestial body under theinfluence of its own gravity. It is one of the basic processes driving evolution ∗ [email protected] † [email protected] o m Vaidya space-time in the exterior region. The time difference between the formation ofapparent horizon and central singularity was also discussed.Using the concept of gravitational lensing (GL), Virbhadra et al. [23] in-troduced a new tool for examining naked singularities. Gravitational lensingis the process of bending of light around a massive object such as a black hole.Virbhadra and Ellis [24] discussed GL by the Schwarzschild black hole. It wasfound that the relativistic images guarantee the Schwarzschild geometry closeto event horizon. The same authors [25] also analyzed GL by a naked singu-larity. Claudel et al. [26] proved that the necessary and sufficient conditionfor the black hole to be surrounded by a photon sphere is that a reasonableenergy condition holds. Virbhadra and Keeton [27] showed that weak cosmiccensorship hypothesis (CCH) can be examined observationally without anyuncertainty. Virbhadra [28] found that Seifert’s conjecture is supported bythe naked singularities forming during Vaidya null dust collapse. The sameauthor developed an improved form of CCH using GL phenomenon [29]In a recent paper, Sharif and Abbas [30] found the effects of electromag-netic field on the gravitational collapse for perfect fluid in the presence ofcosmological constant. It was concluded that charge increases the rate ofcollapse by decreasing the limit of cosmological constant. The same authorsextended this work for 5D collapse [31] and found that the range of appar-ent horizon is greater than 4D case. Di Prisco et al. [32] figured out theconsequences of charge and dissipation for spherical symmetric gravitationalcollapse of a real fluid. It was assumed that heat flow, free streaming ra-diation and shearing viscosity are the causes of dissipation. The dynamicaland transport equations are coupled to observe the effects of dissipation overcollapsing process.This paper extends the work of Di Prisco et al. [32] to plane symmetry.The models exhibiting plane symmetry may swear out as test-bed for nu-merical relativity, quantum gravity and contribute for examining CCH andhoop conjecture among other important issues. The paper is organized in thefollowing pattern. In the next section, the dynamics for non-adiabatic flowis discussed. The Einstein-Maxwell field equations and junction conditionsare found. The dynamical and transport equations are obtained and thencoupled. A relation between energy homogeneity and the Weyl tensor is alsogiven. Section describes dynamical equations for adiabatic case. The lastsection summarizes the results. 3 Dynamics of Viscous Non-adiabatic Case
Here the dynamics for non-adiabatic flow is discussed. We formulate dynam-ical and transport equations and then finally couple these equations.
We consider a plane symmetric distribution of collapsing fluid bounded bya hypersurface Σ. The line element for the interior region has the followingform ds − = − A ( t, z ) dt + B ( t, z )( dx + dy ) + C ( t, z ) dz , (2.1)where we have assumed co-moving coordinates inside Σ. The interior coor-dinates are taken as χ − = t, χ − = x, χ − = y, χ − = z . It is assumedthat fluid is locally anisotropic and suffering dissipation in the form of shear-ing viscosity, heat flow and free streaming radiation. The energy-momentumtensor has the following form T ab = ( µ + P ⊥ ) V a V b + P ⊥ g ab + ( P z − P ⊥ ) χ a χ b + q a V b + V a q b + ǫℓ a ℓ b − ησ ab , (2.2)where µ is the energy density, P z is the pressure in z -direction, P ⊥ is thepressure perpendicular to z -direction (i.e., x or y direction), V a four velocityof fluid, η is the coefficient of shear viscosity, χ a is a unit vector in z -direction, q a is the heat flux, ǫ is the radiation density and ℓ a is a null four vector in z -direction. Furthermore, these quantities satisfy the relations V a V a = − , χ a χ a = 1 , χ a V a = 0 ,V a q a = 0 , ℓ a V a = − , ℓ a ℓ a = 0 . (2.3)Since the metric defined in Eq.(2.1) is co-moving, we can take V a = A − δ a , χ a = C − δ a , q a = qC − δ a ,ℓ a = A − δ a + C − δ a . (2.4)In the standard irreversible thermodynamics by Eckart, we have the fol-lowing relation [33] π ab = − ησ ab , Π = − ζ Θ , (2.5)where η and ζ stand for coefficients of shear and bulk viscosity, σ ab is theshear tensor, Θ is the expansion, Π is the bulk viscosity and π ab is the shear4iscosity tensor. The algebraic nature of Eckart constitutive equations causesseveral problems but we are concerned with the causal approach of dissipativevariables. Thus we would not assume (2.5) rather we shall resort to transportequations of M¨ u ller-Israel-Stewart theory.The non-zero component of acceleration and expansion scalar are givenby a = A ′ A ,
Θ = 1 A ( 2 ˙ BB + ˙ CC ) , (2.6)where dot and prime denote differentiation with respect to t and z respec-tively. The non-vanishing components of shear tensor are σ = − B F = σ , σ = 23 C F ; F = 1 A ( − ˙ BB + ˙ CC ) . (2.7)The magnitude of the shear tensor, i.e., the shear scalar σ is defined as σ = 12 σ ab σ ab = 19 F (2.8)which implies that F = 9 σ . The energy-momentum tensor of an electromagnetic field is defined as E ab = 14 π ( F ca F bc − F cd F cd g ab ) , (2.9)where F ab is the electromagnetic field tensor given by F ab = φ b,a − φ a,b , (2.10) φ a is four potential. The Maxwell field equations are F ab ; b = µ J a , F [ ab ; c ] = 0 , (2.11)where µ = 4 π is the magnetic permeability and J a is the four current. In aco-moving frame φ a and J a are given by φ a = φδ a , J a = ξV a , (2.12)5here ξ and φ are the charge density and electric scalar potential respectivelyand both are functions of t and z . The charge conservation J a ; a = 0 givesthe charge for interior region as s ( z ) = Z z ξB Cdz. (2.13)For a = 0 ,
3, the first of Eq.(2.11) becomes φ ′′ − [ A ′ A + C ′ C − B ′ B ] φ ′ = µ ξAC , (2.14)˙ φ ′ − [ ˙ AA + ˙ CC − BB ] φ ′ = 0 , (2.15)while for a = 1 ,
2, it is trivially satisfied. Also, the second of Eq.(2.11) isidentically satisfied. Integration of Eq.(2.14) with respect to z , assuming φ ′ ( t,
0) = 0, gives φ ′ = µ s ( z ) ACB . (2.16)The Taub’s mass function [34] for plane symmetric spacetime can be gener-alized to include the electromagnetic contributions as m ( t, z ) = B B A − B ′ C ) + s B . (2.17)For the interior spacetime, the Einstein field equations, G ab = 8 π ( T ab + E ab ), yield the following set of equations8 π ( µ + ǫ ) A + ( µ sA ) B = ˙ BB ( 2 ˙ CC + ˙ BB ) + ( AC ) [ − B ′′ B +( 2 C ′ C − B ′ B ) B ′ B ] , (2.18) − πAC ( q + ǫ ) = − ˙ B ′ B + A ′ ˙ BAB + ˙ CB ′ CB , (2.19)8 π ( P ⊥ + 23 ηF ) B + ( µ sB ) = − ( BA ) [ ¨ BB + ¨ CC − ˙ AA ( ˙ BB + ˙ CC ) + ˙ B ˙ CBC ] + B C [ A ′′ A + B ′′ B − A ′ A ( C ′ C − B ′ B ) − B ′ C ′ BC ] , (2.20)8 π ( P z + ǫ − ηF ) C − ( µ sC ) B = − ( CA ) [ 2 ¨ BB + ( ˙ BB ) − A ˙ BAB ] + ( B ′ B ) + 2 A ′ B ′ AB . (2.21)6n view of Eqs.(2.6) and (2.7), Eq.(2.19) can be written as4 π ( q + ǫ ) C = 13 (Θ − F ) ′ − F B ′ B . (2.22)
Here we formulate junction conditions for the general plane symmetric space-time in the interior and charged Vaidya plane symmetric spacetime in theexterior. The line element for the exterior region is [35] ds = ( 2 M ( ν ) Z − e ( ν ) Z ) dν − dZdν + Z ( dX + dY ) , (2.23)where χ +0 = ν, χ +1 = X, χ +2 = Y, χ +3 = Z . The metric for hypersurfaceis defined as ( ds ) Σ = − ( dτ ) + f ( τ ) ( dx + dy ) , (2.24)where w i = ( τ, x, y )( i = 0 , ,
2) are the intrinsic coordinates of Σ. Theequations of hypersurface in terms of interior and exterior coordinates are k − ( t, z ) = z − z Σ = 0 , (2.25) k + ( ν, Z ) = Z − Z Σ ( ν ) = 0 , (2.26)where z Σ is constant. Using Eqs.(2.25) and (2.26), we get the interior andexterior metrics over the hypersurface as( ds − ) Σ = − A ( t, z Σ ) dt + B ( t, z Σ )( dx + dy ) , (2.27)( ds ) Σ = − [( − M ( ν ) Z Σ + e ( ν ) Z ) + 2 dZ Σ dν ] dν + Z ( dX + dY ) . (2.28)Now we use the junction conditions proposed by Darmois [5], the firstcondition is ( ds ) Σ = ( ds − ) Σ = ( ds ) Σ (2.29)which yields the following equations dtdτ = 1 A , (2.30) Z Σ = B, (2.31) dνdτ = [( − M ( ν ) Z Σ + e ( ν ) Z ) + 2 dZ Σ dν ] − . (2.32)7he second junction condition is the continuity of extrinsic curvature (thesecond fundamental form) K ab = K − ab = K + ab . (2.33)The unit normal in terms of interior and exterior coordinates are given re-spectively as n − a = C (0 , , , , n + a = ( − ˆ Z Σ , , , ˆ ν ) , (2.34)here hat denotes differentiation with respect to τ . The surviving componentsof the extrinsic curvature for the interior spacetime are K − = − [ A ′ AC ] Σ , K − = [ BB ′ C ] Σ = K − . (2.35)The non-null components of the extrinsic curvature for the exterior spacetimeare given by K +00 = [ d νdτ ( dνdτ ) − − ( MZ − e Z )( dνdτ )] Σ . (2.36) K +11 = [ Z dZdτ + ( e Z − M ) dνdτ ] Σ = K +22 . (2.37)Thus the second junction condition yields the following equations − [ A ′ AC ] Σ = [ d νdτ ( dνdτ ) − − ( MZ − e Z )( dνdτ )] Σ , (2.38)[ BB ′ C ] Σ = [ Z dZdτ + ( e Z − M ) dνdτ ] Σ . (2.39)After some algebra, it follows that M ( ν ) Σ = m ( t, z ) ⇔ s Σ = e, (2.40) q Σ = P z − ηF − s B ( µ − . (2.41)These equations give necessary and sufficient conditions for the matching ofinterior and exterior spacetimes. Equation (2.41) describes a relationshipbetween heat flux, effective pressure in z -direction and charge over the hy-persurface. It shows that if the fluid has no charge then effective pressureand heat flux are equal over the hypersurface.8 .4 Dynamical Equations Here we develop equations that govern the dynamics of non-adiabatic planesymmetric collapsing process by using Misner-Sharp formalism [4]. Theproper time derivative and proper derivative in z -direction are defined re-spectively [16] as D ˜ T = 1 A ∂∂t , (2.42) D ˜ Z = 1˜ Z ′ ∂∂z , (2.43)where ˜ Z = B . The velocity of the collapsing fluid is the proper time deriva-tive of ˜ Z defined as U = D ˜ T ( ˜ Z ) = ˙ BA (2.44)which is always negative. Using this expression, Eq.(2.17) implies that E = B ′ C = [ U − mB + s B ] . (2.45)When we make use of Eq.(2.43) in (2.22), we have4 πC ( q + ǫ ) = E [ 13 D ˜ Z (Θ − F ) − F ˜ Z ] . (2.46)The rate of change of mass (given in Eq.(2.17)) with respect to proper timeis given by D ˜ T m = − π [( P z + ǫ − ηF ) U + E ( q + ǫ )] ˜ Z + s U Z ( µ − . (2.47)This equation shows how mass is varying within the plane hypersurfaceunder the influence of matter variables. The first term represents effectivepressure in z -direction and radiation density. When collapse takes place,this term is positive implying that energy increases by this factor. Thesecond term in square brackets shows that energy is going out from theplane hypersurface while the last term is the charge contribution. Duringcollapse, energy decreases due to these terms. Similarly, we calculate D ˜ Z m = 4 π [ µ + ǫ + ( q + ǫ ) UE ] ˜ Z + s ˜ Z D ˜ Z s + s Z ( µ − . (2.48)9his equation describes how different quantities influence the mass betweenneighboring hypersurfaces in the fluid distribution. The term ( µ + ǫ ) indicatesthe effects of energy density and radiation density. Similarly, the second termshows the amount of heat and radiation which is getting out. The remainingtwo terms represent contribution of electric charge. Integration of Eq.(2.48)yields m = Z ˜ Z π [ µ + ǫ + ( q + ǫ ) UE ] ˜ Z d ˜ Z + s Z + 12 Z ˜ Z ( s ˜ Z ) d ˜ Z + ( µ − Z ˜ Z ( s ˜ Z ) d ˜ Z. (2.49)The dynamical equations can be obtained from the contracted Bianchiidentities ( T ab + E ab ) ; b = 0. Consider the following two equations( T ab + E ab ) ; b V a = ( T b ; b + E b ; b ) V = 0 , (2.50)( T ab + E ab ) ; b χ a = ( T b ; b + E b ; b ) χ = 0 (2.51)which yield( µ + ǫ ) · A + ( µ + 2 ǫ + P z − ηF ) ˙ CAC + 2( µ + ǫ + P ⊥ + 23 ηF )˙ BAB + ( q + ǫ ) ′ AC + 2 A ′ AC ( q + ǫ ) + 2 B ′ BC ( q + ǫ ) = 0 , (2.52)( q + ǫ ) · A + 1 C ( P z + ǫ − ηF ) ′ + 2( q + ǫ )( BC ) · ABC + ( µ + P z + 2 ǫ − ηF ) A ′ AC + 2( P z − P ⊥ + ǫ − ηF ) B ′ BC − µ ss ′ πCB = 0 . (2.53)The acceleration of the collapsing fluid is defined as D ˜ T U = 1 A ∂U∂t = ¨ BA − ˙ A ˙ BA . (2.54)Using Eqs.(2.21), (2.42) and (2.17), we have D ˜ T U = − π ( P z + ǫ − ηF ) ˜ Z − m ˜ Z + s Z ( µ + 1) + EA ′ AC (2.55)10hich gives the value of A ′ A A ′ A = CE [ D ˜ T U + 4 π ( P z + ǫ − ηF ) ˜ Z ] + mCE ˜ Z − Cs E ˜ Z ( µ + 1) . (2.56)Substituting this value in Eq.(2.53), it follows that( µ + P z + 2 ǫ − ηF ) D ˜ T U = − ( µ + P z + 2 ǫ − ηF )[ m ˜ Z + 4 π ( P z + ǫ − ηF ) ˜ Z − s Z ( µ + 1)] − E [ D ˜ Z ( P z + ǫ − ηF ) + 2˜ Z ( P z − P ⊥ + ǫ − ηF ) − µ sD ˜ Z s π ˜ Z ] − E [ D ˜ T ( q + ǫ ) + 4( q + ǫ ) U ˜ Z +2( q + ǫ ) F ] . (2.57)This equation yields the effect of different forces on the collapsing process.It can be interpreted in the form of Newton’s second law of motion i.e., force= mass density × acceleration. The term within round brackets on LHSrepresents the inertial or passive gravitational mass density. This term showsthat effective pressure, energy density and density of null fluid have effectson mass density while heat flux and charge have no contribution here. Byequivalence principle, the round brackets factor on RHS is taken as activegravitational mass density. The quantities within square brackets in the firstterm show the influence of effective pressure, radiation density and electriccharge on active gravitational mass. Using Eq.(2.49) in (2.57), it follows thatcharge increases the active gravitational mass if s Z + µ Z Z ˜ Z ( s Z ) d ˜ Z − s Z ( µ + 1) > ⇒ s ˜ Z > D ˜ Z s. (2.58)If this inequality holds then charge regeneration phenomenon analogous topressure regeneration occurs [36]. The pressure regeneration means that thepressure which is trying to keep hydrostatic equilibrium by balancing gravi-tational attraction, at the same time contributes to the active gravitationalmass. This implies that it promotes gravitational collapse. Otherwise, ifthe above inequality is not satisfied, charge will decrease active gravitationalmass and consequently the Coulomb repulsion may prevent the gravitationalcollapse.The first term in the second square brackets is the gradient of effectivepressure in z -direction and radiation density. Since this gradient is negative,11t increases the rate of collapse. The second term is due to local anisotropyof pressure, radiation density and contribution of viscosity. If this term ispositive then it contributes to increase collapse and vice versa. The lastterm depicts Coulomb repulsion that opposes gravitation implying that itdecelerates the collapsing process.Finally, the last square brackets is entirely due to dissipation. To see therole of D ˜ T q , this equation is coupled with causal transport equation. Theconsequences of D ˜ T ǫ have been discussed by Misner [37]. The outward fluxof radiation accelerates collapsing process by increasing gravitational force.The third term is positive as U <
0, so it slows down rate of collapse. The lastterm shows the combine effect of viscosity and dissipation. From Eq.(2.57),the condition for hydrostatic equilibrium can be obtained by replacing U = q = ǫ = η = 0 as D ˜ Z P z = − ( P z + µ ) E [ m ˜ Z − s Z ( µ + 1)] + µ sD ˜ Z s π ˜ Z − Z ( P z − P ⊥ ) . (2.59) The transport equation for heat flux derived from the M¨ u ller-Israel-Stewarttheory of dissipative fluids [33] is given by τ h ab V c q b ; c + q a = − κh ab ( T ,b + T a b ) − κT ( τ V b κT ) ; b q a , (2.60)where h ab is the projection tensor, κ denotes thermal conductivity, T is tem-perature and τ stands for relaxation time. This equation has only one inde-pendent component D ˜ T q = − κT q τ D ˜ T ( τ κT ) − q [ 3 U Z + F τ ] − κEτ D ˜ Z T − κTτ E × D ˜ T U − κTτ E [ m + 4 π ( P z + ǫ − ηF ) ˜ Z − s ˜ Z ] 1˜ Z . (2.61)We now couple this equation with dynamical Eq.(2.57) to see the effects ofheat flux or dissipation on collapsing process. Replacing Eq.(2.61) in (2.57),12e obtain( µ + P z + 2 ǫ − ηF )(1 − α ) D ˜ T U = (1 − α ) F grav + F hyd + κE τ D ˜ Z T + E [ κT q τ D ˜ T ( τ κT ) − D ˜ T ǫ ] − Eq ( 5 U Z + 32 F − τ ) − Eǫ ( 2 U ˜ Z + F ) , (2.62)where F grav , F hyd and α are given by the following equations F grav = − ( µ + P z + 2 ǫ − ηF )[ m + 4 π ( P z + ǫ − ηF ) ˜ Z − s Z × ( µ + 1)] 1˜ Z , (2.63) F hyd = − E [ D ˜ Z ( P z + 2 ǫ − ηF ) + 2˜ Z ( P z − P ⊥ + ǫ − ηF ) − sD ˜ Z s π ˜ Z ] , (2.64) α = κTτ ( µ + P z + 2 ǫ − ηF ) − . (2.65)The consequence of coupling transport and dynamical equations is that theinertial and active gravitational mass densities are affected by a factor α given by Eq.(2.65). The gravitational force term defined in Eq.(2.63) is alsoaffected by α but the hydrodynamical forces Eq.(2.64) are not influenced bythis term. Here we find some relationship between the Weyl tensor and matter variables.The Weyl scalar C in terms of Kretchman scalar R , the Ricci tensor R ab andthe Ricci scalar R is given by C = R − R ab R ab + 13 R . (2.66)The Kretchman scalar R = R abcd R abcd becomes R = 4[ 2 A B ( R ) + 1 A C ( R ) + 1 B ( R ) + 2 B C ( R ) − A B C ( R ) ] . (2.67)13he non-zero components of the Riemann tensor can be written in terms ofthe Einstein tensor and mass function as R = ( AB ) [ 12 C G + 1 B ( m − s B )] = R ,R = ( AC ) [ 12 A G − C G + 1 B G − B ( m − s B )] ,R = 2 B ( m − s B ) ,R = ( BC ) [ 12 A G − B ( m − s B )] = R ,R = − B G . Substituting these values in Eq.(2.67), after some algebra, we obtain R = 48 B ( m − s B ) − B ( m − s B )[ G A − G C + G B ] − A C G + 3[( G A ) + ( G C ) ] + 4 B G − G G A C + 4( G A − G C ) G B . (2.68)Now we calculate the remaining part of the Weyl scalars which need Riccitensor and Ricci scalar in terms of the Einstein tensor. These are R = A [ G A + G C + G B ] , R = G ,R = B G A − G C ] = R , R = C [ G A + G C − G B ] ,R = G A − G C − G B ,R ab R ab = G A + G C + 2 G B − G A C . Thus the remaining part of the Weyl scalar becomes13 R − R ab R ab = − G A − G C − G B + 4 G A C − G G A C + 43 G G C B − G G A B . (2.69)14sing Eqs.(2.68) and (2.69), the Weyl scalar takes the form C = 48 B ( m − s B ) − B ( m − s B )[ G A − G C + G B ] − A C G + 3[( G A ) + ( G C ) ] + 4 B G − G G A C + 4( G A − G C ) G B − G A − G C − G B + 4 G A C − G G A C + 43 G G C B − G G A B . (2.70)After some algebra, it leads to the following equation C B (48) = ( m − s B ) − B G A + G B − G C − s B ] . (2.71)Using the field equations, we have G A + G B − G C − µ s B = 8 π ( µ + P ⊥ − P z + 2 ηF ) . (2.72)In view of the above equation and using ˜ Z = B , Eq.(2.71) becomes C ˜ Z (48) = [ m − π µ + P ⊥ − P z + 2 ηF ) ˜ Z − s Z ( µ + 1)] . (2.73)The derivatives of ( C ˜ Z (48) ) with respect to ˜ T and ˜ Z are given by D ˜ T ( C ˜ Z (48) ) = − π [ 13 ˜ Z D ˜ T ( µ + P ⊥ − P z + 2 ηF ) + ( µ + P ⊥ + ǫ + 23 ηF ) × ˜ Z U + ( q + ǫ ) E ˜ Z ] + s U Z ( µ + 1) , (2.74) D ˜ Z ( C ˜ Z (48) ) = 4 π [( q + ǫ ) ˜ Z UE −
13 ˜ Z D ˜ Z ( µ + P ⊥ − P z + 2 ηF ) + ( ǫ − P ⊥ + P z − ηF ) ˜ Z ] − sD ˜ Z s ˜ Z ( µ + 1) + s Z ( µ + 1) . (2.75)These equations give relationship between the Weyl scalar and the fluid prop-erties like density, viscosity and pressure (anisotropy). For perfect and non-charged fluid, Eq.(2.75) reduces to the following form D ˜ Z ( C ˜ Z (48) ) = − π Z D ˜ Z µ. (2.76)15sing the regularity condition, it is concluded that D ˜ Z µ = 0 if and onlyif C = 0. This means that if energy density is homogeneous, the metric isconformally flat and vice versa.We would like to mention here that a particularly simple relation betweenthe Weyl tensor and density inhomogeneity such as (2.76), for perfect non-charged fluids, is at the origin of Penrose’s proposal to provide a gravitationalarrow of time in terms of the Weyl tensor [38]. The rationale behind this ideais that tidal forces tend to make the gravitating fluid more inhomogeneousas the evolution proceeds, thereby indicating the sense of time. However,the fact that such a relationship is no longer valid in the presence of localanisotropy of the pressure and/or dissipative processes and/or electric charge.This has already been discussed [32, 39] explaining its failure in scenarioswhere the above-mentioned factors are present [40]. Here we see how theelectric charge distribution affects the link between the Weyl tensor anddensity inhomogeneity, suggesting that electric charge (whenever present)should enter into any definition of a gravitational arrow of time. In this case, heat flux vanishes, also, we assume that radiation density iszero and hence dissipation is only due to shearing viscosity. The energy-momentum tensor is obtained by replacing q a = ǫ = 0 in Eq.(2.2). Similarly,the Einstein-Maxwell field equations are found by using q = ǫ = 0 in thecorresponding equations derived for non-adiabatic case. For junction condi-tions, the line element for the exterior region is taken as plane symmetricReissner-Nordstr¨ o m spacetime given by ds = − ( − MZ + e Z ) dT + ( − MZ + e Z ) − dZ + Z ( dX + dY ) , (3.77)where ( χ +0 , χ +1 , χ +2 , χ +3 ) = ( T, X, Y, Z ). The equation of hypersurface interms of exterior coordinates is k + ( T, Z ) = Z − Z Σ ( T ) = 0 . (3.78)Using Eq.(3.78), the exterior metric over the hypersurface becomes( ds ) Σ = − [( − MZ Σ + e Z ) − ( − MZ Σ + e Z ) − ( dZ Σ dT ) ] dT (3.79)+ Z ( dX + dY ) . dtdτ = 1 A , Z Σ = B, (3.80) dTdτ = ( − MZ Σ + e Z ) [( − MZ Σ + e Z ) − ( dZ Σ dT ) ] − . (3.81)Equation (3.81) implies that dτ = N dT − N dZ , N = ( − MZ Σ + e Z ) . (3.82)The unit normal in terms of exterior coordinates is given by n + a = ( − ˆ Z Σ , , , ˆ T ) , (3.83)The non-null components of the extrinsic curvature for the exterior spacetimeare K +00 = [ dZdτ d Tdτ − d Zdτ dTdτ − N dNdZ ( dTdτ ) + 32 N dNdZ dTdτ ( dZdτ ) ] Σ . (3.84) K +11 = [ ZN dTdτ ] Σ = K +22 . (3.85)The second junction condition yields M Σ = m ( t, z ) ⇔ s Σ = e. (3.86)The rate of change of mass with respect to ˜ T and ˜ Z are given by the followingequations D ˜ T m ( t, z ) = − π ( P z − ηF ) U ˜ Z + s U Z ( µ − , (3.87) D ˜ Z m ( t, z ) = 4 πµ ˜ Z + s ˜ Z D ˜ Z s + s Z ( µ − . (3.88)The description of these equations is the same as for the non-adiabatic case.Similarly, the dynamical equations can be obtained using q = ǫ = 0 inEqs.(2.52)-(2.57). For this case, Eq.(2.57) becomes( µ + P z − ηF ) D ˜ T U = − ( µ + P z − ηF )[ m ˜ Z + 4 π ( P z − ηF ) ˜ Z − s ˜ Z ] − E [ D ˜ Z ( P z − ηF ) + 2˜ Z ( P z − P ⊥ − ηF ) − µ sD ˜ Z s π ˜ Z ] . (3.89)17he inequality given in Eq.(2.58) remains the same. The interpretation of theterms in Eq.(3.89) is similar to that of the non-adiabatic case just excludingthe effects of heat flux and radiation density. As there is no heat flux, so notransport equation is needed. The relationship between the Weyl tensor andenergy homogeneity also remains the same. Gravitational collapse is an outstanding phenomenon in gravitation theory.The aim of this work is to analyze the dynamics of gravitational collapse forplane symmetric configuration of real fluid. The conclusions are given in thefollowing1. The junction conditions for both cases yield that masses of the interiorand exterior regions are equal if and only if their corresponding chargesare equal. For the non-adiabatic case, junction conditions also givea relationship between heat flux, effective pressure in z -direction andcharge over the hypersurface (2.41). This equation implies that if s = 0,then the effective pressure in z -direction and heat flux are equal overthe hypersurface.2. For the non-dissipative case, Eq.(2.46) reduces to D ˜ Z ( U ˜ Z ) = 0 whichshows that collapse is homologous, i.e., all the matter falls inward in asimilar pattern.3. Condition for hydrostatic equilibrium is given by Eq.(2.59). If thiscondition holds, then the collapsing process will stop and matter attainsan equilibrium state.4. It is observed that charge will increase the rate of collapse if s ˜ Z > D ˜ Z s .Thus the chance of becoming a black plane increases in this case.5. In the non-adiabatic case, the radiation density increases inertial andactive gravitational masses. Also, the outflow of radiation causes anincrease in the rate of collapse and hence the collapsing process is ex-pected to be faster than the adiabatic case.6. The substitution of transport equation (2.61) in dynamical equation(2.57) yields an additional factor α . This α term affects the inertial18ass density and gravitational force term. As α increases, the affectedterms are decreased by the same amount and vice versa. The corre-sponding terms for α come from the term a b T in Eq.(2.60) which isthe Tolman’s inertial term. Hence the inertia of heat by increasing α ,causes a decrease in inertial mass and gravitational force [41]. Thus wecan conclude that • If α →
0, then inertial density and gravitational force are notaffected by coupling. • If 0 < α <
1, then inertia of heat causes a decrease in inertial andgravitational mass densities. • If α →
1, then mass densities approach to zero. • If α >
1, then the gravitational force term becomes negative im-plying that reversal of collapse occurs due to the inertia of heat.7. Under certain conditions homogeneity in energy density and conformalflatness of spacetime are necessary and sufficient condition for eachother.8. A relation (2.75) has been obtained exhibiting the way in which electriccharge affects the link between the Weyl tensor and density inhomo-geneity.
References [1] Penrose, R.: Phys. Rev. Lett. (1965)57.[2] Oppenheimer, J.R. and Snyder, H.: Phys. Rev. (1939)455.[3] Markovic, D. and Shapiro, S.L.: Phys. Rev. D61 (2000)084029.[4] Misner, C.W. and Sharp, D.: Phys. Rev. (1964)B571.[5] Darmois, G.: Memorial des Sciences Mathematiques (Gautheir-Villars,Paris, 1927) Fasc. 25.[6] Lichnerowicz, A.:
Theories Relativistes de le Gravitation et de l’ Electromagne’tisme , Masson, (1955), Chap
I, III .197] Brien, S.O. and Synge, J.L.: Comm. of the Dublin Institute for Ad-vanced Studies, A9 (1952).[8] Bonnor, W.B. and Vickers, P.A.: Gen. Relativ. Grav. (1981)29.[9] Herrera, L. and Santos, N.O.: Gen. Relativ. Grav. (to appear, 2010);gr-qc/0907.2253.[10] Sharif, M. and Iqbal, K.: Mod. Phys. Lett. A24 (2009)1533.[11] Sharif, M. and Ahmad, Z.: Mod. Phys. Lett.
A22 (2007)1493; ibid . 2947.[12] Herrera, L., Santos, N.O. and Wang, A.: Phys. Rev.
D78 (2008)084026.[13] Herrera, L. and Santos, N.O.: Phys. Rev.
D70 (2004)084004.[14] Di Prisco, A., Herrera, L., MacCallum, M.A.H., Santos, N.O.: Phys.Rev.
D80 (2009)064031.[15] Nakao, K. and Morisawa, Y.: Class. Quantum Grav. (2004)2101.[16] Sharif, M. and Rehmat, Z.: Gen. Relativ. Grav. (2010)1795.[17] Sharif, M. and Ahmad, Z.: Int. J. Mod. Phys. A23 (2008)181.[18] Chakraborty, S. and Debnath, U.: Mod. Phys. Lett.
A20 (2005)1451.[19] Chakraborty, S., Chakraborty, S. and Debnath, U.: Int. J. Mod. Phys.
D16 (2007)833.[20] Kurita, Y. and Nakao, K.: Phys. Rev.
D73 (2006)064022.[21] Wang, A., Wu, Y. and Wu, Z.C.: Gen. Relativ. Grav. (2004)1225.[22] Nath, S., Debnath, U. and Chakraborty, S.: Astrophys. Space Sci. (2008)431.[23] Virbhadra, K.S., Narasimha, D. and Chitre, S.M.: Astron. Astrophys. (1998)1.[24] Virbhadra, K.S. and Ellis, G.F.R.: Phys. Rev. D62 (2000)084003.[25] Virbhadra, K.S. and Ellis, G.F.R.: Phys. Rev.
D65 (2002)103004.2026] Claudel, C.M., Virbhadra, K.S. and Ellis, G.F.R.: J. Math. Phys. (2001)818.[27] Virbhadra, K.S. and Keeton, C.R.: Phys. Rev. D77 (2008)124014.[28] Virbhadra, K.S.: Phys.Rev.
D60 (1999)104041.[29] Virbhadra, K.S.: Phys.Rev.
D79 (2009)083004.[30] Sharif, M. and Abbas, G.: Mod. Phys. Lett.
A24 (2009)2551.[31] Sharif, M. and Abbas, G.: J. Korean Phys. Society (2010)529.[32] Di Prisco, A., Herrera, L., Denmat, G. Le, MacCallum, M.A.H. andSantos, N.O.: Phys. Rev. D76 (2007)064017.[33] Maartens, R.:
Causal Thermodynamics in Relativity , astro-ph/9609119.[34] Zannias, T.: Phys. Rev.
D41 (1990)3252.[35] Kramer, D., Stephani, H., MacCallum, M.A.H. and Herlt, E.:
Exact So-lutions to Einstein Field Equations (Cambridge University Press, 2003).[36] Bekenstein, J.: Phys. Rev. D4 (1971)2185.[37] Misner, C.: Phys. Rev. (1965)B1360.[38] Penrose, R.: General Relativity, An Einstein Centenary Survey , eds.Hawking, S.W. and Israel, W. (Cambridge University Press, 1979)581-638.[39] Herrera, L., Di Prisco, A., Martin, J. Ospino, J., Santos, N.O. andTroconis, O.: Phys. Rev.
D69 (2004)084026.[40] Bonnor, W.B.: Phys. Lett.