Dissipative Cylindrical Collapse of Charged Anisotropic Fluid
DDissipative Cylindrical Collapse of Charged Anisotropic Fluid
Sarbari Guha and Ranajoy Banerji ∗ Department of Physics, St. Xavier’s College (Autonomous), 30 Mother Teresa Sarani, Kolkata, India Saha Institute of Nuclear Physics, Kolkata, India
We have studied the dynamics of a cylindrical column of anisotropic, charged fluid which isexperiencing dissipation in the form of heat flow, free-streaming radiation, and shearing viscosity,undergoing gravitational collapse. We calculate the Einstein-Maxwell field equations and, using theDarmois junction conditions, match the interior non-static cylindrically symmetric space-time withthe exterior anisotropic, charged, cylindrically symmetric space-time. The behavior of the density,pressure and luminosity of the collapsing matter has been analyzed. From the dynamical equations,the effect of charge and dissipative quantities over the cylindrical collapse are studied. Finally, wehave derived the solutions for the collapsing matter which is valid during the later stages of collapseand have discussed the significance from a physical standpoint.
PACS numbers: 04.20.-q, 04.40.Dg, 97.10.Cv
I. INTRODUCTION
Gravitational collapse with realistic astronomical matter distribution is an important problem in relativistic gravityand astrophysics [1–3]. Over the years, there has been an extensive study of collapse of dust and fluids under gravitystarting from the works of Chandrasekhar, Zwicky, Oppenheimer and Snyder [4–6]. Vaidya [7, 8] studied the externalgravitational field of a stellar body giving out radiations. Misner and Sharp [9, 10] studied spherically symmetriccollapse. Others [11–20] studied different cases of spherically symmetric fluids undergoing collapse.Although classical considerations rule out the existence of physical objects with large amounts of charge, yet thereare mechanisms which give rise to huge amount of electric charge in objects collapsing under the effect of self gravity.Rosseland [21] indicated that in the stellar ensemble the atoms are strongly ionized and since the forces between thefree particles should follow the inverse square law, it should be of higher order of magnitude than the residual forcesacting between neutral atoms. For a star with 1.5 times solar mass and mean molecular weight 2.8, the effect ofelectrical forces is substantial if the star is built of heavy elements. Eddington [22] showed that a star has an internalelectric field for which the electric potential φ depends on the gravitational potential ψ , the mass m p and charge e ofa proton, and a scalar parameter α which in turn depends on the density n i of the ions, atomic weight A i of the ions,and the effective charge eZ i .Raychaudhuri and De [23] considered the Einstein-Maxwell equations for a charged dust without imposing anyspecial symmetry restrictions. If the magnetic field vanishes, the electric flux through any element of area boundedby particles of the dust is a constant of motion, the vorticity and electric field being orthogonal. For irrotationalmotion in the absence of magnetic fields, the electric field vector is orthogonal to the surfaces with constant valuesof (cid:15)/ρ . They showed the impossibility of isotropic expansion and that for a charged dust in irrotational motion inabsence of magnetic fields, the expansion (or contraction) cannot be shear-free. Further, the electric field and alongwith it the charge density would vanish if the spatial expansion were shear-free and non-vanishing. Olson and Bailyn[24] considered stars with central mass densities larger than those reasonable for a white dwarf and found that thedeviations from the Chandrasekhar model were large. They found that the charge-to-mass ratio of the star was directlyproportional to the average mass density. For large central mass densities, the central charge density increased andeventually produced large internal electric fields.Bally and Harrison [25] showed that for a star of total charge Q and mass M , the charge-to-mass ratio is given by Q/M = Gαm p /e and with α ∼ Q/M of the order of 100 coulombs per solar mass. The positive charge within astar is not automatically screened by a negatively charged atmosphere. The scale length L always exceeds the Debyelength λ D in stellar atmospheres and the interstellar medium, and both are therefore positively charged and haveapproximately the same ratio of charge and mass densities as stars. The Debye length λ D depends on the electrondensity n e in a gas of temperature T . Thus an entire galaxy can be positively charged. Even elliptical galaxies havea size that is large compared with the Debye length of their interstellar media. Oliveira and Santos [26] have studiedthe junction conditions of a collapsing non-adiabatic charged body producing radiation and have observed important ∗ RB did portions of this work during a summer program as a master’s student of the Dept. of Physics and Astrophysics, University ofDelhi. a r X i v : . [ g r- q c ] D ec physical consequence due to the presence of charge. It is possible that very high electric fields may exist in strangestars with quark matter [27, 28] under equilibrium configurations. However these do not apply to phases of intensedynamical activity with time scales of the order of (or smaller than) the hydrostatic time scale, and for which thequasistatic approximation is not reliable (e.g. the collapse of very massive stars or the quick collapse phase precedingneutron star formation).Gravitational collapse is known to be a highly dissipative phenomenon [29–31]. The evolution of massive stars ischaracterized by dissipation due to the emission of photons or neutrinos, or both. The diffusion approximation isbased on the assumption that the energy flux of radiation is proportional to the gradient of temperature. During theprocess of emission, the radiative transport is closer to the diffusion approximation and not to the free-streaming limit.But there are many other situations in which the mean free path of particles transporting energy are so large that thefree-streaming approximation is the viable choice. Hence in a realistic model of collapse we need to consider radiativetransport with both diffusion and free streaming. The study of the collapse of a strongly elongated axisymmetricbody is important since such type of collapse could occur in a real astrophysical situations. Moreover, according tonumerical simulations [32], it is a possible candidate for the violation of the cosmic censorship conjecture [33], and itgives insight into the hoop conjecture [34]. A realistic model of collapse should also include radial heat flux.The study of non-spherical gravitational collapse has gained in momentum following the discovery of cylindrical andplane gravitational waves. Cylindrical gravitational waves were first studied by Einstein and Rosen [35, 36]. Thorne[37] proposed a definition of energy for systems invariant under rotations about and translations along a symmetryaxis. This is the ”cylindrical energy” or ”C energy” which obeys the conservation law and is locally measurable. Theunique static universe of Melvin [38] gives an absolute minimum of the C energy contained inside any cylinder. TheC energy is also used to demonstrate the resistance of magnetic field lines to cylindrical gravitational collapse. Chiba[39] studied the case of cylindrical dust collapse. Others [40–42] investigated various aspects of cylindrical collapseof counter rotating dust and rotating cylindrical shells. Hayward studied gravitational waves, black holes and cosmicstrings in cylindrical symmetry [43]. Considering the most general vacuum cylindrical spacetimes, Goncalves [44]presented a formal derivation of Thorne’s C-energy, based on a Hamiltonian reduction approach. For the cylindricalcollapse of counter-rotating dust, Goncalves and Jhingan [41] showed that generic regular initial data could be specifiedfor which there were no trapped surfaces in the spacetime, and a line-like singularity was inevitably developed. DiPrisco et al. [20] studied nonadiabatic charged, dissipative, spherically symmetric gravitational collapse with shear.They [45] also studied shear-free cylindrical gravitational collapse for an interior non-rotating fluid with anisotropicpressures and exterior vacuum Einstein-Rosen spacetime. The case of the collapse of a heat conducting chargedanisotropic fluid cylinder have been studied by Sharif and Abbas [46] and that of a charged fluid cylinder with shearviscosity by Sharif and Fatima [47].In this work, we have have examined the effect of charge, heat flow, radiation and shear viscosity on the gravi-tational collapse of a cylindrical column of fluid, which is locally anisotropic. Local anisotropy is relevant for thedescription of relativistic compact objects and viscous effects are important in the formation of neutron stars. Theresult of Raychaudhuri and De, that the shear cannot vanish in the evolution of irrotational charged dust, underliesthe importance of shear in the collapse of charged fluids. After describing the gravitational source along with thecorresponding physical parameters like the expansion scalar, acceleration, shear tensor and the Einstein-Maxwell fieldequations in the next section, we discuss the junction conditions for the exterior Vaidya metric in presence of chargein the retarded time coordinate in section III. Subsequently the dynamical equations and the solutions in presence ofshear are derived in section IV. The summary of this whole exercise is presented in section V. II. THE INTERIOR METRIC AND THE FIELD EQUATIONS
We consider a collapsing cylinder filled with an anisotropic, charged fluid and undergoing dissipation in the form ofheat flow, free-streaming radiation, and shearing viscosity, bounded by a timelike cylindrical three-surface Σ, whichdivides the space-time into two distinct 4-dimensional manifolds V + and V − . A. The interior spacetime
For the interior V − space-time we take the general non-static cylindrically symmetric metric in the comovingcoordinates given by [47] ds − = − A ( t, r ) dt + B ( t, r ) dr + C ( t, r )( dθ + dz ) (1)In order to represent cylindrical symmetry, the range of coordinates is required to be as follows: −∞ < t < + ∞ , ≤ r < + ∞ , ≤ θ ≤ π, −∞ < z < + ∞ , with the coordinate labels, x = t , x = r , x = θ and x = z . The interior energy-momentum tensor is given,according to relativistic hydrodynamics, as T αβ = ( µ + P ⊥ ) V α V β + P ⊥ g αβ + ( P r − P ⊥ ) χ α χ β + V α q β + V β q α + (cid:15)l α l β − ησ αβ , (2)where, µ → energy density, P ⊥ → tangential pressure, P r → radial pressure, q α → heat flux, (cid:15) → the radiationdensity, V α → χ α → unit 4-velocity in the radial direction, l α → a null 4-vector and η → coefficient of shearing viscosity > σ αβ , the 4-acceleration a α and the expansion Θ are defined as σ αβ = 12 (( V α ; β + V β ; α ) + ( a α V β + a β V α )) −
13 Θ( g αβ + V α V β ) , (3) a α = V α ; β V β , (4)and Θ = V α ; α . (5)Since we have assumed comoving coordinates for the interior metric, we have V α = A − δ α , χ α = B − δ α , l α = A − δ α + B − δ α , q α = B − qδ α , (6)such that V α V α = − , χ α χ α = 1 , χ α V α = 0 , q α V α = 0 , l α V α = − , l α l α = 0 . (7)In view of the equations (4), (5) and (6), we obtain the acceleration and the expansion scalar as follows: a α = A (cid:48) A δ α , (8)Θ = 1 A (cid:32) ˙ BB + 2 ˙ CC (cid:33) . (9)Using (3) to (6), we obtain the non-zero components of the shear tensor as σ = 2 √ B σ, σ = σ = − √ C σ. (10)The shear scalar σ defined by σ = 12 σ ij σ ij (11)is therefore obtained as σ = 1 √ A (cid:32) ˙ BB − ˙ CC (cid:33) . (12)As defined by Chiba [39], a cylindrically symmetric spacetime may be defined locally by the existence of twocommuting, spacelike, Killing vectors, such that the orthogonal space is integrable. For such a spacetime, there existcoordinates ( θ, z ) such that the Killing vectors are ( ξ θ , ξ z ) = ( ∂/∂θ, ∂/∂z ). The existence of cylindrical symmetryabout an axis implies that the orbits of one of these vectors are closed but those of the other is open. Each of theseKilling vectors must be hypersurface orthogonal. The norms of these Killing vectors are invariants [48], namely thecircumferential radius ζ = (cid:113) ξ θ · ξ (cid:91)θ = (cid:113) ξ (2) a ξ a (2) , and the specific length (cid:96) = (cid:113) ξ z · ξ (cid:91)z = (cid:113) ξ (3) a ξ a (3) with ξ (2) = ∂ θ , ξ (3) = ∂ z , under the sign convention that spatial metrics are positive definite, the dot representingcontraction and the flat (cid:91) represents the covariant dual with respect to the space-time metric. The gravitationalenergy per specific length in a cylindrically symmetric system (also known as C-energy) as defined by Thorne [37]and modified by him to render it finite in space-time, is given by E = 18 (1 − l − ∇ a ˜ r ∇ a ˜ r ) , (13)for which ˜ r = ζl, where the areal radius is ˜ r and E is the gravitational energy per unit specific length of the cylinder.Analogous to the Misner and Sharp energy for spherical symmetry [49], the specific energy of the cylinder due tothe electromagnetic field is therefore given by E (cid:48) = l C (cid:32) ˙ C A − C (cid:48) B (cid:33) + s C . (14)
B. Electromagnetic energy tensor and Maxwell’s equations
The electromagnetic energy-momentum tensor for the charged fluid is given by T ( em ) αβ = 14 π (cid:18) F γα F βγ − F γδ F γδ g αβ (cid:19) . (15)and the corresponding Maxwell’s equations are F αβ = ψ β,α − ψ α,β , (16) F αβ ; β = 4 πJ α , (17)where F αβ is the electromagnetic field tensor, ψ α is the corresponding four potential and J α is the four current densityvector. Since the charge is comoving with the fluid, the charge per unit length of the cylinder is at rest with respectto the fluid and there is no magnetic field, so that the four current density is proportional to the four velocity i.e. wehave ψ α = ψδ α = ψ ( t, r )(1 , , , , J α = ρV α , (18)where ψ ( t, r ) is an arbitrary function and ρ ( t, r ) is the charge density. So the only non-zero component of theelectromagnetic field tensor is F = − F = − ∂ψ∂r . (19)Thus from the Maxwell’s equations we obtain ψ (cid:48)(cid:48) − (cid:18) A (cid:48) A + B (cid:48) B − C (cid:48) C (cid:19) ψ (cid:48) = 4 πρAB , (20)˙ ψ (cid:48) − (cid:32) ˙ AA + ˙ BB − CC (cid:33) ψ (cid:48) = 0 , (21)where the first equation is for α = 0 and the second is for α = 1. Here the dot and the prime represent the partialderivatives with respect to t and r respectively. Integrating (20) we obtain ψ (cid:48) = 2 sABC , (22)where s ( r ) = 2 π (cid:90) r ρBC dr (23)is the total charge distributed per unit length of the cylinder. Equation (22) is in conformity with the law ofconservation of charge and satisfies Eq. (21). C. The Field Equations
We now find the field equations for this distribution of fluid. The Einstein field equations for the interior metriccan be written as G − αβ = 8 π ( T − αβ + T ( em ) − αβ ) (24)where G − αβ is the Einstein tensor for the interior metric. There are five non-zero components of (24) for the metric(1) with energy-momentum tensor (2), which are G − = 8 π ( T − + T ( em ) − )i.e. 8 π ( µ + (cid:15) ) A + 4 s A C = ˙ CC (cid:32) BB + ˙ CC (cid:33) + (cid:18) AB (cid:19) (cid:18) − C (cid:48)(cid:48) C + C (cid:48) C (cid:18) B (cid:48) B − C (cid:48) C (cid:19)(cid:19) . (25)Similarly, G − = 8 π ( T − + T ( em ) − ) , which yields 8 π ( q + (cid:15) ) AB = 2 (cid:32) ˙ C (cid:48) C − ˙ BC (cid:48) BC − ˙ CA (cid:48) CA (cid:33) . (26)The remaining equations are G − = 8 π ( T − + T ( em ) − )= 8 π (cid:18) P r + (cid:15) − √ ησ (cid:19) B − s B C = 8 π ( P r eff + (cid:15) ) B − s B C = − (cid:18) BA (cid:19) CC + (cid:32) ˙ CC (cid:33) − A ˙ CAC + (cid:18) C (cid:48) C (cid:19) + 2 A (cid:48) C (cid:48) AC , (27)where the effective radial pressure is defined as P r eff = P r − √ ησ and G − = 8 π ( T − + T ( em ) − ) = 8 π (cid:18) P ⊥ + 2 √ ησ (cid:19) C + 4 s C = 8 πP ⊥ eff C + 4 s C = − (cid:18) CA (cid:19) (cid:32) ¨ BB + ¨ CC − ˙ AA (cid:32) ˙ BB + ˙ CC (cid:33) + ˙ B ˙ CBC (cid:33) + (cid:18) CB (cid:19) (cid:18) A (cid:48)(cid:48) A + C (cid:48)(cid:48) C − A (cid:48) A (cid:18) B (cid:48) B − C (cid:48) C (cid:19) − B (cid:48) C (cid:48) BC (cid:19) , (28)with the effective tangential pressure as P ⊥ eff = P ⊥ + 2 √ ησ. III. EXTERIOR METRIC AND THE JUNCTION CONDITIONS
Exterior to the hypersurface Σ in the 4D manifold V + , we consider Vaidya’s metric [8] in presence of charge inthe retarded time coordinate as considered by Chao-Guang [50], but with a signature flip. The introduction of theretarded time coordinate removes the singularities of the original line element. Let M ( u ) and Q ( u ) be the mass andcharge of the fluid respectively inside the hypersurface Σ, where u is the retarded time coordinate. Then the exteriorfield in this cylindrically symmetric spacetime can be defined as ds = − (cid:18) − M ( u ) R + Q ( u ) R (cid:19) du − dRdu + R ( dθ + dz ) . (29)The intrinsic metric for the hypersurface Σ which enables a description in comoving coordinates of the interiorspacetime, is given by [10] ( ds ) Σ = − dτ + R ( τ )( dθ + dz ) , (30)where () Σ means the value of () on Σ and ξ i ≡ ( τ, θ, z ) represents the coordinates on Σ, i.e.( ds ) Σ = g ij dξ i dξ j . To match the interior and the exterior space-time, we follow the prescription of Darmois and Israel [51] whichdemands: • The first fundamental form must be continuous over the hypersurface Σ i.e., the continuity of the metrics as V ± approaches Σ: ( ds ) Σ = ( ds − ) Σ = ( ds ) Σ . (31) • The continuity of the second fundamental form. This gives the continuity of the extrinsic curvature K ij at thehypersurface Σ: [ K ij ] = K + ij − K − ij = 0 . (32)According to Eisenhart [52], the extrinsic curvature of Σ is given by K ± ij = − n ± σ (cid:18) ∂ χ σ ± ∂ξ i ∂ξ j + Γ σµν ∂χ µ ± ∂χ ν ± ∂ξ i ∂ξ j (cid:19) , ( σ, µ, ν = 0 , , , . (33)where n ± σ are the outward unit normal vectors to the hypersurface Σ, χ ± µ are the coordinates of V ± .On the r = constant hypersurface, we have dr = 0. Using this condition in (1) and comparing with (30) keeping inmind the junction condition (31), we get, dtdτ = A ( t, r Σ ) − , (34) R Σ ( τ ) = C ( t, r Σ ) . We may also write the exterior metric (29) as,( ds ) Σ = − (cid:20)(cid:18) − M ( u ) R Σ + Q ( u ) R (cid:19) + 2 dR Σ du (cid:21) du + R ( dθ + dz ) . (35)Now, using the junction condition (31) and matching with the metric on the hypersurface Σ, we get dudτ = (cid:20) − M ( u ) R Σ + Q ( u ) R + 2 dR Σ du (cid:21) − / . (36)To apply the junction conditions, we require that Σ has the same parametrisation whether it is considered as embeddedin V + or in V − . In the coordinates of the interior spacetime V − , the bounding surface Σ will have the equation f ( t, r ) = r − r Σ = 0 , (37)where r Σ is a constant.Since the vector ∂f /∂χ α − is orthogonal to Σ, so the unit normal vector to Σ in the χ α − coordinate system is, n − α = [0 , B ( t, r Σ ) , , . (38)In the coordinate system of V + , the equation for the surface Σ may be written as, f ( u, R ) = R − R Σ ( u ) = 0 . (39)The vector ∂f /∂χ α + , orthogonal to the hypersurface Σ is therefore given by, ∂f∂χ α + = (cid:18) − dR Σ du , , , (cid:19) . (40)So the unit normal to Σ in the V + coordinate system is, n + α = (cid:18) − M ( u ) R Σ + Q ( u ) R + 2 dR Σ du (cid:19) − / (cid:18) − dR Σ du , , , (cid:19) . (41)The extrinsic curvature for the hpersurface Σ in the V + coordinates as calculated by using (33) is given by K − = − (cid:18) A (cid:48) AB (cid:19) Σ , (42) K − = K − = (cid:18) CC (cid:48) B (cid:19) Σ , (43) K +00 = (cid:34) d udτ (cid:18) dudτ (cid:19) − − (cid:18) MR − Q R (cid:19) (cid:18) dudτ (cid:19)(cid:35) Σ . (44) K +22 = K +33 = (cid:20) R dRdτ + (cid:18) Q R − M (cid:19) dudτ (cid:21) Σ . (45)On account of the continuity of the second fundamental form given by (32), we obtain the following relations onmatching (42) to (44) and (43) to (45) (cid:34) d udτ (cid:18) dudτ (cid:19) − − (cid:18) MR − Q R (cid:19) (cid:18) dudτ (cid:19)(cid:35) Σ = − (cid:18) A (cid:48) AB (cid:19) Σ , (46) (cid:20) R dRdτ + (cid:18) Q R − M (cid:19) dudτ (cid:21) Σ = (cid:18) CC (cid:48) B (cid:19) Σ . (47) IV. RESULTS
We now use the relations obtained above and simplify them to find useful results. From (36) we have by rearranging, (cid:18) dudτ (cid:19) (cid:18) Q R Σ − M (cid:19) = R Σ (cid:18) dudτ (cid:19) − − R Σ (cid:18) dR Σ dτ (cid:19) . (48)Putting this value in (47) and using (34), we have (cid:18) dudτ (cid:19) − = (cid:32) ˙ CA + C (cid:48) B (cid:33) . (49)Again using (34) and squaring (49) we obtain the total energy entrapped inside the surface Σ as follows: M = C (cid:32) ˙ CA (cid:33) − (cid:18) C (cid:48) B (cid:19) + Q C . (50)Taking the interior and exterior charge to be the same on the hypersurface Σ (i.e. Q = s ) and using (50) and (14),we obtain E (cid:48) = l M, (51)which indicates that the difference between the two masses is equal to l/
8, as obtained in [46] and [47], which is aconsequence of the least unsatisfactory definition of C-energy due to Thorne [37].Using the expressions (8), (9) and (12) we can reconstruct (26) as follows:4 π ( q + (cid:15) ) = 1 B (cid:18)
13 (Θ − √ σ ) (cid:48) − √ σ C (cid:48) C (cid:19) . (52)Differentiating (49) with respect to τ and substituting in (46), we obtain the following expression with the help of(49) and (34), CA (cid:32) ¨ CC − ˙ C ˙ ACA (cid:33) + CAB (cid:32) ˙ C (cid:48) C − ˙ BC (cid:48) BC (cid:33) − AB (cid:32) ˙ CA (cid:48) A + C (cid:48) A (cid:48) B (cid:33) = 1 C (cid:18) Q C − M (cid:19) . (53)Using (50), (26) and (27) in (53) and rearranging terms, we arrive at the result q = (cid:18) P r − √ ησ (cid:19) − s πC , (54)on account of the fact that Q = s on the hypersurface Σ. This equation gives the relation between the heat flux,radial pressure, shear viscosity and the charge per unit length of the cylinder, over the hypersurface Σ. The resultshows that for an uncharged radiating fluid without any shear viscosity, the radial pressure equals the heat flux allover the boundary of the collapsing cylinder. Equations (52) and (54) are generalizations over the results obtainedearlier in [46] and [47].The total luminosity of the collapsing matter visible to an observer at rest at infinity is [53] L ∞ = − (cid:18) dMdu (cid:19) Σ = − (cid:32) dMdt dtdτ (cid:18) dudτ (cid:19) − (cid:33) Σ . (55)Differentiating (50) with respect to t and using (34), (49), (26) and (27), we obtain L ∞ = 4 πC (cid:32) ˙ CA (cid:18)(cid:18) P r + (cid:15) − √ ησ (cid:19) − s πC (cid:19) + C (cid:48) B ( q + (cid:15) ) (cid:33) (cid:32) ˙ CA + C (cid:48) B (cid:33) , (56)which, in view of (54) leads us to the expression L ∞ = 4 π C ( q + (cid:15) ) (cid:32) ˙ CA + C (cid:48) B (cid:33) Σ . (57)Thus the total luminosity of the collapsing matter as visible to a distant observer, depends on the energy flux associatedwith the collapse. For an observer on the boundary Σ, the luminosity is [54] L Σ = − (cid:34)(cid:18) dudτ (cid:19) dMdu (cid:35) Σ . (58)The boundary redshift of the radiation emitted by the collapsing matter can be written as Z Σ = (cid:114) L Σ L ∞ − dudτ − (cid:32) ˙ CA + C (cid:48) B (cid:33) − − (cid:32) ˙ CA + C (cid:48) B (cid:33) = 0the boundary redshift attains unlimited value (i.e., Z Σ → ∞ ). A. Dynamical Equations for the Collapse
The dynamical equations for non-adiabatic charged anisotropic fluid with shear viscosity undergoing cylindricalcollapse can be obtained from the Bianchi identities ( T αβ + T ( em ) αβ ) ; β = 0 for energy-momentum conservation.Using (2), (6), (7) and (15), we have (cid:16) T αβ + T ( em ) αβ (cid:17) ; β V α = − A ( ˙ µ + ˙ (cid:15) ) − ˙ BAB (cid:18) µ + P r + 2 (cid:15) − √ ησ (cid:19) − CAC (cid:18) µ + P ⊥ + (cid:15) + 2 √ ησ (cid:19) − q + (cid:15) ) B (cid:18) A (cid:48) A + C (cid:48) C (cid:19) − B ( q (cid:48) + (cid:15) (cid:48) ) = 0 (60)and (cid:16) T αβ + T ( em ) αβ (cid:17) ; β χ a = 1 B (cid:18) P r + (cid:15) − √ ησ (cid:19) (cid:48) + A (cid:48) AB (cid:18) µ + P r + 2 (cid:15) − √ ησ (cid:19) + 2 C (cid:48) BC (cid:16) P r − P ⊥ + (cid:15) − √ ησ (cid:17) + 1 A ( ˙ q + ˙ (cid:15) ) + 2( q + (cid:15) ) A (cid:32) ˙ BB + ˙ CC (cid:33) − ss (cid:48) πBC = 0 . (61)To discuss the dynamics of the collapsing system, it is customary to introduce the proper time derivative D T = 1 A ∂∂t , (62)and the proper radial derivative D R constructed from the circumference radius of a cylinder inside Σ D R = 1 R (cid:48) ∂∂r , (63)where R = C. (64)The fluid velocity for the corresponding collapse is given by U = D T ( R ) = ˙ CA , (65)which must be negative to ensure collapse to occur. Defining new variable ε = C (cid:48) B (note that (cid:15) and ε are differentquantities) and using (14), we have ε = (cid:20) U + s C − MC (cid:21) / . (66)Consequently, Eq. (52) can be re-written as follows:4 π ( q + (cid:15) ) = ε (cid:20) D R (Θ − √ σ ) − √ σR (cid:21) . (67)The time rate of variation of the total energy inside the collapsing cylinder is given by D T E (cid:48) = − πR (cid:20)(cid:18) P r + (cid:15) − √ ησ − πR (cid:19) U + ε ( q + (cid:15) ) (cid:21) + 3 s U R . (68)In the case of collapse, since ( U < U inside the square brackets, will increase the C-energy of thecylinder if P r + (cid:15) − √ ησ > πR , i.e. the effective radial pressure is greater than a certain value. The work doneby the effective radial pressure leads to the increase of C-energy. The second term in the square brackets, due to theoverall negative sign, describes the outflow of energy in the form of heat flux and radiation during the collapse. Sincethe collapsing cylinder contains the same species of the charges, the last term will decrease the energy of the systemas s R plays the role of Coulomb force of repulsion and U < D R E (cid:48) = 4 πR (cid:18) µ + (cid:15) + Uε ( q + (cid:15) ) (cid:19) + l sR D R s + 3 s R . (69)The first term on the right hand side gives the contribution of the energy density of the element of fluid inside acylindrical shell, along with heat flux and radiation. Since U <
0, the factor Uε ( q + (cid:15) ) decreases the energy of thesystem during the collapse of the cylinder. In the remaining terms, the constant l/ D T U = − R (cid:18) E (cid:48) − l (cid:19) − πR (cid:18) P r + (cid:15) − √ ησ (cid:19) + εA (cid:48) AB + 5 s R . (70)Substituting for A (cid:48) A from Eq.(70) into Eq.(61), we obtain the equivalent of Newton’s second law of motion for thecollapsing matter in the form (cid:18) µ + P r + 2 (cid:15) − √ ησ (cid:19) D T U = − (cid:18) µ + P r + 2 (cid:15) − √ ησ (cid:19) (cid:20) R (cid:18) E (cid:48) − l (cid:19) + 4 π (cid:18) P r + (cid:15) − √ ησ (cid:19) R − s R (cid:21) − ε (cid:34) D T ( q + (cid:15) ) + 4( q + (cid:15) ) UR + 2( q + (cid:15) ) 1 A (cid:32) ˙ BB − ˙ CC (cid:33)(cid:35) − ε (cid:20) D R (cid:18) P r + (cid:15) − √ ησ (cid:19) + 2 (cid:16) P r − P ⊥ + (cid:15) − √ ησ (cid:17) R − sπR D R s (cid:21) , (71)which can be simplified as follows: (cid:0) µ + P r eff + 2 (cid:15) (cid:1) D T U = − (cid:0) µ + P r eff + 2 (cid:15) (cid:1) (cid:20) R (cid:18) E (cid:48) − l (cid:19) + 4 π ( P r eff + (cid:15) ) R − s R (cid:21) − ε (cid:34) D T ( q + (cid:15) ) + 4( q + (cid:15) ) UR + 2( q + (cid:15) ) A (cid:32) ˙ BB − ˙ CC (cid:33)(cid:35) − ε (cid:20) D R ( P r eff + (cid:15) ) + 2 (cid:0) P r eff − P ⊥ eff + (cid:15) (cid:1) R − sπR D R s (cid:21) . Here we assume that in general A ( ˙ BB − ˙ CC ) (cid:54) = 0. The left hand side of (71) represents force. The factor ( µ + P r +2 (cid:15) − √ ησ ) represents the inertial mass density, which gives the effect of dissipation but there is no contribution ofthe electric charge, nor of heat flux. The remaining term on the left hand side is acceleration. Thus, we can say thatthe dynamical system will evolve radially outward or inward according as D T U < D T U >
0. The terms with anegative contribution in (71), favors the collapse while the other contribution prevents the collapse. If both of theseterms cancel each other, then a condition of hydrostatic equilibrium will be encountered.The first term on the right hand side represents the gravitational force. The factor within the first square bracketsshows the effects of specific length, effective radial pressure and the electric charge on the term ( µ + P r + 2 (cid:15) − √ ησ )representing the active gravitational mass. The second term represents the contribution due to radiation and heat flux,which will leave the system (if there is an overall negative sign) through the outward radially directed streamlines. Thusit is in the same direction of pressure and would prevent the collapse. The third term has three main contributions:the first is the effective pressure gradient which is always negative, thereby preventing the collapse. The second isthe local anisotropy of the fluid which will be negative for P r eff < P ⊥ eff , in which case it will decrease the rate ofcollapse. The third is the electromagnetic field term. The third term contributes negatively [20] if sR > D R s . Underthese conditions, the term in the third square brackets, with negative sign, contributes positively by reducing theattractive nature of the force appearing on the left hand side of this equation and hence this term will prevent thegravitational collapse. B. Solution of the Field Equations
In their work, Di Prisco et al. [45] have derived the solutions for the shearfree and isotropic case of cylindricalcollapse. Sharif and Abbas [55] have found analytical solutions for charged perfect fluid cylindrical gravitational1collapse. Keeping in mind the result obtained by Raychaudhuri and De [23] in the case of the evolution of irrotationalcharged dust, we derive the solutions in presence of shear for the evolution of charged fluids. The solution valid forthe entire duration of collapse in presence of dissipation, should be of the following form: A ( t, r ) = A ( r ) f ( t ) ,B ( t, r ) = B ( r ) f ( t ) , (72) C ( t, r ) = C ( r ) f ( t ) , where A ( r ), B ( r ) and C ( r ) are solutions of a static fluid having µ as the energy density and p r and p ⊥ as theradial and tangential pressure. Rescaling the coordinate time leads to A ( t, r ) = A ( r ). Then taking a cue from [17]for the spherically symmetric case with shear, we propose solutions of the field equations (25) to (28) in the form A ( t, r ) = A ( r ) ,B ( t, r ) = B ( r ) , (73) C ( t, r ) = A ( r ) f ( t ) . The expression (12) for the shear scalar becomes σ = − √ A ˙ ff . (74)The field equations (25) to (28) are reduced to8 π ( µ + (cid:15) ) = 8 πµ + ˙ f A f − s A f , (75)8 π ( q + (cid:15) ) = 0 , (76)8 π ( P r + (cid:15) ) = 8 πP r − A ff + (cid:32) ˙ ff (cid:33) − πη ˙ f A f + 4 s A f , (77)8 πP ⊥ = 8 πP ⊥ − ¨ fA f + 16 πη ˙ f A f − s A f (78)where 8 πµ = 1 B (cid:20) A (cid:48) B (cid:48) A B − (cid:18) A (cid:48)(cid:48) A + A (cid:48) A (cid:19)(cid:21) , (79)8 πP r = 3 A (cid:48) B A (80)and 8 πP ⊥ = 1 B (cid:18) A (cid:48)(cid:48) A − A (cid:48) B (cid:48) A B + A (cid:48) A (cid:19) . (81)Equations (75) to (78) represent the static anisotropic fluid configuration in the limit f ( t ) →
1. Substituting (74),(76) and (77) into (54) and assuming that P r ( r Σ ) = 0, we obtain the following differential equation:2 f ¨ f + ˙ f − af = 0 , (82)where a depends on the charge enclosed inside the cylinder and the static fluid conditions, i.e. a = s A . (83)Equation (82) can be solved using maple program assuming that the system represents the static configuration at t → −∞ , when ˙ f ( t ) → f ( t ) →
1. In view of earlier works [23, 25], we assume the charge enclosed inside thecylinder to be positive for the collapse in presence of shear. We find that the solutions represent the configurationof the collapsing matter when the luminosity of the collapsing matter as visible to a distant observer, vanishes andtherefore represents the later stages of the collapse. Sample plots are shown in the following figures. Analysing (82),we find that ˙ f ( t ) becomes infinite as t →
0, which is in agreement with the trend visible in the sample plots.2
FIG. 1: Diagram showing the plot of f ( t ) vs t for the charged case with a as squares of integers i.e a = 1(red), a = 4(black)and a = 9(blue) for different ranges of time. V. SUMMARY
Here we have formulated the general relativistic field equations for the case of dissipative cylindrical collapse inpresence of heat flow, free-streaming radiation, and shear viscosity and have obtained a few results. We have derivedthe relation between the expansion Θ, the shear σ and the energy flowing out of the cylinder in the form of heatflux q and free-streaming radiation. By employing the Darmois-Israel junction condition for the smooth matching ofinterior and exterior spacetimes at the boundary Σ, we have verified the relation between the specific energy of thecylinder due to the electromagnetic field and the mass of the collapsing matter. The total luminosity of the collapsingmatter as visible to a distant observer, depends on the energy flux associated with the collapse. This energy fluxover the hypersurface Σ bounding the cylinder, is dependent on the effective radial pressure and the charge per unitlength of the cylinder. The dynamical equations for the collapse is derived from the Bianchi identities with the helpof Misner-Sharp formalism for this non-adiabatic, anisotropic and dissipative fluid and the equation for the effectiveNewton’s second law of motion is constructed. Finally, we have derived the solution to the field equations for thegiven matter distribution at later stages of collapse. Under this condition the energy flux out of the boundary of thecollapsing matter vanishes and the luminosity for distant observer also vanishes. The collapse is bounded by the eventhorizon.As future work we are considering the solutions which represent the viscous collapsing matter from the onset ofcollapse to the final state of singularity and the status of the corresponding energy conditions. Acknowledgments
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