Effects of Electromagnetic Field on Gravitational Collapse
aa r X i v : . [ g r- q c ] M a y Effects of Electromagnetic Field onGravitational Collapse
M. Sharif ∗ and G. Abbas † Department of Mathematics, University of the Punjab,Quaid-e-Azam Campus, Lahore-54590, Pakistan.
Abstract
In this paper, the effect of electromagnetic field has been investi-gated on the spherically symmetric collapse with the perfect fluid inthe presence of positive cosmological constant. Junction conditionsbetween the static exterior and non-static interior spherically sym-metric spacetimes are discussed. We study the apparent horizons andtheir physical significance. It is found that electromagnetic field re-duces the bound of cosmological constant by reducing the pressureand hence collapsing process is faster as compared to the perfect fluidcase. This work gives the generalization of the perfect fluid case to thecharged perfect fluid. Results for the perfect fluid case are recovered.
Keywords : Electromagnetic Field, Gravitational Collapse, Cosmologi-cal Constant.
PACS numbers:
General Relativity (GR) predicts that gravitational collapse of massive ob-jects (having mass = 10 M ⊙ − M ⊙ , where M ⊙ is mass of the Sun) results ∗ [email protected] † [email protected]
1o the formation of spacetime singularities in our universe [1]. The singularitytheorems [2] of Hawking and Penrose reveal that if a trapped surface formsduring the collapse of compact object, such a collapse will develop a space-time singularity. According to these theorems, the occurrence of spacetimesingularity (can be observed or not) is the generic property of the spacetimein GR. An observable singularity is called naked singularity while other iscalled black hole or covered singularity .An open and un-resolved problem in GR is to determine the final fate ofthe gravitational collapse (i.e., end product of collapse is either covered ornaked singularity). To resolve this problem, Penrose [3] suggested a hypoth-esis so-called Cosmic Censorship Conjecture (CCC). This conjecture statesthat the singularities that appear in the gravitational collapse are alwayscovered by the event horizon. It has two versions, i.e., weak and strongversion [4]. The weak version states that the gravitational collapse fromthe regular initial conditions never creates spacetime singularity visible todistant observer. On the other hand, the strong version says that no singu-larity is visible to any observer at all, even some one close to it. There is nomathematical or theoretical proof for either of the version of the CCC.The singularity (at the end stage of the gravitational collapse) can beblack hole or naked depending upon the initial data and equation of thestate. To prove or disprove this hypothesis, many efforts have been madebut no final conclusion is drawn. It would be easier to find the counterexample that would enable us to claim that the hypothesis is not correct.For this purpose, Virbhadra et al. [5] introduced a new theoretical tool usingthe gravitational lensing phenomena. Also, Virbhadra and Ellis [6] studiedthe Schwarzschild black hole lensing and found that the relativistic imageswould confirm the Schwarzschild geometry close to the event horizon. Thesame authors [7] analyzed the gravitational lensing by a naked singularityand classified it into two kinds: weak naked singularity (those containedwithin at least one photon sphere) and strong naked singularity (those notcontained within any photon sphere).Claudel et al. [8] showed that spherically symmetric black holes, withreasonable energy conditions, are always covered inside at least one photonsphere. Virbhadra and Keeton [9] studied the time delay and magnificationcentroid due to gravitational lensing by black hole and naked singularity. Itwas found that weak CCC can be tested observationally without any ambi-guity. Virbhadra [10] explored the useful results to investigate the Seifert’sconjecture for naked singularity. He found that naked singularity forming in2he Vaidya null dust collapse supports the Seifert’s conjecture. In a recentpaper [11], the same author used the gravitational lensing phenomena to findthe improved form of the CCC. This work is a source of inspiration for manyleading researchers.Oppenheimer and Snyder [12] are the pioneers who investigated gravi-tational collapse long time ago in 1939. They studied the dust collapse bytaking the static Schwarzschild spacetime as exterior and Friedmann like so-lution as interior spacetime. They found black hole as end product of thegravitational collapse. To study the gravitational collapse, exact solutions ofthe Einstein field equations with dust provide non-trivial examples of nakedsingularity formation. Since the effects of pressure cannot be neglected inthe singularity formation, therefore dust is not assumed to be a good matter.There has been a growing interest to study the gravitational collapse inthe presence of perfect fluid and other general physical form of the fluid. Mis-ner and Sharp [13] extended the pioneer work for the perfect fluid. Vaidya[14] and Santos [15] used the idea of outgoing radiation of the collapsing bodyand also included the dissipation in the source by allowing the radial heatflow. The cosmological constant Λ affects the properties of spacetime as itappears in the field equations. It is worthwhile to solve the field equationswith non-zero cosmological constant for analyzing the gravitational collapse.Markovic and Shapiro [16] generalized the pioneer work with positive cos-mological constant. Lake [17] extended it for both positive and negativecosmological constant.Sharif and Ahmad [18]-[21] extended the spherically symmetric gravita-tional collapse with positive cosmological constant for perfect fluid. Theydiscussed the junction conditions, apparent horizons and their physical sig-nificance. It is concluded that apparent horizon forms earlier than singular-ity and positive cosmological constant slows down the collapse. The sameauthors also investigated the plane symmetric gravitational collapse usingjunction conditions [22]. In a recent paper [23], Sharif and Iqbal extendedplane symmetric gravitational collapse to spherically symmetric case.Although a lot of work has been done for dust and perfect fluid collapseof spherically symmetric models. However, no such attempt has been madeby including the electromagnetic field. We would like to study the gravita-tional collapse of charged perfect fluid in the presence of positive cosmologicalconstant. For this purpose, we discuss the junction conditions between thenon-static interior and static exterior spherically symmetric spacetimes. Themain objectives of this work are the following:3
To study the effects of electromagnetic field on the rate of collapse. • To see whether or not CCC is valid in this framework.The plan of the paper is as follows: In the next section, the junctionconditions are given. We discuss the solution of the Einstein-Maxwell fieldequations in section . The apparent horizons and their physical significanceare presented in section . We conclude our discussion in the last section.We use the geometrized units (i.e., the gravitational constant G =1 andspeed of light in vacuum c =1 so that M ≡ MGc and κ ≡ πGc = 8 π ). All theLatin and Greek indices vary from 0 to 3, otherwise it will be mentioned. We consider a timelike 3 D hypersurface Σ which separates two 4 D manifolds M − and M + respectively. For the interior manifold M − , we take sphericallysymmetric spacetime given by ds − = dt − X dr − Y ( dθ + sin θ dφ ) , (2.1)where X and Y are functions of t and r . For the exterior manifold M + , wetake Reissner-Nordstr¨ o m de-Sitter spacetime ds = N dT − N dR − R ( dθ + sin θ dφ ) , (2.2)where N ( R ) = 1 − MR + Q R − Λ3 R , (2.3) M and Λ are constants and Q is the charge. The Israel junction conditionsare the following [24]:1. The continuity of line element over Σ gives( ds − ) Σ = ( ds ) Σ = ds . (2.4)2. The continuity of extrinsic curvature over Σ yields[ K ij ] = K + ij − K − ij = 0 , ( i, j = 0 , ,
3) (2.5)where K ij is the extrinsic curvature defined as4 ± ij = − n ± σ ( ∂ x σ ± ∂ξ i ∂ξ j + Γ σµν ∂x µ ± ∂x ν ± ∂ξ i ∂ξ j ) , ( σ, µ, ν = 0 , , , . (2.6)Here ξ i correspond to the coordinates on Σ, x σ ± stand for coordinates in M ± , the Christoffel symbols Γ σµν are calculated from the interior or exteriorspacetimes and n ± σ are components of outward unit normals to Σ in thecoordinates x σ ± .The equation of hypersurface in terms of interior spacetime M − coordi-nates is f − ( r, t ) = r − r Σ = 0 , (2.7)where r Σ is a constant. Also, the equation of hypersurface in terms of exteriorspacetime M + coordinates is given by f + ( R, T ) = R − R Σ ( T ) = 0 . (2.8)When we make use of Eq.(2.7) in Eq.(2.1), the metric on Σ takes the form( ds − ) Σ = dt − Y ( r Σ , t )( dθ + sin θ dφ ) . (2.9)Also, Eqs.(2.8) and (2.2) yield( ds ) Σ = [ N ( R Σ ) − N ( R Σ ) ( dR Σ dT ) ] dT − R ( dθ + sin θ dφ ) , (2.10)where we assume that N ( R Σ ) − N ( R Σ ) ( dR Σ dT ) > R Σ = Y ( r Σ , t ) , (2.12)[ N ( R Σ ) − N ( R Σ ) ( dR Σ dT ) ] dT = dt. (2.13)Also, from Eqs.(2.7) and (2.8), the outward unit normals in M − and M + ,respectively, are given by n − µ = (0 , X ( r Σ , t ) , , , (2.14) n + µ = ( − ˙ R Σ , ˙ T , , . (2.15)5he components of extrinsic curvature K ± ij become K − = 0 , (2.16) K − = csc θK − = ( Y Y ′ X ) Σ , (2.17) K +00 = ( ˙ R ¨ T − ˙ T ¨ R − N dNdR ˙ T + 32 N dNdR ˙ T ˙ R ) Σ , (2.18) K +22 = csc θK +33 = ( N R ˙ T ) Σ , (2.19)where dot and prime mean differentiation with respect to t and r respectively.From Eq.(2.5), the continuity of extrinsic curvature gives K +00 = 0 , (2.20) K +22 = K − . (2.21)Using Eqs.(2.16)-(2.21) along with Eqs.(2.3), (2.12) and (2.13), the junctionconditions become ( X ˙ Y ′ − ˙ XY ′ ) Σ = 0 , (2.22) M = ( Y − Λ6 Y + Q Y + Y Y − Y X Y ′ ) Σ . (2.23) The Einstein field equations with cosmological constant are given by G µν − Λ g µν = κ ( T µν + T ( em ) µν ) . (3.1)The energy-momentum tensor for perfect fluid is T µν = ( ρ + p ) u µ u ν − pg µν , (3.2)where ρ is the energy density, p is the pressure and u µ = δ µ is the four-vectorvelocity in co-moving coordinates. T ( em ) µν is the energy-momentum tensor forthe electromagnetic field defined [25] as T ( em ) µν = 14 π ( − g δω F µδ F νω + 14 g µν F δω F δω ) . (3.3)6ith the help of Eqs.(3.2) and (3.3), Eq.(3.1) takes the form R µν = 8 π [( ρ + p ) u µ u ν + 12 ( p − ρ ) g µν + T ( em ) µν − g µν T ( em ) ] − Λ g µν . (3.4)To solve this equation, we need to calculate the non-zero components andtrace free form of T ( em ) µν . For this purpose, we first solve the Maxwell’s fieldequations F µν = φ ν,µ − φ µ,ν , (3.5) F µν ; ν = − πJ µ , (3.6)where φ µ is the four potential and J µ is the four current. As the chargedfluid is in co-moving coordinate system, the magnetic field will be zero inthis case. Thus we can choose the four potential and four current as follows φ µ = ( φ ( t, r ) , , , , (3.7) J µ = σu µ , (3.8)where σ is charge density.Now for the solution of the Maxwell’s field Eq.(3.6), µ and ν are treatedas local coordinates. Using Eqs.(3.5) and (3.7), the non-zero components ofthe field tensor are given as follows: F = − F = − ∂φ∂r . (3.9)Also, from Eqs.(3.6) and (3.8), we have1 X ∂ φ∂r − ∂φ∂r X ′ X = − πσX, (3.10)1 X ∂ φ∂r∂t − ˙ XX ∂φ∂r = 0 . (3.11)Equation (3.11) implies that ( 1 X ∂φ∂r ) = K, (3.12)where K = K ( r ) is an arbitrary function of r . Equations (3.10) and (3.12)yield K ′ ( r ) = − πσX. (3.13)7he non-zero components of T ( em ) µν and its trace free form turn out to be T ( em )00 = 18 π K , T ( em )11 = − π K X , T ( em )22 = 18 π K Y ,T ( em )33 = T ( em )22 sin θ, T ( em ) = 0 . When we use these values, the field equations (3.4) for the interior space-time takes the form R = − ¨ XX − YY = 4 π ( ρ + 3 p ) + K − Λ , (3.14) R = − ¨ XX − XX ˙ YY + 2 X [ Y ′′ Y − Y ′ X ′ XY ]= 4 π ( p − ρ ) + K − Λ , (3.15) R = − ¨ YY − ( ˙ YY ) − ˙ XX ˙ YY + 2 X [ Y ′′ Y + ( Y ′ Y ) − X ′ X Y ′ Y − ( XY ) ]= 4 π ( p − ρ ) − K − Λ , (3.16) R = sin θR , (3.17) R = − Y ′ Y + 2 ˙ XX Y ′ Y = 0 . (3.18)Now we solve Eqs.(3.14)-(3.18). Integration of Eq.(3.18) with respect to t yields X = Y ′ H , (3.19)where H = H ( r ) is an arbitrary function of r . The energy conservationequation T νµ ; ν = 0 , (3.20)for the perfect fluid with the interior metric shows that pressure is a functionof t only, i.e., p = p ( t ) . (3.21)Substituting the values of X and p from Eqs.(3.19) and (3.21) in Eqs.(3.14)-(3.18), it follows that2 ¨ YY + ( ˙ YY ) + (1 − H ) Y = Λ + K − p ( t ) . (3.22)8e consider p as a polynomial in t as given by [19] p ( t ) = p c t − s , (3.23)where p c and s are positive constants. Further, for simplicity, we take s = 0so that p ( t ) = p c . (3.24)Replacing this value in Eq.(3.22), we get2 ¨ YY + ( ˙ YY ) + (1 − H ) Y = Λ + K − πp c . (3.25)Integrating this equation with respect to t , it follows that˙ Y = H − K − πp c ) Y mY , (3.26)where m = m ( r ) is an arbitrary function of r and is related to the mass ofthe collapsing system. Substituting Eqs.(3.19), (3.26) into Eq.(3.14), we get m ′ = 2 K ′ K Y + Y ′ Y [4 π ( p c + ρ ) + 2 K ] . (3.27)For physical reasons, we assume that pressure and density are strictlypositive. Integrating Eq.(3.27) with respect to r , we obtain m ( r ) = 4 π Z r ( ρ + p c ) Y ′ Y dr + 2 Z r K Y ′ Y dr + 23 Z r K ′ KY dr. (3.28)The function m ( r ) must be positive because m ( r ) < M = Q Y + m + 16 (Λ + K − πp c ) Y . (3.29)The total energy ˜ M ( r, t ) up to a radius r at time t inside the hypersurfaceΣ can be evaluated by using the definition of mass function [13] given by˜ M ( r, t ) = 12 Y (1 + g µν Y ,µ Y ,ν ) . (3.30)9or the interior metric, it takes the form˜ M ( r, t ) = 12 Y (1 + ˙ Y − ( Y ′ X ) ) . (3.31)Replacing Eqs.(3.19) and (3.26) in Eq.(3.31), we obtain˜ M ( r, t ) = m ( r ) + (Λ + K − πp c ) Y . (3.32)Now we take (Λ + K − πp c ) > H ( r ) = 1 . (3.33)In order to obtain the analytic solutions in closed form, we use Eqs.(3.19),(3.26) and (3.33) so that Y = ( 6 m Λ + K − πp c ) sinh α ( r, t ) , (3.34) X = ( 6 m Λ + K − πp c ) [ { m ′ m − KK ′ K − πp c ) } sinh α ( r, t )+ { t s ( r ) − t ) KK ′ p K − πp c ) + t ′ s ( r ) r Λ + K − πp c }× cosh α ( r, t ) ] sinh − α ( r, t ) , (3.35)where α ( r, t ) = p K − πp c )2 [ t s ( r ) − t )] . (3.36)Here t s ( r ) is an arbitrary function of r and is related to the time of formationof singularity of a particular shell at coordinate distance r .In the limit (8 πp c − K ) → Λ, the above solution corresponds to theTolman-Bondi solution [26]lim (8 πp c − K ) −→ Λ X ( r, t ) = m ′ ( t s − t ) + 2 mt ′ s [6 m ( t s − t )] , (3.37)lim (8 πp c − K ) −→ Λ Y ( r, t ) = [ 9 m t s − t ) ] . (3.38)10 Apparent Horizons
Here we discuss the apparent horizons for the interior spacetime. The bound-ary of two trapped spheres whose outward normals are null is used to findthe apparent horizons. This is given as follows: g µν Y ,µ Y ,ν = ˙ Y − ( Y ′ X ) = 0 . (4.1)Replacing Eqs.(3.19) and (3.26) in this equation, we get(Λ + K − πp c ) Y − Y + 6 m = 0 . (4.2)When we take Λ = 8 πp c − K , it gives Y = 2 m . This is called Schwarzschildhorizon. For m = p c = K = 0, we have Y = q , which is called de-Sitterhorizon. Equation (4.2) can have the following positive roots. Case (i) : For 3 m < √ (Λ+ K − πp c ) , we obtain two horizons Y = 2 p (Λ + K − πp c ) cos ϕ , (4.3) Y = − p (Λ + 8 πK − p c ) (cos ϕ − √ ϕ , (4.4)where cos ϕ = − m p (Λ + K − πp c ) . (4.5)If we take m = 0, it follows from Eqs.(4.3) and (4.4) that Y = q K − πp c ) and Y = 0. Y and Y are called cosmological horizon and black hole horizonrespectively. For m = 0 and Λ = 8 πp c − K , Y and Y can be generalized[27] respectively. Case (ii):
For 3 m = √ (Λ+ K − πp c ) , there is only one positive root whichcorresponds to a single horizon i.e., Y = Y = 1 p (Λ + K − πp c ) = Y. (4.6)11his shows that both horizons coincide. The range for the cosmological andblack hole horizon can be written as follows0 ≤ Y ≤ p (Λ + K − πp c ) ≤ Y ≤ s K − πp c ) . (4.7)The black hole horizon has its largest proper area 4 πY = π (Λ+ K − πp c ) andcosmological horizon has its area between π (Λ+ K − πp c ) and π (Λ+ K − πp c ) . Case (iii):
For 3 m > √ (Λ+ K − πp c ) , there are no positive roots and con-sequently there are no apparent horizons.We now calculate the time of formation for the apparent horizon usingEqs.(3.33), (3.34) and (4.2) t n = t s − p K − πp c ) sinh − ( Y n m − , ( n = 1 , . (4.8)When 8 πp c − K −→ Λ, this corresponds to Tolman-Bondi [26] t ah = t s − m. (4.9)From Eq.(4.8), we can write Y n m = cosh α n , (4.10)where α n ( r, t ) = √ K − πp c )2 [ t s ( r ) − t n )]. Equations (4.7) and (4.8) implythat Y ≥ Y and t ≥ t respectively. The inequality t ≥ t indicates thatthe cosmological horizon forms earlier than the black hole horizon.The time difference between the formation of cosmological horizon andsingularity and the formation of black hole horizon and singularity respec-tively can be found as follows. Using Eqs.(4.3)-(4.5), it follows that d ( Y m ) dm = 1 m ( − sin ϕ sin ϕ + 3 cos ϕ cos ϕ ) < , (4.11) d ( Y m ) dm = 1 m ( − sin ( ϕ +4 π )3 sin ϕ + 3 cos ( ϕ +4 π )3 cos ϕ ) > . (4.12)12he time difference between the formation of singularity and apparent hori-zons is T n = t s − t n . (4.13)It follows from Eq.(4.10) that dT n d ( Y n m ) = 1sinh α n cosh α n p K − πp c ) . (4.14)Using Eqs.(4.11) and (4.14), we get dT dm = dT d ( Y m ) d ( Y m ) dm = 1 m p K − πp c ) sinh α cosh α × ( − sin ϕ sin ϕ + 3 cos ϕ cos ϕ ) < . (4.15)It shows that T is a decreasing function of mass m . This means that timeinterval between the formation of cosmological horizon and singularity isdecreased with the increase of mass. Similarly, from Eqs.(4.12) and(4.14),we get dT dm = 1 m p K − πp c ) sinh α cosh α × ( − sin ( ϕ +4 π )3 sin ϕ + 3 cos ( ϕ +4 π )3 cos ϕ ) > . (4.16)This indicates that T is an increasing function of mass m indicating thattime difference between the formation of black hole horizon and singularityis increased with the increase of mass. This paper is devoted to study the effects of electromagnetic field on gravi-tational collapse with the positive cosmological constant. The cosmologicalconstant acts as Newtonian potential. The relation for the Newtonian poten-tial is φ = (1 − g ). Using Eqs.(2.12) and (3.29), for the exterior spacetime,the Newtonian potential turns out to be φ ( R ) = mR + (Λ + K − πp c ) R . (5.1)13he corresponding Newtonian force is F = − mR + (Λ + K − πp c ) R . (5.2)Now we discuss the consequence of the Newtonian force. This force is zerofor the fixed values of m = √ (Λ+ K − πp c ) and R = √ (Λ+ K − πp c ) and will bepositive (repulsive) if the values of m and R are taken larger than these values.If we take m = √ (Λ+ K − πp c ) and R = √ (Λ+ K − πp c ) , then F = K − πp c )9 which gives positive value if (Λ + K − πp c ) >
0, i.e., Λ > (8 πp c − K ) suchthat 8 πp c > K . Thus we conclude that the repulsive force can be generatedfrom Λ if Λ > (8 πp c − K ) such that 8 πp c > K over the entire range ofthe collapsing sphere. For the perfect fluid and dust cases, Λ can play therole of the repulsive force for Λ > πp c and Λ > K = K ( r ) gives the electromagnetic field contribution. From Eq.(3.26), therate of collapse turns out be¨ Y = − mY + (Λ + K − πp c ) Y . (5.3)This shows that we have re-formulated the Newtonian model which repre-sents the acceleration of the collapsing process. The analysis of positive andnegative acceleration would give the same results as for the Newtonian force.It is worthwhile to mention that the electromagnetic field reduces thebound of the positive cosmological constant by reducing the pressure. Thusthe positive cosmological constant is bounded below as compared to theperfect fluid case. This would decrease the repulsive force which slows downthe collapsing process. Making the analysis of the smaller values of m and R than the values used for the repulsive force, we find that the attractiveforce is larger than the perfect fluid case. Since the attractive force favorsthe collapse while the repulsive force resists against the collapse, thus thecollapsing process is faster as compared to perfect fluid case when we includethe electromagnetic field.Further, we have found two apparent horizons (cosmological and blackhole horizons) whose area decreases in the presence of electromagnetic field.It is found that the cosmological horizon forms earlier than the black holehorizon. Also, Eq.(4.8) shows that apparent horizon forms earlier than sin-gularity. In this sense, we can conclude that the end state of gravitationalcollapse is a singularity covered by the apparent horizons (i.e., black hole).14t is interesting to mention here that our study supports the CCC andwould be considered as one of its counter example. Also, it would be possiblethat the electromagnetic field reduces the range of apparent horizons to ex-treme limits and singularity would be locally naked. Thus the weak versionof the CCC seems to be valid in this case. Acknowledgment
We would like to thank the Higher Education Commission, Islamabad,Pakistan for its financial support through the
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