Generalised isothermal models with strange equation of state
aa r X i v : . [ g r- q c ] F e b Generalised isothermal models with strange equation of state
S. D. Maharaj and S. Thirukkanesh † Astrophysics and Cosmology Research Unit,School of Mathematical Sciences,University of KwaZulu-Natal,Private Bag X54001,Durban 4000,South Africa.
Abstract
We consider the linear equation of state for matter distributions that may be applied tostrange stars with quark matter. In our general approach the compact relativistic body al-lows for anisotropic pressures in the presence of the electromagnetic field. New exact solutionsare found to the Einstein-Maxwell system. A particular case is shown to be regular at thestellar centre. In the isotropic limit we regain the general relativistic isothermal universe. Weshow that the mass corresponds to values obtained previously for quark stars when anisotropyand charge are present.
Keywords:
Exact solutions; Einstein-Maxwell spacetimes; relativistic stars.
PACS Nos: † Permanent address: Department of Mathematics, Eastern University, Sri Lanka, Chenkalady,Sri Lanka. 1
Introduction
In an early and seminal treatment the existence of quark matter in a stellar configurationin hydrostatic equilibrium was suggested by Itoh . Subsequently the analysis of strangestars consisting of quark matter has been considered in a number of investigations. Strangestars are likely to form in the period of collapse of the core regions of a massive star after asupernova explosion which was pointed out by Cheng et al . The core of a neutron star orproto-neutron star is a suitable environment for conventional barotropic matter to convertinto strange quark matter. Regions of low temperatures and sufficiently high temperatures arerequired for a first or second order phase transition which results in deconfined quark matter.Another possibility suggested by Cheng and Dai to explain the formation of a strange staris the accretion of sufficient mass in a rapidly spinning dense star in X-ray binaries whichundergoes a phase transition. The behaviour of matter at ultrahigh densities for quark matteris not well understood: in an attempt to study the physics researchers normally restrict theirattention to the MIT bag model (see the treatments of Chodos et al , Farhi and Jaffe andWitten ). The strange matter equation of state is taken to be p = 13 ( ρ − B ) (1)where ρ is the energy density, p is the pressure and B is the bag constant. The vacuumpressure B is the bag model equilibrates the pressure and stabilises the system; the constant B determines the quark confinement. The studies of Bombaci , Li et al , Dey et al ,Xu et al , Pons et al and Usov directed at particular compact astronomical objectssuggest that these could be strange stars composed of quark matter with equation of state(1).Mak and Harko found an exact general relativistic model of a quark star that admitsa conformal Killing vector. This was shown by Komathiraj and Maharaj to be a part of amore general class of exact analytical models in the presence of the electromagnetic field withisotropic pressures. The role of anisotropy was investigated by Lobo , Mak and Harko and Sharma and Maharaj for strange stars with quark matter with neutral anisotropicdistributions. It is our intention to study the Einstein-Maxwell system with a linear equationof state with anisotropic pressures; this treatment would be applicable to a strange stars whichare charged and anisotropic which is the most general case. In §
2, we write the Einstein-Maxwell system in an equivalent form using a coordinate transformation. A new exactsolution, in terms of simple elementary functions, is given in §
3. In addition, we demonstratethat it is possible to find a particular model which is nonsingular at the stellar origin. Thelimit of vanishing anisotropy is studied in § §
5, we consider the physical features of the new solutions, plot the mattervariables for particular parameter values and show that the quark star mass is consistentwith earlier treatments. Some concluding remarks are made in § Basic equations
It is our intention to model the interior of a dense realistic star with a general matter distri-bution. On physical grounds we can take the gravitational field to be static and sphericallysymmetric. Consequently, we assume that the gravitational field of the stellar interior isrepresented by the line element ds = − e ν ( r ) dt + e λ ( r ) dr + r ( dθ + sin θdφ ) (2)in Schwarzschild coordinates ( x a ) = ( t, r, θ, φ ) . We consider the general case of a matterdistribution with both anisotropy and charge. Therefore we take the energy momentumtensor for the interior to be an anisotropic charged imperfect fluid; this is represented by theform T ij = diag (cid:18) − ρ − E , p r − E , p t + 12 E , p t + 12 E (cid:19) , (3)where ρ is the energy density, p r is the radial pressure, p t is the tangential pressure and E isthe electric field intensity. These physical quantities are measured relative to the comovingfluid velocity u i = e − ν δ i . The line element (2) and the imperfect matter distribution (3)generate the Einstein field equations; the field equations can be written in the form1 r h r (1 − e − λ ) i ′ = ρ + 12 E , (4) − r (cid:16) − e − λ (cid:17) + 2 ν ′ r e − λ = p r − E , (5) e − λ (cid:18) ν ′′ + ν ′ + ν ′ r − ν ′ λ ′ − λ ′ r (cid:19) = p t + 12 E , (6) σ = 1 r e − λ ( r E ) ′ , (7)where primes denote differentiation with respect to r and σ is the proper charge density.We are utilising units where the coupling constant πGc = 1 and the speed of light c = 1.The Einstein-Maxwell system of equations (4)-(7) describes the gravitational behaviour foran anisotropic charged imperfect fluid. For matter distributions with p r = p t (isotropicpressures) and E = 0 (no charge) we regain Einstein’s equations for an uncharged perfectfluid from (4)-(7).An equivalent form of the field equations is obtained if we introduce new variables: theindependent variable x and new functions y and Z . These are given by x = Cr , Z ( x ) = e − λ ( r ) and A y ( x ) = e ν ( r ) , (8)which was earlier used by Durgapal and Bannerji to describe neutron stars. In terms ofthe new variables the line element (2) becomes ds = − A y dt + 14 CxZ dx + xC ( dθ + sin θdφ ) , (9)3here A and C are arbitrary constants. The transformation (8) simplifies the field equations,and we find that the system (4)-(7) can be written as1 − Zx − Z = ρC + E C , (10)4 Z ˙ yy + Z − x = p r C − E C , (11)4 xZ ¨ yy + (4 Z + 2 x ˙ Z ) ˙ yy + ˙ Z = p t C + E C , (12) σ C = 4 Zx (cid:16) x ˙ E + E (cid:17) , (13)where dots denote differentiation with respect to the variable x .The definition m ( r ) = 12 Z r ω ρ ( ω ) dω (14)represents the mass contained within a radius r which is a useful physical quantity. The massfunction (14) has the form m ( x ) = 14 C / Z x √ wρ ( w ) dw, (15)in terms of the new variables introduced in (8).On physical grounds we expect that the matter distribution for realitic stellar mattershould satisfy a barotropic equation of state p r = p r ( ρ ). For the investigations in this paperwe assume the particular equation of state p r = αρ − β, (16)where α and β are constants. This is a simple linear relationship with desirable physicalfeatures and contains models investigated previously. Now it is possible to rewrite (10)-(13)as the system ρC = 1 − Zx − Z − E C , (17) p r = αρ − β, (18) p t = p r + ∆ , (19)∆ = 4 CxZ ¨ yy + 2 C (cid:20) x ˙ Z + 4 Z (1 + α ) (cid:21) ˙ yy + (1 + 5 α )(1 + α ) C ˙ Z − C (1 − Z ) x + 2 β (1 + α ) , (20) E C = 1 − Zx − α ) (cid:20) α ˙ Z + 4 Z ˙ yy + βC (cid:21) , (21) σ C = 4 Zx ( x ˙ E + E ) , (22)where the quantity ∆ = p t − p r is defined as the measure of anisotropy. The Einstein-Maxwellequations as expressed in (17)-(22) is a system of six nonlinear equations in terms of eightvariables ( ρ, p r , p t , ∆ , E, σ, y, Z ). The system (17)-(22) is under-determined so that there aredifferent ways in which we can proceed with the integration process. Here we show that it4s possible to specify two of the quantities and generate an ordinary differential equation inonly one dependent variable in the integration process. This helps to produce a particularexact model. In this paper, we choose physically reasonable forms for the gravitational potential Z andelectric field intensity E and then integrate the system (17)-(22) to generate exact models.We make the specific choices Z = 1 a + bx n , (23) E C = 2 k ( d + 2 x ) a + bx n , (24)where a, b, d, n and k are real constants. The potential Z is regular at the origin and con-tinuous in the stellar interior for a wide range of values for the parameters a, b and n . Theelectric field intensity E is a bounded and decreasing function from the origin to the surface ofthe sphere. Therefore the forms chosen in (23)-(24) are physically acceptable. These specificchoices for Z and E simplify the integration process. Equation (21) can be written as˙ yy = ( a − α )4 x + α bnx n − ( a + bx n ) + (1 + α ) b x n − − β C ( a + bx n ) − (1 + α ) k d + 2 x )(25)where we have used (23) and (24). This has the advantage of being a first order linearequation in the gravitational potential y .Equation (25) can be integrated in closed form to give y = Dx ( a − α )4 ( a + bx n ) α exp[ F ( x )] , (26)where we have defined F ( x ) = − βx C (cid:20) a + bx n n + 1 (cid:21) + (1 + α )4 (cid:20) bx n n − k ( dx + x ) (cid:21) and D is a constant of integration. Now from (23), (24) and (26) we can generate an exact5odel for the system (17)-(22) as follows e λ = a + bx n , (27) e ν = A D x ( a − α )2 ( a + bx n ) α exp[2 F ( x )] , (28) ρC = ( a −
1) + bx n x ( a + bx n ) + 2 bnx n − ( a + bx n ) − k ( d + 2 x )( a + bx n ) , (29) p r = αρ − β, (30) p t = p r + ∆ , (31)∆ =14 (cid:26) C (1 − a − bx n ) x ( a + bx n ) − bCn (1 + 5 α ) x n − (1 + α )( a + bx n ) + 8 β (1 + α ) − a − bx n ( n (1 + α ) − (1 + α ) x ( a + bx n ) × (cid:0) a ( βx − C (1 + α )) + bx n (cid:0) C (cid:2) (1 + α ) (1 − bx n + dkx + 2 kx ) − nα (cid:3) + bβx n +1 (cid:1) + a [ C (1 + α )(1 + dkx + 2 kx − bx n ) + 2 bβx n +1 ] (cid:1) + Cx ( a + bx n ) (cid:20) x (( a − α ) × (cid:18) bnαx n ( a + bx n ) + ( a (1 + α ) − − α ) (cid:19) + 4 bnαx n (2 a ( n −
1) + ( nα − bx n )( a + bx n ) (cid:19) − a (1 + α )( a − bx n ) + bx n (2 nα − (1 + α ))] Cx ( a + bx n ) (cid:0) C (1 + α )( k ( dx + 2 x ) − bx n ) (cid:1) + βx ( a + bx n )) + 4 (cid:18) (1 + α )( b ( n − x n − − k ) − bnβx n − C +[ C (1 + α )( k ( d + 2 x ) − bx n − ) + β ( a + bx n )] C (cid:19)(cid:21)(cid:27) (32) E C = 2 k ( d + 2 x ) a + bx n . (33)The equations (27)-(33) represent an exact solution to the Einstein-Maxwell system (17)-(22)for a charged imperfect fluid with the linear equation of state p r = αρ − β . Exact solutionswith the equation of state p r = αρ − β have been used to model compact objects such asstrange stars, as shown by Sharma and Maharaj , and dark energy stars which are stable, asdemonstrated by Lobo . The exact solution (27)-(33) may be regarded as a generalisationof an isothermal universe model as we indicate in § a = 1 and n = 1 then we generatethe line element ds = − A D (1 + bx ) α exp[2 F ( x )] dt + (1 + bx ) dr + r ( dθ + sin θdφ ) , where F ( x ) = − βx C (2 + bx ) + (1+ α )4 [ bx − k ( dx + x )], with energy density ρC = b (3 + bx )(1 + bx ) − k ( d + 2 x )(1 + bx ) . (34)When k = 0 then the mass function (15) has the form m ( x ) = bx / C / (1 + bx ) . (35)6he expression of the mass function given in (35) represents an energy density which ismonotonically decreasing in the interior of the uncharged sphere and has a finite value at thecentre x = 0. This is physically reasonable and similar mass profiles appear in the treatmentsof general relativistic equilibrium configurations of Matese and Whitman , neutron starmodels of Finch and Skea , anisotropic stellar solutions of Mak and Harko , and the darkenergy stars of Lobo . The gravitational potentials for charged imperfect fluid solutioncorresponding to the line element (34) are nonsingular at the origin. The energy density ρ , themeasure of anisotropy ∆, and associated quantities p r and p t are also regular at the centre forour choice of E . Note that the quantity ∆ vanishes at the centre and is a continuous functionin the stellar interior. Hence the model (27)-(33) admits a particular case corresponding tothe parameter values a = 1 and n = 1 which is regular and well behaved at the origin. It is possible to consider the special case of isotropic pressures with p r = p t in the unchargedlimit for neutral matter. When k = 0 ( E = 0) equation (32) becomes∆ =14 (cid:26) C (1 − a − bx n ) x ( a + bx n ) − bCn (1 + 5 α ) x n − (1 + α )( a + bx n ) + 8 β (1 + α ) − a − ( n (1 + α ) − bx n ](1 + α ) x ( a + bx n ) (cid:2) a × ( βx − C (1 + α )) − bx n ( C ((1 + α )( bx n −
1) +2 nα ) − bβx n +1 (cid:1) + a (cid:0) C (1 + α )(1 − bx )+2 bβx n +1 (cid:1)(cid:3) + Cx ( a + bx n ) [4 b ( n − α ) x n + ( a − α )( a (1 + α ) − − α )+ 4( a − bnα (1 + α ) x n ( a + bx n ) − bnβx n +1 C + 4 bnα [2 a ( n −
1) + b ( nα − x n ] x n ( a + bx n ) + [ aβx − b ( C (1 + α ) − βx ) x n ] C + 2 C ( a + bx n ) ( a (1 + α )( a − bx n )+ (2 nα − (1 + α )) bx n ) (( C (1 + α ) − βx ) bx n − aβx )] } . (36)Equation (36) shows that the model remains anisotropic even for the uncharged case ingeneral. However for particular parameter values we can show that ∆ = 0 in the relevantlimit in the general solution (27)-(33). If we set b = 0 and β = 0 then (36) becomes∆ = C ( a − ax (cid:8) a (1 + α ) − (cid:2) α + (1 + α ) (cid:3)(cid:9) . From the above equation we easily observe that when a = 1 and a = 1 + α (1+ α ) the measureof anisotropy ∆ vanishes. When a = 1 we note from (29) that ρ = 0 since b = k = 0.Consequently we cannot regain an isotropic model when a = 1; to avoid vanishing energydensities we must have a = 1 when ∆ = 0. When a = 1 + α (1+ α ) we obtain the expressions ds = − Br α α dt + (cid:20) α (1 + α ) (cid:21) dr + r ( dθ + sin θdφ ) , (37) ρ = 4 α [4 α + (1 + α ) ] r , p r (= p t ) = αρ, (38)7here we have set A D a α = B and C = 1.The above solution was obtained by Saslaw et al in their investigation of general rela-tivistic isothermal universes. Since ρ ∝ r − we may interpret (37) as a relativistic cosmologicalmetric since there is an analogy that can be made with the well known Newtonian solutionas pointed out by Chandrasekhar . In the Newtonian case the density ρ is finite in the coreregions and decreases according to r − in the rest of the model. The total mass and regionof the isothermal model are infinite. Therefore the line element (37) may be interpreted asa relativistic inhomogeneous universe where the nonzero pressure balances gravity. Saslaw et al point out that (37) may be viewed as the asymptotic state of the Einstein-de Sittercosmological model, in an expansion-free state as t → ∞ , where the hierarchial distributionof matter has clustered over large scales. As t → ∞ the Einstein-de Sitter model tends tothe line element (37) and the universe evolves into an isothermal static sphere given by theexact solution (37) with equation of state p r (= p t ) = αρ . To move from the pressure-freeEinstein-de Sitter model to the static isothermal metric requires a phase transition for conden-sation through clustering of galaxies. It is interesting to observe that Chaisi and Maharaj and Maharaj and Chaisi have obtained generalised anisotropic static isothermal spheres byutilising a known isotropic metric to produce a new anisotropic solution of the Einstein fieldequations. Govender and Govinder have found simple nonstatic generalisations of isother-mal universes which describe an isothermal sphere of galaxies in quasi-hydrostatic equilibriumwith heat dissipation driving the system to equilibrium. Hence the imperfect relativistic fluid,in the presence of electromagnetic field, with the linear equation of state p r = αρ − β , is ageneralisation of the conventional isothermal model with a clear physical basis. We observe that the exact solution (27)-(33) may be singular at the origin in general. Thesolution should be used to describe the gravitational field of the envelope in the outer regionsof a quark star or a dark energy star with equation of state p r = αρ − β . To avoid thesingularity at the centre another solution is required to model the stellar core. Examplesof core-envelope models in general relativity are provided by Thomas et al , Tikekar andThomas and Paul and Tikekar . We observe that if a = 1 then the imperfect chargedsolution (27)-(33) admits the line element ds = − A D (1 + bx n ) α exp[2 F ( x )] dt + (1 + bx n ) dr + r ( dθ + sin θdφ ) , (39)where F ( x ) = − βx C h bx n n +1 i + (1+ α )4 (cid:2) bx n n − k ( dx + x ) (cid:3) . It is clear from (39) that thegravitational potentials are nonsingular at the origin for all values of n . If n = 1 then thereis no singularity in the energy density ρ and the model is nonsingular throughout the stellarinterior. However if n = 1 then the gravitational potential potentials (39) may continue tobe well behaved but singularities may appear in the matter variables at the origin r = 0.By considering a particular example we can demonstrate graphically that the matter vari-ables are well behaved outside the origin. Figures 1-5 represent the energy density, the radial8ressure, the tangential pressure, the electric field intensity and the measure of anisotropy,respectively. Note that solid lines represent uncharged matter ( E = 0) and dashed linesinclude the effect of charged matter E = 0. To plot the graphs we choose the parameters n = 2 , a = 1 , b = 40 , k = 2 , d = 1 , α = and β = 0 . r = 1. From Figures 1-2 we see that both the energy density ρ and the radial pressure p r are continuous throughout the interior, increasing from the centre to r = 0 .
32 and thendecreasing. Note that the radial pressure is zero at the boundary r = 1 for the unchargedcase E = 0. We observe from Figure 3 that the tangential press p t is continuous and wellbehaved in the interior regions. From Figure 4 we observe that the electric field intensity E is decreasing smoothly throughout the stellar interior. We can observe from Figure 5 thatthe measure of anisotropy is continuous throughout the stellar interior. The behaviour of ∆outside the centre is likely to correspond to physically realistic matter in the presence of theelectromagnetic field. Figure 5 has a profile similar to the anisotropic boson stars studied byDev and Gleiser and the compact anisotropic relativistic spheres of Chaisi and Maharaj .From the figures we can see that the effect of electric field intensity E is to produce lowervalues for ρ, p r , p t and ∆. r - Ρ E ¹ = Figure 1: Energy density. r - p r E ¹ = Figure 2: Radial pressure.9 .2 0.4 0.6 0.8 1.0 r - - p t E ¹ = Figure 3: Tangential pressure. r E Figure 4: Electric field intensity.We now show that the solutions generated in this paper can be used to describe realisticcompact objects for the case n = 1 as an example from §
3. In our model, when n = 1, theparameters b has the dimension of length − , k has the dimension of length − and d has thedimension of length . For simplicity, we introduce the transformations˜ b = bR , ˜ k = kR , ˜ d = dR − , where R is a parameter which has the dimension of a length. Under these transformationsthe energy density becomes ρ = 1 R " ˜ b by + 2˜ b (1 + ˜ by ) − ˜ k ( ˜ d + 2 y )1 + ˜ by , r - - D E ¹ = Figure 5: Measure of anisotropy.10here we have set C = 1 and y = r R . Then the mass contained within a radius s has theform M = 12 ( ˜ b s R b s R k b s R + ˜ k (2 − ˜ b ˜ d ) (cid:20) s ˜ b − R ˜ b / arctan hp ˜ b sR i(cid:21)) . For simplicity we set ˜ b ˜ d = 2 so that these expressions reduce to ρ = 1 R " ˜ b by + 2˜ b (1 + ˜ by ) − k ˜ b , (40) M = 12 " ˜ b b s R − k b s R , (41)which are simple forms. It is now easy to calculate the density and mass for particularparameter values from (40) and (42). For example, when s = 7 . R = 1km, ˜ b = 0 .
03 and˜ k = 0 . M E =0 = 1 . M ⊙ for uncharged matter M E =0 = 0 . M ⊙ for charged matter. Note that the value of the mass for uncharged matter is consistent withthe strange star models previously found by Sharma and Maharaj and Dey et al . Wehave shown that the inclusion of the electromagnetic field affects the value for the mass M .It is also possible to to relate our results to other treatments. If we set s = 9 . R = 1km,˜ b = 0 .
35 and ˜ k = 0 . M E =0 = 3 . M ⊙ for uncharged matterand M E =0 = 2 . M ⊙ for charged matter. These values for the mass are similar to chargedquark stellar models generated by Mak and Harko which describe a unique static chargedconfiguration of quark matter admitting a one-parameter group of conformal symmetries.We have shown that the presence of anisotropy and charge in the matter distribution yieldsmasses which are consistent with other investigations. We have investigated the Einstein-Maxwell system of field equations with a strange mat-ter equation of state for anisotropic matter distributions when the electromagnetic field ispresent. A new class of exact solutions was found to this system of nonlinear equations. Aparticular case is nonsingular at the centre of the star and the gravitational potentials andmatter variables are well behaved. When the anisotropy vanishes we regain the general rela-tivistic isothermal universe of Saslaw . This is an extension of the conventional Newtonianisothermal universe with density ρ ∝ r − . The plots of the matter variables indicate that thesolution may be used to model quark stars, at least in the envelop if singularities are presentat the origin. We calculated the mass in a special case ( a = 1 , n = 1) and showed thatthis value is consistent with strange matter distributions of Dey et al , Mak and Harko and Sharma and Maharaj . Ours is a particular solution of the Einstein-Maxwell systemwith the nice feature of containing the isothermal universe in the isotropic limit. A morecomprehensive study of other possible solutions admitted, with the strange matter equationof state, is likely to produce other interesting results.11e now comment on some physical aspects of the solutions found which are of interest.Firstly, there are a number of free parameters in the solution which may be determined byimposing boundary conditions. The solutions may be connected to the Reissner-Nordstromexterior spacetime ds = − (cid:18) − Mr + Q r (cid:19) dt + (cid:18) − Mr + Q r (cid:19) − dr + r (cid:0) dθ + sin θdφ (cid:1) (42)with the interior spacetime (39) across the boundary r = s where M and Q represent thetotal mass and the charge of the star, respectively. This gives the conditions1 − Ms + Q s = A D (cid:8) [1 + b ( Cs ) n ] α exp[2 F ( Cs )] (cid:9)(cid:18) − Ms + Q s (cid:19) − = 1 + b ( Cs ) n . Therefore the continuity of the metric coefficients across the boundary r = s is maintained.The number of free parameters ( A, C, D, b and n ) ensures that these necessary conditions aresatisfied. Secondly, from the graphs generated for the energy density and the pressure weobserve that these quantities are negative in regions close to the centre. We believe that thisfeature arises because of the strange equation of state that we have imposed on the model.To avoid negative matter variables we would need to use another solution to describe the corewith our solution serving as the envelope. This is the situation that arises in the gravastarmodel of Mazur and Mottola where the singular core is replaced by a de Sitter condensatethrough a phase transition near the location of the event horizon. This idea has been utilisedby Lobo to demonstrate that dark energy stars are stable. Thirdly, the example of thispaper and other exact solutions found for the Einstein-Maxwell system suggests that thenegativity of the energy density and the pressure close to the centre may be a generic featureof charged stellar models. It appears that the existence of the electric field always makes theenergy density negative close to the centre of the sphere. This suggests the conjecture thata charged fluid cannot satisfy the energy conditions near the stellar centre is possible. Thiswill be the topic of a separate investigation. Acknowledgements
We are grateful to the referee for insightful comments relating to the physical properties ofthe models. ST thanks the National Research Foundation and the University of KwaZulu-Natal for financial support, and is grateful to Eastern University, Sri Lanka for study leave.SDM acknowledges that this work is based upon research supported by the South AfricanResearch Chair Initiative of the Department of Science and Technology and the NationalResearch Foundation. 12 eferences [1] N Itoh,
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