Analytical models for compact stars with a linear equation of state

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1 Avilble online t WSN 0 (0) - ISSN 9-9 Anlytil models for ompt strs with liner eqution of stte Mnuel Mlver Deprtment of Bsi Sienes, Mritime University of the Cribben, Cti l Mr, Venezuel -mil ddress: mmf.um@gmil.om ABSTRACT In this pper, we found two new lsses of solutions to the instein-mwell system of equtions for ompt strs ssuming n nisotropi pressure nd liner eqution of stte for the mtter distribution within the frmework of MIT-Bg Model with prtiulr form of the metri funtion. The et solutions n be written in terms of elementry funtion in presene of n eletromgneti field. All the obtined models hve singulrity in the hrge density but not dmit singulrities in the mtter nd metri funtions t the entre. Keywords: instein-mwell system; liner eqution of stte; hrge density; MIT-Bg Model; metri funtion; et solution. INTRODUCTION One of the fundmentl problems in the generl theory of reltivity is finding et solutions of the instein field equtions [,]. Some solutions found fundmentl pplitions in strophysis, osmology nd more reently in the developments inspired by string theory []. Different mthemtil formultions tht llow to solve instein s field equtions hve been used to desribe the behvior of objets submitted to strong grvittionl fields known s neutron strs, qusrs nd white dwrfs [-].

2 In the onstrution of the first theoretil models of reltivisti strs re importnt the works of Shwrzshild [], Tolmn [], Oppenheimer nd Volkoff [8]. Shwrzshild [] found nlytil solutions tht llowed desribing str with uniform density, Tolmn [] developed method to find solutions of stti spheres of fluid nd Oppenheimer nd Volkoff [8] used Tolmn's solutions to study the grvittionl blne of neutron strs. It is importnt to mention Chndrsekhr's ontributions [9] in the model prodution of white dwrfs in presene of reltivisti effets nd the works of Bde nd Zwiky [0] who propose the onept of neutron strs nd identify stronomi dense objets known s supernovs. The physis of ultrhigh densities is not well understood nd mny of the strnge strs studies hve been performed within the frmework of the MIT bg model []. In this model, the strnge mtter eqution of stte hs simple liner form given by p B where is the energy density, p is the isotropi pressure nd B is the bg onstnt. However, theoretil works of relisti stellr models [-] it hs been suggested tht superdense mtter my be nisotropi, t lest in some density rnges. The eistene of nisotropy within str n be eplined by the presene of solid ore, phse trnsitions, type III super fluid, pion ondenstion [] or nother physil phenomen. In suh systems, the rdil pressure is different from the tngentil pressure. This generliztion hs been very used in the study of the blne nd ollpse of ompt spheres [-0]. Mny reserhers hve used gret vriety of mthemtil tehniques to try to obtin et solutions for qurk strs within the frmework of MIT bg model, sine it hs been demonstrted by Komthirj nd Mhrj [], Mlver [], Thirukknesh nd Mhrj [] nd Thirukknesh nd Rgel []. With the use of instein field equtions, importnt dvnes hs been mde to model the interior of str. Feroze nd Siddiqui [] nd Mlver [,] onsider qudrti eqution of stte for the mtter distribution nd speify prtiulr forms for the grvittionl potentil nd eletri field intensity. Mf Tkis nd Mhrj [] obtined new et solutions to the instein-mwell system of equtions with polytropi eqution of stte. Thirukknesh nd Rgel [8] hve obtined prtiulr models of nisotropi fluids with polytropi eqution of stte whih re onsistent with the reported eperimentl observtions. More reently, Mlver [9,0] generted new et solutions to the instein-mwell system onsidering Vn der Wls modified eqution of stte with nd without polytropil eponent. Rghoonundun nd Hobill [] found new nlytil models for ompt strs with the use of Tolmn VII solution. Our objetive in this pper is to generte new lss for hrged nisotropi mtter with the brotropi eqution of stte tht presents liner reltion between the energy density nd the rdil pressure in stti spherilly symmetri spetime using prtiulr forms for the metri funtion y (). We hve obtined some new lsses of stti spherilly symmetril models of hrged mtter where the vrition of metri funtion modifies the rdil pressure, hrge density nd the mss of the ompt objets. This rtile is orgnized s follows, in Setion, we present instein s field equtions. In Setion, we mke prtiulr hoie of metri potentil y () tht llows solving the field equtions nd we hve obtined new models for hrged nisotropi mtter. In Setion, physil nlysis of the new solutions is performed. Finlly in Setion, we onlude. --

3 . INSTIN FILD QUATIONS Consider spherilly symmetri four dimensionl spe time whose line element is given in Shwrzshild oordintes by ds ν(r) λ(r) = e dt +e dr +r (dθ + sin θdφ ) () Z()= e λ(r) ()= e Using the trnsformtions, = r, nd with rbitrry onstnts A nd, suggested by Durgpl nd Bnnerji [], the instein field equtions s given in () re A y ν(r) Z Z = ρ () Z y y Z = pr () Z y +( Z+ Z ) y y + Z = y pt () p p () t r Z () pr is the rdil pressure, is eletri field intensity, is where is the energy density, the hrge density, pt pr is the nisotropi ftor, p t is the tngentil pressure nd dot is the derivtive with respet to. With the trnsformtions of [], the mss within rdius r of the sphere tke the form M()= / 0 ρ()d () In this pper, we ssume the following linel eqution of stte within the frmework of MIT-Bg Model p r = ρ (8) --

4 . TH MODLS.. Model Following Komthirj nd Mhrj [], we hoose the metri funtion in the prtiulr form y () s y( ) ( ) (9) where is onstnt. In this pper, we hve onsidered the form of the eletril field proposed for Feroze nd Siddiqui [] ( Z) (0) Substituting (0) in eq. () we obtin Z () Using (0) in eq. (), the rdil pressure n be written s p r y Z () y With (8), the eq. () beomes Z y Z () y Repling (9) in (), we obtin the first order eqution Z Z () Integrting (), we hve for the grvittionl potentil Z () Z ( ) () With eq. (9) nd eq. (), the metri funtions ( r) e nd ( r) e n be written s e ( r ) () --

5 -8- ) ( A e r () for the energy density we hve (8) Repling (8) in (8), we obtin for the rdil pressure p r (9) Using (8) in () the epression of the mss funtion is rtg M ) ( ) ( (0) for the eletri field intensity ) 0 ( () nd for hrge density () The tngentil pressure n be written s p t ()

6 The metri for this model is ds = A dt + d d sin d ().. Model Motivted for Sunzu et l [], nother new et solution of the instein-mwell system of equtions (-) is gives by the metri funtion With (), eqution () beomes y( ) () Z Z () q. () n be integrted to give the grvittionl potentil Z () Therefore we n find the following nlytil model e ( r) (8) e ( r) A (9) (0) p r 8 () -9-

7 M ( ) ( ) 9 8 rtg () 0 8 () () p t () 8 The metri for this model is ds = A dt d + d sin d (). PHYSICAL ANALYSIS Any physilly eptble solutions must stisfy the following onditions [8] : (i) (ii) (iii) (iv) Regulrity of the grvittionl potentils in the origin. Rdil pressure must be finite t the entre. P r > 0 nd >0 in the origin. Monotoni derese of the energy density nd the rdil pressure with inresing of rdius The presented models onstitute nother new fmily of solutions for ompt str with nisotropi pressure. The metri funtions n be written in terms of polynomil nd elementry funtions nd the vribles energy density, rdil pressure, hrge density nd r tngentil pressure lso re represented nlytil. For the model, the metri funtions e r nd e r behves well inside the str nd hve finite vlue of e A nd e r -0-

8 ( r) ( r) in r = 0. In this se e r0 e r0 0 in the origin. This nlysis demonstrted tht the grvittionl potentil is regulr t the entre r=0. The energy density is positive throughout the interior of the str, regulr t the entre with vlue. The rdil pressure p r is regulr t the entre with vlue p r.in this model, the eletri field intensity, hrge density nd tngentil pressure dmit singulrity t the entre of the stellr objet. r For the model, the funtions e r nd e quire finite vlues t the entre nd ( r) ( r) e r0 e r0 0 s in the model. The energy density, rdil pressure nd eletri field tke the vlues of, p r 8 nd in r = 0, respetively. In ll the new lsses of models, the mss funtion is ontinuous nd behves well inside the str nd the hrge density hs singulrity t the entre.. CONCLUSIONS We hve generted new et solutions to the instein-mwell system of equtions speifying the form of the metri funtion y() nd liner eqution of stte whih is relevnt in the desription of hrged nisotropi mtter. The reltivisti solutions presented re physilly resonble. The first solution orrespond to model with finite vlues for the energy density nd the rdil pressure t the origin but present singulrity t the entre of stellr objet for the eletri field, hrge density nd tngentil pressure. In the seond solution, the rdil pressure nd energy density re regulr nd positive throughout the stellr interior. In the new obtined models the grvittionl potentils re regulr t the origin r=0 nd well behved. The hrge density dmits singulrity t the entre nd the mss funtion is n inresing funtion, ontinuous nd finite. The models presented in this rtile my be useful in the desription of reltivisti ompt objets with hrge, qurk strs nd onfigurtions with nisotropi mtter. Referenes [] Kuhfitting, P.K. (0). Some remrks on et wormhole solutions, Adv. Stud. Theor. Phys.,, -. [] Bik, J. (00). instein equtions: et solutions, nylopedi of Mthemtil Physis,, -. [] Mlver, M. (0). Blk Holes, Wormholes nd Drk nergy Strs in Generl Reltivity. Lmbert Ademi Publishing, Berlin. ISBN:

9 [] Komthirj, K., nd Mhrj, S.D. (008). Clsses of et instein-mwell solutions, Gen. Rel. Grv., 9, [] Shrm, R., Mukherjee, S nd Mhrj, S.D. (00). Generl solution for lss of stti hrged strs, Gen. Rel. Grv.,, [] Shwrzshild, K. (9). Über ds Grvittionsfeld einer Kugel us inkompressibler Flüssigkeit, Mth. Phys. Teh, -. [] Tolmn, R.C. (99). Stti Solutions of instein's Field qutions for Spheres of Fluid, Phys. Rev.,, -. [8] Oppenheimer, J.R. nd Volkoff, G. (99). On mssive neutron ores, Phys. Rev.,, -8. [9] Chndrsekhr, S. (9). Mss of Idel White Dwrfs, Astrophys. J.,, 8-8. [0] Bde, W., nd Zwiky, F. (9). Cosmi Rys from Super-Nove, Pro. Nt. Ad. Si. U. S., (0), 9-. [] Komthirj, K., nd Mhrj, S.D.(00). Anlytil models for qurk strs, Int. J. Mod. Phys., D, pp [] Herrer, L., nd Sntos, N.O. (99), Phys. Rep.8,. [] Cosenz, M., Herrer, L., sulpi, M. nd Witten, L.(98), J. Mth. Phys., (), 8. [] Gokhroo, M.K., nd Mehr. A.L. (99). Anisotropi spheres with vrible energy density in generl reltivity, Gen. Relt. Grv., (), -8. [] Herrer, L. (99), Phys. Lett., A, 0. [] Sokolov. A.I. (980), Sov. Phys. JTP.,, [] Herrer, L., Ruggeri, G.J nd Witten. L. (99), Astrophys. J.,, 09. [8] Herrer, L., nd Pone de Leon. J. (98), J. Mth. Phys.,, 08. [9] Herrer, L., nd Sntos N.O. (998), J. Mth. Phys., 9, 8. [0] Bondi.H.(99), Mon.Not. R. Astron. So., 9,. [] Mlver, M. (009). Análisis omprtivo de lgunos modelos nlítios pr estrells de qurks, Revist Integrión,, -. [] Thirukknesh, S., nd Mhrj, S.D. (008). Chrged nisotropi mtter with liner eqution of stte, Clss. Quntum Grvity,, 00. [] Thirukknesh, S., nd Rgel, F.C. (0). A lss of et strnge qurk str model, PRAMANA Journl of physis, 8(), -8. [] Feroze, T. nd Siddiqui, A. (0). Chrged nisotropi mtter with qudrti eqution of stte, Gen. Rel. Grv.,, 0-0. [] Mlver, M. (0). Strnge Qurk Str Model with Qudrti qution of Stte, Frontiers of Mthemtis nd Its Applitions., (), 9-. [] Mlver, M. (0). Reltivisti Modeling of Qurk Strs with Tolmn IV Type Potentil, Interntionl Journl of Modern Physis nd Applition., (), -. --

10 [] Tkis, P.M., nd Mhrj, S.D. (0). Some hrged polytropi models, Gen. Rel. Grv.,, [8] Thirukknesh, S., nd Rgel, F.C. (0). t nisotropi sphere with polytropi eqution of stte, PRAMANA Journl of physis, 8(), 8-9. [9] Mlver, M. (0). Anlytil model for hrged polytropi strs with Vn der Wls Modified qution of Stte, Amerin Journl of Astronomy nd Astrophysis, (), -. [0] Mlver, M. (0). Regulr model for qurk str with Vn der Wls modified eqution of stte, World Applied Progrmming.,, 09-. [] Rghoonundun, A., nd Hobill, D. (0). Possible physil reliztions of the Tolmn VII solution, Physil Review D 9, 00. [] Durgpl, M.C., nd Bnnerji, R. (98). New nlytil stellr model in generl reltivity, Phys. Rev. D, 8-. [] Feroze, T,. nd Siddiqui, A. (0). Some et solutions of the instein-mwell equtions with qudrti eqution of stte, Journl of the Koren Physil Soiety, (), 9-9. [] Sunzu, J.M, Mhrj, S.D nd Ry, S. (0). Astrophysis. Spe. Si., -. ( Reeived 0 My 0; epted 8 June 0 ) --