Relativistic Modeling of Quark Stars with Tolman IV Type Potential

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Intenational Jounal of Moden Physics Application 05; (): -6 Published online Januay 30, 05 (http://www.aascit.

Email addess mmf.umc@gmail.com Citation Manuel Malave. Relativistic Modeling of Quak Stas with Tolman IV Type Potential. Intenational Jounal of Moden Physics Application. Vol., No., 05, pp. -6.

objects with anisotopic matte distibution consideing Tolman IV fom fo the gavitational potential Z. The equation of state pesents a quadatic elation between the enegy density the adial pessue.

1 Intenational Jounal of Moden Physics Application 05; (): -6 Published online Januay 30, 05 ( ISSN: Relativistic Modeling of Quak Stas with Tolman IV Type Potential Manuel Malave Univesidad Maítima del Caibe, Depatamento de Ciencias Básicas, Catia la Ma, Venezuela addess Citation Manuel Malave. Relativistic Modeling of Quak Stas with Tolman IV Type Potential. Intenational Jounal of Moden Physics Application. Vol., No., 05, pp. -6. Keywods Relativistic Objects, Electic Field, Gavitational Potential, Tolman IV Type Potential, Einstein-Maxwell System, Chage Density Abstact In this pape, we studied the behavio of elativistic objects with anisotopic matte distibution consideing Tolman IV fom fo the gavitational potential Z. The equation of state pesents a quadatic elation between the enegy density the adial pessue. New exact solutions of the Einstein-Maxwell system ae geneated. A physical analysis of electomagnetic field indicates that is egula in the oigin well behaved. We show as the pesence of an electical field modifies the enegy density, the adial pessue the mass of the stella object geneates a singula chage density.. Intoduction Received: Decembe 9, 0 Revised: Januay 8, 05 Accepted: Januay 9, 05 Fom the development of Einstein s theoy of geneal elativity, the modelling of supedense mate configuations is an inteesting eseach aea [,]. In the last decades, such models allow explain the behavio of massive objects as neuton stas, quasas, pulsas, black holes white dwafs [3,,5]. In theoetical woks of ealistic stella models, is impotant include the pessue anisotopy [6-8]. Bowes Liang [6] extensively discuss the effect of pessue anisotopy in geneal elativity. The existence of anisotopy within a sta can be explained by the pesence of a solid coe, phase tansitions, a type III supe fluid, a pion condensation [9] o anothe physical phenomena as the pesence of an electical field [0].The physics of ultahigh densities is not well undestood many of the stange stas studies have been pefomed within the famewok of the MIT bag model []. In this model, the stange matte equation of state has a simple linea fom given by p ( ρ B) whee ρis the enegy density, p is the isotopic pessue B is the 3 bag constant. Many eseaches have used a geat vaiety of mathematical techniques to ty to obtain exact solutions fo quak stas within the famewok of MIT bag model, since it has been demonstated by Komathiaj Mahaaj [], Malave [,3], Thiukkanesh Mahaaj [], Mahaaj et al. [5], Thiukkanesh Ragel [6] Sunzu et al. [7]. With the use of Einstein s field equations, impotant advances has been made to model the inteio of a sta. In paticula, Feoze Siddiqui [8] Malave [9] conside a quadatic equation of state fo the matte distibution specify paticula foms fo the gavitational potential electic field intensity. Mafa Takisa Mahaaj [0] obtained new exact solutions to the Einstein-Maxwell system of equations with a polytopic equation of state. Thiukkanesh Ragel [] have obtained paticula models of anisotopic fluids with polytopic equation of state which ae consistent with the epoted expeimental obsevations. Moe ecently, Malave [,3] geneated new exact solutions to the Einstein-Maxwell system consideing Van de Waals modified equation of state with without polytopical exponent Thiukkanesh Ragel

2 Manuel Malave: Relativistic Modeling of Quak Stas with Tolman IV Type Potential [] pesented a anisotopic stange quak matte model by imposing a linea baotopic equation of state with Tolman IV fom fo the gavitational potential. Mak Hako [5] found a elativistic model of stange quak sta with the suppositions of spheical symmety confomal Killing vecto. The objective of this pape is to obtain new exact solutions to the Maxwell-Einstein system fo anisotopic matte with an equation of state that pesents a quadatic elation between the enegy density the adial pessue in static spheically symmetic spacetime using Tolman IV fom fo the gavitational potential Z. We have obtained some new classes of static spheically symmetical models whee the pesence of an electical field modifies the adial pessue, chage density the mass of the compact objects. This aticle is oganized as follows, in Section, we pesent Einstein s field equations. In Section 3, we make a paticula choice of gavitational potential Z(x) that allows solving the field equations we have obtained new models fo chaged anisotopic matte. In Section, a physical analysis of the new solutions is pefomed. Finally in Section 5, we conclude.. Einstein Field Equations of Anisotopic Fluid Distibution We conside a spheically symmetic, static homogeneous anisotopic spacetime in Schwazschild coodinates given by ds ν() λ() e dt +e d + (dθ + sin θdφ ) () whee ν () λ() ae two abitay functions. The Einstein field equations fo the chaged anisotopic matte ae given by T T E 00 ρ () E p (3) T t + E T 33 p () whee ρ is the enegy density, p is the adial pessue, E is electic field intensity p t is the tangential pessue, espectively. Using the tansfomations, xc, λ() Z(x)e ν() Ay(x)e with abitay constants A c>0, suggested by Dugapal Banneji [6], the metic () take the fom x ds A y ( x) dt + dx + (dθ + sin θdφ ) cxz c (5) the Einstein field equations can be witten as Z ρ E Z& + (6) x c c y& Z p Z E y x c c (7) && y y p Z xz t E xz +( + & & ) +Z & + y y c c (8) cz σ xe & + E (9) x σ is the chage density dots denote diffeentiation with espect to x. With the tansfomations of [6], the mass within a adius of the sphee take the fom M(x) 3/ c x 0 xρ(x)dx In this pape, we assume the following equation of state Hee α is abitay constant. (0) p αρ () 3. A New Class of Solutions Following Tolman [7] Thiukkanesh Ragel [], we take the fom of the gavitational potential, Z(x)as ( + ax )( bx ) ( + ax ) Z( x) () whee a b ae eal constants. This potential is egula at the oigin well behaved in the inteio of the sphee. We have consideed the paticula cases fo E 0 E 0. Case I: Fo E 0, using Z(x) in eq.() we obtain [ 3( a + b) + ( 7ab + a ) x + 6a bx ] ρ c (3) ( + ax ) Substituting (3) in eq.(), the adial pessue can be witten in the fom α P c 3 a + b + 7ab + a x + 6a bx ax Using (3) in (0), the expession of the mass function is M(x) 3/ ( + + ) x a b abx c( + ax) () (5) The equations (3) (5) have been deduced fo

3 Intenational Jounal of Moden Physics Application 05; (): -6 3 Thiukkanesh Ragel [] with a linea baotopic equation of state. The tangential pessue is given fo ( a + b + abx + a bx ) ax ) ax ) 3 xc + ax bx && y + 5a 3b x + a a b x 6a bx y& Pt + c + ax y y c (6) Substituting () () in (7), we have ( a + b + abx ) ( ) y& y ax bx α c 3 a + b + 7abx + a x + 6a bx + ax ax bx Integating (7), we obtain 3 ( ) (7) Fig. Radial Pessue whee A B C [ ] y(x) c + bx + ax + ax exp D(x) (8) 3 9b α c + 6α ca b + α cab + ab + a + b A a b b a, α ca + αcab a + αcb B a b 7α cb + α ca + 8αbc C 8 b a, Fig. Radial speed of sound αac + 3αbc + αabcx D( x) ( + ax) (9) The metic functions e λ ax)( bx) λ + ax e e ν can be witten as A B C [ ] (0) e ν A c + bx + ax + ax exp D(x) () Figues, 3 epesent the gaphs of p, v s, M( x ) with eq. () E 0. v M( x) ae the adial speed of sound mass function, espectively. The gaphs has been plotted fo a paticula choice of paametes a 0.069, b 0.005, α/3 with a stella adius of 0 km pesented fo Thiukkanesh Ragel []. The metic fo this model is s A B C [ ] ax ) x + ( ) c ds Ac + bx + ax + ax exp D(x) dt + dx (dθ +sin θdφ ) xc ax bx () Fig 3. Mass. Case II: Fo E 0 we have consideed the fom of the electical field poposed fo Feoze Siddiqui [8] ( ) ( + + ) c Z c a b abx E x + ax (3) With eq.(3), we have found the following expessions fo ρ,p,m ( x ), σ P t

4 Manuel Malave: Relativistic Modeling of Quak Stas with Tolman IV Type Potential ρ c ( a + b + abx + a bx ) ( + ax ) ( a + b + abx + a bx ) α P c ax () (5) E F G [ ] ax) x + ( ) c ds A c + bx + ax + ax exp H(x) dt + dx (dθ +sin θdφ ) xc ax bx (3) 3 3 6a bx + 8a bx a 6ab ax + a + 6a b x + 6a + 3ab actag ax M(x) 8a ac ax (6) ( ) + + c ax bx a bx a a 5b x a b σ x ax a b abx ( ) xc ax bx && y Pt ax y ( + + ) ax ) 3 + 5a 3b x + 8a a b x a bx y& + c y a + b + a a + 5b x + a bx c ax ) Substituting (3) (5) in (7), we have Integating (9), we obtain y& α c a + b + abx + a bx y ax ax bx 3 ( ) (7) (8) (9) Fig. Radial Pessue. E F G [ ] y(x) c + bx + ax + ax exp H(x) (30) Again fo convenience we have let α a + b cb E a + b G H x The metic functions, F c ( a b ) α +, 6ab + a + 3b cα a + b ( + ax ) 8a x + a b cα e λ ax)( bx) λ + ax e e ν can be witten as (3) (3) Fig 5. Enegy density E F G [ ] e ν A c + bx + ax + ax exp H(x) (33) Figues, 5, 6, 7 8 epesent the gaphs of p, ρ, σ, M( x) vs with eq., espectively a 0.069, b 0.005, α/3 with 0 km. The metic fo this model is Fig 6. Mass

Intenational Jounal of Moden Physics Application 05; (): -6 5 the squae of sound speed defined as v dp should be d ρ s within the limit 0 v s in the inteio of the sta. In fig.

5 Intenational Jounal of Moden Physics Application 05; (): -6 5 the squae of sound speed defined as v dp should be d ρ s within the limit 0 v s in the inteio of the sta. In fig. 8 this condition is maintained inside the stella inteio. In fig. 5, that epesent enegy density fo the case E 0, we obseve that is continuous, finite monotonically deceasing function. In fig. 7, the chage density is singula at the oigin, non-negative deceases. In fig.3 6, the mass function is stictly inceasing function, continuos finite. 5. Conclusion Fig 7. Chage density Fig 8. Radial speed of sound. Physical Popeties of the New Solutions In this section we discuss the physical popeties that have to be satisfied by the ealistic sta [8]. With E 0, the gavitational potentials ae egula at the oigin since α ac+ 3αbc ν( 0) e A c λ( 0) e e ae constants λ ν 3c ( e ) ( e ) 0 at 0. In the cente ( a + b) ρ (0) 9α c ( a + b) P both ae positive if a > 0 b > 0. Fo the case c ( Z ) λ(0) E, e, x ( a b) αc c e ν ( 0) e A, in the oigin 0, λ( ) ν( ) ( e ) 0 ( e ) 0 0. This shows that the potential gavitational is egula in the oigin. In the cente ρ (0) c ( a + b), P c ( a b) α + the chage density pesents a singulaity. Fo both cases, the mass function is stictly inceasing function, continuos M ( x) 0 at 0. In fig. fig., the adial pessue is finite deceasing fo two studied cases. To maintain of causality, In this pape, we have geneated new exact solutions to the Einstein-Maxwell system consideing Tolman IV fom fo the gavitational potential Z an equation of state that pesents a quadatic elation between the enegy density the adial pessue. The new obtained models may be used to model elativistic stas in diffeent astophysical scenes. The elativistic solutions to the Einstein-Maxwell systems pesented ae physically easonable. The chage density σ is singula at the oigin fo the case E 0 the mass function is an inceasing function, continuous finite inside the stella inteio. The condition 0 v s inside the stella inteio. The gavitational potentials ae egula at the cente well behaved. The models pesented in this aticle may be useful in the desciption of elativistic compact objects with chage, stange quak stas configuations with anisotopic matte. Refeences [] Kuhfitting, P.K.(0). Some emaks on exact womhole solutions, Adv. Stud. Theo. Phys., 5, [] Bicak, J. (006). Einstein equations: exact solutions, Encyclopedia of Mathematical Physics,, [3] Malave, M. (03). Black Holes, Womholes Dak Enegy Stas in Geneal Relativity. Lambet Academic Publishing, Belin. ISBN: [] Komathiaj, K., Mahaaj,S.D. (008). Classes of exact Einstein-Maxwell solutions, Gen. Rel. Gav., 39, [5] Shama, R., Mukhejee, S Mahaaj, S.D.(00). Geneal solution fo a class of static chaged stas, Gen.Rel. Gav., 33, [6] Bowes, R. L., Liang, E. P. T. (97). Astophys J., 88, 657 [7] Cosenza, M., Heea, L., Esculpi, M. Witten, L.(98), J.Math.Phys., (), 8. [8] Gokhoo, M.K., Meha. A.L. (99). Anisotopic sphees with vaiable enegy density in geneal elativity, Gen.Relat.Gav., 6(), [9] Sokolov. A.I. (980), Sov. Phys.JETP., 5, 575 [0] Usov, V. V. (97). Phys. Rev. D, 70, [] Komathiaj, K., Mahaaj, S.D.(007). Analytical models fo quak stas, Int. J. Mod. Phys., D6, pp

6 6 Manuel Malave: Relativistic Modeling of Quak Stas with Tolman IV Type Potential [] Malave, M. (009). Análisis compaativo de algunos modelos analíticos paaestellas de quaks, Revista Integación, 7, [3] Malave, M. (0). AASCIT Communications,, 8-5. [] Thiukkanesh, S., Mahaaj, S.D. (008). Chaged anisotopic matte with linea equation of state, Class. Quantum Gavity, 5, [5] Mahaaj, S.D., Sunzu, J.M. Ray, S. (0). Eu. Phys. J.Plus., 9, 3. [6] Thiukkanesh, S., Ragel, F.C. (03). A class of exact stange quak sta model, PRAMANA-Jounal of physics, 8(), [7] Sunzu, J.M, Mahaaj, S.D Ray, S.(0). Astophysics. Space. Sci. 35, 57-5 [8] Feoze, T.. Siddiqui, A. (0). Chaged anisotopic matte with quadatic equation of state, Gen. Rel. Gav., 3, [9] Malave, M. (0). Stange Quak Sta Model with Quadatic Equation of State, Fonties of Mathematics Its Applications., (), 9-5. [0] Takisa, P.M., Mahaaj, S.D. (03). Some chaged polytopic models, Gen.Rel.Gav., 5, [] Thiukkanesh, S., Ragel, F.C. (0). Exact anisotopic sphee with polytopic equation of state, PRAMANA-Jounal of physics, 78(5), [] Malave, M. (03). Analytical model fo chaged polytopic stas with Van de Waals Modified Equation of State, Ameican Jounal of Astonomy Astophysics, (), - 6. [3] Malave, M. (03). Regula model fo a quak sta with Van de Waals modified equation of state Wold Applied Pogamming., 3, [] Thiukkanesh, S., Ragel, F.C. (0).Stange sta model with Tolmann IV type potential, Astophysics Space Science, 35(), [5] Mak, M.K., Hako, T. (00). Quak stas admitting a one-paamete goup of confomal motions, Int. J. Mod. Phys, D3, [6] Dugapal, M.C., Banneji, R. (983). New analytical stella model in geneal elativity, Phys. Rev. D7, [7] Tolman, R.C. (939). Static Solutions of Einstein's Field Equations fo Sphees of Fluid, Phys. Rev., 55, [8] Feoze, T,. Siddiqui, A. (0). Some exact solutions of the Einstein-Maxwell equations with a quadatic equation of state, Jounal of the Koean Physical Society, 65(6), 9-97.

RegularExactModelswithVanishingAnisotropyGeneratedusingVanDerWaalsEquationofState

RegularExactModelswithVanishingAnisotropyGeneratedusingVanDerWaalsEquationofState Global Jounal of Science Fontie Reseach: A Physics and Space Science Volume 8 Issue Vesion.0 Type: Double Blind Pee Reviewed Intenational Reseach Jounal Publishe: Global Jounals Online ISSN: 49-466 & Pint