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Gravitation & Cosmology, Vol. 8 (2002), Supplement II,pp. 10–11 Proceedingsofthe5thInernational ConferenceonGravitationandAstrophysicsofAsian-PacificCountries(ICGA-2001), Moscow,1-7October2001 (cid:13)c 2002 RussianGravitational Society INTERIOR STATIC STELLAR MODEL WITH ELECTRIC CHARGE AS AN OSCILLATOR 1 A.M.Baranov2 Dep. of Theoretical Physics, Krasnoyarsk State University, 79 Svobodny Av., Krasnoyarsk, 660041, Russia 2 1 0 A model approach to the description of static stars filled with a charged Pascal perfect fluid within the framework 2 of general relativity is investigated. The metric is written in Bondi’s radiation coordinates. The gravitational n equations are reduced to a nonlinear oscillator equation after transfomation to a new variable as a function of the a radial coordinate. It is shown that in this case exact solutions of the Einstein-Maxwell equations for a concrete J energydensitydistributionlawofthechargedfluidmaybeobtainedassolutionoftheharmonicoscillator equation. 1 2 ] In general relativity, together with Schwarzschild’s energy-momentumtensor ofthe chargedsubstance T αβ c exterior solution (see, e.g., [1]), which describes the is taken here as a sum of the energymomentum tensor q -gravitational static field of spherical objects, there is of the Pascal neutral perfect fluid r a solution of the gravitational equations for an exte- g [rior field of a massive spherical body with an elec- Tfluid =(µ+p)u u pg (3) αβ α β − αβ trical charge, well known as the Reissner-Nordstro¨m 1 solution (see, e.g., [2]). However, as to a similar prob- and the electromagnetic energy-momentum tensor v 9lem of finding of the solution for an interior field of an 2electrified body, there are quite clear difficultes of the Tel−mag = 1 ( F Fµ+g F Fµν), (4) 4solution of the Einstein-Maxwell nonlinear differential αβ 4π − αµ β αβ µν 4equations in a medium. where µ(r) is the density, p is the pressure, uα = . 1 Let us consider an interior static stellar model hav- dxα/ds is the 4-velocity, F is the electromagnetic αβ 0ing some distribution of the electric charge, within the fieldtensor,allfunctionsdependonlyontheradialvari- 2 framework of general relativity. The exterior gravita- able. 1 tionalfield of sucha star is the Reissner-Nordstro¨m so- TheMaxwellequations(thesecondpair)inourcase : v lution [2]. The metric is written in Bondi’s radiation become i Xcoordinates: 1 r Fµν = (√ gFµν) = 4πjµ, (5) a ;ν √ g − ,ν − ds2 =F(r)dt2+2L(r)dtdr r2(dθ2+sinθ2dϕ2), (1) − − where g is the determinant of the metric tensor, jµ is the electrical current density. where t isatimecoordinate, r, isaradialcoordinate, θ, The staticsnature ofthe systemallowsone tointro- and ϕ are angular variables;the Greek indices take the duce a fixed comoving frame of reference with values 0,1,2,3; F(r) and L(r) are metric functions to be found. They depend only on the radial variable. The velocity of light and the Newtonian gravitational δµ δµ g g constant are chosen to be equal to unity. uµ = 0 = 0 ; uµ = µ0 = µ0 (6) √g00 F(r) √g00 F(r) The Einsteinequationsfor the interiorfielf arewrit- p p ten with a source for a charged substance as and the physically observable quantities: the mass- energy density, the electrical charge density, the elec- 1 Gαβ =Rαβ gαβR= æTαβ, (2) tric intensity (radial component) and the electric field − 2 − energy density will be written accordingly as where G isthe Einsteintensor, R is the Ricciten- αβ αβ sor, R = R·αα is the scalar curvature, κ = 8π is Ein- stein’sconstant, gαβ isthemetrictensor. Theresulting µphys =µ(r); ρphys ρ=jµuµ =j0√g00; (7.1) ≡ 1Talkpresentedatthe5thInt. Conf. onGrav. andAstrophys. F E E2 ofAcian-PacificCountries(ICGA-5), Moscow,2001. E E = 01 = ; W = . (7.1) 2e-mail: alex m [email protected] phys ≡ 1 √g00 √g00 el 8πg012 Interior Static Stellar Model with Electric Charge as an Oscillator 11 As a result, the set of Einstein-Maxwell equations Choosing the mass density µ and the electric field is reduced to the following set of nonlinear differential energy density inside the star W as square-lawfunc- el equations: tions of x (as was made in [3]), F ′ µ=µ0(1 x2)=µ0(1 y); (14) (lnL) =æ(p+µ); (8.1) − − xL2 F 1 2 λ2R2x2 Λ2y ′ ′′ ′ ′ ′ (lnL) F + F F (lnL) W = = , (15) xL2 − 2L2 (cid:20) x − (cid:21) el 8π 8π = κ(p+W ); (8.2) we obtain from (11) the equation for a linear oscillator el − G′′ +Ω2G=0 (16) ζζ 0 ′ ′ 1 F xF xF(lnL) ( 1+ + with a constant value of the natural frequency x2 − L2 L2 − L2 χ 11Λ2 = κ (µ−p) +Wel ; (8.3) Ω20 = 5(µ0− 8π ), (17) − (cid:20) 2 (cid:21) The continuity of the electric energy density on the ′ x2E L stellar surface requires Λ = Q/R2, where Q is the =4πRρx2 , (8.4) (cid:18) L (cid:19) √F integral electric charge of the star and µ0 is its central mass density. where the prime denotes a derivative with respect to In this case the function Φ may be written as x = r/R, 0 x 1, R is the radius of the star, ≤ ≤ Wel =E2/8πL2, E is the electric field intensity. µ0 1 Λ2 Bythereplacement ε=F/L2 itispossibletoderive Φ=χy( + ( µ0)y). (18) 3 5 8π − alinearsecond-orderordinarydifferentialequationwith variable coefficients from the first three Einstein equa- The radialdependence of the electric chargedensity in the interior field of the star can be found for the tions by eliminating the presure and the mass density: chosenfunctionallawoftheobservableelectricalenergy ′′ ′ density from Maxwell’s equation (8.4): G +f(x)C +g(x)G=0, (9) where f(x) = (lnϕ)′, ϕ = √ε/x, g(x) = [2(1 ε)+ ρ(x)=Q ε(x)/V, (19) xε′]/(2x2ε) 2χW /ε; χ=κR2. − ·p − el where V =4πR3/3 is the Euclidean volume of 3-space. For the new variable ζ = ζ(x) introduced by the ThegeneralsolutionofEq.(16)fortherequiredfunc- relation tion G will be written as xdx dy dζ = = , (10) ε(x) 2 ε(y) G=G cos(Ω ζ+α), (20) 0 0 p p Eq.(9)isrewritteninaself-conjugateformandbecomes and for the metric function g =F =G2 00 the equation for a nonlinear oscillator F =G2cos2(Ω ζ(x)+α), (21) G′′ +Ω2(ζ(x))G=0, (11) 0 0 ζζ where ζ(x) is a quadrature depending on the relation where y =x2; Ω2 is the squared ”frequency”. between the parameters µ and Λ. 0 Eq.(10) not always can be integrated in elementary functions, and more simply the function Ω2 is written References as a function of the variable y: [1] J.L.Synge,”Relativity: thegeneraltheoryofrelativity”, Ω2 = d(Φ/y)dy 2χWel/y. (12) North-Holland Pub. Company,Amsterdam, 1960. − − [2] A.S.Eddington, ”The mathematical theory of relativ- Taking into account the contribution of the electric ity”, Cambrige, 1924. fieldenergy,thefunction Φ playingtheroleoftheNew- tonian gravitationalpotentialinside the star canbe ob- [3] A.M.Baranov //Editorial Staff of J. Izv. Vuz.(Fizika), tained from the gravitationalequations as 1973. - Dep.in VINITI,No.6729-73 (in Russian). χ Φ=1 ε= (µ(x)+W )x2dx el − x Z χ = (µ(y)+W )√ydy. (13) el 2√y Z