Interior Static Stellar Model with Electric Charge as an Oscillator

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by  A. M. Baranov

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