Dynamical stability of fluid spheres in spacetimes with a nonzero cosmological constant

0.1 MB·English

by  Stanislav Hledik

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DYNAMICAL STABILITY OF FLUID SPHERES IN SPACETIMES
WITH A NONZERO COSMOLOGICAL CONSTANT∗
STANISLAVHLEDÍK†,ZDENĚKSTUCHLÍK‡ andKRISTINAMRÁZOVÁ⊙
Institute of Physics, Faculty of Philosophy and Science, Silesian UniversityinOpava,
Bezručovo nám. 13, Opava, CZ-74601, Czech Republic
E-mail: †[email protected], ‡[email protected], ⊙[email protected]
7
0
0 TheSturm–Liouvilleeigenvalueequation foreigenmodesof theradialoscillationsisde-
2 termined for spherically symmetric perfect fluid configurations in spacetimes with a
nonzero cosmological constant and applied in the cases of configurations with uniform
n
distributionofenergydensityandpolytropicspheres.Itisshownthatarepulsivecosmo-
a
J logicalconstantrisesthecriticaladiabaticindexanddecreases thecriticalradiusunder
whichthedynamicalinstabilityoccurs.
5
1 Keywords: Perfectfluidconfigurations;Cosmologicalconstant; Dynamicalstability.

2
v
1. Introduction
1
5 The internal Schwarzschildspacetimes with nonzero cosmologicalconstant(Λ6=0)
0
and uniform distribution of energy density were given for star-like configurations1
1
0 andextendedtomoregeneralsituations.2Thepolytropicandadiabaticsphereswere
7 preliminary treated and compared,3,4 neutron star models with regions of nuclear
0
matter described by different relativistic equations of state that are matched were
/
c also treated.5 Their stability can be considered on energetic grounds6 or it can be
q
treated in dynamical way.7 Here we determine the dynamical stability conditions
-
r for the uniformdensity and polytropic spheresusing the approachofMisner et al.8
g
:
v
i 2. Sturm–Liouville Equation
X
r Interior of a spherically symmetric configuration is described (in standard
a
Schwarzschildcoordinates) by the line element
ds2 =−e2Φdt2+e2Ψdr2+r2(dθ2+sin2θdϕ2), (1)
with metric coefficients taken in the general form
Ψ=Ψ(r,t), Φ=Φ(r,t). (2)
Theperfectfluiddistributionis givenbyenergydensity andpressureprofilesρ(r,t)
and p(r,t). The static equilibrium configuration is given by Φ (r), Ψ (r), ρ (r),
0 0 0
p (r) satisfying the Euler equations. The radially pulsating configuration is then
0
determined by
q(r,t)=q (r)+δq(r,t), δq ≡(δΦ,δΨ,δρ,δp,δn), (3)
0
∗ThisresearchhasbeensupportedbyCzechgrantMSM4781305903.
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where n is the number density. The pulsation is given by the radial displacement
ξ =ξ(r,t). The Euler perturbationsδq are related to the Lagrangianperturbations
measured by an observer who moves with the pulsating fluid by the relation
′
∆q(r,t)=q(r+ξ(r,t),t)−q (r)≈δq+q ξ. (4)
0 0
Introducing a new variable ζ ≡r2e−Φ0ξ, the radial pulsations are governedby
Wζ¨=(Pζ′)′+Qζ (5)
with the functions W(r), P(r), Q(r) determined for the equilibrium configuration
1
W ≡(ρ +p ) e3Ψ0+Φ0, (6)
0 0 r2
1
P ≡γp eΨ0+3Φ0, (7)
0r2
(p′)2 1 4p′ 8πG e2Ψ0
Q≡eΨ0+3Φ0 0 − 0 −(ρ +p ) p −Λ . (8)
(cid:20)ρ +p r2 r3 0 0 (cid:18) c4 0 (cid:19) r2 (cid:21)
0 0
The linear stability analysis can be realized by the standard assumption of the
displacement decomposition
ζ(r,t)=ζ(r)eiωt. (9)
ThenthedynamicequationreducestotheSturm–Liouvilleequationandtherelated
boundary conditions in the form
(Pζ′)′+(Q+ω2W)ζ =0, (10)
ζ eΦ0
′
finite as r→0, γp ζ →0 as r →R. (11)
r3 0 r2
The Sturm–Liouville equation (10) and the boundary conditions determine eigen-
frequenciesω andcorrespondingeigenmodesζ (r),wherei=1,2,...,n.Theeigen-
j i
value Sturm–Liouville (SL) problem can be expressed in the variational form8 as
the extremal values of
R Pζ′2−Qζ2 dr
ω2 = 0 (12)
R (cid:0) R (cid:1)
Wζ2dr
0
R
determine the eigenfrequencies ω and the corresponding functions ζ (r) are the
i i
eigenfunctions that have to satisfy the orthogonality relation
R
e3Ψ0−Φ0(p +ρ )r2ξ(i)ξ(j)dr =0. (13)
0 0
Z
0
3. Results and Conclusions
We applied the Sturm–Liouvilleapproachto spheres with uniform energydensity,1
andpolytropicspheres.3Thecaseofuniformspherescanbeproperlytakenasatest
bedofthedynamicalinstabilityproblem—althoughthesesolutionsoftheEinstein
equationsareofratherartificialcharacter,theyreflectquitewellthebasicproperties
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3
6
4
c
Γ
2
0
1 2 3 4
9(cid:144)8
R(cid:144)r
g
Figure1. Dependence ofthecriticalvalueofadiabatic indexγc onsphereradiusR. Full curve:
vanishingcosmologicalconstantλ=0;thenγc divergesasR→9rg/8fromabove.Dashed curve:
positive cosmological constant λ = 0.1, the point of divergence is shifted to 1.18421 > 9/8, and
γc,λ>0 > γc,λ=0. Dashed-dotted curve: negative cosmological constant λ = −0.1, the point of
divergenceisshiftedto1.07143<9/8,andγc,λ<0<γc,λ=0.
of very compact objects.9 The dependence of the critical value of adiabatic index
γ ≡ (∂lnp/∂lnn) = (n/p)(∆p/∆n) on configuration radius R is for the uniform
s
casegiveninFig.1(seealso12).ThepolytropiccaseistreatedindetailsbyStuchlík
and Hledík.10
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