Transverse Wave Propagation in Relativistic Two-fluid Plasmas around Schwarzschild-de Sitter Black Hole

0.41 MB·English

by  M. Hossain Ali

Transverse Wave Propagation in Relativistic Two-fluid
Plasmas around Schwarzschild-de Sitter Black Hole
M. Hossain Ali 1 and M. Atiqur Rahman2
Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh.
9
0
0
2

n
a
Abstract
J

0
1
We investigate transverseelectromagnetic wavespropagatingin a plasma influenced by the gravitational

]
field of the Schwarzschild-de Sitter black hole. Applying 3+1 spacetime split we derive the relativistic
c
q
two-fluid equations to take account of gravitationaleffects due to the event horizon and describe the set
-
r
g of simultaneous linear equations for the perturbations. We use a local approximation to investigate the
[
one-dimensional radial propagation of Alfv´en and high frequency electromagnetic waves. We derive the

2
v dispersion relation for these waves and solve it for the wave number k numerically.
5
9
5 PACS number(s): 95.30.Sf, 95.30.Qd, 97.60.Lf
4
.
7
0
8
0
:
v
i
X
r
a
1The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy. E-mail:
m−hossain−ali−[email protected] (Corresponding author).
2E-mail: [email protected]
1
1 Introduction
Black holes are still mysterious [1]. Physicists are grappling the theory of black holes, while astronomers
are searching for real-life examples of black holes in the universe [2]. There still exists no convincing
observationaldata whichcanproveconclusively the existence of blackholes inthe universe. Blackholes,
however, are not objects of direct observing. They must be observed indirectly through the effects they
exert on their environment. It is provedthat black holes exist on the basis of study of effects which they
exert on their surroundings.
Black holes greatly affect the surrounding plasma medium (which is highly magnetized) with their
enormous gravitationalfields. Hence plasma physics in the vicinity of a black hole has become a subject
ofgreatinterestinastrophysics. Intheimmediateneighborhoodofablackholegeneralrelativityapplies.
It is therefore of interest to formulate plasma physics problems in the context of general relativity.
Acovariantformulationofthe theorybasedonthe fluidequationsofgeneralrelativityandMaxwell’s
equationsincurvedspacetimehassofarprovedunproductivebecauseofthecurvatureoffour-dimensional
spacetime in the region surrounding a black hole. Thorne, Price, and Macdonald (TPM) [3, 4, 5, 6]
developed a method of a 3 + 1 formulation of general relativity in which the electromagnetic equations
and the plasma physics at least look somewhat similar to the usual formulations in flat spacetime while
taking accurate account of general relativistic effects such as curvature. In the TPM formulation, work
connected with black holes has been facilitated by replacing the holes event horizon with a membrane
endowedwithelectriccharge,electricalconductivity,andfinitetemperatureandentropy. Themembrane
paradigm is mathematically equivalent to the standard, full general relativistic theory of black holes so
far as physics outside the event horizon is concerned. But the formulation of all physics in this region
turns out to be very much simpler than it would be using the standard covariant approach of general
relativity.
Buzzi, et.al. [7, 8], using the TPM formalism described a general relativistic version of two-fluid
formulation of plasma physics and investigated the nature of plasma waves (transverse waves in [7], and
longitudinal waves in [8]) near the horizon of the Schwarzschild black hole. In this paper we apply
the formalism of Buzzi et.al. [7] to investigate the transverse electromagnetic waves propagating in a
plasma close to the Schwarzschild black hole in the de Sitter (dS) space, called the Schwarzschild-de
Sitter (SdS) black hole. The study is interesting because of the presence of cosmological constant (Λ).
Therehasbeenarenewedinterestincosmologicalconstantasitisfoundtobe presentintheinflationary
scenario of the early universe. In these scenario the universe undergoes a stage where it is geometrically
similar to dS space [9]. Among other things inflation has led to the cold dark matter. According to the
cold dark matter theory, the bulk of the dark matter is in the form of slowly moving particles (axions
or neutralinos). If the cold dark matter theory proves correct, it would shed light on the unification
of forces [10, 11]. Comprehensive review works of the cosmological constant or dark energy, including
the observational evidence of it and the problems associated with it, have been done by many authors
[12, 13, 14, 15, 16, 17, 18, 19, 20].
2
Inrecentyears,considerableattentionhasbeenconcentratedonthestudyofblackholesindSspaces.
This is motivated, basically, by the following two aspects: first, several different types of astrophysical
observations indicate that our universe is in a phase of accelerating expansion [21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. Associated with this
acceleration is a positive cosmological constant. Our universe therefore might approach a dS phase
in the far future. Second, similar as the AdS/CFT correspondence,an interesting proposal,the so-called
dS/CFT correspondence,has been suggestedthat there is a dual relationbetween quantum gravityon a
dS space and Euclidean conformal field theory (CFT) on a boundary of dS space [47, 48, 49, 50, 51, 52].
In view of the above reasons, our study of transverse wave propagation in relativistic two-fluid plasma
in the environment close to the event horizon of the SdS black hole is interesting. The result we have
obtained reduces to that of the Schwarzschild black hole [7] when the cosmologicalconstant vanishes.
This paper is organized as follows. In section 2, we summarize the 3+1 formulation of general rela-
tivity. In section 3, we describe the nonlinear two-fluidequations expressing continuity and conservation
ofenergyandmomentum. The two-fluidsarecoupledtogetherthroughMaxwell’sequationsfor the elec-
tromagnetic fields. In section 4, we restrict one-dimensional wave propagation in the radial z (Rindler
coordinate system) direction, and linearize the equations for wave propagation in section 5 by giving a
small perturbation to fields and fluid parameters. We express the derivatives of the unperturbed quan-
tities with respect to z. In section 6, we discuss the local or mean-field approximation used to obtain
numerical solutions for the wave dispersion relations. We describe the dispersion relation for the trans-
verse waves. We give the numerical procedure for determining the roots of the dispersion relation in
section 7. In section 8 we present the numerical solutions for the wave number k. Finally, in section 9,
we present our remarks. We use units G=c=k =1.
B
2 Formalism in 3+1 Spacetime
InthissectionweapplyTPMformulationtosplittheSchwarzschild-deSitterblackholespacetime,which
is the solution of Einstein equations with a positive Λ(= 3/ℓ2) term corresponding to a vacuum state
spherical symmetric configuration. The metric of the spacetime is asymptotically de Sitter and has the
form
ds2 = g dxµdxν
µν
1
= ∆2dt2+ dr2+r2(dθ2+sin2θdϕ2), (1)
− ∆2
where the metric function is
2M r2
∆2 =1 , (2)
− r − ℓ2
M being the mass of the black hole and the coordinates are defined such that t , r 0,
−∞ ≤ ≤ ∞ ≥
0 θ π, and 0 φ 2π. At large r, the metric (1) tends to the dS space limit. The explicit dS case
≤ ≤ ≤ ≤
is obtainedby setting M =0 while the explicit Schwarzchildcase is obtainedby taking the limit ℓ .
→∞
When ℓ2 is replace by ℓ2, the metric (1) describes an interesting nonrotating AdS black hole called the
−
3
SchwarzschildAnti-de Sitter (SAdS) black hole. The AdSblack holes playa majorrolein the AdS/CFT
correspondence[53,54,55,56]andtheyhavealsoreceivedinterestinthecontextofbrane-worldscenarios
based on the setup of Randall and Sundrum [57, 58].
The horizons of the SdS black hole are located at the real positive roots of ∆2(r) 1 (r r )(r
≡ ℓ2r − h −
r )(r r)=0, and there are more than one horizon if 0<Ξ<1/27 where Ξ=M2/ℓ2. The black hole
c
−−
(event) horizon r and the cosmologicalhorizon r are located, respectively, at
h c
2M π+ψ
r = cos , (3)
h
√3Ξ 3
2M π ψ
r = cos − , (4)
c
√3Ξ 3
where
ψ =cos 1(3√3Ξ). (5)
−
The negative root
2M ψ
r = cos (6)
− −√3Ξ 3
has no physical meaning. In the limit Ξ 0, one finds that r 2M and r ℓ, and it is obvious that
h c
→ → →
r > r , i.e., the event horizon is the smallest positive root. The spacetime is dynamic for r < r and
c h h
r > r . The two horizons coincide: r = r = 3M (extremal), when Ξ = 1/27, and the spacetime then
c h c
becomes the well known Nariai spacetime. Expanding r in terms of M with Ξ<1/27, we obtain
h
4M2
r 2M 1+ + , (7)
h ≈ ℓ2 ···
(cid:18) (cid:19)
thatis,theeventhorizonoftheSdSblackholeisgreaterthantheSchwarzschildeventhorizon,r =2M.
Sch
For Ξ>1/27, the spacetime is dynamic for all r >0, that is, the metric (1) then represents not a black
hole but an unphysical naked singularity at r=0.
The hypersurfaces of constant universal time t define an absolute three-dimensional space described
by the metric
1
ds2 =g dxidxj = dr2+r2(dθ2+sin2θdϕ2), (8)
ij ∆2
where the indices i,j range over 1,2,3 and refer to coordinates in absolute space. The fiducial observers
(FIDOs), the observers at rest with respect to this absolute space, measure their proper time τ using
clocks that they carry with them and make local measurements of physical quantities. Then all their
measuredquantitiesaredefinedasFIDO locallymeasuredquantitiesandallratesmeasuredby themare
measured using FIDO proper time. The FIDOs use a local Cartesian coordinate system with unit basis
vectors tangent to the coordinate lines
∂ 1 ∂ 1 ∂
er =∆∂r, eθˆ= r∂θ, eϕˆ = rsinθ∂ϕ. (9)
For a spacetime viewpoint this set of orthonormal vectors also includes the basis vector for the time
coordinate, given by
d 1 ∂
e = = , (10)
ˆ0 dτ α∂t
4
where α is the lapse function (or redshift factor) defined by
1
dτ 2M r2 2
α(r) = 1 . (11)
≡ dt − r − ℓ2
(cid:18) (cid:19)
The FIDO proper time τ functions as a local laboratory time, where the FIDOs have the role of “local
laboratories”. The coordinate time t slices spacetime in the way that the FIDOs would do physically.
The lapse function α, which governs the ticking rates of clocks and redshifts, plays the role of a
gravitationalpotential. It also provides the gravitationalacceleration felt by a FIDO [3, 4, 5, 6]:
1 M r
a= lnα= e . (12)
−∇ −α r2 − ℓ2 rˆ
(cid:18) (cid:19)
Therateofchangeofanyscalarphysicalquantityoranythree-dimensionalvectorortensor,asmeasured
by a FIDO, is defined by the derivative
D 1 ∂
+v , (13)
Dτ ≡ α∂t ·∇
(cid:18) (cid:19)
v being the fluid velocity measured locally by a FIDO.
3 Two-Fluid Equations in 3+1 Formalism
We consider two-component plasma such as an electron-positron plasma or electron-ion plasma. The
continuity equation for each of the fluid species in the 3+1 notation is
∂
(γ n )+ (αγ n v )=0, (14)
s s s s s
∂t ∇·
where s is 1 for electrons and 2 for positrons (or ions). For a perfect relativistic fluid of species s in
three-dimensions, the energy density ǫ , the momentum density S , and stress-energy tensor Wjk are
s s s
given by
ǫ =γ2(ε +P v2), S =γ2(ε +P )v , Wjk =γ2(ε +P )vjvk+P gjk, (15)
s s s s s s s s s s s s s s s s s
where v is the fluid velocity, n is the number density, P is the pressure, and ε is the total energy
s s s s
density defined by
ε =m n +P /(γ 1), (16)
s s s s g
−
the gas constant γ being 4/3 as T and 5/3 as T 0.
g
→∞ →
The ion temperature profile is closely adiabatic and it approaches 1012K near the horizon [59]. Far
from the event horizon, electron (positron) temperatures are essentially equal to the ion temperatures.
Buttheelectronsclosertothehorizonareprogressivelycooledtoabout108 109K bymechanismssuch
−
as multiple Compton scattering and synchrotron radiation. Using the conservation of entropy one can
express the equation of state in the form:
D P
s
=0, (17)
Dτ nγg
(cid:18) s (cid:19)
5
where D/Dτ =(1/α)∂/∂t+v . For a relativistic fluid, the full equation of state as measured in the
s
·∇
fluid’s rest frame, is given by [60, 61]:
(1)′
P iH (im n /P )
ε=m n +m n s 2 s s s , (18)
s s s s"msns − H2(1)(imsns/Ps) #
where the H(1)(x) are Hankel functions.
2
The fluid quantities in (15) take the following form in the electromagnetic field:
1 1
ǫ = (E2+B2), S = E B,
s s
8π 4π ×
1 1
Wjk = (E2+B2)gjk (EjEk+BjBk). (19)
s 8π − 4π
Energy and momentum conservation equations are given by [3, 4, 5]
1 ∂
ǫ = S +2a S , (20)
s s s
α∂t −∇· ·
1 ∂ 1
S = ǫ a (αW↔ ). (21)
s s s
α∂t − α∇·
When the two-fluid plasma couples to the electromagnetic fields, Maxwell’s equations take the following
3+1 form:
B = 0, (22)
∇·
E = 4πσ, (23)
∇·
∂B
= (αE), (24)
∂t −∇×
∂E
= (αB) 4παJ, (25)
∂t ∇× −
where the charge and current densities are respectively defined by
σ = γ q n , J= γ q n v . (26)
s s s s s s s
s s
X X
Using (16) and (22–25), the energy and momentum conservation equations (20) and (21) can be
rewritten for each species s in the form
1 ∂ 1 ∂
P [γ2(ε +P )] [γ2(ε +P )v ]+γ q n E v +2γ2(ε +P )a v =0, (27)
α∂t s− α∂t s s s −∇· s s s s s s s · s s s s · s
and
1 ∂
γ2(ε +P ) +v v + P γ q n (E+v B)
s s s α∂t s·∇ s ∇ s− s s s s×
(cid:18) (cid:19)
1 ∂
+v γ q n E v + P +γ2(ε +P )[v (v a) a]=0, (28)
s s s s · s α∂t s s s s s s· −
(cid:18) (cid:19)
respectively. Although these equations are valid in a FIDO frame, they reduce for α = 1 to the corre-
sponding special relativistic equations [62] which are valid in a frame in which both fluids are at rest.
The transformation from the FIDO frame to the comoving (fluid) frame involves a boost velocity, which
is simply the freefall velocity onto the black hole, given by
vff =(1 α2)12. (29)
−
6
The relativistic Lorentz factor then becomes γ (1 v2) 1/2 =1/α.
boost ≡ − ff −
The two-fluid equations in SdS coordinates form the basis of the numerical procedure for solving the
linear two-fluid equations; but they cannot be evaluated analytically. However, the Rindler coordinate
system, in which space is locally Cartesian, provides a good approximation to the SdS metric near
the event horizon. Without the complication of explicitly curved spatial three-geometries, the essential
features of the horizon and the 3+1 split are preserved.
The SdS metric (1) in Rindler coordinates is approximated by
ds2 = α2dt2+dx2+dy2+dz2, (30)
−
where
π
x=r θ , y =r ϕ, z =2r ∆. (31)
h h h
− 2
(cid:16) (cid:17)
The standard lapse function in Rindler coordinates becomes α=z/2r , where r is the event horizon of
h h
the black hole.
4 Radial Wave Propagation in One-Dimension
We consider one-dimensional wave propagation in the radial z direction. Introducing the complex vari-
ables
v (z,t)=u (z,t), v (z,t)=v (z,t)+iv (z,t),
sz s s sx sy
B(z,t)=B (z,t)+iB (z,t), E(z,t)=E (z,t)+iE (z,t), (32)
x y x y
and setting
i
v B v B = (v B v B),
sx y− sy x 2 s ∗− s∗
i
v E v E = (v E v E), (33)
sx y − sy x 2 s ∗− s∗
where the denotes the complex conjugate, one could write the continuity equation (14) in the form
∗
∂ ∂
(γ n )+ (αγ n u )=0, (34)
s s s s s
∂t ∂z
and Poisson’s equation (23) as
∂E
z
=4π(q n γ +q n γ ). (35)
1 1 1 2 2 2
∂z
From the e and e components of (24) and (25), one could obtain
xˆ yˆ
1 ∂B ∂
= i a E, (36)
α ∂t − ∂z −
(cid:18) (cid:19)
∂E ∂
i = α a B i4πeα(γ n v γ n v ). (37)
2 2 2 1 1 1
∂t − ∂z − − −
(cid:18) (cid:19)
Differentiating (37) with respect to t and using (36), we obtain
∂2 3α ∂ ∂2 1 ∂
α2 + + E =4πeα (n γ v n γ v ). (38)
∂z2 2r ∂z − ∂t2 (2r )2 ∂t 2 2 2− 1 1 1
(cid:18) h h (cid:19)
7
The transverse component of the momentum conservation equation is obtained from the e and e
xˆ yˆ
components of (28) as follows:
Dv 1 ∂P
ρ s =q n γ (E iv B +iu B) u v ρ a v q n γ E v + s , (39)
s s s s s z s s s s s s s s s
Dτ − − − · α ∂t
(cid:18) (cid:19)
where
1
E v = (Ev +E v )+E u ,
· s 2 s∗ ∗ s z s
and ρ is the total energy density defined by
s
ρ =γ2(ε +P )=γ2(m n +Γ P ) (40)
s s s s s s s g s
with Γ =γ /(γ 1).
g g g
−
5 Linearization
We apply perturbationmethod to linearize the equations derivedin the preceding section. We introduce
the quantities
u (z,t) = u (z)+δu (z,t), v (z,t)=δv (z,t),
s os s s s
n (z,t) = n (z)+δn (z,t), P (z,t)=P (z)+δP (z,t),
s os s s os s
ρ (z,t) = ρ (z)+δρ (z,t), E(z,t)=δE(z,t),
s os s
B (z,t) = B (z)+δB (z,t), B(z,t)=δB(z,t), (41)
z o z
where an applied magnetic field has been chosen to lie along the radial e direction. The relativistic
zˆ
Lorentz factor is also linearized such that
1
γs =γos+δγs, where γos = 1−u2os −2 , δγs =γo3suos·δus. (42)
(cid:0) (cid:1)
Near the eventhorizonthe unperturbed radialvelocity for eachspecies as measuredby a FIDO along e
zˆ
is assumed to be the freefall velocity so that
uos(z)=vff(z)=[1 α2(z)]21. (43)
−
It follows, from the continuity equation (34), that
r2αγ n u =const.=r2α γ n u ,
os os os h h h h h
where the values with a subscript h are the limiting values at the event horizon. The freefall velocity at
the event horizonbecomes unity so that u =1. Since u =v , γ =1/α; andhence αγ =α γ =1.
h os ff os os h h
Also, because v =(r /r)1/2χ, the number density for each species can be written as follows:
ff h
1
n (z)= n v3(z), (44)
os χ4 hs ff
where
1 1
r3 2 1 1 −2
χ= 1 1+r + . (45)
h
− r r r r r
(cid:18) h c −(cid:19) (cid:20) (cid:18) c −(cid:19)(cid:21)
8
When ℓ2 , (44) reduce to the Schwarzschild result [7]. The equation of state (17) and (44) lead to
→ ∞
write the unperturbed pressure in terms of the freefall velocity as
1
P (z)= P v3γg(z). (46)
os χ4γg hs ff
Since P =k n T , then with k =1 the temperature profile is
os B os os B
1
Tos(z)= χ4(γg−1)Thsvff3(γg−1)(z). (47)
The unperturbed magnetic field, chosen to be in the radial z direction, is parallel to the infall fluid
velocity u (z)e for each fluid. It thus does not experience effects of spatial curvature along with the
os zˆ
infall fluid velocity. The magnetic field depends on the radial coordinate r because of flux conservation.
Since B =0,itfollowsthatr2B (r)=const.,fromwhichonecouldobtaintheunperturbedmagnetic
o o
∇·
field in terms of the freefall velocity as
1
B (z)= B v4(z). (48)
o χ4 h ff
Since
dv α 1
ff
= , (49)
dz −2r v
h ff
we have
du α 1 dB 4α B
os o o
= , = ,
dz −2r v dz −2r v2
h ff h ff
dn 3α n dP 3α γ P
os os os g os
= , = . (50)
dz −2r v2 dz −2r v2
h ff h ff
When the linearized variables from (41) and (42) are substituted into the continuity equation and
products of perturbation terms are neglected, it follows that
∂ ∂ u du ∂ 1
γ +u α + os +γ2 α os δn + α + (n γ u )
os ∂t os ∂z 2r os dz s ∂z 2r os os os
(cid:18) h (cid:19) (cid:18) h(cid:19)
∂ ∂ 1 1 dn du
+n γ3 u +α + +α os +3γ2 u os δu =0. (51)
os os os∂t ∂z 2r n dz os os dz s
(cid:20) h (cid:18) os (cid:19)(cid:21)
The conservation of entropy, (17), on linearization, gives
γ P
g os
δP = δn . (52)
s s
n
os
Then from the total energy density, (40), it follows that
ρ γ2 γ P
δρ = os 1+ os g os δn +2u γ2 ρ δu , (53)
s n ρ s os os os s
os (cid:18) os (cid:19)
whereρ =γ2 (m n +Γ P ). Linearizingthetransversepartofthemomentumconservationequation,
os os s os g os
differentiating it with respect to t, and then substituting from (36), we derive
∂ ∂ u iαq γ n B ∂δv αq γ n ∂ ∂ u
os s os os o s s os os os
αu + + αu + + δE =0. (54)
os os
∂z ∂t − 2r ρ ∂t − ρ ∂z ∂t 2r
(cid:18) h os (cid:19) os (cid:18) h(cid:19)
When linearized, Poisson’s equation (35) and (38), respectively, give
∂δE
z =4πe(n γ n γ )+4πe(γ δn γ δn )+4πe(n u γ3 δu n u γ3 δu ), (55)
∂z o2 o2− o1 o1 o2 2− o1 1 o2 o2 o2 2− o1 o1 o1 1
∂2 3α ∂ ∂2 1 ∂δv ∂δv
α2 + + δE =4πeα n γ 2 n γ 1 . (56)
∂z2 2r ∂z − ∂t2 (2r )2 o2 o2 ∂t − o1 o1 ∂t
(cid:18) h h (cid:19) (cid:18) (cid:19)
9
6 Dispersion Relation
We restrict our consideration to effects on a local scale for which the distance from the horizon does
not vary significantly. We apply a local (or mean-field) approximation for the lapse function α(z) and
hence forthe equilibriumfields andfluidquantities. Ifthe plasmais situatedrelativelyclosetothe event
horizon, α2 1, then a relatively small change in distance z will make a significant difference to the
≪
magnitude of α. Thus it is important to choose a sufficiently smallrange in z so that α(z) does not vary
much.
We consider thin layers in the e direction, each layer with its own α , where α is some mean value
zˆ o o
ofαwithinaparticularlayer. Byconsideringalargenumberoflayerswithinachosenrangeofα values,
o
amorecompletepicturecanthenbe builtup. However,suchalocalapproximationimposesarestriction
onthemagnitudeofthewavelengthand,hence,onthewavenumberk. Itisassumedthatthewavelength
is small in comparison with the range over which the equilibrium quantities change significantly. Then
the wavelengthmust be smaller in magnitude than the scale of the gradient of the lapse function α, i.e.,
1
∂α −
λ< =2r ζ5.896 105cm,
h
∂z ≃ ×
(cid:18) (cid:19)
or, equivalently,
2π
k> ζ 11.067 10 5cm 1, 1 ζ 1.5,
− − −
2r ≃ × ≤ ≤
h
for ablackhole ofmass 1M . The valuefor ζ =1.5correspondsto the extremalSdSblack hole,while
∼ ⊙
the value for ζ =1 corresponds to the Schwarzschild black hole.
The hydrodynamical approach used in this work has some disadvantages, such as it is essentially
a bulk, fluid approach. So the microscopic behavior of the two-fluid plasma is treated in a somewhat
approximatemannerviatheequationofstate. Itmeansthattheresultsarereallyonlystrictlyvalidinthe
long wavelength limit. However,the restriction, imposed by the local approximation, on the wavelength
is not too severe and permits the consideration of intermediate to long wavelengths so that the small k
limit is still valid.
Theunperturbedfieldandfluidquantitiesareassumednottobeconstantwithrespecttoα(z). Then,
usingthelocalapproximationforα,thederivativesoftheequilibriumquantitiescanbeevaluatedateach
layer for a given α . In the local approximation for α, α α is valid within a particular layer. Hence,
o o
≃
the unperturbed fields and fluid quantities and their derivatives, which are functions of α, take on their
corresponding “mean-field”values for a given α . Then the coefficients in (51), (54), and (55) become
0
constants within eachlayer,evaluated at eachfixed mean-fieldvalue, α=α . So it is possible to Fourier
o
transform the equations with respect to z, using plane-wave-type solutions for the perturbations of the
form ei(kz ωt) for each α layer.
− o
∼
When Fourier transformed, (54) and (56) turn out to be
iu α q γ n B q γ n iu
os o s os os o s os os os
ω α ku ω+ + δv iα α ku ω δE =0, (57)
o os s o o os
− 2r ρ − ρ − − 2r
(cid:18) h os (cid:19) os (cid:18) h(cid:19)
i4πeα ω(n γ δv n γ δv )
o o2 o2 2 o1 o1 1
δE = − . (58)
α k(α k i3/2r ) ω2 1/(2r )2
o o h h
− − −
10