Resonant interaction between gravitational waves,
electromagnetic waves and plasma flows
arXiv:gr-qc/0302039v2 27 Aug 2003
Martin Servin and Gert Brodin
Department of Physics, Umeå University, S-901 87 Umeå, Sweden
Accepted for publication in Physical Review D
Scheduled issue: 15 August 2003.
In magnetized plasmas gravitational and electromagnetic waves may interact coherently and exchange energy between themselves and with plasma flows. We derive the wave interaction equations
for these processes in the case of waves propagating perpendicular or parallel to the plasma background magnetic field. In the latter case, the electromagnetic waves are taken to be circularly
polarized waves of arbitrary amplitude. We allow for a background drift flow of the plasma components which increases the number of possible evolution scenarios. The interaction equations are
solved analytically and the characteristic time scales for conversion between gravitational and electromagnetic waves are found. In particular, it is shown that in the presence of a drift flow there
are explosive instabilities resulting in the generation of gravitational and electromagnetic waves.
Conversely, we show that energetic waves can interact to accelerate particles and thereby produce a
drift flow. The relevance of these results for astrophysical and cosmological plasmas is discussed.
PACS numbers:
I.
INTRODUCTION
In empty and flat space-time linear gravitational wave
(GW) perturbations and electromagnetic waves (EMWs)
do not interact when propagating in the same direction.
If, however, there is a background electromagnetic field
or a background curvature present the two waves may
couple, interact and exchange energy [1]-[3].
Although GWs typically interact weakly with matter,
it is of interest to consider how GW-EMW interaction
may be altered by the presence of a magnetized plasma
– which is a common state of matter in astrophysical
and cosmological scenarios. The interaction may; lead to
production or modification of observable EMWs, as considered in the Refs. [4]-[5], energize the plasma by exciting Alfvén and magnetosonic waves [6]-[7] which may be
of importance in more complex processes such as supernovae explosions, as discussed in Ref. [8]. Furthermore,
it may even modify the expected GW signals [9] that are
currently under attempted detection and will, hopefully,
provides us with a new window through which the universe can be observed. Studying nonlinearly interacting
waves may also reveal new types of instabilities that cannot be found using conventional linear stability theory,
see e.g. Ref. [10].
Wave interactions are most efficient if they are resonant
(coherent), i.e. if the frequencies satisfy certain matching conditions and the relative wave phase remains unchanged for a long time. We will thus not consider incoherent (non-resonant) wave interaction, in which case energy conversion takes place on a much longer time scale.
Resonant interaction in vacuum or in the presence of a
background electromagnetic field requires only that the
frequencies should coincide since both GWs and EMWs
propagate along null geodesics and therefore have identical phase velocity equal to c, the speed of light. Consider-
ing GW-EMW interaction in a “medium” the occurrence
of such resonances is more rare because the phase velocity
for GW is (in the high frequency approximation and to
a very high accuracy) still equal to c, whereas for EMWs
this occurs only for particular wave frequencies (or in the
limit of very high frequencies). For this reason resonant
GW-EMW interaction has been studied in Ref. [6] for
the case of multiple EMWs, where two or more EMWs –
that seperately do not propagate along null geodesics –
together produce perturbations that are resonant with a
GW. See also Ref. [11], where generation of GWs with
the same mechanism was considered but for interacting
sound waves.
Ref. [12] showed that resonant interaction between a
GW and a single low-frequency EMW (a ”magnetosonic
wave”, in the terminology of their formalism) can be realized for “incompressible” relativistic magnetofluids, i.e.
with sound velocity equal to c, but no calculations regarding the coupling strength where made. In Ref. [7]
the excitation of magnetosonic waves by GWs was considered with particular focus on almost coherent waves.
In Ref. [13] kinetic theory was used to derive, among
other things, dispersion relations governing the coupling
between a GW and an EMW propagating perpendicular
to a background magnetic field. From the dispersion relations in Ref. [13] it is clear that the waves are resonant
if the wave frequencies coincide with the electron plasma
frequency (defined below) but the strength of the interaction cannot easily be deduced from those results.
In this paper we study the interaction between GWs of
small amplitude and EMWs in a magnetized plasma modelled by multifluid equations. In the case of perpendicular propagation with respect to the background magnetic
field, we take the EMW to be a high-frequency extraordinary electromagnetic wave (using standard terminology of plasma wave theory), the only wave in this case
2
for which we may have phase velocity equal to c (without introducing a more complicated background state).
The main focus of our study is on the case of parallel
propagation, where we make use of an exact (when neglecting gravitational effects) EMW solution that can be
reduced to e.g. high-frequency electromagnetic waves,
whistler waves, low-frequency Alfvén waves or waves in
electron-positron plasmas. This solution also allows the
presence of a relative drift flow of the fluids constituting
the plasma. The inclusion of a drift in the background
state is important for at least three reasons, when considering GW-EMW perturbations. Firstly, it increases the
number of ways that resonant interaction can occur, secondly, it alters the coupling strength between the waves
and, most importantly, it supplies the system with free
energy. This later fact means, as we will demonstrate,
that the background configuration may be unstable, leading to simultaneous generation of GWs and EMWs.
The paper is organized as follows. In section II we
give an overview of certain principles of resonant wave
interactions. The basic equations are presented in section III, i.e. the Einstein-Maxwell system together with
multifluid equations. We demonstrate in section IV that,
provided the conditions for the high-frequency approximation are fulfilled, GWs can be taken to be in the transverse traceless gauge also in the presence of matter and
we derive the corresponding evolution equations, describing how the GWs are coupled to the matter perturbation.
In section V we derive evolution equations for the EMWs,
including the effect of GWs on the EMWs and expressions
for the wave energy density. It is shown that circularly
polarized EMWs can have negative wave energy for certain background parameter values and simultaneously be
resonant with a GW. Resonant interaction between the
GWs and EMWs is studied in section VI. Wave interaction equations are derived and their solutions are examined, whereby the characteristic time scales for conversion between gravitational and electromagnetic waves
are found. For the case of negative energy EMWs the
coupling to GWs give rise to explosive EMW-GW instabilities or, with different initial conditions, acceleration
of plasma flow. The results are summarized and further
discussed in section VII, including the relevance for astrophysical and cosmological plasmas.
II.
OVERVIEW OF RESONANT WAVE
INTERACTIONS
The natural wave modes of the gravitational field in
the high-frequency approximation [16] and of the electromagnetic field in magnetized plasmas [17] are wellknown. Based on these solutions, we choose a set of wave
perturbations, Fn , that we represent with the following
complex notation: Fn = fn + c.c., where fn = fn eiθ(i) ,
fn is the complex amplitude, θn the (real) wave phase
and c.c. stands for complex conjugation of the preceding
term. Letting f1 denote the GW (e.g. some linear combi-
nation of components of the perturbed metric tensor) and
f2 , say, the electric field of the EMW. From the governing equations, the Einstein-Maxwell-plasma fluid system
presented in section III, we derive evolution equations for
these wave perturbations. As can be found from section
IV and V, these equations take the form
D̂1 f1 = S1 (f2 , f1 ) + ...
(1)
D̂2 f2 = S2 (f1 , f2 ) + ...
(2)
where D̂n is the linear wave propagator and the interaction source term Sn (fm , fn ) is an algebraic expression [14] involving fm (and possibly also fn ) such that
Sn ≡ Sn eiφn is resonant with fn , i.e. the relative phase
θn −φn is constant. The dots in Eqs. (1) and (2) indicates
that there are in general also terms that are non-resonant
with fn . Non-resonant terms are discarded, as they only
have an effect on a much larger time-scale. The source
term S1 have the structure such that in the absence of
EMWs, f2 = 0, S1 vanishes and Eq. (1) reduces to the
free wave equation for GWs, [∂t2 − ∂z2 ]f1 = 0. Conversely,
if there are no GWs then S2 vanishes and Eq. (2) reduces to the plasma EMW equation that corresponds to
the assumed perturbation.
Our immediate purpose is to determine the coupling
strength and thereby the characteristic time-scales for
the wave interactions. We do this by studying spatially uniform monochromatic waves propagating in a
(locally) static and uniform background (the relevance
for less idealized situations will be discussed in Section VII). The interaction will then result in time dependent wave amplitudes, governed by the derived evolution equations. Since the source terms are small –
either proportional to the gravitational coupling constant or to the small GW amplitude – we assume that
|ωn fn | ≫ |∂t fn |. This means that, to lowest order, the
waves will propagate freely according to the dispersion
relation Dn (−iωn , ikn ) = 0 and, to the next level of accuracy, have a slowly evolving amplitude. Applying this,
the differential operators in Eqs. (1) and (2) can be approximated by D̂ ≈ D(−iω, ik) + (∂ω D)∂˜t , where the
tilde notation on ∂˜t means that the derivative acts only
on the amplitude. Higher order derivatives are neglected.
The evolution equations (1) and (2) thus becomes
−1
∂˜t f1 = (∂ω1 D1 ) S1 (f2 , f1 )
−1
∂˜t f2 = (∂ω2 D2 ) S2 (f1 , f2 )
(3)
(4)
and we refer to these as the interaction equations. Determining S1 and S2 and studying the solutions of these
equations are the main purposes of this paper.
III.
BASIC EQUATIONS
We take the gravitational and electromagnetic field to
be governed by the Einstein field equations
Gab = κTab
(5)
3
Maxwell and fluid equations can be formulated as [4], [15]
and the Maxwell equations
∇a F ab = j b
∇a Fbc + ∇b Fca + ∇c Fab = 0
(6)
(7)
where Gab is the Einstein tensor, κ = 8πG, Tab is
the energy-momentum tensor, Fab is the electromagnetic
field tensor, j a is the total four-current density and ∇
denotes covariant derivative. We use units where the velocity of light in vacuum is c = 1. The metric signature is
(− + ++) and for the indices we use a, b, c, ... = 0, 1, 2, 3
and i, j, k, ... = 1, 2, 3. The matter present in the interaction region is a magnetized plasma for which we choose
a multifluid description. This means that we take the
plasma to consist of a number of interpenetrating charged
fluids, one for each species of particles that constitutes
the plasma. The fluids interact only through the electromagnetic and gravitational field, i.e. we neglect effects
of particle collisions as can be done for most plasmas.
The appropriate fluid equations are then, for each fluid
component (i), the equation of mass conservation
∇a (m(i) n(i) ua(i) )
=0
(8)
and the momentum equation
m(i) n(i) ua(i) ∇a ub(i) = −∇b p(i) + q(i) n(i) ua(i) Fab
(9)
where n(i) is the proper particle number density, ua(i) is
the fluid four-velocity, m(i) is the particle mass, p(i) is
the pressure and q(i) is the particle electric charge. For
closure this should be supplemented by some equation of
state which we do not specify here.
The total current
P
q
n(i) ua(i) and the
density is then given by j a =
(i)
(i)
(f l)
(i)
and the electromagnetic field contribution is
(em)
= Fac Fbc − 14 gab F cd Fcd
mn(e0 + v · ∇)γv = −γ −1 ∇p
+ qn(E + v × B) + mng
(12)
(13)
(14)
(15)
(16)
(17)
where we have introduced an Euclidean three-vector notation E = (E 1 , E 2 , E 3 ) etc., ∇ = (e1 , e2 , e3 ), ρ = j 0
and coordinate velocity, v a , so that ua = γv a with
1
γ = (1 − vi v i )− 2 . The effect of the gravitational field is
included in the form of effective charges, currents, gravitational forces etc. that were introduced above, and they
are given by [15]
ρE ≡ −Γiji E j − ǫijk Γ0ij Bk
ρB ≡ −Γiji B j + ǫijk Γ0ij Ek
h
jE ≡ − Γi0j − Γij0 E j + Γj0j E i
ei
−ǫijk Γ0j0 Bk + Γm
jk Bm
h
jB ≡ − Γi0j − Γij0 B j + Γj0j B i
ei
+ǫijk Γ0j0 Ek + Γm
jk Em
i
i
i
∆n ≡ −γn Γ 0i + Γ 00 vi + Γ ji v j
g ≡ −γ Γi00 + Γi0j + Γij0 v j + Γijk v j v k ei
where Γabc are the Ricci rotation coefficients.
(em)
energy-momentum tensor by Tab = Tab + Tab , where
the fluid contribution is
X
(f l)
(m(i) n(i) + p(i) )u(i)a u(i)b + p(i) gab
(10)
Tab =
Tab
∇ · E = ρ + ρE
∇ · B = ρB
e0 E − ∇ × B = −j − jE
e0 B + ∇ × E = −jB
e0 (γn) + ∇ · (γnv) = ∆n
(11)
From now on we will omit the species index, (i), unless
there can be any confusion.
It is practical to rewrite the Maxwell and fluid equations in a more convenient form. In particular we intend to introduce an orthonormal frame so that the interpretation of the quantities becomes more direct. First,
we note that by introducing an observer four velocity,
V a , the electromagnetic field can be decomposed, relative to V a , into an electric part, E a , and a magnetic
part, B a , according to F ab = V a E b − V b E a + ǫabc Bc ,
where Ea = Fab V b , Ba = 21 ǫabc Fbc , ǫabc = V d ǫabcd
and ǫabcd
p is the 4 dimensional volume element with
ǫ0123 = | det g|. If one chooses an orthonormal frame
(in which V a = (1, 0)) with contravariant basis {ea }, the
IV.
GRAVITATIONAL WAVE EVOLUTION
In this section we show that, in the high-frequency
approximation [16] for GWs propagating in matter, the
GWs can be taken to be in the transverse trace-less (TT)
gauge and we give the corresponding evolution equation.
In flat space-time linearized gravitational waves can always be taken to be in the TT gauge. In vacuum spacetime with a background curvature this is in general only
possible in the high frequency limit, where the ratio of
the GW wavelength and the characteristic background
length scale (radius of curvature) tends to zero. The
advantages of having GWs in the TT-gauge is that the
evolution equation becomes more simple, the polarization state is more clear and, most important for us here,
it reduces the amount of algebra when calculating the effect on a plasma, e.g. when calculating the gravitational
source terms in the EMW evolution equation. However,
the (perturbed) energy-momentum tensor should have
the same properties as the perturbed Einstein tensor and
this will in general not be the case if one just assumes
the TT-gauge. Nevertheless, we demonstrate that the
TT gauge can also be applied in the presence of matter
4
provided the conditions for the high frequency approximation is fulfilled. This result seems to have been overlooked in the litterature.
Let the background gravitational field and the unperturbed plasma fulfill the Einstein field equations,
(0)
(0)
Gab = κTab , and introduce a perturbation so that
(0)
(0)
Gab = Gab + δGab and Tab = Tab + δTab . We assume
the high frequency approximation so that the background
space-time can be taken to be Minkowski space-time [16]
and put gab = ηab + hab , where ηab is the Minkowski
metric and hab the small metric perturbation that should
obey
δGab = κδTab
(18)
We limit ourself to linear gravitational perturbations.
The perturbed Einstein tensor, linearized in hab , is given
by
δGab [h̄] = − 12 ∂ c ∂c h̄ab + ∂ c ∂(b h̄a)c − 12 ηab ∂ c ∂ d h̄cd (19)
h̄ab ≡ hab − 21 ηab h and the brackets ( ) stands for symmetrization with respect to the enclosed indices. In vacuum, the next step would be to apply the Lorentz gauge
condition, ∂ b h̄ab = 0. Instead of this, we split h̄ab according to
¯
h̄ab = h̄L
ab + fab
(20)
where h̄L
ab is the (maximal) part of h̄ab that fulfills the
Lorentz gauge condition and the other part, f¯ab , containing terms that cannot be fitted into h̄L
ab (both terms
should be symmetric). With this splitting Eq. (18) reads
¯
− 21 ∂ c ∂c h̄L
ab + δGab [f ] = κδTab
(21)
Next we assume that the metric perturbation (and the
perturbed energy-momentum
tensor) is of the form hab =
c
hab (xd )eikc x + c.c. with the dispersion relation ka k a =
0. The dependency of hab on xd is assumed weak, i.e.
|∂c hab | ≪ |kc hab |. Computing δGab [f¯] gives to lowest
order
δGab [f¯] = −k c k(b f¯a)c + 12 ηab k c k d f¯cd
(22)
where terms involving slow derivatives (derivatives on the
weakly varying amplitude) have been neglected. From
this expression it follows, for perturbations with four
wave vector k a = (ω, 0, 0, ω), that δG12 [f¯] = δG21 [f¯] = 0
and δG11 [f¯] = δG22 [f¯]. Furthermore, nonzero components of f¯ab should be of order f¯ ∼ κδT /ω 2 (which must
be much smaller than unity in order not to invalidate
the linearization in hab ) for the field equations (21) to be
fulfilled.
We now turn to the terms in h̄L
ab . A gauge transformab
a′
a
a
a
tion x = x +ξ with ξ ≪ 1 and ξ a = −iξ˜a (xc )eikb x +
c.c. alters the metric perturbation by
h̄a b = h̄ab − ξ̃ a k b − ξ̃ b k a + η ab ξ˜c kc
′ ′
(23)
where it is understood that the amplitude, ξ˜a (xc ), is
weakly dependent on xc in the same way as h̄ab . We can,
however, neglect slow derivatives on ξ a as such terms
only produce small corrections to the f¯ab part or, finally
in the wave evolution equations, terms of second order
slow derivatives. Consequently, ξ a can, as in the vacuum
case, be chosen as linear combinations of h̄L
ab such that
′ ′
¯ab
h̄a b = h̄ab
TT + f
(24)
where h̄ab
T T is “in the transverse traceless gauge”, i.e.
12
21
h̄11
=
−
h̄22
TT
T T ≡ hA and h̄T T = h̄T T ≡ hB . The evolution equations for hA and hB follows from Eq. (21) by
subtracting the 11 and the 22 components and by – for
aestethical reasons only – adding the 12 and 21 components. The result is
∂ c ∂c hA = −κ (δT11 − δT22 )
∂ c ∂c hB = −κ (δT12 + δT21 )
(25)
(26)
which depends critically on the fact that δG12 [f¯] =
δG21 [f¯] = 0 and δG11 [f¯] = δG22 [f¯].
To summarize, for the part of the metric perturbation that represents GWs propagating in the x3 -direction,
h̄ab
T T , we have derived evolution equations that describes
how the waves are coupled to the perturbed energymomentum tensor. Clearly, nonvanishing δT11 − δT22
and δT12 can potentially act to drive or damp GWs.
The remaining part of the perturbed metric, given by
f¯ab , should be included for the sake of consistency. The
perturbed energy-momentum that is not accounted for
in Eqs. (25) and (26) produces some gravitational response, which is given by δGab [f¯] and found from the
remaining components of Eqs. (21) (due to Eqs. (21)
and (22) these are purely algebraic equations). As f¯ab
represents the self-gravitation of the energy-momentum
perturbation it is in general important for the evolution of the matter and electromagnetic field, but since
f¯ ∼ κδT /ω 2 it is negligible when we consider waves in
the high-frequency limit. We conclude that in the highfrequency limit and for slowly evolving wave amplitudes
the GWs can be taken to be in the TT-gauge and the
metric perturbation hab = hTabT evolves according to Eqs.
(25) and (26). In this limit the energy density carried by
GWs follows directly from the Landau-Lifshitz pseudo
energy-momentum tensor [18]
EGW =
1
2κ
(∂t hA )2 + (∂t hB )2
(27)
When studying the effect of GWs on a magnetized
plasma it is practical to use tetrad formalism and introduce an orthonormal frame. A contravariant basis that
corresponds to GWs in the TT gauge with propagation
in the x3 ≡ z direction is given by
e0 = ∂t , e1 = (1 − 21 hA )∂x − 21 hB ∂y ,
e2 = (1 + 12 hA )∂y − 12 hB ∂x , e3 = ∂z
5
and the gravitationally induced terms in Eqs. (12)-(17)
are (displaying the nonzero terms only)
g = − 21 γ(1 − vz ) [vx ∂t hA + vy ∂t hB )] e1
− 12 γ(1 − vz ) [vx ∂t hB − vy ∂t hA ] e2
− 12 γ (vx2 − vy2 )∂t hA + 2vx vy ∂t hB e3
jE = − 21 [(Ex − By )∂t hA + (Ey + Bx )∂t hB ] e1
−
jB =
−
1
2
[−(Ey + Bx )∂t hA + (Ex − By )∂t hB ] e2
− 21 [(Ey + Bx )∂t hA − (Ex − By )∂t hB ] e1
1
2 [(Ex − By )∂t hA + (Ey + Bx )∂t hB ] e2
(30)
In this section we present the evolution equations for
the EMWs that we are considering. Since the GW fourwave vector is assumed to fulfill the dispersion relation
ka k a = 0, they have phase velocity equal to unity. The
EMWs that can interact resonantly with the GWs should
then also have phase velocity equal to unity. For simplicity, we restrict our study to the case of perpendicular and parallel propagation to the background magnetic
field, respectively.
The plasma is assumed neutral and uniform, with a
constant background magnetic field, B(0) . The plasma
2
frequency is denoted by ωp(i) ≡ (n(i) q(i)
/m(i) )1/2 and the
cyclotron frequency by ωc(i) ≡ q(i) B(0) /γ(i) m(i) , where
n(i) is the unperturbed proper particle number density.
We also allow the presence of a background drift flow
in the direction of the magnetic field. The inclusion of
the drift flow provides the system with free energy that
may be transferred to the GWs and EMWs during the
interaction.
In magnetized plasmas there are many electromagnetic
wave modes and their properties are well known for flat
space-time and for linear (small amplitude) waves (see
e.g. [17]). In order to obtain the effect of GWs on the
EMWs we derive their evolution equations from the basic
equations (12)-(17), including the effect of the gravitational field. For later reference, we note that the total
energy density of an EMW is given by
(31)
EEM =E∗ · ω −1 ∂ω ω 2 ǫ · E
i(kz−ω)t
where E ∝ e
is the electric field of the wave, ω
is the frequency and ǫ is the effective dielectric tensor
defined such that D = ǫ·E gives the electric displacement
field. For the cold (i.e. zero pressure) linearized plasma
fluid equations ǫ is given by[19]
S −iD 0
ǫ = iD S 0
0
0 P
X
R≡1−
X
(29)
ELECTROMAGNETIC WAVE EVOLUTION
P ≡1−
(i)
(28)
where we have used ∂z ≈ −∂t .
V.
where there is a background magnetic field and drift flow
vz in the z-direction, S ≡ 12 (R + L), D ≡ 21 (R − L) and
(32)
L≡1−
2
ωp(i)
ω − kvz(i)
2
2
ωp(i)
ω − kvz(i)
ω 2 (ω − kvz(i) + ωc(i) )
2
X ωp(i)
ω − kvz(i)
(i)
(i)
ω 2 (ω − kvz(i) − ωc(i) )
Although this result concerns linear waves in a cold
plasma it also applies for the large amplitude waves in
a moderately warm plasma that we will consider below.
This is because they are circularly polarized and purely
transverse, making them effectively linear [20] and with
no density (nor pressure) perturbations.
A.
Perpendicular propagation
In the case of perpendicular propagation the plasma
equations are difficult to treat without resorting to perturbation theory. We restrict ourself in this case to linear perturbations and to this level of approximation the
wave perturbations do not produce any effective force,
g, that is not rapidly oscillating. This implies that the
waves cannot exchange energy with the drift flow in an
efficient way. Consequently, we discard the drift flow entirely and only considers one particular wave mode in an
electron-ion plasma, namely the extraordinary electromagnetic wave – using standard plasma physics terminology. This is the only wave that can be resonant with
a GW in the simplest plasma model, the cold plasma fluid
model.
We take the background magnetic field to be B(0) =
B(0) e1 . The extraordinary EMW is a high-frequency
wave, ω ≫ ωpi (such that only the electron fluid is perturbed), and has, when propagating in the z-direction,
the following non-zero linear perturbations: δn, v =
vy e2 + vz e3 , E = Ey e2 + Ez e3 and δB = δBx e1 , where
the density and velocity perturbation referres to the electron fluid only. For a cold plasma, the Eqs. (12)-(17) reduces to the following equations, linearized in the small
perturbations
∂z Ez = qδn
∂t Ey − ∂z δBx = −qnvy − jE y
∂t Ez = −qnvz
∂t δBx − ∂z Ey = − jB x
q
∂t vy = (Ey + vz B(0) )
m
q
∂t vz = (Ez − vy B(0) )
m
∂t δn + n∂z vz = 0
(33)
(34)
(35)
(36)
(37)
(38)
(39)
6
which can be combined into the following evolution equation for the wave magnetic field (after one time integration)
D̂X δBx = SX
(40)
where the wave propagator, D̂X , and the source term,
SX , are
D̂X ≡ ∂t4 + (ωp2 + ωh2 )∂t2 − ∂t2 ∂z2 − ωh2 ∂z2 + ωp4
SX ≡ 21 B(0) ∂t4 + (ωp2 + ωh2 )∂t2 + ωp4
+∂z ∂t (∂t2 + ωp2 + ωc2 ) hA
and we have made use of Eqs. (28)-(30), linearized in the
wave perturbations and introduced the hybrid frequency
ωh2 ≡ ωp2 + ωc2 . In the absence of GWs Eq. (40) reduces
to D̂X δBx = 0 and, in particular, for wave perturbations
δBx ∝ ei(kz−ωt) it produces the dispersion relation for
the extraordinary wave
DX ≡ ω 4 − (ωh2 + ωp2 + k 2 )ω 2 + ωh2 k 2 + ωp4 = 0
δT12 = δT21 = 0
(42)
(43)
The dispersion relation (41) implies that the condition
for having phase velocity ω/k = 1 is ω = ωp . We also
note that from Eq. (31) and Eqs. (33)-(39) it follows,
after some algebra, that the energy density of a free extraordinary wave with ω = ωp is given by
EX = 2ωc−2 ωh2 |δBx |2
B.
We take the background magnetic field to be B(0) =
B(0) e3 and suppose there is a velocity drift, vz , in the
z-direction. The above mentioned circularly polarized
EMW, propagating in the z-direction, has the following
perturbations (of arbitrary amplitude) v = vx e1 + vy e2 ,
E = Ex e1 + Ey e2 and δB = δBx e1 + δBy e2 , where v and
δB are parallel and all perturbations are functions of z
and t alone. Note that circular polarization implies that
γ depends only on the wave amplitude and on vz , not on
the rapidly varying phase. In the following we treat γ as
being constant and vz as a constant uniform background
flow. Small (linear) deviations in γ and vz can still be
described (see Section VI and VII). For convenience we
introduce the variables E± = Ex ± iEy , and similarly for
all other vector variables. In these variables, suitable for
circularly polarized waves (the plus/minus variables corresponds to the amplitudes of the right/left hand polarization), those equations of Eqs. (12)-(17) that governs
the given perturbations can be rewritten as
(41)
It should be pointed out that the extraordinary mode,
with the given polarization, only couples to hA and not to
hB . Computing the perturbed energy-momentum tensor
gives, linear in the perturbations,
δT11 = −δT22 = −δBx B(0)
for electron-ion type of plasmas as well as for electronpositron type of plasmas.
(44)
Parallel propagation
If one neglects general relativistic effects, there exist
exact solutions of the multifluid equations that represents
EMWs propagating parallel to a background magnetic
field (see e.g. [21]). These solutions describe circularly
polarized waves of arbitrary amplitude and arbitrary frequency and can thus, taking the appropriate limits, be reduced to high-frequency electromagnetic waves, whistler
waves, low-frequency Alfvén waves or waves in electronpositron plasmas. The solution can also be extended to
include a background drift flow [22]. We derive the evolution equation for this wave for a two-component plasma,
including the effects of GWs. The indices e and i will now
stand for negatively charged particles (e.g. electrons)
and positively charged particles (e.g. positive ions), respectively, but we make no assumptions for the mass ratio me /mi . The calculations are therfore equally valid
q
(E± ∓ iB0 v± ± ivz B± ) + g± (45)
m
∂t E± = ±i∂z B± − Σqnγv± − jE±
(46)
∂t B± = ∓i∂z E± − jB±
(47)
(∂t + vz ∂z )γv± =
where the meaning of g± is g± = gx ±igy and similarly for
jE± and jB± . Recall that there are two sets of Eqs. (45)(47), one set for each particle species. These equations
imply the following evolution equation for E±
D̂E E± = SE
(48)
where the wave propagator is
2
2
Ĝi dˆe
Ĝe dˆi + ωpe
D̂E ≡ Ĝe Ĝi + ωpi
≡
∂t2
−
(49)
∂z2
dˆ(i) ≡ ∂t + vz(i) ∂z
Ĝ(i) ≡ ∂t + vz(i) ∂z ± iωc(i)
and the source term is
SE ≡ −qi nĜe ∂t gi± − qe nĜi ∂t ge±
2
2
vez Ĝi − Ĝe Ĝi ∂z jB±
± i ωpi
viz Ĝe + ωpe
− Ĝe Ĝi ∂t jE±
(50)
In the absence of the gravitational source terms the evolution equation (48) reduces to D̂E E± = 0 and for this
7
be calculated by means of Eq. (31). For the given wave
perturbations, the wave energy density is given by
case we in particular note the solution [21]
E± = Eei(kz−ωt)
k
B± = ±i E±
ω
v± = iΛE±
q
ω − kvz
Λ≡
γm ω(ω − kvz ∓ ωc )
γ = (1 − v± v∓ −
(51)
1
∂ω (ω 2 A± )|E± |2
2ω
X ωp2 ω − kvz
A± ≡ 1 −
ω 2 ω − kvz ∓ ωc
EE =
(52)
(53)
(54)
1
vz2 )− 2
(55)
Applying the dispersion relation (57) with ω = k, the
wave energy density can, after some algebra, be formulated as
with the dispersion relation
2
0 = DE ≡ (ω − kvze ∓ ωce ) (ω − kvzi ∓ ωci ) ω − k
2
− ωpi
(ω − kvze ∓ ωce ) (ω − kvzi )
2
− ωpe
(ω − kvzi ∓ ωci ) (ω − kvze )
2
2
1
2
(1 − ve + αi )
2ω αe αi − ωpe
2ω 2 αe αi
2
2
−ωpi
(1 − vi + αe ) |E± |
EE =
(56)
(i)
ω − kvz(i)
ω − kvz(i) ∓ ωc(i)
(57)
We again emphasize that the solution is valid for arbitrarily large amplitudes. The condition for the plasma
to be electrically neutral, j 0 = 0, implies γe ne = γi ni .
As γ(i) n(i) is a preserved quantity (by Eq. (16)), the
proper density, n(i) , is not. Accordingly, both ωp(i) and
ωc(i) depend on γ(i) . This makes the dispersion relation
amplitude dependent.
For future reference we devote the remainder of this
section to establish some properties of the free EMWs.
As we will focus on EMWs with ω = k we note that
the dispersion relation in this case implies the following
relation between the background magnetic field and the
drift velocities
Ωce = ±(γe +
mi
(1 − vze )(1 − vzi )
γi )
me
vze − vzi
(58)
where Ωce ≡ γe ωce /ω = ω −1 qe B(0) /me is the normalized
(and preserved) “electron” cyclotron frequency and we
have made use of the neutrality condition γe ne = γi ni .
Note that for any given background plasma parameters (except the absence of any drifts), there always exist a wave frequency fulfilling Eq. (58). The components of the perturbed energy-momentum tensor, for this
large amplitude circularly polarized EMW, that couples
to GWs has the values
X
γmnvx2 (59)
δT11 = − 21 Ex2 + Bx2 − Ey2 − By2 +
X
δT22 = 21 Ex2 + Bx2 − Ey2 − By2 +
γmnvy2
(60)
X
δT12 = − (Ex Ey + Bx By ) +
γmnvx vy
(61)
Because the wave is circularly polarized and transversal
it is effectively of linear nature (the Lorentz force is linear
in the wave amplitude) and has no density (nor pressure)
perturbations. Therefore, the wave energy density can
(64)
1
0.5
E
X
α(i) ≡ 1 − v(i) ∓ ωc(i) /ω
ε
ω2 − k2 =
(63)
where we have introduced the abbreviation
that can be written more compact as
2
ωp(i)
(62)
0
−0.5
−1
−1
−0.5
0
v
0.5
1
FIG.1. The dependence of wave energy EE (normalized
2
by |E± | and rescaled) on drift flow v ≡ vi = −ve for
null geodesic small amplitude waves in an electron-ion
plasma (me /mi ∼ 1000). The different curves, from the
top and down, corresponds to the plasma frequencies
Ωpe = 2, 5, 10, 20, 40, 60, 100, 1000. As the sign of the
energy is the main focus here, the energy in the different
cases are rescaled by suitable factors.
It is important to note here that EE is, in contrast
to EGW and EX , not positive definite. For some background parameters the wave energy density may be negative [10]. This is due to the presence of the velocity
drifts, providing the system with free energy. Negative
energy waves are wave perturbations such that the total energy of the plasma is less than the energy of the
unperturbed state [23]. To show that there are waves
fulfilling ω = k and EE < 0 simultaneously, we consider
an electron-ion plasma (me /mi ∼ 1000) with drift flows
2
ve = −vi . In FIG 1 the wave energy factor EE / |E± | (for
small amplitude waves) is presented, as a function of ion
8
drift velocity, for a number of different values of the normalized “electron” plasma frequency Ωpe ≡ γe ωpe /ω and
with the cyclotron frequency fulfilling the condition (58).
As the sign of the energy is the main focus here, the energy factor in the different cases are rescaled by suitable
factors. Negative wave energy occurs for positive ion velocities [24]. The minimum required ion velocity depends
inversely on the plasma frequency. For Ωpe = 1000,
negative wave energy requires only vi & 10−4 whereas
for Ωpe . 20 it requires highly relativistic velocities,
vi & 0.95.
VI.
A.
RESONANT WAVE INTERACTION
The wave evolution equations (25), (26), (40) and (48)
where derived with no assumptions on the specific wave
forms. In this section we assume the wave frequencies
and the (uniform) background plasma to be in a state
where wave resonance occurs, since this gives the most
efficient energy transfer, and we derive the corresponding
wave interaction equations. We let
hA,B = h̃A,B + c.c.
δBx = B̃ + c.c.
E± = Ex ± iEy
a
(65)
From the dispersion relation (41) it follows that in
order for the extraordinary EMW to propagate along
null geodesics the wave frequencies must be equal to the
plasma frequency, ω ≡ ωpe . Although this is a special case, a GW or an EMW propagating in a slowly
varying background density may often reach such resa
onance regions. Assuming resonant waves, we take kG
=
a
kX
= (ω, 0, 0, ω). The wave evolution equations (25) and
(40) then imply the following wave interaction equations
(dropping the tilde notation on the waves and denoting
h ≡ hA and B ≡ δB)
∂˜t h = −iCG B
∂˜t B = −iCX h
(66)
Here we have used that the extraordinary EMW contributes linearly to the perturbed energy momentum tensor (42) and the circular polarized large amplitude EMW
quadratically in (59)-(61) and thus produces terms proportional to ei2(kE z−ωE t) . Consistently, the source terms
for the extraordinary EMW is linear (in the GW amplitude) whereas for the circular polarized large amplitude
EMW there are terms proportional to E± hA,B . Since (in
the high frequency limit) ωG /kG = 1, also the EMWs
should have ωX /kX = 1 and ωE /kE = 1 in order to remain in phase with the GW. The interaction is weak in
the sense that the coupling terms in the evolution equations (25), (26), (40) and (48) are either proportional
to the small GW amplitude or the gravitational coupling constant. This leads to a weak time dependence
in the wave amplitudes h, B and E, i.e. ∂t h ≪ ωG h
(67)
(68)
where CG ≡ ω −1 κB(0) and CX ≡ 21 ωωh−2 ωc2 B(0) . From
Eqs. (27) and (44) the total wave energy is EGW + EX =
2
ω 2 κ−1 |h|2 + 2ωc−2 ωh2 |B| and it follows from Eq. (67)
and (68) that this is a conserved quantity. The interaction equations has the following solution for the wave
amplitudes
B = B0 cos(ψt) + iCX ψ −1 h0 sin(ψt)
h = h0 cos(ψt) + iCG ψ
and in the case of parallel propagation
a
a
kG
= 2kE
Perpendicular propagation
a
with h̃A,B = hA,B ei kG a x , B̃ = Bei kX a x and E± =
a
a
Eei kE a x and the four-wave vectors kG
= (ωG , 0, 0, kG ),
a
a
kX = (ωX , 0, 0, kX ), and kE (ωE , 0, 0, kE ) are assumed
a
to fulfill the dispersion relations kG a kG
= 0, Eq. (41)
and Eq. (57), respectively. As before, c.c. stands for the
complex conjugate of the preceding term. The conditions
for the EMWs to be resonant with a GW are, in the case
of perpendicular propagation
a
a
kG
= kX
etc. We will use the notation of “slow derivatives”,
∂˜t , that acts only on the wave amplitude, so that e.g.
∂t hA = iωG hA + ∂˜t hA , and we will neglect slow derivatives that are of second or higher order. Consistent with
this level of approximation, we make use of the relations
between the perturbations valid for free waves, i.e. Eqs
(51)-(53), in the computation of the source terms. Corrections to this approximations produces higher order
source terms.
We now treat the two cases of propagation direction
separately.
(69)
−1
B0 sin(ψt)
(70)
√
where B0 ≡ B(t = 0) and h0 ≡ h(t = 0), ψ ≡ CX CG =
p
−1
κ
2 |ωh ωc B(0) |. The solution shows that the total wave
energy alternates between the GW and the EMW with
the frequency ψ. In most applications ψ is very small
and it is then meaningful to linearize the trigonometric
functions in ψ and it is clear that, for time scales smaller
than ψ −1 , GWs are converted into EMWs as h = CG B0 t
or, alternatively, EMWs are converted into GWs as B =
CX h0 t. It should be noted that in the later case the
time tnl for the EMW perturbation to reach the nonlinear
stage, B ∼ B(0) , is typically much smaller than total
energy conversion time, ψ −1 .
B.
Parallel propagation
For resonant interaction between EMWs and GWs in
a
a
the parallel case we take kG
= 2kE
= (2ω, 0, 0, 2ω). As
9
the EMWs that we are considering are circularly polarized
waves, we also introduce variables for circularly polarized
GWs h± ≡ hA ± ihB . If the GW and the EMW are
oppositely polarized the coupling vanishes, so from here
on we assume identical polarization. The evolution Eqs.
(25) and (26) combines to
X
2
κmnγ 2 v±
(71)
−∂t2 + ∂z2 h± = −
κ X
mnγ 2 Λ2
4ω
mi
2
CE ≡ ω 2 γi Λi ωpi
(1 − vzi )αe
qi
me
2
+γe
Λe ωpe
(1 − vze )αi
qe
2
2
× 2ω αe αi − ωpi
(1 − vzi + αe )
−1
2
−ωpe
(1 − vze + αi )
(75)
and we also note that we may, with no loss in generality,
assume s2 = s3 = 1 (it is only the sign of s1 s2 = s1 s3
that matters because an overall sign can be removed by a
renormalization). The general solution can then be found
in litterature, e.g. in Ref. [10]. The strongest coupling
occurs for cos φ = ±1 and in this case there are some
particularly simple solutions that are listed below (recall
that Ψ2 = Ψ3 ).
(73)
In the derivation we have noted that the effective currents, jE and jB , vanishes and used that g± = − 21 γ(1 −
∗
vz )v±
∂t h± . The coupling coefficients can be related to
the wave energy densities, Eqs. (27) and (63), according
to
2
CGW = CE −1
GW |h|
CE =
CE −1
E
2
|E|
(80)
(81)
(82)
(74)
(72)
where
CGW ≡
∂t ψ1 = s1 c1 ψ2 ψ3 cos(φ)
∂t ψ2 = s2 c2 ψ1 ψ3 cos(φ)
∂t ψ3 = s3 c3 ψ1 ψ2 cos(φ)
where we have taken ih = ψ1 eiφ1 , En = ψn eiφn ,
−4CGW = s1 c1 , CE = sn cn (for n = 2, 3) with ψ1 = |h|,
ψn = |En |, c1 = |CGW |, cn = |CE |, φ = φ1 − φ2 − φ3 and,
s1 , s2 and s3 are the signs of −CGW and CE , respectively. The coupling coefficients c1 , c2 and c3 are made
unity by the renormalization
√
ψ1 → Ψ1 ≡ c2 c3 ψ1
(83)
√
ψ2 → Ψ2 ≡ c3 c1 ψ2
(84)
√
ψ3 → Ψ3 ≡ c1 c2 ψ3
(85)
and together with the EMW evolution equation (48) this
implies the following interaction equations for the variables h ≡ h± and E ≡ E±
∂˜t h = iCGW E 2
∂˜t E = iCE E ∗ h
convert the obtained three-wave interaction equations to
a real system and renormalize the amplitudes such that
the coupling coefficients become unity. The real system
is [25] (dropping the tilde notation)
(76)
(77)
a) EMW to GW conversion with positive EMW energy
s1 = −1, Ψ1 (0) = 0, Ψ2,3 (0) = Ψ and cos φ = −1
Ψ1 = Ψtanh(Ψt)
Ψ2,3 = Ψsech(Ψt)
(86)
b) EMW-GW instability with no initial GW
where
2
2
1 X ωp(i)
1 − vz(i)
C≡
2
2ω
α(i)
s1 = 1, Ψ1 (0) = 0, Ψ2,3 (0) = Ψ and cos φ = 1
(78)
It is then straightforward to confirm that Eqs. (72) and
(73) implies that the total wave energy density
2
2ω 2 2
1
|h| +
2ω αe αi
2
κ
2ω αe αi
2
2
−ωpi
(1 − vzi + αe ) − ωpe
(1 − vze + αi ) |E|2
E=
Ψ1 = Ψtan(Ψt)
Ψ2,3 = Ψsec(Ψt)
(87)
c) EMW-GW instability with equal initial amplitudes
s1 = 1, Ψ1 (0) = Ψ2,3 (0) = Ψ and cos φ = 1
(79)
is conserved.
The solution to the wave interaction equations (72) and
(73) can be given in terms of Jacobi elliptic functions.
We exploit the fact that the system can be reformulated
as a three-wave interacting system (see e.g. [10]) with
two identical EMWs and one GW. This is done by taking
E2 = E3 ≡ 21 E such that Eq. (73) can be splitted into
two (identical, but differently labeled) equations and the
right hand side of Eq. (72) reads i4CGW E2 E3 . Next we
Ψ1 = Ψ−1 − t
−1
Ψ2,3 = Ψ−1 − t
−1
(88)
d) EMW-GW interaction with decaying amplitudes
s1 = 1, Ψ1 (0) = Ψ2,3 (0) = Ψ and cos φ = −1
Ψ1 = Ψ−1 + t
−1
Ψ2,3 = Ψ−1 + t
−1
(89)
10
Also conversion from GW to EMW energy occurs (given
that there is a small initial EMW perturbation) the solution is somewhat more complicated than the ones listed
above [10]. Note that Ψ1 (0) = Ψ and Ψ2,3 (0) = 0 only
has the trivial solution Ψ1 = Ψ and Ψ2,3 = 0. The solution a) corresponds to an EMW being converted into a
GW on the characteristic time scale π/2Ψ. The solutions
b) and c) are explosive solutions, i.e. the wave amplitudes diverges on a finite time t = π/2Ψ and t = 1/Ψ,
respectively. These explosive instabilities can only occur
for s1 = s2 = s3 , which is equivalent to saying that the
GW and EMW energy must be of different sign. The explosive instability thus relies on the existence of negative
energy EMWs and it is the free energy connected with
the background drift flow that feeds the instability. We
return to these interesting cases in section VII. The solution d) also involves a negative energy EMW, but in
this case both the EMW and the GW amplitudes tends
to zero as (Ψ−1 + t)−1 . This means that wave energy is
converted into free energy of the background state, i.e.
acceleration of the drift flow. As for the cases when the
wave phases are not fulfilling cos φ = ±1 the behaviour
is similar [25] but with somewhat larger characteristic
times.
We now explore the possibility, indicated by the solution d), that the waves may interact to accelerate the
background drift flow. For the wave perturbations we
are considering here, the fluid equation of motion (17)
has the property vi ∂t γv i = gz (where i = 1, 2, 3). This
implies that ξ ≡ γ(1 − vz ) is an exactly conserved quantity, as is seen by making the computation
∂t ξ ≡ ∂t γ − ∂t γvz = vi ∂t γv i − gz = 0
(90)
Making use of this, the z-component of the fluid momentum equation (Eq. (17)) may be rewritten as
2
∂t vz = iω
E 2 h∗ − E ∗2 h
2 ξΛ
= −ωξΛ2 |E|2 |h| cos(φ)
(91)
where the phase angle φ has the same meaning as in Eqs.
(80)-(82). From this equation it is clear that the drift flow
may be accelerated or decellerated in the z-direction by
the interacting waves. Maximum acceleration occurs for
cos φ = −1. In the pump-wave approximation, where
the waves are assumed energetic and can be taken to be
of constant amplitude (an approximation breaking down
when the energy of the drift flow becomes comparable
with the wave energies), the drift flow vz grows linearly
in time as (for cos φ = −1)
vz = vz (0) + ωξΛ2 |E|2 |h|t
(92)
and we point out that for relativistic EMW amplitudes
Λ2 |E|2 = |v± |2 ≈ 1.
VII.
SUMMARY AND DISCUSSION
In this paper we have studied the interaction between
GWs and EMWs in magnetized plasmas for perpendicular
and parallel propagtion with respect to the background
magnetic field. The wave evolution equations were derived in section IV and V. An important point here
which, as far as we know, has not been fully recognized
previously is that, given the high frequency approximation, the GWs can be taken to be in the TT gauge even in
the presence of matter. Consequently, the deduction of
the GW source term associated with the EMW perturbed
energy-momentum tensor is simplified, see Eqs. (25) and
(26). The GW, on the other hand, produces effective
currents in the Maxwell equations and an effective gravitational force on the plasma, resulting in a source term
in the EMW evolution equations (40) and (48). Furthermore, it was shown in section V that, provided there is
a velocity drift in the background state, the wave energy
may be negative [10] for EMWs with phase velocity equal
to c = 1 (when propagating parallel to the background
magnetic field).
Resonant interaction (involving a single EMW) may
occur when the frequencies matches and the phase velocity of the EMW coincide with that of the GW, which to
a very good precision equals c = 1 in the high-frequency
approximation. The wave interaction equations governing the resonant interaction process were derived and analyzed in section VI for the cases of perpendicular (highfrequency extraordinary EMWs) and parallel propagation
(finite amplitude circular polarized EMWs) with respect
to the background magnetic field. In both cases the interaction equations were shown to imply conservation of
total wave energy.
i) Perpendicular propagation. The wave interaction
equations for this case are Eqs. (67) and (68). The solution, given by Eqs. (69) and (70), reveals that the
conversion rate [26] is of order ψ = (κ/2)1/2 ωc ωh−1 B(0) .
The effect of the plasma is to diminish the coupling
strength. The conversion rate becomes ψ = (κ/2)1/2 B(0)
for strong magnetic fields, i.e. ωc ≫ ωp (similar to having no plasma present), whereas for weak magnetic fields,
ωc ≪ ωp , the conversion rate is a factor ωc /ωp smaller.
Although ψ is typically small, GWs may still generate
EMWs with significant amplitude. For small ψt, the
EMW are converted into GWs as B = 21 ωp ωc2 ωh−2 B(0) h0 t.
ii) Parallel propagation with positive EMW energy.
The solutions of the wave interaction equations (72) and
(73) were found by reformulating them as a three-wave
interacting system (with two identical EMWs) for which
the general solutions are well known. From the solution
(86) it follows that the conversion
rate [26] for conversion
√
from EMW to GW is Ψ = CGW CE |Eini |, where Eini is
the initial electric field amplitude. The conversion rate
increases with increasing initial amplitude and with increasing drift velocity (through γ). In order to have more
transparent formulas we consider a special case, namely
low-frequency waves, such that ω ≪ ωci , ωce . The conversion rate can be formulated as
2 −1
Ψ = CGW CE
C EE,ini
12
(93)
11
and in the case of low-frequency waves
κ
C
2ω 2
CE = CD−1
CGW =
2
EE = D |E± |
where
C=
ω X ωp2
(1 − vz )2 ,
2
ωc2
D =1±
1 X ωp2
2ω
ωc
and the final GW amplitude is given by hf inal =
(EE,ini κ/2ω 2 )1/2 .
iii) Parallel propagation with negative EMW energy.
As was remarked previously, the EMW energy density
(given by Eq. (63)) may be negative. Such waves are
perturbations with the property that the energy of the
perturbed system is less than for the unperturbed system [23]. As is clear from FIG 1, this occurs most commonly for low-frequency waves in electron-ion plasmas,
i.e. ω ≪ γe ωpe . The wave energy conservation then
allows simultaneous growth of the GW and EMW amplitudes. The physical interpretation is that the background
flow is unstable with respect to this type of perturbations
and the energy associated with the flow is converted into
wave energy. Given that there is an ion flow in the direction of wave propagation and a large enough plasma
frequency, this instability will always produce waves with
a unique EMW frequency determined by Eq. (58) and
GW frequency given by the matching condition (66). It
should be noted that this is a nonlinear instability and
would not have been found using conventional linear stability analysis. The solutions (87) and (88) shows that
the background is explosively unstable with respect to
GW-EMW perturbations, the amplitudes reaches infinity
in a finite time t ∼ Ψ−1 (the interaction equations are of
course invalidated before this time is reached). The solution (89) is an example of the inverse process, where both
waves decreases in amplitude. This means that wave energy is converted into free energy of the background state,
i.e. the drift flow is accelerated. An evolution equation
for the background flow was also derived, see Eq. (91).
The response of the plasma on the interacting waves is an
acceleration (if cos φ > 0) or deceleration (if cos φ < 0).
The conclusion is that GWs and EMWs can interact to
produce a drift flow along the background magnetic field
with initial linear growth rate as large as ωγ(1 − vz )|h|.
The magnitude of the produced drift flow can be estimated from the initial wave energies and the time scale
for all the wave energy to be converted into drift flow is
at least of the order given by Eq. (89), t ∼ Ψ−1 .
Finally we discuss the relevance of the results presented
in this paper to various astrophysical and cosmological
processes. The phenomena studied in this paper involve
energy transfer between GWs, EMWs and a background
flow. This could in principle be an important feature
in any scenario involving magnetized plasmas, energetic
GWs and EMWs and/or strong background plasma flows,
e.g. supernova explosions, gamma-ray bursts, jets of accreting condensed objects, phenomena in the vicinity of
neutron stars and in the primordial fluctuations of the
cosmological plasma. The energy conversion must take
place within reasonable times and/or volumes for the
considered processes to be of significance. As a specific numerical example let us first consider graviational
to electromagnetic conversion for perpendicular propagation. For the case of GWs from a binary pulsar close
to collapse (corresponding to a wavelength of, let us
say, λ ∼ 105 m) we may have h ∼ 0.001 a few wavelengths away from the pulsar. A rough estimation based
on Eq. (69) then gives a distance L ∼ λ/h ∼ 108 m
for the induced magnetic field to be comparable to the
unperturbed field [28]. A significant transfer of GWenergy is not realistic in this example, however, since
the background energy density is too low. As another
example, let us consider EMW to GW conversion for
parallell propagation and positive wave energy in the
low-frequency regime. Let us take B0 ∼ 10−3 T and
n0 ∼ 1018 m−3 . Naturally, the beam velocities are assumed to fulfill (58), and the wave frequency is assumed
to obey ω ≪ ωci ∼ 2 × 105 s−1 . Using Eq. (93) the characteristic time scale for energy conversion then becomes
−1/2
t ∼ Ψ−1 ∼ 107 (EE,ini )
s, where the electromagnetic
wave energy density should be given in SI-units. Appar3
ently we need wave energy densities EE,ini ∼ 1014 J/m
to get conversion within 1 second. Note that EMW energy densities in the laboratory can be magnitudes larger
than this [29] and the value is also modest as compared to
some astrophysical events, like gamma-ray bursts. However, we note that the required wave energy content is
still very large, since the EMW must be distibuted in a
volume not less than a cube with a side of 1 light second,
for efficient GW conversion to take place.
The model we have presented is idealized in several
ways, e.g. we have assumed a uniform background and
monochromatic waves. The model can be extended to
include weakly nonuniform backgrounds, where the interaction can be treated using mode conversion theory,
see e.g. Ref. [27] and Refs. therein. It should also be
noted that the background drift flow is in general also unstable with respect to other types of perturbations, e.g.
ion-accoustic perturbations which typically may have a
higher growth rate. With a more realistic velocity distribution among the particles, including a thermal spread,
the conditions and growth rates for various instabilities
change. It is an open question whether GW-EMW instabilities (of the type presented in this paper) can be the
only ones (or the dominating ones) in certain regions of
parameter space. We have also treated the gamma factors and the background flow as constants in time. This
is not a valid approximation for the long term evolution
when the change in energy becomes comparable to the
initial energy of the source. An exception to this case is
interaction between waves with “moderate” positive energies in the presence of a highly energetic background
1
flow, such that γ ≈ (1 − vz2 )− 2 and vz are approximately
12
constant for all times. Variations in γ ad vz can, as these
occurs in the dispersion relation (57), also affect the frequency of the EMW, thereby invalidate the assumed perfect resonance and thus reducing the coupling strength.
In conclusion, energy transfer between GWs, EMWs
and a background plasma flow should occur also in less
idealized situations than considered in this paper, but
in a more involved way. An improved model that correctly describes the long term evolution of the interaction between a GW, an EMW and the background flow
in an inhomogeneous background is a project for future
research.
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[14] In general, the source terms may contain derivatives on
f(n) and f(m) . As we restrict ourself to weak interaction
these derivatives are later approximated by algebraic expressions, i.e. involving frequencies and wave numbers.
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[20] The dispersion relation, however, depends on the wave
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circular polarization the gamma factors are constant (for
free waves).
[21] L. Stenflo, Phys. Scr. 14, 320 (1976).
[22] Formally, the drift velocities, vz(i) , cannot be taken arbitrarily. They should fulfill jz = 0 for the given solution
to be valid, or else the drifts produce a background magneic field that is not accounted for in the solution. On
the other hand, one can always divide a fluid component
into several species, each with a different drift velocity
and thereby regain freedom in the relative drifts. Thus
we discard this restriction because it should not influence
the final result qualitatively.
[23] From a mathematical point of view, our large EMW solution (together with the background drift) contains more
energy than the state with only the pure drift flow. However, from energy conservation it is clear that the electromagnetic field amplitude cannot be physically altered
without simultaneously affecting the drift velocity. When
confirming that the wave energy is negative, one should
compare with the state where this change in drift flow
has been taken into account.
[24] One could expect that the curves in FIG 1. should be
even functions of the velocity. The symmetry is, however, broken by the definite propagation direction. The
important point is that the relation between the wave polarization and the particle gyration (determined by Ωce )
is fixed by the condition ω = k, which depends on the
flow velocities, as seen by Eq. (58).
[25] In general also the phase φ evolves in time. Eqs. (80)(82) are still valid but should be supplemented with an
evolution equation for the phase, see e.g. Ref. [10]. For
our purposes, however, it suffices to consider the special
cases where the phase is constant.
[26] By conversion rate we mean the inverse time scale for
one wave to be converted into another type of wave.
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[28] This example makes use of the substitution ∂t → vg ∂z
appropriate for a boundary value problem. Effects of an
inhomogeneous background is neglected.
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