Exact solution of the relativistic magnetohydrodynamic equations in the background of a plane gravitational wave with combined polarization

0.11 MB·English

by  A. A. Agathonov

Grav. Cosmol. No.1, 2011
Exact solution of the relativistic magnetohydrodynamic equations
in the background of a plane gravitational wave
with combined polarization1
A. A. Agathonov and Yu. G. Ignatyev
Kazan State Pedagogical University, Mezhlauk str. 1, Kazan 420021, Russia
1
1
0 We obtain an exact solution of the self-consistent relativistic magnetohydrodynamic equations for an anisotropic
2 magnetoactiveplasmainthebackgroundofaplanegravitationalwavemetric(PGW)withanarbitrarypolarization.
n It is shown that, in the linear approximation in the gravitational wave amplitude, only the e+ polarization of the
PGW interacts with a magnetoactive plasma.
a
J
9 1 Introduction Maxwell equations of the first group

] ∗
c Inaseriesofpreviousarticlesbyoneoftheauthors(see, Fik =0 (2)
,k
q e.g.,[1–3]atheoryofgravimagnetic shockwavesinaho-
- mogeneous magnetoactive plasma has been developed. with the necessary and sufficient condition
r
g The essence of this phenomenon is that a magnetized
Inv = F Fij =2H2 >0, (3)
[ plasma in anomalouslystrong magnetic fields drifts un- 1 ij
1 der the actionof gravitationalwaves (GWs) in the GW Inv2 = F∗ij Fij =0, (4)
v propagationdirectionundertheconditionthatthewave
4 amplitude islargeenough,and,onacertainwavefront, the Maxwell equations of the second group3:
5
6 tehrgeypdlaesnmsiatyvaenlodcitthyetienntdesnstiotythoef tshpeeefrdozoefnl-iginhtm.aIgtsneetnic- Fik,k =−4πJdir (5)
1
. fieldthentendtoinfinity. Inthesubsequentpapersthis with a spacelike drift current
1
effect was provedonthe basisof the kinetic theory,and
0 p
1 the possibility of using this mechanism as an effective 2Fik Tl
1 tool for detecting GWs from astrophysical sources was Jdir =− F Fjmk,l, (Jdr,Jdr)<0 (6)
jm
: also shown. However, in all cited papers, a monopolar-
v
ized gravitational wave was considered. In the present and a conservation law for the total energy-momentum
i
X paper we consider the action of a GW with combined of the system
r polarization on a magnetoactive plasma.
a p f
Tik =Tik +Tik =0. (7)
,k ,k ,k
2 Self-consistent RMHD equations
The energy-momentum tensor (EMT) of the elec-
in a gravitational field tromagnetic field, in the case of a coincidence of the
plasma’s and the field’s dynamic velocities (1), is ex-
In [1], under the assumption that the dynamic velocity pressed through a pair of vectors, v and H [1]:
oftheplasma(vi)isequaltothatoftheelectromagnetic
f 1
field2 Tik =−8π (δki −2vivk)H2+2HiHk) . (8)
(cid:2) (cid:3)
p f
Tij vj =εpvi; Tij vj =εfvi, (v,v)=1, (1) The EMT of a relativistic anisotropic magnetoactive
plasma in gravitationaland magnetic fields is (see, e.g.,
a full self-consistent set of relativistic magnetohydrody- [3])
namic equations for a magnetized plasma in arbitrary
p
gravitational field has been obtained. It consists of the Tij =(ε+p⊥)vivj p⊥gij +(pk p⊥)hihj, (9)
− −
1Talk given at the International Conference RUDN-10, June where hi = Hi/H is the spacelike unit vector of the
28—July3,2010,PFUR,Moscow
2Theindex“p”referstotheplasma,theindex“f”tothefield, magneticfield((h,h)= 1); p⊥ and pk aretheplasma
−
the comma denote a covariant derivative. The dynamic velocity pressuresinthedirectionsorthogonalandparalleltothe
ofanykindofmatteris,bydefinition,atimelikeuniteigenvector magnetic field, respectively.
oftheenergy-momentum tensorofthismatter [4].
3c=G=~=1
2
3 Solving the RMHD equations
where LTij is aLie derivativeinthe directionof ξ. We
ξ
in the PGW metric
p
further require that the EMTs of the plasma Tij and
f
Consider a solution of the Cauchy problem of the self- the electromagneticfield Tij inherit the symmetry sep-
consistentRMHDequationsinthebackgroundofavac- arately. Thus allobservedphysicalquantities P inherit
uum gravitational-wavemetric (see, e.g., [5])4: the symmetry of the metric (10):
ds2 =2dudv L2 cosh2γ e2β(dx2)2 LP=0 (α=1,3), (16)
− (cid:20) (cid:18) ξα
+2e−2β(dx3)2 sinh2γdx2dx3 , (10) i.e., taking into account the explicit form of the Killing
(cid:19)− (cid:21) vectors (14),
with homogeneous initial conditions on the null hyper- p=p(u), ε=ε(u), vi =vi(u); (17)
surface u=0:
F =F (u), H =H (u), h =h (u). (18)
β(u 0)=0; β′(u 0)=0; L(u 0)=1, ik ik i i i i
≤ ≤ ≤
The vector potential agreeingwith the initial condi-
(11)
tions (13) is
We assume the following:
A =A =A =0;
v u 2
the plasma is homogeneous and at rest: 0
• A3 =H (x1sinΩ x2cosΩ); (u 0). (19)
− ≤
vv(u 0)=vu(u 0)=1/√2;
≤ ≤ InthepresenceofaPGW,thevectorpotentialbecomes
v2 =v3 =0; ε(u 0)=ε0;
≤ A2 =Av =Au =0;
0 0
p (u 0)=p ; p (u 0)=p ; (12) 0 1
k ≤ k ⊥ ≤ ⊥ A3 =H (v ψ(u))sinΩ x2cosΩ , (20)
(cid:18)√2 − − (cid:19)
a homogeneous magnetic field is directed in the
• where ψ(u) isanarbitraryfunctionoftheretardedtime,
(x1,x2) plane:
satisfying the initial condition
0
H1(u≤0)=H cosΩ; ψ(u≤0)=u. (21)
0
H2(u 0)=H sinΩ; Thus the magnetic field freezing-in condition in the
≤
plasma reduces to the two equalities
H (u 0)=0, E (u 0)=0, (13)
3 i
≤ ≤
v3 =0,
where Ω is the angle between the axis 0x1 (the
1
PGW propagation direction) and the magnetic (v ψ′ v )sinΩ+v2cosΩ=0. (22)
v u
√2 −
field H.
Thecovariantcomponentsofthevectorofmagneticfield
The metric (10) admits the group of isometries G ,
5 intensity relative to the Maxwell tensor are
associated with three linearly independent (at a point)
Killing vectors 0
H 1
H = v cosΩ+ v2sinΩ (23)
ξi =δi ; ξi =δi; ξi =δi. (14) v −L2 (cid:18) v √2 (cid:19)
v 2 3
(1) (2) (3) 0
H 1
H = v cosΩ v2ψ′sinΩ , (24)
Due to their existence in the metric (10), all geometric u L2 (cid:18) u − √2 (cid:19)
objects, including the Christoffelsymbols, the Riemann 1 0
tensor,the Ricci tensorand consequentlythe EMT of a H2 = H cosh2γe2βsinΩ(vvψ′+vu), (25)
−√2
magnetoactive plasma, are automatically conserved at
1 0
motions along the Killing directions: H3 = H sinh2γsinΩ(vvψ′+vu). (26)
√2
Lgij =0 LRij =0 LTij =0, (15) The magnetic field intensity squared is
ξα ⇒ ξα ⇒ ξα
4β(u) and γ(u) are the amplitudes of the polarizations e+ H0 2
and e×, respectively; u = (t x1)/√2 is the retarded time, H2 = (L2ψ′cosh2γe2βsin2Ω+cos2Ω). (27)
v = (t+x1)/√2 is the advanc−ed time. The PGW amplitudes L4
are arbitrary functions of the retarded time u, and L(u) is a
background factorofthePGW.
3
Using (23)-(27), the normalization relation for the of the retarded time:
velocity vector can be written in the equivalent form
00
(εp)
vvcosΩ+v2 1 sinΩ 2 vv2 = 2L2(ε+pk)∆(u)
(cid:20) √2 (cid:21)
0
= H02vv2L4− sin22ΩL2cosh2γe2β. (28) + ((pεk+−ppk⊥))HH22cosh2L2γ2e2β. (40)
H2
From (35), (36) we get:
The components of the drift current are
1 v2 =v3 =0. (41)
Ji = ∂ (L2Fiu). (29)
dr −4πL2 u
We obtain the component v from the normalization
u
Then, relation for the velocity vector, using (40) and (41):
Jdvr =Jdur =0, (30) 1
v = , (42)
u
2v
0 v
H sinΩ
J2 = cosh2γ γ′, (31)
dr −2√2πL2 · andfromthe freezing-incondition(22)we getthe value
of the derivative of potential ψ′:
0
Jd3r =−H2√sin2πΩLe22β(sinh2γ·γ′+cosh2γ·β′). (32) ψ′ = 21v2. (43)
v
Because of existence of the isometries (14), we obtain
Using it, the scalar H2 is determined from the relation
the following integrals [1]:
(27):
L2 ξ iT =C =const (α=1,3). (33)
vi a
0
(α) H2cosh2γe2β
H2 = . (44)
We consider only the case of transverse PGW prop- L2 2v2
v
agation (Ω = π/2). Then, substituting the expressions
FromtheRMHDsetofequationsitispossibletoobtain
for the plasma and electromagnetic field EMT into the
the following differential equation in the PGW metric:
integrals(33), using the relations(26)-(28)and alsothe
initial conditions (11), we bring the integrals of motion L2ε′v +(ε+p )(L2v )′
v k v
to the form
1
+ L2(p p )v (lnH2)′ =0. (45)
H0 2 2 k− ⊥ v
2L2(ε+p )v2 (p p ) cosh2γe2β
k v − k− ⊥ H2 To solve this equation, it is necessary to impose two
=(ε0 +p0)∆(u), (34) additional relations between the functions ε, pk, and
p , i.e., an equation of state:
⊥
L2(ε+p )v v =0, (35)
k v 2
p =f(ε); p =g(ε). (46)
k ⊥
L2(ε+p )v v =0, (36)
k v 3
4 Barotropic equation of state
where
0 0
p=p , (37) Considerabarotropicequationofstateoftheanisotropic
⊥
plasma, where the relations (46) are linear:
and the so-called governing function of the GMSW is
introduced: p =k ε; p =k ε. (47)
k k ⊥ ⊥
∆(u)=1 α2(cosh2γe2β 1), (38) Equation (45) is easily integrated under the conditions
− −
(47), and we get one more integral:
with the dimensionless parameter α2,
H0 2 ε(√2L2vv)(1+kk)H(kk−k⊥) =ε0H0 (kk−k⊥). (48)
α2 = . (39)
4π(ε0+p0) In the case of a barotropic equation of state under the
conditions (47), substitution of (44) into (40) results in
Solving (34) with respect to v , we obtain expres-
v
sions for the components of the velocity vector as func- 1 ε0
tions ofthe scalars ε, pk, p⊥, ψ′ andexplicitfunctions vv2 = 2L2ε∆(u). (49)
4
g
Substituting (44) and (49) into (48), we obtain a where T 41(β,γ) is the energy flow of a weak GW in
closed equation with respect to the variable ε, whose the direction 0x1 (see [7]).
solution gives: In the case of transversal PGW propagation and
with a barotropic equation of state of an anisotropic
ε=ε0 ∆1+k⊥L2(1+kk)(cosh2γe2β)kk−k⊥ −g⊥, (50) plasma, using the solutions of magnetohydrodynamics
h i
and Eqs.(50), (51), (52) with the dimensionless param-
vv = 1 ∆L(kk+k⊥)(cosh2γe2β)kk−2k⊥ g⊥, (51) eter α2 (39),onecanobtaintheenergybalanceequation
√2(cid:20) (cid:21) in the form
H =H0 ∆L(1+kk)(cosh2γe2β)−1−2kk −g⊥ , (52) H0 2 ∆−4g⊥ 1 1 +1
(cid:20) (cid:21) 4 − (cid:18)α2 (cid:19)
(cid:0) (cid:1)
where +(γ′)2+(β′)2 =(γ′)2+(β′)2. (61)
∗ ∗
1 Since,inalinearapproximationbysmallnessoftheam-
g = [1,2]. (53)
⊥ 1 k ∈ plitudes β and γ, the governing function (38) does not
⊥
−
depend on the function γ(u),
In particular, for an ultrarelativistic plasma with zero
parallel pressure, ∆(u)=1 2α2β+O(β2,γ2), (62)
−
1
k 0; k (54) and the functions β(u), γ(u) are arbitrary and func-
k → ⊥ → 2 tionally independent, then, up to β2,γ2, the relation
(61) can be split into two independent parts:
we obtain from (50)–(53):
vv = 1 L∆2(cosh2γe2β)−1/2, (55) 2H0 2g⊥(1+α2)β+(β′)2 =(β∗′)2, (63)
√2
(γ′)2 =(γ′)2. (64)
ε=ε0L−4∆−3(cosh2γe2β), (56) ∗
Here, according to the meaning of the local energy bal-
0
H =H L−2∆−2(cosh2γe2β). (57) ance equation, we consider short gravitational waves
(59), so we can neglect the squares of the PGW ampli-
tudes as compared with the squares of their derivatives
5 The energy balance equation
with respect to the retarded time. Thus, according to
(64),
In [1], it has been shown that the singular state, which
exists in a magnetized plasma under the condition γ (u)=γ(u), (65)
∗
2β α2 >1 on the hypersurface
0
i.e., in the linear approximation, a weak gravitational
∆(u∗)=0, (58) wave with the polarization e× does not interact with a
magnetized plasma. This coincides with the conclusion
isremovedusingthebackreactionofthemagnetoactive
ofthepaper[8]. Thustheenergybalanceequationtakes
plasmaontheGW.Thatleadstoefficientabsorptionof
the form obtained in [3]:
GW energy by the plasma and a restrictionon the GW
amplitude. A qualitative analysis of this situation can ∆˙2+ξ2Υ2 ∆−4g⊥ 1 =Υ2sin2(s), (66)
be carried out using a simple model of energy balance h − i
proposed in [2]. The energy flow of the magnetoactive
where ξ2 is the so-called first parameter of the GMSW
plasma is directed along the PGW propagation direc-
[2]:
tion, i.e., along the x1 axis. Let β (u) and γ (u) be
∗ ∗
thevacuumPGWamplitudes. IntheWKBapproxima- 0
H2
tion, ξ2 = , (67)
4β2ω2
0
8πε ω2, (59)
≪ Υ=2α2β (68)
0
where ω is the characteristic PGW frequency and ε is
—thesecondGMSWparameter. Thedotdenotesdiffer-
the matter energy density, all functions still depend on
entiationwithrespecttothedimensionlesstimevariable
theretardedtimeonly(see[6]). Thus β(u) and γ(u) are
s,
the PGW amplitudes subject to absorption in plasmas.
The local energy conservation law should be satisfied: s=√2ωu. (69)
g g
T41(β,γ)+T 41(β,γ)=T 41(β∗,γ∗), (60)
5
6 Conclusion
Thus we have obtained a generalization of the results
of [1]- [3] to gravitational waves with two polarizations
andshowedthat,inthelinearapproximation,thepolar-
ization e doesnotinteractwith amagnetizedplasma.
×
Thisjustifiestheapplicabilityofthepreviouslyobtained
results for arbitrarily polarized gravitationalwaves.
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