Magnetohydrodynamic Equations in a Gravitational Field and Excitation of Magnetohydrodynamic Shock Waves by a Gravitational Wave

0.29 MB·English

by  Yu. G. Ignatyev

Magnetohydrodynamic Equations in a
Gravitational Field and Excitation of
Magnetohydrodynamic Shock Waves by a
Gravitational Wave
Yu.G.Ignatyev
Kazan State PedagogicalUniversity,
Mezhlauk str., 1, Kazan 420021,Russia
1
1 Abstract
0
2 On the basis of simple principles we derive and investigate the equa-
tionsofrelativisticplasmamagnetohydrodynamics(MHD)inanarbitrary
n
gravitationalfield. Anexactsolutiondescribingthemotionofmagnetoac-
a
tiveplasma against thebackgroundofthemetricofaplanegravitational
J
wave(PGW)withanarbitraryamplitudeisobtained. Itisshownthatin
5
strongmagneticfieldsevenasufficientlysmallamplitudePGWcancreate

a shock MHD wave, propagating at a subluminal velocity. Astrophysical
]
c consequences of theanomalous plasma acceleration are considered.
q
-
r
1 Introduction
g
[

In[14]the effectofPGWonplasmalikemediawasinvestigatedby the methods
1
v of relativistic kinetic theory in the approximation when the back reaction of
3 matter on the PGW is negligible:
6
8 ε ω2, (1)
0 ≪
.
1 where ω is the characteristic frequency of a PGW, ε is the matter energy den-
0 sity (G = ~ = c = 1). These papers have revealed a number of phenomena of
1 interest, consisting in the induction of longitudinal electric oscillations in the
1
plasma by PGW. In spite of the strictness of the results obtained in [1.4], the
:
v effects discovered in these papers have very little to do with the real problem
i of GW detection. Moreover, the above calculations show lack of any prospect
X
for GW detectors based on dynamic excitation of electric oscillations by grav-
r
a itational radiation. There are two reasons for that: the smallness of the ratio
(m2G/e2)=10 43 and the small relativistic factor v2 /c2 of standardplasma-
−
h i
likesystems. TheGWenergytransformationcoefficienttoplasmaticoscillations
is directly proportionalto a product of these factors.
However, the situation may change radically if strong electric or magnetic
fields are present in the plasma. In Ref. [5], where the induction of surface
currentsatametal-vacuuminterfacebyaPGWwasstudied,itwasshownthat
the values of currents thus induced can be of experimental interest. In [6], on
thebasisofrelativistickineticequations,asetofMHDequationswasobtained,
whichdescribedthemotionofcollisionlessmagnetoactiveplasmainthefieldofa
PGWofanarbitrarymagnitudeinadriftapproximationanditwasshownthat,
providedthe propagationofthe PGWis transversal,therearisesaplasma drift
inthePGWpropagationdirection. Thesetofequationsobtainedin[6]israther
1
complex and unwieldy: it is a set of nonlinear partial differential equations. In
[7],however,it wasshownthat,providedthe plasmais originallyelectroneutral
anduniform,thesolutionoftheabovesetofequationsisstrictlystationary,i.e.,
itdepends only onretardedtime. This factpermits us to substantiallysimplify
the problem and to find its exact solution, possessing a number of remarkable
peculiarities.
2 The conditions of magnetic field embedding
in the plasma
As pointed out above, in [6], on the basis of a selfconsistent set of collisionless
kinetic equationsandthe Maxwellequations(i.e., onthe basisofgeneralrelati-
vistic Vlasov equations [8]), a set of MHD equations describing the motion of
magnetoactiveplasmainthefieldofaPGW,wasobtained. Thissetofequations
is obtained in the so-called drift approximation, i.e., in the first approximation
in the small parameter ξ :
ω
ξ = 1, (2)
ω ≪
B
whereω =eH/m cisthe Larmorfrequency. However,theequationsobtained
B e
in [6] are applicable only in the case of a strictly transverse PGW propagation,
where the original magnetic field is perpendicular to the GW propagation di-
rection. Itisnotdifficulttoverifythatifthe conditionsofthe stricttransvesity
of PGW propagation are not met, the equations of [6] violate the energy and
momentum conservation laws. For our purposes it is necessary to consider a
moregeneralcase,soinwhatfollowsweshallobtaintheMHDequationsonthe
basis of other principles.
It is not difficult to see that a consequence of the MHD equations from [6]
isthe magneticfieldembeddinginthe plasma(MFEP).Thisreflectthegeneral
nature of magnetoactive plasma provided that the condition (2) is met. There-
fore, in order to describe the motion of the plasma in a drift approximation, it
issimplertodemandatoncethattheMFEPconditionbemet. Mathematically
this requirement means a coincidence between the timelike eigenvectors of the
p
plasma energy-momentum tensor (EMT), T , and that of the electromagnetic
ik
f
field, T , i.e., according to Synge [9] , a coincidence of the dynamic velocities
ik
of the plasma and the electromagnetic field:
p
Ti uk =ε ui, (3)
k p
f
Ti uk =ε ui, (4)
k H
where
(u,u)D=f g uiuk =1. (5)
ik
2
We shall consider in this paper the EMT of the plasma as that of a perfect
isotropic fluid
p
Tik=(ε+p)vivk pgik, (6)
−
where
(v,v)=1, (7)
and ε and p are the fluid energy density and pressure connected by a certain
equation of state:
p=p(ε). (8)
Thus vi is the timelike eigenvector of the plasma EMT (vi = ui), while is
p
the eigenvalue Tik (ε =ε), and there remain the conditions (4):
p
f
Ti vk =ε vi. (9)
k H
Thus (9) are precisely the conditions of magnetic field embedding in the
plasma (MFEP). It is our purpose to clarify all the restrictions imposed by
these conditions on the Maxwell tensor F . Using the plasma velocity vector
ik
vi, we shall introduce the vectors of the electric field E and magnetic field H
i i
as observed in the frame of reference (FR) comoving with the plasma [10]:
E =vkF ; H =vk F∗ , (10)
i ki i ki
where F∗ is a tensor dual to the Maxwell antisymmetric tensor F :
ki ki
1
F∗ = η Flm, (11)
ki kilm
2
andη isthecovarianlyconstantdiscriminanttensor(see,e.g.,[9])satisfying
kilm
the identity
δq δq δq
η ηiqps δqps D=f δkp δlp δmp . (12)
iklm ≡− klm −(cid:12) k l m (cid:12)
(cid:12) δs δs δs (cid:12)
(cid:12) k l m (cid:12)
(cid:12) (cid:12)
Due to (10), the vectors E and H are space(cid:12)like and orthog(cid:12)onal to the velocity
(cid:12) (cid:12)
vector:
(v,E)=0; (v,H)=0. (13)
The relations (10) can be resolved with respect to the Maxwell tensor [10]:
F =v E v E η vlHm;
ik i k k i iklm
− −
F∗ =v H v H +η vlEm. (14)
ik i k k i iklm
−
Let us represent the EMT of the electromagnetic field as follows:
f 1 1
Ti= (FiFl + δiFlmF ) (15)
k 4π l k 4 k lm
3
using three vectors (v;E;H), one of which, (v), is timelike, while two others,
(E and H), are spacelike:
f 1
Ti= δi(E2+H2) 2vivk(E2+H2)+
k −8π k −
h
+2EiE +2HiH +2viη EpvqHs+2vkηipqsE v H , (16)
k k kpqs p q s
i
where the following notations are introduced:
E2 D=f (E,E); H2 D=f (H,H). (17)
− −
We shallrequire that the vector v be an eigenvectorof the EMT (16). Con-
tracting (16) with vk and taking into account the identities (12) and (13), we
get:
1
[vi(E2+H2) 2ηipqsE v H ]=ε vi. (18)
p q s H
8π −
From (18) we shall find the necessary condition for vi being an eigenvector of
the electromagnetic field EMT:
ηipqsE v H =λvi, λ R. (19)
p q s
∀ ∈
Contracting this relation with vi, we get λ = 0. Thus the necessary condition
of compatibility (3) and (4) is:
ηipqsE v H =0. (20)
p q s
The necessary and sufficient condition for the fulfilment of Eq. (20) is, as we
know, the complanarity of the vectors E,H,v, i.e.,
αv +βE +γH =0.
i i i
Contracting this relation with vi and taking into account (7), (13) we obtain
α=0. Thus, Eq. (20) is equivalent to the condition:
βE +γH =0.
i i
Since we consider magnetoactive plasma, we shall further assume:
F Fik 2(H2 E2)>0. (21)
ik
≡ −
Due to (21), the necessary and sufficient condition for the fulfilment of (20) is:
E =λH ; λ R (22)
i i
∀ ∈
Besides, according to (18),
E2+H2
ε = . (23)
H
8π
4
However, we have not yet extracted all the algebraic information contained in
(3) and (4). With due account of the definitions of the vectors E and H (10),
Eqs. (10) can be written as:
vk =O, (24)
ik
F
where a new antisymmetric tensor has been introduced:
ik
F
=F λF∗ . (25)
ik ik ik
F −
The relations (24) can be regarded as a set of linear homogeneous algebraic
equations with respect to the velocity vector vi. The necessary and sufficient
condition for a nontrivial compatibility of these equations is:
Det =0. (26)
ik
kF k
As is an even order antisymmetric matrix,
ik
kF k
1
Det = (√ g ik ∗ )2. (27)
ik ik
kF k 16 − F F
Therefore, the condition (26)) reduces to the following:
ηijkl =0, (28)
ij kl
F F
– in this case
rank =2, (29)
ik
kF k
i.e., the set (24) admits two linearly independent solutions for the eigenvector
vi.
Substituting into (28) the expressions for Fik and F∗ik from (14), we get:
ηijkl =4λ(1+λ2)H2 =0,
ij kl
F F
hence follows the only possible solution under the condition (21):
λ=0.
Thus we come to the followingrigorousconclusion. For the EMT ofelectro-
magneticfield (15)to permitas its eigenvectorthe dynamicalvelocityvectorof
theisotropicperfectfluidunderthecondition(21),itis necessaryandsufficient
that the electric field intensity vector in the comoving FR should be equal to
zero:
E =0. (30)
i
In this case the conditions (24)-(29) give:
F∗ik F =0; (31)
ik
Det F =0= rank F =2, (32)
ik ik
k k ⇒ k k
5
whiletheeigenvectorofthefluidEMTmustsatisfythesetoflinearhomogeneous
algebraic equations:
F vk =0. (33)
ik
NotethatifEq. (32)isfulfilled,thensimilarconditionsforthedualMaxwell
tensor are automatically fulfilled as well:
Det F∗ =0= rank F∗ =2. (34)
ik ik
k k ⇒ k k
With(30)writedowntheMaxwelltensorandtheelectromagneticfieldEMT
(15):
F = η vlHm;
ik iklm
−
F∗ik=viHk vkHi; (35)
−
f 1
Ti = (2H2viv 2HiH δiH2). (36)
k 8π k− k− k
Note that due to Eq. (35) a more severe condition than (31) is fulfilled:
F∗ Flk =0. (37)
ik
The summed EMT of the magnetoactive plasma
p f
Tik =Tik +Tik
takes the form:
T =( +P)v v Pg 2P n n , (38)
ik i k ik H i k
E − −
where
H2
P = ; =ε+ε ; P =p+P , (39)
H H H
8π E
P and being the summed pressure and energy density of the magnetoactive
E
plasma and
H
i
n = (40)
i
H
– is the spacelike unit vector of magnetic field direction:
(n,n)= 1, (41)
−
with
(n,v)=0. (42)
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3 MHD equations for plasma in a gravitational
field
TheMHDequationsaretobe obtainedonthe basisofthevanishingdivergence
requirement for the summed EMT of the magnetoactive plasma (38) supple-
mented by the first group of the Maxwell equations:
Tk =0, (43)
i,k
F∗ik =0. (44)
,k
Thissetofequationswithdueaccountoftheequationofstate(8),thedefinition
of the plasma EMT (8) and the algebraic relations (7), (13), (14), (30), (35)
and (38) completely describes the self-consistent motion of the magnetoactive
plasma with an embedded magnetic field in a prescribed gravitational field.
Indeed, Eqs. (43), (44) represent a set of 8 differential equations with respect
to 10 quantities ε,p,Hi,vi. However, the equation of state (8), the velocity
vector normalization (7), the ortogonality condition (13) and (30) raise the
total number of equations up to 12. Nevertheless, it turns out that not all of
these relations are independent, as we shall see below.
To deduce the MHD equations, let us take into account the well-known
relationship (see, e.g., [11]):
f 1
Tk = F Fkl. (45)
i,k −4π il ,k
Thus, Eq. (43) can be presented in the form:
F Φk =τ , (46)
ik i
where
Φk D=f Fkl; (47)
,l
p
τ D=f 4π Tk . (48)
i i,k
−
Eqs. (46) can be regardedas a set of linear inhomogeneous algebraic equations
with respect to Φk. If Det F = 0, then the equations are solved in a simple
ik
k k6
straightforwardway:
k F;τ
Φk = A k k, (49)
Det F
k k
where k F;τ isacofactoroftheaugmentedmatrixoftheset(46). Inpartic-
A k k
ular, for the case of vacuum (τ = 0) we obtain a trivial solution: Φk = 0, and
i
the set of equations (43), (44) reduces to the Maxwell equations in vacuum.
Now we pose the problem of solving the set of equations (46) with respect
to Φk, i.e., the problem of reducing Eqs. (43, (44) to the form of the Maxwell
equations with minimal requirements upon the electomagnetic field invariants:
Fik F∗ik=0, (50)
7
F Fik >0. (51)
ik
The existence of a positive invariant (51) means that we can choose a local FR
where the electric field is absent [11].
Provided (50) is fulfilled, as before (see (32)),
Det F =0= (52)
k k ⇒
rank F =2. (53)
k k
Thus for the consistency of the algebraic set of equations (46) under the condi-
tion (50) it is necessary and sufficient that:
rank F;τ =2. (54)
k k
Calculating allthe 3rdorder minors ofthe augmented matrix F;τ with (50),
k k
we obtain a condition equivalent to (54):
F∗ik τk =0 rank F;τ =2. (55)
⇐⇒ k k
To solve the set of equations (46), consider the eigenvectors of the matrix
F :
ik
k k F uk =λu . (56)
ik i
Due to the antisymmetry of F , it follows from (56) that either λ = 0, or u
ik
is a null vector. It can be demonstrated that provided Eqs. (50) and (51) are
valid, u cannot be a null vector. Thus, the Maxwell tensor admits only nonnull
eigenvectors with zero eigenvalues:
F uk =0. (57)
ik
By(50),(52)and(53),thiseigenvalueisdoublydegenerate,andthus,according
to a well-known algebraic theorem, two lineary independent eigenvectors cor-
respond to it: u and u.
(1) (2)
Underthecondition(51)wecanalwayschoosealocalFRwhereF =0. In
i4
thisFRtherealwaysexistsaneigenvectoroftheMaxwelltensoroftheformuk =
δk. Therefore,oneoftheeigenvectorsofthematrix F ,e.g., u,istimelikeand
4 k k (1)
the second one, u, is spacelike. Using the standard orthogonalization process,
(2)
we normalize them as follows:
(u, u)=1; (u, u)= 1; (u, u)=0. (58)
(1) (1) (2) (2) − (1) (2)
Then the general solution of (57) can be written in the form
uk =αuk +β uk, (59)
(1) (2)
where α and β are arbitary scalars.
8
Letusnowinvestigatetherelations(55),whichcanberegardedasalgebraic
equations with respect to τ. Since
FikFik =F∗ikF∗ik<0, (60)
−
and Eq. (50) is invariant under the substitution F F∗, as well as the expres-
↔
sion for Det F (27), we conclude that the dual matrix F∗ also admits
ik
k k k k
two and only two linearly independent spacelike eigenvectors w and w, which
(1) (2)
correspondtoazeroeigenvalue. Itisnotdifficulttoverify(e.g.,turningtoaFR
where Fα4 =0) that the rank of the unified matrix F,F∗ under the condition
k k
(50) is equal to 4. Hence the eigenvectors of the matrices F and F∗ are
k k k k
linearly independent and we can choose the following normalization for them:
(w, w)= 1; (w, w)= 1;
(1) (1) − (2) (2) −
(w, w)=0; (w, u)=0, (α,β =1,2). (61)
(1) (2) (α) (β)
Thus, the general solution to Eq. (55) is:
τ =λw +µw , (62)
i i i
(1) (2)
where λ and µ are arbitary scalars and due to (61):
(τ, u)=0. (63)
(α)
By (53) and (55), the Maxwell tensor and its dual can be represented in
terms of the eigenvectors of the matrices F and F∗ :
k k k k
F = ση ulum; (64)
ik iklm
− (1) (2)
F∗ik=̺ηiklm wlwm, (65)
(1) (2)
whereσ and̺ arecertainscalars. Contractingthese relationswiththe discrim-
inant tensor, we obtain the dual relations:
F∗ik=σ(uiuk uiuk); (66)
(1) (2) −(2) (1)
Fik = ̺(wiwk wiwk). (67)
− (1) (2) −(2) (1)
In particular, the following relation stems from (64) and (66):
F∗ Fkl =0. (68)
ik
9
Using the Maxwell tensor representation (64) and the orthonormality rela-
tions (58), we get the useful formula:
F Fil =σ2( u ul +u ul +δl). (69)
ik −(1k)(1) (2k)(2) k
Contracting (69), we get:
1
F Fik =σ2 >0. (70)
ik
2
ContractingEqs(46)withFil andtakingintoaccount(69)and(70),weobtain
an equation equivalent to Eq. (46):
σ2[Φl ul (u,Φ)+ ul (u,Φ)]=Filτ . (71)
i
−(1) (1) (2) (2)
A special solution to Eq. (71) is:
1
Φi = Filτ . (72)
(1) σ2 i
Therefore the general solution of Eqs. (46) can be presented in the form:
p
8πFik Tj
Fik Φi = k,j +α ui +β ui . (73)
,k ≡ FlmFlm (1) (2)
This exhausts the problem of reducing the set of equations (43),(44) to the
standard Maxwell form.
If the external currents are absent (α = β = 0), then Eq. (73) reduces to
the form of the second group Maxwell equations:
Fik = 4πJi , (74)
,k − dr
where:
p
2Fik Tl
Ji = k,l (75)
dr − F Fjm −
jm
is the drift current, which by (68) satisfies the relation:
F∗ij Jdir =0, (76)
Hence, due to (55) and (63:
(J , u)=0, (77)
dr
(α)
(J ,J )<0 (78)
dr dr
−
i.e., the drift current is spacelike.
Calculating the covariantdivergence of Eq. (68) with the aid of the first
l
∇
group Maxwell equations (44), we get the differential implication:
F∗kl Fkl,i =Fkl F∗kl,i=0. (79)
10