The general relativistic MHD dynamo equation

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by  Mattias Marklund

Mon.Not.R.Astron.Soc.000,??–??(2004) Printed2February2008 (MNLATEXstylefilev2.2)
The general relativistic MHD dynamo equation
M. Marklund1,2 and C. A. Clarkson3
1Department of Physics, Ume˚a University,SE–901 87 Ume˚a, Sweden
2Centre for Fundamental Physics, Rutherford Appleton Laboratory, Chilton Didcot, Oxfordshire, OX11 0QX, UK
3Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 2EG, UK
5 Email:[email protected] [email protected]
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ABSTRACT
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The magnetohydrodynamic dynamo equation is derived within general relativity, us-
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ing the covariant 1+3 approach, for a plasma with finite electric conductivity. This

2 formalism allows for a clear division and interpretation of plasma and gravitational
v effects, and we have not restricted to a particular spacetime geometry. The results
0 shouldbeofinterestinastrophysicsandcosmology,andtheformulationiswellsuited
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togaugeinvariantperturbationtheory.Moreover,thedynamoequationispresentedin
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some specific limits. In particular, we consider the interaction of gravitational waves
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1 with magnetic fields, and present results for the evolution of the linearly growing
4 electromagnetic induction field, as well as the diffusive damping of these fields.
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/ Key words: MHD — Gravitation — Gravitationalwaves
h
p
-
o
r 1 INTRODUCTION
t
s
Magnetic fields play a vital role in many astrophysical systems and in cosmology, and the possible sources for the fields on
a
: differentscaleshasbeenthefocusofimmenseresearch overtheyears(Kronberg1994;Han & Wielebinski2002;Parker1979;
v
Zeldovich, Ruzmaikin & Sokoloff 1983;Beck et al. 1996).Inparticular, theorigin oflarge scale cosmological fieldsisstill not
i
X clearly understood. Although there are well understood mechanisms for amplifying a given seed field (Grasso & Rubinstein
r 2001; Widrow 2002), the explanation of the seed sources is more cumbersome, especially in the case of an inflationary sce-
a
nario. A wide range of possible physical explanations for establishing a seed field of high enough field strength has been pro-
posedintheliterature(see,e.g.,Harrison(1970);Hogan(1983);Vachaspati(1991);Gasperini, Giovanini & Veneziano(1995);
Sigl, Jedamzik & Olinto(1997);Joyce & Shaposhnikov(1997);Davis et al.(2001);Calzetta & Kandus(2002);Dimopoulos et al.
(2002);Betschart, Dunsby& Marklund(2004);Matarrese et al.(2004)).Themainmechanismbehindtheamplificationofthe
seed fields is the dynamo (Parker 1979; Zeldovich, Ruzmaikin & Sokoloff 1983; Beck et al. 1996), for which the fluid motion
managestoamplifyevenveryweakfieldsonveryshorttimescales.Thedynamomechanismderivesfromthemagnetohydrody-
namic(MHD)approximation,andithasbeenanalysedvigorouslywithin,e.g.,expandingcosmologicalmodels(foraselection,
see Fennelly (1980); Sil, Banerjee & Chatterjee (1996); Gailis, Frankel& Dettmann (1995); Jedamzik, Katalini´c & Olinto
(1998); Brandenburg, Enqvist & Olesen (1996); Subramanian & Barrow (1998) and references therein).
Cosmic magnetic fields are observed on all but the largest scales. On the other hand, gravitational waves have yet to
be directly detected. Although strong evidence exists in their favour (such as the PSR 1913+16 binary pulsar observations
(Hulse& Taylor 1975)), the very weak interaction of gravitons with light and matter makes an earth based detection diffi-
cult. Currently, gravitational wave detectors, both of interferometer and bar type, are coming on-line, and are expected to
start collecting important astronomical data, opening up a new window for observational astrophysics and cosmology. As
gravitational wave information is gathered, the detailed study of typical wave signatures will be of major importance. Thus,
knowledgeofbothdirectgravitationalwavesignature,andindirectsignatures,e.g.,electromagneticwavesinducedbygravity
waves (Clarkson et al. 2004), will facilitate the study of gravitational waves, and is likely to bring new understanding to the
observations.
Theinteraction ofgravitationalwaves,andothergeneralrelativistic gravityeffects,withelectromagnetic fieldsandplas-
mashaveattractedabroadinterestoverthelastfewyears,becauseitspossibleapplicationtobothastrophysicsandcosmology
(e.g., Blandford & Znajek (1977); Thorne & Macdonald (1982); Macdonald & Thorne (1982); Marklund,Dunsby& Brodin
(2000);Tsagas, Dunsby& Marklund(2003);Clarkson et al.(2004);Tomimatsu(2000);Tsagas & Barrow(1997);Tsagas & Maartens
2 M. Marklund and C. Clarkson
(2000); Tsagas (2001)). Here, we will derive the general form of the dynamo equation, using the 1+3 covariant formalism
(Ellis & van Elst 1998), and analyse its coupling to matter and spacetime geometry. This formalism allows for a clear cut
interpretation of gravitational vs. purely MHD effects, and also presents the influenceof the gravitational kinematics on the
MHD modes for straightforward comparison. Specifically, we will consider the propagation of gravitational waves through a
resistive magnetohydrodynamic plasma, and derive covariant equations for the propagation of a resulting magnetohydrody-
namic wave. Here, a word of caution concerning the use of the phrase ‘dynamo’ is in place. While speaking of the dynamo
equation,ageneralevolutionequationforthemagneticfieldindiffusivemediaisassumed,whilethedynamomechanism is a
particularapplication of thisgeneral equation.Indeed,sincethedynamomechanism iswithout question themost prominent
feature of the general dynamo equation, we would like to stress that although we derive the general dynamo equation, we
will not apply this to the dynamo mechanism. Instead, we will investigate what could appropriately be termed induction
fields. The waves presented here represents linearly growing induction fields, and are thus not due to the proper dynamo
mechanismwhichinsteadgivesrisetorapidamplificationofseedmagneticfieldduetothevorticalfluidmotion.Furthermore,
theeffects of a finite conductivity is presented. The equations are analysed numerically, and we discuss possible applications
to astrophysical and cosmological systems.
2 BASIC EQUATIONS
When analysing low frequency magnetised plasma phenomena, MHD gives an accurate and computationally economical
description. Specifically, a simple plasma model is obtained if the characteristic MHD time scale is much longer than both
the plasma oscillation and plasma particle collision time scales, and the characteristic MHD length scale is much longer
than the plasma Debye length and the gyro radius. These assumptions will make it possible to describe a two-component
plasma in terms of a one-fluid description, as the electrons can be considered inertialess (see also, e.g., Servin & Brodin
(2003); Moortgat & Kuijpers (2003, 2004); Papadopoulos et al. (2001); Papadopoulos, Vlahos & Esposito. (2002)). The one-
fluid description means a tremendous computational simplification, especially for complicated geometries. Moreover, if the
mean fluid velocity, the mean particle velocity, and the Alfv´en velocity is much smaller than the velocity of light in vacuum,
thedescription becomes non-relativistic and simplifies further.
2.1 Covariant theory
The1+3covariantapproachreliesontheintroductionofatimelikevectorfieldua:u ua =−1,withwhichalltensorialobjects
a
are split into their invariant ‘time’ and ‘space’ parts: scalars, 3-vectors, and projected, symmetric, trace-free (PSTF) tensors
(seeEllis & van Elst(1998)foracomprehensivereview).Thisallows ustowritethecoupledEinstein-Maxwellequationsina
relatively simple and intuitive fashion. Here, we introduce consistent 3-vector notation of Euclidean vector calculus, making
thefully general relativistic equations relatively easy to read.
Inanarbitrarycurvedspacetime,inunitssuchthatc=1andG=1/8π,Maxwell’sequationstaketheform(Tsagas & Barrow
1997; Ellis & van Elst 1998; Marklund et al. 2003)
B˙hai+curlEa = − 2Θhab−σab+ǫabcω B −ǫabcu˙ E , (1)
(cid:16)3 c(cid:17) b b c
−E˙hai+curlBa = µ jhai+ 2Θhab−σab+ǫabcω E −ǫabcu˙ B , (2)
0 (cid:16)3 c(cid:17) b b c
D Ea = 2ω Ba, (3)
a a
D Ba = −2ω Ea, (4)
a a
where curlXa ≡ǫabcD X and X˙hai ≡habX˙ for any spatial vector Xa, and h =g +u u is the metric on the local rest
b c b ab ab a b
space orthogonal to the observer four-velocity ua. Moreover, we have assumed quasi-neutrality, i.e., the total charge density
satisfies ρ≈0sincetheelectronsandionsfollow thesame motion.Furthermore,wewill assumethatOhm’slawin theform
1
jhai = Ea+ǫabcv B , (5)
µ λ(cid:16) b c(cid:17)
0
holds,wherejhai ≡habj isthethree-current,(µ λ)−1 istheelectric conductivity,λ isthemagneticdiffusivity,andva isthe
b 0
plasmathree-velocity.Wenotethathighconductivityimpliessmallmagneticdiffusivity.Thecaseofnon-idealMHDincurved
spacetimes has been treated sparsely in the literature (with some exceptions (Fennelly 1980; Jedamzik, Katalini´c & Olinto
1998)), and the possible effects of a finite conductivity due to gravity–electromagnetic field or fluid couplings has therefore
largely gone unnoticed.
Thus, taking thecurl of the curlof Ba and using the commutator relation
curl(curlB)a =−D2Ba+Da(D Bb)+2ǫabcB˙ ω +RabB , (6)
b hbi c b
The general relativistic MHD dynamo equation 3
where R is defined by therelation
ab
R ≡ 2 E +Λ− 1Θ2−σ2+ω2 h +Π −σ˙ +D u˙
bd 3(cid:0) 3 (cid:1) bd bd hbdi hb di
−1Θ(σ −ω )−u˙ u˙ −2 σ kσ +ω ω −2σ ωk , (7)
3 bd bd hb di (cid:16) hb dik hb di(cid:17) k[b d]
where E = E + 1(E2+B2) is the total energy density, and Πab = Πab −(EhaEbi +BhaBbi) is the total anisotropic
fluid 2 fluid
pressure. Taking thecurl of Eq. (2), using Eq. (1) and (5) to replace Ea by Ba and jhai,we obtain
curl(curlB)a =λ−1ǫabcǫ D (vdBe)− 2Θ+λ−1 B˙hai+ 2Θhab−σab+ǫabcω B −λµ ǫabcu˙ j +ǫabcǫ u˙ vdBe
cde b (cid:0)3 (cid:1)h (cid:16)3 c(cid:17) b 0 b hci cde b i
−2µ λǫabcj D Θ+ 2ǫabcǫ vdBeD Θ−ǫabcD σ µ λjhdi−ǫdefv B −ǫabcD ǫ ωd µ λjhei−ǫefgv B
3 0 hbi c 3 bde c bh cd(cid:16) 0 e f(cid:17)i bh cde (cid:16) 0 f g(cid:17)i
−ǫabcD ǫ u˙dBe +curlE˙hai. (8)
b(cid:16) cde (cid:17)
Furthermore,the last term in Eq. (8), dueto thedisplacement current E˙hai, may be written
curlE˙hai =−B¨hai+Ξa (9)
where
Ξa ≡− 2Θ˙hab−σ˙habi+ǫabcω˙ B − Θhab−σab+ǫabcω B˙ − 1Θ 2Θhab−σab+ǫabcω B
(cid:16)3 hci(cid:17) b (cid:16) c(cid:17) hbi 3 (cid:16)3 c(cid:17) b
−ǫabcu˙ curlB −µ j +ǫ u˙dBe− 1Θh −σ +ǫ ωe Ed + 1ǫabcE q + 2u˙haωbi+Dhaωbi−(curlσ)ab E
bh c 0 hci cde (cid:0)3 cd cd cde (cid:1) i 2 b c h i b
−ǫabcu¨ E +ǫabc(σ +ǫ ωe)DdE (10)
hbi c bd bde c
by commuting spatial and time like covariant derivatives and using Maxwell’s equations. The commutation of derivatives
introducescurvatureeffectsintotheexpression.Hereqa =qa +ǫabcE B istheenergyfluxduetofluidandelectromagnetic
fluid b c
(i.e., Poynting flux) contributions. The latter derives from the gravitational self-interaction of the electromagnetic field, and
can in many cases safely be neglected. We note that all terms in Ξa defined by Eq. (10) are curvature contributions. The
displacement current is normally neglected in the MHD approximation, since it corresponds to high frequency phenomena.
Here we see that the presence of gravity alters this interpretation of the displacement current. We will be interested in low
frequency phenomena,and we therefore assume curlE˙hai≈Ξa.
Thus, equating (6) with the expression (8), we obtain the general relativistic dynamo equation (see also Tsagas (2004)
for the source free general relativistic electromagnetic wave equations)
B˙hai−ǫabcǫ D (vdBe)−λD2Ba =−2λΘB˙hai+2λǫabcω B˙ +2λDa ω µ λjhbi−ǫbcdv B −λRbaB
cde b 3 b hci h b(cid:16) 0 c d(cid:17)i b
− 1+ 2λΘ 2Θhab−σab+ǫabcω B −µ λǫabcu˙ j +ǫabcǫ u˙ vdBe
(cid:0) 3 (cid:1)h(cid:16)3 c(cid:17) b 0 b hci cde b i
−2µ λ2ǫabcj D Θ+ 2λǫabcǫ vdBeD Θ−λǫabcD σ µ λjhdi−ǫdefv B
3 0 hbi c 3 bde c bh cd(cid:16) 0 e f(cid:17)i
−λǫabcD ǫ ωd µ λjhei−ǫefgv B −λǫabcD ǫ u˙dBe +λΞa (11)
bh cde (cid:16) 0 f g(cid:17)i b(cid:16) cde (cid:17)
where we have used Maxwell’s equation (4) and Ohm’s law (5). The terms on the right hand of the dynamo equation (11)
representstheinfluenceof general relativistic gravity,while theleft handsidegives thenormal dynamoaction from thefluid
vorticity ǫ Dbvc.
abc
SometimesitmaybeadvantageoustokeeptheelectricfieldinthegravitationalrighthandsideofEq.(11),inwhichcase
we have
B˙hai−ǫabcǫ D (vdBe)−λD2Ba =−2λΘB˙hai+2λǫabcω B˙ +2λDa ω Eb −λRbaB
cde b 3 b hci (cid:16) b (cid:17) b
− 1+ 2λΘ 2Θhab−σab+ǫabcω B −ǫabcu˙ E − 2ǫabcE D Θ−λǫabcD σ Ed
(cid:0) 3 (cid:1)h(cid:16)3 c(cid:17) b b ci 3 b c b(cid:16) cd (cid:17)
−λǫabcD ǫ ωdEe −λǫabcD ǫ u˙dBe +λΞa. (12)
b(cid:16) cde (cid:17) b(cid:16) cde (cid:17)
2.2 Three-dimensional vector notation
The general relativistic dynamo equation may also be formulated in terms of three-dimensional vector notation. For every
spacelikevectorXa orPSTFtensorYab wedenotethecorresponding three-dimensionalvectorortensoraccordingtoX~ and
Y¯¯. With thesedefinitions, Eq. (12) takes theform
B~˙ −∇~ ×(~v×B~)−λ∇2B~ =−2λΘB~˙ +2λ~ω×B~˙ +2λ∇~(~ω·E~)−λB~ ·R¯¯− 1+ 2λΘ 2ΘB~ −σ¯¯·B~ +B~ ×~ω−~a×E~
3 (cid:0) 3 (cid:1)(cid:16)3 (cid:17)
−2λE~ ×∇~Θ−λ∇~ ×(σ¯¯·E~)−λ∇~ ×(~ω×E~)−λ∇~ ×(~a×B~)+λΞ~, (13)
3
4 M. Marklund and C. Clarkson
where we have introduced the three-dimensional acceleration vector~a corresponding to u˙a, and ∇~ corresponds to Da. Dot
and cross productsare defined in theobvious way.Furthermore, theterms dueto thedisplacement current takesthe form
~Ξ=− 2Θ˙B~ −σ¯¯˙ ·B~ +B~ ×~ω˙ − 2ΘB~˙ −σ¯¯·B~˙ +B~˙ ×ω~ −~a˙ ×E~ + 1Θ∇~ ×E~ +(σ¯¯·∇~)×E~ −(~ω×∇~)×E~
(cid:16)3 (cid:17) (cid:16)3 (cid:17) 3
−~a× ∇~ ×B~ −µ ~j+~a×B~ − 2ΘE~ −σ¯¯·E~ +E~ ×~ω + 1E~ ×~q−H¯¯ ·E~, (14)
h 0 (cid:16)3 (cid:17)i 2
where we have introduced the magnetic part of the Weyl tensor Hab = −2u˙haωbi−Dhaωbi+(curlσ)ab, which signifies the
presence of gravitational waves or frame dragging effects.
2.3 The equations of motion
The dynamo equation contains the centre of mass fluid velocity va ≡ (E va + E va )/(E + E ) and the current
(e) (e) (i) (i) (e) (i)
jhai ≡ ρ va +ρ va , e (i) denoting the electrons (ions). Using quasi-neutrality, the evolution of the total fluid energy
(e) (e) (i) (i)
density (droppingtheindex denoting thefluid) E ≡E +E and thecentre of mass velocity is given by
(e) (i)
E˙+D (Eva)=−(Θ+u˙ va)E, (15)
a a
and
E v˙hai+vbD va =− 1Θva+u˙a+σavb+ǫabcω v E +ǫabcj B , (16)
(cid:16) b (cid:17) (cid:16)3 b b c(cid:17) b c
respectively,in thecold plasma limit. Moreover,thecurrentja canbeexpressedin termsofthemagnetic fieldviaMaxwell’s
equation (2) and Ohm’s law (5).
WenotethattheEqs.(15) and(16)can easily begeneralised toincorporate anisotropic and/orviscouseffects(see, e.g.,
Subramanian & Barrow (1998)).
3 SPECIAL CASES
Inordertoextractsomeinformation fromtherathercomplicated relativisticdynamoequation,weshallpresentsomespecial
cases.
3.1 Infinite conductivity
The large bulk of literature on general relativistic MHD has treated the case of ideal MHD, i.e., infinite conductivity. As
shown below, thegeneral dynamoequation greatly simplifies when this assumption is made.
As σ→∞,the diffusion of themagnetic field lines decreases, i.e., λ→0, and we are left with
B˙hai−ǫabcǫ D (vdBe)=− 2Θhab−σab+ǫabcω B −ǫabcǫ u˙ vdBe, (17)
cde b (cid:16)3 c(cid:17) b cde b
or
B~˙ −∇~ ×(~v×B~)=− 2ΘB~ −σ¯¯·B~ +B~ ×~ω+~a× ~v×B~ , (18)
h3 (cid:16) (cid:17)i
whichofcoursealsocanbefounddirectlyfromMaxwell’sequationsandthevanishingoftheLorentzforce.Infiniteconductivity
is a common assumption in cosmology, for example.
3.2 Effects of expansion/collapse
As in the case of infinite conductivity, the case of curved spacetime MHD has largely resorted to analysing the effects of
expansion/collapse, at which the gravitational–electromagnetic field coupling is kept at a minimum. It also gives a good
description of some fundamental aspects of cosmological MHD phenomena.
Thesimplest case wherethenontrivial effectsofexpansion occuris in homogeneousand isotropic fluidbackground.The
homogeneityandisotropysignificantlysimplifiestheEinsteinequations,andthespacetimeisdescribedintermsofthescalar
quantitiesE, p, Θ,and Λ. Therefore, equations (11) and (13) simplify to
B˙a−ǫabcǫ D (vdBe)−λD2Ba =−2λΘB˙a−λRbaB − 1+ 2λΘ 2ΘBa+λΞa, (19)
cde b 3 b (cid:0) 3 (cid:1)3
and
B~˙ −∇~ ×(~v×B~)−λ∇2B~ =−2λΘB~˙ −λB~ ·R¯¯− 1+ 2λΘ 2ΘB~ +λΞ~, (20)
3 (cid:0) 3 (cid:1)3
respectively,where R = 2(E +Λ− 1Θ2)h is theRicci three-curvature.Theeffects of thedisplacement current becomes
ab 3 3 ab
Ξa =−2Θ˙Ba−ΘB˙a− 2Θ2Ba, (21)
3 9
The general relativistic MHD dynamo equation 5
i.e.,
Ξ~ =−2Θ˙B~ −ΘB~˙ − 2Θ2B~ (22)
3 9
Herewe haveneglected thegravitational self-interaction of theelectromagnetic field. Thus, from Eq. (19) and (20), together
with (21) and (22), we obtain
B˙a−ǫabcǫ D (vdBe)−λD2Ba =−1Θ(2Ba+5λB˙a)− 1λ E −3p+4Λ+ 2Θ2 Ba, (23)
cde b 3 3 (cid:0) 3 (cid:1)
and
B~˙ −∇~ ×(~v×B~)−λ∇2B~ =−1Θ(2B~ +5λB~˙)− 1λ E −3p+4Λ+ 2Θ2 B~, (24)
3 3 (cid:0) 3 (cid:1)
respectively.Thus, we see that for collapsing solutions (Θ<0), there will bea growing magnetic field.
3.3 The effects of rotation
In a rotating space time, i.e., ωa 6= 0, Einstein’s equations shows that the vorticity has as its source the acceleration u˙a.
G¨odel’s universe does not require this, as it is spacetime homogeneous, but for the sake of completeness we will keep both
spacetime acceleration and vorticity, while neglecting the expansion and shear (thusonly considering only rigid rotation), in
theequations presented below. They read
B˙hai−ǫabcǫ D (vdBe)−λD2Ba =2λǫabcω B˙ +2λDa ω µ λjhbi−ǫbcdv B −λRbaB −λǫabcD ǫ u˙dBe
cde b b hci h b(cid:16) 0 c d(cid:17)i b b(cid:16) cde (cid:17)
− −ǫabcω B −µ λǫabcu˙ j +ǫabcǫ u˙ vdBe −λǫabcD ǫ ωd µ λjhei−ǫefgv B +λΞa (25)
h b c 0 b hci cde b i bh cde (cid:16) 0 f g(cid:17)i
or
B~˙ −∇~ ×(~v×B~)−λ∇2B~ =2λ~ω×B~˙ +2λ∇~ ω~ · µ λ~j−~v×B~ −λB~ ·R¯¯− B~ ×~ω−~a× µ λ~j−~v×B~
h (cid:16) 0 (cid:17)i h (cid:16) 0 (cid:17)i
−λ∇~ × ω~ × µ λ~j−~v×B~ −λ∇~ × ~a×B~ +λΞ~. (26)
h (cid:16) 0 (cid:17)i (cid:16) (cid:17)
We note that the contribution Ξa from the displacement current is only slightly simplified compared to the generic case.
Thus,asinmanyproblemsofgeneralrelativity,rotationofspacetimegivesrisetohighlycomplexequations,andthedynamo
equation is noexception.
4 GRAVITATIONAL WAVES
LineargravitationalwavesonahomogeneousbackgroundcanbecovariantlydefinedastensorperturbationssatisfyingD σab=
a
D Eab = D Hab = 0, thus being transverse and traceless. In order to clearly see the effects of gravitational waves on the
a a
induction of electromagnetic fields, we consider this perturbation on a Minkowski background. In this case, the dynamo
equation becomes
B˙hai−ǫabcǫ D (vdBe)−λD2Ba =−λRbaB +σabB −λǫabcD (σ Ed)+λΞa, (27)
cde b b b b cd
or
B~˙ −∇~ ×(~v×B~)−λ∇2B~ =−λB~ ·R¯¯+σ¯¯·B~ −λ∇~ ×(σ¯¯·E~)+λΞ~, (28)
where we haveneglected all productsof gravitational variables.
The contribution from Ξa becomes
Ξa =σ˙habiB +σabB˙ +ǫabcσ DdE −(curlσ)abE . (29)
b hbi bd c b
Equation(7)givesanexpressionfortheRiccithree-curvatureintermsoftheshearoftheperturbation.Thus,using(29),
Eqs. (27) and (28) give us
B˙hai−ǫabcǫ D (vdBe)−λD2Ba =σabB +λ 2σ˙habiB −ǫabcD (σ Ed)+σabB˙ +ǫabcσ DdE −HabE , (30)
cde b b h b b cd hbi bd c bi
or
B~˙ −∇~ ×(~v×B~)−λ∇2B~ =σ¯¯·B~ +λ 2σ¯¯˙ ·B~ −∇~ ×(σ¯¯·E~)+σ¯¯·B~˙ +(σ¯¯·∇~)×E~ −H¯¯ ·E~ , (31)
h i
respectively, where Hab = (curlσ)ab. The terms on the rhs describe the effect of the curvature, and given a GW they will
drive the magnetic field evolution. We see that the finite electric conductivity gives rise to new couplings between GWs and
magnetic fields.
6 M. Marklund and C. Clarkson
4.1 Gravitational waves in a constant magnetic field
The case of a GW passing through an initially constant magnetic field will help illustrate these equations. We treat all
disturbances∇B inthemagneticfieldaswellasthegravitationalwavevariablesasfirstorderperturbations,andwetherefore
neglect all terms in the equations which are a product of derivatives of the magnetic field with GW terms (σ ), and terms
ab
O([∇B]2). We may also neglect disturbances in the energy density of the fluid.With these approximations, we find that the
momentum conservation equation (16) becomes
Ev˙a =ǫabcj B , (32)
b c
while the current is given by Eq. (2):
ja =µ−1curlBa. (33)
0
The dynamoequation (30) simplifies to
B˙a−ǫabcǫ BeD vd−λD2Ba =2λσ˙abB +σabB . (34)
cde b b b
Wemay decouple Eqs. (32)–(34) by taking thetime derivative of Eq.(34):
1
B¨a−λD2B˙a+ Bb −BaD2B +Bc(DaD B −D D Ba) =2λσ¨abB +σ˙abB (35)
µ0E (cid:2) b b c b c (cid:3) b b
where (neglecting backreaction from the electromagnetic field) the shear perturbation is determined by the source-free wave
equation
σ¨ab−D2σab=0. (36)
Wesplit theperturbed magnetic field parallel and perpendicular to the‘background’ magnetic field according to
Ba =B [(1+B)ea+Ba], Ba ≡NabB /B , and B≡(e Ba−1)/B . (37)
0 b 0 a 0
whereN =h −e e andea isaspacelikeunitvector:e ea =1(seeClarkson & Barrett(2003)foradetaileddiscussion of
ab ab a b a
thissplit).Here,B isaconstant,denotingthemagnitudeofthemagneticfieldwhentheGWiszero(inwhichcaseBa =B ea).
0 0
The perturbations in the static magnetic field are given by B, parallel to the background field, and Ba perpendicular to it.
Weneglect all productsof B and Ba, and terms of theform Bσ (notethat ∇e=O(σ)),since these are of higher order.
The shear perturbation can be decomposed according to
σ =Σ +2Σ e + e e − 1N Σ (38)
ab ab (a b) (cid:0) a b 2 ab(cid:1)
byintroducing thegravitational wave variables
Σ≡σabe e , Σ ≡N σbce , and Σ ≡ N cN d− 1N Ncd σ , (39)
a b a ab c ab (cid:16) (a b) 2 ab (cid:17) cd
and theoperators
d
D ≡eaD = and Da ≡NabD , (40)
k a dz ⊥ b
allow us to split thespatial derivativealong, and perpendicular to, the unperturbedmagnetic field.1
Using these definitions, and the frame choice2 e˙a =0=Dae , we may split Eq. (35) into a longitudinal scalar equation
⊥ a
along ea:
B¨−λD2B˙ −C2D2B =2λΣ¨+Σ˙, (41)
A
where C ≡(ǫ B2/E)1/2 is theAlfv´en velocity of theMHD plasma, and a transverse vectorequation perpendicular toea:
A 0 0
B¨a−λD2B˙a−C2 D2Ba−DaD B =2λΣ¨a+Σ˙a. (42)
A(cid:0) k ⊥ k (cid:1)
We also have the D Ba = 0 constraint, which takes the simple form D B+DaB =0 using our new variables. Neglecting
a k ⊥ a
backreaction,gravitationalwavespropagatingalongthebackgroundmagneticfieldwillnotaffectthemagneticfieldevolution,
duetothecondition of vanishingdivergenceD σab=0=e σab.This can also beseen from Eq.(38),where(if ea represents
a a
the propagation direction of the gravitational wave) Σ is the only non-vanishing part of the decomposition of σ , thus
ab ab
removingthecoupling tothemagnetic field viaΣ and Σa.On theotherhand,includingthebackreaction from themagnetic
field on thegravitational wave will in general makethe gravitational wave non-transverse, by the introduction of small scale
tidal forces as well as large scale curvatureeffects, but this issue is left for future research.
Next we introduce spatial harmonics for the perturbed variables (see Clarkson & Barrett (2003)). Define the scalar
harmonics
D2Q =−k2Q , (43)
⊥ (k⊥) ⊥ (k⊥)
1 NotethatthedefinitionofDa⊥ fortensorshasprojectionwithNab oneachfreeindex(Clarkson&Barrett2003).
2 WerefertoClarkson&Barrett(2003)forafulldiscussionofframechoicevs.trueGWdegreesoffreedom.
The general relativistic MHD dynamo equation 7
wherek istheharmonic numberfor thepart ofthefieldsperpendiculartoea.Wecan thendefinevectorharmonicsof even
⊥
and odd parity,
Qa =(k )−1DaQ Q¯a =(k )−1ǫaDbQ , (44)
(k⊥) ⊥ ⊥ (k⊥) (k⊥) ⊥ b ⊥ (k⊥)
respectively,where ǫ =ǫ ec =−ǫ isthevolumeelement ontwo-surfaces orthogonaltoea.Thesevectorharmonicsobey
ab abc ba
D2Qa =−k2Qa (similarly for Q¯a ).
⊥ (k⊥) ⊥ (k⊥) (k⊥)
Using these definitions, we may expand themagnetic field variables as
B = BS Q , (45)
X (k⊥) (k⊥)
k⊥
Ba = BV Qa +B¯V Q¯a , (46)
X (k⊥) (k⊥) (k⊥) (k⊥)
k⊥
and similarly for theGW shear variables
Σ= ΣS Q , and Σa = ΣV Qa +Σ¯V Q¯a . (47)
X (k⊥) (k⊥) X (k⊥) (k⊥) (k⊥) (k⊥)
k⊥ k⊥
Using theseharmonic expansions, Eq. (41) can now bewritten as a partial differential equation for each k , using ′ =∂ :
⊥ z
B¨ −C2B′′−λB˙′′+k2λB˙ +k2C2B =2λΣ¨ +Σ˙ , (48)
S A S S ⊥ S ⊥ A S S S
while the even part of (42) may be derived from k B = B′ (the div-B constraint), and is in fact the same equation, but
⊥ V S
with S replaced byV; these give thedynamics of the‘even sector’, while for theodd sector we have
B¨¯ −C2B¯′′ −λB¯˙′′ +k2λB¯˙ +k2C2B¯ =2λΣ¨¯ +Σ¯˙ . (49)
V A V V ⊥ V ⊥ A V V V
The GW variables obey Σ¨ −D2Σ +k2Σ = 0, and similarly for Σ and Σ¯ ; Dbσ = 0 implies k Σ = D Σ . We
k S ⊥ S V V ab ⊥ V k S
havedropped k subscripts,with theunderstandingthattheseequations applyfor each valueof k .Wemay nowintroduce
⊥ ⊥
harmonics for ∂ and ∂ in theusual way if desired.
z t
4.2 Integration
As a means of understanding the effect of conductivity on a magnetic field in a GW spacetime we shall consider the case of
a Gaussian pulseof GW travelling in thepositive z-direction through a static magnetic field such that at t=0,
Σ(k) = Σe−z2/L2cos(ωz), (50)
V
Σ˙(k) = Σe−z2/L2 ωsin(ωz)+2z cos(ωz) , (51)
V h L i
where we set Σ = −1 without loss of generality (alternatively, this could be Σ¯ because the decoupled equation for B¯ is
V V
identical to that of B ). This means that B is measured in units of the GW amplitude in the plots below. We will now
V
integrateEq:(48)foravarietyofsituationsassumingthatatt=0,B =B =0–i.e.,theinteractionswitchesonatt=0.
S V
Wetakeω=π/3, k =0 (so B =0) and L=8, below, all in units where c=1.
⊥ S
The qualitative features of finiteconductivity are shown in figures1 and 2.
In Fig. 1 we consider the case of C =1, that is waves in a very tenuous plasma, giving rise to resonant magnetic field
A
amplification. The figures shown apply equally to B or B¯ . In thecase of infiniteconductivity,shown on the left, we have
V V
a linearly growing magnetic field pulse, which is continually sourced by the GW as it travels through the background field.
When conductivity is present, however, this amplification is inhibited in that the induced magnetic field can only grow to a
certainstrength(comparethescalesofthetwoplots)beforeasaturation amplitudeismet,atanamplitudeconsiderablyless
than if the magnetic diffusivity is zero.
If the magnetic field is strong, or the fluid dense, then the Alfv´en velocity can be small, resulting in weak non-resonant
amplification of GW. The plot on the left of Fig. 2 shows the modes with C = 0.5 when λ = 0. Dispersion shows up here
A
as a trail of small amplitude waves, trailing behind the initial pulse, due to the mismatch in the GW and plasma dispersion
relations. The plot on the right demonstrates what happens when the conductivity is finite. As in the resonant case the
forward driven pulse is damped and can only reach a certain amplitude, which is considerably less than when C = 1. The
A
trailing part of thewave has nothingto sustain it, so diffusesand rapidly decays away.
5 DISCUSSION
Finiteconductivitycanalter thebehaviourofconductingfluids,plasmas andelectromagnetic fieldsinavastvarietyofways.
When spacetime curvaturemay be neglected these effects are well understood in a significant numberof physical situations,
ranging from the early universe, to astrophysical and laboratory plasmas. When spacetime curvature effects are present and
8 M. Marklund and C. Clarkson
60
15
40 10
20 5
B B
0 0
–5
–20
–10
–40
–15
–60
0 0
50 50
t 100 0 20 40 60 z80 100 120 140 t 100 0 20 40 60 z80 100 120 140
Figure1.Theeffectofλonlongwavelengthmodes,withCA=1.Theplotonthelefthasλ=0showinglineargrowthofthemagnetic
field as the GW passes through the background field. On the right, we have the same situation but with finite conductivity, λ=0.05.
Whilethereissomeearlyamplificationofthemagneticfield,thisquicklysaturates,withlatetimebehaviourjustabriefdisturbanceas
theGWpasses.Notethedifferenceoftheverticalscalebetweenthetwoplots.
1.5
2
1
1
0.5
B B
0 0
–0.5
–1
–1
–2
0 0
20 20
40 40
60 60
80 120 140 80 120 140
t 100 80 100 t 100 80 100
120140 0 20 40 60 z 120140 0 20 40 60z
Figure 2. The effect of λ on slow Alfv´enwaves, with CA =0.5. The plot on the left has λ=0 showing little growth of the magnetic
field as the GW passes through the background field, trailed by a slow wave. The dispersion takes the form of a slower wave trailing
theoscillationsinducedbytheGW.Ontheright,wehavethesamescenariobutwithfiniteconductivity, λ=0.05.ThetrailingAlfv´en
wavesaredissipatedveryquickly.Themainpulse,asinthepreviousplot,isfromadirectforcingfromtheGW.
significant,however,verylittleisknownabouttheroleplayedbyfiniteconductivityinastrophysicsandcosmology.Wepresent
here, for thefirst time, the dynamoequation in its full generality in an arbitrary curved spacetime.
The MHD approximation, describing low frequency charged fluid phenomena, allows us to neglect the displacement
current in Amp`ere’s Law, from which we may derive a diffusion equation for the magnetic field – the dynamo equation (11)
or (12). For easy comparison with non-general relativistic results we have utilised the intuitive 1+3 covariant approach to
relativistic analysis (Ellis & van Elst 1998). This approach splits spacetime into space and time in a covariant way by use of
a timelike vector field, which then also plays the role of the convective derivative, but now on an arbitrary curved manifold.
Spacetimedynamicsisthencovariantlydescribedbytheuseofinvariantlydefinedscalars,3-vectors,andPSTFtensors,made
upfromtheRiemanncurvaturetensor,andthedynamicalquantitiesassociatedwithua.ThesequantitiesfeedintoMaxwell’s
equationsinanon-trivialway,makingtheresultantelectromagneticfieldexperience‘gravitationalcurrents’ofaverydifferent
naturefrom flat space physics.
In the relativistic dynamo equation we havederived, these gravitational currents come in a huge variety of terms which
The general relativistic MHD dynamo equation 9
areverydifficulttoanalyseingeneral.Wehavepresentedinsomesimplifyingassumptionswhichareoftenusedinspacetime
modelling, in order tobetter gain an intuition about the kindsof effects conductivity coupled to gravity can produce.
As a specific example of analysing the relativistic dynamo equation, we considered in detail the case of a plane GW
interactingwithastaticmagneticfield.Webelievethismayplayansignificantrolearoundcompact objects,wheremagnetic
fields and gravity waves can interact strongly. By introducing covariant vector harmonics adapted to the magnetic field we
reduced the dynamoequation to two fairly simple coupled PDEs with a constraint, and one decoupled PDE, with the space
dimension along the background field. These PDEs for the induced magnetic field are sourced by the GW in an intuitive
fashion,andmaybeanalysedbystandardmeans.WeintegratedtheseequationsnumericallyforaGaussianpulseofradiation
passingthrough thebackgroundstatic magnetic field;thesummary ofthisintegration ispresentedin thefigures. Inthecase
of infiniteconductivity,there is linear amplification of themagnetic field which mimics theGW; thisis followed bya trail of
oscillations resultingfrom wavesinthemagnetic fieldtravelling withtheAlfv´enspeed.Whenmagneticdiffusivityispresent,
these trailing modes diffuse quickly, and the overall amplification of the magnetic field is inhibited; a ripple in the magnetic
field is forced as theGW passes through.
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