Axion and dilaton + metric emerge from local and linear electrodynamics

0.33 MB·

by  Friedrich W. Hehl

Axion and dilaton + metric emerge jointly from
an electromagnetic model universe
with local and linear response behavior
FriedrichW.Hehl
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AbstractWe takea quicklookatthe differentpossible universallycoupledscalar
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fieldsinnature.Then,wediscusshowthegaugingofthegroupofscaletransforma-
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tions(dilations),togetherwiththePoincare´group,leadstoaWeyl-Cartanspacetime

] structure.Therethedilatonfieldfindsanaturalsurrounding.Moreover,wedescribe
c
shortlythephenomenologyofthehypotheticalaxionfield.—Inthesecondpartof
q
- ouressay,weconsideraspacetime,thestructureofwhichisexclusivelyspecifiedby
r
g thepremetricMaxwellequationsandafourthrankelectromagneticresponsetensor
[ densityc ijkl = c jikl = c ijlk with36independentcomponents.Thistensorden-
− −
sityincorporatesthepermittivities,permeabilities,andthemagneto-electricmoduli
2
ofspacetime.Nometric,noconnection,nofurtherpropertyisprescribed.Ifwefor-
v
0 bidbirefringence(double-refraction)inthismodelofspacetime,weeventuallyend
2 upwiththefieldsofanaxion,adilaton,andthe10componentsofametrictensor
3 withLorentzsignature.Ifthedilatonbecomesaconstant(thevacuumadmittance)
0
andtheaxionfieldvanishes,werecovertheRiemannianspacetimeofgeneralrela-
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tivitytheory.Thus,themetricisencapsulatedinc ijkl,itcanbederivedfromit.
.
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FriedrichW.Hehl
Univ.ofCologneandUniv.ofMissouri-Columbia,email:[email protected].
InvitedcontributiontoT.Asselmeyer-Maluga(ed.),AttheFrontierofSpacetime:Scalar-Tensor
Theory,Bell’sInequality,ExoticSmoothness.FestschriftontheoccasionofCarlBrans’80thbirth-
day,tobepublishedbySpringerinMay2016.
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2 Axionanddilaton+metricemergefromlocalandlinearelectrodynamics
Contents
1. Dilatonandaxionfields
1.1 Scalarfields
1.2 Einsteingravityandtheenergy-momentumcurrent
1.3 Einstein-Cartangravity:theadditionalspincurrent
1.4 Dilatonfieldanddilationcurrent
1.5 TheWeyl-Cartanspacetimeasanaturalhabitatofthedilatonfield
1.6 Axionfield
2. Anelectromagneticmodeluniverse
2.1 ThepremetricMaxwellequations
2.2 Alocalandlinearelectromagneticresponse
2.3 Propagationofelectromagneticdisturbances
2.4 Fresnelwavesurface
2.5 Suppressionofbirefringence:thelightcone
2.6 Axion,dilaton,metric
3. Discussion
“Universally coupled, thus gravitational, scalar fields are still
active players in contemporarytheoreticalphysics. So, what is
therelationshipbetweenthescalarofscalar-tensortheories,the
dilatonandtheinflaton?Clearlythisisanunansweredandim-
portantquestion.Thescalarfieldisstillaliveandactive,ifnot
alwayswell,incurrentgravityresearch.”
CarlH.Brans(1997)
1 Dilatonandaxionfields
1.1 Scalar fields
TheJordan-Brans1-Dickescalar, the dilaton,the axion,the inflaton—scalarfields
everywhere—andeventuallyevenone,thescalar,thatis, thespinlessHiggsboson
1CarlBransisoneofthepioneersofthescalar-tensortheoryofgravitation.Thisessayisdedicated
toCarlontheoccasionofhis80thbirthdaywithallbestwishestohimandhisfamily.Duringthe
yearof1998,wehadtheprivilegetohostCarl,asanAlexandervonHumboldtawardee,forseveral
monthsattheUniversityofCologne.Irememberwithpleasurethemanylivelydiscussionswehad
onscalars,onstructuresofspacetime,onphysicsingeneral,andonvariousothertopics.
Axionanddilaton+metricemergefromlocalandlinearelectrodynamics 3
H0, which has been found experimentally as heavy as some 134 protons. These
differentscalarfields2 arenotnecessarilyindependentfromeachother,itcouldbe,
for example, that the JBD-scalar can be identified with the dilaton (see [30]) or
theHiggsbosonwith theinflaton(see [6,7]).Thus,thelistofpotentiallyexisting
universallycoupledscalarfieldscouldbesomewhatsmaller.Forthehistoryofthe
JBD-scalar,onecancompareBrans[11]andGoenner[32]and,fortheroleofthe
inflatonindifferentmodels,Venninetal.[98].
1.2 Einsteingravityand the energy-momentumcurrent
As remarked by Brans [10] in the quotation above, if universally coupled, the
scalar fields are intrinsically related to the gravitationalfield. In Einstein’s theory
ofgravity,generalrelativity(GR),thegravitationalpotentialisthemetricg ,with
ij
i,j =0,1,2,3 as (holonomic) coordinate indices. As its source acts the symmet-
ric energy-momentumtensor T of matter. This is a second rank tensor, which is
ij
generatedalreadyinspecialrelativity(SR)withthehelpofthegroupT(4)oftrans-
lations in space and in time. Together with the Lorentz transformationsSO(1,3),
thetranslationsT(4)buildupthePoincare´ groupP(1,3)asasemi-directproduct:
P(1,3)=T(4)⋊SO(1,3).Thisisthegroupofmotionin theMinkowskispace of
specialrelativity,see[31].
Accordingly,if one desiresto understandgravityfrom the pointof view of the
gaugeprinciple,theT(4)isanindispensablepartoftheseconsiderations.However,
beingonlyonepieceoftheP(1,3),itissuggestiveto gaugethe completeP(1,3).
Thisis exactlywhatSciama andKibble didduringthe beginning1960s,see [73],
[9,Chap.4],and[16].
1.3 Einstein-Cartangravity:theadditionalspincurrent
This gauging of the P(1,3) extends the geometrical framework of gravity. The
4 translational potentials ea and the 6 Lorentz potentials G ab = G ba span a
i i i
−
Riemann-Cartanspacetime,enrichingtheRiemannianspacetimeofGRbythepres-
ence of Cartan’s torsion; here a ,b = 0,1,2,3 are (anholonomic) frame indices.
Whereasthetranslationalpotentialscoupletothecanonicalenergy-momentumten-
2Weskipheretheplethoraofscalarmesons,
p ±,p 0,h ,f0(500),h ′(958),f0(980),a0(980),...,
K±,K0,KS0,KL0,K0∗(1430),D±,D0,D∗0(2400)0,D±s,...;
theyareallcomposedoftwoquarks. Thus,thescalarmesonsdonotbelongtothefundamental
particles.
4 Axionanddilaton+metricemergefromlocalandlinearelectrodynamics
sor of matter S a i, the Lorentz potentials couple to the canonical spin current of
mattertab i= tba i.
−
The simplest version of the emerging Poincare´ gauge models is the Einstein-
Cartantheory(EC),aviablegravitationaltheorycompetingwithGRifhighestmat-
terdensitiesareinvolved.Ifl denotesthePlancklengthandl theComp-
Planck Compton
tonwavelengthofaparticle,thendeviationsoftheEinstein-CartanfromEinstein’s
theory are expected at length scales of below (l2 l )1/3; for protons,
∼ Planck Compton
prevalentintheearlycosmos,itisabout10 29meter.AccordingtoMukhanov[65],
−
itisexactlythisorderofmagnitudedowntowhich,accordingtorecentcosmologi-
caldata,GRisknowntobevalid.
From a gauge theoretical point of view, the EC-theory looks more reasonable
than GR since the Einsteinian principles of how to heuristically derive a gravi-
tational theory were followed closely: they were just applied to fermionic matter
insteadoftomacroscopicpointparticlesorEulerfluidsortoclassicalelectromag-
netism,asEinsteindid.
Incidentally,intheEC-theoryand,moregenerally,inthePoincare´gaugetheory,
thePoincare´and,inparticular,theLorentzcovariancearevalidlocallybyconstruc-
tion,similarasinaSU(2)Yang-Millstheory,wehavelocalSU(2)covariance.Kost-
elecky´ (priv. comm., Jan. 2016)agrees that the “Einstein-Cartan theory maintains
localLorentzinvariance.”ThenthesameistrueforaPoincare´gaugetheory,which
actslikewiseinaRiemann-Cartanspacetimewithtorsionandcurvature.However,
in an experimental set-up, according to Kostelecky´ [48], torsion must be consid-
eredasanexternalfieldand,accordingtothe“standardloreforbackgrounds,”local
Lorentzinvarianceisbroken.Bythesametoken,anexternalmagneticfieldinelec-
trodynamics breaks local Lorentz invariance. This is, in my opinion, an abuse of
language, which conveys the wrong message that the existence of a torsion field
violateslocalLorentzinvariance.
If, for theoretical reasons, one wants to evade the emergence of the Riemann-
Cartan spacetime, then one can manipulate, in the underlying Minkowski space,
the intrinsic or spin part of the total angular momentum of matter in such a way
that it vanishes on the cost of increasing the orbital part of it by the correspond-
ing amount, see [63]. This procedure is called Belinfante-Rosenfeld symmetriza-
tionofthecanonicalenergy-momentumcurrent,which,ingeneral,isdefinedasan
asymmetric tensor by the Noether procedure.Accordingly,by symmetrizationthe
energy-momentumcurrentismadefittoactasasourceoftheEinsteinfieldequa-
tion. In thisway, one can effectivelysweep the spin andthe torsionunderthe rug
andcanlivehappilyforeverintheparadiseoftheRiemannianspacetimeofGR.
Ofcourse,inthe endobservationsand/orexperimentswilldecidewhichofthe
twotheories,GRorEC,willsurvive.Weoptforthelatter.
Axionanddilaton+metricemergefromlocalandlinearelectrodynamics 5
1.4 Dilatonfield and dilationcurrent
Thedilatonfieldf enteredlifeasaNambu-Goldstonebosonofbrokenscaleinvari-
ance,seeFujiiin[30].Thus,f isrelatedtodilat[at]ionsorscaletransformationsin
spaceandtime.Butthedilatonalsooccursintheoriesofgravity(JBD)andinstring
theory,seeDiVecchiaetal.[21].TheP(1,3),ifmultiplied(semi-directly)withthe
scalegroup,becomestheWeylgroupW(1,3).This11-parametergroupisaninvari-
ancegroupformasslessparticlesinspecialrelativity.Thetranslations,viaNoether’s
theorem,generatetheconservedenergy-momentumtensorS ,theLorentztransfor-
ij
mationstheconservedtotalangularmomentumtensorJ k:=t +x S k= J k,
ij ijk [i j] − ji
andthedilationtheconservedtotaldilationcurrent¡ k:=D k+xlS k,
l
¶ S k =0, (1)
k i
¶ J k =¶ t k S =0, (2)
k ij k ij [ij]
−
¶ ¡ k = ¶ D k+S k=0, (3)
k k k
see [59, 47] particularly for¡ k. Thus, if a universal coupling is assumed, then f
shouldhavetheintrinsicdilationcurrentD kasitssource;fortheoriesinWeylspaces
inwhichD k doesnotplayarole,seeScholz[87,88].
Therearenumerousfieldtheoreticalmodelsunderwaywhich,ifscaleordilation
invarianceisimplemented,haveconformalinvarianceasaconsequence;foramore
recentreviewseeNakayama[66].Hence,jumpingtoconformalinvariance,before
oneunderstoodscaleinvariance,isprobablynotaverygoodstrategy.Forthisreason
weconfineourselvesheretoscaleinvariance,tothedilaton,andtothe11-parametric
Weylgroup.Butitshouldbeunderstoodthatthelightconeisalsoinvariantunder
the15-parametricconformalgroup,seeBarut&Ra¸czka[5]andBlagojevic´[8]and,
forahistoricalaccount,Kastrup[45].
Bothcurrents,theintrinsicdilationcurrentD kandtheenergy-momentumcurrent
S karerelatedtoexternalgroups,tothedilation(scale)andtothetranslationgroups,
i
respectively.Thisisthereasonfortheiruniversality.
1.5 TheWeyl-Cartanspacetimeas anatural habitatofthedilaton
field
We only tried to make a strong case in favor of the EC-theory in order to repeat
thecorrespondingargumentsforthedilationgroup.GaugingtheWeylgroupyields
a Weyl-Cartan spacetime. The classical paper in that respect is the one of Charap
andTait,see[9,Chap.8].AuniversallycoupledmasslessscalarfieldinducesaWeyl
covector Q as the correspondingdilation potential willy nilly. This is the type of
i
spacetimeWeylused(withvanishingtorsion)forhisfailedunifiedtheoryof1918.
HeretheWeylspacewiththeconnectionWG isresurrectedforthedilationcurrent,
insteadoffortheelectriccurrent,see[8]:
6 Axionanddilaton+metricemergefromlocalandlinearelectrodynamics
W 1
(cid:209) g = Qg , WG = RCG + (Qg +Q g Q g ); (4)
i jk i jk ijk ijk i jk j ki k ij
− 2 −
RCG is the connection of the Riemann-Cartan space. Again, as in the case of the
Lorentzgroup,onecanmanipulatethetotaldilationcurrent¡ k andcantransform
its intrinsic partinto an orbitalpartby modifyingin this case the trace S k of the
k
energy-momentumcurrent.Then,again,onecanstay withinthe realmofthe Rie-
mannianspaceofGR,seeCallan,Coleman,andJackiw[13].
Aswementionedalready,thegaugetheoreticalanswerwasgivenbyCharapand
Tait [15]. Again, which approach will succeed is eventually a question to experi-
mentalverification.
We see, if theJBD-scalar isinterpretedasadilaton,thenwewouldexpectthat
theWeyl-Cartanspacetimeisitsarena.Clearlythisdoesonlyprovidethekinemat-
ics of the theory. The dynamics would depend on the exact choice of the dilaton
Lagrangian.
Recently, Lasenby and Hobson [53] wrote an in-depth review of gauging the
Weyl group and, moreover, formulated an “extended Weyl gauge theory.” Also
withintheirframework,theWeyl-Cartanspace,andastraightforwardextensionof
it, play an importantrole, see also Haghani et al. [33]. Definite progress has also
beenachievedinthestudyofequationsofmotionwithinthescalartensortheories
of gravity, see Obukhov and Puetzfeld [76, 81, 82]. The breaking of scale invari-
anceinthemoregeneralapproachofmetric-affinegravitywasstudiedin[34],for
example;forsomewhatanalogousbreakingmechanisms,see[60,61,62].
1.6 Axionfield
Dicke did notonly introducein 1961,togetherwith Brans[12], a scalar field into
gravity,buthealsodiscussed,in1964,andpseudoscalaroraxialscalarfieldj 2 in
thecontextofgravitationaltheory,see[20,Appendix4,p.51,Eq.(7)].
Subsequently,intheearly1970s,Ni[67]investigatedmattercoupledtothegrav-
itationalfield and to electromagnetismandlookedfor consistencywith the equiv-
alence principle. He found it possible to introduce in this context a new neutral
pseudoscalar field accompanying the metric field, see also [68, 69, 4, 70]. Later,
in the context of the vacuum structure of quantum chromodynamics,a light neu-
tralpsedoscalar,subsequentlydubbed“axion”washypothesized,seeWeinberg[99,
pp.458–461]. Similar as Ni’s field, the axion couples also to the electromagnetic
field,seeWilczek’spaper[100]on“axionelectrodynamics”.
The axion field is of a similar universality as the gravitational field. In other
words,theaxionbelongstotheuniversallycoupledscalarfields. Letinelectrody-
namics,Hij =(D,H)= H ji andFij=(E,B)= Fji denotetheexcitationand
− −
the field strength, respectively. The constitutive relation characterizing the axion
fielda (x)(inelementaryparticleterminologyitiscalledA0)reads[35],
Axionanddilaton+metricemergefromlocalandlinearelectrodynamics 7
1 Da= a Ba,
Hij= 2ae ijklFkl or (Ha= a Ea, (5)
−
see also [37] for the corresponding formalism; here e is the totally antisymmet-
ric Levi-Civita symbol with e ijkl = 1, moreover, a=1,2,3. Clearly, the axion
±
embodies the magnetoelectric effect par excellence. It is a pseudoscalar under 4-
dimensionaldiffeomorphisms.
In electrotechnical terms, the axion behaves like the (nonreciprocal Tellegen)
gyrator of network analysis, see [49, 85]; also the perfect electromagnetic con-
ductor (PEMC) of Lindell & Sihvola [58, 94] represents an analogous struc-
ture. Metaphorically speaking, as we see from (5), the axion “rotates” the volt-
ages (B,E) into the currents(D,H). In SI, we have the units [B]=Vs/m2,[E]=
V/m; [D]=As/m2,[H]=A/m.Thus,[a ]=1/ohm=1/W carriesthephysicaldi-
mensionofanadmittance.Now,intheMaxwellLagrangian,wefindanadditional
piece a (x)e ijklF F a (x)E B, a term, which was perhapsfirst discussed by
ij kl
Schro¨d∼inger[90, pp.25∼to 26].If·a were a constant,thefield equationswouldnot
change.
Aswealreadyremarked,a isa4-dimensionalpseudoscalar.Thesameistruefor
thevonKlitzingconstantR 25813W .Andthiscovarianceisaprerequisitefor
K
≈
itsuniversalmeaning.Phenomenologically,thequantumHalleffect(QHE)canalso
bedescribedbyaconstitutivelawofthetype(5),see[35,Eq.(B.4.60)].
It is possible to apply the constitutive relation (5) directly to a solid, too. By
the evaluation of experiments we have shown [37] that in the multiferroic Cr O
2 3
(chromium sesquioxide) we have a nonvanishing axion piece of up to 10 3l ,
− 0
wherel is thevacuumadmittanceof about1/377W .Thisfactdemons∼tratesthat
0
thereexistmaterialswithanonvanishing,ifsmall,(pseudoscalar)axionpiece.This
maybeconsideredasaplausibilityargumentinfavorofasimilarstructureemerging
in fundamental physics. If the A0 were found, it would not be an unprecedented
structure,seeinthiscontextalsoNietal.[72].
In matter-coupled N =2 supergravity models, there are examples in which a
dilatonandanaxionarecontainedsimultaneouslyintheallowedparticlespectrum,
see Freeman and Van Proeyen [29, p.451]. However, in the next section we will
demonstratethatinafairlysimpleclassicalmodelofanelectromagneticuniverse,
theaxioncanemergejointlywithadilatonandthemetric.
More recently, there have been attempts to relate the axion field to the torsion
of spacetime, see, for example Mielke et al. [64] and Castillo-Felisola et al. [14].
Tous,thisassumedlinkbetweentheinternalsymmetryU(1)oftheaxionwiththe
externaltranslationsymmetryT(4)relatedtothetorsionappearstobeartificialand
notsupportedbyphysicalarguments.
8 Axionanddilaton+metricemergefromlocalandlinearelectrodynamics
2 An electromagnetic model universe
2.1 ThepremetricMaxwellequations
We considera 4-dimensionaldifferentiablemanifold.The electromagneticfield is
specifiedbyitsexcitationHij,a2ndrankantisymmetriccontravarianttensorden-
sity, and by its field strength F , a 2nd rank antisymmetric covariant tensor; the
ij
electric currentJk is a contravariantvector density,see Post [80]. On this mani-
fold,theMaxwellequationsread
¶ kHik=Jk, ¶ [iFjk]=0; (6)
thebrackets denoteantisymmetrizationofthecorrespondingindiceswith1/3!as
[]
afactor,see[89];fortheTonti-diagramof(6),compare[96,p.315].
In none of these equations the metric tensor g nor the connectionG k are in-
ij ij
volved.Still,theseequationsarevalidandaregenerallycovariantintheMinkowski
space of special relativity, in the Riemann space of general relativity, and in the
Riemann-CartanorWeyl-Cartanspaceofgravitationalgaugetheories.TheMaxwell
equations(6)assuch,apartfromahistoricalepisodeupto1916,see[22,23],have
nospecificrelationtothePoincare´ortheLorentzgroup.
Perlick [78] has shown that the initial value problem in electrodynamics can
be particularly conveniently implemented by means of the premetric form of the
Maxwellequations.
Incontrasttomosttextbookrepresentations,no“commagoestosemicolonrule”
is required. The Maxwell equations (6) are just universally valid for all forms of
electrically charged matter. Incidentally, this represents also a simplifying feature
fornumericalimplementations.Thepriceonehastopayistointroduce,asMaxwell
did, the excitation Hij, besides Fij, as an independent field quantity and to note
thatitisatensordensity.Fromaphenomenologicalpointofview,thisisdesirable
anyway, since the excitation has an operational definition of its own, namely as
charge/length2(D)andcurrent/length(H),respectively,whichisindependentfrom
the definitionof the field strengthas force/charge(E) and force/current(B). For a
renditioninthecalculusofexteriordifferentiableforms,onecancomparewiththe
axiomaticschemein[35]and[18],seealso[19].
LetusstressadditionallythatHij,Fij,andJk canbedefinedinabackground
independentway.
TheMaxwellequations(6)arebasedontheconservationlawsofelectriccharge
Q:= ds ijkeijklJl (unitinSI“coulomb”)andmagneticfluxF := ds ijFij(unit
in SI “weber”). Charge Q and flux F are 4-dimensionalscalars. They induce the
R R
structure of the excitation Hij and the field strength Fij. In this context, the field
strengthisoperationallydefinedviatheLorentzforcedensityfi=FijJj,thecur-
rentbeingdirectlyobservableandtheforceanditsmeasurementknownfromme-
chanics.
Axionanddilaton+metricemergefromlocalandlinearelectrodynamics 9
Thechargeanditsconservationistheanchorofelectrodynamics.ItscurrentJk
defines,bymeansoftheLorentzforcedensityf,thefieldstrengthF ,whichallows
i ij
todefinethemagneticfluxF .Faraday’sinductionlawisanincarnationofmagnetic
fluxconservation.
Some people have no intuition about the conservation of a quantity that is de-
fined in 3 dimensions by integration over a 2-dimensional area ds aB , since
a
∼
we usually associate conservation with a quantity won by 3-dimensional volume
R
integration,namely dVr .Somemathematicseducationaboutdimensionswill
∼
enableustounderstandtheinductionlawasa“continuityequation.”
R
Summingup:thepremetricMaxwellequationsareaclose-knitstructure,the4-
dimensionaldiffeomorphismscovarianceholdsit all together. Clearly,a metric as
wellasaconnectionarealientotheMaxwellequations.
2.2 Alocal and linearelectromagneticresponse
InordertofilltheMaxwellequationswithlife,onehastorelateFij toHij:
Hij=Hij(Fkl) (7)
Ifweassumethisfunctionaltobelocal,thatis,Hij(x)dependsonlyonFkl(x),and
linearhomogeneously,thenwefind
1
Hij= c ijklFkl with c ijkl = c ijlk= c jikl; (8)
2 − −
herethefieldc ijkl(x)representstheelectromagneticresponsetensordensityofrank
4andweight+1,withthephysicaldimension[c ]=1/resistance.Anantisymmetric
pair of indicescorresponds,in 4 dimensions,to 6 independentcomponents.Thus,
c ijkl canbeunderstoodasa6 6matrixwith36independentcomponents.
×
We want to characterizethe electromagneticmodelspacetime by this response
tensor field c ijkl(x) with 36 independent components.3 This is the tensor density
definingthestructureofspacetime.Ittranscendsthemetricand/ortheconnection.
Wedecomposethe6 6matrixintoits3irreduciblepieces.Onthelevelofc ijkl,
×
thisinduces[35,17]
c ijkl = (1)c ijkl+(2)c ijkl+(3)c ijkl. (9)
36= 20 15 1.
⊕ ⊕
3 Schulleretal.[91]tookthec ijkl–tensordensity,whicharisessonaturallyinelectrodynamics,
calledthetensorproportionaltoit“areametric”,andgeneralizedittondimensionsandtostring
theory.Forreconstructingavolumeelement,theyhave,dependingonthecircumstances,twodif-
ferentrecipes,like,forexample,takingthesixthrootofadeterminant.Fromthepointofviewof
4-dimensionalelectrodynamics,theprocedureofSchulleretal.lookscontrivedtous.
10 Axionanddilaton+metricemergefromlocalandlinearelectrodynamics
The third part, the axion part, is totally antisymmetric (3)c ijkl := c [ijkl] =ae ijkl,
with the pseudoscalar a , see also [83]. The skewon part is defined according to
(2)c ijkl := 1(c ijkl c klij). Under reversible conditions, (8) can be derived from
2 −
a Lagrangian, then (2)c ijkl =0. The principal part (1)c ijkl fulfills the symmetries
(1)c ijkl=(1)c klij and(1)c [ijkl]=0.
Thelocalandlinearresponserelationnowreads
1
Hij= (1)c ijkl+(2)c ijkl+ae ijkl Fkl, (10)
2
(cid:16) (cid:17)
and,splitinspaceandtime[35,37],
Da=(e ab e abcn )E +( g a +s a d as c)Bb+a Ba, (11)
− c b b b − b c
H = (m 1 e mc)Bb+( g b +s b d bs c)E a E ; (12)
a a−b − abc − a a − a c b− a
heree abc=e = 1,0arethe3-dimensionalLevi-Civitasymbols.The6permit-
abc
tivitiese ab=e ba,t±he6permeabilitiesm =m werealreadyknowntoMaxwell.
ab ba
The8magnetoelectricpiecesg a (itstracevanishes,g c =0)werefoundsince1961,
b c
see Astrov[2].Eventually,thehypotheticalskewonpiece[35]carries3 permittiv-
ities n , the 3 permeabilitiesma, and the 9 magnetoelectricpieces s b. Equivalent
a a
response relations were formulated by Serdyukov et al. [92, p.86] and studied in
quitesomedetail,seealsodeLangeandRaab[52].
SupposewehaveasspecialcaseavacuumspacetimedescribedbyaRiemannian
metricg .Thentheresponsetensorturnsouttobe
ij
c ijkl = (1)c ijkl=2l 0√ ggi[kgl]j and Hij=l 0√ gFij, (13)
− −
withthevacuumadmittancel 1/377W .Thus,werecoverknownstructures,and
0
≈
werecognizethattherelation(8)representsanaturalgeneralizationofthevacuum
case.Themetricgij canbeconsideredassomekindofasquarerootoftheelectro-
magneticresponsetensorc ijkl.
Weshouldkeepinmindthatalocalandhomogeneouselectromagneticresponse
like (8) can be, if the circumstances require it, generalized to nonlocal and/or to
nonlinearlaws.ExamplesofnonlocallawshavebeenproposedbyBoppandPodol-
sky4 and by Mashhoon.5 Nonlinear laws are due to Heisenberg and Euler,6 Born
andInfeld,7 andPleban´ski.8 Fresnelsurfacesforthenonlinearcasewerefoundby
ObukhovandRubilar[77],forexample.Morerecently,La¨mmerzahletal.[51]and
Itin et al. [44] investigated electrodynamics in Finsler spacetimes. In the premet-
ricframework,thiscorrespondstoanonlocalconstitutivelaw,see[44,Eq.(3.29)],
somewhatreminiscentoftheBopp-Podolskyscheme.
4See[28,Sec.28-8].
5See[35,Sec.E.2.2].
6See[35,Sec.E.2.3].
7See[35,Sec.E.2.4].
8See[35,Sec.E.2.5].