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Metric-independence of vacuum and force-free electromagnetic fields ∗ Abraham I. Harte School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland ElectromagneticfieldswhichsolvethevacuumMaxwellequationsinonespacetimearewell-known toalsobesolutionsinallspacetimeswithconformally-relatedmetrics. Thisprovidesasenseinwhich electromagnetismalonecannotbeusedtomeasurecertainaspectsofgeometry. Weshowthatthere isactuallymuchmorewhichcannotbesomeasured;relativelylittleofaspacetime’sgeometryisin factimprintedinanyparticularelectromagneticfield. Thisisdemonstratedbyfindingamuchlarger class of metric transformations—involving five free functions—which preserve Maxwell solutions bothin vacuum,without local currents,and also for theforce-free electrodynamics associated with a tenuous plasma. One consequence of this is that many of the exact force-free fields which have previouslybeenfoundaroundSchwarzschildandKerrblackholesarealsosolutionsinappropriately- 7 identifiedflatbackgrounds. Asamoredirectapplication,weuseourmetrictransformationstowrite 1 downalargeclassofelectromagneticwaveswhichremainunchangedbyalargeclassofgravitational 0 waves propagating “in the same direction.” 2 n a Beginningwiththe1915prediction,andthenthe1919 Maxwell’sequationsintheircommonly-giventensorial J observation of starlight deflected by the Sun [1], elec- form appear to involve the spacetime metric g both ab 8 tromagnetic fields in nontrivial spacetimes have played explicitly, via an index-raising operation, and also im- 1 an essential role in the development of general relativ- plicitly, via the covariant derivative ∇ ; see e.g., [4]. It a ity. Essentially all observations of general relativistic is known, however, that these equations simplify con- ] c phenomena—whethertheyinvolvelightdeflection,inten- siderably using structures natural to the study of differ- q sity modulation, frequency shifts, time delays, or some- ential forms. In this context, the electromagnetic field - thing else—involve electromagnetic fields in one way or F =F isreinterpretedasa2-formF,andMaxwell’s r ab [ab] g another. Even the recent detections of gravitational equations reduce to [ wavesbyLIGO[2,3]havedependedinpartontheinter- ∗ ∗ actionsofthosewaveswithlightcirculatinginitsinterfer- dF =0, d F =−4π J. (1) 1 v ometers. In these measurements and others,appropriate 7 properties of electromagnetic radiation are often inter- Here, d denotes the exterior derivative, J the current ∗ 5 pretedascontaininganimprintofthe geometrythrough density 1-form,and the Hodge dual operator. Exterior 2 which they’ve traveled. Here,we askto whatextent this derivativesaredefinedindependently ofthemetric,soat 5 is reallytrue. Whatcanreallybe learnedabouta space- least in vacuum where J = 0, the local geometry can 0 time from an electromagnetic (test) field? affect an electromagnetic field only via its contributions . ∗ 1 The usual approach to understanding electromagnetic to the Hodge dual F 7→ F. In particular, if a 2-form 0 fieldsingeneralrelativityistoconsideraparticularclass F solves the vacuum Maxwell equations in a metric gab, 7 of physical systems and to directly compute how vari- and if ∗F is preserved under some metric transforma- 1 ous observables depend on the properties of those sys- tion gab 7→ gãb, it follows immediately that F is also a : v tems. This “forward” approach is useful, even essen- solution to the vacuum Maxwell equations in the met- Xi tial, in many contexts, but does not provide a partic- ric gãb. We may thus characterize at least some of the ularly broad understanding. We instead focus on the “metric-independence” ofaparticularvacuumsolutionF r a inverse problem—finding which metrics are compatible by finding the class ofmetric transformationswhichpre- ∗ with a given field—and indeed only on the uniqueness serve F. Reverting to tensorial notation, we look for of those metrics. Given one metric compatible with a those metric transformations which do not affect given electromagnetic field, which others are also compatible? Somewhat more precisely, we identify classes ∗F = 1ǫ gcegdfF , (2) ab abcd ef 2 of metric transformations for which Maxwell fields remain Maxwellfields. Although these transformations do where ǫ denotes a volume form compatible with g . abcd ab notexhaustallpossibilities,theyalreadyinvolvefivefree Rather remarkably, this problem is purely local and al- functions. That there exists such a broadfreedomin the gebraic; it does not depend on derivatives of the met- allowablemetricssuggeststhatrelativelyfewaspectsofa ric transformation. Although these observations are ele- geometrycanbeinferredfromknowledgeofasingleelec- mentary, their consequences do not appear to have been tromagneticfield(althoughconsiderablymoreispossible thoroughly examined. One special case which has been using matter or families of fields). explored is the conformal invariance of Maxwell’s equations [4–8]: Any F which solves the vacuum Maxwell equations in one metric g is also solution in all met- ab rics g˜ = Ω2g for which Ω2 > 0. We show that much ab ab ∗ [email protected] more general transformations are allowed when the re- 2 quirementthatall fields be preservedis weakenedto the Recalling that ℓ must also be null, and that null eigen- a requirement that only a single field be preserved. vectorsremainnulleigenvectorswithrespecttoarbitrary We also note that metric transformations which pre- rescalings, 1-forms picked out in this way define a prin- ∗ serve F are relevant even in non-vacuum electromag- cipal null direction [16, 17], or PND of F , with respect ab netism. Itisclearfrom(1)thattheynecessarilypreserve to g . Any nonzero multiple of ℓ generates the same ab a ∗ J, and this is physically interesting in at leastone non- PND. vacuum context: force-free electrodynamics. In many Using this terminology, it follows from (4) that any astrophysicalenvironments,electromagneticfieldsdonot electromagnetic field is preserved by any Kerr-Schild existinvacuum,butareinsteadsourcedbycurrentsinan transformationgeneratedbyaPNDofthatfield. IfF isa ambientplasma. Ifthisplasmaissufficientlytenuousand vacuumorforce-freesolutioninametricg ,andifℓ de- ab a the magnetic field sufficiently strong, matter responds fines a PND with respectto g , F remains a solutionin ab almost instantaneously to the electromagnetic field, and allmetricsΩ2(g +Vℓ ℓ ). ThescalarV isarbitraryand ab a b theLorentzforcedensitygbcF J maybetakentovanish Ω2 is constrained only to be positive. Many important ab c [9–11]. Using Maxwell’s equations to replace the current metrics are known to be related via Kerr-Schild trans- densityinthisconstraintbythedivergenceofF results formations to simple, and even flat, spacetimes: Kerr- ab in the nonlinear field equation gbdgcfF ∇ F = 0. Al- Newman black holes, plane-fronted gravitational waves ab c df though this appears to have a complicated dependence with parallel rays (pp-waves), de Sitter spacetime, and onthemetric,considerablesimplificationsareagainpos- more [17]. Moreover,every static, spherically-symmetric sible using the language of differential forms. First, it is metric can be related to a flat metric by a combination known that any force-free field in which J 6= 0 may be ofKerr-Schildandconformaltransformations[18]. Kerr- written,atleastlocally,asF =dα∧dβ,forsomescalars Schildtransformationsarealsoknownto interactinpar- α and β. Fields with this form are automatically solu- ticularlysimplewayswithEinstein’sequation,effectively tions to dF = 0, and the remaining force-free equation linearizing it [15, 19, 20]. may be shown to be equivalent to [12] The result that Kerr-Schild transformations preserve Maxwellfieldsmaybeconsiderablygeneralized. Mostdi- ∗ ∗ dα∧d F =0, dβ∧d F =0. (3) rectly,onemayconsidersuccessivecompositionsofKerr- Schild transformations generated by each PND. This is Once again, the metric enters only via the Hodge dual ∗ nontrivialonlyifF isnon-null,whichmeansthatatleast F 7→ F: If some potentials α and β are known to gen- one of the scalars erate a force-free field F in the metric g , it remains a ab ∗ force-free field in all metrics which preserve F. In both 1 gacgbdF F =|B|2−|E|2, (7a) the vacuum and force-free settings, this single algebraic ab cd 2 criWteerionnowisisdueffintciifeynmt teotrpicretsrearnvsefoerlmecattrioomnsagwnheitcihc fiaeclcdosm.- 1gacgbdFab∗Fcd =E·B, (7b) 2 plish this. Although their most general forms depend on the algebraic characterof F, we begin by considering is nonzero. Non-null fields are known to admit exactly a special case which does not. The first non-conformal two real PNDs, represented here by ℓa and ka, and also transformations we discuss are of Kerr-Schild type [13– two complex PNDs, represented by ma and m¯a [16, 21]. 15], meaning that Thelatter1-formsarecomplexconjugatesofoneanother, and additionally, ℓ·m = k·m = 0, with dot products gab 7→gãb =gab+Vℓaℓb (4) andothermetric-dependentconceptstobeunderstoodas withrespecttog . Applyingaconformaltransformation for some nonzero ℓ which is null in the sense that ab a togetherwiththeKerr-Schildtransformationsassociated gabℓ ℓ =0. GivenanyV,the exactinverseofthe trans- a b with these four PNDs may be shown to result in g 7→ formed metric is known to be gãb = gab−Vℓaℓb, where ab g˜ , where ℓa =gabℓ . Itfollowsthatℓ isnullnotonlywithrespect ab b a toof Kgaebr,r-bSucthialdlsotrwanitshforrmesapteicotnstoisg˜tahb.atAthseeycopnrdesperrvoepevrotly- gãb =Ω2(cid:20)gab+ Vℓaℓb+(1k−·ℓ1)(VkW·ℓℓ)(2aVkbW)+Wkakb ume elements: A volume 4-form ǫ associated with g is 4 ab equal to a volume form ǫ˜ associated with gãb. Substi- + Ymamb+(m·m¯)|Y|2m(am¯b)+Y¯m¯am¯b . (8) tuting these results into (2) now shows that under any 1− 1(m·m¯)2|Y|2 (cid:21) 4 Kerr-Schild transformation, The scalarsV andW are real,while Y may be complex. ∗Fab 7→∗Fab+ǫabcdVℓcFdfℓf, (5) They are arbitrary so long as 1− 1(k·ℓ)2VW and 1− 4 1(m·m¯)2|Y|2 remain nonzero and Ω2 > 0. Note that where indices have been raised using gab. If VFab 6= 0, 4 g˜ involvesnotonly sums ofordinaryKerr-Schildterms the second term on the right-hand side vanishes if and ab proportionaltoℓ ℓ ,k k ,andsoon,butalsocrossterms only if ℓ is an eigenvector of F with respect to g , in a b a b a ab ab such as ℓ k . These arise because, e.g., k may fail to the sense that (a b) a be a null eigenvector with respect to g +vℓ ℓ even if ab a b ℓ F gcdℓ =0. (6) it is a null eigenvector with respect to g . [a b]c d ab 3 Regardless, it follows from our above result on indi- some spacelike s which is orthogonal to ℓ recovers the a a vidualKerr-Schildtransformationsthatif F satisfies the “extended Kerr-Schild metrics” discussed in [26–28]. vacuum or force-free Maxwell equations and admits the Perhaps the simplest descriptions of electromagnetic non-parallel null eigenvectors ℓ , k , m , and m¯ with (andmanyother)fieldsarethosewhichariseinthe limit a a a a respect to g , this field is also a vacuum or force-free so- of geometric optics. In certain cases—waves of suffi- ab lutionwithrespecttoanyg˜ givenby (8). Attemptingto cientlyhighfrequency[22]oratasufficientdistancefrom ab find metrics compatible with a non-null electromagnetic theirsource[7,23]—the partialdifferentialequationsde- field thus results in an ambiguity involving at least five scribing the underlying fields may be ignored in favor of realfunctions,whichisclosetothesixgauge-independent ordinarydifferentialequationsdescribingthetrajectories functionsthatcharacterizegeneralfour-dimensionalmet- oflightrays,amplitudesalongthoserays,andsoon[29]. rics (although we do not attempt to precisely describe There is a sense in whichfields become nullin this limit, the sense in which our transformations are independent andsoonemightexpectthatsomestructuresrelevantto of ordinary gauge transformations). geometricoptics,andhencegravitationallensing,arealso The allowable metric transformations are somewhat preserved by metric transformations which preserve null different if F is null, meaning that the quadratic field electromagnetic fields. This is indeed the case. Suppose scalars (7) both vanish. Physically, null fields appear to that a collection of light rays in a metric g is tangent ab anyobservertoinvolveelectricandmagneticfieldswhich to a vector field ℓa. This must be null and geodesic with areorthogonalandofequalmagnitude;asimpleexample respect to g , meaning that ab is a plane wave in flat spacetime. Null fields arise gener- g ℓaℓb =0, (ℓc∇ ℓ[a)ℓb] =0, (12) ically in the limit of geometric optics, and also far from ab c sources (see, e.g., [7, 23] and references therein). Real and a direct calculation shows that these equations are null fields may be shown to have the form [16] preserved under any metric transformation g 7→ g˜ ab ab where g˜ is given by (10); light rays remain light rays ab F =ℓ∧(m+m¯), (9) under the same transformations as those which preserve null electromagnetic fields. where ℓ generatesa realPND, m is complex, null, and a a Now that we have established metric transformation orthogonaltoℓ ,andm¯ isthecomplexconjugateofm . a a a laws which preserve both null and non-null electromag- Null fields admit only one PND, so interesting composi- neticfields,itisnaturaltoaskifvariousobservablescon- tions of Kerr-Schild transformations are not possible. structedfromthesefieldsarealsopreserved. Perhapsthe We progress instead by writing down an ansatz and simplest possibilities are the quadratic field scalars (7). verifying that it works. Recalling that a Kerr-Schild In the null case where these both vanish with respect to transformation generated by ℓ adds a rank-1 pertur- a the metric g , they also vanish for all metrics g˜ with ab ab bation to g , consider a more general rank-1 transfor- ab the form (10); null fields remain null. For the non-null mation generated by both ℓ and some other 1-form ξ . a a case,both scalarsareaffectedonlyby the conformalfac- Additionally including a conformal transformation, let tor in (8), gab 7→gãb =Ω2(gab+Vℓ(aξb)). (10) |B|2−|E|2 7→Ω−4(|B|2−|E|2), (13a) E·B 7→Ω−4E·B. (13b) We now show that these transformations preserve any null, vacuum or force-free Maxwell solution F with PND Non-null fields thus remain non-null. Electrically or generatedbyℓa. Firstnotethatinvertingthetransformed magnetically-dominated fields remain so dominated. In metric yields gãb =Ω−2(gab−Vℓcgc(aΞb)), where both null and non-null cases, the electromagnetic stress- energy may be shown to be preserved up to scaling, but Ξa = gab ξ − 14(ξ·ξ)V ℓ . (11) only in its mixed-index form gabTbc. Eigenvectors of the 1+ 1(ℓ·ξ)V (cid:20) b (cid:18)1+ 1(ℓ·ξ)V (cid:19) b(cid:21) stress-energy are thus preserved, possibly with rescaled 2 2 eigenvalues. It may additionally be noted that since F Furthermore, the volume element transforms as ǫ 7→ and ∗F remain unchanged by our transformations, so Ω4[1+ 1(ℓ·ξ)V]ǫ. It follows from these formulas and do their integrals through arbitrary 2-surfaces. In par- 2 ∗ ∗ also(2) and(9)that F 7→ F forallgab 7→gãb withthe ticular, these transformations cannot change electric or form (10). As claimed, these transformations preserve magnetic charges inferred to lie inside closed 2-surfaces. nullMaxwellfieldsinbothvacuumandforce-freeelectro- Despitetheseremarks,itmustbecautionedthatiden- dynamics. Similarlytothenon-nullcase(8),theyinvolve tical fields do not necessarily have identical physical in- five free functions—one for Ω and four for Vξa. These terpretations in different metrics. Frequencies of elec- functions are restricted only to satisfy 1+ 1(ℓ·ξ)V 6=0 tromagneticwavesgenericallydiffer,forexample;proper 2 and Ω2 > 0, conditions which ensure that g˜ does not timesassociatedwithanobserver’stimelikeworldlineare ab become degenerate. If ξ is null, g˜ has the form of not necessarily preserved. Even more dramatically, an a ab the metrics considered in [24, 25]. Ordinary Kerr-Schild observer which is physically-acceptable—i.e., timelike— transformationsariseasspecialcasesinwhichΩ=1and with respect to one metric can be unacceptably space- ξ ℓ = 0. Generalizing these to allow ξ ℓ = s ℓ for like with respect to other metrics in our class. It should [a b] [a b] [a b] 4 also be noted that although our transformations take Theseresultshaveallbeenobtainedusingonlyconfor- solutions into solutions, they do not necessarily pre- malandKerr-Schildtransformations. Wenowuseamore servephysically-interestinginitialdataorboundarycon- complicatedtransformationto demonstrate that there is ditions. asenseinwhichalargeclassofgravitationalwavesprop- Movingontoapplications,consideraforce-freeF with agatinginagivendirectiondonotaffectelectromagnetic nonvanishingcurrentdensityinametricg . Ifthatden- testwavespropagatinginthatsamedirection. Introduc- ab sity is null, the force-free condition implies that it may inginertialcoordinatesxµ =(t,x,y,z)associatedwitha beidentifiedwithaPND,represented,say,byℓa. Awide flat metric ηab, the fields F = dα(u,x,y)∧du describe, class of such solutions have been found in the presence among other things, electromagnetic waves propagating of Schwarzschild and Kerr black holes [12, 30]. In all along the ±z direction, where u = t∓z and α(u,x,y) cases, they admit a PND for F which coincides with a is an arbitrary potential. These are force-free null solu- PND associated with the spacetime’s Weyl tensor. It is tions. Vacuum examples also exist, however, in which known however [17] that any Kerr metric gKerr may be case (∂x2 + ∂y2)α = 0. The null 1-form ℓ = du gener- ab deformedintoaflatmetricη usingaKerr-Schildtrans- ates the lone PND. Also introducing the null coordinate ab formation generated by either of its two Weyl-PNDs: v = t±z, it follows from (10) that F is a solution for, η = gKerr +Vℓ ℓ . Our results on Kerr-Schild defor- e.g., all line elements ab ab a b mations thus imply that force-free solutions of this type ds2 =A2(dx2+dy2)−du(dv+Bdu+Cdx+Ddy), (14) must also be solutions in an appropriately-identifiedflat metric. Similar results have been recognized essentially with A,B,C,D essentially arbitrary. Comparing with byinspection,atleastintheSchwarzschildcase,although Eq. (31.6) of [17], the so-called Kundt metrics are seen the underlying reasonwas notexplained. That these are to be special cases. Physically, Kundt metrics can de- reallyflatsolutions“indisguise” providesasimpleexpla- scribe a wide range ofphysicalscenarios,including quite nation for why there is no apparent scattering from the generaltypes of gravitationalradiation (with or without spacetimecurvature(seealso[31]foradifferentexplana- a cosmological constant) [17, 34–36]. The plane-fronted tion). waveswith parallelrays,pp-waves,are containedas spe- A more general result may be deduced for electro- cialcases,and gravitationalplane wavesare in turn spe- magnetic fields which have been found in Schwarzschild cial pp-waves. Regardless, the gravitational radiation and which admit a PND coinciding with one of the described by any Kundt metric may be said to propa- spacetime’s PNDs: These PNDs are radially-ingoing gate along the direction associatedwith ℓ, at least when or radially-outgoing, and it is known that all static, it is appropriate to interpret it as containing radiation. spherically-symmetric metrics can at least locally be de- This is similarly true for the electromagnetic radiation formedintoflatmetricsusingacombinationofconformal associated with F (again, where appropriate). That the transformations and Kerr-Schild transformations gener- fieldremains a solutionin a wide variety ofgravitational ated by radial 1-forms [18]. There is therefore a sense in wave metrics—independent of waveformor anything be- which solutions found in [12, 30] in Schwarzschild back- side propagation direction—suggests that these types of grounds are actually solutions in all static (and many waves do not interact. Despite this, examination of cer- non-static) spherically-symmetric spacetimes. The addi- tain observables, such as the electric field components tionofacosmologicalconstanthasnoeffect,forexample. seenbyafreely-fallingobserver,wouldinsteadappearto Going in the opposite direction—from flat spacetime suggest that these waves do interact; correct interpreta- to curved—flat-spacetime, force-free, null-current solutions can be subtle. tions have been found in which the field’s PND is tan- Regardless, many more applications are possible. Ad- gent to light cones whose vertices lie on a (possibly ac- ditionally,theclassesofMaxwell-preservingmetrictrans- celerating) timelike worldline [32]. Applying Kerr-Schild formations(8)and(10)maybeinterestinglyenlargedby ∗ ∗ transformations with such a PND is known to gener- allowing F 7→ γ F +dλ, where λ is arbitrary and γ ∗ ate a wide variety of “rocket”-type and other geometries satisfies dγ∧ F = 0. We have thus far considered only [17, 33]. 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