Purely Magnetic Spacetimes Alan Barnes University of Aston, Birmingham, B4 7ET, U.K. Spacetimes in which the electric part of the Weyl tensor vanishes (relative to some timelike unit vector field) are said to be purely magnetic. Examples of 4 purely magnetic spacetimes are known and are relatively easy to construct, if 0 norestrictionsareplacedontheenergy-momentumtensor. Howeverithaslong 0 been conjectured that purely magnetic vacuum spacetimes (with or without a 2 cosmological constant) do not exist. The history of this conjecture is reviewed n andsomeadvancesmadeinthelastyeararedescribedbriefly. Ageneralisation a ofthisconjecturefirstsuggestedfortypeDvacuumspacetimesbyFerrandoand J S´aez is stated and proved in a number of special cases. Finally an approach 5 to a general proof of the conjecture is described using the Newman-Penrose 1 formalism based on a canonical nulltetrad of theWeyltensor. 1 v 8 1 Introduction 6 0 1 TheelectricandmagneticpartsoftheWeyltensorwithrespecttosomeunittimelike a 0 vector field u are defined by 4 0 Eab =Cacbducud Hab =Ca∗cbducud (1) / c ∗ q respectively, where Cabcd is the dual of the Weyl tensor. Using this decomposition, - whichwasfirstintroduced(forthevacuumRiemanntensor)byMatte[1],theBianchi r g identitiestakeaformanalagoustoMaxwell’selectromagneticfieldequations. Space- : times for which Hab =0 are said to be purely electric whilst those in which Eab =0 v are saidto be purely magnetic. If the electric and magnetic parts ofthe Weyl tensor i X are proportional, that is if r νEab =µHab (2) a for some scalar fields ν and µ (not both zero), we will say that the electric and magnetic parts are aligned. The aligned case, of course, includes the purely electric and magnetic fields as the special cases ν = 0 and µ=0 respectively. In all aligned casesthecomplextensorQab =Eab+iHab isacomplexmultiple ofarealsymmetric a tensor and so may be diagonalised by a tetrad rotation leaving u fixed. Thus the Petrov type is I, D or O (see, for example [2]) and the eigenvalues αA of Eab and βA of Hab are also proportional, that is ναA = µβA where uppercase Latin indices run from 1 to 3. The purely electric and magnetic cases correspond to βA = 0 and αA = 0 respectively. Although the condition in (2) appears to depend on choice of 1 2 A. Barnes a a a vector field u , in fact this is not the case. If u is such that (2) holds then it a is a Weyl principal vector. Thus for Petrov type I, u is, like the rest of the Weyl principal vectors, determined uniquely (up to sign) by Cabcd and in the type D case it is determined up to a boost in the plane of the repeated principal null directions of the Weyl tensor. There is an alternative characterisation of type I fields satisfying (2): namely that the four principal null directions of the Weyl tensor are linearly dependent and span the three-dimensional vector space orthogonal to the eigenvector of Qab corresponding to the eigenvalue λA = αA +iβA of smallest absolute value [3, 4, 5]. For type I fields the Weyl tensor invariant M = I3/J2 6 is real and positive or − infinite, where I =λ2+λ2+λ2 J =λ3+λ3+λ3 =3λ λ λ (3) 1 2 3 1 2 3 1 2 3 Thus the Petrov type is I(M+) or I(M∞) in the extended Petrov classification of Arianrhod&McIntosh[6]. Infactinthevacuumcase(withorwithoutacosmological constant) a result first proved by Szekeres [7], but usually attributed to Brans [8], ∞ showsthatthecaseI(M ),thatiswhereoneoftheeigenvaluesofQabiszero,cannot occur. For non-vacuum spacetimes some authors use the term purely electric to mean ∗ c d R u u = 0. As shown in [3], this condition is equivalent to the two conditions: acbd a Hab = 0 and u is a Ricci eigenvector. For vacuum spacetimes (Rab = Λgab), the twodefinitionsofpurelymagneticareequivalent. Almostinvariablyinstudiesofthe a non-vacuumcase,u isassumedtobeaRiccieigenvectorandsothereislittledanger ofconfusionarisingfromthetwodifferentdefinitionsofthetermpurelyelectric. This c d is not so for the purely magnetic case which some authors define as Racbdu u = 0. This condition does not imply Eab =0 unless the Ricci tensor satisfies c Rab =u(aqb) u qcgab (4) − b d for some vector field qa. In fact, any two of the conditions Racbdu u = 0, Eab = 0 and (4) imply the third [9]. The non-equivalence of these two definitions of purely magnetic has led to some confusionin the literature; see [10] for a discussionof this. Note that even for vacuum spacetimes, the two definitions are not equivalent unless the cosmological constant Λ vanishes. In this paper the term purely magnetic will always be used to mean Eab =0. Many examples of purely electric spacetimes, both vacuum and non-vacuum, are known in the literature. For example all static spacetimes are necessarily purely electric [11] as are all shear-free and hypersurface othogonalperfect fluid spacetimes [12]. However, relatively few purely magnetic spacetimes are known although the situation has improved in recent years (see for example [13, 14]). To date none of the purely magnetic solutions that have been found, satisfy the vacuum field Purely Magnetic Spacetimes 3 equations: namely Rab = Λgab. This has led researchers to conjecture that there are no purely magnetic vacuum spacetimes (excluding the trivialconstant curvature case). A number of special cases of this conjecture have been proved. For example Hall[15]showedthattherewerenopurelymagnetictypeDvacuummetricsandthis result was rediscovered in [5]. Note that Hall also proved a related result: namely that there are no vacuum type II spacetime in which the eigenvaluesλA =αA+iβA of Qab are purely imaginary. For type I spacetimes the conjecture has been proved a under the additional assumption that the vector field u is shear-free by Barnes [16] and this result was rediscoveredby Haddow [9]. More recently van der Berg [17, 18] has proved the conjecture in a further two special cases: namely when the vector a a a field u is either hypersurface orthogonal(ω =0) or geodesic (u˙ =0). 2 A Generalised Conjecture Recently Ferrando & Sa´ez [19] considered special type D fields in which Eab =RHab (5) where R is a real constant. This class of spacetimes is clearly a specialisation of the aligned case defined in (2). Alternatively this class may be characterised by the assumption that the eigenvalues λA of Qab are all real multiples of a single complex constantorequivalently thatallthese eigenvalueshavethe sameconstantargument. For conciseness below these fields will be referred to as having constant argument. Notethatthepurelymagneticfieldsareaspecialcaseoftheconstantargumentfields with R = 0, but that the purely electric case is excluded (it corresponds informally to R= ). We will also use the term constant argument for Petrov type II fields in ∞ which the eigenvalues of Qab have constant argument although (5) is not now valid. In[19]itwasshownthattherearenovacuumtypeDfieldsofconstantargument. This leadsoneto considerwhether there areanyPetrovtype IorII vacuumfields of constant argument. Below it is shown that I. there are no vacuum constant argumentPetrovtype I fields in which the a vector field u is shear-free; II. there are no vacuum constant argument Petrov type II fields. These two results generalise those for the purely magnetic case in [16, 9] and [15, 5] respectively. Furthermore,intheintervalbetweentheconferenceandtheappearance of these proceedings, Ferrando & Sa´ez [20] have shown that there are no vacuum a constant argument Petrov type I fields in which the vector field u is hypersurface orthogonal;thisisadirectgeneralisationoftheresultforthepurelymagneticcasein [17]. Thisnaturallyleadstotheconjecturethattherearenovacuumfieldswhatsoever of constant argument. 4 A. Barnes The proof of result I above is now presented. Putting σab = 0 in the vacuum Bianchi identities (4.21a) and (4.21c) of [21], one obtains a bc d a b a bc d a b h E h +3H ω =0 h H h 3E ω =0 (6) b ;d c b b ;d c − b where hab = gab +uaub is the projection tensor into the three-space orthogonal to a a a the vector field u and ω is the vorticity vector of u . Using (5) it follows that (1+R2)Hbaωb =0 (7) b andthus either ω is zero or it is an eigenvectorofHab (and hence of Qab) with zero eigenvalue. The latter case is excluded by the theorem of Szekeres [7, 8] discussed a in the Introduction. In the former case the congruence defined by u is shear-free andhypersurfaceorthogonalandsobyawell-knownresultHab =0 (seeforexample [12, 2]), hence Eab =0 and the spacetime is conformally flat and so not of type I. In both cases a contradiction is obtained and the required result I is proved. 3 An Approach using the NP Formalism In this section vacuum constant argument spacetimes will be investigated using the a Newman-Penrose(NP) formalism [22, 2]. When the timelike vector field u satisfies certain kinematic restrictions, an orthonormal tetrad approach is natural and has beenusedinmanypreviousinvestigationsofpurelymagneticandconstantargument spacetimes (for example [12, 9, 17, 18, 19, 20]). However,when there are no a priori a restictionsonthevectorfieldu ,thenanulltetradformalismalsobecomesattractive. b b Suppose (u ,e ) is an orthonormal Weyl principal tetrad of the spacetime. If A a a a a we introduce an associated null tetrad (k ,l ,m ,m¯ ) defined by ka =1/√2(ua+ea) la =1/√2(ua ea) ma =1/√2(ea+iea) (8) 3 − 3 1 2 then, if the spacetime is of Petrov type I or D, the NP Weyl curvature components satisfy (see [2] p. 51) Ψ =Ψ =0 Ψ =Ψ =(λ λ )/2 Ψ = λ /2 (9) 1 3 0 4 2 1 2 3 − − where the λA’s are the eigenvaluesof Qab =Eab+iHab. For Petrovtype D, without loss of generality, λ =λ and so Ψ =Ψ =0. For Petrov type II we have instead 1 2 0 4 Ψ =Ψ =Ψ =0 Ψ = 2 Ψ = λ /2 (10) 0 1 3 4 2 3 − − From (9) & (10) for Petrovtype I, II & D fields, a null tetrad may be chosen so that Ψ =Ψ =0. In such a frame the vacuum Bianchi identities reduce to (see [2] 1 3 Purely Magnetic Spacetimes 5 p. 81) δ¯Ψ = (4α π)Ψ +3κΨ (11) 0 0 2 − ∆Ψ = (4γ µ)Ψ +3σΨ (12) 0 0 2 − DΨ = (ρ 4ǫ)Ψ 3λΨ (13) 4 4 2 − − δΨ = (τ 4β)Ψ 3νΨ (14) 4 4 2 − − DΨ = λΨ +3ρΨ (15) 2 0 2 − ∆Ψ = σΨ 3µΨ (16) 2 4 2 − δ¯Ψ = κΨ 3πΨ (17) 2 4 2 − δΨ = νΨ +3τΨ (18) 2 0 2 − We may now proceed to prove proposition II of the previous section: namely thattherearenovacuumconstantargumentspacetimesofPetrovtypeII(oroftype D). For constant argument type II and D fields, the NP Weyl tensor components satisfy Ψ = Ψ = Ψ = 0 and Ψ = Aexp(iB) where A > 0 is a real scalar field 0 1 3 2 andB is a realconstant(andB =0,π as the purely electric case is excluded). With 6 these assumptions it is easy to deduce from the Bianchi identies (11–18) that κ=σ =τ +π¯ =0 ρ=ρ¯ µ=µ¯ (19) DA=3ρA ∆A= 3µA (20) − With the restrictions (19) on the spin coefficients the commutator relation (7.6a) of [2] becomes (∆D D∆)=(γ+γ¯)D+(ǫ+ǫ¯)∆ (21) − Applying this commutator to A it may be deduced with the aid of (20) that ∆ρ+Dµ=(γ+γ¯)ρ (ǫ+ǫ¯)µ (22) − butfromtheRicciidentities(7.21h)&(7.21q)of[2],onmakinguseof(19),itfollows that iB −iB ∆ρ+Dµ=(γ+γ¯)ρ (ǫ+ǫ¯)µ+A(e e ) (23) − − It follows immediately that either A = 0 or that B = 0,π. In either case we have a contradiction and so vacuum type D and type II spacetimes of constant argument cannot exist. This proof extends the result of [15] from the purely magnetic to the constantargumentcaseandextendstheresultof[19]fortheconstantargumentcase from type D to type II fields. The proof is somewhat more direct than the original proofs of these two results and is valid without modification when the cosmological constant Λ is non-zero. For type I vacuum fields of constant argument the NP Weyl tensor components satisfy Ψ =Ψ =0, Ψ =Ψ =A exp(iB) and Ψ =A exp(iB) where A andA 1 3 0 4 0 2 2 0 2 6 A. Barnes are positive scalar fields and B =0,π is a real constant. The Bianchi identities (11– 6 18) then lead to a number of algebraic relations, quadratic in the spin-coefficients, which must be satisfied. Currently workis in progressinvestigatingthe integrability conditions of these algebraic relations in the hope of proving the conjecture that no vacuum constant argument and/or purely magnetic fields can exist or of integrating the NP equations to find some counter-examples to these conjectures. References [1] A. Matte, Canadian Jour. Math. 5, 1 (1953). [2] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers & E. 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