Einstein-Maxwell fields with vanishing higher-order corrections
aa r X i v : . [ g r- q c ] M a r EINSTEIN-MAXWELL FIELDS WITH VANISHING HIGHER-ORDERCORRECTIONS
MARTIN KUCHYNKA † AND MARCELLO ORTAGGIO ⋆ † ,⋆ Institute of Mathematics of the Czech Academy of SciencesˇZitn´a 25, 115 67 Prague 1, Czech Republic † Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University inPrague, V Holeˇsoviˇck´ach 2, 180 00 Prague 8, Czech Republic
Abstract.
We obtain a full characterization of Einstein-Maxwell p -form solutions ( g , F ) in D dimensions for which all higher-order corrections vanish identically. These thus simultaneouslysolve a large class of Lagrangian theories including both modified gravities and (possibly non-minimally coupled) modified electrodynamics. Specifically, both g and F are fields withvanishing scalar invariants and further satisfy two simple tensorial conditions. They describea family of gravitational and electromagnetic plane-fronted waves of the Kundt class and ofWeyl type III (or more special). The local form of ( g , F ) and a few examples are also provided. Introduction and summary
While the Einstein-Maxwell Lagrangian is generally considered to describe the prototype the-ory of gravity coupled to electromagnetism, there is also a long history of so-called “alternativetheories”. The long-standing problem of the electron’s self-energy led to a modified electrody-namics already in 1912 [1] and subsequently to the well-known non-linear theory of Born andInfeld [2, 3] (see, e.g., [4] for more general non-linear electrodynamics (NLE)). Soon after thebirth of General Relativity, the quest for a unified description of gravity and electromagnetismalso inspired several modifications of Einstein’s theory – see, e.g., the early works [5, 6] and thereviews [7, 8] for more references. In subsequent years, further motivation to take into accountdeviations from the Einstein-Maxwell theory came from considering effective Lagrangians whichinclude various type of quantum corrections (cf., e.g., [9, 10] and the original references quotedthere) or low-energy limits of string theory [11–14].Not surprisingly, adding higher-order corrections to the Einstein and Maxwell equations makesthose generically more difficult to solve. However, it is remarkable that there exist theory-independent solutions, i.e., solutions “immune” to (virtually) any type of corrections. One canthus employ known solutions of the Einstein-Maxwell equations to explore more complicatedtheories, at least in certain regimes. This was first pointed out in the context of NLE bySchr¨odinger, who showed that all null fields which solve Maxwell’s theory also automaticallysolve any NLE in vacuum [15, 16]. The inclusion of backreaction on the spacetime geometry inthe full Einstein-Maxwell theory was later discussed in [17]. Subsequently, it was noticed thatelectromagnetic plane waves solve not only NLE but also higher-order theories [9] (in flat space-time; see also [18]), and that a similar property is shared by Yang-Mills and gravitational plane
E-mail address : † kuchynkm(at)gmail(dot)com, ⋆ ortaggio(at)math(dot)cas(dot)cz . Date : March 7, 2019. aves [9]. Backreaction was taken into account in [19], whereas extensions of these results tomore general (electro)vacuum pp - and AdS-waves were obtained in [20–22] and [23], respectively.This was used, in particular, to discuss spacetime singularities in string theory [21, 22].Recently, a more systematic analysis of D -dimensional Einstein spacetimes immune to purelygravitational corrections (“universal spacetimes”) was initiated in [24] and further developedin [25–27] (see also [28–30] for related results in the case of Kundt (AdS-)Kerr-Schild metrics).From a complementary viewpoint, a study of test Maxwell fields which simultaneously solvealso generalized theories of ( p -form) electrodynamics (“universal electromagnetic fields”) hasbeen performed in [31–33]. In spite of considerable progress, a full characterization of (i.e., anecessary and sufficient condition for) universal spacetimes and universal electromagnetic fieldsis, in general, still lacking (but see the above references for various results in special cases).In the present contribution we investigate solutions of the coupled (possibly also non-minimally)Einstein-Maxwell equations for which all higher-order corrections vanish identically in arbitrarydimension D and for any rank p of the Maxwell form. We show that a full characterizationis possible, which we formulate as theorems 3.1 and 3.4. Essentially (up to technicalities tobe explained in the following), we prove that for a solution ( g , F ) of the Einstein-Maxwelltheory, all higher-order corrections vanish if, and only if, both ( g , F ) are fields with vanishingscalar invariants ( V SI ) and additionally satisfy the two tensorial conditions C acde C cdeb = 0 and ∇ c F ad...e ∇ c F d...eb = 0. This implies, in particular, that the spacetime is Kundt and possesses arecurrent null vector field (but is not necessarily a pp -wave) and that the cosmological constantvanishes. This characterization of a large class of exact solutions make those relevant in contextsmore general than the Einstein-Maxwell theory, with possible applications, e.g., in string theoryalong the lines of [19–22]. Moreover, the methods used in this work are suitable also for furtherextensions of the results obtained here, for example to Yang-Mills solutions.The structure of the paper is as follows. In section 2 we define the theories under considera-tions and in what sense those can be considered as corrections to the Einstein-Maxwell theory.Section 3 contains the main results of this paper, namely theorems 3.1 and 3.4 (in the case of min-imally and non-minimally coupled theories, respectively) and their proofs. A simpler result forthe special case of Einstein gravity coupled to algebraically corrected electrodynamics (relevantfor theories similar to NLE) is also obtained (theorem 3.3). In section 4, we present the explicitform of the solutions ( g , F ) in adapted coordinates, which is more suitable for practical applica-tions, along with a few examples. The relation of the solutions to universal spacetimes [24–27]and universal electromagnetic fields [31–33] is also discussed, along with the overlap with Kerr-Schild spacetimes. Some additional comments are provided in the special case of four spacetimedimensions. The four appendices contain various technical results used throughout the paper(in particular, in the proofs of the main theorems). Most of those are new and of some interestin their own, and we believe they will be useful also in future investigations (we have quoted therelevant references in the few cases in which we simply summarize previously known results). Notation
Throughout the paper, we employ the boost-weight classification of tensors [34] (cf. also thereview [35]) – this relies on setting up a frame of D real vectors m ( a ) which consists of twonull vectors ℓ ≡ m (0) , n ≡ m (1) and D − m ( i ) (with a, b . . . =0 , . . . , D − i, j, . . . = 2 , . . . , D − g = ℓ ⊗ n + n ⊗ ℓ + m ( i ) ⊗ m ( i ) . (1.1) he range of lowercase Latin indices when indicating an order of differentiation (e.g., in ∇ ( k ) R )will be specified as needed. Furthermore, R , C , S denote the Riemann and Weyl tensors and thetracefree part of the Ricci tensor (cf. (B.1)), respectively. A p -form is denoted by F . A “Maxwell p -form” is a p -form which obeys the sourcefree Maxwell equations, i.e., d F = 0 = d ⋆ F .2. Higher order theories of gravity and electromagnetism
Form of the Lagrangian
In the paper, we take into account virtually all classical Lagrangian theories of gravity coupledto electromagnetism, described by the electrovacuum Einstein-Maxwell equations with higher-order corrections. More precisely, we consider a theory of gravity and p -form electromagnetism,in spacetime dimensions D ≥ ≤ p ≤ D − characterized by the action S [ g , A ] = Z d D x √− g L , (2.1)with a Lagrangian L of the form L ≡ L grav ( R , ∇ R , . . . ) + L elmag ( F , ∇ F , . . . ) + L int ( R , ∇ R , . . . , F , ∇ F , . . . ) . (2.2)Here, the individual parts of L are scalars constructed from the corresponding tensors: R denotesthe Riemann tensor of the metric g , and F denotes the field strength of the electromagneticpotential ( p − A , i.e. F = d A . We assume that the individual parts of L satisfy: • L grav is a function of scalar polynomial curvature invariants { I i } constructed from R and its covariant derivatives ∇ ( k ) R of arbitrary order (suitably contracted with the themetric and, possibly, the volume element). Moreover, L grav ( I , I , . . . ) is analytic atzero with a Taylor expansion of the form L grav = L EH + L GC , (2.3)where 16 π L EH = R −
2Λ (2.4)defines the Einstein-Hilbert Lagrangian (we have set G = 1 = c ), and L GC (“GravityCorrections”) consists strictly of higher order (i.e., greater than two) curvature mono-mials. This means that the possible monomials are at least quadratic in R or containderivatives ∇ ( k ) R . • L elmag is a function of scalar polynomial electromagnetic invariants { J j } constructedfrom F and ∇ ( k ) F of arbitrary order. Moreover, L elmag ( J , J , . . . ) is analytic at zerowith a Taylor expansion of the form L elmag = L M + L EC , (2.5) As well-known, a Maxwell D -form reduces to the spacetime volume element (up to a multiplicative constant)and simply gives rise to an effective positive cosmological constant, so that a spacetime with vanishing higher-order corrections must be Einstein (for D = 2 this simply fixes the value of Λ in terms of F ). The cases p = D and, by duality, p = 0, are thus of little interest in our work. We also exclude the case D = 2 with p = 1, sinceEinstein’s equations imply the trivial condition F = 0. This is why we restrict ourselves to D ≥ Following the terminology of [36], throughout the paper by “order” we indicate the number of differentiationsof the metric/vector potential (so, for example, in terms containing the curvature, each factor R contributes aterm 2 and each explicit covariant derivative a term 1 [36]). Two quantities of the same order have thus the samephysical dimensions. Most importantly, the field variation of an invariant of order n (in our case, w.r.t. g or A )yields a tensor again of the same order n . here 16 π L M = − κ p F ( F = F ab...c F ab...c ) , (2.6)defines the source-free Maxwell Lagrangian, and L EC (“Electromagnetic Corrections”)consists strictly of higher order (i.e., greater than two) monomials. • L int is a function of mixed invariants { K k } (i.e., scalar monomials each containing both R , ∇ R , . . . and F , ∇ F , . . . ) and satisfies L int (0) = 0.The above assumptions ensure that when the invariants entering L are small, L approachesthe standard Einstein-Maxwell p -form Lagrangian, i.e., 16 π L ≈ R − − κ p F . However, L isnot assumed to be analytic everywhere – as is the case for some of the theories mentioned inremark 2.1 below. Remark 2.1 (Theories contained in our definition) . The class of theories encompassed by(2.1), (2.2) (with (2.3)–(2.6)) is rather broad. It naturally includes Einstein’s gravity coupledto NLE [17] for arbitrary D and p (see section 3.1.1 below). Obviously, it also contains theorieswith arbitrary polynomial higher-order corrections, such as generic Lovelock [37] or any quadraticgravity [6, 38–40] in the gravitational sector, or Bopp-Podolsky electrodynamics [41, 42] in theelectromagnetic sector. Nonlinear theories such as f ( R ) [43] and, more generally, f (Riemann)[44], or Born-Infeld inspired modifications of gravities [45] coupled to generalized electrodynamics(such as NLE and their various generalizations) are also encompassed. Another special classof theories covered by (2.1) are then non-minimally extended Einstein-Maxwell theories (see,e.g., [46] for an early discussion).Also some theories not encompassed by our assumptions are worth mentioning. These aretypically theories without the Einstein term in the gravity sector, such as conformal gravity [47]or any Lovelock gravity containing only quadratic or higher powers of R (e.g., pure Gauss-Bonnet gravity). We observe that also theories containing an electromagnetic Chern-Simons(CS) term (possible for D = p ( k + 1) −
1, where k ≥ F satisfies identically F ∧ F = 0, CS corrections to theMaxwell equations with k ≥ k = 1, CS corrections to the Maxwell equations are instead linearand therefore a non-zero solution of standard Maxwell’s theory cannot solve those.2.2. Field equations
Variation of action (2.1) with respect to the fields g and A yields the following equations ofmotion G gravab + G intab = 8 πT elmagab , (2.7) ∇ a H elmagab...c + ∇ a H intab...c = 0 . (2.8)From the Taylor expansion (2.3) and (2.5) of L grav and L elmag , respectively, the followingexpressions for the individual tensors in (2.7), (2.8) follow (within the radii of convergence ofthe Taylor series): G gravab = G ab + Λ g ab + G GCab , G
GCab ≡ π √− g δ ( √− g L GC ) δg ab , (2.9) T elmagab = T Mab + T ECab , T
ECab ≡ − √− g δ ( √− g L EC ) δg ab , (2.10) ∇ a H elmagab...c = ∇ a F ab...c + ∇ a H ECab...c , ∇ a H ECab...c ≡ πκ δL EC δA b...c , (2.11) here G ab ≡ π √− g δ ( √− g L EH ) δg ab = R ab − Rg ab , (2.12) T Mab ≡ − √− g δ ( √− g L M ) δg ab = κ π (cid:18) F ac...d F c...db − p g ab F (cid:19) , (2.13)are the Einstein tensor and the part of the energy-momentum tensor coming from the standardMaxwell term. The interaction tensors G int , div H int are then a symmetric and skew-symmetrictensor obtained by the field variation of L int with respect to g and A , respectively. The explicitform of variations of L GC , L EC and L int evaluated on V SI fields (which suffices for our purposes)is given in appendix A (expressions (A.3)–(A.7)).
Remark 2.2 (Simplifications of CSI and VSI fields) . When evaluated on fields ( g , F ) with con-stant scalar invariants ( CSI ), variations of Lagrangians L grav , L elmag and L int being functionsof the corresponding scalar polynomial invariants { I i } , { J j } and { K k } , respectively, reduce to alinear combination (with constant coefficients) of variations of these scalar invariants (see appen-dix A). This means that the fields equations reduce considerably for such fields – in particular, itenables one to study (in general complicated) higher-order theories in the context of CSI fieldsjust by studying field variations of the individual scalar polynomial invariants, independently ofthe specific functional dependence of the Lagrangian. Further simplification occurs in the caseof VSI fields (clearly a subset of CSI fields). In the next section, this strategy will be employedin the proofs of the main results.3.
Solutions with vanishing higher-order corrections
We will show that under certain assumptions on a solution ( g , F ) of the Einstein-Maxwell equa-tions, the tensors G GC , G int , T EC , div H EC and div H int , representing higher-order correctionsto the Einstein-Maxwell theory, vanish identically. Minimally coupled ( L int = 0) and non-minimally coupled ( L int = 0) theories will be treated separately.3.1. Minimally coupled theories
In the minimally coupled case one has L int = 0 , (3.1)so that the interaction tensors G int and H int are not present in the field equations (2.7), (2.8).Consequently, we shall deal with simpler higher-order corrections to the Einstein-Maxwell sys-tem. Theorem 3.1 (Solutions with vanishing corrections) . Let ( g , F ) be a solution of the Einstein-Maxwell theory with a non-vanishing F and (2.1) , (2.2) be a minimally coupled theory (i.e. with L int = 0 ) satisfying the assumptions outlined in section 2.1. Then, the following statements areequivalent:(i) All higher-order corrections of (2.1) to the Einstein-Maxwell theory vanish for ( g , F ) .(ii) ( g , F ) are V SI fields and satisfy C acde C cdeb = 0 and ∇ c F ad...e ∇ c F d...eb = 0 . Remark 3.2.
First of all, let us note that the VSI property in condition (ii) of theorem 3.1requires the cosmological constant Λ to be zero. Condition (ii) also implies that the spacetimeis of Weyl type III [51] and admits a recurrent multiple Weyl aligned null direction (mWAND) ℓ aligned with F (see remark D.10), thus being Kundt. Note also that the condition C acde C cdeb = 0can be traced back to the vanishing of the Gauss-Bonnet term in the gravitational field equations(as such, it has been discussed in related contexts, e.g., in [25, 52–54]). We further emphasize hat it is satisfied identically by VSI spacetimes in D = 4 dimensions, thanks to the well-knownfour-dimensional identity C acde C bcde = ( C cdef C cdef ) δ ba . For D = 3 it is also trivial since C abcd = 0 identically. Proof.
Let us first show that ( i ) implies ( ii ). Consider the 2 N -th order Lagrangian L EC ≡ J N ( N > J = F a...b F a...b . The condition T EC = 0 implies that the trace Tr T EC = − N p − D/ J N has to vanish and hence necessarily J = 0, since N can be chosen arbitrarily.Now, one can take L EC ≡ J I , where I is an arbitrary scalar polynomial invariant of F and itscovariant derivatives. Thanks to J = 0, the corresponding correction reduces to T EC ∝ I T M for our field F , where T M is the standard Maxwell energy-momentum tensor (2.13) (which isnecessarily non-zero since F = 0). Hence, also I has to vanish and, since it was an arbitraryinvariant, F is V SI . In particular, it is null and the metric g is (degenerate) Kundt of tracelessRicci type N with constant Ricci scalar [31]. The condition G GC = 0 then implies that also g has to be V SI . Indeed, considering L GC = R , we get R = 0. This suffices to conclude that g is CSI , as immediately follows from (the proof of) theorem 3.2 of [25] (using Tr G GC = 0).Then, varying L GC = RI with I being an arbitrary scalar polynomial curvature invariant (alsousing R = 0 and the CSI property of g ), one obtains that I has to vanish as well, i.e. g is truly V SI . In particular, g is of aligned Weyl type III and Ricci type N [51] (in additionto being degenerate Kundt). In view of the results obtained so far, varying the higher-orderinvariants R ab R ab and R abcd R abcd and demanding that such corrections also vanish, we obtainthat S ab , and consequently also C acde C cdeb , vanishes. Under the given conditions on ( g , F ),the Weitzenb¨ock identity implies F = 0 (cf. eq. (12) of [33]). Since here S ab = κ F ac...d F c...db (by Einstein’s equations with null F ), we have that S ab = 0 iff ∇ c F ad...e ∇ c F d...eb = 0, whichcompletes the first part of the proof.Now we will prove that ( ii ) implies ( i ). First, both fields are V SI , thus, as pointed out inremark 2.2, all higher-order corrections of (2.1) reduce to a linear combination of variations ofthe individual polynomial invariants I k , J k , K i (see expressions (A.3)–(A.7) and the discussionbelow those). Hence, the discussion can be without loss of generality restricted to polynomialhigher-order corrections G GC , T EC and H EC . Now, according to theorem 1 of [51], g is ofaligned Weyl type III and Ricci type N, and thus also aligned with the V SI form F (thanks toEinstein’s equations). Theorem 2.5 of [33] then implies div H EC = 0. In view of theorem D.9(with remark D.10), ∇ F is 1-balanced, all conditions of lemma B.7 are satisfied and consequently T EC = 0 (recall that T EC has order greater than two). It remains to show that G GC vanishesas well. Since F is a null Maxwell field aligned with a Kundt null direction ℓ , from remark D.10we get τ i = 0, i.e. ℓ is recurrent. Theorem C.1 thus guarantees that G GC takes the form G GCab = N X n =0 a n n S ab . (3.2)As noticed above, here S ab = κ F ac...d F c...db . Hence, 1-balancedness of ∇ F implies S = 0 andwe are left with G GCab = a S ab . But since L GC is a higher-order scalar, a must be a non-trivialcurvature invariant and hence vanishes due to the V SI property of g . (cid:4) Algebraic corrections to the Maxwell Lagrangian.
A subclass of theories of particular in-terest consists of standard General Relativity coupled to generalized electrodynamics, for which We do not reproduce here the tensors produced by variation of such kinds of Lagrangians w.r.t. the metricsince they have been well-known for some time [40, 55]. The same comment applies also to the other quadraticterms mentioned in the following. he higher-order corrections are assumed to be only algebraic . This includes, in particular,the well-known case of NLE in four dimensions [17]. Let us thus consider Einstein-generalizedMaxwell theories with algebraic corrections, i.e. a subclass of theories (2.1), (2.2) for which theexpansion (2.3), (2.5) reduces to L = L EH + L M + L EC , (3.3)where L EC is a (higher-order) function of the algebraic invariants { J j } only (i.e. those con-structed solely from F and its dual, and not their covariant derivatives). Theorem 3.3 (Einstein gravity with algebraically corrected electrodynamics) . Let ( g , F ) be asolution of the Einstein-Maxwell equations with non-vanishing F . Then, ( g , F ) solves Einsteingravity coupled to any generalized Maxwell theory with higher-order algebraic corrections if andonly if F is null.Proof. To prove that F is necessarily null, we can proceed similarly as in the proof of theorem3.1. By considering L EC ≡ J N , where J ≡ F a...b F a...b , one obtains J = 0 for a suitable choiceof N . Now, the Lagrangian L EC ≡ J I , where I is an arbitrary algebraic polynomial invariantof F , is clearly an admissible correction. Since J vanishes for F , we have T EC ∝ I T M , whichimplies that also I = 0. Therefore, all algebraic invariants of F must vanish, i.e., F is null [31,56].On the other hand, since all algebraic invariants { J j } of any null field F vanish, the tensors H EC and T EC again effectively reduce to polynomial higher-order corrections (cf. appendix A).Therefore, we have div H EC = 0 thanks to Proposition 2.4 of [33]. In addition, since any higher-order algebraic polynomial T EC has to be at least cubic in F , one also immediately obtains T EC = 0, i.e. all algebraic higher-order corrections vanish trivially. (cid:4) Hence, we observe that null Einstein-Maxwell fields are indeed of particular importance inthe context of higher-order theories. It is worth emphasizing that, in this case, the metric isrestricted neither to be of Weyl type III nor Kundt, and Λ can be non-zero, thus allowing formore general spacetimes. Many such solutions are known in the case D = 4 = 2 p (cf. [57] andreferences therein). In higher dimensions, some non-Kundt solutions have been presented, e.g.,in [58] (when D = 2 p ). A simple Weyl type D example with D = 6 = 2 p is given by [58]d s = r δ ij d x i d x j +2d u d r + (cid:18) Λ10 r + µ ( u ) r (cid:19) d u ( i, j, . . . = 2 , . . . ,
5) (3.4) F = 12 f ij ( u )d u ∧ d x i ∧ d x j , µ ( u ) = µ + κ Z ( f ij f ij )d u, (3.5)where µ is a constant, which describes (for Λ <
0) the formation of asymptotically locally AdSblack holes by collapse of electromagnetic radiation with non-zero expansion.Note also that, for the case D = 4 = 2 p , it was already known to Schr¨odinger that all nullMaxwell fields automatically solve any NLE [15, 16], while the fact that all null solutions of theEinstein-Maxwell theory solve also General Relativity coupled to any NLE was pointed out inthe early 60’s [17] (see also [59, 60]).3.2. Non-minimally coupled theories
In this section, we show that the Einstein-Maxwell solutions studied in section 3.1 are free fromhigher order corrections also in the context of a wider class of non-minimally coupled theories –that is, also the interaction part of the field equations (2.7), (2.8) amounting to L int vanishesidentically for these Einstein-Maxwell fields. heorem 3.4. The Einstein-Maxwell fields with vanishing higher-order corrections of sec-tion 3.1 solve also all non-minimally coupled theories (2.1) , (2.2) .Proof. It is sufficient to show that the tensors G int , div H int arising from L int vanish – sincethe vanishing of the other terms in (2.7), (2.8) clearly follows by the same arguments as in theproof of ( ii ) ⇒ ( i ) in theorem 3.1. Again, we can without loss of generality restrict ourselves topolynomial higher-order corrections. Clearly, when both G int and H int consist of monomialscontaining ∇ ( k ) R ∗ ∇ ( l ) F with k ≥ , l >
0, then a trivial boost weight (b.w.) counting showsthat they have to vanish (recall lemmas B.2 and B.3 and the fact that ∇ F is 1-balanced). Thisargument does not apply to G int in the case l = 0 – which however is covered by lemma B.6.However, different forms of G int and H int are also possible. Namely, if L int ∝ ∇ ( k ) R ∗ ∇ ( l ) F ,variations with respect to g and A may yield terms of type ∇ ( k +2) ∗ ∇ ( l ) F and ∇ ( l +1) ∗ ∇ ( k ) R ,respectively. Fortunately, even these two types of terms are safe – the first one is zero bylemma B.7 and the second one vanishes thanks to lemma B.5. Hence, we conclude that alsointeraction terms necessarily vanish for ( g , F ). (cid:4) Explicit form of the solutions and discussion
The local form of general
V SI fields ( g , F ) solving the Einstein-Maxwell equations is knownin standard Kundt coordinates (see [61] and [31] ). For solutions with vanishing higher-ordercorrections (theorem 3.1) the multiple null direction (of both R and F ) must be recurrent(remark 3.2), which gives (in the metric (1.1)) ℓ = d u, n = d r + h H (1) ( u, x ) r + H (0) ( u, x ) i d u + W k ( u, x )d x k , m ( i ) = d x i , (4.1) F = 1( p − f i...j ( u )d u ∧ d x i ∧ · · · ∧ d x j ( i, j, k, . . . = 2 , . . . , D − , (4.2)where we have also used the condition ∇ c F ad...e ∇ c F d...eb = 0 in ( ii ) of theorem 3.1 to constraintthe form of F (cf. remarks D.10, D.11). The functions H (0) , H (1) , W i and f i...j are then subjectto the following equations W [ i,j ] k W [ i,j ] k = 2 W m [ k,m ] W [ k,n ] n , (4.3) H (1) ,j = W k [ j,k ] , (4.4)∆ H (0) = 2 H (1) ,k W k + H (1) W k,k + W [ m,n ] W [ m,n ] + W m,um − κ F , (4.5)where ∆ is the Laplace operator in the (flat) transverse space and F ≡ f i...j f i...j was defined.Equation (4.3) is equivalent to the condition C acde C cdeb = 0 (( ii ) of theorem 3.1), while equations(4.4) and (4.5) correspond to the Einstein equations of negative boost weight (cf. [31, 61]). Therest of projections of the Einstein-Maxwell equations is already satisfied [31, 61]. For the sake ofdefiniteness, an explicit example with D ≥ p = 3 (building on an example given in [54]) isgiven by (4.1) with F = d u ∧ ( f d x ∧ d x + f d x ∧ d x ) , (4.6) W = ax , H (1) = ax , (4.7) H (0) = a x + bx + cx , b + c = − κ ( f + f ) , (4.8)where f , f , a , b and c are arbitrary functions of u and the remaining W i ( i >
2) are understoodto be zero. There is a typo in (6, [31]): the factorial p ! should be simply p . he above spacetimes are generically of Weyl type III [61]. The Weyl type N subclass ofsolutions takes the form (4.1), (4.2) with the constraints (after using some coordinate freedom)[62] H (1) = 0 , W i = 0 . (4.9)Eqs. (4.3) and (4.4) are thus automatically satisfied, while (4.5) reduces to∆ H (0) = − κ F , (4.10)where the RHS (the “source” term) depends only on u . An example is given by (4.6)–(4.8) with a = 0.Finally, conformally flat solutions (i.e., Weyl type O) can be cast in the form W i = 0 , H (1) = 0 , H (0) = − κ F D − X i ( x i ) , (4.11)where a permitted term linear in (or independent of) the x i in H (0) has been removed by atransformation of the form x i x i + h i ( u ), r r − ˙ h i x i + g ( u ) (cf. section 24.5 of [57]).The solution (4.1), (4.2) can be understood as a gravitational and electromagnetic plane-fronted wave propagating in a flat spacetime (recovered for H (1) = H (0) = W i = 0). Since f i...j in (4.2) depends only on u , every admissible electromagnetic field F is constant over its wavesurfaces and hence gives rise to a pure radiation with (transversely) homogeneous energy density κ F / π . Solutions of type N and O belong to the class of pp -waves [63], already discussed ina similar context (for particular values of D and p ) in [19, 21, 22]. Remark 4.1 (Relation to universal spacetimes and electromagnetic fields) . According to theo-rem 3.1, Einstein-Maxwell solutions with vanishing higher-order corrections are defined by
V SI fields ( g , F ) that satisfy τ i = 0 , C acde C cdeb = 0 . (4.12)This ensures that, in the limit of a test electromagnetic field (i.e., a “small” F with negligiblebackreaction), the solution (4.1), (4.2) gives rise to a universal electromagnetic field (theorem 1.5of [33]) propagating in a Ricci-flat universal spacetime (theorem 1.4 of [25]). However, let usemphasize that the vacuum solutions of [25] (and [26,27]) are more general than the backgroundsallowed by our theorem 3.1. One reason for this is that we required all higher-order curvaturecorrections to the Einstein tensor to vanish (and not just be proportional to the metric, asin [25–27] – cf. [24] for related comments), which implied that g (as well as F ) is V SI . Thesecond reason is that we needed to ensure that also corrections constructed out of F vanish,which led to the first of (4.12) (cf. again the proof of ( i ) ⇒ ( ii ) in theorem 3.1 for more details).Similarly, also the test electromagnetic fields on a fixed background obtained in [32,33] are moregeneral than those allowed by our theorem 3.1, and examples are known in which g and/or F are not VSI [32, 33].In addition, it is worth observing that the metric g defined in (4.1) can be related to (a subsetof the) Ricci-flat universal spacetimes of [25] also by a generalized Kerr-Schild transformation with a suitable function H ( u, x ), under which both (4.12) are automatically preserved – in theKundt coordinates (4.1), this amounts to a change H (0) H (0) + H with ∆ H = κ F ( H (0)5 The first of (4.9) was obtained in [61] and means that these solutions belong to the class of pp -waves (i.e., ℓ is covariantly constant). Then, for pp -waves, the “ebenfrontiger Symmetrie” condition (2.1) of [62] is equivalentto imposing the Riemann type N, which allows one to use theorem 2.1 of [62] to arrive at the second of (4.9) (cf.also [22]). oes not appear in ℓ a ; b nor in Riemann components of b.w. 0 , −
1, which explains why (4.12)are preserved).
Remark 4.2 (Kerr-Schild form) . Note that
Weyl type N solutions are Kerr-Schild metrics with ℓ (of (4.1) ) being the Kerr-Schild vector, while genuine type III solutions are not (not even ifthe Kerr-Schild vector is allowed to be a geodetic null vector different from ℓ ). The type N partof this statement is manifest using (4.9). The type III part follows from section 4.2.1 of [64](which implies that a spacetime with a Kerr-Schild, Kundt vector is necessarily of Weyl type N,provided the Ricci tensor is N (aligned) or zero) and from Proposition 2 of [64] (which impliesthat a spacetime of Weyl type III cannot posses a geodesic Kerr-Schild vector distinct from the(unique) mWAND). Remark 4.3 ( D = 4 solutions) . As noticed in Remark 3.2, when D = 4 the condition C acde C cdeb = 0 can be dropped from theorems 3.1 and 3.4, and there are no additional con-straints on the spacetime apart from being VSI (and thus Kundt) with a recurrent PND (andsatisfying Einstein’s equations). Thanks to known results [57], all solutions admitted by theo-rem 3.1 can thus be reduced to the compact formd s = 2d ζ d¯ ζ − u (cid:0) d r + W d ζ + ¯ W d¯ ζ + H d u (cid:1) , F = d u ∧ [ f ( u )d ζ + ¯ f ( u )d¯ ζ ] , (4.13)where W = W ( u, ¯ ζ ) , H = 12 ( W , ¯ ζ + ¯ W ,ζ ) r + H (0) ( u, ζ, ¯ ζ ) , (4.14) H (0) ,ζ ¯ ζ − (cid:16) W , ¯ ζ + ¯ W ,ζ + W W , ¯ ζ ¯ ζ + ¯ W ¯ W ,ζζ + W , ¯ ζu + ¯ W ,ζu (cid:17) = κ f ¯ f . (4.15)These spacetimes are in general of Petrov type III. They are of type N iff W , ¯ ζ ¯ ζ = 0, in whichcase W can be gauged away [57] and one is left with the standard form of electrovac pp -wavesd s = 2d ζ d¯ ζ − u d r − H (0) d u , with H (0) = κ f ( u ) ¯ f ( u ) ζ ¯ ζ + h ( u, ζ ) + ¯ h ( u, ¯ ζ ). These solutionswere mentioned in a related context in [65].Above we discussed the standard case p = 2. When p = 1 (or p = 3 up to duality), the onlydifference is that the electromagnetic field is given by F = f ( u )d u , where f is now real, and theRHS of (4.15) should be replaced by κ f . Remark 4.4 ( D = 4 example of Petrov type III) . In the special case of Einstein gravity coupledto generalized higher-derivative electrodynamics (i.e., L GC = 0 = L int ), the fact that (ii) implies(i) was already pointed out in [31] (but without presenting a proof of this statement). Thanksto theorem 3.1, a simple four dimensional example of Petrov type III constructed there is alsofree of corrections in the more general theory (2.2). This readsd s = 2d u (cid:20) d r + 12 (cid:0) xr − xe x − κ e x c ( u ) (cid:1) d u (cid:21) + e x (d x + e u d y ) , (4.16) F = e x/ c ( u )d u ∧ (cid:18) − cos ye u x + e u sin ye u y (cid:19) . (4.17)It is contained in the more general family (4.13), although here it is expressed in slightly differentcoordinates. cknowledgments We thank Sigbjørn Hervik for useful comments. M.O. has been supported by research plan RVO:67985840 and by the Albert Einstein Center for Gravitation and Astrophysics, Czech ScienceFoundation GA ˇCR 14-37086G.
Appendix A. Variations of L evaluated on V SI fields
Varying the action (2.1) with L ( I i , J j , K k ) ≡ L grav ( I i ) + L elmag ( J j ) + L int ( K k ), one has δS = Z d D x √− g − L g ab δg ab + X i ∂ L grav ∂I i δI i + X j ∂ L elmag ∂J j δJ j + X k ∂ L int ∂K k δK k . (A.1)Let us take a closer look at variation of the individual invariants. Taking e.g. the n -th term ofthe first sum and assuming the boundary terms vanish, integration by parts yields Z d D x √− g ∂ L grav ∂I n δI n = Z d D x √− g ∂ L grav ∂I n δI n δg ab δg ab + (cid:26) terms involving ∇ ( k ) ∂ L grav ∂I n (cid:27) . (A.2)For a CSI metric g , the derivatives ∂ L grav /∂I n are just some constants. Hence, when evaluatedon a CSI metric g , the bracketed term in (A.2) does not contribute to the resulting variation.A similar argument holds also for the rest of the terms in (A.1).Hence, we conclude that, when evaluated on CSI fields ( g , F ), variations of L GC , L EC (recall(2.3) and (2.5) ) and L int reduce to a linear combination of variations w.r.t. δg ab or δF ab...c ofthe individual polynomial invariants. If, moreover, ( g , F ) are V SI , their polynomials invariants I k , J k , K i vanish and so do L GC , L EC and L int . Hence, we arrive at the following expressions(evaluated on VSI fields) G GCab [ g ] = 16 π X i ∂ L GC ∂I i (0) δI i δg ab [ g ] , (A.3) T ECab [ F ] = − X j ∂ L EC ∂J j (0) δJ j δg ab [ F ] , (A.4) G intab [ g , F ] = 16 π X k ∂ L int ∂K k (0) δK k δg ab [ g , F ] , (A.5) ∇ a H ECab...c [ F ] = − πpκ X j ∂ L EC ∂J j (0) ∇ a δJ j δF ab...c [ F ] , (A.6) ∇ a H intab...c [ F ] = − πpκ X k ∂ L int ∂K k (0) ∇ a δK k δF ab...c [ F ] . (A.7)The above results are used in the proof of theorem 3.1 (see also remark 2.2). Similar conclusions(for the metric variations) were obtained in section 4 of [66].For other applications, it may also be useful to note that, since the Lagrangian corrections areof higher-order ( > { I i , J j , K k } is of order 2 (such as I = R , J = F a...b F a...b , . . . ), then necessarily the partial derivative of L GC , L EC and L int withrespect to that invariant vanishes at zero and hence the corresponding term does not contributeto the variation, when evaluated on V SI field. Thus, for example, Einstein’s equations for VSI To avoid possible confusion, let us emphasize that the argument does not really need to assume that (2.3)and (2.5) come from a Taylor expansions – one could alternatively simply define L GC ≡ L grav − L EH and L EC ≡ L elmag − L M (under the assumption that the Taylor expansions of L GC and L EC consist only of termsof higher order, but with no need to take such an expansion). pacetimes are unaffected by higher-order corrections of the form R or RR abcd R abcd , but maycontain corrections coming, e.g., from R ab R ab (cf. [66] in the special case of 4D pp -waves). Appendix B. Curvature/electromagnetic rank-2 tensors and p -forms B.1.
Preliminaries and previous results
Let us start with some preliminary comments. For the definition of degenerate Kundt spacetimes(needed in the following) we refer the reader to [67, 68] (see also appendix A of [31]), while thedefinition of balanced and 1-balanced tensors can be found in [51,69] and [25], respectively. TheGHP (Geroch-Held-Penrose) notation in arbitrary dimension is defined in [70].A null p -form is defined by (D.1). The traceless part of the Ricci tensor is given by S ab ≡ R ab − RD g ab . (B.1)In the following, we will mostly consider spacetimes with constant Ricci scalar. It is thus usefulto recall Lemma B.1 (Bianchi identity when R =const [29]) . In a D -dimensional spacetime ( D ≥ )with R = const., the following identities hold ∇ b R abcd = ∇ d S ac − ∇ c S ad , (B.2) ∇ b C abcd = D − D − ∇ d S ac − ∇ c S ad ) . (B.3) Proof.
Just use the contracted Bianchi identity, the definition of the Weyl tensor and (B.1). (For D = 2 this lemma would be trivial since all the involved quantities vanish identically.) (cid:4) Furthermore, we will restrict ourselves to Kundt spacetimes. A Kundt spacetime with con-stant R is necessarily degenerate Kundt (cf. Proposition A.2 of [31]), for which we have theuseful result
Lemma B.2 (Derivatives of 1-balanced tensors in degenerate Kundt spacetimes [33]) . In adegenerate Kundt spacetime, the covariant derivative of a 1-balanced tensor is a balanced 1-tensor.
In particular,
V SI spacetimes coincide with the Kundt spacetimes of Riemann type III (ormore special) [51, 69], and are therefore a subset of the degenerate Kundt metrics. Recall that
Lemma B.3 ( ∇ ( k ) R in V SI spacetimes [51]) . In a
V SI spacetime, the covariant derivatives ∇ ( k ) R are balanced for any k ≥ . In the rest of this appendix, we will only consider Kundt spacetimes of traceless Ricci type N ,i.e., S ab = ω ′ ℓ a ℓ b ( ℓ a ℓ a = 0) , (B.4)where ω ′ is a function. .2. New results useful in the proof of theorems 3.1 and 3.4
With a mild assumption on R (i.e., not necessarily constant) one can prove Lemma B.4.
Let g be a traceless Ricci type N Kundt metric with þ ′ R = 0 in a frame adaptedto ℓ . Then ∇ ( k ) S is 1-balanced for any k ≥ .Proof. Under the assumptions, the contracted Bianchi identity implies þ ω ′ = 0 (cf. the primedversion of (2.50, [70]), or (2.35, [71])). Therefore, the definition of 1-balanced tensors, togetherwith lemma B.2, implies that the traceless part of the Ricci tensor and its covariant derivativesof arbitrary order are 1-balanced. (cid:4) Lemma B.5.
Let g be a Weyl type III, Ricci type N Kundt metric. There is no non-vanishing p -form constructed from R and its covariant derivatives for p ≥ .Proof. Before starting, let us note that, under the assumptions, ∇ ( k ) R is balanced for any k ≥ ∇ ( l ) S is 1-balanced for any l ≥ p = 0, such form would be a curvature scalar, which here vanishes since g is V SI . The dual case p = D is treated analogously. Let us thus discuss the 0 < p < D case.If there was a non-vanishing p -form H [ R , ∇ R , . . . ] (with boost order at least ( − ∇ ( k ) C , k ≥
0, and from the Ricci identity, it follows that covariantderivatives in ∇ ( k ) C effectively commute (i.e., up to terms of b.w. -2).Consider the p ≤ C is traceless, there are necessarily at least two contractionsof a derivative index with a Weyl tensor index within ∇ ( k ) C . After commuting derivatives andemploying (B.3), we observe that such a contraction is of boost order ( −
2) and H vanishes.Now, let us discuss the p > p contraction of ∇ ( k ) C . In order to produce a p -form, it has to be antisymmetrized overall remaining p ≥ R a [ bcd ] = 0 , R ab [ cd ; e ] = 0 . (B.5)Since covariant derivatives in ∇ ( k ) C effectively commute, the antisymmetrization has to beperformed over at most one derivative index and at least two Weyl tensor indices. But from(B.5), it follows that (after shuffling the derivatives if needed) the result is zero anyway. (cid:4) Lemma B.6.
Let g be a Weyl type III, Ricci type N Kundt metric and F be an aligned null p -form. There is no non-vanishing symmetric rank-2 contraction of ∇ ( k ) C ⊗ F and ∇ ( k ) R ⊗ F for k ≥ .Proof. Both F and ∇ ( k ) C are of boost order ( − F , at most one ofits indices can be left uncontracted, while each of the rest of the indices of F has to be contractedwith some index of ∇ ( k ) C . Moreover, covariant derivatives of C again effectively commute.If p >
3, this necessarily yields antisymmetrization of ∇ ( k ) C over at least 3 indices, which iszero due to Bianchi identities and effective commutativity of covariant derivatives of C , as wesaw in the proof of lemma B.5.For p ≤
3, there is either one index of F left uncontracted (and hence there is necessarilya contraction of indices within ∇ ( k ) C ) or each of the indices of F is contracted with someindex of ∇ ( k ) C . However, any contraction within ∇ ( k ) C will eventually (after commuting thederivatives) vanish, since ∇ a C abcd , and consequently also C abcd , are (recall (B.3)) of boostorder ( − ne can easily verify that the corresponding contraction vanishes again due to skew-symmetryof F and Bianchi identities (B.5).That the same result holds also for ∇ ( k ) R ⊗ F follows from the Ricci tensor being of type N(and a trivial b.w. counting). (cid:4) Lemma B.7.
Let g be a Weyl type III, Ricci type N Kundt spacetime and F an aligned nullMaxwell p -form. If ∇ F is -balanced, then all non-vanishing symmetric rank-2 tensors con-structed from F and its covariant derivatives are of second order.Proof. By simple b.w. counting, terms cubic in F and quadratic in ∇ ( k ) F ( k >
0) cannotcontribute (and similarly for higher powers), while terms quadratic in F are obviously of secondorder. Terms linear in F cannot contribute because of its total antisymmetry. It remainsto be shown that also terms linear in ∇ ( k ) F do not contribute. Let us first discuss the case1 < p < D −
1. By the symmetry of the indices there must be at least one contraction of an indexof F with one derivative index. The idea is thus to use commutators of covariant derivativesand the Maxwell equations to show that all such terms vanish. By the Ricci identity and 1-balancedness of ∇ ( k ) F , commutators [ ∇ , ∇ ] ∇ ( k ) F with k > − ∇ , ∇ ] F (and its derivatives). This gives terms which are contractions of ∇ ( l ) C ⊗ F for l ≥ − < p < D −
1. When p = 1 (or, by duality, p = D − ∇ ( k ) F even without contracting an index of F with one derivative index.For k = 1 this gives the term ∇ ( a F b ) , which is of order 2. For k > k ≥
3) thereis at least a contraction between two derivative indices. Similarly as above, derivatives in suchterms can thus be shuffled to obtain ∇ k − F , which vanishes thanks to Maxwell’s equationsand the Weitzenb¨ock identity (cf. eq. (12) of [33]). (cid:4) Remark B.8 (Terms of second order) . For completeness, let us observe that, under the as-sumptions of lemma B.7, terms quadratic in F generically reduce to (constant multiples of) F ac...d F c...db = F ℓℓ . In the special case n = 2 p with p odd, another possible term is F ac...d ⋆F c...db , which is in general non-zero and different from F ac...d F c...db (but still ∝ ℓℓ ). For n = 2 p with p even, instead, such a term vanishes identically thanks to the identity F ac...d ⋆F c...db [1 +( − p ] = p ( F cd...e ⋆F cd...e ) g ab (since F is VSI and thus F cd...e ⋆F cd...e = 0). We further notethat, for p = 1, the term ∇ ( a F b ) is also proportional to ℓ a ℓ b (just by b.w. counting), but ingeneral different from F a F b . Appendix C. Rank-2 curvature tensors in recurrent spacetimes of Weyl type IIIand traceless Ricci type N with C acde C cdeb = 0In [28] (cf. also [25]), it was shown that if g is a Weyl type N and traceless Ricci type N Kundtmetric with a constant Ricci scalar (in which case g is necessarily CSI , see Corollary A.5 of [72]and Remark A.9 of [33]), then any symmetric rank-2 tensor constructed from the Riemann tensorand its covariant derivatives takes the form T ab = λg ab + P Nn =0 a n n S ab , where λ and a n aresome constants and N ∈ N . It has been recently shown that the assertion can be extended alsoto Weyl type III, provided the Weyl tensor satisfies certain conditions [73]. A special subcase ofProposition 6 of [73] (cf. also (14) therein), useful for our purposes, can be formulated as Throughout the proof we did not discuss explicitly terms constructed using the dual ( D − p )-form ⋆ F .However, all the steps still apply, since ⋆ F is automatically aligned with F and inherits from it all the essentialproperties. heorem C.1 (On symmetric rank-2 tensors [73]) . Let g be a Weyl type III and traceless Riccitype N metric such that: (i) the mWAND is recurrent; (ii) C acde C cdeb = 0 . Then any symmetricrank-2 tensor constructed from the Riemann tensor and its covariants derivatives of arbitraryorder takes the form T ab = N X n =0 a n n S ab . (C.1)For self-containedness, let us present a proof tailored to this special case. Proof.
Before starting we observe that the Weyl and Ricci tensors are necessarily aligned thanksto proposition 3.1 of [71]. Then, the line of the proof will be similar to that of [28]. However, incontrast with the Weyl type N case, also various contractions of the Weyl tensor and its covariantderivatives can in principle contribute to T [25]. But under the additional conditions, we willshow that any of these actually vanishes, so that one is left with T of the form (C.1).First, τ i = 0 implies that the Ricci scalar vanishes (cf., e.g., Remark A.9 of [33]). Also, C isbalanced and S is 1-balanced (lemma B.4), and hence the only possible contributions to T comefrom contractions of ∇ ( k ) R and of ∇ ( k ) C ⊗ ∇ ( l ) C with k, l ≥
0. In particular, T is traceless.Now, let us focus on contractions of ∇ ( k ) R (clearly, k has to be even). From the Ricci identity,it is obvious that any change in the order of covariant derivatives in ∇ ( k ) R produces only termsof type ∇ ( m C ⊗∇ ( l ) C . Following the procedure sketched in [28] with use of (B.2) and ∇ b S ab = 0(since S is 1-balanced), any contraction of ∇ ( k ) R can be cast in the form linear in k/ S plusterms quadratic in the Weyl tensor and its covariant derivatives.At this moment, to finish the proof of the assertion, it is sufficient to show that all rank-2contractions of ∇ ( k ) C ⊗ ∇ ( l ) C vanish. This can be done employing (B.3) and following step bystep the procedure of section 5.1 in [25]. In this manner, one obtains an extension of proposition5.6 of [25] to the Ricci type N case, which completes the proof. (cid:4) Appendix D. On covariant derivatives of null p -forms in Kundt spacetimes In this section, we will provide some useful results on null p -forms and their covariant derivatives.A p -form F is null iff it can be written as [31] F = ℓ ∧ f , ℓ a ℓ a = 0 = f a...b ℓ a , (D.1)where f is a ( p − F possesses only components of b.w. − ≤ p ≤ D − Remark D.1 (Maxwell’s equations) . If one assumes that ℓ in (D.1) is Kundt, in a null frameadapted to ℓ the GHP Maxwell equations reduce to [70] (cf. also eqs. (2.16)–(2.18) of [72] – f ij...k is denoted ϕ ′ ij...k in [70, 72]) k i f ij...k = τ i f ij...k , (D.2) k [ i f j...k ] = τ [ i f j...k ] , (D.3) þ f i...j = 0 . (D.4)If ℓ in (D.1) is Kundt, ∇ F possesses generically non-zero components of b.w. 0 , − , −
2. Moreprecisely, defining the standard directional derivatives D ≡ ℓ a ∇ a , △ ≡ n a ∇ a , δ i ≡ m ( i ) a ∇ a , Lemma D.2.
Let F = ℓ ∧ f be a null p -form and ℓ a Kundt vector field. Then, in a null frameadapted to ℓ (i.e., with m (0) = ℓ but otherwise arbitrary)(i) D F is null with frame components ( DF ) i...j = þ f i...j ; ii) δ i F is null with frame components ( δ i F ) j...k = k i f j...k ;(iii) △ F is of type II with frame components ( △ F ) j...k = τ i f ij...k , ( △ F ) ij...k = pτ [ i f j...k ] and ( △ F ) i...j = þ ′ f i...j .Proof. The result follows by a direct calculation of the various frame components of ∇ F . (cid:4) Lemma D.3.
Let F = ℓ ∧ f be a null p -form and ℓ a Kundt vector field. If þ f i...j = 0 , then ∇ c F ad...e ∇ c F d...eb = ( k i f j...k )( k i f j...k ) ℓ a ℓ b . (D.5) Proof.
Thanks to lemma D.2, we know that ∇ F has only components of negative b.w.. Thecontraction over c in (D.5) further ensures that only the components (ii) of lemma D.2 contribute,and the result thus follows. (cid:4) Remark D.4.
The assumption þ f i...j = 0 in lemma D.3 is satisfied identically if F is a Maxwellfield (eq. (D.4)).The special case when ∇ F has only components of b.w. − Lemma D.5.
Let F be a non-vanishing null p -form. Then, ∇ F is of type N (necessarily aligned)iff ℓ is Kundt and the scalars þ f i...j , k i f j...k , τ i vanish.Proof. The type N condition means that ∇ F possesses only components of b.w. − ∇ F is aligned with F , since F is null). Proposition C.1 of [31] implies that ℓ isKundt and þ f i...j = 0. Using lemma D.2 further gives k i f j...k = 0 and τ i f ij...k = 0 = τ [ i f j...k ] .Since f i...k = 0, the last two equations imply τ i = 0. (For p = 1 the equation τ i f ij...k = 0 doesnot appear, but the conclusion is unchanged.) The other direction of the lemma can be provenby just reversing the above steps. (cid:4) Remark D.6.
The fact that ℓ is Kundt and τ i = 0 is equivalent to saying that ℓ is recurrent .In addition, note that, in particular, a null F with ∇ F of type N satisfies Maxwell’s equationsidentically (cf. eqs. (D.2)–(D.4)). Lemma D.7.
Let g be a spacetime of Weyl type III and F an aligned null p -form F such that ∇ F is of type N. If ( g , F ) is a solution of the Einstein-Maxwell equations, necessarily Λ = 0 and both ( g , F ) are V SI .Proof.
From lemma D.5 we have that ℓ is recurrent (and thus Kundt). The Einstein equationsimply that the traceless Ricci type is N (and that R is proportional to Λ), therefore the spacetimeis Kundt degenerate. A non-vanishing Ricci scalar would require τ i = 0 (cf., e.g., Remark A.9of [33]), therefore Λ = 0. The Ricci type is thus N and the VSI property of g then followsimmediately from theorem 1 of [51]. Finally, the VSI property of F follows from theorem 1.5of [31] (since þ f i...j = 0 by lemma D.5). (cid:4) Lemma D.8.
Let F be a non-vanishing null Maxwell field. Then ∇ F is of type N iff k i f j...k = 0 and ℓ is Kundt.Proof. Maxwell’s equations (D.2), (D.3) guarantee that, if a non-vanishing null solution F in aKundt spacetime satisfies k i f j...k = 0, then automatically also τ i = 0. By lemma D.5, the “if”assertion follows. The same lemma ensures that also the “only if” direction holds. (cid:4) Lemma D.9.
Let F be a non-vanishing null Maxwell field in an aligned Weyl and tracelessRicci type III spacetime. Then ∇ F is 1-balanced iff k i f j...k = 0 and ℓ is Kundt. roof. Thanks to lemma D.8, we know that k i f j...k = 0 and ℓ Kundt are necessary conditionsfor 1-balancedness of ∇ F . To show that these conditions are also sufficient, it remains toverify (lemma D.2) that D þ ′ f i...j = 0 in an affinely parametrized, parallely propagated frame.Since Df i...j = DL = DM ij = 0 and [ △ , D ] = L D (thanks to the Kundt and curvatureassumptions, cf., e.g., appendix A.1 of [31]), the assertion follows. (cid:4) Remark D.10.
Thanks to lemma D.3 and remark D.4, the condition k i f j...k = 0 in theorem D.9can equivalently be written in a covariant form as ∇ c F ad...e ∇ c F d...eb = 0 . (D.6)Let us emphasize that for a null Maxwell field aligned with a Kundt direction, this conditionimplies that ℓ is recurrent (as observed in the proof of lemma D.8). Remark D.11.
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