Classification of the Weyl Tensor in Higher Dimensions and Applications
aa r X i v : . [ g r- q c ] J a n Classi(cid:28) ation of the Weyl Tensor in Higher Dimensions andAppli ations.A. Coley ‡ ‡ Department of Mathemati s and Statisti s, Dalhousie University, Halifax, Nova S otia, CanadaAbstra t.We review the theory of alignment in Lorentzian geometry and apply it to the algebrai lassi(cid:28) ation of the Weyl tensor in higher dimensions. This lassi(cid:28) ation redu es to the thewell-known Petrov lassi(cid:28) ation of the Weyl tensor in four dimensions. We dis uss the algebrai lassi(cid:28) ation of a number of known higher dimensional spa etimes. There are many appli ations ofthe Weyl lassi(cid:28) ation s heme, espe ially when used in onjun tion with the higher dimensional frameformalism that has been developed in order to generalize the four dimensional Newman(cid:21)Penroseformalism. For example, we dis uss higher dimensional generalizations of the Goldberg-Sa hstheorem and the peeling theorem. We also dis uss the higher dimensional Lorentzian spa etimeswith vanishing s alar urvature invariants and onstant s alar urvature invariants, whi h are ofinterest sin e they are solutions of supergravity theory.lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 21. Introdu tionWe review the dimension-independent theory of alignment, using the notions of an aligned nulldire tion and alignment order, in Lorentzian geometry [1℄. We then apply it to the tensor lassi(cid:28) ation problem for the Weyl tensor in higher dimensions [2℄. In parti ular, it is possibleto ategorize algebrai ally spe ial tensors in terms of their alignment type, with in reasingspe ialization indi ated by a higher order of alignment. In essen e, one tries to normalize theform of the tensor by hoosing null ve tors ℓ and n so as to indu e the vanishing of the largestpossible number of leading and trailing (with respe t to boost weight) null-frame s alars. The tensor an then be ategorized by the extent to whi h su h a normalization is possible. The approa hreviewed is inspired by the work of Penrose and Rindler used in general relativity [3℄, whi h isreinterpreted and generalized in terms of frame (cid:28)xing. The omponents of a Lorentzian tensor an be naturally ordered a ording to boost weight. The approa h is equivalent to the well-knownPetrov lassi(cid:28) ation of Weyl tensors in 4 dimensions [4, 1℄.In pra ti e, a omplete tensor lassi(cid:28) ation in terms of alignment type is possible only forsimple symmetry types and for small dimensions N [2℄. However, partial lassi(cid:28) ation intobroader ategories is still desirable. We note that alignment type su(cid:30) es for the lassi(cid:28) ationof 4-dimensional Weyl tensors, but the situation for higher-dimensional Weyl tensors is more ompli ated. For example, a 4-dimensional Weyl tensor always possesses at least one aligneddire tion; for higher dimensions, in general, a Weyl tensor does not possess any aligned dire tions.The end result of this approa h is a oarse lassi(cid:28) ation, in the sense that it does not always leadto anoni al forms for higher-dimensional Weyl tensors (unlike the 4-dimensional lassi(cid:28) ation, inwhi h ea h alignment type admits a anoni al form), and hen e is simply a ne essary (cid:28)rst step inthe investigation of ovariant tensor properties.In the higher dimensional lassi(cid:28) ation, the se ondary alignment type is also of signi(cid:28) an e. Foralgebrai ally spe ial types, even though an aligned null ve tor ℓ exists there does not, generi ally,exist a se ond aligned null ve tor n , again in ontrast to 4 dimensions. Hen e, in higher dimensionswe an distinguish algebrai ally spe ial sub lasses that possess an aligned n . Therefore, byde(cid:28)nition, lassi(cid:28) ation a ording to alignment type orresponds to a normal form where the omponents of leading and trailing boost weight vanish. It would be desirable to re(cid:28)ne the lassi(cid:28) ation to obtain true anoni al forms. For example, this is possible in the ase of the mostalgebrai ally spe ial Weyl tensors (e.g., type N an always be put into a anoni al form). For moregeneral Weyl tensors, the generi situation is that a number of relevent Weyl tensor omponents(a ording to boost weight) are non-zero whi h an be further simpli(cid:28)ed by performing spins andboosts (so that hopefully the Weyl tensor an onsequently be put into a anoni al form). Finally,the Ri i tensor an be further lassi(cid:28)ed a ording to its eigenvalue stru ture (i.e., Segré type). Itmay also be of interest to study the algebrai properties of the orresponding Weyl bive tors.The lassi(cid:28) ation outlined is a general s heme whi h is appli able to the lassi(cid:28) ation ofarbitrary tensor types in arbitrarily high dimensions. Indeed, mu h of the analysis for higherdimensional Weyl tensors an be applied dire tly to the lassi(cid:28) ation of higher dimensional Riemann urvature tensors. In parti ular the higher-dimensional alignment types give well de(cid:28)ned ategorieslassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 3for the Riemann tensor (although there are additional onstraints oming from the extra non-vanishing omponents). The most important algebrai lassi(cid:28) ation results in 4-dimensional generalrelativity is the lassi(cid:28) ation of the Weyl tensor a ording to Petrov type and the lassi(cid:28) ation ofthe Ri i tensor a ording to Segré type. We an also use alignment to lassify the se ond-ordersymmetri Ri i tensor (whi h we refer to as Ri i type). We an extend the lassi(cid:28) ation of the Ri itensor by des ribing a number of additional algebrai types for the various alignment on(cid:28)gurationsand hen e obtain a omplete lassi(cid:28) ation; for example, by onsidering the geometri properties ofthe orresponding alignment variety or by studying the Segré type [1℄.The lassi(cid:28) ation of algebrai tensor types in Lorentzian geometry has a purely mathemati alimportan e. The inde(cid:28)nite signature makes Lorentzian geometry profoundly di(cid:27)erent fromRiemannian geometry. The representation of the Weyl tensor is also of parti ular interest fromthe physi al standpoint. First, as in four dimensions, tensor lassi(cid:28) ation is important for physi alappli ations and, in parti ular, for the study of exa t solutions of the Einstein (cid:28)eld equations [20℄.In addition, beyond the lassi al theory, there has been great interest in higher dimensional Lorentzmanifolds as models for generalized (cid:28)eld theories that in orporate gravity [5, 6℄.The main fo us of the lassi(cid:28) ation s heme is purely algebrai (i.e., we onsider tensors at asingle point of a Lorentzian manifold rather than tensor (cid:28)elds). There is, however, a ri h interplaybetween algebrai type and di(cid:27)erential identities. Indeed, as in 4 dimensions, in higher dimensionsit is possible to use the Bian hi and Ri i equations to onstru t algebrai ally spe ial solutionsof Einstein's (cid:28)eld equations, at least for the simplest algebrai ally spe ial spa etimes. The vastmajority of urrently known higher-dimensional exa t solutions are simple generalizations of 4-dimensional solutions. This approa h may lead to new, genuinely higher dimensional exa t solutions.Type N and D solutions may be of parti ular physi al interest.1.1. OverviewIn the next se tion we shall review the algebrai lassi(cid:28) ation of the Weyl tensor in higherdimensional Lorentzian manifolds, by hara terizing algebrai ally spe ial Weyl tensors by meansof the existen e of aligned null ve tors of various orders of alignment. We shall dis uss theprin ipal lassi(cid:28) ation in terms of the prin ipal type of the Weyl tensor in a Lorentzian manifold,and the se ondary lassi(cid:28) ation in terms of the alignment type. For irredu ible representationsthe lassi(cid:28) ation problem essentially redu es to the study of the orresponding variety of all topalignment s hemes; we brie(cid:29)y dis uss the alignment variety. We dis uss further lassi(cid:28) ation interms of redu ibility, the dimension of the alignment variety, and the multipli ity of prin ipledire tions. We then des ribe how the higher dimensional lassi(cid:28) ation redu es to the lassi alPetrov lassi(cid:28) ation in 4 dimensions.We next dis uss the di(cid:27)erent methods of lassi(cid:28) ation, in luding a more pra ti al method interms of ne essary onditions. We then review the lassi(cid:28) ation of a number of known higherdimensional spa etimes, in luding many va uum solutions of the Einstein (cid:28)eld equations whi h anrepresent higher dimensional bla k holes.In se tion 3 we review a higher dimensional frame formalism that has been developed in orderlassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 4to generalize the Newman(cid:21)Penrose formalism for any N > . In parti ular, we des ribe the frame omponents of the Bian hi equations and the Ri i identities. There are many appli ations of theWeyl lassi(cid:28) ation s heme, espe ially when used in onjun tion with the higher dimensional frameformalism. For example, we shall dis uss higher dimensional generalizations of the Goldberg-Sa hstheorem, the type-D onje ture that asserts that stationary higher dimensional bla k holes are ofWeyl type D, and the peeling theorem. We also brie(cid:29)y dis uss the (cid:28)ve dimensional ase and typeD spa etimes.In se tion 4 we shall dis uss the higher dimensional Lorentzian spa etimes with vanishing s alar urvature invariants ( V SI spa etimes) and onstant s alar urvature invariants (
CSI spa etimes).Higher dimensional
V SI spa etimes are ne essarily of Weyl type III, N or O, and higher dimensional
CSI spa etimes that are not lo ally homogeneous are at most of type II. We begin with a dis ussionof higher dimensional Kundt spa etimes and spa etimes that admit a ovariantly onstant null ve tor(whi h are Kundt and generi ally of Ri i and Weyl type II).All of the higher dimensional
V SI spa etimes are known expli itly. Various lasses of
CSI spa etimes are dis ussed, and it has been onje tured that if a spa etime is
CSI then that spa etimeis either lo ally homogeneous or it belongs to the higher dimensional Kundt lass and an be onstru ted from lo ally homogeneous spa es and
V SI spa etimes. The various
CSI onje turesare dis ussed. The
V SI and
CSI spa etimes are of parti ular interest sin e they are solutionsof supergravity or superstring theory, when supported by appropriate bosoni (cid:28)elds. It is knownthat the higher-dimensional
V SI spa etimes with (cid:29)uxes and dilaton are solutions of type IIBsupergravity, and it has been argued that they are exa t solutions in string theory.
V SI solutionsof IIB supergravity with NS-NS and RR (cid:29)uxes and dilaton have been expli itly onstru ted. Wedis uss
CSI spa etimes that are solutions of supergravity. The supersymmetry properties of the
V SI and
CSI spa etimes are also dis ussed.In the (cid:28)nal se tion we brie(cid:29)y dis uss some of the outstanding problems and outline possiblefuture work.lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 52. Classi(cid:28) ation of the Weyl Tensor in Higher DimensionsLet us review the algebrai lassi(cid:28) ation of the Weyl tensor in higher N-dimensional Lorentzianmanifolds [2, 1℄, by hara terizing algebrai ally spe ial Weyl tensors by means of the existen e ofaligned null ve tors of various orders of alignment. Further lassi(cid:28) ation is obtained by spe ifyingthe alignment type and utilizing the notion of redu ibility. The present lassi(cid:28) ation redu es to the lassi al Petrov lassi(cid:28) ation in 4 dimensions [4℄.We shall onsider a null frame ℓ = m , n = m , m , ... m i ( ℓ , n null with ℓ a ℓ a = n a n a =0 , ℓ a n a = 1 , m i real and spa elike with m ia m ja = δ ij ; all other produ ts vanish) in an N -dimensionalLorentzian spa etime with signature ( − + ... +) , so that g ab = 2 l ( a n b ) + δ jk m ja m kb . Indi es a, b, c rangefrom to N − , and spa e-like indi es i, j, k also indi ate a null-frame, but vary from to N − only. We note that all notation and onventions (e.g., for the de(cid:28)nitions of the Riemann and Ri itensors) follow that of [1, 2, 7℄.The frame is ovariant relative to the group of real ortho hronous linear Lorentztransformations, generated by null rotations, boosts and spins. A null rotation about n is a Lorentztransformation of the form ˆ n = n , ˆm i = m i + z i n , ˆ ℓ = ℓ − z i m i − δ ij z i z j n . (1)A null rotation about ℓ has an analogous form. A spin is a transformation of the form ˆ n = n , ˆm i = X ji m j , ˆ ℓ = ℓ , (2)where X ji is an orthogonal matrix. Finally, and most importantly, a boost is a transformation ofthe form ˆ n = λ − n , ˆm i = m i , ˆ ℓ = λ ℓ , λ = . (3)Let T a ...a p be a rank p tensor. For a (cid:28)xed list of indi es A , ..., A p , we all the orresponding T A ...A p a null-frame s alar. These s alars transform under a boost (3) a ording to ˆ T A ...A p = λ b T A ...A p , b = b A + ... + b A p , (4)where b = 1 , b i = 0 , b = − . We all b the boost-weight of the s alar. We de(cid:28)ne the boostorder of the tensor T to be the boost weight of its leading term (i.e., the maximum value for b A ...A p for all non-vanishing T A ...A p ). The result of applying null rotations, boosts and spins on the omponents of the Riemann, Ri i and Weyl tensors and the Ri i rotation oe(cid:30) ients are given in[1, 7, 8℄.We introdu e the notation T { pqrs } ≡
12 ( T [ pq ][ rs ] + T [ rs ][ pq ] ) . (5)We an de ompose the Weyl tensor and sort the omponents of the Weyl tensor by boost weight[1℄: C abcd = z }| { C i j n { a m ib n c m j d } + z }| { C i n { a ℓ b n c m id } + 4 C ijk n { a m ib m jc m kd } + lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 6 ( C n { a ℓ b n c ℓ d } + 4 C ij n { a ℓ b m ic m jd } +8 C i j n { a m ib ℓ c m j d } + C ijkl m i { a m j b m kc m ld } ) + (6) z }| { C i ℓ { a n b ℓ c m id } + 4 C ijk ℓ { a m ib m j c m kd }− + z }| { C i j ℓ { a m ib ℓ c m j d }− . − − C i j C i , C ijk C , C ij , C i j , C ijkl C i , C ijk C i j Table 1. Boost weights of the Weyl s alars.The Weyl tensor is generi ally of boost order . If all C i j vanish, but some C i , or C ijk donot, then the boost order is , et . The Weyl s alars also satisfy a number of additional relations,whi h follow from urvature tensor symmetries and from the tra e-free ondition: C i i = 0 , C j = C iji , C ijk ) = 0 , C = C i i , C i ( jkl ) = 0 ,C i j = − C ikjk + 12 C ij , C j = − C ij i , C ijk ) = 0 , C i i = 0 . (7)A real null rotation about ℓ (cid:28)xes the leading terms of a tensor, while boosts and spins subje t theleading terms to an invertible transformation. It follows that the boost order (along ℓ ) of a tensoris a fun tion of the null dire tion ℓ (only). We shall therefore denote boost order by B ( ℓ ) [1℄. Wede(cid:28)ne a null ve tor ℓ to be aligned with the Weyl tensor whenever B ( ℓ ) ≤ (and we shall refer to ℓ as a Weyl aligned null dire tion (WAND)). We all the integer − B ( ℓ ) ∈ { , , , } the orderof alignment. The alignment equations are N ( N − degree-4 polynomial equations in ( N − variables, whi h are in general overdetermined and hen e have no solutions for N > .Any tensor, in luding the Riemann tensor, an be algebrai ally lassi(cid:28)ed a ording to boostweight in arbitrary dimensions in a similar way [1℄. However, the value of the maximum boostorder will depend on the rank and symmetry properties of the tensor. In parti ular, we an alsouse alignment to lassify the se ond-order symmetri Ri i tensor in higher dimensions. We anextend the lassi(cid:28) ation of the Ri i tensor by des ribing a number of additional algebrai typesfor the various alignment on(cid:28)gurations and hen e obtain a omplete lassi(cid:28) ation (for example,by onsidering the geometri properties of the orresponding alignment variety or by studying theSegré type [1℄).2.1. Prin ipal Classi(cid:28) ation:Following [1℄, we say that the prin ipal type of the Weyl tensor in a Lorentzian manifold is I,II, III, N a ording to whether there exists an aligned ℓ of alignment order , , , (i.e. B ( ℓ ) =1 , , − , − ), respe tively. If no aligned ℓ exists we will say that the manifold is of (general) typeG. If the Weyl tensor vanishes, we will say that the manifold is of type O. The algebrai ally spe ialtypes are summarized as follows: T ype I : C i j = 0 lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 7 T ype II : C i j = C ijk = 0 T ype
III : C i j = C ijk = C ijkl = C ij = 0 T ype N : C i j = C ijk = C ijkl = C ij = C ijk = 0 (8)Note that the number of independent frame omponents of various boost weights is , − z }| { N ( N − ! + , − z }| { ( N − N − N − ! + z }| { ( N − ( N − N − N − N − , where the number of independent omponents of the Weyl tensor is ( N + 2)( N + 1) N ( N − . A 4-dimensional Weyl tensor always possesses at least one aligned dire tion. For higher dimensions,in general, a Weyl tensor does not possess any aligned dire tions. Indeed, in [1℄ it was shown thatif N ≥ , then the set of Weyl tensors with alignment type G is a dense, open subset of the set(ve tor spa e) of all N -dimensional Weyl tensors.2.2. Se ondary Classi(cid:28) ation:Further ategorization an be obtained by spe ifying alignment type [1℄, whereby we try to normalizethe form of the Weyl tensor by hoosing both ℓ and n in order to set the maximum number of lead-ing and trailing null frame s alars to zero. Let ℓ be a WAND whose order of alignment is as large aspossible. We then de(cid:28)ne the prin ipal (or primary) alignment type of the tensor to be b max − b ( ℓ ) .Supposing su h a WAND ℓ exists, we then let n be a null ve tor of maximal alignment subje t to ℓ a n a = 1 . We de(cid:28)ne the se ondary alignment type of the tensor to be b max − b ( n ) . The alignmenttype of the Weyl tensor is then the pair onsisting of the prin ipal and se ondary alignment type[1℄. In general, for types I , II , III there does not exist a se ondary aligned n (in ontrast to thesituation in 4 dimensions), when e the alignment type onsists solely of the prin ipal alignmenttype. Alignment types (1,1), (2,1) and (3,1) therefore form algebrai ally spe ial sub lasses of types I , II , III , respe tively (denoted I i , II i , III i ). There is one (cid:28)nal sub lass possible, namely type (2,2)whi h is a further spe ialization of type (2,1); we shall denote this as type II ii or simply as type D . Therefore, a type D Weyl tensor in anoni al form has no frame omponents of boost weights , , − , − (i.e., all frame omponents are of boost weight zero).lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 82.3. The alignment varietyIn the lassi(cid:28) ation s heme we essentially study the lo us of aligned null dire tions, where thesingular points orrespond to null dire tions of higher alignment (for whi h multiple leadingorders vanish). For irredu ible representations the lassi(cid:28) ation problem is redu ed to the studyof the orresponding variety of all top alignment s hemes quotiented by the group of Möbiustransformations. Let us dis uss this in more mathemati al detail.It an be shown that the set of null dire tions aligned with a (cid:28)xed N -dimensional tensor is avariety ( alled the top alignment variety); i.e., it an be des ribed by a ertain set of polynomialequations in N − variables [1℄. These alignment equations are given with respe t to a parti ularnull-frame. A hange of null-frame transforms the equations in a ovariant fashion. The ne essarymathemati al framework needed to des ribe su h ovariant ompatibility is a s heme.The null dire tions of higher alignment order are subvarieties of the top alignment variety.These higher-order dire tions have an important geometri hara terization related to singularities.Indeed, it an be proven that for irredu ible representations of the Lorentz group, the equations forhigher order alignment are equivalent to the equations for the singular points of the top alignmentvariety [1℄. Thus, for the Weyl tensor, the algebrai ally spe ial tensors an be hara terized by thesingularities of the orresponding alignment variety.Indeed, it an be shown that the set of aligned dire tions is a Möbius variety, by exhibiting the ompatible equations for this set. By de(cid:28)nition, a null ve tor k = ℓ −
12 z i z i n + z i m i , is aligned with a rank p tensor T , with alignment order q , if the orresponding q th order alignmentequations (polynomial in z i ) are satis(cid:28)ed. The ideal generated by the above polynomials is alledthe alignment ideal of order q . The alignment s heme of order q , A q = A q ( T ) , is the s hemegenerated by the alignment ideals A q ( T ) . The top alignment s heme, denoted A ≡ A ( T ) ( q = 0 ),has a distinguished role. The orresponding varieties V ( A q ) onsist of aligned null dire tions ofalignment order q or more. It is possible that the alignment equations are over-determined anddo not admit a solution, in whi h ase the variety is the empty set. In those ases where V ( A q ) onsists of only a (cid:28)nite number of null dire tions, we all these dire tions prin ipal and speak ofthe prin ipal null dire tions (PNDs) of the tensor.A point of the variety z i ∈ V ( I ) will be alled singular if the (cid:28)rst-order partial derivativesof the polynomials in the a(cid:30)ne ideal I also vanish at z i . It will be alled singular of order q ifall partials of order q and lower vanish at that point. Geometri ally, a singular point represents aself-interse tion su h as a a node or a usp, or a point of higher multipli ity. It was shown [1℄ that if k spans an aligned dire tion of order q , then [ k ] is a q th order singular point of the top variety V ( A ) .Furthermore, if T belongs to an irredu ible representation of the Lorentz group and k is a q th -ordersingular element of the top alignment variety, then k spans a q th order aligned null dire tion (and onsequently the q th order alignment s heme A q des ribes the q th -order singular points of the tops heme A ).lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 93. Further lassi(cid:28) ation3.1. De omposability and Redu ibilityThe Weyl tensor C abcd is redu ible if and only if it is the sum of two Weyl tensors, one of whi h isa Weyl tensor of an (irredu ible) Lorentzian spa e of dimension M , and the other is a Weyl tensorof a Riemannian spa e of dimension N − M . It turns out that sub lassi(cid:28) ation of the Weyl tensoris most easily a omplished by de omposing the Weyl tensor and lassifying its irredu ible parts.Let us onsider a spa etime, M , whi h is a para ompa t, Hausdor(cid:27), simply onne ted smoothLorentzian manifold of arbitrary dimension N . We an dis uss the lo al de omposability andredu ibility of manifolds in terms of holonomy groups and their asso iated holonomy algebras. Letus work in lo al oordinates, where lower ase Latin indi es range from , . . . , N − , and Greekand upper ase Latin indi es have the ranges α, β, γ = 0 , . . . , M − and A, B, C = M, . . . , N − ,respe tively. We onsider the Lorentzian manifold M to be an N-dimensional produ t manifold: M N = M M ⊗ ˜ M ˜ M ; N = M + ˜ M , (9)where M M is an M-dimensional Lorentzian manifold and ˜ M ˜ M an ˜ M -dimensional Riemannianmanifold, with metri g N = g M ⊕ ˜ g ˜ M , (10)or in oordinates ds = g αβ ( x γ ) dx α dx β + ˜ g AB ( y C ) dy A dy B . (11)A produ t spa e M N = M M ⊗ ˜ M ˜ M is said to be de omposable and the metri tensor g anbe written in terms of (10) and (11). An obje t in a de omposable M N is alled breakable if its omponents with respe t to the oordinates de(cid:28)ned in (11) are always zero when they have indi esfrom both ranges. A breakable obje t is said to be de omposable if and only if the omponentsbelonging to M M (resp., ˜ M ˜ M ) depend on x γ (resp., y C ) only. The Riemann tensor in a de omposablemanifold is de omposable (it follows immediately that the Riemann tensor is breakable [9℄, and ifthe Riemann tensor is breakable it follows from the Bian hi identities that it is de omposable).Symboli ally this an be written as: R N = R M ⊕ ˜ R ˜ M ; R abcd = R αβγδ + ˜ R ABCD . (12)Alternatively, we an write the Riemann tensor in the suggestive blo k diagonal form: R abcd = blockdiag { R αβγδ ( x ǫ ) , ˜ R ABCD ( y E ) } . Sin e the manifold is a produ t spa e, if R αβγδ is de omposable,then so is R αβγδ . This an be trivially generalized to: M N = M M ⊗ ˜ M ˜ M ⊗ ˜ M ˜ M ⊗ . . . ; N = M + ˜ M + ˜ M + . . . ; g N = g M ⊕ ˜ g ˜ M ⊕ ˜ g ˜ M ⊕ . . . (13)The onne tion in a de omposable manifold is also de omposable, and ea h of M M and ˜ M ˜ M are onsequently totally geodesi [9, 10, 11℄. Conversely, if a manifold M is onstru ted from two totallygeodesi submanifolds M M and ˜ M ˜ M , it is de omposable. In addition, if a manifold is de omposable,then there exists a real non-trivial ovariantly onstant geometri al (cid:28)eld (not proportional to thelassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 10metri g ). Su h a ovariantly onstant (cid:28)eld is ne essarily invariant under the holonomy group,and hen e de omposable manifolds are asso iated with manifolds with a redu ible holonomy group.The generators of the holonomy group span the de omposable parts of the Riemann tensor (and its ovariant derivatives). M is said to be redu ible if the holonomy group is redu ible as a linear group a ting on thetangent bundle (otherwise it is said to be irredu ible). Let M be redu ible and let T ′ denote thedistribution asso iated with the proper (non-trivial) subspa e invariant under the holonomy group.Then T ′ is di(cid:27)erentiable and involutive and the maximal integral manifold M ′ of T ′ through apoint of M is a totally geodesi submanifold of M [12℄. In addition, if T ′′ is a omplementary andorthogonal distribution to T ′ at ea h point of M , and M ′ and M ′′ are their asso iated maximalintegral manifolds, then ea h point p of M has an open neighbourhood V = V ′ × V ′′ , where V ′ (resp., V ′′ ) is an open neighbourhood of p in M ′ (resp., M ′′ ), and the metri in V is the dire tprodu t of the metri s in V ′ and V ′′ [12℄. The onverse is also true [9℄. The de Rham de ompositiontheorem then asserts that if M is onne ted, simply onne ted and omplete then M is isometri to the dire t produ t M ′ ⊗ M ′′ [12, 13℄.The ase N = 4 has been studied in detail. A 4-dimensional Lorentzian spa etime M isredu ible if the holonomy group of M is redu ible; i.e., the holonomy algebra of M is a propersubalgebra of the Lorentz algebra and of (holonomy) type R − R ( R is the trivial ase, R isthe full Lorentz algebra) [14℄. Therefore, the holonomy group is redu ible in all ases ex ept forthe general type R , and M onsequently has a produ t stru ture in all ases where the redu tionis non-degenerate [9℄. If the Riemann tensor is de omposable, then the 4-dimensional spa etimeis or de omposable. Let x α and y A be the asso iated lo al oordinates. Suppose that R = R ( x α ) ⊕ ˜ R ( y A ) . Then there is a produ t manifold with metri g ( x α ) ⊕ ˜ g ( y A ) where R ( x α ) and ˜ R ( y A ) are the Riemann tensors asso iated with g ( x α ) and ˜ g ( y A ) , respe tively. Suppose that g isalso a metri with Riemann tensor R . Then g is expli itly related to g in terms of eigenve tors thatlive in M or ˜ M [14℄. The metri g still has a produ t stru ture, but g is not ne essarily isometri to g (i.e., although all metri s ompatible with R are produ ts, the metri is not unique).3.1.1. The Weyl tensor: In the ase in whi h the Weyl tensor is redu ible, it is possible to obtainmore information by de omposing the Weyl tensor and algebrai ally lassifying its irredu ible parts.In the ase of va uum, it follows from above that if C abcd is breakable, it is also de omposable andthe manifold has a produ t stru ture. In general, it does not follow that if C abcd is breakable thenit is also de omposable. For a produ t spa e manifold, if C abcd is de omposable then so is C abcd .However, in general the manifold is not a produ t spa e and hen e the de omposibility of C abcd isnot equivalent to the de omposibility of C abcd .The Weyl tensor C abcd is redu ible if there exists a null frame and a onstant M < N su h that: C N = C M ⊕ ˜ C ˜ M ; C abcd = C αβγδ + ˜ C ABCD . (14)If the Weyl tensor is (orthogonally) redu ible, the only non-vanishing omponents of N C abcd are C αβγδ = 0 , ≤ α, β, γ, δ ≤ M −
1; ˜ C ABCD = 0 M ≤ A, B, C, D ≤ N − lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 11(i.e., in a (cid:28)xed frame, there are no Weyl s alars N C abcd of mixed type). Therefore, C abcd is(algebrai ally) redu ible if and only if it is the sum of two Weyl tensors (and the redu ed alignmenttype of C αβγδ an be de(cid:28)ned to be the alignment type of the redu ed tensor). A redu ible Weyltensor is said to be (geometri ally) de omposable if and only if the omponents belonging to M M (resp., ˜ M ˜ M ) depend on only x γ (resp., y C ) [12℄: M C αβγδ is a Weyl tensor of an (irredu ible)Lorentzian spa etime of dimension M , ˜ M ˜ C ABCD is a Weyl tensor of a Riemannian spa e of dimension ˜ M ≡ N − M . Note that non-trivial Weyl tensors do not exist if N < . Thus, redu ibility be omesan issue only in dimensions N ≥ , and even then one of the omplementary summands mustne essarily vanish for N ≤ . Therefore, true de omposability an only o ur for N ≥ .We an also write a de omposable Weyl tensor in the suggestive blo k diagonal form: blockdiag { C αβγδ ( x ǫ ) , C ABCD ( y E ) } . This an be trivially generalized to the ase where ˜ M ˜ M isfurther redu ible. Writing out the Weyl tensor in terms of boost weights we obtain (6), withindi es α, β, γ, .. running from − ( M − , and an additional term (+) { C IJKL m I { A m J B m K C m L D } } . (15)This term, orresponding to the Riemannian part of the Weyl tensor ˜ C , is either identi ally zero(of type O ) or has terms of boost weight zero only and hen e is of type D .We note that the Weyl tensor is learly redu ible for a manifold that is onformal to a produ tmanifold. In parti ular, the redu ibility property for the Weyl tensor applies to a warped produ tmanifold with metri : ds = g αβ ( x γ ) dx α dx β + F ( x γ )˜ g AB ( y C ) dy A dy B . (16)This is onformal to the metri ds = F ( x γ ) { F − ( x γ ) g αβ ( x γ ) dx α dx β + ˜ g AB ( y C ) dy A dy B } , (17)and hen e the two onformally related manifolds have the same Weyl tensor; i.e., the warpedmanifold has the same Weyl tensor as the onformally related produ t manifold. A onformallyde omposable manifold (doubly warped or twisted manifold) is of the form: ds = F ( x γ , y C ) { g αβ ( x γ ) dx α dx β } + ˜ F ( x γ , y C ) { ˜ g AB ( y C ) dy A dy B } (18)(whi h is di(cid:27)erent to a manifold that is onformal to a produ t manifold).We note that almost all higher dimensional manifolds of physi al interest are either produ t orwarped produ t manifolds [15℄. Let us assume that the Weyl tensor is redu ible as in (14), where M C and ˜ C are the M - dimensional irredu ible Lorentzian and ( N − M ) - dimensional Riemannian parts.Then asso iated with ea h part would be a prin ipal type. The prin ipal type of the Lorentzian M C would be G , I , II , III , N , or O . However, the prin ipal type of the Riemannian ˜ C is either D or O . We ould denote a se ondary type of a redu ible Weyl tensor (14) as T M × ˜ T ˜ M , or simplyby T ˜ T if the dimensions M, ˜ M are lear. For example, I D would denote a redu ible Weyl tensor inwhi h the irredu ible Lorentzian part of the Weyl tensor is of prin ipal type I , and the irredu ibleRiemannian part of the Weyl tensor is non-zero.lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 123.2. Full Classi(cid:28) ation:Alignment type by itself is insu(cid:30) ient for a omplete lassi(cid:28) ation of the Weyl tensor. It is ne essaryto ount aligned dire tions, the dimension of the alignment variety, and the multipli ity of prin ipledire tions. We note that unlike in 4 dimensions, it is possible to have an in(cid:28)nity of aligned dire tions.If a WAND is dis rete, for onsisten y with 4-dimensional nomen lature we shall refer to it as aprin ipal null dire tion (PND). We an introdu e extra normalizations and obtain further sub lasses.Compli ations in attempting to (cid:28)nd anoni al forms arise due to gauge (cid:28)xing (i.e., ertain terms an be hosen to be zero by an appropriate hoi e of frame through boosts and spatial rotations).The full type of an irredu ible Weyl tensor C abcd is de(cid:28)ned by its prin ipal and se ondarytypes, and in ludes all of the information on sub ases and multipli ities. We do not lassify thesein full detail here (indeed, it may be ne essary to onsider di(cid:27)erent spe i(cid:28) dimensions on a aseby ase basis), but rather des ribe some of the key algebrai ally spe ial subtypes below.First, there are additional onditions for algebrai spe ializations: (i) Type I (a) C i = 0 . (ii)Type II (a) C = 0 , (b) the tra eless Ri i part of C ijkl = 0 , ( ) the Weyl part of C ijkl = 0 , (d) C ij = 0 . (iii) Type III (a) C i = 0 . For example, there are two sub ases of type III, namely typeIII (general type) and type III (a) in whi h C i = 0 . Se ond, there are further spe ializationsdue to multipli ities. In types III and N all WANDs are ne essarily PNDs. For type III tensors,the PND of order 2 is unique. There are no PNDs of order 1, and at most 1 PND of order 0. For N = 4 there is always exa tly PND of order 0. For
N > this PND need not exist. For type Ntensors, the order 3 PND is the only PND of any order.We an write a anoni al form for Weyl type N. From (6)(cid:21)(8), we have that for type N: C abcd = 4 C i j ℓ { a m ib ℓ c m j d } ; C i i = 0 . (19)The general form of the Weyl tensor for type III is given by (6) subje t to (7) and (8). In thesub lass III i we have that C i j = 0 and hen e C abcd = 8 C i ℓ { a n b ℓ c m id } + 4 C ijk ℓ { a m ib m j c m kd } , (20)where C j = − C ij i , C ijk ) = 0 , and not all of C ijk are zero (else it redu es to Weyl type O ). Inthe sub lass III(a) we have that C j = 0 , so that C abcd = 4 C ijk ℓ { a m ib m j c m kd } + 4 C i j ℓ { a m ib ℓ c m j d } , (21)where C ij i = C ijk ) = C i i = 0 . There is a further sub ase obtained by ombining the twosub lasses above.The omplete lassi(cid:28) ation for N = 4 is relatively straightforward, due to the fa ts thatthere always exists at least one aligned dire tion, that all su h aligned dire tions are dis rete,normalization redu es the possible number of sub lasses (leading to unique sub ases) and sin eredu ibility is not an issue be ause a Weyl tensor of a manifold of dimension 3 or less must ne essarilyvanish. The present lassi(cid:28) ation redu es to the lassi al Petrov lassi(cid:28) ation in 4 dimensions [1℄.In most appli ations [15, 16, 17℄ the Weyl lassi(cid:28) ation is relatively simple and the details ofthe more omplete lassi(cid:28) ation are not ne essary. It would be useful to be able to (cid:28)nd a morepra ti al way of determining the Weyl type, su h as for example employing ertain s alar higherlassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 13dimensional invariants. We also note that it may be more pra ti al in some situations to lassifythe Riemann tensor sin e it is not onsiderably more di(cid:30) ult to lassify the Riemann tensor ratherthan the Weyl tensor in higher dimensions [1℄.3.3. Petrov lassi(cid:28) ation in four dimensionsThe Petrov lassi(cid:28) ation [4℄ (see also [3, 18℄), invariantly determines the algebrai type of the Weyltensor at a given point of a 4-dimensional spa etime. A spa etime is said to be of a given Petrovtype if the type is the same at all points. There are several equivalent methods of obtaining the lassi(cid:28) ation. The most widely used method is the two omponent spinor approa h, developedby Penrose [3, 18℄, in whi h the Weyl tensor is represented by a totally symmetri spinor and onsequently an be de omposed in terms of four prin ipal spinors. Petrov types now orrespondto various multipli ities of prin ipal spinors. Null ve tors orresponding to various prin ipal spinorsare alled prin ipal null ve tors and the orresponding dire tions are alled prin ipal null dire tions(PNDs). Thus there are at most four distin t PNDs at ea h point of a spa etime.In four dimensions an equivalent but distin t method leading to the Petrov lassi(cid:28) ation interms of alignment type was given in [1℄. In four dimensions, in the standard omplex null tetrad( ℓ , n , m , ¯ m ) [3, 19, 20℄, the Weyl tensor has (cid:28)ve omplex omponents. Components of the Weyltensor have di(cid:27)erent boost weights (as de(cid:28)ned by eqns. (3) and (4)). When ℓ oin ides with thePND, then the +2 boost weight terms vanish. For the algebrai ally spe ial types I, II, III, N, omponents with boost weight greater or equal to 2,1,0,-1, respe tively, an be transformed away(see Table 2). Type D is a spe ial sub ase of type II in whi h all omponents with non-zero boostweight an be transformed away.In this approa h the top alignment equations an be rewritten in terms of an asso iated fourthdegree polynomial of one omplex variable, Ψ( z + ) = 0 . It follows that the equations for alignmentorder q have a solution if and only if Ψ( z + ) possesses a root of multipli ity q + 1 or more. In thisway the usual Petrov lassi(cid:28) ation, whi h ounts the root multipli ities of the polynomial Ψ( z + ) ,is re overed. Of ourse, the two versions are equivalent in four dimensions [1℄. However, it is thisalignment type version of the Petrov lassi(cid:28) ation that we have generalized to higher dimensions.3.3.1. Classi(cid:28) ation of the Weyl tensor in higher dimensions: The WANDS are a naturalgeneralization of the PNDs in higher dimensions, and the lassi(cid:28) ation of the Weyl tensor is basedon the existen e of WANDs of various orders of alignment. We again note that on e we (cid:28)x ℓ asa WAND with maximal order of alignment, we an sear h for n with maximal order of alignmentsubje t to the onstraint n · ℓ = 1 and similarly de(cid:28)ne se ondary alignment type. Alignment typeis a pair onsisting of primary and se ondary alignment types. The possible alignment types aresummarized in Table 2 (taken from [1, 21℄); the link with the four dimensional Petrov lassi(cid:28) ationis emphasised. In (cid:28)ve dimensions and higher, the generi Weyl tensor does not have any aligneddire tions. Thus, unlike the ase in four dimensions, type I tensors are an algebrai ally spe ial ategory.lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 14D > i (1,1) I (1,1,1,1)II (2)II i (2,1) II (2,1,1)D (2,2) D (2,2)III (3)III i (3,1) III (3,1)N (4) N (4)Table 2. Algebrai lassi(cid:28) ation of the Weyl tensor in four and higher dimensions. Note that infour dimensions alignment types (1), (2) and (3) are ne essarily equivalent to the types (1,1), (2,1)and (3,1), respe tively, and sin e there is always at least one PND type G does not exist.3.4. Methods of Classi(cid:28) ationThere are essentially three methods urrently available to determine the Weyl type. In astraightforward approa h the alignment equations are studied, whi h are N ( N − degree-4polynomial equations in ( N − variables (and are generally overdetermined and hen e have nosolutions for N > ), to determine if there exist non-trivial solutions. A se ond method, in whi h thene essary onditions are investigated [21℄, is more pra ti al (and results in studying essentially thesame equations but perhaps in a more organized form and, in the ase of type N for example, redu esto the study of linear equations); this approa h is followed in lassifying the bla k ring solutions (seebelow). Finally, in many appli ations whi h are simple generalizations of 4-dimensional solutions inwhi h the preferred 4-dimensional null frame is expli itly known, the 5-dimensional null frame anbe determined (guessed) dire tly. Some examples of this latter approa h will be presented below.It is lear that these urrent methods for (cid:28)nding WANDs are very umbersome. It is onsequentlyimportant to derive a more pra ti al method for determining Weyl type; for example, by utilizinginvariants as in the ase of 4 dimensions [20℄.A urvature invariant of order n is a s alar obtained by ontra tion from a polynomial inthe Riemann tensor and its ovariant derivatives up to the order n . The Krets hmann s alar, R abcd R abcd , is an example of a zeroth order invariant. In Riemannian geometry the vanishing of theKrets hmann s alar invariant implies (cid:29)at spa e. Invariants have been extensively used in the studyof V SI and
CSI spa etimes.In 4 dimensions, demanding that the zeroth order quadrati and ubi Weyl invariants I and J vanish ( I = J = 0 ) implies that the Petrov type is III, N, or O. In higher dimensions, all of thezeroth order invariants vanish in type III, N and O spa etimes. It would be parti ularly useful to(cid:28)nd ne essary onditions in terms of simple invariants for a type I or type II spa etime.lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 153.4.1. Ne essary onditions for WANDs: The Weyl lassi(cid:28) ation is based on the existen e ofpreferred WANDs. The ne essary onditions for various lasses, whi h an signi(cid:28) antly simplify thesear h for WANDs, are as follows [21℄: ℓ b ℓ c ℓ [ e C a ] bc [ d ℓ f ] = 0 ⇐ = ℓ is WAND, at most primary type I; ℓ b ℓ c C abc [ d ℓ e ] = 0 ⇐ = ℓ is WAND, at most primary type II; ℓ c C abc [ d ℓ e ] = 0 ⇐ = ℓ is WAND, at most primary type III; ℓ c C abcd = 0 ⇐ = ℓ is WAND, at most primary type N. (22)In higher dimensions ( N > ) the onditions given on the left-hand-side of (22) are onlyne essary (but not ne essarily su(cid:30) ient) onditions for the statements on the right-hand-side of(22) [21℄. For type I, equivalen e holds in arbitrary dimension (see below), but this is not so formore spe ial types. (In four dimensions, all of these relationships are equivalen es). For example,a spa etime satisfying ℓ c C abcd = 0 an be of type II [21℄; su h a spa etime has in prin iple a non-vanishing urvature invariant C abcd C abcd and for type N and III spa etimes in arbitrary dimensionall polynomial urvature invariants onstru ted from omponents of the Weyl tensor vanish.There are N ( N − independent s alars of maximal boost weight 2 for a Weyl tensor. Thetop alignment equations, C i j = 0 , are a system of N ( N − , fourth order equations in N − variables. In [1℄ it was shown that, in any dimension N , a null ve tor k satis(cid:28)es the (cid:28)rst equationof (22); i.e., k b k [ e C a ] bc [ d k f ] k c = 0 , k a k a = 0 , (23)if and only if k is aligned with C abcd (i.e., the above system of equations is equivalent to the Weylalignment equations). In 4 dimensions, these equations de(cid:28)ne the PNDs of the Weyl tensor [18, 20℄.3.5. Classi(cid:28) ation of spa etimesMany higher dimensional spa etimes are now known, in luding a number of va uum solutions of theEinstein (cid:28)eld equations whi h an represent higher dimensional bla k holes. These N-dimensionalbla k holes are of physi al interest, parti ularly in view of the development of string theory. Let usreview some of the higher dimensional spa etimes that have been lassi(cid:28)ed algebrai ally (many ofthese results are taken from [22℄); the results are summarized in Table 3.Higher dimensional generalizations of the S hwarzs hild solution, the S hwarzs hild-Tangherlini(ST) solutions [23℄, whi h are spheri ally symmetry on spa elike ( N − -surfa es, are of algebrai Weyl type D. In 5 dimensions this solution is the unique asymptoti ally (cid:29)at, stati bla k holesolution with spheri al symmetry on spa elike 3-surfa es. Higher dimensional generalizations ofReissner-Nordstrom bla k holes are also of type D.A lass of 5-dimensional Kaluza-Klein va uum soliton solutions [16℄, the Sorkin-Gross-Perry-Davidson-Owen soliton (GP), are also of physi al interest. The non-bla k hole solutions (i.e., allsolutions ex ept the 5-dimensional generalized S hwarzs hild (S ∗ ) solution) are of type I [22℄. Thereis a spe ial ase, GP s , whi h is of type D.lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 16The Myers-Perry solution (MP) in (cid:28)ve and higher dimensions [24℄, a dire t generalization ofthe 4-dimensional aymptoti ally (cid:29)at, rotating Kerr bla k hole solution, is also of type D. A lassof higher dimensional Kerr-(anti) de Sitter (K(A)S) solutions, whi h are given in N-dimensions andhave ( N − / independent rotation parameters, have been given in Kerr-S hild form [25℄. Theserotating bla k hole solutions with a non-zero osmologi al onstant redu e to the 5-dimensionalsolution of [26℄ and the Kerr-de Sitter spa etime in 4 dimensions, and the Myers-Perry solutionin the absen e of a osmologi al onstant. The K(A)S spa etimes were shown to be of type D[22℄. Re ently, it has been shown that the family of N-dimensional rotating bla k holes with a osmologi al onstant and a NUT parameter (LNUT) are of type D [27℄.In all of the above examples a anoni al null frame was found [22℄, in whi h the Weyl basis omponents had the appropriate form. In the following example the ne essary onditions (22) werestudied dire tly to lassify the spa etimes [21℄.Non-rotating un harged bla k string Randall-Sundrum braneworlds were (cid:28)rst dis ussed in [28℄.The rotating (cid:16)bla k ring(cid:17) solutions are va uum, asymptoti ally (cid:29)at, stationary bla k hole solutions oftopology S × S [29℄. These solution have subsequently been generalized to the non-supersymmetri bla k ring solutions of minimal supergravity in [30℄.The ne essary onditions (22) an be used to lassify the bla k ring solution (BR). The methodis to solve the ne essity onditions and then he k that these solutions do indeed represent WANDsby al ulating the omponents of the Weyl tensor in an appropriate frame [21℄. In order to solve the(cid:28)rst equation in (22), ℓ a is denoted by ( α, β, γ, δ, ǫ ) . A set of fourth order polynomial equations in α . . . ǫ is then obtained. An additional se ond order equation follows from ℓ a ℓ a = 0 . From a dire tanalysis of these equations, it was shown that the bla k ring solution is algebrai ally spe ial andof type I i [21℄. On the horizon, a transformation leads to a metri regular on the horizon so thatthere is se ond solution to (22). It an be he ked that the boost order of the Weyl tensor in theframe with this solution is zero, and thus the bla k ring is of type II on the horizon (BR H ). By anappropriate hoi e of parameters we obtain the Myers-Perry metri (MP) [24℄ with a single rotationparameter. It turns out that the se ond equation in (22) then admits two independent solutions,and the spa etime is thus of type D.There are also supersymmetri rotating bla k holes that exist in (cid:28)ve dimensions. There isthe extremal harged rotating BMPV bla k hole of [31℄ in minimal supergravity, with a horizon ofspheri al topology (see also [31℄). The BMPV metri is of Weyl type I i . Re ently, the stati hargedbla k ring (CBR) [32℄ has been shown to be of type G [33℄; this is the only expli itly known exa tsolution that is algebrai ally general.There are many other higher dimensional spa etimes of interest. In arbitrary dimensions it isknown that the Weyl tensor of a spheri ally symmetri and stati spa etime is "boost invariant",and thus of type D [34℄. Indeed, re ently it has been shown that all stati spa etimes (STAT) (anda lass of stationary spea times) in dimensions N > are ne essarily of Weyl types G, I i , D or O,and that spheri ally symmetri spa etimes (SS) are of type D or O [35℄. V SI spa etimes are N -dimensional Lorentzian spa etimes in whi h all urvature invariants ofall orders vanish [36, 37℄. Higher-dimensional V SI spa etimes are of Weyl and Ri i types N, III orlassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 17O [36℄. The Ri i type N
V SI spa etimes in lude the higher-dimensional Weyl type N (generalized)pp-wave spa etimes (PP). A lass of higher dimensional relativisti gyratons (RG) [38℄, whi h areva uum solutions of the Einstein equations of the generalized Kundt lass (representing a beampulse of spinning radiation), are of type III. A lass of higher dimensional Robinson-Trautmanspa etimes (RT) are of algebrai type D [40℄.In general, the Weyl and Ri i types of non-lo ally homogeneous
CSI spa etimes, in whi hall urvature invariants of all orders are onstant [55℄, is II [41℄. There are a number of spe ialexamples of
CSI spa etimes;
AdS × S spa etimes (AdSXS), su h as for example AdS × S [42℄,are of Weyl type D (or O) [41℄, generalizations of AdS × S based on di(cid:27)erent V SI seeds (AdSXG)are of Weyl type III if the se tional urvatures are of equal magnitude and opposite sign [41℄, andthe higher-dimensional
AdS gyratons (AdSG) [38℄ are of Weyl type III.Supergravity solutions with a ovariantly onstant null ve tor (CCNV) are of interest in thestudy of supersymmetry [6℄. These CCNV spa etimes are a produ t manifold with a CCNV-
V SI
Lorentzian pie e of Ri i and Weyl type III and a lo ally homogeneous transverse Riemannianspa e of Ri i and Weyl type II [37℄.lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 184. Higher Dimensional Frame FormalismA higher dimensional frame formalism has been developed [7, 8, 36℄ in order to ompletely generalizethe Newman(cid:21)Penrose (NP) formalism [43, 20, 3℄ for any
N > . The main fo us of the review ofthe lassi(cid:28) ation of the Weyl tensor so far has been algebrai . However, there is a ri h interplaybetween algebrai type and di(cid:27)erential identities. Indeed, in higher dimensions it is possible touse the Bian hi and Ri i equations to onstru t algebrai ally spe ial solutions of Einstein's (cid:28)eldequations, at least for the simplest algebrai ally spe ial spa etimes.The Riemann tensor is a 4-rank tensor with the symmetries R abcd = 12 ( R [ ab ][ cd ] + R [ cd ][ ab ] ) , (24)su h that R a { bcd } ≡ R abcd + R acdb + R adbc = 0 . We an also de ompose the Riemann tensor in itsframe omponents and sort them by their boost weights in the same way as is done for the Weyltensor [1℄. The e(cid:27)e ts of null rotations, boosts and spins on the omponents of the Riemann tensorare given in [1, 7℄.Constraints on the Riemann tensor and the Ri i rotation oe(cid:30) ients are obtained from the ommutation relations [36℄, the Bian hi identities (or more per isely the Bian hi equations) [7℄ R ab { cd ; e } ≡ R abcd ; e + R abde ; c + R abec ; d = 0 , (25)and the Ri i identities V a ; bc = V a ; cb + R sabc V s , where V is an arbitrary ve tor [8℄.In 4 dimensions, for algebrai ally spe ial va uum spa etimes some of the tetrad omponentsof the Bian hi identities in the Newman-Penrose formalism [43℄ lead to simple algebrai equations(i.e., equations with no derivatives). In higher dimensions these algebrai equations are mu h more omplex and the number of independent equations, as well as the number of unknowns, dependon the dimension of the spa etime. In addition, in 4 dimensions it is possible to use the Bian hiand Ri i equations to onstru t many algebrai ally spe ial solutions of Einstein's (cid:28)eld equations.It is on eivable that a similar thing is possible in higher dimensions, at least for the simplestalgebrai ally spe ial spa etimes. Indeed, the vast majority of today's known higher dimensionalexa t solutions are simple generalizations of 4-dimensional solutions.4.1. PreliminariesLet us denote the omponents of the ovariant derivatives of the frame ve tors ℓ , n , m ( i ) by L ab , N ab and i M ab (the Ri i rotation oe(cid:30) ients), respe tively: ℓ a ; b = L cd m ( c ) a m ( d ) b , n a ; b = N cd m ( c ) a m ( d ) b , m ( i ) a ; b = i M cd m ( c ) a m ( d ) b (where we follow the notation of [7℄). Sin e the norm of all frame ve tors is onstant, it follows that L a = N a = i M ia = 0 . lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 19Name Ref Comments Type Dim Spe ial ases TypeST [23℄ va uum BH D N R × S SS [35℄ Sph. Sym. D/O NSTAT [35℄ Stati G/I i /D NGP [16℄ va uum soliton I 5 R × R × S S ∗ DK(A)S [25℄ Rotating BH D N Λ = 0 [24℄ N MP DLNUT [27℄ D NBR [29℄ Rotating BR I i R × R × S [21℄ BR H II[24℄ 5 MP DBMPV [31℄ Supersymmetri I i Λ , its topology et . In the "Dim" olumn, it is indi ated whether there are arbitrary dimensional generalizations of the solution ( N )or the solution is for a spe i(cid:28) dimension (e.g., N = 5 ).lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 20Also from the fa t that all s alar produ ts of the frame ve tors are onstant, we get N a + L a = 0 , i M a + L ia = 0 , i M a + N ia = 0 , i M ja + j M ia = 0 . (26)Therefore, there are N ( N − independent rotation oe(cid:30) ients. (In N = 4 dimensions the L ab , N ab and i M ab are equivalent to the 12 omplex NP spin oe(cid:30) ients.)The ovariant derivatives of the frame ve tors an then be expressed as [7℄: ℓ a ; b = L ℓ a ℓ b + L ℓ a n b + L i ℓ a m ( i ) b + L i m ( i ) a ℓ b + L i m ( i ) a n b + L ij m ( i ) a m ( j ) b , (27) n a ; b = − L n a ℓ b − L n a n b − L i n a m ( i ) b + N i m ( i ) a ℓ b + N i m ( i ) a n b + N ij m ( i ) a m ( j ) b , (28) m ( i ) a ; b = − N i ℓ a ℓ b − N i ℓ a n b − L i n a ℓ b − L i n a n b − N ij ℓ a m ( j ) b + i M j m ( j ) a ℓ b − L ij n a m ( j ) b + i M j m ( j ) a n b + i M kl m ( k ) a m ( l ) b . (29)The transformation properties of the rotation oe(cid:30) ients under Lorentz transformations (nullrotations, spins and boosts) are given in [8℄.The dire tional derivatives D , ∆ , and δ i an then be introdu ed: D ≡ ℓ a ∇ a , △ ≡ n a ∇ a , δ i ≡ m ia ∇ a , ∇ a = n a D + ℓ a ∆ + m ia δ i . (30)The ommutators then have the form [36℄ ∆ D − D ∆ = L D + L ∆ + L i δ i − N i δ i , (31) δ i D − Dδ i = ( L i + N i ) D + L i ∆ + ( L ji − M i j ) δ j , (32) δ i ∆ − ∆ δ i = N i D + ( L i − L i )∆ + ( N ji − M i j ) δ j , (33) δ i δ j − δ j δ i = ( N ij − N ji ) D + ( L ij − L ji )∆ + ( M j ki − M i kj ) δ k . (34)4.1.1. The frame omponents of the Bian hi equations and the Ri i identities: Contra tions of theBian hi equations and the Ri i identities with various ombinations of the frame ve tors lead to aset of (cid:28)rst order di(cid:27)erential equations presented in [7, 8℄. For algebrai ally general spa etimes theseequations are quite ompli ated. However, for algebrai ally spe ial ases they are mu h simpler.For N = 4 these equations are equivalent to the standard Bian hi and Ri i identities arising in theNP formalism [20℄.From a study of the algebrai Bian hi equations, it was shown [7℄ that in va uum type III andN spa etimes of arbitrary dimension the multiple prin ipal null dire tion (PND) is geodesi . It wasalso shown that for type N and III va uum spa etimes the expansion and twist matri es S and A (de(cid:28)ned in the next subse tion) have very spe i(cid:28) properties (des ribed in detail in [7℄; however,note that the shear does not vanish for N > ). Conversely, it was shown that for a va uumspa etime the properties of the S and A matri es mentioned above imply that the spa etime isalgebrai ally spe ial. The omplete analysis, in luding all possible degenerate ases, was presentedfor 5-dimensional spa etimes in [7℄.The opti al properties of the gravitational (cid:28)eld in higher dimensions were further explored in[8℄. It was shown that it follows immediately from the Ri i identities and the Sa hs equations inlassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 21arbitrary dimensions [8℄ that (under the assumption R = 0 = R i on the matter (cid:28)elds) N ≥ dimensional Kundt spa etimes are of type II (or more spe ial), and for odd N a twisting geodeti WAND must also be shearing (in ontrast to the ase N = 4 ).4.2. Null geodesi ongruen esThe ongruen e orresponding to ℓ is geodesi if ℓ a ; b ℓ b ∝ ℓ a ⇔ L i = 0 . It is always possible to res ale ℓ in su h a way that ℓ a ; b ℓ b = 0 and thus also L = 0 (i.e., ℓ is a(cid:30)nelyparametrized). Using this parametrization, the ovariant derivative of the ve tor ℓ is then ℓ a ; b = L ℓ a ℓ b + L i ℓ a m ( i ) b + L i m ( i ) a ℓ b + L ij m ( i ) a m ( j ) b . (35)We an de ompose L into its symmetri and antisymmetri parts, S and A , L ij = S ij + A ij , S ij = S ji , A ij = − A ji , where S ij = ℓ ( a ; b ) m ( i ) a m ( j ) b , A ij = ℓ [ a ; b ] m ( i ) a m ( j ) b . We an then de(cid:28)ne the expansion (tra e) θ and the shear (tra efree symmetri ) matrix σ ij asfollows: θ ≡ N − ℓ a ; a , σ ij ≡ (cid:16) ℓ ( a ; b ) − θδ kl m ( k ) a m ( l ) b (cid:17) m ( i ) a m ( j ) b = S ij − θδ ij (36)(i.e., L ij = σ ij + θδ ij + A ij ). For simpli ity, let us all A the twist matrix and S the expansionmatrix, although S ontains information about both expansion and shear. In addition to θ , we an onstru t other (opti al) s alar quantities from ℓ a ; b whi h are invariant under null and spatialrotations with (cid:28)xed ℓ ; e.g., the shear and twist s alars given by the tra es σ ≡ σ ii = σ ij σ ji and ω ≡ − A ii = − A ij A ji (note that σ ij = 0 ⇔ σ = 0 and A ij = 0 ⇔ ω = 0 ).We note that it is always possible to hoose n and m i to be parallely propagated along thegeodesi null ongruen e ℓ ; i.e., we an an set i M j = 0 and N i = 0 by performing an appropriatespin transformation and then a null rotation. This, for example, simpli(cid:28)es the Ri i identities onsiderably and may be onvenient for ertain al ulations.Finally, for a geodeti null ongruen e, splitting the equations into their tra efree symmetri ,tra e and antisymmetri parts, we obtain the set of N -dimensional Sa hs equations for Dσ ij , Dθ and DA ij [8℄.lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 224.3. Appli ationsThere are many appli ations of the Weyl lassi(cid:28) ation s heme. Higher dimensional V SI spa etimesare ne essarily of Weyl type III, N or O, and higher dimensional
CSI spa etimes that are notlo ally homogeneous are at most of type II; these results have been extremely useful in the studyof
V SI and
CSI spa etimes (see the next se tion). In addition, in many appli ations the higherdimensional frame formalism has proven to be ru ial.4.3.1. Goldberg-Sa hs theorem: In 4 dimensions an important onne tion between geometri opti s and the algebrai stru ture of the Weyl tensor is provided by the Goldberg-Sa hs theorem[43, 44, 45℄, whi h states that a va uum metri is algebrai ally spe ial if and only if it ontainsa geodesi shearfree null ongruen e [20, 3℄. This theorem has played a fundemental role in the onstru tion and lassi(cid:28) ation of exa t solutions of Einstein's equations in four dimensions. Inaddition, given an NP tetrad and assuming that this result applies to both ℓ and n , then theWeyl tensor is of type D with repeated PNDs ℓ and n that are geodesi and shear-free. There aregeneralizations of the Goldberg-Sa hs theorem to non-va uum spa etimes in 4 dimensions [20℄.A higher-dimensional version of the Goldberg-Sa hs theorem is of onsiderable interest.Unfortunately, the Goldberg-Sa hs theorem does not have a straightforward generalization forhigher dimensions [39, 7℄, and any higher dimensional version of the Goldberg-Sa hs theorem willne essarily be more ompli ated. However, a partial extension of the Goldberg-Sa hs theorem to N > has been obtained by de omposing the (cid:28)rst ovariant derivatives of ℓ as in equation (35)(where ℓ is assumed to be geodesi and a(cid:30)nely parametrized), utilysing the algebrai type of theWeyl tensor and studying the onsequen es following from the Bian hi and Ri i identities [7℄.It has been found that in va uum, where the Riemann tensor is equal to the Weyl tensor,the Bian hi identities for type N spa etimes imply that in arbitrary dimension the ongruen e orresponding to the WAND is geodesi [7℄. It also follows that in higher dimensions for va uumtype N spa etimes with non-vanishing expansion the matri es S ij , A ij and the Weyl tensor have ertain spe i(cid:28) forms and that the shear does not vanish. Similar results apply for the multipleWAND of a type III va uum spa etime [7℄. Therefore, in N > va uum spa etimes of type IIIor N, a multiple WAND with expansion ne essarily also has non-zero shear [7℄. This onstitutesa non-zero shear `violation' of the 4-dimensional Goldberg-Sa hs theorem in higher dimensions. Inaddition, there exist expli it examples of spe ial va uum type D spa etimes in N ≥ admittingnon-geodeti multiple WANDs [35℄; this shows that there are `violations' of the geodeti part of theGoldberg-Sa hs theorem in higher dimensions (in addition to the non-zero shear `violations').For example, the WANDs of 5-dimensional va uum rotating bla k hole spa etimes (whi h arealgebrai ally spe ial and of type D) are geodeti but shearing [39, 7℄. This result onstitutes anexpli it ounterexample to a omplete higher dimensional extension of the Goldberg-Sa hs theorem;in parti ular, this ounterexample in ludes the higher dimensional Weyl type D Kerr shearingva uum metri and the N = 5 type D Myers-Perry twisting and shearing va uum bla k holespa etime [39, 7℄. In addition, we note that a non-zero σ ij does not imply a non-zero A ij ; for example,stati bla k strings and stati bla k branes are type D spa etimes with expanding, shearing but non-lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 23twisting geodeti multiple WANDs [35℄. Note that the shear in arbitrary dimensional Myers-Perryspa etimes was studied in the Appendix of [35℄.4.3.2. The type-D onje ture: Let us (cid:28)rst summarize the 4-dimensional stati and stationarybla k hole solutions (with topology S ); there is the S hwarsz hild solution, the more general Kerr-Newman solution, the non-va uum Reissner-Nordstrom spa etime, and the non-asymptoti ally (cid:29)atva uum solutions su h as the S hwarsz hild-de Sitter spa etime. All of these solutions are highlysymmetri and are of Weyl (Petrov) type D and hen e algebrai ally spe ial [20℄. All of the knownhigher dimensional bla k holes also have a great deal of symmetry [46℄. It is anti ipated thatthis will again be re(cid:29)e ted in their having spe ial algebrai properties. Indeed, all of the higherdimensional bla k holes lassi(cid:28)ed so far are of algebrai ally spe ial (Weyl) type. In addition, as isthe ase in 4 dimensions [20℄, in dimensions N > it is known that the Weyl tensor of a spheri allysymmetri spa etime is of type D [35, 34℄. This has led to a onje ture that asserts that stationaryhigher dimensional bla k holes, perhaps with the additional onditions of va uum and/or asymptoti (cid:29)atness, are ne essarily of Weyl type D [22, 21℄.This onje ture has re eived support re ently from a study of lo al (so that the results may beapplied to surfa es of arbitrary topology) non-expanding null surfa es (generi isolated horizons)[47℄. Assuming the usual energy inequalities, it was found that the vanishing of the expansion ofa null surfa e implies the vanishing of the shear so that a ovariant derivative is indu ed on ea hnon-expanding null surfa e. The indu ed degenerate metri tensor, lo ally identi(cid:28)ed with a metri tensor de(cid:28)ned on the ( N − -dimensional tangent spa e, and the indu ed ovariant derivative,lo ally hara terized by the rotation 2-form in the va uum ase, onstitute the geometry of anon-expanding null surfa e. The remaining omponents of the surfa e ovariant derivative lead to onstraints on the indu ed metri and the rotation 2-form in the va uum extremal isolated nullsurfa e ase. This leads to the ondition that at the non-expanding horizon the boost order of thenull dire tion tangent to the surfa e is at most , so that the Weyl tensor is at most of type II[47℄ (where the aligned null ve tors tangent to the surfa e orrespond to a double PND of the Weyltensor in the 4-dimensional ase). For example, va uum bla k rings [29℄, whi h are of type I i in thestationary region, are of Weyl type II on the horizon [21℄.There has been some interesting re ent work on hidden symmetries that may be related tothe type-D onje ture [48℄. Killing tensors are dire tly asso iated with onserved quantities forgeodesi motion. More fundamental hidden symmetries are onne ted with antisymmetri Killing-Yano tensors. Hidden symmetries play an important role in the separability of the Hamilton-Ja obiand Klein-Gordon equations. In 4 dimensions the relations between hidden symmetries, the spe ialalgebrai type of spa etimes, and the separation of variables for various (cid:28)eld equations are wellknown [20℄. It is of interest to study whi h of these properties arry over to spa etimes withmore than four dimensions; that is, to study the relationship between the lass of spa etimes withhidden symmetries and their algebrai type in higher dimensions. The origin of hidden symmetriesin higher dimensions is related to the existen e of a so- alled prin ipal Killing-Yano tensor [48℄.The most general Kerr-NUT-(A)dS bla k hole spa etimes are of Weyl type D [27℄. It has re entlylassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 24been shown that, starting with the prin ipal Killing-Yano tensor, it is possible to prove the ompleteintegrability of geodesi motion in these spa etimes and, moreover, to demonstrate this integrabilityby expli itly separating the Hamilton-Ja obi and Klein-Gordon equations [48℄.4.4. Peeling theorem in higher dimensionsThere is interest in the asymptoti (cid:16)peeling properties(cid:17) of the Weyl tensor in higher dimensions.In 4-dimensional spa etimes it has proven very useful to utilize the onformal stru ture of spa e-time in studying asymptoti questions in general relativity, whi h in lude a geometri al de(cid:28)nitionof asymptoti (cid:29)atness and ovariant de(cid:28)nitions of in oming and outgoing gravitational waves inva uum gravitational (cid:28)elds due to an isolated gravitating system [43℄. The (cid:16)peeling property(cid:17) inhigher dimensions in the ase of even dimensions (and with some additional assumptions) wasdemonstrated in [49℄, thereby providing a (cid:28)rst step towards an understanding of the general peelingbehaviour of the Weyl tensor, and the aymptoti stru ture at null in(cid:28)nity, in higher dimensions. Amore rigorous analysis of the peeling theorem in higher dimensions would be desirable.Let us onsider an N -dimensional spa etime ( M , g ab ), N even, that is weakly asymptoti allysimple at null in(cid:28)nity [50℄. The metri of an unphysi al manifold ( ˜ M , ˜ g ab ) with boundary ℑ ,is related to the physi al metri by a onformal transformation, ˜ g ab = Ω g ab , where Ω = 0 and ˜ n a ≡ − Ω ; a = 0 , null, at ℑ . Let us make the further natural assumption in higher dimensions that omponents of the unphysi al Weyl tensor with respe t to the unphysi al tetrad are of order O (Ω q ) (with q ≥ ) in the neighbourhood of ℑ (in 4 dimensions q = 1 , whi h follows from Einstein'sequations). Let ˜ γ ⊂ ( ˜ M , ˜ g ab ) be a null geodesi in the unphysi al manifold that has an a(cid:30)neparameter ˜ r ∼ − Ω near ℑ and a tangent ve tor ℓ and let γ ⊂ ( M, g ab ) be a orresponding nullgeodesi in the physi al manifold with an a(cid:30)ne parameter r ∼ / Ω near ℑ and a tangent ve tor ˜ ℓ .We hoose the frame in the physi al spa etime to be parallelly propagated along γ with respe t to g ab and the orresponding frame ˜ ℓ , ˜ n , ˜ m i in the unphysi al spa etime to be related by ˜ ℓ a = ℓ a , ˜ m a = Ω m a , ˜ n a = Ω n a → ˜ ℓ a = Ω − ℓ a , ˜ m a = Ω − m a , ˜ n a = n a . (37)The Weyl tensor in its frame omponents an then be sorted a ording to their boost weight: ˜ C abc d = ˜ g de ˜ C abce = ˜ g de [4 ˜ C i j ˜ n { a ˜ m ( i ) b ˜ n c ˜ m ( j ) e } + · · · ]= Ω − g de (cid:20) Ω C i j n { a m ( i ) b n c m ( j ) e } + · · · (cid:21) = C abc d . Sin e all unphysi al omponents of the Weyl tensor ˜ C i j , ˜ C ijk , ˜ C i , ˜ C ijkl , ˜ C i j , ˜ C ij , ˜ C , ˜ C ijk , ˜ C i , ˜ C i j are of order O (Ω q ) , ea h physi al omponent is of order O (Ω b +2+ q ) (where b denotesboost weight), and we obtain the peeling property [49℄ C i j = O (Ω q ) ,C ijk , C i = O (Ω q +1 ) ,C ijkl , C i j , C ij , C = O (Ω q +2 ) ,C ijk , C i = O (Ω q +3 ) , lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 25 C i j = O (Ω q +4 ) , and thus C abc d = Ω q C ( N ) abc d + Ω q +1 C ( III ) abc d + Ω q +2 C ( II ) abc d + Ω q +3 C ( I ) abc d + Ω q +4 C ( G ) abc d + O (Ω q +5 ) . (38)Re ently, this higher-dimensional formalism has been utilized to establish a formal analogue ofthe Weyl tensor peeling theorem in arbitrary dimensions in the Penrose limit ontext [51℄.lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 264.5. Further appli ationsIt is of interest to study the 5-dimensional ase and type D spa etimes in more detail. In parti ular,we ould look for exa t 5-dimensional solutions that are generalizations of parti ular 4-dimensionaltype D solutions of spe ial interest.4.5.1. Classi(cid:28) ation of the Weyl tensor in 5-dimensions: A spinorial formulation of 5-dimensionalgeometry was given in [52℄ in order to lassify the Weyl tensor in 5 dimensions (i.e., in an attemptto introdu e su(cid:30) iently many normalizations so that orresponding normal forms be ome truely anoni al).Similar to the ase of 4 dimensions, N=5 is spe ial and an be equipped with a ompatiblesymple ti stru ture so that in 5 dimensions spinors an be used to reate a ve tor respresentation.Indeed, in 5 dimensions the (rank 4) Weyl tensor an thus be represented by the Weyl spinor,whi h is equivalent to a Weyl polynomial (homogeneous quarti polynomial in 3 variables). The lassi(cid:28) ation is then realised by putting the Weyl polynomial into a normal form; this amountsto determining the aligned spinors or prin iple spin dire tions (PSDs) by solving 3 homogeneousquarti polynomials in 4 variables, and lassifying a ording to the nature of the solutions (i.e.,the singularities of the quarti , planar urves up to proje tive transformations) and hen e themultipli ities of the PSDs.This 5-dimensional spinorial formulation is related to De Smet's polynomial representation ofthe Weyl tensor [53℄. We re all that the De Smet lassi(cid:28) ation is restri ted to stati spa etimesand (cid:28)ve dimensions (only). In addition, note that in (cid:28)ve dimensions the De Smet lassi(cid:28) ation [53℄is not equivalent to our Weyl tensor lassi(cid:28) ation. For example, the simple 5-dimensional va uumRobertson-Trautman solution is algebrai ally spe ial and of type III in our lassi(cid:28) ation but isalgebrai ally general in De Smet's lassi(cid:28) ation [54℄.4.5.2. Type D spa etimes: In general, the Weyl tensor has ( N − N − N − N + 8) independent omponents in type D spa etimes in N dimensions. Type D Einstein spa es werestudied in [35℄, with an emphasis on an investigation of the properties of WANDs using the Bian hiidentities (whi h be ome algebrai equations in this ase). In parti ular, it was shown that themultiple WAND in a va uum type D spa etime is geodeti in the `generi ' ase (as de(cid:28)ned in [35℄)in higher dimensions. However, spe ial (i.e., not `generi ') va uum type D spa etimes an admitnon-geodeti multiple WANDs; expli it examples in N ≥ were onstru ted in [35℄. Shear-freeva uum type D spa etimes were also studied.Va uum type D spa etimes have also been studied in the parti ular ase of (cid:28)ve dimensions. In5 dimensions, the type D Weyl tensor is fully determined by a × matrix Φ ij . It is possible tofurther algebrai ally lassify the type D (or type II) spa etimes in terms of Φ ij in this ase [35, 52℄.The relationship between shear-free spa etimes, twist-free spa etimes and the properties of the Φ ij in va uum and Einstein spa es were investigated in [35℄; in parti ular, the multiple WANDs aregeodeti in 5-dimensional type D spa etimes.lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 275. VSI and CSI Spa etimes in Higher DimensionsLet us dis uss the lasses of higher dimensional Lorentzian spa etimes with vanishing s alar urvature invariants ( V SI spa etimes) [37, 36℄ and onstant s alar urvature invariants (
CSI spa etimes) [55℄. There are two important appli ations of
V SI and
CSI spa etimes. (i)The equivalen e problem and the lassi(cid:28) ation of spa etimes; the hara terization of the
V SI and
CSI spa etimes may be a useful (cid:28)rst step toward addressing the important question of when aspa etime an be uniquely hara terized by its urvature invariants [56℄. (ii) Physi al appli ations;for example, many
V SI and
CSI spa etimes are exa t solutions in supergravity and string theory(to all perturbative orders in the string tension) [57℄.5.0.3. Higher dimensional Kundt spa etimes: The generalized N-dimensional Kundt metri (orsimply Kundt metri [20℄) an be written [36℄ ds = 2 du h dv + H ( v, u, x k ) du + W i ( v, u, x k ) dx i i + g ij ( u, x k ) dx i dx j . (39)The metri fun tions H , W i and g ij satisfy the Einstein equations ( i, j = 2 , ..., N − ). It is onvenientto introdu e the null frame ℓ = du, n = dv + Hdu + W i m i , m i = m ij dx j , (40)su h that g ij = m ki m kj and where m i j an be hosen to be in upper triangular form by anappropriate hoi e of frame. The metri (39) possesses a null ve tor (cid:28)eld ℓ = ∂∂v whi h is geodesi ,non-expanding, shear-free and non-twisting; i.e., L ij ≡ ℓ i ; j = 0 [36℄. The Ri i rotation oe(cid:30) ientsare thus given by: L ab = ℓ ( a ; b ) = Lℓ a ℓ b + L i ( ℓ a m ib + ℓ b m ia ) . (41)5.0.4. Covariantly onstant null ve tor: In general, the generalized Kundt metri has the non-vanishing Ri i rotation oe(cid:30) ients L and L i . From L = 0 , we obtain H ,v = 0 . By making useof the upper triangular form of m i j , it follows that L i = 0 implies W i,v = 0 [58℄. The remainingtransformations an be used to further simplify the remaining non-trivial metri fun tions.Thus, the aligned, repeated, null ve tor ℓ of (39) is a null Killing ve tor (KV) if and only if H v = 0 and W i,v = 0 (when e the metri no longer has any v dependen e). Furthermore, it followsthat if ℓ is a null KV then it is also ovariantly onstant. Without any further restri tions, thehigher dimensional metri s admitting a null KV have Ri i and Weyl type II. Therefore, the mostgeneral metri that admits a ovariantly onstant null ve tor (CCNV) is (39) with H = H ( u, x k ) and W i = W i ( u, x k ) [6, 37℄.5.1. Spa etimes with vanishing s alar urvature invariants in higher dimensionsThe following Theorem was proven in [36℄ (generalizing a theorem in 4 dimensions [56℄):lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 28Theorem 1 All urvature invariants of all orders vanish in an N -dimensional Lorentzian spa etimeif and only if there exists an aligned non-expanding ( S ij = 0 ), non-twisting ( A ij = 0 ), shearfreegeodesi null dire tion ℓ a along whi h the Riemann tensor has negative boost order.An analyti al form of the onditions in this Theorem are as follows: R abcd = 8 A i ℓ { a n b ℓ c m id } + 8 B ijk m i { a m j b ℓ c m kd } + 8 C ij ℓ { a m ib ℓ c m j d } (42)(i.e., the Riemann tensor, and onsequently the Weyl and Ri i tensors, are of algebrai type III,N or O [2℄), and ℓ a ; b = L ℓ a ℓ b + L i ℓ a m ib + L i m ia ℓ b ; (43)hen e, V SI spa etimes belong to the generalized Kundt lass [20℄. It follows that any
V SI metri an be written in the generalized Kundt form (39), where lo al oordinates an be hosen so thatthe transverse metri is (cid:29)at; i.e., g ij = δ ij [55℄. The metri fun tions H and W i in the Kundtmetri (whi h an be obtained by substituting σ = 0 in eqns. (46) and (47)), satisfy the remainingvanishing s alar invariant onditions and the Einstein equations (whi h gives rise to two ases,whi h an be hara terized by W (1)1 = − x ǫ ; W (1) i = 0 , i = 1 , where ǫ = 0 or ).Further progress an be made by lassi(cid:28)ng the V SI metri s in terms of their Weyl type (III, Nor O) and their Ri i type III, N or O), and the form of the non-vanishing Ri i rotation oe(cid:30) ients L ab . The Ri i tensor an be written as R ab = Φ ℓ a ℓ b + Φ i ( ℓ a m ib + ℓ b m ia ) . (44)The Ri i type is N if Φ i = 0 = R i (otherwise the Ri i type is III; Ri i type O is va uum),when e H (1) ( u, x k ) is determined in terms of the fun tions W (0) i ( u, x k ) . The remaining non-zero omponents an be simpli(cid:28)ed and hosen to be onstant by an appropriate hoi e of frame. TheWeyl tensor an be expressed as C abcd = 8Ψ i ℓ { a n b ℓ c m id } + 8Ψ ijk m i { a m jb ℓ c m kd } + 8Ψ ij ℓ { a m ib ℓ c m jd } . (45)The ase Ψ ijk = 0 is of Weyl type III, while Ψ ijk = 0 (and onsequently also Ψ i = 0 ) orresponds totype N. Note that Ψ ij is symmetri and tra eless. Ψ ijk is antisymmetri in the (cid:28)rst two indi es with Ψ i = 2Ψ ijj , and in va uum also satis(cid:28)es Ψ { ijk } = 0 . Further sub lasses an be onsidered [2℄. Thesub lass III(a) is de(cid:28)ned by C n +1) = 0 ; this sub ase in ludes the higher dimensional generalizedKundt waves and the higher dimensional generalized pp-waves [15, 59℄. Additional onstraints on Ψ ij , Ψ ijk , et . an be obtained by employing the Bian hi and Ri i identities [7℄. In addition, thespatial tensors (indi es i, j ) an be written in anoni al form when the remaining oordinate andframe freedom is utilized. In Table in [37℄, all of the V SI spa etimes supported by appropriatebosoni (cid:28)elds were presented and the metri fun tions expli itly listed.As noted earlier, the aligned, repeated, null ve tor ℓ of (39) is a null KV and onsequently ovariantly onstant if and only if H ,v = 0 and W i,v = 0 (these metri fun tions are de(cid:28)ned in eqns.(46) and (47)), when e the metri no longer has any v dependen e. Therefore, in higher dimensionsa V SI spa etime whi h admits a CCNV is, in general, of Ri i type III or N and Weyl type III orN. The sub lass of Ri i type N CCNV spa etimes are related to the ( F = 1 ) hiral null modelslassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 29of [59℄. The sub lass of Ri i type N and Weyl type III (a) spa etimes (in whi h the fun tions W i satisfy further restri tions [37℄) are related to the relativisti gyratons [38℄. The sub lass of Ri itype N and Weyl type N spa etimes are the generalized pp-wave spa etimes.lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 305.2. Spa etimes with onstant s alar urvature invariants in higher dimensionsLorentzian spa etimes for whi h all polynomial s alar invariants onstru ted from the Riemanntensor and its ovariant derivatives are onstant ( CSI spa etimes) were studied in [55℄. The set ofall lo ally homogeneous spa etimes is denoted by H . Clearly V SI ⊂ CSI and H ⊂ CSI . Let usdenote by
CSI R all redu ible CSI spa etimes that an be built from
V SI and H by (i) warpedprodu ts (ii) (cid:28)bered produ ts, and (iii) tensor sums. Let us denote by CSI F those spa etimes forwhi h there exists a frame with a null ve tor ℓ su h that all omponents of the Riemann tensorand its ovariants derivatives in this frame have the property that (i) all positive boost weight omponents (with respe t to ℓ ) are zero and (ii) all zero boost weight omponents are onstant.Finally, let us denote by CSI K those CSI spa etimes that belong to the higher-dimensional Kundt lass; the so- alled Kundt
CSI spa etimes. In parti ular, the relationship between
CSI R , CSI F , CSI K and CSI \ H was studied in arbitrary dimensions (and onsidered in more detail in the four-dimensional ase) in [55℄. We note that by onstru tion CSI R is at most of Weyl type II (i.e., oftype II, III, N or O [2℄), and by de(cid:28)nition CSI F and CSI K are at most of Weyl type II (morepre isely, at most of Riemann type II). In 4 dimensions, CSI R , CSI F and CSI K are losely related,and it is plausible that CSI \ H is at most of Weyl type II.For a Riemannian manifold, every CSI is lo ally homogeneous ( CSI ≡ H ) [60℄. This isnot true for Lorentzian manifolds. However, for every CSI spa etime with parti ular onstantinvariants there is a homogeneous spa etime (not ne essarily unique) with pre isely the same onstant invariants. This suggests that
CSI an be onstru ted from H and V SI (e.g.,
CSI R ).Indeed, from the work in [55℄ it was onje tured that if a spa etime is CSI then there exists a nullframe in whi h the Riemann tensor and its derivatives an be brought into one of the followingforms: (i) the Riemann tensor and its derivatives are onstant, in whi h ase we have a lo allyhomogeneous spa e, or (ii) the Riemann tensor and its derivatives are of boost order zero with onstant boost weight zero omponents at ea h order, whi h implies that the Riemann tensor (andhen e the Weyl tensor) is of type II, III, N or O (the
CSI F onje ture). This then suggests that if aspa etime is CSI , the spa etime is either lo ally homogeneous or belongs to the higher dimensionalKundt
CSI lass (the
CSI K onje ture) and that it an be onstru ted from lo ally homogeneousspa es and V SI spa etimes (the
CSI R onje ture). The CSI onje tures were proven in threedimensions in [61℄. Partial results were obtained in four dimensions in [55℄.In a higher dimensional Kundt spa etime, there exists a lo al oordinate system in whi h themetri takes the form (39). In [55℄ it was shown that for Kundt
CSI ( CSI K ) spa etimes thereexists (lo ally) a oordinate transformation ( v ′ , u ′ , x ′ i ) = ( v, u, f i ( u ; x k )) that preserves the form ofthe metri su h that ˜ g ′ ij,u ′ = 0 (i.e., ˜ g ′ ij is independent of u ) and that ds Hom = ˜ g ′ ij dx ′ i dx ′ j is a lo allyhomogeneous spa e. The remaining CSI onditions then imply that W i ( v, u, x k ) = vW (1) i ( u, x k ) + W (0) i ( u, x k ) , (46) H ( v, u, x k ) = v h σ + ( W (1) i )( W (1) i ) i + vH (1) ( u, x k ) + H (0) ( u, x k ) , (47)where σ is a onstant [55℄. For CSI spa etimes in whi h the metri fun tions are independent of v (i.e., W (1) i = H (1) = σ = 0 ), the null ve tor ℓ is a CCNV; indeed, CCNV- CSI spa etimes arelassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 31ne essarily Kundt and are onsequently of this form [58℄. CCNV-
CSI spa etimes were studied in[41, 58℄.5.3. Supergravity and SupersymmetryThe
V SI and
CSI spa etimes are of parti ular interest sin e they are solutions of supergravityor superstring theory, when supported by appropriate bosoni (cid:28)elds [41, 62℄. The supersymmetryproperties of these spa etimes have also been dis ussed.5.3.1. Dis ussion: In 4 dimensions it is known that pp-wave spa etimes, in whi h all of the s alar urvature invariants vanish [64℄, are exa t va uum solutions to string theory to all orders in thestring tension s ale α ′ , even when the dilaton (cid:28)eld and antisymmetri tensor (cid:28)elds (whi h are alsomassless (cid:28)elds of string theory) are in luded [57℄. It has been shown that in 4 dimensions all of the V SI spa etimes are lassi al solutions of the string equations to all orders in σ -model perturbationtheory by showing that all higher order orre tion terms vanish [63℄.The lassi al equations of motion for a metri in string theory an be expressed in terms of σ -model perturbation theory [65℄, through the Ri i tensor R µν and higher order orre tions in powersof α ′ and terms onstru ted from derivatives and higher powers of the Riemann urvature tensor.The only non-zero symmetri se ond-rank tensor ovariantly onstru ted from s alar invariants andpolynomials in the urvature and their ovariant derivatives in V SI spa etimes is the Ri i tensor,and hen e all higher-order terms in the string equations of motion automati ally vanish [57℄. Moreimportantly, other bosoni massless (cid:28)elds of the string an be in luded. For example, a dilaton Φ and an antisymmetri (massless (cid:28)eld) H µvρ an be in luded. Assuming appropriate forms for Φ and H µvρ in V SI spa etimes, where H = ∇ Φ = ( ∇ Φ) = 0 , when the (cid:28)eld equation R µν − H µρσ H ν ρσ − ∇ µ ∇ ν Φ = 0 , is satis(cid:28)ed (typi ally this equation onstitutes a single di(cid:27)erential equation for the fun tions Φ , H µνρ and the metri fun tions), all of the (cid:28)eld equations [65℄ are satis(cid:28)ed to leading order in σ -model perturbation theory (i.e., to order α ′ ). Consequently, all higher order orre tions in σ -modelperturbation theory, whi h are of the form of se ond rank tensors and s alars onstru ted from ∇ µ Φ , H µνρ , the metri and their derivatives, vanish for V SI spa etimes. That is,
V SI spa etimesare solutions to string theory to all orders in σ -model perturbation theory [63℄. It is plausible thata wide lass of V SI solutions are exa t solutions to string theory non-perturbatively [63, 57℄.In the ontext of string theory, it is of onsiderable interest to study Lorentzian spa etimes inhigher dimensions. String theory in higher dimensional generalizations of pp-wave ba kgrounds hasbeen studied by many authors. It is known that su h spa etimes are exa t solutions in string theory[57, 59, 15℄, and type-IIB superstrings in higher dimensional pp-wave ba kgrounds were shown tobe exa tly solvable even in of the presen e of the RR (cid:28)ve-form (cid:28)eld strength [15℄. In parti ular,supergravity theories have been studied in eleven and ten-dimensions, and a lass of 10-dimensionalRi i and Weyl type N pp-wave string spa etimes supported by non- onstant NS-NS or R-R formlassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 32(cid:28)elds (whi h depend on arbitrary harmoni fun tions of the transverse oordinates) were shown tobe exa t type II superstring solutions to all orders in the string tension [15℄. In addition, a numberof lassi al solutions of branes [17℄ in higher dimensional pp-wave ba kgrounds have been studiedin order to better understand the non-perturbative dynami s of string theories.5.3.2. VSI Supergravity theories: In [62℄ it was shown that the higher-dimensional
V SI spa etimeswith (cid:29)uxes and dilaton are solutions of type IIB supergravity, and it was argued that they are exa tsolutions in string theory.
V SI solutions of IIB supergravity with NS-NS and RR (cid:29)uxes and dilatonhave been onstru ted. The solutions are lassi(cid:28)ed a ording to their Ri i type (N or III). TheRi i type N solutions are generalizations of pp-wave type IIB supergravity solutions. The Ri itype III solutions are hara terized by a non- onstant dilaton (cid:28)eld.In parti ular, it was shown that all Ri i type N
V SI spa etimes are solutions of supergravity(and it was argued that Ri i type III
V SI spa etimes are also supergravity solutions if supported byappropriate sour es). A number of new
V SI
Ri i type III and Ri i type N type IIB supergravitysolutions were presented expli itly. It was also argued that, in general, the
V SI spa etimes areexa t string solutions to all orders in the string tension. As noted above, it is the higher dimensionalgeneralized pp-wave spa etimes, that are known to be exa t solutions in string theory [57, 59, 15℄,that have been most studied in the literature.5.3.3. CSI Supergravity theories: A number of
CSI spa etimes are also known to be solutionsof supergravity theory when supported by appropriate bosoni (cid:28)elds [41℄. It is known that
AdS d × S ( N − d ) (in short AdS × S ) is an exa t solution of supergravity (and preserves the maximalnumber of supersymmetries) for ertain values of (d,N) and for parti ular ratios of the radii of urvature of the two spa e forms. Su h spa etimes (with d = 5 , N = 10 ) are supersymmetri solutions of IIB supergravity (and there are analogous solutions in N = 11 supergravity) [66℄. AdS × S is an example of a CSI spa etime [55℄. There are a number of other
CSI spa etimes knownto be solutions of supergravity and admit supersymmetries; namely, generalizations of
AdS × S (forexample, see [67℄) and (generalizations of) the hiral null models [59℄. The AdS gyraton (whi his a
CSI spa etime with the same urvature invariants as pure
AdS ) [38℄ is a solution of gaugedsupergravity [68℄ (the
AdS gyraton an be ast in the Kundt form [41℄).More general
CSI spa etimes an be investigated to determine whether they are solutionsof supergravity. For example, produ t manifolds of the form M × K an be onsidered, where M is an Einstein spa e with negative onstant urvature and K is a ( ompa t) Einstein-Sasakispa etime (in [69℄ supersymmetri solutions of N = 11 supergravity, where M is the squashed S ,were given). The warped produ t of AdS with an 8-dimensional ompa t (Einstein-Kahler) spa e M with non-vanishing 4-form (cid:29)ux are supersymmetri solutions of N=11 supergravity [67℄.A lass of supergravity CSI solutions have been built from a
V SI seed and lo ally homogeneous(Einstein) spa es by warped produ ts, (cid:28)bered produ ts and tensor sums [55℄, yielding generalizationsof
AdS × S or AdS gyratons [41℄. In parti ular, solutions obtained by restri ting attention to Ri itype N CCNV spa etimes were onsidered. Some expli it examples of
CSI supergravity spa etimeslassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 33were onstru ted by taking a homogeneous (Einstein) spa etime, ( M Hom , ˜ g ) , of Kundt form andgeneralising to an inhomogeneous spa etime, ( M , g ) , by in luding arbitrary Kundt metri fun tions(by onstru tion, the urvature invariants of ( M , g ) will be identi al to those of ( M Hom , ˜ g ) ); anumber of 5-dimensional examples were given expli itly, in whi h ds Hom was taken to be Eu lideanspa e or hyperboli spa e [70℄.5.3.4. Supersymmetry: The supersymmetri properties of
V SI and
CSI spa etimes have also beenstudied. It is known that in general if a spa etime admits a Killing spinor, it ne essarily admitsa null or timelike Killing ve tor (KV). Therefore, a ne essary (but not su(cid:30) ient) ondition for aparti ular supergravity solution to preserve some supersymmetry is that the spa etime possess su ha KV. The existen e of null or timelike KVs in
V SI spa etimes was studied in [62℄. Subsequently,the supersymmetry properties of
V SI type IIB supergravity solutions admitting a CCNV wereinvestigated; in parti ular, the previously studied Weyl type N spa etimes were dis ussed extensivelyand new expli it examples of Weyl type III(a) NS-NS (one-half) supersymmetri solutions (whenthe axion and metri fun tions are appropriately related) were presented [62℄. Supersymmetryhas also been studied in
CSI supergravity solutions, parti ularly in the CCNV sub lass of
CSI spa etimes [41℄.lassi(cid:28) ation of the Weyl Tensor in Higher Dimensions and Appli ations. 346. Con lusions and OutlookWe have reviewed the algebrai lassi(cid:28) ation of the Weyl tensor in higher dimensional Lorentzianmanifolds. This lassi(cid:28) ation is a hieved by hara terizing algebrai ally spe ial Weyl tensors bymeans of the existen e of aligned null ve tors of various orders of alignment. In most appli ations this lassi(cid:28) ation su(cid:30) es and the details of a more omplete lassi(cid:28) ation are not ne essary. However,further lassi(cid:28) ation an be obtained using redu ibility and by determining the alignment type.One outstanding issue is that it would be very useful to be able to (cid:28)nd a more pra ti al way ofdetermining the Weyl type of a spa etime su h as, for example, by employing higher dimensionals alar invariants. In parti ular, it would be useful to (cid:28)nd ne essary onditions for a type I or typeII spa etime in terms of simple invariants.We have dis ussed a number of appli ations of this lassi(cid:28) ation s heme, used in onjun tionwith the higher dimensional frame formalism developed. In future work it would be importantto prove higher-dimensional versions of the Goldberg-Sa hs theorem and the peeling theorem. Itwould also be of interest to study the (cid:28)ve dimensional ase and type D spa etimes in more detail.We also dis ussed the higher dimensional Lorentzian spa etimes with vanishing s alar urvatureinvariants (
V SI spa etimes) and onstant s alar urvature invariants (
CSI spa etimes). It has been onje tured that if a spa etime is
CSI , then the spa etime is either lo ally homogeneous or belongsto the higher dimensional Kundt lass and an be onstru ted from lo ally homogeneous spa es and
V SI spa etimes. It remains to prove the various
CSI onje tures, parti ularly in the ase of fourdimensions. It is also of importan e to further study supergravity and supersymmetry in
V SI and