Algebraic classification of spacetimes using discriminating scalar curvature invariants
AAlgebraic classification of spacetimes usingdiscriminating scalar curvature invariants
Alan Coley ♥ and Sigbjørn Hervik l ♥ Department of Mathematics and Statistics,Dalhousie University, Halifax, Nova Scotia,Canada B3H 3J5 l Faculty of Science and Technology,University of Stavanger,N-4036 Stavanger, Norway [email protected], [email protected]
November 2, 2018
Abstract
The Weyl tensor and the Ricci tensor can be algebraically classifiedin a Lorentzian spacetime of arbitrary dimensions using alignment the-ory. Used in tandem with the boost weight decomposition and curvatureoperators, the algebraic classification of the Weyl tensor and the Riccitensor in higher dimensions can then be refined utilizing their eigenbivec-tor and eigenvalue structure, respectively. In particular, for a tensor ofa particular algebraic type, the associated operator will have a restrictedeigenvector structure, and this can then be used to determine necessaryconditions for a particular algebraic type. In principle, this analysis canbe used to study all of the various algebraic types (and their subclasses)in more detail. We shall present an analysis of the discriminants of theassociated characteristic equation for the eigenvalues of an operator todetermine the conditions on (the associated) curvature tensor for a givenalgebraic type. We will describe an algorithm which enables us to com-pletely determine the eigenvalue structure of the curvature operator, upto degeneracies, in terms of a set of discriminants. Since the charac-teristic equation has coefficients which can be expressed in terms of thescalar polynomial curvature invariants of the curvature tensor, we expressthese conditions (discriminants) in terms of these polynomial curvatureinvariants. In particular, we can use the techniques decribed to study thenecessary conditions in arbitrary dimensions for the Weyl and Ricci curva-ture operators (and hence the higher dimensional Weyl and Ricci tensors)to be of algebraic type II or D , and create syzygies which are necessaryfor the special algebraic type to be fulfilled. We are consequently ableto determine the necessary conditions in terms of simple scalar polyno-mial curvature invariants in order for the higher dimensional Weyl and a r X i v : . [ g r- q c ] N ov A. Coley and S. Hervik
Ricci tensors to be of type II or D . We explicitly determine the scalarpolynomial curvature invariants for a Weyl or Ricci tensor to be of type II or D in five dimensions. A number of simple examples are presentedto illustrate the calculational method and the power of the approach. Inparticular, we will present a detailed analysis of the important exampleof a 5 dimensional rotating black ring. iscriminating scalar invariants 3 Higher dimensional Lorentzian spacetimes are of considerable interest in currenttheoretical physics. Lorentzian spacetimes for which all polynomial scalar in-variants constructed from the Riemann tensor and its covariant derivatives areconstant are called
CSI spacetimes [1]. All curvature invariants of all ordersvanish in an D -dimensional Lorentzian V SI spacetime [2]. The higher dimen-sional
V SI and
CSI degenerate Kundt spacetimes are examples of spacetimesthat are of fundamental importance since they are solutions of supergravity orsuperstring theory, when supported by appropriate bosonic fields [3]. Higherdimensional black hole solutions are also of current interest [4].The introduction of alignment theory [5] has made it possible to algebraicallyclassify any tensor in a Lorentzian spacetime of arbitrary dimensions by boostweight. In particular, the dimension-independent theory of alignment, using thenotions of an aligned null direction and alignment order in Lorentzian geometry,can be applied to the tensor classification problem for the Weyl tensor in higherdimensions [5] (thus generalizing the Petrov classifications in four dimensions(4D or D = 4)). Indeed, it is possible to categorize algebraically special tensorsin terms of their alignment type, with increasing specialization indicated bya higher order of alignment. In practice, a complete tensor classification interms of alignment type is possible only for simple symmetry types and forsmall dimensions [5]. However, partial classification into broader categories isstill desirable. We note that alignment type suffices for the classification of4D Weyl tensors, but the situation for higher-dimensional Weyl tensors is morecomplicated (and different classifications in 4D are not equivalent in higherdimensions). In the higher dimensional classification, the secondary alignmenttype is also of significance. Further refinement using bivectors is also useful (seebelow).The analysis for higher dimensional Weyl tensors can also be applied di-rectly to the classification of higher dimensional Riemann curvature tensors. Inparticular the higher-dimensional alignment types give well defined categoriesfor the Riemann tensor (although there are additional constraints coming fromthe extra non-vanishing components). We can also use alignment to classify thesecond-order symmetric Ricci tensor (which we refer to as Ricci type). The Riccitensor can also be classified according to its eigenvalue structure. In addition,using alignment theory, the higher dimensional Bianchi and Ricci identities havebeen computed [6] and a higher dimensional generalization of Newman-Penroseformalism has been presented [5].Another classification can be obtained by introducing bivectors [7]. Thealgebraic classification of the Weyl tensor using bivectors is equivalent to thealgebraic classification of the Weyl tensor by boost weight in 4D (i.e., the Petrovclassification [8]). However, these classifications are different in higher dimen-sions. In particular, the algebraic classification using alignment theory is rathercourse, and it may be useful to develop the algebraic classification of the Weyltensor using bivectors to obtain a more refined classification.The bivector formalism in higher dimensional Lorentzian spacetimes was de-veloped in [7]. The Weyl bivector operator was defined in a manner consistentwith its boost weight decomposition. The Weyl tensor can then be algebraicallyclassified (based, in part, on the eigenbivector problem), which gives rise to arefinement in dimensions higher than four of the usual alignment (boost weight) A. Coley and S. Hervikclassification, in terms of the irreducible representations of the spins. In partic-ular, the classification in five dimensions was discussed in some detail [7]. A scalar polynomial curvature invariant of order k (or, in short, a scalar in-variant) is a scalar obtained by contraction from a polynomial in the Riemanntensor and its covariant derivatives up to the order k . The Kretschmann scalar, R abcd R abcd , is an example of a zeroth order invariant. Scalar invariants havebeen extensively used in the study of V SI and
CSI spacetimes [1, 2, 3]. In [9] itwas proven that a four-dimensional Lorentzian spacetime metric is either I -non-degenerate , and hence completely locally characterized by its scalar polynomialcurvature invariants, or degenerate Kundt .In arbitrary dimensions, demanding that all of the zeroth polynomial Weylinvariants vanish implies that the Weyl type is III , N , or O (similarly for theRicci type). It would be particularly useful to find necessary conditions in termsof simple scalar invariants in order for the Weyl type (or the Ricci type) to be II or D . The main goal of this work is the determination of necessary conditionsin higher dimensions for algebraic type, and particularly type II (or D ), usingscalar curvature invariants. In 4D, demanding that the complex zeroth order quadratic and cubic Weylinvariants I and J vanish ( I = J = 0) implies that the Weyl (Petrov) type is III , N , or O [10]. In addition, the Weyl tensor is of type II (or more special;e.g., type D ) if 27 J = I .It is useful to express the Weyl type II conditions in non-NP form. Thesyzygy I − J = 0 is complex, whose real and imaginary parts can be ex-pressed using invariants of Weyl not containing duals. The real part is equivalentto: − W + 33 W W − W = 0 , (1)and the imaginary part is equivalent to:( W − W )( W + W ) + 18 W (6 W − W − W W + 3 W ) = 0 , (2)where W = 18 C abcd C abcd , (3) W = 116 C abcd C cdpq C pqab ,W = 132 C abcd C cdpq C pqrs C rsab ,W = 1128 C abcd C cdpq C pqrs C rstu C tuvw C vwab . The Ricci type II conditions are : s (4 s − s s + s ) − s (3 s − s ) = 0 , (4)iscriminating scalar invariants 5where S ab is the trace-free Ricci tensor R ab − Rg ab and s = 112 S ba S ab , (5) s = 124 S ba S cb S ac ,s = 148 [ S ba S cb S dc S ad −
14 ( S ba S ab ) ] . If a spacetime is Riemann type II , then not only do the Weyl type II andRicci type II syzygies hold, but there are additional alignment conditions (e.g., C abcd R bd , C abcd R be R de are of type II ). The Weyl tensor and the Ricci tensor can be algebraically classified in a Lorentzianspacetime of arbitrary dimensions by boost weight (using alignment theory). Abivector formalism in higher dimensional Lorentzian spacetimes has been de-veloped to algebraically classify the Weyl bivector operator. Used in tandemwith the boost weight decomposition, the algebraic classification of the Weyltensor and the Ricci tensor (based on their eigenbivector and eigenvalue struc-ture, respectively) can consequently be refined. The purpose of this paper isthe determination of necessary conditions for the algebraic type of a higher di-mensional Weyl tensor or Ricci tensor, and particularly type II (or D ), usingscalar curvature invariants.For a tensor of a particular algebraic type, the associated operator [11] willhave a restricted eigenvector structure. For a given curvature operator in ar-bitrary dimensions, we can consider the eigenvalues of this operator to obtainnecessary conditions. In principle, the analysis can be used to study the varioussubclasses in more detail. In particular, requiring the algebraic type to be II or D will impose restrictions on the eigenvalues on the operator.In this paper we shall present an analysis of the discriminants of the associ-ated characteristic equation to determine the conditions on a tensor for a givenalgebraic type. Since the characteristic equation has coefficients which can beexpressed in terms of the scalar polynomial curvature invariants of the operator,we can consequently give conditions on the eigenvalue structure expressed man-ifestly in terms of these polynomial curvature invariants. We will describe thealgorithm which will enable us to completely determine the eigenvalue structureof the curvature, up to degeneracies, in terms of a set of discriminants n D i . Theresulting syzygies (discriminants) can then be written as special scalar polyno-mial invariants.In particular, we use the technique to study the necessary conditions in ar-bitrary dimensions for the Weyl and Ricci curvature operators (and hence thehigher dimensional Weyl and Ricci tensors) to be of algebraic type II or D . Weare consequently able to determine the necessary condition(s) in terms of sim-ple scalar polynomial curvature invariant for the higher dimensional Weyl andRicci tensors to be of type II or D . We explicitly display the scalar polynomialcurvature invariants for a Weyl or Ricci tensor to be of type II (or D ) in 5D. Note the different index notation on the s i to that in [10], in order to be self-consistentin this paper. A. Coley and S. HervikA number of specific results are obtained, which are summarized at the endof the paper. In addition, a number of simple examples are presented, includingEinstein spaces, the 5D Schwarzschild spacetime, and 5D space with complexhyperbolic sections. We will also present a detailed analysis of the importantexample of a 5D rotating black ring [12] which is generally of type I i , but can alsobe of type II or D . This example serves to illustrate the calculational methodand the power of the approach. In particular, we shall show that the rotatingblack ring is of type II (or type D ) on the black hole horizon, by showing thata number of discriminants (the CHP invariants) vanish there.We briefly discuss the utility of using these methods to study classificationproblems that also involve differential scalar polynomial curvature invariantsconstructed from the Riemann tensor and its covariant derivatives, and presenta simple illustration. We also make some brief comments on possible futurework. In the Appendices we review the Weyl bivector operator (particularly fortype II or D ) and present some important discriminants (or syzygies) that areused in the paper.iscriminating scalar invariants 7 We can use an analysis of the discriminants of the associated characteristicequation to determine the conditions on a tensor for a given algebraic type.In particular, we shall seek necessary conditions for a higher dimensional Weyltensor or Ricci tensor to be type II or D . In principle, the analysis can be usedto study the various subclasses in more detail.For a tensor of a particular algebraic type, the associated operator (acting ona vector space of dimension n ) will have a restricted eigenvector structure. Fora given curvature operator, R , we can consider the eigenvalues of this operatorto obtain necessary conditions. In particular, requiring the algebraic type to be II or D ( II / D ) will impose restrictions of the eigenvalues on the operator (e.g.,,the eigenvalue type (‘Segre type”) will have to be of a particular form). Crucialin this discussion is the eigenvalue equation or characteristic equation [11]:det( R − λ ) = 0 . (6)This equation is a polynomial equation in λ and the eigenvalues are the rootsof this equation. Since the characteristic equation has coefficients which can beexpressed in terms of the invariants of R , we can give conditions of the eigen-value structure expressed manifestly in terms of the invariants of R . Since theinvariants of R are polynomial curvature invariants of spacetime, these condi-tions will be referred to syzygies. Henceforth, we will describe an algorithmwhich enables us to completely determine the eigenvalue structure of R usingthe invariants Tr( R k ), up to degeneracies which will be explained later. The characteristic equation can be expanded to a polynomial equation: f ( λ ) = det( λ − R ) = a λ n + a λ n − + . . . a i λ n − i + · · · + a n . (7)In our case, the coefficients are expressed in terms of invariants of R , using New-ton’s identities. However, the algorithm which follows applies to any polynomialequation.Defining the polynomial invariants R ≡ Tr( R ) , R ≡ Tr( R ) , R ≡ Tr( R ) , etc , (8)we can generally write the coefficients a i as a determinant of an i × i matrix as Notation: For a Lorentzian spacetime of dimension D , non-capitalized Latin indicesrun over 1 , ..., D − n is the dimension of the vector space (or the order of the associatedcharacteristic equation for the eigenvalues) of the curvature operator (for the Ricci curvaturetensor n = D and for the Weyl curvature tensor n = D ( D − / Since the coefficients in the characteristic equations are written in terms of invariantsof the form Tr( R k ), we do not need to consider Bianchi identities or dimensional dependentidentities to simplify the resulting polynomial expressions obtained. We will also use the notation R R , since this is how it will be presented in MAPLEexpressions. We have also omitted any numerical coefficients in the definitions of the R i forconvenience here (see later). A. Coley and S. Hervikfollows: a i = ( − i i ! det R · · · R R R R R . . . 0... . . . . . . . . . ( i − R i . . . R R R , (9)where a ≡ i = 1 , . . . , n ). Explicitly, the first six are given by: a = 1 ,a = − R ,a = 12 R − R ,a = − R + 12 R R − R ,a = 124 R − R R + 13 R R + 18 R − R ,a = − R + 112 R R − R R − R R + 14 R R + 16 R R − R . (10)Also note that the order of a i is i ; i.e., O ( a i ) ∼ R i . Note that if the invariant of highest order, a n , is zero, i.e., a n = 0, then theeigenvalue equation trivially factorises and we have a zero eigenvalue. Therefore,it is convenient to first check the existence of zero-eigenvalues. In particular, if a n = a n − = ... = a n − k = 0 , then there exists a zero eigenvalue of multiplicity k + 1. If this the case thenthe polynomial factorises trivally and the order can be reduced. The followingprocedure can then be simplified accordingly.In general the polynomial, eq.(7), can be analysed and criteria for the var-ious ’Segre types’ can be given. The resulting syzygies are special polynomialinvariants which can be used to characterise the various eigenvalue cases; i.e.,they are discriminants . A complete set of discriminats can be algorithmicallyfound and in the following we will give the algorithm which is found in [13] (seealso [14]). The resulting discriminants will be denoted n D i , n E i , n F i etc., where n denotes order of the polynomial, and i is a running index. These discriminantscan be given in terms of the coefficients a i ; however, using Newton’s identitieswe can express them explicitely in terms of the polynomial invariants R , R ,etc.Given the polynomial f ( x ) = a x n + a x n − + . . . a i x n − i + · · · + a n , It is often convenient to analyse the algebraic structure of the trace-free part of the cur-vature operator R , S , where S = 0 and the expressions above simplify; e.g., a = − S , a = − S , a = − S + S , . . . . iscriminating scalar invariants 9we define the (2 n + 1) × (2 n + 1) discrimination matrix Disc ( f ): a a a · · · a n · · · na ( n − a · · · a n − · · · a a · · · a n − a n na · · · a n − a n − · · · na ( n − a · · · a n −
00 0 · · · a a · · · a n − a n (11)Consider now the principal minor series, { d , d , d , ..., d n +1 } defined as thedeterminants: d k = det the submatrix of Disc ( f )formed by the first k rows and k columns (12)Let n D i = d i , i = 1 , ..., n , then the discriminant sequence of the polynomial f ( x ) is given by { n D , n D , n D , ..., n D n } . (13)By expressing the n D i in terms of the curvature invariants, R , R , etc, we canobtain the primary syzygies n D i for the operator R . Note that the order of n D i is O ( n D i ) = R i ( i − . Sign List.
We call [sign( n D ) , sign( n D ) , . . . , sign( n D n )], wheresign( x ) = , x > , , x = 0 , − , x < , the sign list of the sequence { n D , n D , n D , ..., n D n } . Revised Sign list.
Given a sign list [ s , s , ..., s n ]. If this contains any “in-ternal zeros”, i.e., if there is a subsequence [ s i , , , · · · , , s j ], where s i (cid:54) = 0 and s j (cid:54) = 0, then we replace this subsequence with:[ s i , − s i , − s i , s i , s i , − s i , − s i , s i , s i . . . , s j ] . The revised sign list will therefore contain no “internal” zeros, but may havezeros at the end. The revised sign list will give us the number of distinct realand complex roots. To distinguish between the descriminants of different curvature operators (for example, theWeyl operator and the Ricci operator) in a particular application, we shall (where necessary)add an additional index; e.g., W D k , k = 1 , ...,
10, denote the descriminants of the 5D Weylcurvature operator.
Number of real and complex roots.
Consider the revised sign list of { n D , n D , n D , ..., n D n } . Let: K = (number of sign changes) , L = (number of non-zero members) , of the revised sign list. Then for f ( x ): • the number of distinct pairs of complex conjugate roots is K ; and • the number of distinct real roots is L − K .If we are not interested in the multiplicities of the eigenvalues, the discriminantsequence, { n D , n D , n D , ..., n D n } is sufficient. In some cases, this is enoughto determine the eigenvalue structure of R , but not always. Example: trace-free Ricci tensor in 3D.
Let R be the 3-dimensionaltracefree Ricci tensor; i.e., R = ( S αβ ) , S αβ = R αβ − Rδ αβ . (14)This implies, R = 0 , R = S αβ S βα , R = S αβ S βδ S δα . Consequently, a = 1 , a = 0 , a = − S αβ S βα , a = − S αβ S βδ S δα . Using the procedure above, we get the discriminants: D = 3 , (15) D = − a = 3 S αβ S βα , (16) D = − a − a = 12 ( S αβ S βα ) − S αβ S βδ S δα ) . (17)We clearly have D >
0. The possible signs of the discriminants D and D can now be used to determine the number of real/complex eigenvalues of S αβ .1. D > D >
0: 3 distinct real eigenvalues.2. D > D ≤
0: 2 pairs of complex eigenvalues, which is impossible.3. D <
0: 1 real and 2 complex eigenvalues.4. D = 0, D >
0: 2 real eigenvalues (one of them must be of muliplicity2).5. D = 0, D <
0: 2 complex eigenvalues (impossible, since the lasteigenvalue must be real and hence, distinct).6. D = 0, D = 0: 1 real eigenvalue (which must be equal to zero since S αβ is tracefree).Note that in terms of the Segre type, a (tracefree) Ricci type II / D is of type { } , { (1 , } , { } , or simpler. Consequently, if the (tracefree) Ricci is of type II / D , or simpler, then D = 0. For illustrative purposes we explicitly repeat the definitions and notation here. iscriminating scalar invariants 11
Multiple factor sequence.
We note that for polynomials of order 4 andmore, the discriminants n D i may not be sufficient to determine the completeeigenvalue structure. For example, for quartics, if we have 2 distinct real roots,then we cannot, using n D i only, distinguish the cases ( x − λ ) ( x − λ ) and( x − λ ) ( x − λ ) . Therefore, we need to go a step further in order to distinguishthese cases.Consider the discriminant matrix Disc ( f ). Define the submatrices: M ( k, l ) ≡ the submatrix of Disc ( f ) formed bythe first 2 k rows andfirst (2 k −
1) columns + (2 k + l )th column (18)Then, construct the polynomials:∆ k ( f ) = k (cid:88) i =0 det[ M ( n − k, i )] x k − i , (19)for k = 0 , , ..., n −
1. The sequence { ∆ ( f ) , ∆ ( f ) , ..., ∆ n − ( f ) } is called themultiple factor sequence of f ( x ) due to the following result [13]: Lemma 2.1.
If the number of zeros in the revised list of the discriminant se-quence of f ( x ) is k , then ∆ k ( f ) = g . c . d . ( f ( x ) , f (cid:48) ( x )) . The greatest common devisor (g.c.d.) of f ( x ) and f (cid:48) ( x ) is thus always in themultiple factor sequence. Indeed, the polynomial ∆( f ) ≡ g . c . d . ( f ( x ) , f (cid:48) ( x )) isthe repeated part of f ( x ), because if ∆( f ) has k real roots of multiplicities n , n , ..., n k , and f has m distinct real roots, then f has k real roots of multiplicities n + 1, n + 1, ..., n k + 1, and m − k simplie real roots (similar arguement forcomplex roots). Therefore, by considering ∆( f ) we reduce the multiplicities ofall the roots by 1 .We can now consider the discriminants of the polynomial ∆( f ) in the sameway as we computed the discriminant sequence of f . We will call the discrim-inant sequence of ∆( f ) for { n E , n E , n E , ..., n E k } . We can now use these todetermine the sign list of the E -sequence, etc. We can repeat this procedureand consider ∆(∆( f )) = ∆ ( f ), ∆ ( f ) etc. These have the relation:∆ j − ( f ) ∝ ( x − λ ) n +1 ( x − λ ) n +1 ... ( x − λ k ) n k +1 ( x − λ k +1 ) ... ( x − λ m ) , ∆ j ( f ) ∝ ( x − λ ) n ( x − λ ) n ... ( x − λ k ) n k , (20)This gives us the following algorithm for determining the root structure (oreigenvalue structure) [13]: Algorithm for Root Classification
1. Find the discriminant sequence of f ( x ): { n D , n D , n D , ..., n D n } , and the revised sign list. Find the number of distinct roots by countingsign changes and non-zero elements of the revised sign list. If the revisedsign list contains no 0’s, stop.2 A. Coley and S. Hervik2. If the revised sign list contains k zeros, then compute the ∆( f ) = ∆ k ( f )by the definition for the multiple factor sequence. Then repeat step 1 for∆( f ).3. Continue considering the multiple factor sequence ∆ ( f ), ∆ ( f ),..., untilfor some j , the revised sign list of ∆ j ( f ) contains no zeros.4. We now compute the number of real/complex distinct roots of ∆ j ( f ). Wecan now determine the roots and multiplicities of ∆ j − ( f ), which againenables us to determine the roots and multiplicites of ∆ j − ( f ) etc. At theend of this process, we have a complete root classification for f ( x ).Note that this procedure will provide us with the discriminants (or syzy-gies) which gives us a complete eigenvalue classification of any operator R . Asexplained, these discriminants can be expressed in terms of polynomials of theinvariants, R = Tr( R ), R = Tr( R ), R = Tr( R ) etc., of R . In principle, wecan use this method to study the necessary conditions on any curvature operatorof any specific eigenvalue type.In particular, we can use the technique to study the necessary conditionsof the Weyl and Ricci curvature operators for it to be of algebraic type II / D .We note that the condition n D n = 0 will signal a double eigenvalue since thenumber of eigenvalues is maximum ( n − n D n − = 0 also, then we havemaximum ( n −
2) eigenvalues, etc. We can utilise this to create syzygies whichare necessary for the special algebraic type to be fulfilled.
Types G and I are both of equal generality with respect to their possible (eigen-bivector/eigenvalue) roots structure. Now, for a tensor to be of type II (or D ) then the eigenvalues of the corre-sponding operator need to be of a special form. Since the invariants of a type II are the same as for type D , we will assume type D . The type D case possessesan important symmetry, namely a boost isotropy (the tensor, not necessarilythe complete spacetime). This is what we will utilise in the following. This im-plies specific structure for (a) particular discriminant(s), which then gives riseto necessary condition(s) in terms of scalar polynomial curvature invariants.For the Ricci tensor, we note that a type D tensor is of Segre type { (1 , ... } ,or simpler. This implies that the Ricci operator has at least one eigenvalue of(at least) multiplicity 2. Furthermore, all the eigenvalues are real.
For the Weyl tensor in D dimensions we can use the bivector operator in [7]where the canonical form of a Weyl type D tensor is given (see also Appendix).In particular, for type D , C = blockdiag( M, Ψ , M t )where M is a ( D − × ( D −
2) matrix and Ψ is a square matrix (see Appendixfor the explicit form of this matrix). Since the eigenvalues of M and M t are thesame, we have that the Weyl operator has at least ( D − eigenvalues of (atleast) multiplicity 2. iscriminating scalar invariants 13These observations connect the algebraic types to the eigenvalue structureand enables us to construct the necessary discriminants. Applying the condition D = 0 for the complex three dimensional Weyl tensorwe obtain the complex syzygy I − J = 0. This is equivalent to the (12thorder) real syzygies given by eqns. (1)-(2) from the six dimensional system with D = 0 and D = 0. Applying the condition D = 0 for the four dimensionaltrace-free Ricci tensor we obtain the (12th order) syzygy given by eqn. (4). We can also apply these conditions to the full Riemann tensor (to be oftype II / D , which implies both the Weyl and Ricci tensor are of type II / D andaligned). Alternatively, we note that these will also give us syzygies for mixedtensors. For example requiring that Riemann tensor is type II / D , implies thatboth Ricci and Weyl is type II / D , but also mixed tensors, like: L µν = C µανβ R αβ , M µν = C µανβ R αδ R δβ , N µν = C αµλπ C λπβν R αβ . The type II / D condition therefore implies that we have the syzygy D = 0 for all of L = ( L µν ), M = ( M µν ) and N = ( N µν ). For the trace-free Ricci tensor, we note that type D has to be of Segre type { (1 , } or simpler. This implies that 2 eigenvalues are equal, while theremaining has to be real. Therefore, we get the necessary (20th order) syzygyfor the trace-free Ricci tensor to be of type II / D : D = 0 , D ≥ , D ≥ , D ≥ . Result:
The necessary condition for the trace-free Ricci tensor, S, to be ofalgebraic type II (or D ) in 5D is that the discriminant S D is zero, so that therelated scalar polynomial curvature invariant D ≡ S D = 0. For the Weyl tensor, we consider the bivector operator C . Since the bivectorspace is 10-dimensional, we get a condition involving a syzygy of order 90! Inparticular, the type II operator has 3 eigenvalues of (at least) multiplicity 2.Therefore, we get the syzygies: D = D = D = 0 . Since these polynomial invariants are of particular importance, we will denotethem by
C ≡ W D , H ≡ W D , P ≡ W D , the CHP
Weyl invariants.
Result:
The necessary condition for the Weyl tensor to be of type II (or D )in 5D is that the scalar polynomial curvature invariants C = H = P = 0 . Note that the numerical coefficients in the definitions of the polynomial invariants in eqns.(1) - (2) and (4) are different; for example, R i ∝ W i (i.e., R = 12 W , etc., in (2)). D ≡ S D is given explicitly in Appendix B.3. but its probably more useful to consider specific metrics .A necessary condition can also be found from considering combinations ofthe Weyl tensor; for example, the operator T αβ = C αµνρ C βµνρ . This gives again T D = 0 , T D ≥ , T D ≥ , T D ≥ . Note that T D = 0 is now a 40th order syzygy (in the Weyl tensor; a 20th ordersyzygy in the square of the Weyl tensor). We have stressed that the conditions determined are necessary conditions. In-deed, these conditions may not be sufficient. The reason for this is that thecharacteristic equation for different algebraic types may be identical and con-sequently the scalar invariants are also identical. For example, its not possibleto distingush the Segre types { (1 , } (of Ricci type D ) and { , } (ofRicci type I ). This implies also that the CHP conditions may be fulfilled in spiteof the fact that the spacetime is of type G or I . An explicit example of this isthe following result: Proposition 2.2.
Assume a 5D spacetime has a Weyl tensor with SO (2) isotropy. Then it fulfills the CHP syzygies; i.e., C = H = P = 0 . This result can be seen from using the bivector operator and imposing the SO (2)-symmetry. One then sees that this forces that there must be 3 pairs ofequal eigenvalues, which implies that there are maximum 7 distinct eigenvalues;hence, the result follows.This degeneracy in the classification is a fundamental problem when consid-ering scalar invariants only. Sometimes these cases can be resolved by consider-ing other invariants; however, there is no guarantee that this can be achieved.Using the invariants only, we can only determine the eigenvalue type of the oper-ator. For example, if we find the eigenvalue type to be { (11)111 } , then this cancorrespond to three Ricci types: { (1 , } , { , } , and { } becauseall of these Ricci types have the same eigenvalue type. This is a fundamentaldegeneracy in the classification of tensors using scalar invariants only and hasbeen discussed in earlier papers [9]. Therefore, we need to keep this in mindwhen using the discriminants. Note that this degeneracy is a discrete degen-eracy, unlike the notion of I -degenerate metrics [9] which require a continuous deformation. We present the expression for P in Appendix B.4. Using MAPLE it was possible to compute some of these analytically but in practise theexpressions are not very useful. However, for specific metrics these are still computable andmay give useful results. This syzygy is presumably not independent of the syzygies involving C , H , P , whichperhaps suggests that an appropriate algebraic combination of C , H and P either simplifies orfactors. We shall return to this in future work. On the other hand, the same degeneracy implies that exactly the same procedure (andequations) can be used for spaces of other metric signatures; that is, the procedure to give thediscriminants for other signature metrics is therefore identical to the one given here, see [11]. iscriminating scalar invariants 15
In higher dimensions we will obtain similar syzygies for type II / D tensors. In n dimensions, the Ricci and Weyl type II / D conditions are the correspondingsyzygies ( m = n ( n − / n D n = 0 , (21)Weyl: m D m = m D m − = ... = m D m − n +2 = 0 . (22)Note that the Ricci syzygy is of order n ( n − n ( n − n − / The
CHP conditions are non-trivial:
Let us consider a simple metric whichshows the
CHP conditions are non-trivial conditions. Consider the solvmanifold:d s = − d t + e p t d x + e p t d y + e p t d z + e p t d w (23)The computation for this metric is a bit lengthy (even for MAPLE); however,by choosing randomly some values of the parameters , for example, p = 1, p = 2, p = 5, and p = −
7, MAPLE quickly calculates the values of thediscriminants for the Weyl operator, obtaining: D > , D > , D > , ..., D > , (24)showing that this metric is not type II or D (and that the CHP invariants arenot trivial).
In an Einstein space all of Tr( R k ) ∝ Λ k , and hence all of thediscriminants for the trace-free Ricci eigenvalue equation are zero, and the onlynon-trivial scalar invariant is the Ricci scalar R ∝ Λ.
5D Schwarzschild spacetime:
For the Weyl operator C we get D = D = · · · = D = 0 , D > , D > . This implies that the Weyl operator has 3 distinct real eigenvalues which agreeswith the results of [7]. In fact, this spacetime is of type D .
5D space with complex hyperbolic sections.
Let us consider the examplein [7] with complex hyperbolic spatial sections:d s = − d t + a ( t ) (cid:104) e − w (cid:0) d x + ( y d z − z d y ) (cid:1) + e − w (cid:0) d y + d z (cid:1) + d w (cid:105) . (25) For special values of the parameters this metric has some symmetries; however, this is notgenerally the case. C we get D = D = · · · = D = 0 , D > , D > . This again implies that the Weyl operator has 3 distinct real eigenvalues, likethe 5D Schwarzschild spacetime. However, unlike the Schwarzschild case, wenote that the some of the invariants a i , defined in eq.(9), are zero: a = a = a = ... = a = 0 . As explained, this is a signal that there is a zero-eigenvalue of multiplicity 7!Hence, since there are 3 distrinct eigenvalues of which one of must be zero withmultiplicity 7, the eigenvalue structure is { (1111111)(11)1 } . In particular, sincethe Weyl operator is trace-free, we explicitly get eigenvalues:0 [ × , λ [ × , − λ, which agrees with the results of [7].However, this spacetime is not of type II / D ; indeed, it is I -non-degenerate.This can be seen by computing the operator T αβ = C αµνρ C βµνρ which is of“Segre” type { , (1111) } . However, due to the fundamental degeneracy in theeigenvalue type, we need to compute differential invariants to delineate this casecompletely. The 5D rotating black ring [12] is generally of type I i , but can also be of type II or D at different locations and for particular values of the parameters λ, µ .Assuming that the form of the metric is given by eqn. (9) in [12] (in terms ofthe parameters λ, µ , where R has been set to unity), we consider the coordinateranges − ≤ x ≤ ≤ y < ∞ (and hence 0 ≤ µ ≤ ≤ λ ≤ B, A , A in [12] in order to retain the correct(Lorentzian) signature. We consider the algebraic type of the 5D Weyl tensor.Calculating the polynomial invariants Tr( C k ) and evaluating at the ‘target’ point x = 0 and y = 2 in the region under consideration, all of the R i and hence allof the resulting discriminants are functions of the parameters λ, µ only. Then, atthe ‘target’ point, in general the metric is of type I i , the case λ = 1 correspondsto the Myers-Perry metric (type D ), µ = 1 / y = 1 /µ , type II ), µ = 0 corresponds to the static subcase, and y = 1 /λ corresponds to a curvature singularity.Let us first consider the trace-free part operator T αβ = C αµνρ C βµνρ , whichgives us the discriminant: T D = λ ( λ − µ ) (2 µ − (1 − λ ) (1 + λ ) (1 − λ ) F ( µ, λ ) (26)where F ( µ, λ ) is a polynomial which is generically not zero. On the horizon µ = 1 /
2, we see that T D = 0, and computing T D we get T D > λ . This is a signal that the metric is of type II on the horizon. Note that as x → y → iscriminating scalar invariants 17Indeed, at the horizon, µ = 1 /
2, the computation simplifies and we can computethe
CHP invariants: C = H = P = 0 , while: W D ∝ λ (2 λ − λ − λ − λ − λ − λ + 1)(2 λ − × ( λ − ( λ + 2) ( λ + 1) (4 λ − λ − (2 λ − λ − × (4 λ − λ − λ + 6) (2 λ − λ + 9 λ + 16 λ − × (4 λ − λ + λ + 12) ( λ − λ + λ + 6) (2 λ + 1) , (27)where a (postive) numerical factor has been ignored. Since, the CHP syzygiesare satisfied, this gives further evidence that the the metric is of type II onthe horizon. Note that we actually get further contraints from the secondarydiscriminants, as can be seen from the table in Appendix B.4. Indeed, bycalculating the secondary discriminant W F on the horizon, we get: W F ∝ λ ( λ − λ − λ + 1)(2 λ − λ − (2 λ − , (28)which we see is non-zero as long as W D (cid:54) = 0. Consequently, as long as W D (cid:54) = 0,then the eigenvalue type is { (11)(11)(11)1 .. } . This is consitent with type II .Another interesting special case is λ = 1 (Myers-Perry), for which both T D = T D = 0, and: T D = 67108864 µ ( µ − ( µ − (5 µ − ( µ + 1) , T D > . We can also here compute the
CHP invariants, which are all zero.
Note:
We note that T D = 0 is a 40th order syzygy in the Weyl tensor.Therefore, a useful strategy in practical computations (for example, determiningthe algebraic type of a 5D Weyl tensor), as illustrated by this example, mightbe to test for necessicity using an operator like T , which is relatively simple.If the syzygy is not satisfied we have a definitive result. It is possible that thesyzygy can only be satified for certain coordinate values (or parameter values),whence the CHP syzygies can be tested in these simpler particular cases.8 A. Coley and S. HervikFigure 1: A flow diagram indicating how we can attempt to determine thealgebraic type for tensors. The degeneracy indicated is due to the fact thatseveral types may have the same eigenvalue type (sometimes this can be resolvedby considering other invariants, but not always).
For a curvature tensor of a particular algebraic type, the associated operatorwill have a restricted eigenvector structure. For a given curvature operator inarbitrary dimensions, we can thus consider the eigenvalues of this operator toobtain necessary conditions in order for the tensor to be of a particular alge-braic type. In principle, this analysis can be used to study all of the varioussubclasses of particular algebraic types in more detail. In particular, requiringthe algebraic type to be II or D will impose useful restrictions. In this paper wehave used an analysis of the discriminants of the associated characteristic equa-tion to determine the conditions on a tensor for a given algebraic type. Sincethe characteristic equation has coefficients which can be expressed in terms ofthe scalar polynomial curvature invariants of the operator, we can give con-ditions, or syzygies, on the eigenvalue structure expressed manifestly in termsof these polynomial scalar curvature invariants. Indeed, we have described analgorithm which enables us to completely determine the eigenvalue structure ofthe curvature, up to degeneracies, in terms of a set of discriminants n D i , n E i ,etc.. The resulting syzygies (discriminants) can then be written as special scalarpolynomial invariants.In particular, we have used the technique to study the necessary conditionsin arbitrary dimensions for the Weyl and Ricci curvature operators (and hencethe higher dimensional Weyl and Ricci tensors) to be of algebraic type II / D ,and created syzygies which are necessary for the special algebraic type to be ful-filled. We are consequently able to determine the necessary conditions in termsiscriminating scalar invariants 19of simple scalar polynomial curvature invariant for the higher dimensional Weyland Ricci tensors to be of type II or D . We have explicitly determined thescalar polynomial curvature invariants for a Weyl or Ricci tensor to be of type II (or D ) in 5D. This will be of considerable utility in classifying higher di-mensional solutions obtained in supergravity or superstring theory [3] or higherdimensional black hole solutions [4]. A number of specific results have been obtained in this work. The necessarycondition for the trace-free Ricci tensor, S , to be of algebraic type II (or D ) in5D is that the discriminant S D is zero, so that the related scalar polynomialcurvature invariant D ≡ S D = 0. The necessary condition for the Weyl tensorto be of type II (or D ) in 5D is that the scalar polynomial curvature invariants C = H = P = 0. In principle, we can repeat a similar analysis for other algebraictypes (and making more use of the secondary discriminants).A number of simple examples were presented, including Einstein spaces, the5D Schwarzschild spacetime, and 5D space with complex hyperbolic sections. Inaddition, a simple solvmanifold was considered to show that the CHP conditionsare non-trivial.We also presented a detailed analysis of the important example of a 5Drotating black ring [12] which is generally of type I i , but can also be of type II or D for particular values of the parameters. This example serves to illustratethe calculational method and the power of the approach. In particular, weshowed that the rotating black ring is of type II (or type D ) on the blackhole horizon ( y = 1 /µ ), by showing that C = H = P = 0 on the horizon (andstudying some of the secondary discriminants). The example also illustrates theutility in practical computations of employing rather more simple discriminantslike T D = 0 (which is a 40th order syzygy in the Weyl tensor). In Lorentzian spacetimes, identical metrics are often given in different coordinatesystems, which disguises their true equivalence. Perhaps the easiest way ofdistinguishing metrics is through their scalar polynomial curvature invariants ,due to the fact that inequivalent invariants implies inequivalent metrics. In [9]the notion of an I -non-degenerate spacetime metric in the class of 4D Lorentzianmanifolds, which implies that the spacetime metric is locally determined byits scalar polynomial curvature invariants, was introduced. By determiningan appropriate set of projection operators from the Riemann tensor and itscovariant derivatives, it was proven that a 4D Lorentzian spacetime metric iseither I -non-degenerate or degenerate Kundt [9]. Therefore, a metric that isnot characterized by its curvature invariants must be of degenerate Kundt form.These results were generalized to higher dimensions in [15]. The I -non-degenerate theorem contains not only zeroth order invariants butalso differential scalar polynomial curvature invariants constructed from theRiemann tensor and its covariant derivatives. For example, if the spacetime is0 A. Coley and S. Hervikof Weyl type N , then the differential invariants I and I vanish if the space-time is degenerate Kundt [6] (the definitions of the invariants I and I aregiven therein). Similar results follow for Weyl type III spacetimes (in termsof invariants ˜ I and ˜ I ) and in the conformally flat (but non-vacuum) case (interms of similar invariants I and I constructed from the Ricci tensor [6].These conditions are necessary conditions in order for a spacetime not to be I -non-degenerate [9]. In the case that 27 J = I (cid:54) = 0 (Weyl types II or D ), in[9] two higher order invariants S and S were given as sufficient conditions for I -non-degeneracy (if 27 J = I , but S (cid:54) = 0 or S (cid:54) = 0, then the spacetime is I -non-degenerate). This analysis in 4D can be repeated using discriminants. Let us focus onthe Ricci tensor for illustrative purposes. The necessary condition for the Riccitensor to be of type II or D is given by eqn. (4) which, as noted earlier, followsfrom a discrimant analysis. Now, if we consider the covariant derivatives ofthe Ricci tensor, R ab ; cd... , then for the spacetime to be I -non-degenerate eachcovariant derivative term must also be of type II or D . Hence we could studythe eigenvalue structure of the operators associated with the R ab ; cd... and applythe type II / D necessary conditions in turn. For example, considering the trace-free parts of the tensors T ab = R ac : d R c ; db , R ; ab , (cid:3) R ab , . . . , we obtain necessaryconditions of the form of eqn. (4) but with the s i ≡ Tr( T i ) , i = 2 , , This can be repeated for the Weyl tensor and in higher dimensions [15].
Example.
The class of vacuum type D spacetimes which are I - non-degenerate ,are invariantly classified by their scalar polynomial curvature invariants [16]. Forexample, for the Kinnersley class I type D vacuum spacetime [17] (the othercases work in a similar way), there are 4 algebraically independent (complex)Cartan invariants, which can be written in terms of 4 independent (complex)scalar polynomial invariants (e.g., I , and invariants such as T µµ , where T µν ≡ C αβγδ ; µ C αβγδ ; ν which include first and second covariant derivatives). The Schwarzschild vac-uum type D spacetime belongs to the Kinnersley class I and, as discussed in [9],all of the algebraically independent Cartan scalars are related to the two func-tionally independent polynomial scalar curvature invariants C αβγδ C αβγδ and T µµ (which are equal to 48 M r − and 720( r − M ) M r − , respectively, as func-tions of the two parameters r and M in canonical coordinates) [18]. The Kerrsolution belongs to Kinnersley class IIA; this spacetime has been invariantlycharacterized intrinsically [19]. Kerr metric.
For illustration let us consider the example of the Kerr metric.The Kerr metric is of Petrov type D and we are thus interested in whetherthe covariant derivative, ∇ C , is of type D or not. As noted above, the meth-ods discussed earlier can also be used for differential invariants. The covariantderivative of the Weyl tensor, C αβγδ ; µ , has no natural operator associated with If the spacetime is I -non-degenerate, then essentially we can construct positive boostweight terms in the derivatives of the curvature and determine an appropriate set of differentialscalar curvature invariants. In practice it may be advantageous to work with operators involving second covariantderivatives. iscriminating scalar invariants 21it. However, we can, for example, consider the second order operator T µν definedabove. If T µν is not of type D / II , then ∇ C cannot be of type D / II either.For the Kerr metric, we obtain the syzygy: T D = m a G − G ( r + a − mr ) ( r + a cos θ − mr ) sin θ ( r + a cos θ ) f f , where G ± = r ± ar cos θ − a r cos θ ∓ a r cos θ + a cos θ, (29)and f = f ( a, m, r, cos θ ) and f = f ( a, m, r, cos θ ) are some complicated poly-nomials. We note that away from the horizon, the ergosphere, and some otherspecial points, this syzygy is non-zero and hence ∇ C is not of type D / II (gener-ically), outside the horizon. The Kerr metric is therefore I -non-degenerate bythe results of [9]. Recently, there has been considerable interest in black holes in more than fourdimensions [4]. While the study of black holes in higher dimensions was per-haps originally motivated by supergravity and string theory, now the physicalproperties of such black holes are of interest in their own right. Indeed, studieshave shown that even at the classical level gravity in higher dimensions exhibitsmuch richer dynamics than in 4D, and one of the most remarkable features ofhigher dimensions is the non-uniqueness of black holes [4]There now exist a number of different higher dimensional black hole solu-tions [4], including the rotating black rings [12], that are the subject of ongoingresearch in classical relativity and string theory. Some of these new space-times have be classified algebraically [5, 12]. However, in order to make furtherprogress it is absolutely crucial to be able to develop new techniques for solvingthe vacuum field equations in higher dimensions and to be able to comprehen-sively classify such solutions, and the algebraic techniques recently introduced[5, 7] will be of fundemental importance in this development. However, thealgebraic techniques used to date now are rather difficult to apply, and the de-velopment of simpler criteria, including the use of necessary conditions in termsof scalar curvature invariants introduced here, will hopefully prove to be of greatutility.Therefore, the analysis presented in this paper will be of considerable impor-tance for analysing higher dimensional black hole solutions [4] (and solutions insupergravity or superstring theory [3]). Indeed, the detailed analysis of the 5Drotating black ring [12] serves to illustrate the power of the approach.In future work we hope to extend the analysis presented in this paper andfurther generalize it to the study of differential operators. In addition, we intendto discuss a number of other applications, including the algebraic classificationof some other known higher dimensional black hole solutions.2 A. Coley and S. Hervik
Acknowledgements
The main part of this work was done during a visit to Dalhousie UniversityApril-June 2010 by SH. The work was supported by NSERC of Canada (AC)and by a Leiv Eirikson mobility grant from the Research Council of Norway,project no: (SH).
A The Weyl Bivector operator
Given a vector basis k µ we can define a set of (simple) bivectors F A ≡ F µν = F [ µν ] = k µ ∧ k ν , spanning the space of antisymmetric tensors of rank 2. Consider a D = (2 + n )-dimensional Lorentzian space with the following null-frame { (cid:96) , n , m i } so thatthe metric is d s = 2 (cid:96) n + δ ij m i m j . Let us consider the following bivector basis: (cid:96) ∧ m i , (cid:96) ∧ n , m i ∧ m j , n ∧ m j , or for short: [0 i ], [01], [ ij ], [1 i ]. The Lorentz metric also induces a metric, η MN ,in bivector space. If m = n ( n − /
2, then( η MN ) = 12 n − m n , where n , and m are the unit matrices of size n × n and m × m , respectively,and we have assumed the bivector basis is in the order given above. This metriccan then be used to raise and lower bivector indices.Let V ≡ ∧ T ∗ p M be the vector space of bivectors at a point p . Then consideran operator C = ( C MN ) : V (cid:55)→ V . We will assume that it is symmetric in thesense that C MN = C NM . With these assumptions, the operator C can bewritten in the following ( n + 1 + m + n )-block form [7]: C = M ˆ K ˆ L ˆ H ˇ K t − Φ − A t − ˆ K t ˇ L t A ¯ H ˆ L t ˇ H − ˇ K ˇ L M t (30)Here, the block matrices H (barred, checked and hatted) are all symmetric.Checked (hatted) matrices correspond to negative (positive) boost weight com-ponents.The eigenbivector problem can now be formulated as follows. A bivector F A is an eigenbivector of C if and only if C MN F M = λF N , λ ∈ C . Such eigenbivectors can now be determined using standard results from linearalgebra. The Lorentz transformations (boosts, spins and null rotations) in (2 + n )-dimensions are explicitely written down in [7].iscriminating scalar invariants 23 A.1 Weyl operator
In particular, for the Weyl tensor we can make the following identifications(indices
B, C, .. should be understood as indices over [ ij ]):ˆ H ij = C i j , ˇ H ij = C i j , (31)ˆ L iB = C ijk , ˇ L iB = C ijk , (32)ˆ K i = C i , ˇ K i = − C i , (33) M ij = C i j , Φ = C , (34) A B = C ij , ¯ H BC = C ijkl . (35)The Weyl tensor is also traceless and obeys the Bianchi identity: C µαµβ = 0 , C α ( βµν ) = 0 . These conditions translate into conditions on our block matrices. We can con-sider each boost weight in turn, and use this to express these matrices intoirreducible representations of the spins [7].
A.1.1 Boost-weight 0 components
Here we have C = C i i , C i j = − C ikjk + C ij , C i ( jkl ) = 0 . (36)Starting with the latter, this means that the matrix ¯ H BC fulfills the reducedBianchi identies. It is also symmetric which means that it has the same sym-metries as an n -dimensional Riemann tensor. Hence, we can split this intoirreducible parts over SO ( n ) using the “Weyl tensor”, “trace-free Ricci” and“Ricci scalar” as follows ( n > H BC = ¯ C ijkl + 2 n − (cid:0) δ i [ k ¯ R l ] j − δ j [ k ¯ R l ] i (cid:1) − n − n −
2) ¯ Rδ i [ k δ l ] j , (37)¯ R ij = ¯ S ij + n ¯ Rδ ij . (38)The remaining Bianchi identities now imply: M ij = − n ¯ Rδ ij − ¯ S ij − A ij (39)Φ = − ¯ R. (40)This means that the boost weight 0 components can be specified using theirreducible compositions above ( ¯ R, ¯ S ij , A ij , ¯ C ijkl ). We note that in lower di-mensions we have the special cases for the n -dimensional Riemann tensor: (i)Dim 4 ( n = 2): ¯ S ij = ¯ C ijkl = 0, (ii) Dim 5 ( n = 3): ¯ C ijkl = 0, (iii) Dim 6( n = 4): ¯ C ijkl = ¯ C + ijkl + ¯ C − ijkl , where ¯ C + and ¯ C − are the self-dual, and theanti-self-dual parts of the Weyl tensor, respectively. The same can be done withthe antisymmetic tensor A ij = A + ij + A − ij .A spin G ∈ SO ( n ) acts as follows on the various matrices:( M, Φ , A, ¯ H ) (cid:55)→ ( GM G − , Φ , ¯ GA, ¯ G ¯ H ¯ G − ) . (41)4 A. Coley and S. HervikIf C µναβ is the Weyl tensor, the type D case is therefore completely characterisedin terms of a n -dimensional Ricci tensor, a Weyl tensor, and an antisymmetrictensor A ij . Therefore, the spins are first used to diagonalise the “Ricci tensor”¯ R ij . This matrix can then be described in terms of the Segre-like notationcorresponding to its eigenvalues. There is a degeneracy in the eigenvalues whichoccurs when two, or more, eigenvalues are equal. Using a Segre-like notation,we therefore get the types for ¯ R ij : { .. } , { (11)11 .. } , { (11)(11) ... } , etc. , { .. } , { .. } , { ... } , etc. , (42)where a zero indicates a zero-eigenvalue. Regarding the antisymmetric matrix A ij , this must be of even rank and can be put into canonical block-diagonalform, and we can characterise an antisymmetric matrix using the rank. Theantisymmetric matrix A may also possess further degeneracies. Finally, charac-terisation of the “Weyl tensor” ¯ C ijkl reduces to characterisating the Weyl tensorof the corresponding fictitious n -dimensional Riemannian manifold. A.2 The algebraic classification
Let us consider the classification in [5], and investigate the different algebraictypes in turn. In general, there will be algebraically special cases of type G .The type I , III and N ’s were delineated in [7]. It is of interest to explicitlyreview the type II / D ’s here. A.2.1 Type II/D
The tensor C µναβ is of type II / D if and only if there exists a null frame suchthat the operator C takes the form: C = M K t − Φ − A t L t A ¯ H H − ˇ K ˇ L M t (43)For type D there exists a null frame such that, in addition, ˇ K t = 0 , ˇ L t = 0 , ˇ H =0 , ˇ K = 0 , ˇ L = 0. Then there will be algebraic subcases according to whether some of theirreducible components of boost weight 0 are zero or not. A complete charac-terisation of all such subcases is very involved in its full generality. However, arough classification in terms of the vanishing the irreducible components underspins can be made: (a) Type II / D (a): A = 0, (b) Type II / D (b): ¯ R ij = 0, (c)Type II / D (c): ¯ C ijkl = 0. Note that we can also have a combination of these;for example, type II (ac), which means that A = 0 and ¯ C = 0 (i.e., furtheralgebraically special subcases can arise). For type D tensors, which are invariant under boosts, all Lorentz transformations hasbeen utilised except for the spins. iscriminating scalar invariants 25 A.3 Type D in 4D ( n = 2 ) In 4D, the Weyl operator can always be put into type I form by using a nullrotation (hence, ˆ H = 0). Furthermore, the irreducible representations underthe spins are: ˆ v i , ¯ R, A, ˇ v i , ˇ H . Utilizing the unused freedom of one spin, oneboost and two null-rotations, in each of the algebraically special cases we canuse these to simplify the Weyl tensor even further.Let us only consider type D for illustration. For n = 2, the Weyl tensorreduces to specifying two scalars, namely ¯ R and A . We now get M = (cid:34) − ¯ R − A A − ¯ R (cid:35) , (cid:34) − Φ − A t A ¯ H (cid:35) = (cid:34) ¯ R − A A
34 12 ¯ R (cid:35) ; (44)consequently, the Weyl operator C has eigenvalues: λ , = λ , = −
14 ( ¯ R ± iA ) , λ , = 12 ( ¯ R ± iA ) . (45)We note that this is in agreement with the standard type D analysis in 4D(see [8]). The type D case is boost invariant, and also invariant under spins,consequently the isotropy is 2-dimensional.The two subcases A = 0 and ¯ R = 0 (type D (a) and D (b), respectively)are in 4D referred to the purely “electric” and “magnetic” cases, respectively.In 4D, there is a duality relation, (cid:63) , which interchanges these two cases; i.e., C (cid:55)→ (cid:63)C interchanges the electic and magnetic parts. A.4 Type II/D in 5D ( n = 3 ) The 5D case is considerably more difficult than the 4D case. The complexitydrastically increases and hence the number of special cases also increases. How-ever, the 5D case is still managable and some simplifications occur (comparedto the general case). Most notably, ¯ C ijkl = 0, and ˇ T ijk can be written, using amatrix ˇ n ij , as follows (similarly for ˆ T ijk ):ˇ T ijk = ε jkl ˇ n li , (46)where the conditions on ˇ T ijk imply that ˇ n ij is symmetric and trace-free. There-fore, we can use the spins to diagonalise ˇ n ij . Thus the general case is { } (all eigenvalues different), with the special cases { (11)1 } , { } and { } .Furthermore, in the general case, the vector ˇ v i needs not be aligned with theeigenvectors of ˇ n ij . There would consequently be special cases where ˇ v i is aneigenvector of ˇ n ij . The components in 5D are displayed in Table 1. For type D we have that ˇ n ij = 0, and hence ˇ T ijk = 0. A.4.1 Type D
For a type D Weyl tensor only the following components can be non-zero:¯ R, ¯ S ij , A ij , H ij C i j = ˆ H ij +1 ˆ v i , ˆ n ij C ijk = δ ij ˆ v k − δ ik ˆ v j + ε jkl ˆ n li , C i = 2ˆ v i R , ¯ S ij , A ij (cid:40) C i j = − ¯ R ij − A ij , C ij = A ij ,C = − ¯ R, C ijkl = ¯ R ijkl − v i , ˇ n ij C ijk = δ ij ˇ v k − δ ik ˇ v j + ε jkl ˇ n li , C i = − v i − H ij C i j = ˇ H ij Table 1: Dimension D = 5: Here ¯ R kikj = ¯ R ij = ¯ Rδ ij + ¯ S ij . Also see [16]where i, j = 3 , ,
5. Let us use the spins to diagonalise ( S ij ) = diag( S , S , S ).Without any further assumptions, the Weyl blocks take the form: M = − ¯ R − S − A
34 12 A A − ¯ R − S − A − A
53 12 A − ¯ R − S , (cid:34) − Φ − A t A ¯ H (cid:35) = ¯ R − A − A − A A
45 16 ¯ R − S A ¯ R − S A ¯ R − S (47)The general type D tensor thus has this canonical form.There are two special cases where we can use the extra symmetry to getthe simplified canonical form: (i) S = S = − S : A = 0. (ii) S = S = S = 0: A = A = 0. We note that case (ii) will, without furtherassumptions, be invariant under spatial rotations in the [34]-plane (in additionto the boost). Assuming, in addition, that A = 0, then case (i) is also invariantunder a rotation in the [34]-plane. Assuming that A ij vanishes completely, wenote that case (ii) enjoys the full invariance under the spins (i.e., SO (3)). B Some Discriminants
For convenience, let us consider a trace-free operator S so that S = Tr( S ) = 0.We recall that S i ≡ Tr( S i ). We also will give the table that gives the eigenvaluetype using these discriminants. Here, 1 C means a pair of complex conjugateeigenvalues.To translate into Ricci/Weyl type we need to consider degeneracies. Forexample, the Eigenvalue type { (11)11 } , corresponds to the three Ricci types { (1 , } , { , } , { } because all of these have the same eigenvalue type. B.1 Dimension 3 Operator
For a 3-dimensional trace-free operator ( S = 0), the syzygies are given by: D = 3 S D = 12 S − S (48)iscriminating scalar invariants 27 D D Eigenvalue type+ + { }− { C } { (11)1 } { (111) } B.2 Dimension 4 Operator
For a 4-dimensional trace-free operator ( S = 0), the syzygies are given by: D = 4 S D = − S + 4 S S − S D = 18 S − S S − S S +4 S S + 2 S S S − S − S E = S + 2 S − S S (49) D D D E Eigenvalue type+ + + { } + ± / ∓ / { C C }− ± / ± / { C } { (11)11 } ± ∓ { (11)1 C } { (11)(11) } { (111)1 } − { (1 C C ) } { (1111) } B.3 Dimension 5 Operator
For a 5-dimensional trace-free operator ( S = 0), the syzygies are given by: D = 5 S D = − S + 5 S S − S D = 18 S − S S + 724 S S +2 S S S + 194 S S − S S S − S S − S − S + 10 S S S D = 212 S S S − S S S − S S S − S S S + 4196 S S S − S S S + 118 S S S − S S S + 1148 S S S + 94 S S S S − S S S S + 445 S S − S S S − S S − S S S + 1512 S − S S − S S − S S + 151192 S S − S S + 19128 S S + 53 S S S − S S + 12 S S − S + 5 S + 4312 S S S S E = 9124 S S − S S S + 254 S S − S S S + 252 S S S S − S S + 254 S S − S + 78 S S − S S + 1256 S S F = 13 S + 12 S − S S (50) D D D D E F Eigenvalue type+ + + + { } + ≤ ∗ ≤ ∗ ≤ ∗ { C C }− { C } { (11)111 } − { (11)1 C } { (111)11 } − { (111)1 C } (cid:54) = 0 { (11)(11)1 } − (cid:54) = 0 { (1 C C )1 } (cid:54) = 0 (cid:54) = 0 { (111)(11) } (cid:54) = 0 0 { (1111)1 } { (11111) } iscriminating scalar invariants 29 ∗ One of these conditions is sufficient.
B.4 Dimension 10 Operator
For a 10-dimensional trace-free operator ( S = 0), we only give a partial tableindicating when the CHP invariants are zero (we will also ignore whether theroots are real or complex). D
10 10 D D D E F F Eigenvalue type (cid:54) = 0 { .. } (cid:54) = 0 { (11)11 ... } (cid:54) = 0 (cid:54) = 0 { (11)(11)1 ... } (cid:54) = 0 0 { (111)1 ... } (cid:54) = 0 (cid:54) = 0 { (11)(11)(11)1 .. } (cid:54) = 0 (cid:54) = 0 0 { (111)(11)1 .. } (cid:54) = 0 0 0 { (1111)11 .. } B.4.1 The discriminant P The simplest of the three syzygies, the discriminant P (which is of 56th orderin terms of S k , k ≤
10, and contains 13377 terms) is too lengthy to write downexplicitly here, but it has the symbolic form: P = 156623104000 ( S S − S ) S + ( · · · ) S +...+ · · · + · · · +...+ ( · · · ) S + (10 S S − S − S ) S . (51) The explicit expression for P is given in [20]. References [1] A. Coley, S. Hervik and N. Pelavas, 2009, Class. Quant. Grav. , 125011[arXiv:0904.4877]; A. Coley, S. Hervik and N. Pelavas, 2006, Class. Quant.Grav. , 3053; A. Coley, S. Hervik and N. Pelavas, 2008, Class. Quant.Grav. , 025008.[2] A. Coley, R. Milson, V. Pravda and A. Pravdova, 2004, Class. Quant. Grav. , 5519 [gr-qc/0410070].[3] A. Coley, A. Fuster and S. Hervik, 2009, Int. J. Mod. Phys. A24 , 1119[arXiv:0707.0957].[4] R. Emparan and H. S. Reall, 2008,
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