A The Weyl Bivector operator

Algebraic classification of spacetimes using discriminating scalar curvature invariants

Abstract

The Weyl tensor and the Ricci tensor can be algebraically classified in a Lorentzian spacetime of arbitrary dimensions using alignment theory. Used in tandem with the boost weight decomposition and curvature operators, the algebraic classification of the Weyl tensor and the Ricci tensor in higher dimensions can then be refined utilizing their eigenbivector and eigenvalue structure, respectively. In particular, for a tensor of a particular algebraic type, the associated operator will have a restricted eigenvector structure, and this can then be used to determine necessary conditions for a particular algebraic type. In principle, this analysis can be used to study all of the various algebraic types (and their subclasses) in more detail. We shall present an analysis of the discriminants of the associated characteristic equation for the eigenvalues of an operator to determine the conditions on (the associated) curvature tensor for a given algebraic type. We will describe an algorithm which enables us to completely determine the eigenvalue structure of the curvature operator, up to degeneracies, in terms of a set of discriminants. Since the characteristic equation has coefficients which can be expressed in terms of the scalar polynomial curvature invariants of the curvature tensor, we express these conditions (discriminants) in terms of these polynomial curvature invariants. In particular, we can use the techniques decribed to study the necessary conditions in arbitrary dimensions for the Weyl and Ricci curvature operators (and hence the higher dimensional Weyl and Ricci tensors) to be of algebraic type II or D, and create syzygies which are necessary for the special algebraic type to be fulfilled. We are consequently able to determine the necessary conditions in terms of simple scalar polynomial curvature invariants in order for the higher dimensional Weyl and Ricci tensors to be of type II or D. We explicitly determine the scalar polynomial curvature invariants for a Weyl or Ricci tensor to be of type II or D in five dimensions. A number of simple examples are presented to illustrate the calculational method and the power of the approach. In particular, we will present a detailed analysis of the important example of a 5 dimensional rotating black ring.

1 Introduction

Higher dimensional Lorentzian spacetimes are of considerable interest in current theoretical physics. Lorentzian spacetimes for which all polynomial scalar invariants constructed from the Riemann tensor and its covariant derivatives are constant are called spacetimes [1]. All curvature invariants of all orders vanish in an -dimensional Lorentzian spacetime [2]. The higher dimensional and degenerate Kundt spacetimes are examples of spacetimes that are of fundamental importance since they are solutions of supergravity or superstring theory, when supported by appropriate bosonic fields [3]. Higher dimensional black hole solutions are also of current interest [4].

The introduction of alignment theory [5] has made it possible to algebraically classify any tensor in a Lorentzian spacetime of arbitrary dimensions by boost weight. In particular, the dimension-independent theory of alignment, using the notions of an aligned null direction and alignment order in Lorentzian geometry, can be applied to the tensor classification problem for the Weyl tensor in higher dimensions [5] (thus generalizing the Petrov classifications in four dimensions (4D or )). Indeed, it is possible to categorize algebraically special tensors in terms of their alignment type, with increasing specialization indicated by a higher order of alignment. In practice, a complete tensor classification in terms of alignment type is possible only for simple symmetry types and for small dimensions [5]. However, partial classification into broader categories is still desirable. We note that alignment type suffices for the classification of 4D Weyl tensors, but the situation for higher-dimensional Weyl tensors is more complicated (and different classifications in 4D are not equivalent in higher dimensions). In the higher dimensional classification, the secondary alignment type is also of significance. Further refinement using bivectors is also useful (see below).

The analysis for higher dimensional Weyl tensors can also be applied directly to the classification of higher dimensional Riemann curvature tensors. In particular the higher-dimensional alignment types give well defined categories for the Riemann tensor (although there are additional constraints coming from the extra non-vanishing components). We can also use alignment to classify the second-order symmetric Ricci tensor (which we refer to as Ricci type). The Ricci tensor can also be classified according to its eigenvalue structure. In addition, using alignment theory, the higher dimensional Bianchi and Ricci identities have been computed [6] and a higher dimensional generalization of Newman-Penrose formalism has been presented [5].

Another classification can be obtained by introducing bivectors [7]. The algebraic classification of the Weyl tensor using bivectors is equivalent to the algebraic classification of the Weyl tensor by boost weight in 4D (i.e., the Petrov classification [8]). However, these classifications are different in higher dimensions. In particular, the algebraic classification using alignment theory is rather course, and it may be useful to develop the algebraic classification of the Weyl tensor using bivectors to obtain a more refined classification.

The bivector formalism in higher dimensional Lorentzian spacetimes was developed in [7]. The Weyl bivector operator was defined in a manner consistent with its boost weight decomposition. The Weyl tensor can then be algebraically classified (based, in part, on the eigenbivector problem), which gives rise to a refinement in dimensions higher than four of the usual alignment (boost weight) classification, in terms of the irreducible representations of the spins. In particular, the classification in five dimensions was discussed in some detail [7].

1.1 Scalar curvature invariants

A scalar polynomial curvature invariant of order (or, in short, a scalar invariant) is a scalar obtained by contraction from a polynomial in the Riemann tensor and its covariant derivatives up to the order . The Kretschmann scalar, , is an example of a zeroth order invariant. Scalar invariants have been extensively used in the study of and spacetimes [1, 2, 3]. In [9] it was proven that a four-dimensional Lorentzian spacetime metric is either -non-degenerate, and hence completely locally characterized by its scalar polynomial curvature invariants, or degenerate Kundt.

In arbitrary dimensions, demanding that all of the zeroth polynomial Weyl invariants vanish implies that the Weyl type is III, N, or O (similarly for the Ricci type). It would be particularly useful to find necessary conditions in terms of 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 conditions in higher dimensions for algebraic type, and particularly type II (or D), using scalar curvature invariants.

Scalar invariants in 4D

In 4D, demanding that the complex zeroth order quadratic and cubic Weyl invariants and vanish () 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 .

It is useful to express the Weyl type II conditions in non-NP form. The syzygy is complex, whose real and imaginary parts can be expressed using invariants of Weyl not containing duals. The real part is equivalent to:

(1)

and the imaginary part is equivalent to:

(2)

where

(3)

The Ricci type II conditions are :

(4)

where is the trace-free Ricci tensor and 1

(5)

If a spacetime is Riemann type II, then not only do the Weyl type II and Ricci type II syzygies hold, but there are additional alignment conditions (e.g., are of type II).

1.2 Overview

The Weyl tensor and the Ricci tensor can be algebraically classified in a Lorentzian spacetime of arbitrary dimensions by boost weight (using alignment theory). A bivector formalism in higher dimensional Lorentzian spacetimes has been developed to algebraically classify the Weyl bivector operator. Used in tandem with the boost weight decomposition, the algebraic classification of the Weyl tensor and the Ricci tensor (based on their eigenbivector and eigenvalue structure, respectively) can consequently be refined. The purpose of this paper is the determination of necessary conditions for the algebraic type of a higher dimensional Weyl tensor or Ricci tensor, and particularly type II (or D), using scalar curvature invariants.

For a tensor of a particular algebraic type, the associated operator [11] will have a restricted eigenvector structure. For a given curvature operator in arbitrary dimensions, we can consider the eigenvalues of this operator to obtain necessary conditions. In principle, the analysis can be used to study the various subclasses 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 associated characteristic equation to determine the conditions on a tensor for a given algebraic type. Since the characteristic equation has coefficients which can be expressed in terms of the scalar polynomial curvature invariants of the operator, we can consequently give conditions on the eigenvalue structure expressed manifestly in terms of these polynomial curvature invariants. We will describe the algorithm which will enable us to completely determine the eigenvalue structure of the curvature, up to degeneracies, in terms of a set of discriminants . The resulting syzygies (discriminants) can then be written as special scalar polynomial invariants.

In particular, we use the technique to study the necessary conditions in arbitrary dimensions for the Weyl and Ricci curvature operators (and hence the higher dimensional Weyl and Ricci tensors) to be of algebraic type II or D. We are consequently able to determine the necessary condition(s) in terms of simple scalar polynomial curvature invariant for the higher dimensional Weyl and Ricci tensors to be of type II or D. We explicitly display the scalar polynomial curvature invariants for a Weyl or Ricci tensor to be of type II (or D) in 5D.

A number of specific results are obtained, which are summarized at the end of the paper. In addition, a number of simple examples are presented, including Einstein spaces, the 5D Schwarzschild spacetime, and 5D space with complex hyperbolic sections. We will also present a detailed analysis of the important example of a 5D rotating black ring [12] which is generally of type , but can also be of type or . This example serves to illustrate the calculational method and the power of the approach. In particular, we shall show that the rotating black ring is of type (or type ) on the black hole horizon, by showing that a number of discriminants (the invariants) vanish there.

We briefly discuss the utility of using these methods to study classification problems that also involve differential scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives, and present a simple illustration. We also make some brief comments on possible future work. In the Appendices we review the Weyl bivector operator (particularly for type II or D) and present some important discriminants (or syzygies) that are used in the paper.

2 Discriminant Analysis

We can use an analysis of the discriminants of the associated characteristic equation to determine the conditions on a tensor for a given algebraic type. In particular, we shall seek necessary conditions for a higher dimensional Weyl tensor or Ricci tensor to be type II or D. In principle, the analysis can be used to study the various subclasses in more detail.

For a tensor of a particular algebraic type, the associated operator (acting on a vector space of dimension ) 2 will have a restricted eigenvector structure. For a given curvature operator, , we can consider the eigenvalues of this operator to 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). Crucial in this discussion is the eigenvalue equation or characteristic equation [11]:

(6)

This equation is a polynomial equation in and the eigenvalues are the roots of this equation. Since the characteristic equation has coefficients which can be expressed in terms of the invariants of , we can give conditions of the eigenvalue structure expressed manifestly in terms of the invariants of . Since the invariants of are polynomial curvature invariants of spacetime, these conditions will be referred to syzygies. Henceforth, we will describe an algorithm which enables us to completely determine the eigenvalue structure of using the invariants , up to degeneracies which will be explained later. 3

2.1 Algorithm

The characteristic equation can be expanded to a polynomial equation:

(7)

In our case, the coefficients are expressed in terms of invariants of , using Newton’s identities. However, the algorithm which follows applies to any polynomial equation.

Defining the polynomial invariants 4

(8)

we can generally write the coefficients as a determinant of an matrix as follows:

(9)

where (and ). Explicitly, the first six are given by:

(10)

Also note that the order of is ; i.e., . 5

Note that if the invariant of highest order, , is zero, i.e., , then the eigenvalue equation trivially factorises and we have a zero eigenvalue. Therefore, it is convenient to first check the existence of zero-eigenvalues. In particular, if

then there exists a zero eigenvalue of multiplicity . If this the case then the polynomial factorises trivally and the order can be reduced. The following procedure can then be simplified accordingly.

In general the polynomial, eq.(7), can be analysed and criteria for the various ’Segre types’ can be given. The resulting syzygies are special polynomial invariants which can be used to characterise the various eigenvalue cases; i.e., they are discriminants. A complete set of discriminats can be algorithmically found and in the following we will give the algorithm which is found in [13] (see also [14]). The resulting discriminants will be denoted , , etc., where denotes order of the polynomial, and is a running index. These discriminants can be given in terms of the coefficients ; however, using Newton’s identities we can express them explicitely in terms of the polynomial invariants , , etc.

Given the polynomial

we define the discrimination matrix :

(11)

Consider now the principal minor series, defined as the determinants:

(12)

Let , , then the discriminant sequence of the polynomial is given by

(13)

By expressing the in terms of the curvature invariants, , , etc, we can obtain the primary syzygies for the operator . Note that the order of is . 6

Sign List. We call , where

the sign list of the sequence .

Revised Sign list. Given a sign list . If this contains any “internal zeros”, i.e., if there is a subsequence , where and , then we replace this subsequence with:

The revised sign list will therefore contain no “internal” zeros, but may have zeros at the end. The revised sign list will give us the number of distinct real and complex roots.

Number of real and complex roots. Consider the revised sign list of . Let:

of the revised sign list. Then for :

  • the number of distinct pairs of complex conjugate roots is ; and

  • the number of distinct real roots is .

If we are not interested in the multiplicities of the eigenvalues, the discriminant sequence, is sufficient. In some cases, this is enough to determine the eigenvalue structure of , but not always.

Example: trace-free Ricci tensor in 3D. Let be the 3-dimensional tracefree Ricci tensor; i.e., 7

(14)

This implies,

Consequently,

Using the procedure above, we get the discriminants:

(15)
(16)
(17)

We clearly have . The possible signs of the discriminants and can now be used to determine the number of real/complex eigenvalues of .

  1. , : 3 distinct real eigenvalues.

  2. , : 2 pairs of complex eigenvalues, which is impossible.

  3. : 1 real and 2 complex eigenvalues.

  4. , : 2 real eigenvalues (one of them must be of muliplicity 2).

  5. , : 2 complex eigenvalues (impossible, since the last eigenvalue must be real and hence, distinct).

  6. , : 1 real eigenvalue (which must be equal to zero since is tracefree).

Note that in terms of the Segre type, a (tracefree) Ricci type II/D is of type , , , or simpler. Consequently, if the (tracefree) Ricci is of type II/D, or simpler, then .

Multiple factor sequence. We note that for polynomials of order 4 and more, the discriminants may not be sufficient to determine the complete eigenvalue structure. For example, for quartics, if we have 2 distinct real roots, then we cannot, using only, distinguish the cases and . Therefore, we need to go a step further in order to distinguish these cases.

Consider the discriminant matrix . Define the submatrices:

(18)

Then, construct the polynomials:

(19)

for . The sequence is called the multiple factor sequence of due to the following result [13]:

Lemma 2.1.

If the number of zeros in the revised list of the discriminant sequence of is , then .

The greatest common devisor (g.c.d.) of and is thus always in the multiple factor sequence. Indeed, the polynomial is the repeated part of , because if has real roots of multiplicities , , …,, and has distinct real roots, then has real roots of multiplicities , , …,, and simplie real roots (similar arguement for complex roots). Therefore, by considering we reduce the multiplicities of all the roots by 1.

We can now consider the discriminants of the polynomial in the same way as we computed the discriminant sequence of . We will call the discriminant sequence of for . We can now use these to determine the sign list of the -sequence, etc. We can repeat this procedure and consider , etc. These have the relation:

(20)

This gives us the following algorithm for determining the root structure (or eigenvalue structure) [13]:

Algorithm for Root Classification
  1. Find the discriminant sequence of :

    and the revised sign list. Find the number of distinct roots by counting sign changes and non-zero elements of the revised sign list. If the revised sign list contains no 0’s, stop.

  2. If the revised sign list contains zeros, then compute the by the definition for the multiple factor sequence. Then repeat step 1 for .

  3. Continue considering the multiple factor sequence , ,…, until for some , the revised sign list of contains no zeros.

  4. We now compute the number of real/complex distinct roots of . We can now determine the roots and multiplicities of , which again enables us to determine the roots and multiplicites of etc. At the end of this process, we have a complete root classification for .

Note that this procedure will provide us with the discriminants (or syzygies) which gives us a complete eigenvalue classification of any operator . As explained, these discriminants can be expressed in terms of polynomials of the invariants, , , etc., of . In principle, we can use this method to study the necessary conditions on any curvature operator of any specific eigenvalue type.

In particular, we can use the technique to study the necessary conditions of the Weyl and Ricci curvature operators for it to be of algebraic type II/D. We note that the condition will signal a double eigenvalue since the number of eigenvalues is maximum . If also, then we have maximum eigenvalues, etc. We can utilise this to create syzygies which are necessary for the special algebraic type to be fulfilled.

Type G and I

Types G and I are both of equal generality with respect to their possible (eigenbivector/eigenvalue) roots structure.

2.2 Type Ii/D

Now, for a tensor to be of type II (or D) then the eigenvalues of the corresponding 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 possesses an important symmetry, namely a boost isotropy (the tensor, not necessarily the complete spacetime). This is what we will utilise in the following. This implies specific structure for (a) particular discriminant(s), which then gives rise to 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 , 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 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,

where is a matrix and is a square matrix (see Appendix for the explicit form of this matrix). Since the eigenvalues of and are the same, we have that the Weyl operator has at least eigenvalues of (at least) multiplicity 2.

These observations connect the algebraic types to the eigenvalue structure and enables us to construct the necessary discriminants.

Type Ii/D in 4D

Applying the condition for the complex three dimensional Weyl tensor we obtain the complex syzygy . This is equivalent to the (12th order) real syzygies given by eqns. (1)-(2) from the six dimensional system with and . Applying the condition for the four dimensional trace-free Ricci tensor we obtain the (12th order) syzygy given by eqn. (4). 8

We can also apply these conditions to the full Riemann tensor (to be of type II/D, which implies both the Weyl and Ricci tensor are of type II/D and aligned). Alternatively, we note that these will also give us syzygies for mixed tensors. For example requiring that Riemann tensor is type II/D, implies that both Ricci and Weyl is type II/D, but also mixed tensors, like:

The type II/D condition therefore implies that we have the syzygy for all of , and .

Type Ii/D in 5D

For the trace-free Ricci tensor, we note that type D has to be of Segre type or simpler. This implies that 2 eigenvalues are equal, while the remaining has to be real. Therefore, we get the necessary (20th order) syzygy for the trace-free Ricci tensor to be of type II/D:

Result: The necessary condition for the trace-free Ricci tensor, S, to be of algebraic type II (or D) in 5D is that the discriminant is zero, so that the related scalar polynomial curvature invariant . 9

For the Weyl tensor, we consider the bivector operator . Since the bivector space is 10-dimensional, we get a condition involving a syzygy of order 90! In particular, the type II operator has 3 eigenvalues of (at least) multiplicity 2. Therefore, we get the syzygies:

Since these polynomial invariants are of particular importance, we will denote them by , the 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 .

These are syzygies of order 90, 72 and 56, respectively. In principle, these can be computed, 10 but its probably more useful to consider specific metrics11.

A necessary condition can also be found from considering combinations of the Weyl tensor; for example, the operator . This gives again

Note that is now a 40th order syzygy (in the Weyl tensor; a 20th order syzygy in the square of the Weyl tensor). 12

Caveat: A fundamental degeneracy.

We have stressed that the conditions determined are necessary conditions. Indeed, these conditions may not be sufficient. The reason for this is that the characteristic equation for different algebraic types may be identical and consequently the scalar invariants are also identical. For example, its not possible to distingush the Segre types (of Ricci type D) and (of Ricci type I). This implies also that the conditions may be fulfilled in spite of the fact that the spacetime is of type G or I. An explicit example of this is the following result:

Proposition 2.2.

Assume a 5D spacetime has a Weyl tensor with isotropy. Then it fulfills the syzygies; i.e., .

This result can be seen from using the bivector operator and imposing the -symmetry. One then sees that this forces that there must be 3 pairs of equal eigenvalues, which implies that there are maximum 7 distinct eigenvalues; hence, the result follows.

This degeneracy in the classification is a fundamental problem when considering scalar invariants only. Sometimes these cases can be resolved by considering 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 operator. For example, if we find the eigenvalue type to be , then this can correspond to three Ricci types: , , and because all of these Ricci types have the same eigenvalue type. This is a fundamental degeneracy in the classification of tensors using scalar invariants only and has been discussed in earlier papers [9]. Therefore, we need to keep this in mind when using the discriminants.13 Note that this degeneracy is a discrete degeneracy, unlike the notion of -degenerate metrics [9] which require a continuous deformation.

Type Ii/D in higher dimensions

In higher dimensions we will obtain similar syzygies for type II/D tensors. In dimensions, the Ricci and Weyl type II/D conditions are the corresponding syzygies (:

Ricci: (21)
Weyl: (22)

Note that the Ricci syzygy is of order , while the highest Weyl syzygy is of order .

2.3 Examples

The conditions are non-trivial: Let us consider a simple metric which shows the conditions are non-trivial conditions. Consider the solvmanifold:

(23)

The computation for this metric is a bit lengthy (even for MAPLE); however, by choosing randomly some values of the parameters14, for example, , , , and , MAPLE quickly calculates the values of the discriminants for the Weyl operator, obtaining:

(24)

showing that this metric is not type II or D (and that the invariants are not trivial).

Some simple examples

Einstein spaces In an Einstein space all of , and hence all of the discriminants for the trace-free Ricci eigenvalue equation are zero, and the only non-trivial scalar invariant is the Ricci scalar .

5D Schwarzschild spacetime: For the Weyl operator we get

This implies that the Weyl operator has 3 distinct real eigenvalues which agrees with the results of [7]. In fact, this spacetime is of type D.

5D space with complex hyperbolic sections. Let us consider the example in [7] with complex hyperbolic spatial sections:

(25)

For the Weyl operator we get

This again implies that the Weyl operator has 3 distinct real eigenvalues, like the 5D Schwarzschild spacetime. However, unlike the Schwarzschild case, we note that the some of the invariants , defined in eq.(9), are zero:

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 with multiplicity 7, the eigenvalue structure is . In particular, since the Weyl operator is trace-free, we explicitly get eigenvalues:

which agrees with the results of [7].

However, this spacetime is not of type II/D; indeed, it is -non-degenerate. This can be seen by computing the operator which is of “Segre” type . However, due to the fundamental degeneracy in the eigenvalue type, we need to compute differential invariants to delineate this case completely.

The rotating black ring.

The 5D rotating black ring [12] is generally of type , but can also be of type or at different locations and for particular values of the parameters . Assuming that the form of the metric is given by eqn. (9) in [12] (in terms of the parameters , where has been set to unity), we consider the coordinate ranges and (and hence and ), corresponding to the regions in [12] in order to retain the correct (Lorentzian) signature. We consider the algebraic type of the 5D Weyl tensor. Calculating the polynomial invariants and evaluating at the ‘target’ point and in the region under consideration, 15 all of the and hence all of the resulting discriminants are functions of the parameters only. Then, at the ‘target’ point, in general the metric is of type , the case corresponds to the Myers-Perry metric (type ), corresponds to the black hole horizon (, type ), corresponds to the static subcase, and corresponds to a curvature singularity.

Let us first consider the trace-free part operator , which gives us the discriminant:

(26)

where is a polynomial which is generically not zero. On the horizon , we see that , and computing we get except for special values of . This is a signal that the metric is of type II on the horizon. Indeed, at the horizon, , the computation simplifies and we can compute the invariants:

while:

(27)

where a (postive) numerical factor has been ignored. Since, the syzygies are satisfied, this gives further evidence that the the metric is of type II on the horizon. Note that we actually get further contraints from the secondary discriminants, as can be seen from the table in Appendix B.4. Indeed, by calculating the secondary discriminant on the horizon, we get:

(28)

which we see is non-zero as long as . Consequently, as long as , then the eigenvalue type is . This is consitent with type II.

Another interesting special case is (Myers-Perry), for which both , and:

We can also here compute the invariants, which are all zero.

Note: We note that is a 40th order syzygy in the Weyl tensor. Therefore, a useful strategy in practical computations (for example, determining the algebraic type of a 5D Weyl tensor), as illustrated by this example, might be 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 the syzygy can only be satified for certain coordinate values (or parameter values), whence the syzygies can be tested in these simpler particular cases.

3 Conclusions

3.1 Discussion

Figure 1: A flow diagram indicating how we can attempt to determine the algebraic type for tensors. The degeneracy indicated is due to the fact that several types may have the same eigenvalue type (sometimes this can be resolved by considering other invariants, but not always).

For a curvature tensor of a particular algebraic type, the associated operator will have a restricted eigenvector structure. For a given curvature operator in arbitrary dimensions, we can thus consider the eigenvalues of this operator to obtain necessary conditions in order for the tensor to be of a particular algebraic type. In principle, this analysis can be used to study all of the various subclasses of particular algebraic types in more detail. In particular, requiring the algebraic type to be II or D will impose useful restrictions. In this paper we have used an analysis of the discriminants of the associated characteristic equation to determine the conditions on a tensor for a given algebraic type. Since the characteristic equation has coefficients which can be expressed in terms of the scalar polynomial curvature invariants of the operator, we can give conditions, or syzygies, on the eigenvalue structure expressed manifestly in terms of these polynomial scalar curvature invariants. Indeed, we have described an algorithm which enables us to completely determine the eigenvalue structure of the curvature, up to degeneracies, in terms of a set of discriminants , , etc.. The resulting syzygies (discriminants) can then be written as special scalar polynomial invariants.

In particular, we have used the technique to study the necessary conditions in arbitrary dimensions for the Weyl and Ricci curvature operators (and hence the 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 fulfilled. We are consequently able to determine the necessary conditions in terms of simple scalar polynomial curvature invariant for the higher dimensional Weyl and Ricci tensors to be of type II or D. We have explicitly determined the scalar 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 dimensional solutions obtained in supergravity or superstring theory [3] or higher dimensional black hole solutions [4].

Summary of results.

A number of specific results have been obtained in this work. The necessary condition for the trace-free Ricci tensor, , to be of algebraic type II (or D) in 5D is that the discriminant is zero, so that the related scalar polynomial curvature invariant . The necessary condition for the Weyl tensor to be of type II (or D) in 5D is that the scalar polynomial curvature invariants . In principle, we can repeat a similar analysis for other algebraic types (and making more use of the secondary discriminants).

A number of simple examples were presented, including Einstein spaces, the 5D Schwarzschild spacetime, and 5D space with complex hyperbolic sections. In addition, a simple solvmanifold was considered to show that the conditions are non-trivial.

We also presented a detailed analysis of the important example of a 5D rotating black ring [12] which is generally of type , but can also be of type or for particular values of the parameters. This example serves to illustrate the calculational method and the power of the approach. In particular, we showed that the rotating black ring is of type (or type ) on the black hole horizon (), by showing that on the horizon (and studying some of the secondary discriminants). The example also illustrates the utility in practical computations of employing rather more simple discriminants like (which is a 40th order syzygy in the Weyl tensor).

3.2 Classification using scalar invariants

In Lorentzian spacetimes, identical metrics are often given in different coordinate systems, which disguises their true equivalence. Perhaps the easiest way of distinguishing metrics is through their scalar polynomial curvature invariants, due to the fact that inequivalent invariants implies inequivalent metrics. In [9] the notion of an -non-degenerate spacetime metric in the class of 4D Lorentzian manifolds, which implies that the spacetime metric is locally determined by its scalar polynomial curvature invariants, was introduced. By determining an appropriate set of projection operators from the Riemann tensor and its covariant derivatives, it was proven that a 4D Lorentzian spacetime metric is either -non-degenerate or degenerate Kundt [9]. Therefore, a metric that is not characterized by its curvature invariants must be of degenerate Kundt form. These results were generalized to higher dimensions in [15].

Differential invariants

The -non-degenerate theorem contains not only zeroth order invariants but also differential scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. For example, if the spacetime is of Weyl type N, then the differential invariants and vanish if the spacetime is degenerate Kundt [6] (the definitions of the invariants and are given therein). Similar results follow for Weyl type III spacetimes (in terms of invariants and ) and in the conformally flat (but non-vacuum) case (in terms of similar invariants and constructed from the Ricci tensor [6].

These conditions are necessary conditions in order for a spacetime not to be -non-degenerate [9]. In the case that (Weyl types II or D), in [9] two higher order invariants and were given as sufficient conditions for -non-degeneracy (if , but or , then the spacetime is -non-degenerate). 16

This analysis in 4D can be repeated using discriminants. Let us focus on the Ricci tensor for illustrative purposes. The necessary condition for the Ricci tensor to be of type II or D is given by eqn. (4) which, as noted earlier, follows from a discrimant analysis. Now, if we consider the covariant derivatives of the Ricci tensor, , then for the spacetime to be -non-degenerate each covariant derivative term must also be of type II or D. Hence we could study the eigenvalue structure of the operators associated with the and apply the type II/D necessary conditions in turn. For example, considering the trace-free parts of the tensors , we obtain necessary conditions of the form of eqn. (4) but with the . 17

This can be repeated for the Weyl tensor and in higher dimensions [15].

Example. The class of vacuum type D spacetimes which are -non-degenerate, are invariantly classified by their scalar polynomial curvature invariants [16]. For example, for the Kinnersley class I type D vacuum spacetime [17] (the other cases 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., , and invariants such as , where

which include first and second covariant derivatives). The Schwarzschild vacuum 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 functionally independent polynomial scalar curvature invariants and (which are equal to and , respectively, as functions of the two parameters and in canonical coordinates) [18]. The Kerr solution belongs to Kinnersley class IIA; this spacetime has been invariantly characterized 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 whether the covariant derivative, , is of type D or not. As noted above, the methods discussed earlier can also be used for differential invariants. The covariant derivative of the Weyl tensor, , has no natural operator associated with it. However, we can, for example, consider the second order operator defined above. If is not of type D/II, then cannot be of type D/II either.

For the Kerr metric, we obtain the syzygy:

where

(29)

and and are some complicated polynomials. We note that away from the horizon, the ergosphere, and some other special points, this syzygy is non-zero and hence is not of type D/II (generically), outside the horizon. The Kerr metric is therefore -non-degenerate by the results of [9].

3.3 Physical applications

Recently, there has been considerable interest in black holes in more than four dimensions [4]. While the study of black holes in higher dimensions was perhaps originally motivated by supergravity and string theory, now the physical properties of such black holes are of interest in their own right. Indeed, studies have shown that even at the classical level gravity in higher dimensions exhibits much richer dynamics than in 4D, and one of the most remarkable features of higher dimensions is the non-uniqueness of black holes [4]

There now exist a number of different higher dimensional black hole solutions [4], including the rotating black rings [12], that are the subject of ongoing research in classical relativity and string theory. Some of these new spacetimes have be classified algebraically [5, 12]. However, in order to make further progress it is absolutely crucial to be able to develop new techniques for solving the vacuum field equations in higher dimensions and to be able to comprehensively classify such solutions, and the algebraic techniques recently introduced [5, 7] will be of fundemental importance in this development. However, the algebraic techniques used to date now are rather difficult to apply, and the development of simpler criteria, including the use of necessary conditions in terms of scalar curvature invariants introduced here, will hopefully prove to be of great utility.

Therefore, the analysis presented in this paper will be of considerable importance for analysing higher dimensional black hole solutions [4] (and solutions in supergravity or superstring theory [3]). Indeed, the detailed analysis of the 5D rotating black ring [12] serves to illustrate the power of the approach.

In future work we hope to extend the analysis presented in this paper and further generalize it to the study of differential operators. In addition, we intend to discuss a number of other applications, including the algebraic classification of some other known higher dimensional black hole solutions.

Acknowledgements

The main part of this work was done during a visit to Dalhousie University April-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: 200910/V11 (SH).

Appendix A The Weyl Bivector operator

Given a vector basis we can define a set of (simple) bivectors