Mahdi Godazgar
email: mmg31@cam.ac.uk

Spinor classification of the Weyl tensor in five dimensions

We investigate the spinor classification of the Weyl tensor in five dimensions due to De Smet. We show that a previously overlooked reality condition reduces the number of possible types in the classification. We classify all vacuum solutions belonging to the most special algebraic type. The connection between this spinor and the tensor classification due to Coley, Milson, Pravda and Pravdová is investigated and the relation between most of the types in each of the classifications is given. We show that the black ring is algebraically general in the spinor classification.

Introduction

Despite growing interest in the study of higher dimensional gravity in recent years, most higher dimensional solutions found to date have been direct generalisations of four dimensional solutions. One way to investigate higher dimensional gravity independent of the 4d case is to attempt a systematic study of d > 4 general relativity. In four dimensions, the Petrov classification [START_REF] Cartan | Sur les espaces conformes généralisés et l'univers optique[END_REF][START_REF] Stephani | Exact solutions of Einsteins field equations[END_REF] of the Weyl tensor has been a useful tool in studying solutions to the Einstein equations (a wellknown example being [START_REF] Kinnersley | Type D vacuum metrics[END_REF]). Thus, it is natural to consider higher dimensional generalisations of the Petrov classification.

A tensorial classification scheme based on a higher dimensional generalisation of the concept of principal null directions of the Petrov classification has been proposed by Coley, Milson, Pravda and Pravdová [START_REF] Coley | Letter to the editor: Classification of the Weyl tensor in higher dimensions[END_REF][START_REF] Milson | Alignment and algebraically special tensors in Lorentzian geometry[END_REF] (henceforth abbreviated to CMPP). The CMPP classification applies to spacetimes of any dimension.

The CMPP classification scheme has been successfully applied to studying many aspects of higher dimensional gravity (see [START_REF] Coley | Classification of the weyl tensor in higher dimensions and applications[END_REF] and references therein), including a partial generalisation of the Goldberg-Sachs theorem to higher dimensions [START_REF] Durkee | A higher-dimensional generalization of the geodesic part of the Goldberg-Sachs theorem[END_REF] (see also [START_REF] Pravda | Bianchi identities in higher dimensions[END_REF]), the asymptotic properties of higher dimensional spacetimes [START_REF] Ortaggio | On asymptotically flat algebraically special spacetimes in higher dimensions[END_REF][START_REF] Ortaggio | Type III and N Einstein spacetimes in higher dimensions: general properties[END_REF] and a classification of axisymmetric solutions to vacuum Einstein equations in higher dimensions [START_REF] Godazgar | Algebraically special axisymmetric solutions of the higher-dimensional vacuum Einstein equation[END_REF].

Another higher dimensional classification scheme has been proposed by De Smet [START_REF] De | Black holes on cylinders are not algebraically special[END_REF]. The De Smet classification generalises the concept of a Petrov Weyl spinor to five dimensions. The classification uses the 5d Clifford algebra to define a totally symmetric 4-spinor, called the Weyl spinor, that is equivalent to the Weyl tensor. A given solution is classified by studying how its Weyl spinor factorises. Or more precisely, how the fourth order quartic homogeneous polynomial formed from its Weyl spinor factorises. The fact that the Weyl spinor is generally complex means that it must satisfy a reality condition and this reduces the number of possible types.

In five dimensions, it is known that the two classification schemes are not equivalent; that is they do not agree on the definition of an "algebraically special" solution. An example is known that is algebraically special in the CMPP classification, but algebraically general in the De Smet classification [START_REF] Hervik | Mathematical Cosmology: Bianchi Models, asymptotics and extra dimensions[END_REF] and vice versa [START_REF] Godazgar | Algebraically special axisymmetric solutions of the higher-dimensional vacuum Einstein equation[END_REF]. The presence of two inequivalent classification schemes in five dimensions presents us with the opportunity of studying solutions that are algebraically general in one scheme and special in the other.

Apart from a classification of static axisymmetric solutions belonging to two particular algebraic types [START_REF] De | Black holes on cylinders are not algebraically special[END_REF][START_REF] De | Five-dimensional metrics of Petrov type 22[END_REF], the De Smet classification has not been studied much. The aim of this paper is to better understand the De Smet classification and its relation to the 4d Petrov and 5d CMPP classifications. We shall find that the previously overlooked reality condition will play an important part in this study.

As a way of highlighting the most important characteristics of the spinor classification of the Weyl tensor, we shall also consider the spinor classification of two-form fields, where it is much easier to appreciate subtle issues such as reality conditions. This is because, we shall be dealing only with a bispinor, rather than a 4-spinor as is the case in the Weyl classification.

Therefore, we begin, in section 2.1, with a derivation of a spinor classification of 2-form fields. We construct a bispinor equivalent of the 2-form and use properties of the Clifford algebra to show that it is symmetric and satisfies a reality condition. This leads to a classification of 2-forms based on whether the equivalent bispinor factorises or not.

Then, in section 2.2, we move on to derive the spinor classification of the Weyl tensor due to De Smet [START_REF] De | Black holes on cylinders are not algebraically special[END_REF] in similar vein to the derivation of the spinor classification of 2-forms in section 2.1. We define the Weyl spinor, and show that it is totally symmetric and satisfies a reality condition. The reality condition reduces the number of algebraically special types.

The De Smet classification is intended to be a generalisation of the spinor formulation of the Petrov classification to five dimensions. It is not clear, though, how the two schemes are related and in what sense the De Smet classification is a generalisation of Petrov's beyond the superficial link that they both deal with the factorisability properties of a totally symmetric spinor quantity. This issue is addressed in section 2.3, where it is shown that one can define an analogue of the De Smet Weyl spinor in 4d and that the classification of the Weyl tensor based on this can be thought of as a classification using Majorana spinors. Recall that in the Petrov classification, one uses chiral spinors.

In section 2.4, we use the results obtained in section 2.3 to study direct product solutions. We find that the De Smet type of solutions with a 4d factor is equal to the De Smet type of the 4d submanifold. Thus, the analysis reduces to that done in section 2.3. For the case with 2d and 3d Lorentzian factors with non-zero cosmological constant, the Weyl spinor factorises into two proportional bispinors that cannot be further factorised. These results are similar to those found in the study of warped product manifolds in the context of the CMPP classification in [START_REF] Pravda | Type D Einstein spacetimes in higher dimensions[END_REF].

In section 3.1, we consider the connection between the tensor and spinor classifications of a 2form, where the tensor classification is based on the CMPP classification. We find that a solution of any spinor algebraic type may be algebraically general in the tensorial sense. For solutions that are algebraically special in the spinorial sense, what determines whether they are algebraically special or general in the tensorial sense is whether the vector that can be formed from the spinor that we have from the factorisation of the bispinor is null or timelike.

We find that similar statements can be made regarding the relation between the De Smet and 5d CMPP classifications of the Weyl tensor in section 3.2. However, in this case we cannot study all types fully. Thus, we begin by assuming that the solution is of type N, III or D and derive the general De Smet polynomials for the respective cases. Considering the factorisability properties of these general polynomials gives the possible spinor types that they can have. We show that type III and D solutions may be algebraically general in the spinor classification, while for type N solutions, the De Smet polynomial is guaranteed to factorise into linear factors, so type N solutions are also algebraically special in the spinor classification. We do not consider more general types due to the complexities of factorising a general polynomial. Then, we go on to consider the reverse case, i.e. assuming a particular De Smet type and examining what this implies about the CMPP type. Since the general form of Weyl tensor is important for this analysis, we can only do this for the case where the Weyl spinor factorises into two bispinors, or a more special case of this, using the general form of the Weyl tensor of such solutions derived in section 2.2. Thus, we do not consider the case where the solution is algebraically general, i.e. the Weyl spinor does not factorise nor the case where it factorises into a rank-3 spinor and a univalent spinor. We find that any spinor type may be algebraically general in the CMPP sense.

An important motivation, given above, for understanding the De Smet classification and its relation to the 5d CMPP classification was that this may allow us to study 5d solutions that are algebraically general in one classification scheme and special in the other. Furthermore, the result found in section 3.2 that any spinor type may be algebraically general in the CMPP sense strengthens this motivation. The black ring [START_REF] Emparan | A rotating black ring solution in five dimensions[END_REF] is a well-known example of a CMPP algebraically general five dimensional solution [START_REF] Pravda | WANDs of the black ring[END_REF] 1 . Therefore, it would be desirable to know the De Smet type of the black ring solution. It is shown in section 3.3 that the black ring is also, unfortunately, algebraically general in the De Smet classification.

In section 4.1, we consider the constraints imposed on a spacetime by the existence of an algebraically special 2-form solving Maxwell-type equations. An algebraically special 2-form is defined by a single spinor. In 4d, the existence of an algebraically special Maxwell field is equivalent to the spacetime being algebraically special. This follows from the Mariot-Robinson [START_REF] Mariot | Le champ électromagnétique singulier[END_REF][START_REF] Stephani | Exact solutions of Einsteins field equations[END_REF] and Goldberg-Sachs [START_REF] Goldberg | A theorem on Petrov types[END_REF] thereoms. Thus, studying the existence of algebraically special fields can shed light on the status of the Goldberg-Sachs theorem in higher dimensions. The 2-form field is assumed to satisfy the Bianchi identity and a general equation of motion that includes Maxwell theory as well as minimal supergravity [START_REF] Gauntlett | All supersymmetric solutions of minimal supergravity in five dimensions[END_REF]. The analysis splits into two cases of whether the vector derived from the spinor that defines the 2-form field is null or timelike. From section 3.1, we know that if it is null then this is equivalent to the field being algebraically special in the CMPP sense. This analysis has already been done in [START_REF] Durkee | Generalization of the Geroch-Held-Penrose formalism to higher dimensions[END_REF]. The null vector defines a geodesic congruence with constraints on its optical properties, which are explained in section 4.1. If the vector is timelike, then the solution admits a timelike geodesic congruence and an almost-Kähler structure.

Finally, in section 4.2, we undertake a classification of solutions belonging to the most special type, that is type 11 11 solutions. These are defined as those for which the Weyl spinor factorises into two proportional bispinors, which factorise further into spinors. The reality condition gives that the Weyl tensor is fully determined from a single spinor. We use the Bianchi identity to find constraints on this spinor for a vacuum Einstein solution. As in section 4.1, the analysis divides into two cases of whether the vector defined from the spinor is null or timelike. If the vector is null, then we have a type N Kundt solution satisfying further conditions that are explained in section 4.2. The timelike case reveals more structure. The spacetime is found to be a cosmological solution with spatial geometry a type (D,O) Einstein solution. The solutions in section 4.2 are more constrained that those found in section 4.1.

The index conventions in this paper are as follows: indices a, b, c . . . refer to orthonormal or null frame basis vectors and generally take values from 0 to 4, although this is not always the case. Indices i, j, k . . . refer to spacelike basis vectors and generally take values from 2 to 4. In section 3.2.1, where we move between orthonormal and null frame bases, indices a, b, c . . . refer to null frame basis vectors, while µ, ν, ρ . . . refer to orthonormal basis vectors. α, β, γ . . . and α, β, γ . . . label left and right-handed chiral spinor indices in four dimensions and run from 1 to 2, while A, B, C . . . and Ȧ, Ḃ, Ċ . . . label Dirac and Dirac complex conjugate indices in four and five dimensions and run from 1 to 4. There are additional index conventions in section 2.4, which are explained separately in that section.

CMPP classification

The CMPP classification relies on the existence of a null frame in which certain components of the Weyl tensor vanish, implying the solution to be a certain type. Given a null frame (ℓ, n, m i ), one can apply four sets of continuous Lorentz transformations: null rotations about ℓ and n, spins (rotating the spacelike basis vectors), and boosts, given by

ℓ ′ = λℓ, n ′ = λ -1 n, m ′ i = m i ,
where λ = 0. Under a boost, a particular component of a p-rank tensor T in the null frame transforms as

T a 1 ...ap -→ λ b T a 1 ...ap ,
where b is the boost weight of T a 1 ...ap , and is equal to the number of a i that are 0 minus the number that are 1.

For the Weyl tensor, the possible boost weights lie in the range -2 ≤ b ≤ 2. For example, boost weight +2 components of the Weyl tensor are C 0i0j .

The solution is type G at a point p if there does not exist a ℓ such that C 0i0j = 0 at p, i.e. C 0i0j = 0 at p for any choice of ℓ. If there does exist a ℓ such that C 0i0j = 0 at p, then ℓ called a Weyl aligned null direction or WAND and the solution is type I or more special at p. The solution is said to be algebraically special at p and of given type The algebraic type of the solution is defined to be the type of its most algebraically general point. If the solution is algebraically special, then ℓ for which C 0i0j = C 0ijk = 0 is said to be a multiple WAND or mWAND. A given mWAND need not be unique. Indeed there may be infinitely many mWANDs in a spacetime. For example, consider dS 3 × S d-3 (for d > 4). Any null vector in dS 3 is a mWAND [START_REF] Godazgar | Algebraically special axisymmetric solutions of the higher-dimensional vacuum Einstein equation[END_REF].

• II ⇐⇒ C 0i0j = C 0ijk = 0, • D ⇐⇒ C 0i0j = C 0ijk = C 1ijk = C 1i1j = 0, • III ⇐⇒ C 0i0j = C 0ijk = C 01ij = C ijkl = 0, • N ⇐⇒ C 0i0j = C 0ijk = C 01ij = C ijkl = C 1ijk = 0, • O ⇐⇒ C abcd = 0.
Note that the definition of type D solutions depends on n being multiply Weyl aligned as well, i.e. C 1i1j = C 1ijk = 0. Thus, the type D definition requires a secondary classification, in which n is chosen such that as many trailing Weyl tensor components as is possible can be set to zero, that is, of course, once a WAND ℓ has been found such that as many leading Weyl tensor components as is possible have been set to zero. For example one defines type I i solutions to be those for which a ℓ and n can be found such that C 0i0j = C 1i1j = 0. We shall not utilise the secondary classification scheme here, except in the definition of type D solutions.

2 Spinor classification of two-form and Weyl tensor

Spinor classification of two-form

Let F ab be a real two-form. We can construct a bispinor ǫ AB that is equivalent to the 2-form2

ǫ AB = i 8 F ab Γ ab AB , (1)
where

Γ ab = Γ [a Γ b] .
As explained in appendix A, for brevity, we omit factors of C and C -1 where it is clear from the index structure that charge conjugation matrices have been used. Thus,

Γ ab AB = (CΓ ab ) AB = C AC Γ ab C B .
It can be shown, using the antisymmetry of the charge conjugation matrix C that Γ ab AB is symmetric in its spinor indices: using the definition of C, we find that

Γ t ab = -CΓ ab C -1 , (2)
which implies

(C t Γ ab ) t = -(CΓ ab ), i.e. (CΓ ab ) t = (CΓ ab ). (3)
Thus, the bispinor ǫ AB is symmetric. Using properties of gamma-matrices, we can invert equation ( 1)

F ab = i tr(Γ ab ǫ), (4)
where tr(

Γ ab ǫ) = tr(Γ ab C -1 ǫ) = Γ ab A B C BC ǫ CA .
Note that while the 2-form is real, the bispinor is generally complex since there is no Majorana representation of the Clifford algebra in five dimensions. A complex bispinor has 20 real independent components, whereas a real 2-form has 10 real components. Therefore, the bispinor must satisfy a reality condition, which halves its number of independent components.

Using the definitions of the Dirac and charge conjugation matrices (see appendix A) one can derive the following relation between Γ * ab and Γ ab

Γ * ab = A Γ ab A -1 , (5)
where A = (CB -1 ) t . Now, taking the complex conjugate of equation (1) and using the equation above gives

ǫ Ȧ Ḃ = - i 8 F ab A ȦA Γ ab AB (A -1 ) B Ḃ ,
where A ȦA = (CAC -1 ) ȦA = -(A -1 ) A Ȧ [START_REF] Lee | Spinors in arbitrary dimensions[END_REF]. Then re-arranging the above equation gives

ǫ AB = ǭAB , (6)
where ǭAB ≡ ǫ Ȧ Ḃ A ȦA A Ḃ B . A 2-form field is said to be algebraically special if the bispinor factorises. If this is the case, then the reality condition, equation [START_REF] Coley | Classification of the weyl tensor in higher dimensions and applications[END_REF], implies that3

ǫ AB = ǫ (A ǭB) . (7)
Then, the 2-form F is of the form

F ab = iǭΓ ab ǫ (8)
We can also form a real scalar and vector

f = ǭǫ, V a = iǭΓ a ǫ. (9)
The Fierz identity can be used to relate the above three quantities [20]

V 2 = -f 2 (10)
F 2 = F ab F ab = 4f 2 (11)
ι V F = 0 (12)
F a c F c b = -f 2 δ a b -V a V b (13)
F ∧ F = 2f ⋆ V (14)
f F = ⋆(V ∧ F ). ( 15
)
Note that the above equations are not independent. In fact, equations ( 10) and ( 15) can be used to derive equations ( 11)-( 14).

De Smet classification

In four dimensions, the Petrov classification is most simply derived by defining a totally symmetric Weyl spinor Ψ αβγδ and considering the Weyl polynomial

Ψ(χ) = Ψ αβγδ χ α χ β χ γ χ δ
formed from the Weyl spinor, where χ α is a general chiral spinor. The fundamental theorem of algebra ensures the factorisability of the polynomial and the Petrov classification reduces to an analysis of the multiplicity of the factors.

In similar vein, the spinor classification of the Weyl tensor in five dimensions [START_REF] De | Black holes on cylinders are not algebraically special[END_REF] uses a spinorial approach to the classification of the Weyl tensor. Define the Weyl spinor, associated with the Weyl tensor, to be

C ABCD = C abcd Γ ab AB Γ cd CD . (16)
The Weyl spinor is symmetric in its first and last pair of indices since CΓ ab is symmetric (see section 2.1). Also, using the symmetries of the Weyl tensor, it is symmetric under interchange of AB and CD. In five dimensions, the Fierz identity can be used to show that it is totally symmetric. The five dimensional Fierz identity is

M AB N CD = 1 4 C AD (N M ) CB + 1 4 Γ eAD (N Γ e M ) CB - 1 8 Γ ef AD (N Γ ef M ) CB . (17)
Letting M = Γ ab and N = Γ cd , and multiplying by C abcd gives

C abcd Γ ab AB Γ cd CD = 1 4 C abcd C AD (Γ cd Γ ab ) CB + 1 4 C abcd Γ eAD (Γ cd Γ e Γ ab ) CB - 1 8 C abcd Γ ef AD (Γ cd Γ ef Γ ab ) CB . (18)
The trace free property of the Weyl tensor implies that

C abcd Γ a Γ b Γ c = C abcd Γ abc = C a[bcd] Γ abc . Thus, the Bianchi identity gives C abcd Γ a Γ b Γ c = C abcd Γ b Γ c Γ d = 0.
Therefore, equation ( 18) reduces to

C ABCD = 1 4 C abcd Γ eAD ([Γ cd , Γ e ]Γ ab ) CB - 1 8 C abcd Γ ef AD ([Γ cd , Γ ef ]Γ ab ) CB . (19)
Using the following identities

[Γ ab , Γ c ] = 2(g bc Γ a -g ac Γ b ), (20)
[Γ ab , Γ cd ] = 2(g bc Γ ad + g ad Γ bc -g ac Γ bd -g bd Γ ac ), (21)
equation ( 19) reduces to

C ABCD = C ADCB .
Therefore, the Weyl spinor is totally symmetric4

C ABCD = C (ABCD) . (22)
As with the case of the 2-form in section 2.1, the Weyl tensor is real, while the Weyl spinor will in general be complex. The complex Weyl spinor has 70 real independent components, while the 5d Weyl tensor has 35 independent components. Thus, the Weyl spinor satisfies a reality condition, which halves its number of independent components.

Taking the complex conjugate of equation ( 16) and using equation [START_REF] Milson | Alignment and algebraically special tensors in Lorentzian geometry[END_REF] gives

C ABCD = C Ȧ Ḃ Ċ ḊA ȦA A Ḃ B A Ċ C A ḊD . (23)
The De Smet classification involves the factorisability properties of the invariant Weyl polynomial

C(ψ) = C ABCD ψ A ψ B ψ C ψ D , ( 24
)
where ψ is a general Dirac spinor. In contrast to the Petrov classification, in general, the polynomial above will not factorise. If it does factorise, the solution is said to be algebraically special. Each polynomial factor in the product is distinguished by its degree and multiplicity. There are 12 possibilities, as depicted in figure 2 [START_REF] De | Black holes on cylinders are not algebraically special[END_REF]. The notation is such made that a number represents the degree of the polynomial factor and an underline represents its multiplicity. For example, type 22 corresponds to the case where the Weyl polynomial factorises into two quadratic factors that are proportional to one another and cannot be further factorised. Type 4 solutions (for which the polynomial does not factorise) are said to be algebraically general.

For type 22 or more special solutions, we can learn more. For all such solutions, the Weyl spinor is of the form

C ABCD = ǫ (AB η CD) . (25)
The reality condition, equation ( 23), reduces to

ǫ (AB η CD) = ǭ(AB ηCD) , (26)
where ǭAB

≡ ǫ Ȧ Ḃ A ȦA A Ḃ B .
It can be shown that this implies that either5

ǫ AB = ǭAB , η AB = ηAB , (27)
or ǫ AB = ηAB .

We can invert equation ( 16), so that the Weyl tensor is given in terms of the Weyl spinor, i.e.

C abcd = 1 64 (Γ ab ) AB (Γ cd ) CD C ABCD . (29)
Using equation ( 25) and the 5d Fierz identity, one can derive the general form of the Weyl tensor of type 22 or more special solutions6

C abcd =A a[c B d]b + B a[c A d]b -A ab B cd -B ab A cd - 1 2 A ef B ef g a[c g d]b -A ae B e [c g d]b -B ae A e [c g d]b + A be B e [c g d]a + B be A e [c g d]a , (30)
where A ab = i tr(Γ ab ǫ) and B ab = i tr(Γ cd η).

From the derivation of the reality condition in section 2.1, we find that reality conditions ( 27) and ( 28) translate to

A * ab = A ab , B * ab = B ab (31)
and

A * ab = B ab , (32)
respectively.

The reality condition constrains the Weyl spinor and we can use the results above to show that some types are not possible.

Type 1111

The Weyl spinor of type 1111 solutions is of the form

C ABCD = ǫ A ǫ B ǫ C ǫ D . (33)
Letting

ǫ AB = ǫ A ǫ B , η AB = ǫ A ǫ B ,
reality conditions ( 27) and ( 28) both reduce to

ǫ A ǫ B = ǭA ǭB .
In appendix B, we show that this implies a Majorana condition on ǫ ǫ A ∝ ǭA , which gives that ǫ = 0 since the Majorana condition has no non-trivial solutions in 5d. ǫ = 0 contradicts the assumption that the solution is not conformally flat.

Type 1111

The Weyl spinor of type 1111 solutions is of the form

C ABCD = η (A ǫ B ǫ C ǫ D) , (34)
where η ∝ ǫ. We can choose

ǫ AB = ǫ A ǫ B , η AB = η (A ǫ B) .
Reality condition [START_REF] Ficken | The Riemannian and affine differential geometry of product-spaces[END_REF] gives two constraints, one of which is

ǫ A ǫ B = ǭA ǭB ,
which contradicts the original assumption, as we showed for type 1111 solutions. Reality condition (28) reduces to

ǫ A ǫ B = ǭ(A ηB) . (35)
Since ǫ = 0, this gives

ǫ A = α ǭA + β ηA .
Substituting this into equation [START_REF] Berger | Sur les groupes d'holonomie des varits a connexion affine et des varits Riemanniennes[END_REF] gives

α 2 ǭA ǭB + (2αβ -1)ǭ (A ηB) + β 2 ηA ηB = 0.
Using similar techniques to those used in appendix B, it is not too difficult to show that this implies α = 0 or β = 0. β = 0 gives a Majorana condition on ǫ, so α = 0. Then equation ( 35) becomes

β 2 ηA ηB = ǭ(A ηB) .
Since η = 0, this implies that ǫ ∝ η. However, this contradicts the assumption that the solution is type 1111.

Type 1111

The Weyl spinor of type 1111 solutions is of the form

C ABCD = η (A κ B ǫ C ǫ D) , (36)
where none of the spinors are proportional to one another. Choose

ǫ AB = ǫ A ǫ B , η AB = η (A κ B) .
As before, reality condition [START_REF] Ficken | The Riemannian and affine differential geometry of product-spaces[END_REF] gives two constraints, one of which is

ǫ A ǫ B = ǭA ǭB ,
which gives a contradiction. Reality condition (28) reduces to

ǫ A ǫ B = η(A κB) . (37)
Since ǫ = 0, this gives

ǫ A = α ηA + β κA .
The arguments used for the analysis of type 1111 solutions apply to give

ǫ ∝ η, or ǫ ∝ κ.
As before, both these conditions give that η ∝ κ, which contradicts the assumption that the solution is type 1111.

Type 211

The Weyl spinor of type 211 solutions is of the form

C ABCD = ǫ (AB η C η D) , (38)
where ǫ AB does not factorise. Reality condition [START_REF] De | Letter: The Petrov type of the five-dimensional Myers-Perry metric[END_REF] gives that ǫ AB factorises, contradicting the assumption that the solution is type 211. Reality condition [START_REF] Ficken | The Riemannian and affine differential geometry of product-spaces[END_REF] gives a Majorana condition on η, since it implies that η A η B = ηA ηB .

Thus, type 211 solutions are also not possible.

In summary, we have shown that, the reality condition on the Weyl spinor means that a solution cannot be of types 1111, 1111, 1111 and 211. Therefore, the number of possible types reduces to eight. A revised version of figure 2 is drawn in figure 3.

Although, we considered the spinor classification of Lorentzian solutions in the analysis above, the spinor classification of Euclidean solutions is identical except that one uses the Euclidean Clifford algebra. The reason for this is that as with the 5d Lorentzian Clifford algebra, the 5d Euclidean Clifford algebra does not admit a Majorana representation. This means that the Weyl spinor of the Euclidean solution must satisfy the same reality condition as the Lorentzian case, i.e. equation [START_REF] Zakharov | Gravitational Waves in Einstein's Theory[END_REF], where A now defines a Majorana condition for the Euclidean Clifford algebra. However, since our arguments do not depend on specific properties of A, the same conclusions as those found above will follow.

Relation to Petrov classification

The five dimensional De Smet classification is a generalisation of the four dimensional Petrov classification, insofar as it is concerned with the factorisability of a totally symmetric 4-spinor that is equivalent to the Weyl tensor. Here, we discuss the relation between the two classification schemes. The 4d analogue of the De Smet Weyl spinor is

C ABCD = C abcd γ ab AB γ cd CD , (39)
where γ a form a representation of the 4d Clifford algebra. We need to show that the 4d De Smet Weyl spinor defined above is totally symmetric. We can do this by using the results found in 5d.

In the definition of the 5d Weyl spinor (equation ( 16)) restrict the indices to take values 0, . . . , 3 so that

C ABCD = C abcd (γ 0 γ 5 γ ab ) AB (γ 0 γ 5 γ cd ) CD , (40)
where we have used the Clifford algebra representation defined in (169), for which C = γ 0 γ 5 . Lower case Latin indices range now from 0 to 3. Using the definition of γ 5 we find that

γ 5 γ ab = 1 2 ε ab ef γ ef ,
where ε abcd is the Levi-Civita or permutation tensor. Given the relation [START_REF] Zakharov | Gravitational Waves in Einstein's Theory[END_REF]

C ef gh = 1 4 ε ab ef ε cd gh C abcd , equation (40) reduces to C ABCD = C abcd (γ 0 γ ab ) AB (γ 0 γ cd ) CD .
Choosing the four-dimensional charge conjugation matrix, C = γ 0 gives the 4d De Smet Weyl spinor

C ABCD = C abcd γ ab AB γ cd CD , (41)
which must be totally symmetric since the spinor we began with is totally symmetric. In 4d, the Weyl tensor has 10 independent components, while a general totally symmetric 4spinor has 35 complex independent components. However, the definition of the 4d De Smet Weyl spinor using the 4d Weyl tensor in equation (39) ensures that it has 10 complex independent components. Put another way, the symmetries of the Weyl tensor in 5d give that the Weyl spinor is totally symmetric, whereas in 4d, the symmetries give more constraints on the spinor, including the condition that it be totally symmetric. The fact that the Weyl tensor is real further constrains the De Smet Weyl spinor via a reality condition that halves its number of real independent components to 10.

In the Petrov classification, the homomorphism between SL(2, C) and the Lorentz group is used to relate chiral spinor and Lorentz indices

X α α = iX a σ a α α, X a = i 2 X α α σa αα , (42)
where α and α are left-handed and right-handed chiral spinor indices, respectively, σ a = (1, σ i ) and σ = (1, -σ).

Using the symmetries of the Weyl tensor and spinor calculus, it can be shown that the spinor equivalent of the Weyl tensor

C αβγδ α β γ δ = σ a α ασ b β β σ c γ γ σ d δ δ C abcd , (43)
is equivalent to a totally symmetric spinor Ψ αβγδ , known as the Weyl spinor in the Petrov classification [START_REF] Penrose | A spinor approach to general relativity[END_REF][START_REF] Stewart | Advanced General Relativity[END_REF]

C αβγδ α β γ δ = Ψ αβγδ ε α β ε γ δ + ε αβ ε γδ Ψ α β γ δ , (44)
where ε αβ = ε α β are the alternating tensors, which can be used to lower undotted and dotted indices, i.e. they act as charge conjugation matrices for chiral spinors. Now, multiplying equation ( 43) with ε α β = -ε α β , and contracting over dotted indices gives

Ψ αβγδ = C abcd ζ ab αβ ζ cd γδ , (45)
where equation ( 44) has been used and

ζ ab αβ = 1 2 ε α β σ a α ασ b β β (46)
are the Lorentz algebra generators. The fundamental theorem of algebra guarantees that Ψ αβγδ factorises

Ψ αβγδ = α (α β β γ γ δ δ) . (47)
The Petrov classification concerns the multiplicity of the factors in (47), with the Petrov types defined in table 1 [START_REF] Stephani | Exact solutions of Einsteins field equations[END_REF].

C ABCD = C abcd γ ab AB γ cd CD ,
we work in a chiral representation given by

γ a = 0 σ a -σ a 0 (48)
in the hope of relating the De Smet Weyl spinor to the Petrov Weyl spinor Ψ αβγδ as defined in (45). The reality condition is

C ABCD = C Ȧ Ḃ Ċ ḊA ȦA A Ḃ B A Ċ C A ḊD , (49)
where

A = 0 -iσ 2 iσ 2 0 . ( 50
)
In the chiral representation, γ ab AB are block diagonal. Thus, using the fact that C ABCD is totally symmetric, we deduce that

C ABCD = (C αβγδ , C α β γ δ ), (51)
where C αβγδ = C abcd γ ab αβ γ cd γδ , i.e. mixed components vanish. C αβγδ and C α β γ δ are related via reality condition (49). But, γ ab αβ =ζ ab αβ , and so

C αβγδ = Ψ αβγδ , (52)
i.e. the undotted part of the De Smet Weyl spinor in four dimensions is the Petrov Weyl spinor. Equivalently, the Petrov polynomial

Ψ αβγδ χ α χ β χ γ χ δ = C ABCD ψ A ψ B ψ C ψ D , (53)
where ψ = χ 0 . We can move to a Majorana representation by performing a similarity transformation such that a Majorana spinor in the chiral representation

III

χ α χ α -→ √ 2 Reχ Imχ ( 54
)
in the Majorana representation. In a Majorana representation, the De Smet Weyl spinor is real. Thus, the De Smet polynomial

C(ψ) = C ABCD ψ A ψ B ψ C ψ D , ( 55
)
where ψ is an arbitrary real (Majorana) spinor given by equation (54), is real. Thus, the De Smet classification in 4d can be viewed as a classification of the Weyl tensor using Majorana spinors, in contrast to the Petrov classification, which uses chiral spinors. Rewriting the polynomial as

C(ψ) = C abcd γ ab (χ)γ cd (χ),
where γ ab (χ) = γ ab AB ψ A ψ B , it can be shown by direct calculation that

γ ab (χ) = ζ ab (χ) + ζ ab (χ) * ,
where

ζ ab (χ) = ζ ab αβ χ α χ β , so that C(ψ) = Ψ(χ) + Ψ(χ) * + 2C abcd ζ ab (χ) * ζ cd (χ),
where

Ψ(χ) = C abcd ζ ab (χ)ζ cd (χ)
is the Petrov polynomial. The tracefree property of the Weyl tensor implies

C abcd ζ ab (χ) * ζ cd (χ) = 0, which implies C(ψ) = Ψ(χ) + Ψ(χ) * . (56)
This can be used to relate De Smet types in four dimensions to Petrov types. For example, assume that the solution is type N. Then,

Ψ(χ) = ω 4 .
(57) Equation (56) gives 39). As discussed before, the symmetries of the 4d Weyl tensor imply not only that the spinor is totally symmetric, but give further conditions.

C(ψ) = ω 4 + ω * 4 = (ω + √ -i ω * )(ω - √ -i ω * )(ω + √ i ω * )(ω - √ i ω * ). (58)
It is these further conditions in addition to the reality condition that constrains the spinor in such a way that it can only admit three types.

For a Euclidean solution, we need to consider the Petrov classification of Euclidean solutions. Starting from the chiral representation of the Lorentzian Clifford algebra (48), we can define a chiral representation for the Euclidean Clifford algebra by setting γ 4 = iγ 0 , i.e. γ a is given by

γ i = 0 σ i σ i 0 , γ 4 = i 0 1 -1 0 ( 59
)
where i = 1, 2 or 3. As with the Lorentzian case, since γ ab AB are block-diagonal and C ABCD is totally symmetric, we conclude that

C ABCD = (Ψ αβγδ , Ψ α β γ δ ), ( 60
)
where Ψ αβγδ and Ψ α β γ δ are the spinor equivalents of the self-dual and anti-self-dual parts of the Weyl tensor. Unlike the Lorentzian case, these two parts are independent of each other and this is manifested in the reality condition. The reality condition on the Weyl spinor is

C ABCD = C Ȧ Ḃ Ċ ḊA ȦA A Ḃ B A Ċ C A ḊD , (61)
where

A = -iσ 2 0 0 iσ 2 , (62)
i.e. for the Euclidean case, A, which defines the reality condition, is block-diagonal (compare this to (50)). Thus, rather than relating the two different parts of the Weyl spinor to one another, the reality condition places conditions on each part separately. Analysing these conditions leads to the Petrov classification of Euclidean geometries [START_REF] Karlhede | Classification of Euclidean metrics[END_REF]. In the Petrov classification of Euclidean metrics, the two independent parts are classified separately and the Petrov type of the geometry is given as a pair consisting of the Petrov type of each part. The reality condition implies that the Petrov types of the self-dual and anti-self-dual parts can only be I, D or O, leading to nine different cases. The relation of these types to the De Smet types of the 4d geometry is given in table 3.

Direct product solutions

In [START_REF] Pravda | Type D Einstein spacetimes in higher dimensions[END_REF], direct and warped product solutions to vacuum Einstein equations are studied in the context of the CMPP classification and it is found that solutions with a one-dimensional Lorentzian factor are necessarily of types G, I, D or O. It is also found that solutions with a two-dimensional Lorentzian factor can only be of types D or O.

Here, we study direct product solutions in the context of De Smet classification. The results below directly generalise to warped product manifolds for which the conformally related product manifold is again an Einstein manifold, since the Weyl tensors are conformally related. Let

g ab (x a ) = g Γ∆ (x Γ ) ⊕ g µν (x µ ), (63)
where g Γ∆ is the metric of a n-dimensional Lorentzian manifold, and g µν is the metric of a (5 -n)dimensional Euclidean manifold. In such a setting, a tensor that splits like the metric-it has no mixed components and components belonging to one submanifold depend only on the coordinates covering that manifold-is known as a product-object or as being decomposable. The Riemann tensor and its contractions are product-objects [START_REF] Ficken | The Riemannian and affine differential geometry of product-spaces[END_REF]. However, the Weyl tensor is in general not decomposable.

Assuming that the spacetime solves the vacuum Einstein equations

R ab = Λg ab ⇐⇒ R Γ∆ = Λg Γ∆ , R µν = Λg µν , (64)
the mixed Weyl tensor components are

C Γ∆Θµ =C Γ∆µν = C Γµνρ = 0, C Γµ∆ν = - Λ 4 g Γ∆ g µν . (65)
The non-mixed Weyl tensor components are

C Γ∆ΘΛ =    0, n = 1, (1)
C Γ∆ΘΛ + (5 -n)Λ 2(n -1) g Γ[Θ g Λ]∆ , n ≥ 2, C µνρσ =    (2) C µνρσ + nΛ 2(4 -n) g µ[ρ g σ]ν , n ≤ 4, 0, n = 4, (66)
where (1) C Γ∆ΘΛ and (2) C µνρσ are the Weyl tensors derived from metrics g Γ∆ and g µν , respectively [START_REF] Pravda | Type D Einstein spacetimes in higher dimensions[END_REF].

For n = 1 and n = 4, the vacuum Einstein equations (64) give that Λ = 0. For n = 4, the Weyl spinor is

C ABCD = (1) C ABCD , (67)
i.e. the De Smet type of the solution is equal to the De Smet type of the four dimensional submanifold. Thus, from section 2.3, we know that the De Smet type can only be one of 4, 22 or 1111. The relation between the De Smet type of the 5d solution and the Petrov type of the 4d factor is given in table 2. The black string is an example of a direct product solution (with n = 4). It is formed from the direct product of 4d type D Schwarzschild solution with a line, which means it is type 22 in De Smet classification [START_REF] De | Five-dimensional metrics of Petrov type 22[END_REF].

For n = 1, the story is similar:

C ABCD = (2) C ABCD , (68)
and the De Smet type can only be one of 4, 22, 1111 or 11 11, as is shown in section 2.3. The relation between the De Smet type of the 5d solution and the Petrov type of the 4d factor is given in table 3.

For n = 2, (1) C Γ∆ΘΛ = (2) C µνρσ = 0. Using a vielbein, the Weyl spinor is

C ABCD = Λ 4 3Γ Γ∆ AB Γ Γ∆CD -4Γ Γµ AB Γ Γµ CD + 2Γ µν AB Γ µν CD . (69)
If Λ = 0, the solution is conformally flat. If Λ = 0, the Weyl polynomial is

C(ψ) = -24Λ(vw -uz) 2 , (70)
where ψ = (u, v, w, z). Hence, such solutions are type 22. This result mirrors that found in [START_REF] Pravda | Type D Einstein spacetimes in higher dimensions[END_REF], where it is shown that n = 2 product solutions can only be types D and O.

For n = 3, as with the n = 2 case, we have (1) C Γ∆ΘΛ = (2) C µνρσ = 0 and the Weyl spinor is

C ABCD = Λ 4 2Γ Γ∆ AB Γ Γ∆CD -4Γ Γµ AB Γ Γµ CD + 3Γ µν AB Γ µν CD . (71)
If the solution is not conformally flat, then Λ = 0 and

C(ψ) = -24Λ(uw -vz) 2 . ( 72
)
Hence, such solutions are also type 22.

3 Connection between tensor and spinor classifications

Connection between tensor and spinor classifications of two-form

A non-vanishing real two-form F can be classified in two different ways. There is the tensorial approach analogous to the CMPP classification of the Weyl tensor whereby one looks for null vector ℓ such that in the null frame (ℓ, n, m i ) with ℓ • n = -1, F 0i = 0. If this is the case, the 2-form is said to be aligned with respect to ℓ and of type I. If no such ℓ exists, then the 2-form is of type G. The 2-form is algebraically special (or type II) if

F 0i = F 01 = F ij = 0.
We also have the spinorial approach outlined in section 2.1, whereby the 2-form is algebraically special if and only if its bispinor equivalent ǫ AB factorises, i.e. ǫ AB = ǫ (A ǭB) . The fact that the factors are conjugates of one another is implied by the reality condition, as shown in section 2.1. We call algebraically special 2-form fields type 11 fields in analogy with the De Smet classification of the Weyl tensor. Type 11 fields can be further classified by considering whether the scalar f = ǭǫ vanishes or not. If the bispinor does not factorise, we say that the field is type 2.

The question we address in this section is how are the different types in the two classification schemes related to one another?

From Fierz identities (10)-( 12), we know that given a 2-form F with bispinor equivalent ǫ AB

ǫ AB = ǫ (A ǭB) =⇒ V a F ab = 0 and F ab F ab = -4V 2
for V a = iǭΓ a ǫ. For the case where V is null, the converse can also be shown by direct computation, i.e. given a null vector V and 2-form F with bispinor equivalent ǫ AB V a F ab = 0 and

F ab F ab = 0 =⇒ ǫ AB = ǫ (A ǭB) .
We prove this by taking a general bispinor, finding its equivalent 2-form F and showing that constraining it as on the left hand side above gives that the bispinor factorises. The reality condition gives that the factorisation must be of the form given on the right hand side and it turns out that V a ∝ iǭΓ a ǫ.

From the above two results, we conclude that the 2-form field F is Type 11

(f = 0) ⇐⇒ Type II, ( 73
) Type 11 (f = 0) =⇒ Type G. ( 74
)
Assume the field F is type 2. From (73), we know that the field cannot be type II. However, there could still exist a null vector ℓ such that in the null frame (ℓ, n, m i ) F 0i = 0, i.e. the field can be type I. If no such ℓ exists, then the 2-form is type G.

These results are summarised in table 4.

Connection between CMPP and De Smet classifications

It is known that the definition of algebraic specialness in the CMPP and De Smet classification schemes do not agree [START_REF] Hervik | Mathematical Cosmology: Bianchi Models, asymptotics and extra dimensions[END_REF][START_REF] Godazgar | Algebraically special axisymmetric solutions of the higher-dimensional vacuum Einstein equation[END_REF]. Indeed, we showed in section 2.4 that the direct product of any type II or III 4d Ricci-flat solution with a line is algebraically general in the De Smet sense, but special in the CMPP sense. Moreover, it is not even the case that the CMPP classification is a refinement, because there are examples that are algebraically special in the De Smet classification and general in the CMPP classification. In this section, we investigate the connection between the two classification schemes.

Relation of CMPP types to De Smet types

We shall proceed by assuming a five dimensional solution to be of particular CMPP type and consider what this means in the De Smet classification. The vielbeins are chosen to be

e 0 = 1 √ 2 (ℓ + n), e 1 = 1 √ 2 (ℓ -n), e î = m i ,
where (ℓ, n, m i ) form a null frame, such that ℓ and n are null, ℓ • n = -1 and m i are a set of d -2 orthonormal spacelike vectors orthogonal to ℓ and n. The implicit Latin letters in the equation above label null frame indices. Now, using equations ( 16) and [START_REF] Penrose | A spinor approach to general relativity[END_REF], where ψ = (u, v, w, z) we derive the Weyl polynomials associated with type D, III and N solutions and consider how they may factorise.

Type N

For type N solutions the only non-zero components of the Weyl tensor are C 1i1j . Using the tracefree property of the Weyl tensor (C i 1i1 = 0), we have five independent Weyl tensor components: C 1313 , C 1414 , C 1213 , C 1214 , and C 1314 . Rotating m i → m ′ i such that C 1i1j is diagonal in the new frame and computing the Weyl polynomial gives

C(ψ) = 8 (2C 1313 + C 1414 )(u 4 + v 4 ) -6C 1414 u 2 v 2 , (75)
which factorises to give

C(ψ) = A(u + av)(u -av)(u + v/a)(u -v/a), (76)
where A = 8(2C 1313 + C 1414 ) and a is given by

a 2 +
C(ψ) =8 (C 1313 -C 1212 )(u 4 + v 4 ) + 6(C 1212 + C 1313 )u 2 v 2 -2 √ 2w[(C 1342 -C 1234 + iC 1343 -iC 1242 )u 3 + 3(C 1434 -iC 1424 )u 2 v + 3(C 1234 + C 1342 + iC 1242 + iC 1343 )uv 2 + (2C 1232 + C 1434 + 2iC 1323 + iC 1424 )v 3 ] + 2 √ 2z[(2C 1232 + C 1434 -2iC 1323 -iC 1424 )u 3 -3(C 1234 + C 1342 -iC 1242 -iC 1343 )u 2 v + 3(C 1434 + iC 1424 )uv 2 -(C 1342 -C 1234 + iC 1242 -iC 1343 )v 3 ] . (77)
Note that if all the coefficients of factors with w or z vanish, then this implies that the solution is type N giving a contradiction. It can be shown that the polynomial may factorise into cubic and linear factors. However, the conditions needed for the polynomial to factorise into two quadratic factors directly imply that one of the quadratic factors must factorise further into linear factors. The polynomial cannot be factorised any further without contradicting the assumption that the solution is type III. Hence, any type III solution must be of De Smet type 4, 31 or 211.

Type D

Finally, we consider type D solutions, for which the non-zero Weyl tensor components are C ijkl , C 0i1j , C 01ij and C 0101 . The nine independent components are chosen to be: C 2323 , C 2424 , C 3434 , C 0123 , C 0124 , C 0134 , C 2324 , C 3234 and C 2434 . We can rotate the spacelike basis vectors m i such that the symmetric part of Φ ij ≡ C 0i1j is diagonal. Then, computing the Weyl polynomial gives

C(ψ) = 16 u 2 [3(Φ 22 -Φ 33 )w 2 + 2(Φ 24 + (2 + i)Φ 34 )wz -(3(Φ 22 + Φ 33 ) + 6iΦ 23 + 4iΦ 24 )z 2 ] + 2uv[-(Φ 24 + (2 + i)Φ 34 )w 2 + 6Φ 44 wz + (Φ 24 + (2 -i)Φ 34 )z 2 ] -v 2 [(3(Φ 22 + Φ 33 ) -6iΦ 23 -4iΦ 24 )w 2 + 2(Φ 24 + (2 -i)Φ 34 )wz -3(Φ 22 -Φ 33 )z 2 ] . (78)
If the polynomial factorises, then, to avoid a contradiction, it does so into two non-factorisable quadratic factors that may or may not be proportional to one another, or into four independent linear factors. Hence, any type D solution must of De Smet type 4, 22, 22 or 1111. As with the previous section, we shall not be able to examine all De Smet types. However, we shall study all algebraically special types, except type 31. All such solutions can be regarded as special cases of type 22 solutions, for which the form of the Weyl tensor is given in equation ( 30)

C abcd =A a[c B d]b + B a[c A d]b -A ab B cd -B ab A cd - 1 2 A ef B ef g a[c g d]b -A ae B e [c g d]b -B ae A e [c g d]b + A be B e [c g d]a + B be A e [c g d]a , (79)
where 2-forms A ab = i tr(Γ ab ǫ) and B ab = i tr(Γ cd η) satisfy one of the following reality conditions

A * ab = A ab , B * ab = B ab (80) or A * ab = B ab . (81)
Also, we shall require results ( 73) and ( 74) and the generalisation of result (73) for 2-forms that do not satisfy any reality condition. If a 2-form F is type I, then it can be shown by direct computation that

ǫ AB = ǫ (A κ B) ⇐⇒ F is type II, ( 82
)
where ǫ AB is the spinor equivalent of F .

Type 22

Working in null frame (ℓ, n, m i ) such that ℓ • n = -1, the +2 boost weight components of the Weyl tensor Ω ij ≡ C 0i0j are of the form

Ω ij = A 0k B 0k δ ij -3A 0(i B |0|j) .
A 0i = 0 or B 0i = 0 is sufficient for Ω ij = 0. However, one can show that it is also necessary.

Ω ij = 0 gives A 0k B 0k δ ij = 3A 0(i B |0|j) . ( 83
)
Assume that neither A 0i nor B 0i vanish. Contracting the above equation with A 0i A 0j gives

A 0k B 0k = 0. ( 84
)
Hence, from equation (83)

A 0(i B |0|j) = 0.
Contracting this with A 0i and using equation (84) gives

A 0i = 0.
But, this contradicts the original assumption that A 0i = 0. Therefore, either A 0i = 0 or B 0i = 0. This means that the solution is type I or more special if and only if A 0i = 0 or B 0i = 0. If there does not exist a ℓ such that A 0i or B 0i vanish then the solution is type G. Now, assume there exists a ℓ such that without loss of generality A 0i = 0.

Given that A 0i = 0, the solution is type II if and only if Ψ ijk ≡ C 0ijk = 0.

Ψ ijk = B 0[j A k]i -A jk B 0i + A 01 B 0[j δ k]i -B 0l A l[j δ k]i .
It can be shown that

Ψ ijk = 0 ⇐⇒ A 01 = A ij = 0, or B 0i = 0. ( 85
)
From the result in equation ( 82), A 01 = A ij = 0 would imply that bispinor ǫ factorises, contradicting the assumption that the solution is type 22. Thus, B 0i must also vanish for the solution to be type II. This would be a further condition if reality conditions (80) are satisfied. However, A 0i = 0 =⇒ B 0i = 0 if reality condition (81) is satisfied.

Given that the solution is type II, then it can be shown that boost weight 0 components vanish if and only if

A 01 = A ij = 0, or B 01 = B ij = 0, ( 86
)
which would imply (using (82)) that one of the bispinors associated with A or B factorises, contradicting the original assumption. However, if there exists an n such that A 1i = B 1i = 0, then the results above apply directly to give that the solution is type D. As before, A 1i = 0 ⇐⇒ B 1i = 0 for reality condition (81).

To summarise, type 22 solutions are of CMPP types G, I, II or D, depending on whether the 2-forms A and B are aligned. An example of a type 22 solution that is algebraically general in the CMPP sense is the 'homogeneous wrapped object' of [12] [11].

Type 22

A special case of type 22 is when A ∝ B. In this case the solution is type 22 and the Weyl tensor is completely determined by 2-form F ; using equation ( 79)

C abcd = 2(F a[c F d]b -F ab F cd -F ae F e [c g d]b + F be F e [c g d]a ) - 1 2 F 2 g a[c g d]b , (87)
where

F 2 = F ef F ef .
The reality condition is simply that F is real. Using the results derived when analysing type 22 solutions, the solution is type II if and only if there exists a ℓ such that F 0i = 0. If no such ℓ exists then the solution is type G. If, in addition, there exists a n such that F 1i = 0, then and only then is the solution type D. Any further constraint on F contradicts the original assumptions. Thus, in summary, type 22 solutions can only be of CMPP type G, II and D. An example of a type 22 solution is the 5d Myers-Perry solution [START_REF] De | Letter: The Petrov type of the five-dimensional Myers-Perry metric[END_REF] (CMPP type D [START_REF] Coley | Algebraic classification of higher dimensional spacetimes[END_REF]). For this solution, the 2-form F that squares to give the Riemann tensor is conformal to a test Maxwell field on it.

Type 211

The solution is type 211 if one of the bispinors, for example η, factorises. Reality condition (81) contradicts the assumption that only one of the spinors factorises. One of the reality conditions in (80) implies that η AB = η (A ηB) (using the results of section 2.1), so that B ab = iηΓ ab η. We can also form a vector from η, V a = iηΓ a η. The result in (73) gives that B 0i = B 01 = B ij = 0 if and only if V is null. Thus, the analysis splits to two cases of V timelike or null.

If V is timelike, then the solution is type I if and only there exists a ℓ such that A 0i = 0. Otherwise, result (74) implies that the solution is type G. Any other constraints on A or B contradict the original assumptions.

If V is null, then choosing ℓ = V gives that the solution is type II. If, in addition, A 0i = 0 then solution is type III. Any further constraints on the 2-forms give contradictions.

To summarise, type 211 solutions can only be of CMPP types G, I, II and III.

Type 1111

For type 1111 solutions, both bispinors ǫ and η factorise, i.e. ǫ AB = ζ (A κ B) and η AB = λ (A µ B) . Using arguments very similar to those used in section 2.2 to show that type 1111 solutions are not possible, one can show that reality condition (81) implies that at least two of the spinors coincide, contradicting the assumption that they are distinct. Type 11 11 solutions are special cases of type 1111 solutions for which two pairs of spinors coincide. However, it is more useful to think of them as special cases of type 22 solutions for which the bispinor ǫ factorises. The reality condition gives that ǫ AB = ǫ (A ǭB) . Thus, the Weyl tensor is determined only from one spinor ǫ. Forming a vector from this, V a = iǭΓ a ǫ, we find that the solution is type G if and only if V is timelike and type N if and only if it is null.

The results found above are summarised in table 6. These results are consistent with those found in section 3.2.1 (see table 5).

De Smet classification of black ring

The results above show that a type G solution could be of any type in the De Smet classification. This can be used to study algebraically general solutions as defined by the CMPP classification. The singly rotating black ring solution [START_REF] Emparan | A rotating black ring solution in five dimensions[END_REF] is a well-known example of a CMPP algebraically general five dimensional solution [START_REF] Pravda | WANDs of the black ring[END_REF]. Therefore, it would be desirable to know the De Smet type of the black ring solution. The metric of the black ring can be written as [START_REF] Emparan | Rotating circular strings, and infinite non-uniqueness of black rings[END_REF][START_REF] Emparan | Black holes in higher dimensions[END_REF]

ds 2 = - F (y) F (x) dt -C R 1 + y F (y) dψ 2 + R 2 (x -y) 2 F (x) dx 2 G(x) - dy 2 G(y) + G(x) F (x) dφ 2 - G(y) F (y) dψ 2 , (88)
where

F (ζ) = 1 + λζ, G(ζ) = (1 -ζ 2 )(1 + νζ), C = λ(λ -ν) 1 + λ 1 -λ ,
The parameters λ and ν are not independent and are related via an equation that will not be given here. Furthermore, they satisfy 0 < ν ≤ λ < 1. The coordinates x and y lie in the ranges -1 ≤ x ≤ 1 and -∞ ≤ y ≤ -1. Asymptotic infinity is at x = y = -1, the ergosurface is at y = -1/λ and inside this is the horizon at y = -1/ν.

Choosing the following vielbein

e 0 = F (y) F (x) dt -C R 1 + y F (y) dψ , e 1 = R x -y F (x) G(x) dx, e 2 = R x -y -F (x) G(y) dy, e 3 = R x -y G(x)dφ, e 4 = R x -y -F (x)G(y) F (y) dψ, (89)
which is well-defined everywhere, except at the horizon and ergosurface, the Weyl polynomial is

C(ψ) = 6(x -y) R 2 F (x) 3 F (y) A 1 (vw + uz)(uw -vz) + i(u 2 -v 2 + w 2 -z 2 )[A 2 (vw + uz) + A 3 (uw -vz)] + A 4 (u 4 + v 4 + w 4 + z 4 ) + A 5 uvwz + A 6 (u 2 z 2 + v 2 w 2 ) + A 7 (u 2 w 2 + v 2 z 2 ) + A 8 (u 2 v 2 + w 2 z 2 ) , (90)
where ψ = (u, v, w, z) and A i are expressions involving x, y, λ and ν given in appendix E. The polynomial above does not factorise, in general. Thus, the solution is algebraically general in the De Smet sense. If we take the static limit ν → λ, for which C = 0, the polynomial is again not factorisable, so the static black ring is also algebraically general.

4 Classification of 2-forms and Weyl tensors defined by a spinor

Algebraically special 2-form fields

In four dimensions, the algebraic specialness of the Weyl tensor of a solution is related to the admittance by the solution of algebraically special electromagnetic fields. The Mariot-Robinson theorem [START_REF] Mariot | Le champ électromagnétique singulier[END_REF][START_REF] Stephani | Exact solutions of Einsteins field equations[END_REF] states that in a null frame defined by null vector field V , a test null electromagnetic field F has only negative boost weight components (F is algebraically special), if and only if V defines a shear-free geodesic null congruence. But the Goldberg-Sachs theorem [START_REF] Goldberg | A theorem on Petrov types[END_REF] states that a vacuum solution admits a shear-free geodesic null congruence if and only if the solution is algebraically special. Thus in 4d the algebraic specialness of a 2-form test field satisfying Maxwell equations and the algebraic specialness of the Weyl tensor coincide. Considering the property of algebraically special p-form fields in higher dimensions could help clarify the status of a higher dimensional generalisation of the Goldberg-Sachs theorem. Some progress has been made in this regard [START_REF] Ortaggio | Higher dimensional spacetimes with a geodesic, shearfree, twistfree and expanding null congruence[END_REF][START_REF] Durkee | Generalization of the Geroch-Held-Penrose formalism to higher dimensions[END_REF] using the CMPP classification.

Here, we consider the algebraic specialness of a 2-form field in the context of the De Smet classification. We define a real 2-form field F to be algebraically special if and only if its bispinor equivalent

ǫ AB = i 8 F ab Γ ab AB , (91) factorises
, i.e. ǫ AB = η (A ǫ B) . (92)
From section 2.1, we find that requiring F to be real implies that η A ∝ ǭA and hence

F ab = iǭΓ ab ǫ. (93)
Furthermore, we can construct a scalar f and vector V (defined in [START_REF] Ortaggio | On asymptotically flat algebraically special spacetimes in higher dimensions[END_REF]) that are related to eachother and F via the Fierz identities (equations ( 10)-( 15)).

The equations of motion for F are given by dF = 0, (94)

d ⋆ F = λF ∧ F, (95)
where λ = 0 corresponds to Maxwell theory and λ = -2/ √ 3 corresponds to minimal supergravity [START_REF] Gauntlett | All supersymmetric solutions of minimal supergravity in five dimensions[END_REF]. The analysis divides naturally into two cases of f zero and non-zero, corresponding to V null and timelike, respectively.

V null

For the null case, we showed in section 3.1 that this is equivalent to the algebraic specialness of F in the CMPP sense. Also, equation ( 14) reduces to

F ∧ F = 0, so that d ⋆ F = 0, (96)
i.e. F solves Maxwell equations. From equation ( 15), we have

F = V ∧ W, (97)
for some 1-form W . Equations ( 10) and ( 13) imply that

V • W = 0 and W 2 = 1,
respectively. Such Maxwell fields have been studied in [START_REF] Durkee | Generalization of the Geroch-Held-Penrose formalism to higher dimensions[END_REF], where it is shown that V is geodesic; W is an eigenvector of the shear matrix and the wedge product of the rotation matrix with W vanishes, i.e.

V • ∇V ∝ V, (98)
and in null frame (V, n, m i ),

ρ ij = ∇ j V i satisfies ρ (ij) W j = ρ 2 W i , (99)
ρ [ij] = 1 2 (Y ∧ W ) ij , (100)
for some 1-form Y .

V timelike

Taking Hodge dual and then exterior derivative of equation [START_REF] Pravda | Type D Einstein spacetimes in higher dimensions[END_REF] gives

λF + d V f ∧ F = 0. (101)
The interior product of the above equation with F implies

4λf 3 + ι F dV F = 2f ι V d V f ∧ V, (102)
and the interior product of this equation with V gives

V • ∇V = V • ∇f f V, (103)
i.e. V defines a timelike geodesic congruence. Taking exterior derivative and then Hodge dual of equation ( 15) and using equations of motion gives

df a V [a F bc] = f ∇ a (V [a F bc] ),
which is equivalent to

f F c[a ∇ c V b] + F c[a ∇ b] V c -df c F c[a V b] -(V • ∇f )F ab = 0.
Interior product of this with F gives

f ∇ • V + V • ∇f = 0. ( 104
)
Introduce coordinates (t, x m ) such that V = ∂/∂t. The metric can then be written as

ds 2 = -f 2 (dt + ω(x p )) 2 + f -1 h mn (t, x p )dx m dx n , (105)
where the manifold with metric h mn will be referred to as the base space. ω is a 1-form with components only on the base space. Equation [START_REF] De | Black holes on cylinders are not algebraically special[END_REF] implies that one can regard F as a 2-form on the base space. Then, equation ( 13) gives

F m p F p n = -δ m n , (106)
where indices have been raised with respect to h mn . Thus, F m n defines an almost complex structure with respect to h mn . Moreover, the Bianchi identity on F implies that F mn is closed, i.e.

(h) ∇ [p F mn] = 0.

(107) Thus, h mn is an almost-Kähler metric. Defining ξ = f -1 V, so that ξ 2 = -1 and ξ • ∇ξ = 0, the expansion, shear and rotation of the timelike congruence are

θ = ∇ a ξ b (g ab + ξ a ξ b ) = -2f -2 V • ∇f, (108)
σ ab = ∇ (a ξ b) - 1 4 θ(g ab + ξ a ξ b ) = f -1 ∇ (a V b) -V (a df b) - 1 4 f -1 θh ab = 1 2 f -2 (L V h ab ), (109)
ω ab = -∇ [a ξ b] = -f -1 ∇ [a V b] , (110)
respectively, where we have used equation (104) in the first line above.. The Bianchi identity and equation [START_REF] De | Black holes on cylinders are not algebraically special[END_REF] give that

L V F = 0. ( 111
)
Rewriting equation (106) as

F mn F pq h np = h mq (112)
and taking Lie derivative with respect to V gives

F m n F q p (L V h np ) = -(L V h mq ) or F m n F q p σ np + σ mq = 0, (113)
i.e. the (1, 1)-part of the shear vanishes. Therefore, for a timelike 2-form field, we have that the solution admits a timelike geodesic congruence and that the base space is almost-Kähler. Furthermore, the (1, 1)-part of the shear vanishes.

Classification of type 11 11 solutions

Type 11 11 solutions are significant in that their Weyl tensors are fully determined from a single Dirac spinor. In this section, we consider type 11 11 solutions and use the Bianchi identity to classify them. The spinor ǫ that defines the type 11 11 solution can be used to form a real scalar, vector and 2-form as follows

f = ǭǫ, V a = iǭΓ a ǫ, F ab = iǭΓ ab ǫ. (114)
As noted in section 2.1, these quantities are related via Fierz identities ( 10)- [START_REF] Pravda | Type D Einstein spacetimes in higher dimensions[END_REF]. Simplifying the general form of the Weyl tensor of type 22 or more special solutions (equation ( 30)) using the Fierz identities, we find that the Weyl tensor of type 11 11 solutions is of the form

C abcd = 2 F a[c F d]b -F ab F cd + V a V [c g d]b -V b V [c g d]a + f 2 g a[c g d]b . ( 115
)
Assuming a vacuum Einstein solution

R ab = Λg ab , (116)
the Bianchi identity reduces to C ab[cd;e] = 0.

We shall use the above equation to find restrictions on the spinor. As in section 4.1, the analysis divides into two cases of V null and timelike.

V null

As shown in section 3.2.2, this class of type 11 11 solutions are type N in the CMPP classification (V is a multiple WAND). Type N solutions to vacuum Einstein equations have been studied before [START_REF] Pravda | Bianchi identities in higher dimensions[END_REF][START_REF] Durkee | Generalization of the Geroch-Held-Penrose formalism to higher dimensions[END_REF]. In [START_REF] Durkee | Generalization of the Geroch-Held-Penrose formalism to higher dimensions[END_REF], it is shown that in the null frame (V, n, m i ), the optical matrix ρ ij = ∇ j V i is of the form

ρ ij = 1 2   ρ a 0 -a ρ 0 0 0 0   ,
and that if the expansion ρ = 0 then a = 0, giving a Kundt solution.

The Fierz identities give

F = V ∧ W, (118)
for some W satisfying V • W = 0 and W 2 = 1. The Weyl tensor is of the form

C abcd = -3F ab F cd + 2(V a V [c g d]b -V b V [c g d]a ). (119)
Then, V b C a bcd;a = 0 and W b C a bcd;a = 0 give

V • ∇V = (∇ • V )V, (120)
i.e. V is geodesic. This has been shown before [START_REF] Pravda | Bianchi identities in higher dimensions[END_REF][START_REF] Godazgar | Algebraically special axisymmetric solutions of the higher-dimensional vacuum Einstein equation[END_REF]. Now, F cd C ab[cd;e] = 0 gives

∇ • V = 0, (121)
i.e. ρ = 0, which means that we have a Kundt solution.

Choosing null frame (V, n, W, m î) where î = 3, 4 and V • n = -1, C a bcd;a = 0 gives

2L 1 ĵ -τ ĵ = 0, (122)
where

∇ a V b = L 11 V a V b + L 1i m (i) a V b + τ i V a m (i) b . (123)
Then, W a C ab[cd;e] = 0 gives

V ∧ dW = 0, ( 124
) which implies V • ∇W ∝ V, and
W • ∇W ∝ V,
where we have used ρ ij = 0. The above equations give that

[V, W ] ∝ V, (125)
which, by Frobenius' theorem, gives that the distribution spanned by {V, W } is integrable. Moreover, using hyper-surface orthogonality of V and equation ( 124), the dual formulation of Frobenius' theorem implies that the distribution spanned by {V, m î} is also integrable. We also have such a structure for the null case in section 4.1.

The Bianchi identity then reduces to

∇ a W b = 1 3 (τ 2 -2L 12 ) m ( î) a m ( î) b -V a n • ∇W b -V a V b (n c n d ∇ c W d ) -n c ∇ a W c V b . (126)
Furthermore, the only non-zero components of the Weyl tensor, Ω ′ ij ≡ C 1i1j are of the form

Ω ′ ij = -3W i W j + δ ij ,
i.e. Ω ′ ij has two equal eigenvalues. The information above does not seem to be strong enough to allow a complete classification of all such solutions. In summary, type 11 11 solutions to vacuum Einstein equations with f = 0 are type N Kundt solutions. In addition, the WAND is in two orthogonal integrable null distributions of dimensions two and three and Ω ′ ij has two equal eigenvalues.

V timelike

Contracting the Bianchi identity with the inverse metric and using the trace-free property of the Weyl tensor gives C a bcd;a = 0. Then F bc C a bcd;a = 0 gives 3f

2 F a d;a + (f df a -V • ∇V a )F ad = 0, (127)
and V c C a bcd;a = 0 gives

f 2 ∇ a V b +(F c a F d b + 2F c b F d a )∇ c V d -∇ • V (f 2 g ab + V a V b ) + f V • ∇f g ab + V a V • ∇V b -f df a V b = 0. ( 128
)
Using the Fierz identities and equation (127) to simplify F ab F ea;b , F ab F cd C ab[cd;e] = 0 reduces to

3 V • ∇f + f ∇ • V = 0, (129)
and a geodesity condition on

V f V • ∇V = (V • ∇f )V . (130)
Using equation ( 128) to simplify V a F cd C ab[cd;e] = 0 gives

∇ (a (f -1 V ) b) = 1 4 ∇ • (f -1 V )(g ab + f -2 V a V b ) (131)
and F ab C ab[cd;e] = 0 reduces to

d(f 1/3 F ) = 2 3 f -3 (V • ∇f ) V ∧ (f 1/3 F ). ( 132
)
To simplify the above equations, define new vector and 2-form

W = f -1 V and J = f 1/3 F. (133)
Then, the equations above give that W forms a shear-free affinely parametrised geodesic congruence

W 2 = -1, W • ∇W = 0, ∇ (a W b) = 1 4 ∇ • W (g ab + W a W b ), (134) while
J satisfies d ⋆ J = 0, dJ = 2 3 f -1 (W • ∇f ) W ∧ J. (135)
Also, we have

dW ab -f -8/3 J c a J d b dW cd = 0. ( 136
)
Introduce coordinates (t, x m ) such that W = ∂/∂t. The metric can then be written as

ds 2 = -(dt + ω(x p )) 2 + f -4/3 k mn (x p , t)dx m dx n , (137)
where the manifold with metric k mn is called the base space, and ω is a 1-form with components only on the base space. J lives on the base space and equation [START_REF] Hervik | Mathematical Cosmology: Bianchi Models, asymptotics and extra dimensions[END_REF] gives that J m n is an almost complex structure on it, i.e.

J m p J p n = -δ m n , (138)
where base space indices have been raised using k mn , and will be done so hereafter. Equation (136) implies that dW is a (1, 1)-form. Written in terms of base space components,

V e C ab[cd;e] = 0 reduces to k m[p dW q]n -k n[p dW q]m -dW mr J r [p J q]n + dW nr J r [p J q]m = 0. ( 139
)
Contracting the above equation with k mp gives dW nq + J m n J p q dW mp + (dW mp J mp )J nq = 0.

Now, contracting this with J nq gives dW mp J mp = 0.

Hence, from equation (140), we have

dW mn + J p m J q n dW pq = 0, (141)
i.e. the (1, 1)-part of dW vanishes. But, we have from before that dW is a (1, 1)-form. This means that dW = 0.

Remaining coordinate freedom can then be used to set

ω(x p ) ≡ 0. (143)
Finally, F bc C ab[cd;e] = 0 gives (h) ∇ p J mn = 0,

i.e. 2-form J is a Kähler form on the base space. Now, taking the exterior derivative of the second equation in (135) and using equations ( 135) and [START_REF] Pravda | Type D Einstein spacetimes in higher dimensions[END_REF] gives

d(f -1 W • ∇f ) a J ab = 0 or d(f -1 W • ∇f ) ∝ W.
Note that equation (129) gives that the expansion of the congruence defined by

W θ = -4f -1 W • ∇f. (145)
Thus, dθ = -(W • ∇θ) W, i.e. the expansion depends only on t: θ = θ(t).

Since W a C abcd = 0, we can think of the Weyl tensor as living on the base space

C mnpq = -2f -2/3 J mn J pq -J m[p J q]n -k m[p k q]n . (147)
Equation [START_REF] De | Five-dimensional metrics of Petrov type 22[END_REF] gives that

ε mnpq = 1 2 (J ∧ J) mnpq , (148)
where ε mnpq = f 8/3 W a ε amnpq is the Levi-Civita tensor on the base space. Equation [START_REF] Pravda | Type D Einstein spacetimes in higher dimensions[END_REF] gives that

J is self-dual ⋆ 4 J = J. (149)
Using the above two equations gives

⋆ 4 C mnpq = 1 2 ε mn rs C rspq = C mnpq , (150)
i.e. the 5d Weyl tensor with components restricted to the base space with metric k is self-dual.

Rewriting the metric as ds 2 = -dt

for some function A(t). Thus, the metric can be written as

ds 2 = -dt 2 + A(t) 2 h mn (x p )dx m dx n , (152)
where the expansion

θ(t) = 4 A ′ (t)/A(t), ( 153
)
and h is a conformally Kähler metric. Defining L by

Λ = 4ǫ L 2 , ǫ ∈ {-1, 0, 1}, (154)
Raychaudhuri's equation

W • ∇θ = - 1 4 θ 2 + Λ, reduces to A ′′ (t) - ǫ L 2 A(t) = 0. The first integral of this equation gives A ′ (t) 2 - ǫ L 2 A(t) 2 = -η, (155)
where we can use the freedom in soaking up constants into metric h to normalise η ∈ {-1, 0, 1}. Using equation (155), Einstein equations reduce to a constraint on the Ricci tensor of metric h

(h) R mn = 3η h mn , (156)
i.e. h is an Einstein metric. Solving equation (155) gives

A(t) η = -1 η = 0 η = 1 ǫ = -1 L sin(t/L) ǫ = 0 t 1 ǫ = 1 L sinh(t/L) e ±t/L L cosh(t/L)
Furthermore, because of the warped product nature of the metric and the duality property of the 5d Weyl tensor derived above, we can learn something about the duality property of the Weyl tensor of metric h. The Weyl tensor of the 5d metric (152) is proportional to the Weyl tensor of the conformally related direct product metric

ds 2 = -d t2 + h mn (x p )dx m dx n ,
where d t = dt/A(t). Using equations derived in [START_REF] Pravda | Type D Einstein spacetimes in higher dimensions[END_REF] and equation (156) one can show that

C mnpq = (h) C mnpq -η h m[p h q]n . (157)
Above, we showed that the 5d Weyl tensor with indices restricted to the 4d base space with metric k is self-dual. The Weyl tensor in the equation above is the 5d Weyl tensor with indices restricted to 4d base space with metric h. By finding the relation between these two Weyl tensors, one can show that the self-duality result above translates to

⋆(C mnpq + η h m[p h q]n ) = C mnpq + η h m[p h q]n , or ⋆ (h) C mnpq = (h) C mnpq , (158)
i.e. the manifold with metric h has self-dual Weyl tensor. Since h is conformal to a Kähler metric k with Kähler form J self-dual (equation ( 149)), this means that the self-dual part of (h) C mnpq is type D [START_REF] Dunajski | Four-dimensional metrics conformal to Kähler[END_REF].

A simpler way of deriving this result is to absorb the Euclidean indices in equation ( 157) by multiplying with gamma-matrices so that

C ABCD = (h) C ABCD -η Γ mn(AB Γ mn CD) , (159)
where we have used the definition of the Weyl spinor given in equation [START_REF] Emparan | A rotating black ring solution in five dimensions[END_REF]. By direct computation, or using the Fierz identity, one can show that Γ mn(AB Γ mn CD) = 0, so that

C ABCD = (h) C ABCD . (160)
Since the solution is type 11 11, we find from the correspondence between the 4d De Smet classification and the Euclidean Petrov classification (table 3) that the solution with metric h is self-dual and of Petrov type (D,O). In summary, all type 11 11 (f = 0) solutions to vacuum Einstein equations are warped product solutions of the form

ds 2 = -dt 2 + A(t) 2 h mn (x p )dx m dx n , (161)
where A(t) is one of the functions in the table above depending on the curvature of the 5d metric and the curvature of the 4d Euclidean metric h. The 4d manifold with metric h is a self-dual Einstein solution 7 . In addition, h is conformal to a Kähler metric k, thus the solution is of Petrov type (D,O). If η = 0, h is a self-dual Ricci-flat metric. Thus, it is hyper-Kähler (see e.g. [START_REF] Dunajski | Solitons, Instantons and Twistors[END_REF]). Note that the ǫ = η = 0 case for which A(t) = 1 corresponds to a direct product solution and agrees with the result found in section 2.4, where it was shown that for type [START_REF] Godazgar | Algebraically special axisymmetric solutions of the higher-dimensional vacuum Einstein equation[END_REF] The Fubini-Study metric on CP 2 is an example of a η > 0 type (D,O) Einstein solution. A η < 0 example is the Bergman metric on complex hyperbolic space CH 2 . An important example of a type D hyper-Kähler metric is the Euclidean Taub-NUT solution. The Λ = 0 5d solution with metric (161), where h is the metric of Euclidean Taub-NUT is a magnetic monopole solution of Kaluza-Klein theory [START_REF] Gross | Magnetic monopoles in Kaluza-Klein theories[END_REF].

A 5d Clifford algebra

In five dimensions, the Clifford algebra is {Γ a , Γ b } = 2g ab .

(162)

The gamma-matrices have spinor index structure (Γ a ) A B . Given a Dirac spinor ψ, we can define its Majorana and Dirac conjugates as

ψ C = ψ t C (ψ B = ψ A C AB ) (163)
and ψ = ψ † B ( ψA = ψ ȦB ȦA ) (164)

respectively, where C is the charge conjugation matrix, defined by

Γ t a = CΓ a C -1 , ( 165
)
and B is the Dirac conjugation matrix, defined by

Γ † a = -BΓ a B -1 . ( 166
)
Note that ψ Ȧ ≡ ψ * A . It follows from Schur's lemma that B and C are unique up to a phase factor. Moreover, B is Hermitian or anti-Hermitian, where we are free to choose which and C is antisymmetric [START_REF] Lee | Spinors in arbitrary dimensions[END_REF]. Also, from the definition of Clifford algebra, we find that Γ 0 is anti-Hermitian, while Γ i are Hermitian. Thus, from equation (166), we find that B is Γ 0 up to a phase. We choose B Hermitian so that assignment of indices is consistent, i.e. B ȦA = (B t ) ȦA = B A Ȧ.

The Majorana condition is ψC -1 = ψ or ψ * = Aψ (ψ Ȧ = A ȦB ψ B ), (167) up to a phase, where A = (CB -1 ) t . In 5d, the Majorana condition has no non-trivial solutions, i.e. it implies that ψ = 0. This is because A * A = -1.

A convenient representation to use for the five dimensional Clifford algebra is to start with the Majorana representation for the four dimensional Clifford algebra

γ 0 = -iσ 2 0 0 iσ 2 , γ 1 = σ 1 0 0 σ 1 , γ 2 = σ 3 0 0 σ 3 , γ 3 = 0 iσ 2 -iσ 2 0 , (168)
where σ î for î = 1, 2, 3 are the usual Pauli matrices and add γ 5 = γ 0 γ 1 γ 2 γ 3 . Thus, Γ a = (γ 0 , γ î, iγ 5 ).

The five dimensional Fierz identity is

M AB N CD = 1 4 C AD (N M ) CB + 1 4 Γ eAD (N Γ e M ) CB - 1 8 Γ ef AD (N Γ ef M ) CB . (171)
For brevity, we omit factors of C and C -1 where it is clear that indices have been lowered or raised.

B Solving reality condition for 2-form

In this appendix, we outline how one can prove that the reality condition Letting B = 1, 2, 3, 4 gives four equations relating ǫ and η. Now, the analysis splits into three cases.

The first case is that some coefficients in the equations are non-zero. This implies ǫ ∝ η, which gives a contradiction as shown above. The second case is that all coefficients in each of the equations vanish. But, it can be shown that this too implies that ǫ ∝ η. Thus, we are left with the final case that the coefficients in the equation above vanish. Since α = 0, we have β = 0. Then, equation (174) gives η ∝ ǫ or η ∝ ǭ.

(176)

C Solving reality condition for type 22

In this appendix, we outline how one can prove that the reality condition for type 22 solutions ǫ (AB η CD) = ǭ(AB ηCD) ,

implies either ǫ AB = ǭAB , η AB = ηAB ,

or

ǫ AB = ηAB , (179)
where ǫ and η are non-zero bispinors, and

ǭAB ≡ ǫ Ȧ Ḃ A ȦA A Ḃ B .
The strategy used for the proof here is similar in nature to that used to prove the result in appendix B, except that there are more cases to consider.

Fig. 1 :

 1 Fig. 1: Penrose diagram of the CMPP classification

Fig. 2 :

 2 Fig. 2: The 12 different algebraic types in the spinor classification.

Fig. 3 :

 3 Fig. 3: Revised figure showing the 8 different algebraic types in the spinor classification.

Fig. 4 :

 4 Fig. 4: Penrose diagram of the Petrov classification

  11 direct product solutions, the Euclidean base space has Petrov type (D,O) or (O,D) depending on the choice of orientation. It is simple to show the converse, i.e. that the uplift of all type (D,O) Einstein solutions with metric (161) are type 11 11. Equation (160) gives that the De Smet type of the 5d solution coincides with the De Smet type of the 4d base space. Table 3 gives that the De Smet type of the 5d solution is 11 11. Therefore, we have that a solution is type 11 11 (f = 0) ⇐⇒ a cosmological solution with spatial geometry a type (D,O) Einstein solution (metric (161)).

  0 , C = γ 0 γ 5 and A = -iγ 5 .

ǫ

  (A η B) = ǭ(A ηB) ǫ and η are non-zero spinors. Letting B = 1, 2, 3, 4 in (172) gives four equations of the formǫ 1 η A + η 1 ǫ A = ǭ1 ηA + η1 ǭA .If all four of the equations are dependent, that is they are proportional to one another, then by considering all possible cases one can show that 8 ǫ ∝ η.Then, (172) becomesǫ (A ǫ B) ∝ ǭ(A ǭB) .Since ǫ is non-zero, assume without loss of generality that ǫ 1 = 0. Then letting B = 1 in the equation above gives ǭ ∝ ǫ, which is a Majorana condition on ǫ. The Majorana condition has no non-zero solutions in five dimensions. Therefore, ǫ ∝ η implies a contradiction. Now, assuming that two of the equation are independent givesη = α ǫ + β η,(174)where α and β are constants and α = 0, otherwise we have a Majorana condition on η. Taking complex conjugate of the above equation and multiplying appropriately byA gives ǭ = -1 α * αβ * ǫ + (1 + |β| 2 ) η .(175)Substituting equations (174) and (175) into equation (172) givesα 2 β * ǫ (A ǫ B) + (α + α * + 2α|β| 2 ) ǫ (A η B) + β(1 + |β| 2 )η (A η B) = 0.

11 2 = ǭ 2 11 .

 1111 First consider the case η ∝ ǫ. Then, equation (177) becomes ǫ (AB ǫ CD) = ǭ(AB ǭCD) . (180) Assume ǫ 11 = 0. This is true unless ǫ AA = 0 for all A (no sum on A), in which case, it can be shown that ǫ AB ∝ ǭAB , as required. Letting A = B = C = D = 1 in equation (180) gives ǫ Now, letting B = C = D = 1 in equation (180) and using the equation above gives ǫ A1 = ±ǭ A1 . Finally, letting C = D = 1 in equation (180) and using the two equations above gives ǫ AB = ǭAB . (181) Thus, we have proved the result above for η ∝ ǫ. Note that this is equivalent to solving the reality condition for type 22 solutions. Now, assume η ∝ ǫ. Letting B, C, D = 1, 2, 3, 4 in equation (177) with at least two of them coinciding gives 16 equations of the form ǫ A1 η 11 + ǫ 11 η A1 = ǭA1 η11 + ǭ11 ηA1 , or ǫ A2 η 11 + ǫ A1 η 21 + ǫ 21 η A1 + ǫ 11 η A2 = ǭA2 η11 + ǭA1 η21 + ǭ21 ηA1 + ǭ11 ηA2 . If three of the 16 equations are not independent, then by considering all possible cases it can be shown that η ∝ ǫ, which contradicts the original assumption that η ∝ ǫ. Therefore, at least 15 of the equations are independent. They can be used to express ǫ A2 , . . . , η A1 , . . . , ǭA1 , . . . , ηA1 , . . . , ηA4 in terms of ǫ A1 . Now, let C = D = 1, 2, 3, 4 in equation (177) to give 4 equations of the form ǫ AB η 11 + ǫ A1 η B1 + ǫ B1 η A1 + ǫ 11 η AB = ǭAB η11 + ǭA1 ηB1 + ǭB1 ηA1 + ǭ11 ηAB .

Table 1 :

 1 The Petrov classification of the Weyl tensor

	Petrov type Multiplicities Diagram
	I	(1,1,1,1)
	II	(2,1,1)
	D	(2,2)
	III	(3,1)
	N	(4)
	O	(C abcd ≡ 0)

table 2 .

 2 What we find is that type I, II and III solutions are all algebraically general in the 4d De Smet classification, while type D and N solutions are type 22 and 1111, respectively. Thus, the De Smet or Majorana spinor classification of the Weyl tensor in 4d is a coarse version of the Petrov or chiral spinor classification.

Hence, type N solutions are type 1111 in the 4d De Smet classification.

We can consider the other Petrov types in a similar manner. The results are summarised in

Table 2 :

 2 Relation of De Smet classification in 4d to Petrov classification

	De Smet type Petrov types
	4	I, II, III
	22	D
	1111	N

The reason why other De Smet types are not possible goes back to the definition of the 4d De Smet Weyl spinor via the 4d Weyl tensor, in equation (

Table 3 :

 3 De Smet classification and Petrov classification of 4d Euclidean metrics

	De Smet type	Petrov types
	4	(I,I), (I,D), (D,I)
	22	(D,D)
	1111	(I,O), (O,I)
	11 11	(D,O), (O,D)

Table 4 :

 4 Relation between spinor and tensor types for 2-form

	Spinor type Possible tensor types
	2	G, I
	11	G, II

  For type III solutions, the non-zero components of the Weyl tensor are C 1i1j , C 1ijk and C 011i . The thirteen independent components are chosen to be: C 1212 , C 1313 , C 1213 , C 1214 , C 1314 , C 1234 , C 1342 , C 1232 , C 1242 , C 1323 , C 1343 , C 1424 and C 1434 . Again, rotating the frame as before so that C 1i1j is diagonal and computing the Weyl polynomial gives

	Type III
	1/a 2 =	6C 1414 2C 1313 + C 1414	,

assuming that 2C 1313 + C 1414 = 0. If 2C 1313 + C 1414 = 0, then C(ψ) ∝ u 2 v 2 ,

which means the solution is type 11 11. Equation (76) implies that type N solutions are type 1111 or more special. If a = ±1, then the solution is of type 11 11. Hence, any type N solution must be of De Smet type 1111 or 11 11.

Table 5 :

 5 Possible De Smet types given CMPP type

	CMPP type Possible De Smet types
	D	4, 22, 22, 1111
	III	4, 31, 211
	N	1111, 11 11
	3.2.2 Relation of De Smet types to CMPP types

Now, we shall consider the different De Smet types and study what possible CMPP types they imply.

Table 6 :

 6 Possible CMPP types given De Smet type AB = ǫ (A ǭB) and η AB = η (A ηB) . As above, we can form two vectors, V a = iǭΓ a ǫ and W a = iηΓ a η.The solution is type G if and only if V and W are timelike. If only one of the vectors is null, then and only then is the solution type II. The solution is type D if and only if both vectors are null but not proportional to one another. Finally, the solution is type N if and only if both vectors are null and proportional to one another.Thus, type 1111 solutions are of CMPP types G, II, D or N.

	De Smet type Possible CMPP types
	22	G, I, II, D
	22	G, II, D
	211	G, I, II, III
	1111	G, II, D, N
	11 11	G, N
	Reality conditions (80) give that ǫ Type 11 11

  2 + hmn (t, x p )dx m dx n , where we have used the fact that W is hyper-surface orthogonal and the result in (146) that the expansion is only time-dependent. The equation above can be integrated for each component to give hmn (t, x p ) = A(t) 2 h mn (x p ),

	the final equation that gives the shear-free property of the congruence defined by W in (134) reduces
	to	∂ t hmn =	1 2	θ(t) hmn ,

In[START_REF] Pravda | WANDs of the black ring[END_REF], it is shown that WANDs can only be found in certain regions for the black ring and it is claimed that the black ring is type I. However, if we take the strict definition of the classification, which states that the algebraic type of the spacetime corresponds to the type of its most algebraically general point, then the black ring is type G. The black ring is an example of a solution that is type G in one open region and type I in another. This kind of behaviour is discussed in[START_REF] Godazgar | Algebraically special axisymmetric solutions of the higher-dimensional vacuum Einstein equation[END_REF].

See appendix A for conventions used for the 5d Clifford algebra.

A sketch of the proof of this result is given in appendix B.

The fact that CABCD as defined by equation (16) is totally symmetric depends very much on properties of the

5d Clifford algebra and the 5d Fierz identity. At least, with regard to the antisymmetry property of C, which is crucial in ensuring that CΓ ab is symmetric, this does not hold in d = 7, 8, 9 mod 8[START_REF] Lee | Spinors in arbitrary dimensions[END_REF]. That is, in these dimensions, a representation of the Clifford algebra for which C is antisymmetric does not exist.

A sketch of the proof of this result is given in appendix C.

See appendix D for the derivation of the form of the Weyl tensor of type 22 or more special solutions.

In 4d, self-dual Einstein solutions are quaternion-Kähler[START_REF] Tod | Cohomogeneity-one metrics[END_REF]. The holonomy group of a quaternion-Kähler solution is a subgroup of Sp(1) 2 ∼ = SU (2) 2[START_REF] Berger | Sur les groupes d'holonomie des varits a connexion affine et des varits Riemanniennes[END_REF].

In fact it is enough for only a pair of the equations above to be proportional to one another for this result to hold. We shall not use this, since we would like the sketch of the proof in this section to mirror that given in appendix C.

Acknowledgements

I would like to thank H. S. Reall for suggesting this project and reading through a draft manuscript. His many comments and suggestions have been invaluable throughout. I would also like to thank M. Dunajski, M. Durkee, D. Kubiznak, H. Godazgar and S. Hervik for discussions. I am especially grateful to G. W. Gibbons for helpful discussions and advice, particularly with regard to the spinor classification and its relation to the Petrov classification. I am supported by EPSRC.

Similar to before, if two of these four equations are not independent then it can be shown by considering all the different possibilities that η ∝ ǫ, which contradicts the original assumption. Thus, three of the equations are independent, which means we can eliminate ǫ A1 , ηAB and use the last equation to show that ǭ = α ǫ + β η,

where α and β are constants and if β = 0, it can be shown that η ∝ η.

Thus β = 0 gives one of the possibilities allowed above: ǭ ∝ ǫ and η ∝ η. Furthermore, α = 0 gives the second possibility: ǭ ∝ η. Using equations ( 182) and (177), one can show that

and

Assume α = 0 and β = 0. Taking complex conjugate of equation ( 182) and multiplying appropriately by a pair of A's gives

Substituting equations ( 182) and (186) into equation (177) gives

where

Using the first equivalence in (185) gives that λ = 0 implies that α = 0 or β = 0, which contradicts the original assumption. This is also trivially true for ν = 0. µ = 0 implies

Multiplying the equation above with its complex conjugate gives

Thus, µ = 0 also contradicts the original assumption that αβ = 0. Letting B = C = D = 1, 2, 3, 4 in equation ( 187) gives four equations of the form

The equations are independent unless the coefficients vanish. If this is the case, then we have

It can be shown that this implies that α = 0, which contradicts the original assumption. Thus, by considering different components of the four equations, one can show that, in general, they imply that η ∝ ǫ, which contradicts the original assumption. Although, one must also consider special cases, where, for example, ǫ A1 = 0 only for A = 1. However, in these cases too, one can show that η ∝ ǫ. Therefore, αβ = 0 contradicts the original assumption, which implies that αβ = 0.

D Weyl tensor of type 22 or more special solutions

In this appendix, we use the 5d Fierz identity to derive the form of the Weyl tensor of type 22 or more special solutions. For all such solutions, the Weyl spinor is of the form

We can invert the definition of the Weyl spinor (equation ( 16)), so that given a Weyl spinor, the associated Weyl tensor is given by

Using the form of the Weyl spinor (188) and equation ( 189), the Weyl tensor is

where A ab = i tr(Γ ab ǫ) and B ab = i tr(Γ cd η), and ǫ and η have been rescaled. Using the 5d Fierz identity (equation ( 171)) with M = Γ ab , N = Γ cd and using the fact that C and CΓ a are antisymmetric in their spinor indices, while CΓ ab and η are symmetric gives

Using the Fierz identity with M = Γ cd , N = Γ ef , contracting two spinor indices between Γ cd and Γ ef , and multiplying by Γ ab gives an expression for (Γ cd Γ ef Γ ab ) CB , which when inserted into the equation above gives

Again, using the Fierz identities in a similar way to that used to derive equation (191) and properties of gamma-matrices in 5d, in particular that

gives

Equation (190) then gives

i.e. the Weyl tensor of type 22 solutions is determined by two 2-forms A and B.

E Weyl polynomial of black ring

The Weyl polynomial of the singly rotating black ring (88), using the tetrad given in (89), is

where ψ = (u, v, w, z) and