2.4. ASIDE: ALTERNATIVE METHODS OF CLASSIFICATION 39 Theorem 2.10 ([132]) Let 𝑔 be a Kerr-Schild metric, i.e. one that can be written in the form
𝑔𝜇𝜈 =𝜂𝜇𝜈+𝑘𝜇𝑘𝜈 (2.29)
for some conformally flat metric 𝜂𝜇𝜈 and null vector 𝑘𝜇. Then 𝑔 is algebraically special with multiple WAND tangent to𝑘.
It is well known that Myers-Perry and Kerr-(A)dS black holes can be written in Kerr- Schild form (indeed, this is how they were originally constructed [62]), and hence they are algebraically special. They have multiple WANDs that are expanding everywhere outside the horizon. Furthermore:7
Theorem 2.11 ([131]) A stationary spacetime admitting an expanding multiple WAND is Type D (or conformally flat).
Hence, such black holes are Type D outside the horizon, and hence also on the horizon (by continuity).
In fact, it is reasonably straightforward to show that a spacetime is algebraically special on any null Killing horizon, with a multiple WAND tangent to the null generators of the horizon.8 However, it is well known that the null generators are non-expanding on the horizon, and hence the conditions of Theorem 2.11 fail there. Hence, there is no inherent reason that a black hole spacetime (e.g. the black ring) that is algebraically special only on the horizon should be Type D there.
2.4.1 Bivector methods
Coley & Hervik [127] generalized the bivector classification to arbitrary dimension. The bivector map C defined by equation (2.13) is valid in any dimension. However, note that it is only in four dimensions that Hodge duality provides a map from bivectors to bivectors, and hence the self-duality structure that we then imposed on bivectors cannot be extended to higher dimensions.
Despite this, one can construct a natural bivector classification in arbitrary dimension by classifying the eigenvalue structures (e.g. Segre types) of the operator C. In fact, the authors of Ref. [127] chose to describe their classification in terms of the CMPP classification for ease of comparison, and found that even in higher dimensions there are still some links between the bivector and boost weight classifications. For example, it can be shown that
Lemma 2.12 ([127]) A spacetime is of CMPP Type III, N or O if and only if the bivector operator is nilpotent.
If a spacetime is CMPP Type II, then the bivector operator has at least 3 pairs of matching eigenvalues.
Recent work [125] has given a concrete way of computing the eigenvector structure of the bivector operator for a given spacetime, in terms of conditions on a particular series of
‘discriminants’, derived from scalar invariants of various curvature operators. However, the potential applications of this approach have not yet been explored in great detail.
2.4.2 Spinorial methods
An entirely different approach to a higher-dimensional generalization of the Petrov clas- sification was given by De Smet [120]. His work attempts to generalize the 4D spinorial approach. However, there are no 2-component spinor representations of the Lorentz group in 5D. For this reason, de Smet’s work uses a particular Dirac spinor representa- tion of the 5D Clifford algebra. A clear exposition of this approach is given by Godazgar [133], who also notes that an analogous approach to algebraic classification can be used in four dimensions, but that it gives a different classification scheme to the others discussed above.
Using such a representation Γ𝑎, a spinor conterpart of the Weyl tensor can be defined as:
𝐶𝐴𝐵𝐶𝐷 =𝐶𝑎𝑏𝑐𝑑Γ𝑎𝑏𝐴𝐵Γ𝑐𝑑𝐶𝐷 (2.30) where Γ𝑎𝑏 = Γ[𝑎Γ𝑏]. The motivation behind the particular choice of representation is that it renders 𝐶𝐴𝐵𝐶𝐷 totally symmetric. It is not possible to make such a choice in all
2.4. ASIDE: ALTERNATIVE METHODS OF CLASSIFICATION 41 spacetime dimensions.
The symmetry allows the construction of a Weyl polynomial
𝐶(𝜓) =𝐶𝐴𝐵𝐶𝐷𝜓𝐴𝜓𝐵𝜓𝐶𝜓𝐷 (2.31) for 4-spinors𝜓𝐴. They have four components, so this is a homogeneous quartic polyno- mial in 4 variables, which is not guaranteed to factorise. If it does, then the spacetime is algebraically special in the de Smet classification.
This notion of algebraically special is distinct from the notion of algebraically spe- cial in the CMPP classification. For example, the product of any 4D Petrov Type III spacetime with a flat direction is Type III in the CMPP classification, but algebraically general in the de Smet sense [133].
The de Smet classification can be refined further, giving a list of possible algebraic types according to the way in which the quartic polynomial factorises. We use notation where a number represents the degree of a polynomial factor, and underlining a set of factors indicates that they are repeats of each other. Naively, there are 12 allowed types:
4 (no factorisation, algebraically general), 22, 31, 211, 22, 1111, 211, 1111, 1111, 1111, 1111, 0 (where the last option corresponds to a conformally flat spacetime).
However, the complex spinor 𝐶𝐴𝐵𝐶𝐷 has 70 independent real components, while the Weyl tensor only has 35 independent components in 5 dimensions. Godazgar [133]
shows how to impose the appropriate reality condition on𝐶𝐴𝐵𝐶𝐷 to halve the number of independent components. After the imposition of this condition, he shows that four of the de Smet types cannot occur, reducing the allowed types to 4, 31, 22, 22, 211, 1111, 1111, 0.
Some examples of spacetimes that are algebraically special in this classification in- clude:
∙ Schwarzchild-Tangherlini black holes [61] are Type 22. [134]
∙ Singly-spinning Myers-Perry black holes [62] are Type 22. [134]
∙ BMPV black holes [135] are Type 22. [136]
∙ Singly-spinning black rings [68] are Type 4 (algebraically general) [133].
2.4.3 Type D spacetimes and hidden symmetry
In four dimensions, there are strong links between Petrov Type D spacetimes, and the hidden symmetry structures discussed in Section 1.3.1. It is known that:
Theorem 2.13 ([26, 137, 27]) In four dimensions, every Petrov Type D vacuum so- lution admits a conformal Killing tensor. All Petrov Type D vacuum solutions with the exception of the generalized C-metric admit a rank-2 Killing tensor, and an associated Killing-Yano 2-form.
Conversely,
Theorem 2.14 ([138, 139, 27]) A vacuum spacetime admitting a non-degenerate con- formal Killing-Yano 2-form is Petrov Type D.
These results have been partially generalized to higher dimensions. It is known that:
Theorem 2.15 ([140]) A 𝑑-dimensional vacuum spacetime admitting a closed, non- degenerate conformal Killing-Yano 2-form is Type D in the CMPP classification.
However, there is no converse result; it is not known whether all Type D vacuum solutions admit a conformal Killing tensor. Attempting to prove this in the same way as the four- dimensional result does not work, as it requires the use of the Goldberg-Sachs theorem (which we will discuss in detail later).
Furthermore, in four dimensions all Type D solutions were constructed explicitly by Kinnersley [141]. In higher dimensions, this has not been done, and it is far from clear that finding all such solutions is likely to be possible. On this basis, it has been suggested [142] that perhaps the natural generalization of the Type D class of metrics to higher dimensions is actually those metrics satisfying the assumptions of Theorem 2.15. There is some merit in this suggestion; Krtouˇset al. [143] (generalizing work of Houri et al.
[144]) are able to explicitly construct all metrics satisfying these conditions. However, we will see later in the thesis that the more general class of metrics that are algebraically special in the CMPP classification also have useful general properties, which seems to motivate this less restrictive definition.