These instabilities are of a qualitatively different nature to those found in [109], appearing at a lower value of angular momentum, and breaking more of the symmetry of the original solution.
Despite this recent progress, performing an analysis of the linearized stability of general Myers-Perry black holes seems to be extremely difficult. Though the principles of doing this are well understood, doing it in practice is not easy. The equations of motion involved in these perturbations are extremely complicated, which hinders attempts to extract information from them analytically, whilst the large parameter space makes numerical approaches time consuming. Things are even worse in the case of black rings [68, 69, 70], for which there are physical arguments for various kinds of instabilities [112, 113] but little in the way of concrete results.
1.8. NEW RESULTS OF THIS THESIS 23 However, the analogous equation does decouple in the near-horizon geometry of any extreme vacuum black hole, and in Chapter 5 we are able to use this equation to conjec- ture information about instabilities of black holes in arbitrary dimension. In particular, we show that the equations for linearized perturbations of the near-horizon geometry can be reduced to the equation of motion for a charged, massive scalar field in 𝐴𝑑𝑆2. A generalized Breitenl¨ohner-Freedman stability bound can be defined for such fields.
We conjecture that if there exist perturbation modes that violate this bound, then the full black hole geometry will be unstable, provided that the unstable modes obey a certain symmetry condition. Although this only allows us to study a limited class of perturbations, it allows progress to be made without resorting to numerics, and offers the possibility of making general statements about stability in arbitrary dimension. We provide evidence for this conjecture by comparing our results with those obtained by numerical work in a few particular cases, and find good agreement.
The final chapter of the thesis has a rather different flavour, studying properties of a particular solution to the Einstein equations in five dimensions: the Pomeransky- Sen’kov doubly spinning black ring [70]. We will see that the Hamilton-Jacobi equation describing geodesic motion admits separable solutions in the case of null, zero energy geodesics. Given the very complicated metric describing such black ring spacetimes, this is something of a surprise. However, we are able to give some insight into this separability by showing that the black ring admits a novel form of hidden symmetry.
While the full spacetime does not admit a conformal Killing tensor, one can make a Kaluza-Klein dimensional reduction to obtain a four-dimensional spacetime that does admit such a tensor.
Chapter 2
Algebraic classification and null frames
2.1 Introduction
When looking to find out more about gravity in higher dimensions, it is natural to try to generalize mathematical methods that have proved powerful in four dimensions.
The algebraic classification of spacetimes, first considered by Petrov [116], is one example of such a method. Such classification played a crucial role in understanding various aspects of four-dimensional GR. For example, Kerr made use of it in order to construct the metric describing a rotating black hole [24], while the asymptotic behaviour of gravitational radiation can be conveniently understood in this language (see, e.g.
[117]).
The basic idea behind algebraic classification is to divide spacetimes into different types, in order to prove general results about the properties of a precisely defined set of spacetimes. The schemes discussed below only say useful things about a few partic- ular spacetimes; the reason that they are useful is that these include various important examples, such as the Kerr black hole and pp-waves.
There are at least four distinct approaches to defining such an algebraic classifica- tion. Roughly speaking, the four approaches make use of null vectors, 2-spinors, scalar invariants and bivectors. In four dimensions, perhaps surprisingly, all of these methods can be used to give different descriptions of the same classification. In Section 2.2 we will briefly review these various approaches.
For each technique, it is possible to define (at least one) generalization to higher dimensions. However, the generalisations are typically not equivalent to each other, and lead to distinct notions of an algebraically special spacetime. The focus of much of
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this thesis will be on a null vector based generalization of these classification schemes to higher dimensions, defined in 2001 by Coley, Milson, Pravda & Pravdov´a (CMPP) [118, 119]. This will be introduced in Section 2.3, after which we will briefly review some other higher-dimensional classification schemes, including the spinorial de Smet classification [120].
As we shall see below, algebraically special spacetimes are partly characterized by the existence of preferred null directions. Therefore, it is useful to introduce computational techniques built around one or two particular null directions. In four dimensions, the Newman-Penrose (NP) [121] and Geroch-Held-Penrose (GHP) [114] formalisms are two related examples of such techniques. Higher-dimensional versions of these approaches will be discussed in detail in Sections 2.5 and 2.6. The higher-dimensional generalisation of the NP formalism was developed by various authors (see e.g. [122, 123, 124]), while myself, Pravda, Pravdov´a & Reall [4] constructed a higher-dimensional version of the GHP formalism.