Black holes, hidden symmetries, and complete integrability
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Abstract
The study of higherdimensional black holes is a subject which has recently attracted vast interest. Perhaps one of the most surprising discoveries is a realization that the properties of higherdimensional black holes with the spherical horizon topology and described by the Kerr–NUT–(A)dS metrics are very similar to the properties of the well known fourdimensional Kerr metric. This remarkable result stems from the existence of a single object called the principal tensor. In our review we discuss explicit and hidden symmetries of higherdimensional Kerr–NUT–(A)dS black hole spacetimes. We start with discussion of the Killing and Killing–Yano objects representing explicit and hidden symmetries. We demonstrate that the principal tensor can be used as a “seed object” which generates all these symmetries. It determines the form of the geometry, as well as guarantees its remarkable properties, such as special algebraic type of the spacetime, complete integrability of geodesic motion, and separability of the Hamilton–Jacobi, Klein–Gordon, and Dirac equations. The review also contains a discussion of different applications of the developed formalism and its possible generalizations.
Keywords
General relativity Higher dimensions Black holes Kerr–NUT–(A)dS Hidden symmetries Principal tensor Complete integrability Separability1 Introduction
1.1 Black holes in four and higher dimensions
The study of fourdimensional black holes has begun long time ago. Their detailed characteristics were obtained in the 1960s and 1970s, also known as the “golden age” of the general relativity. A summary of the obtained results can be found, for example, in the books by Misner et al. (1973), Wald (1984), Hawking and Ellis (1973), Chandrasekhar (1983), Frolov and Novikov (2012), Frolov and Zelnikov (2011). According to the proven theorems, assuming the weak energy condition for the matter, the black hole horizon has to have spherical topology. The most general stationary vacuum black hole solution of the Einstein equations is axially symmetric and can be described by the Kerr metric.
The interest in fourdimensional black holes is connected with the important role these objects play in modern astrophysics. Namely, there exist strong evidences that the stellar mass black holes manifest themselves in several Xray binaries. Supermassive black holes were discovered in the centers of many galaxies, including our own Milky Way. Great discovery made by LIGO on September 14, 2015 gives first direct confirmation that strong gravitational waves have been emitted in the process of the coalescence of two black holes with masses around 30 solar mass (Abbott et al. 2016). Three month later LIGO registered gravitational waves from another merging black hole binary. These events marked the beginning of the gravitational waves astronomy. In all previous observations the information concerning astrophysical black holes was obtained by registering the electromagnetic waves emitted by the matter in the black hole vicinity. Such matter usually forms an accretion disc whose temperature and size are determined by the mass and angular momentum of the black hole. Before reaching a distant observer, the emitted radiation propagates in a strong gravitational field of the black hole; to extract the information contained in astrophysical observations one needs to solve the equations for particle and wave propagation in the Kerr spacetime. Fortunately, the remarkable properties of this geometry, namely the complete integrability of geodesics and the separability of wave equations, greatly simplify the required calculations. Based on these results there were developed powerful tools for studying physical effects in the black hole vicinity and their observational manifestation. Similar tools were also used for the study of quantum evaporation of miniblack holes.
In this review we mainly concentrate on black holes in dimensions greater than four, with a particular focus on their recently discovered remarkable geometric properties. Black holes in higher dimensions, see e.g., Emparan and Reall (2008), Horowitz (2012) for extended reviews, have attracted much attention for several reasons. A first reason is connected with the development of string theory and the demand for the corresponding black hole solutions. In order to make this theory consistent one needs to assume that besides usual four dimensions there exist (at least six) additional spatial dimensions.
A second reason stems from the (in past 20 years very popular) braneworld models (Maartens and Koyama 2010; Pavsic 2002; Raychaudhuri and Sridhar 2016). In these models the usual matter and the nongravitational fields are confined to a fourdimensional brane, representing our world. This brane is embedded in higherdimensional bulk spacetime where only gravity can propagate. Higherdimensional black holes play a very special role in the braneworld models. Being just a clot of gravity, they can ‘live’ both on and outside the brane. Miniblackholes whose size is smaller than the size of extra dimensions thus play a role of probes of extra dimensions. One of the intriguing features of the braneworld models that is intensively discussed in the literature is a possibility of miniblackhole formation in the collision of high energy particles in modern TeV colliders (see e.g., Landsberg 2015; Aad et al. 2014 and references therein). Numerous discussions of this effect generated a great interest in the study of properties of higherdimensional black holes.
A third main reason to study higherdimensional black holes comes from the desire to better understand the nature of gravitational theory and in particular to identify which properties of gravitational fields are specific to four dimensions and which of them are valid more generally irrespective of the spacetime dimension (Emparan and Reall 2008).
1.2 Remarkable properties of the Kerr black hole
The Kerr metric has the following remarkable properties: the equations of motion for a free particle in this geometry are completely integrable and the physically interesting field equations allow for the separation of variables. What stands behind these properties?
In a simpler case, when a black hole does not rotate, the answer is well known. The corresponding Schwarzschild solution is static and spherically symmetric. As a result of this symmetry, the energy of the particle and the three components of its angular momentum are conserved. One can thus construct four integrals of geodesic motion that are functionally independent and mutually Poisson commute, choosing, for example, the (trivial) normalization of the fourvelocity, the particle’s energy, the square of its total angular momentum, and a projection of the angular momentum to an arbitrary ‘axis. According to the Liouville’s theorem, the existence of such quantities makes the geodesic motion in the spherically symmetric black hole case completely integrable.
For rotating black holes the situation is more complicated since the total angular momentum is no longer conserved. Surprisingly, even in this case there exists another integral of motion, nowadays known as the Carter’s constant. Obtained in 1968 by Carter by a method of separation of variables in the Hamilton–Jacobi equation (Carter 1968a, b), this additional integral of motion is quadratic in momentum, and, as shown later by Walker and Penrose (1970), it is in onetoone correspondence with the rank 2 Killing tensor of the Kerr geometry. A rank 2 Killing tensor \(k^{ab}\) is a symmetric tensor whose symmetrized covariant derivative vanishes, \(\nabla ^{(c}k^{ab)}=0\). It was demonstrated by Carter in the same papers (Carter 1968a, b) that not only the Hamilton–Jacobi equation but also the Klein–Gordon equation allows for a complete separation of variables in the Kerr spacetime.
Remark
This fact may not be as surprising as it looks at first sight. In fact, the following 3 problems: complete integrability of geodesic equations, separability of the Hamilton–Jacobi equation, and separability of the Klein–Gordon equation are closely related. Namely, looking for a quasiclassical solution \(\varPhi \sim \exp (i S)\) of the Klein–Gordon equation \((g^{ab}\nabla _a\nabla _b m^2)\varPhi =0\), one obtains the Hamilton–Jacobi equation \(g^{ab}\nabla _a S\nabla _b S +m^2=0\). By identifying \(\nabla _a S\) with the momentum \(p_a\), one reduces the problem of finding the action function S to the problem of integrating the Hamilton equations of motion for a relativistic particle. \(\square \)
Following Carter’s success a boom of discoveries regarding the remarkable properties of the Kerr geometry has taken place. Teukolsky (1972, 1973) decoupled the equations for the electromagnetic and gravitational perturbations and separated variables in the obtained master equations. Equations for massless neutrinos were separated Unruh (1973) and Teukolsky (1973), and the equations for the massive Dirac field were separated by Chandrasekhar (1976) and Page (1976).
Penrose (1973) and Floyd (1973) demonstrated that in the Kerr geometry there exists a new fundamental object, the so called Killing–Yano tensor \(f_{ab}\), which behaves as a ‘square root’ of the Killing tensor. This object is a 2form that obeys the following equation: \(\nabla _{(c}f_{a)b}=0\). If \(f_{ab}\) is nondegenerate, the integrability conditions for this equation imply that the spacetime is algebraically special, of Petrov type D (Collinson 1974). Hughston and Sommers (1973) showed that the existence of such Killing–Yano tensor necessarily implies that the corresponding spacetime admits also two commuting Killing vectors, generating time translation and rotation.
It is interesting to note that some of the above described properties extend beyond the case of the vacuum Kerr geometry. Namely, in 1968 Carter obtained a 6parametric solution of the Einstein–Maxwell equations with a cosmological constant \(\varLambda \) that shares with the Kerr geometry many of the remarkable properties (Carter 1968b, c). Besides the cosmological constant \(\varLambda \), the mass M and the angular momentum J, this solution contains also an electric charge Q, a magnetic monopole P, and the NUT parameter N. The whole class of Carter’s metrics admits the Killing–Yano tensor (Demianski and Francaviglia 1980; Carter 1987).
Carter’s solution is now called the charged Kerr–NUT–(A)dS metric. In the absence of the NUT parameter it is the most general regular solution describing a stationary isolated black hole in the fourdimensional asymptotically flat (\(\varLambda =0\)) or (anti) de Sitter (\(\varLambda \ne 0\)) space. The hidden symmetries of the fourdimensional Kerr–NUT–(A)dS metric and its generalization by Plebański and Demiański (1976) will be discussed in detail in Chap. 3.
1.3 Higherdimensional black objects
With the advent of interest in higherdimensional black holes at the beginning of this century the following questions arose: (i) How far can the results on the fourdimensional black holes be generalized to higher dimensions? (ii) What is the most general solution describing a stationary black hole in asymptotically flat and/or asymptotically (anti) de Sitter space? iii) What can one say about particle motion and field propagation in the gravitational field of such black holes? By now partial answers to some of these questions have been obtained.
The ‘zoo’ of higherdimensional black holes is vast: there exist extended objects such as as black strings and branes, and the topology of the horizon of an isolated stationary higherdimensional black hole needs not to be spherical, see e.g. Emparan and Reall (2002a), Elvang and Figueras (2007), Kunduri and Lucietti (2014). In particular, in 2002 Emparan and Reall obtained an exact solution of 5dimensional vacuum Einstein equation which describes a stationary rotating black hole with toroidal horizon (Emparan and Reall 2002b). Later many new exact 5dimensional vacuum stationary black hole solutions with a more complicated structure of the horizon were found. There are strong arguments that similar solutions do also exist in more than five dimensions, though the stability of all these objects is in question, e.g., Santos and Way (2015). Many useful references on this subject can be found in the remarkable review by Emparan and Reall (2008), see also Emparan et al. (2010), Kunz (2015), Kleihaus and Kunz (2017).
The problem of uniqueness and stability of the higher dimensional black holes is far from its solution—see e.g. a review Hollands and Ishibashi (2012) and references therein.
1.4 Higherdimensional Kerr–NUT–(A)dS black holes
Within this ‘zoo’ of higher dimensional black objects there exists a large important family of black hole solutions which are natural generalizations of the fourdimensional Kerr–NUT–(A)dS solution. Called higherdimensional Kerr–NUT–(A)dS metrics, these solutions will be in the main focus of this review. They have the spherical topology of the horizon, and in the absence of the NUT parameters, describe isolated rotating black holes in either asymptotically flat or asymptotically (A)dS spacetime.
Remark
Let us emphasize, that even if the stationary black hole maintains the spherical horizon topology, its horizon may be ‘distorted’—the sizes of symmetric cycles on the horizon may vary nonmonotonically with the polar angle. Such ‘bumpy’ black holes were conjectured to exist in Emparan and Myers (2003) and later found numerically in Emparan et al. (2014). These black holes do not belong to the Kerr–NUT–(A)dS family and are not studied in this review. However, it might be an interesting problem for future studies to see whether some of the integrability results presented here for ‘smooth’ Kerr–NUT–(A)dS black holes could be extended to bumpy black holes or other black holes as well. \(\square \)
Let us briefly recapitulate a history of study of the Kerr–NUT–(A)dS family of black hole solutions. Denote by \(D=2n+\varepsilon \) a total number of spacetime dimensions, with \(\varepsilon =0\) in even dimensions and \(\varepsilon =1\) in odd dimensions. A higherdimensional generalization of the Schwarzschild black hole solution was readily obtained by Tangherlini (1963). The Tangherlini solution is static and spherically symmetric, it admits \(SO(D1)\) group of rotational symmetries, and contains exactly one arbitrary parameter which can be identified with the gravitational radius and is related to the black hole mass. A task of finding a higherdimensional generalization of the Kerr geometry is much harder and was achieved by Myers and Perry (1986). The general solution contains, besides the mass M, up to \((n1+\varepsilon )\) independent rotation parameters.
In ‘our threedimensional world’ we are used to think about rotations as operations about a given axis and identify the angular momentum with a 3vector \(J^a\). In a general case, however, the angular momentum is described by a rank 2 antisymmetric tensor \(J_{ab}\). In three dimensions one can write \(J^a=\epsilon ^{abc}J_{bc}\), where \(\epsilon ^{abc}\) is the totally antisymmetric tensor, and the usual description is recovered. In higher dimensions such relation no longer exists. Nevertheless, one can always write \(J_{ab}\) in a canonical form by finding a set of mutually orthogonal 2planes such that the components of \(J_{ab}\) vanish unless the two indices ‘belong’ to the same 2plane. Since the number of spatial dimensions is \(D1\), the largest possible number of mutually orthogonal 2planes (and hence the number of independent components of the angular momentum tensor) is \((n1+\varepsilon )\). This is also the number of independent components of the angular momentum of the black hole which enters the general Myers–Perry solution.
It took another 20 years to find a generalization of the Myers–Perry metric which includes the cosmological constant. Hawking et al. (1999) found singlyspinning Kerr–(A)dS metrics in all dimensions. These metrics were then generalized by Gibbons et al. (2005) and Gibbons et al. (2004) to the case of a general multiple spin. After several attempts to include NUT parameters (e.g., Chong et al. 2005; Chen et al. 2007), Chen et al. (2006a) finally found the most general higherdimensional Kerr–NUT–(A)dS metric, generalizing the higherdimensional Carterlike ansatz studied previously in Klemm (1998). It is the purpose of this review to study the most general Kerr–NUT–(A)dS metric (Chen et al. 2006a) and its remarkable symmetries.
1.5 Explicit and hidden symmetries
Despite of being significantly more complicated, the most general Kerr–NUT–(A)dS metrics in all dimensions have very similar properties to their fourdimensional ‘cousin’, the Kerr metric. A discussion of this similarity and its origin is the subject of the present review. Namely, we shall describe a fundamental geometric structure which is responsible for the remarkable properties of the Kerr–NUT–(A)dS metrics. These properties stem from the existence a complete set (‘tower’) of explicit and hidden symmetries that are ‘miraculously’ present in these spacetimes. Moreover, the existence of such a Killing tower of symmetries is also a characteristic property of the Kerr–NUT–(A)dS spacetimes. It is possible that some of the hidden symmetries may also exist in other higherdimensional black object spacetimes and their study is an open interesting problem. But we concentrate on the case when the metric possesses the complete tower of hidden symmetries.
What do we mean by a hidden symmetry? We say, that a spacetime possesses a symmetry if there exists a transformation which preserves its geometry. This means that the metric, as well as all other quantities constructed from it (for example curvature), remain unchanged by such a transformation. Continuous symmetry transformations are generated by Killing vector fields; we call the corresponding symmetries explicit. By famous Noether’s theorem they generate conserved charges. Let us demonstrate this on an example of particle motion.
The motion of a free particle in a curved spacetime can be described using the Hamiltonian formalism. A state of the particle is characterized by a point \((x^a,p_a)\) in the phase space. Its motion is defined by the Hamiltonian, which is quadratic in momenta. The explicit symmetry generated by the Killing vector \(\xi ^a\) implies that the quantity \(p_a \xi ^a\) remains constant along the particle’s worldline, it is an integral of motion. Integrals of motion are phase space observables that Poisson commute with the Hamiltonian.
An important property of integrals of motion generated by spacetime symmetries is that they are linear in momentum. However, this does not exhaust all possibilities. There may exist integrals of motion that are higherorder polynomials in particle momenta. The existence of such integrals implies that the spacetime admits a special geometric structure, known as a Killing tensor. Killing tensors are in onetoone correspondence with constants of geodesic motion that are homogeneous in particle momenta, namely, a rank r Killing tensor gives rise to a homogeneous constant of motion of degree r in momentum. Inhomogeneous polynomial integrals of geodesic motion can be decomposed into their homogeneous parts and are associated with Killing tensors of various ranks.
Perhaps the best known example of a Killing tensor is the spacetime metric itself. The corresponding conserved quantity is the Hamiltonian for the relativistic particle and its value is proportional to the square of particle’s mass. Familiar Killing vectors, associated with the explicit spacetime symmetry, are Killing tensors of rank 1. To distinguish from this case, we call the geometric structure of the spacetime encoded in Killing tensors of rank 2 and higher a hidden symmetry.
1.6 Complete integrability of geodesic motion
The existence of integrals of motion simplifies the study of dynamical systems. There exits a very special case, when the number of independent commuting integrals of motion of a dynamical system with N degrees of freedom, described by a 2Ndimensional phase space, is equal to N. Such a system is called completely integrable and its solution can be written in terms of integrals, a result known as the Liouville theorem (Liouville 1855). Specifically, the equation of motion for a free relativistic particle in a Ddimensional spacetime can be explicitly solved if there exist D independent Killing vectors and Killing tensors, including the metric, which are ‘in involution’.
Remark
For Killing tensor fields there exists an operation, a generalization of the Lie bracket, which allows one to construct from two Killing tensors a new one. This operation, called the Schouten–Nijenhuis commutator, will be defined in Sect. 2.1. ‘In involution’ then means that the Schouten–Nijenhuis commutator of the corresponding tensor fields mutually vanishes. On the level of the phasespace observables this is equivalent to the statement that the corresponding conserved quantities mutually Poissoncommute. \(\square \)
Consider, for example, a fivedimensional Myers–Perry metric with two independent rotation parameters. This metric has three Killing vectors: one generates translations in time, and the other two correspond to rotations in the two independent 2planes of rotation. Together with the normalization of the 5velocity this gives 4 integrals of geodesic motion. For complete integrability, an additional integral corresponding to a Killing tensor is needed. This tensor can be found by Carter’s method, that is by separating the variables in the Hamilton–Jacobi equation written in the standard Boyer–Lindquist coordinates (Frolov and Stojković 2003a, b), making the geodesic motion completely integrable in this case. Interestingly, in more than five dimensions and for general multiplyspinning black holes the Boyer–Lindquist type coordinates are not ‘nice’ anymore and Carter’s method no longer works. This mislead people to believe that the geodesic motion in these spacetimes is no longer integrable and black holes of smaller generality were studied.
1.7 Principal tensor and its Killing tower
It turns out that the restriction on rotation parameters is not necessary and even for the most general multiplyspinning Kerr–NUT–(A)dS black holes one can find special coordinates in which the Hamilton–Jacobi equation separates, proving the geodesic motion completely integrable. A breakthrough in solving this problem occurred in 2007, when it was demonstrated that the Myers–Perry metric as well as the most general Kerr–NUT–(A)dS spacetime in any number of dimensions both admit a nondegenerate closed conformal Killing–Yano 2form (Frolov and Kubizňák 2007; Kubizňák and Frolov 2007). The claim is that the very existence of this single object implies complete integrability of geodesic motion in all dimensions. Let us explain why this is the case.
In four dimensions this does not help much. Already at the first step of this procedure, one obtains a 4form that is proportional to the totally antisymmetric tensor. In higher dimensions, however, the story is quite different: there is enough room to ‘accommodate’ nontrivial higherrank closed conformal Killing–Yano tensors. It is evident, that the smaller is the tensorrank of the original form \({\varvec{h}}\) the larger number of its nontrivial higherrank “successors” one can obtain. This makes the case of a 2form \({\varvec{h}}\) a special one. One can also assume that the matrix rank of this 2form is the largest possible, that is, the 2form is nondegenerate. In \((2n+\varepsilon )\)dimensional spacetime the maximal matrix rank is 2n. By ‘squaring’ the Killing–Yano tensors obtained as Hodge duals of the so constructed ‘successors’ of \({\varvec{h}}\), one obtains the whole Killing tower of n independent Killing tensors (Krtouš et al. 2007a). Supplemented by the \((n+\varepsilon )\) integrals of motion corresponding to explicit symmetries (as we shall see later such symmetries can also be generated from \({\varvec{h}}\)), one obtains a set of \(D=2n+\varepsilon \) (generically independent) mutually Poisson commuting constants of geodesic motion, making such a motion completely integrable. In the following chapters we discuss these results in very details, and give the corresponding references.
Using canonical coordinates, internally connected with and determined by the principal tensor \({\varvec{h}}\), greatly simplifies the study of properties of a spacetime which admits such an object. Namely, we demonstrate that the corresponding spacetime necessarily possesses the following remarkable properties (Houri et al. 2007; Krtouš et al. 2008; Houri et al. 2008b): (i) When the Einstein equations are imposed one obtains the most general Kerr–NUT–(A)dS metric. (ii) The geodesic motion in such a space is completely integrable, and the Hamilton–Jacobi, Klein–Gordon, and Dirac equations allow for complete separation of variables. The separation of variables occurs in the canonical coordinates determined by the principal tensor.
1.8 ‘Hitchhikers guide’ to the review
The review is organized as follows. In Chap. 2 we introduce the Killing vectors, Killing tensors, and the family of Killing–Yano objects and discuss their basic properties. In particular, the principal tensor is defined and its most important properties are overviewed. Chapter 3 contains a summary of the symmetry properties of the fourdimensional Kerr metric and its Kerr–NUT–(A)dS and Plebański–Demiański generalizations. We demonstrate how the explicit and hidden symmetries of the Kerr spacetime arise from the principal tensor. Chapter 4 gives a comprehensive description of the higherdimensional Kerr–NUT–(A)dS metrics. In Chap. 5, starting from the principal tensor which exists in a general higherdimensional Kerr–NUT–(A)dS spacetime, we construct a tower of Killing and Killing–Yano objects, responsible for the explicit and hidden symmetries of this metric. In Chap. 6 we discuss a free particle motion in the higherdimensional Kerr–NUT–(A)dS spacetime and show how the existence of the principal tensor in these metrics leads to a complete integrability of geodesic equations. We also demonstrate the separability of the Hamilton–Jacobi, Klein–Gordon and Dirac equations in these spacetimes. Chapter 7 contains additional material and discusses further possible generalizations of the theory of hidden symmetries presented in this review.
To help the reader with various concepts used in the review, we included some complementary material in appendices. Appendix A summarizes our notation and conventions on exterior calculus; Appendix B reviews the symplectic geometry, the concept of complete integrability, and the requirements for separability of the Hamilton–Jacobi equation. Appendix F covers basic notions of a theory of spinors in a curved spacetime, discusses symmetry operators of the Dirac operator, as well as introduces Killing spinors and reviews their relationship to special Killing–Yano forms. Integrability conditions for the Killing–Yano objects are summarized in Appendix C. Appendix E discusses the Myers–Perry solutions in its original and Kerr–Schild forms and supplements thus material in Sect. 4.4. Finally, various identities and quantities related to the Kerr–NUT–(A)dS metric are displayed in Appendix D.
Before we begin our exploration, let us mention several other review papers devoted to hidden symmetries and black holes that might be of interest to the reader (Frolov and Kubizňák 2008; Kubizňák 2008, 2009a; Yasui and Houri 2011; Cariglia et al. 2012; Frolov 2014; Cariglia 2014; Chervonyi and Lunin 2015).
2 Hidden symmetries and Killing objects
In this chapter we discuss Killing vectors and Killig tensors, which are responsible for explicit and hidden symmetries of spacetime. We also introduce the Killing–Yano tensors which are generators of hidden symmetries and discuss their basic properties. Named after Yano (1952), these new symmetries are in some sense ‘more fundamental’ than the Killing tensors. A special attention is devoted to a subclass of closed conformal Killing–Yano tensors and in particular to the principal tensor which plays a central role for the theory of higherdimensional black holes.
2.1 Particle in a curved spacetime
Geometrical properties of a curved spacetime and its various symmetries can be studied through investigation of geodesic motion. For this reason we start with a short overview of the description of relativistic particle in a curved spacetime, formulated both from a spacetime perspective and in terms of the phase space language.
2.1.1 Phase space description
Let us consider a Ddimensional spacetime (configuration space) M and a pointlike particle moving in it. In the Hamilton approach the motion is described by a trajectory in the 2Ddimensional phase space. A point in the phase space represents a position x and a momentum \({\varvec{p}}\) of the system. The momenta \({\varvec{p}}\) are naturally represented as covectors (1forms) on the configuration space, the phase space \({\varGamma }\) thus corresponds to a cotangent bundle over the configuration space.
Using the symplectic structure we can introduce the standard machinery of the symplectic geometry: we can define symplectic potential \({\varvec{\theta }}\), the Poisson brackets \(\{\ ,\ \}\), or the Hamiltonian vector field \({\varvec{X}}_{F}\) associated with an observable F. The overview of the symplectic geometry and the convention used in this review can be found in Sect. B.1 of the appendix, cf. also standard books Arnol’d (1989), Goldstein et al. (2002).
2.1.2 Nijenhuis–Schouten bracket
2.1.3 Time evolution and conserved quantities
2.1.4 Relativistic particle and propagation of light
Remark
The normalization (2.13) fixes the norm of the momentum. The momentum thus has \({D1}\) independent components. For a massive particle one can identify these with the spatial velocities, while the energy (the time component of the momentum) is computable from the normalization. In the massless case, one cannot chose an arbitrary magnitude of velocity, only a direction of the ray. At the same time, there exists an ambiguity in the choice of the affine parameter along the ray. Its rescaling results in the transformation \(l^a \rightarrow \tilde{l}^a=\alpha l^a\), where \(\alpha \) is constant. Although two such null particles differ just by a scale of their momenta and they follow geometrically the same path in the spacetime, they correspond to two physically different photons: they differ by their energy, or, intuitively, by their ‘color’. Instead of a freedom of choosing an arbitrary magnitude of velocity for massive particles, in the case of null particles we have thus a freedom choosing an arbitrary energy, i.e., an arbitrary ‘color’. \(\square \)
In the description of the relativistic particle above the configuration space is the whole Ddimensional spacetime, suggesting thus D degrees of freedom. However, the correct counting of the physical degrees of freedom is \({D1}\). The difference is related to the existence of the constraint \(H=\text {const}\) and the remaining timereparametrization freedom \(\sigma \rightarrow \sigma +\text {const}\). For more details on the timereparametrization symmetry and related constraints see, e.g., Frolov and Zelnikov (2011), Sundermeyer (1982), Thirring (1992), Rohrlich (2007).
2.2 Explicit and hidden symmetries
If the spacetime has some symmetries they can be always ‘lifted up’ to the phase space symmetries. The corresponding integrals of motion are observables in the phase space which are linear in momenta. However, the contrary is not true: not every phase space symmetry can be easily reduced to the configuration space. Symmetries which have the direct counterpart on the configuration space will be called the explicit symmetries, those which cannot be reduced to the configuration space transformation are called the hidden symmetries.
2.2.1 Killing vectors
The welldefiniteness of the projection requires that it is a quantity solely dependent on the spacetime variables, i.e., independent of the momentum. Clearly it means that the \({{{\varvec{\partial }}}_{x^a}}\)term in the Hamiltonian vector field \({{\varvec{X}}_{I}}\) must not depend on the momentum, which requires that the observable I is linear in momentum. The integrals of particle motion in curved space that correspond to explicit symmetries are thus linear in particle’s momentum.
Remark
2.2.2 Killing tensors
Besides the conserved quantities which are linear in momentum, there might also exist more complicated conserved quantities that indicate the existence of deeper and less evident symmetries. For the motion of relativistic particles these hidden symmetries are encoded in Killing tensors. Namely, the Killing tensors are in onetoone correspondence with the integrals of geodesic motion that are monomial in momenta.
In other words, Killing tensors of order \({s\ge 2}\) represent symmetries that do not generate a spacetime diffeomorphism and in that sense they are not ‘encoded’ in the spacetime manifold. Their presence, however, can be ‘discovered’ by studying the particle dynamics in the spacetime. This is to be compared to the action of Killing vectors, \(s=1\), for which the projection defines a spacetime isometry and the symmetry is explicit, cf. (2.22).
Given two constants of geodesic motion \({K_{(1)}}\) and \({K_{(2)}}\) of the type (2.2), their Poisson bracket \({\{K_{(1)},K_{(2)}\}}\) is also an integral of motion of the same type. This immediately implies that provided \({\varvec{k}}_{(1)}\) and \({\varvec{k}}_{(2)}\) are two Killing tensors, so is their Nijenhuis–Schouten bracket \([{\varvec{k}}_{(1)},{\varvec{k}}_{(2)}]_{\scriptscriptstyle \mathrm {NS}}\). Slightly more generally, an integral of motion that is polynomial in the momentum corresponds to an inhomogeneous Killing tensor, defined as a formal sum of Killing tensors of different ranks. Such objects together form the Lie algebra under the Nijenhuis–Schouten bracket.
Similarly, given two monomial integrals of geodesic motion \({K_{(1)}}\) and \({K_{(2)}}\) of order \(s_1\) and \(s_2\), respectively, their product \(K=K_{(1)}K_{(2)}\) is also a monomial constant of geodesic motion of order \(s=s_1+s_2\). This means that K corresponds to a Killing tensor \({\varvec{k}}\) given by \(k^{a_1\dots a_{s}}=k_{(1)}^{(a_1\dots a_{s_1}}\,k_{(2)}^{a_{s_1+1}\dots a_{s})}\). In other words, a symmetrized product of two Killing tensors is again a Killing tensor.
This hints on the following definition. A Killing tensor is called reducible, if it can be decomposed in terms of the symmetrized products of other Killing tensors and Killing vectors. Otherwise it is irreducible.
Remark
2.2.3 Conformal Killing vectors and Killing tensors
Remark
It is obvious that symmetries generated by conformal Killing tensors (for \(s\ge 2\)) are again hidden. Moreover, by the same arguments as in the case of Killing tensors, it can be shown that the Nijenhuis–Schouten bracket of two conformal Killing tensors is again a conformal Killing tensor. Similarly, a symmetrized product of two conformal Killing tensors is again a conformal Killing tensor.
2.3 Separability structures
Separability structures are classes of separable charts for which the Hamilton–Jacobi equation allows an additive separation of variables. For each separability structure there exists such a family of separable coordinates which admits a maximal number of, let us say r, ignorable coordinates. Each system in this family is called a normal separable system of coordinates. We call the corresponding structure the rseparability structure. Its existence is governed by the following theorem:
Theorem
 (i)all mutually (Nijenhuis–Schouten) commute:$$\begin{aligned} \bigl [{\varvec{k}}_{(\alpha )},{\varvec{k}}_{(\beta )}\bigr ]_{\scriptscriptstyle \mathrm {NS}}=0,\quad \bigl [{\varvec{l}}_{(i)},{\varvec{k}}_{(\beta )}\bigr ]_{\scriptscriptstyle \mathrm {NS}}=0, \quad \bigl [{\varvec{l}}_{(i)}, {\varvec{l}}_{(j)}\bigr ]_{\scriptscriptstyle \mathrm {NS}}=0, \end{aligned}$$(2.36)
 (ii)Killing tensors \({\varvec{k}}_{(\alpha )}\) have in common \(Dr\) eigenvectors \({\varvec{m}}_{(\alpha )}\), such that$$\begin{aligned}{}[{\varvec{m}}_{(\alpha )},{\varvec{m}}_{(\beta )}]=0,\quad [{\varvec{m}}_{(\alpha )},{\varvec{l}}_{(i)}]=0,\quad {\varvec{g}}({\varvec{m}}_{(\alpha )},{\varvec{l}}_{(i)})=0. \end{aligned}$$(2.37)
It is evident, that the existence of a separability structure implies the complete integrability of geodesic motion. Indeed, the requirement of independence means that r linear in momenta constants of motion \(L_{(i)}\) associated with Killing vectors \({\varvec{l}}_{(i)}\) and \((Dr)\) quadratic in momenta constants of motion \(K_{(\alpha )}\) corresponding to Killing tensors \({\varvec{k}}_{(\alpha )}\) are functionally independent. Moreover, equations (2.36) and the discussion in the Appendix B imply that all such constants are in involution, that is obey conditions (B.19). Hence the geodesic motion is completely integrable.
Theorem
The Klein–Gordon equation allows a multiplicative separation of variables if and only if the manifold possesses a separability structure in which the vectors \({\varvec{m}}_{(\alpha )}\) are eigenvectors of the Ricci tensor. In particular, if the manifold is an Einstein space, the Hamilton–Jacobi equation is separable if and only if the same holds for the wave equation.
We will see a particular realization of this structure in Chap. 5 where we write down the metric consistent with a special example of separable structure, namely the offshell Kerr–NUT–(A)dS metric. The separable structure of the Kerr–NUT–(A)dS spacetimes justifies the complete integrability of geodesic motion, as well as the fact that the Hamilton–Jacobi equation and the wave equations allow for a separation of variables, see Chap. 6.
2.4 Defining the Killing–Yano family
2.4.1 Motivation: parallel transport
2.4.2 Decomposition of the covariant derivative
Remark
Note that we have defined here the twistor operator as an operator acting on the space of antisymmetric forms. Perhaps better known is the twistor operator defined on Dirac spinors which naturally complements the Dirac operator. Both twistor operators are closely related, but not identical. In particular, any pform constructed from a twistor spinor (by sandwiching gamma matrices) belongs to the kernel of the above pform twistor operator, see Appendix F. \(\square \)
Decomposition of the covariant derivative of a form \({\varvec{\omega }}\)  

General form  \({{\varvec{\nabla }}}{\varvec{\omega }}=\mathcal {A}{{\varvec{\nabla }}}{\varvec{\omega }}+\mathcal {C}{{\varvec{\nabla }}}{\varvec{\omega }}+\mathcal {T}{{\varvec{\nabla }}}{\varvec{\omega }}\)  
Closed form  \({{\varvec{\nabla }}}{\varvec{\omega }}=\mathcal {C}{{\varvec{\nabla }}}{\varvec{\omega }}+\mathcal {T}{{\varvec{\nabla }}}{\varvec{\omega }}\)  \({{\varvec{d}}}{\varvec{\omega }} = 0\) 
Divergencefree coclosed form  \({{\varvec{\nabla }}}{\varvec{\omega }}=\mathcal {A}{{\varvec{\nabla }}}{\varvec{\omega }}+\mathcal {T}{{\varvec{\nabla }}}{\varvec{\omega }}\)  \({{\varvec{\delta }}}{\varvec{\omega }} = 0\) 
Conformal Killing–Yano form  \({{\varvec{\nabla }}}{\varvec{\omega }}=\mathcal {A}{{\varvec{\nabla }}}{\varvec{\omega }}+\mathcal {C}{{\varvec{\nabla }}}{\varvec{\omega }}\)  \({\mathbf {T}}{\varvec{\omega }}=0\) 
Killing–Yano form  \({{\varvec{\nabla }}}{\varvec{\omega }}=\mathcal {A}{{\varvec{\nabla }}}{\varvec{\omega }}\)  \({{\varvec{\delta }}}{\varvec{\omega }} = 0,{\mathbf {T}}{\varvec{\omega }}=0\) 
Closed conformal Killing–Yano form  \({{\varvec{\nabla }}}{\varvec{\omega }}=\mathcal {C}{{\varvec{\nabla }}}{\varvec{\omega }} \)  \({{\varvec{d}}}{\varvec{\omega }} = 0,{\mathbf {T}}{\varvec{\omega }}=0\) 
Harmonic form  \({{\varvec{\nabla }}}{\varvec{\omega }}=\mathcal {T}{{\varvec{\nabla }}}{\varvec{\omega }}\)  \({{\varvec{d}}}{\varvec{\omega }} = 0,{{\varvec{\delta }}}{\varvec{\omega }} = 0\) 
Covariantly constant form  \({{\varvec{\nabla }}}{\varvec{\omega }}=0 \)  \({{\varvec{d}}}{\varvec{\omega }} = 0,{{\varvec{\delta }}}{\varvec{\omega }} = 0,{\mathbf {T}}{\varvec{\omega }}=0\) 
2.4.3 Alternative definitions
The definition of conformal Killing–Yano forms (2.50) can be reformulated in a slightly modified form:
Theorem
Indeed, by antisymmetrizing (2.53) one obtains the first relation (2.54). Similarly, contracting the first two indices in (2.53) leads to the second relation (2.54). Substituting these two relations back to (2.53) one recovers the definition (2.50).
2.4.4 Killing–Yano objects in a differential form notation
2.5 Basic properties of conformal Killing–Yano forms
2.5.1 Conformal Killing–Yano forms

The Hodge dual of a conformal Killing–Yano tensor is again a conformal Killing–Yano tensor.

The Hodge dual of a closed conformal Killing–Yano tensor is a Killing–Yano tensor and vice versa.
Contrary to conformal Killing tensors, conformal Killing–Yano tensors do not form in general a graded Lie algebra, though they do in constant curvature spacetimes (Kastor et al. 2007). See Cariglia et al. (2011a), Ertem and Acik (2016), Ertem (2016) for attempts to generalize this property using the suitably modified Schouten–Nijenhuis brackets.
Remark
Let us finally mention that conformal Killing–Yano tensors are closely related to twistor spinors, and play a crucial role for finding symmetries of the massless Dirac operator. At the same time the subfamilies of Killing–Yano and closed conformal Killing–Yano tensors are responsible for symmetries of the massive Dirac equation (Carter and McLenaghan 1979; Benn and Charlton 1997; Cariglia 2004; Cariglia et al. 2011a), see the discussion in Sect. 6.4 and Appendix F.
2.5.2 Killing–Yano forms
Remark
2.5.3 Closed conformal Killing–Yano forms
Closed conformal Killing–Yano tensors are conformal Killing–Yano tensors that are in addition closed with respect to the exterior derivative. This additional property implies the following two important results.
The second property of closed conformal Killing–Yano forms which plays a key role in the construction of hidden symmetries in higherdimensional black hole spacetimes is the following statement (Krtouš et al. 2007a; Frolov 2008):
Theorem
2.6 Integrability conditions and method of prolongation
The conformal Killing–Yano equation (2.61) represents an overdetermined system of partial differential equations (Dunajski 2008) and significantly restricts a class of geometries for which nontrivial solutions may exist. For this reason it is very useful to formulate and study integrability conditions for these objects. For example, it was shown in Mason and TaghaviChabert (2010) that the integrability condition for a nondegenerate conformal Killing–Yano 2form implies that the spacetime is necessary of type D of higherdimensional algebraic classification (Coley et al. 2004; Milson et al. 2005). We refer to Appendix C for detailed derivation of integrability conditions for (closed conformal) Killing–Yano tensors.
Taking into account that (closed conformal) Killing–Yano conditions impose severe restrictions on the spacetime geometry, it is natural to ask the following questions: (i) What is the maximum possible number of independent Killing–Yano symmetries that may in principle exist? (ii) Given a spacetime, is there an algorithmic procedure to determine how many Killing–Yano symmetries are present? Fortunately, the answers to both of these questions are known. Given a geometry, there is an effective method, called the method of prolongation, that provides an algorithmic tool for determining how many at most solutions of an overdetermined system of Killing–Yano equations may exist (Houri and Yasui 2015). See also Houri et al. (2017) for a recent study of the prolongation of the Killing tensor equation.
2.6.1 Prolongation of the Killing vector equation
2.6.2 Maximum number of (closed conformal) Killing–Yano forms
The method of prolongation has been readily extended to (closed conformal) Killing–Yano tensors, in which case one can use an elegant description in terms of the so called Killing connection. We refer to the work by Houri and Yasui (2015) for details and state only the formulae for the maximum number of (closed conformal) Killing–Yano tensors, e.g. Kastor and Traschen (2004).
We refer the reader to a recent paper by Batista (2015), where the integrability conditions are studied for a general conformal Killing–Yano tensor and to Appendix C for the overview and derivations of the integrability conditions for Killing–Yano and closed conformal Killing–Yano forms.
2.7 Killing–Yano tensors in maximally symmetric spaces
The maximally symmetric spaces possess the maximum number of Killing–Yano and closed conformal Killing–Yano tensors. Their special properties have been studied in Batista (2015). In what follows let us write explicitly a basis for these tensors in the simple case of a Ddimensional flat space, using the Cartesian coordinates.
2.8 Principal tensor
There exists a very deep geometrical reason why the properties of higherdimensional rotating black holes are very similar to the properties of their fourdimensional ‘cousins’. In both cases, the spacetimes admit a special geometric object which we call the principal tensor. As we shall see, this tensor generates a complete set of explicit and hidden symmetries and uniquely determines the geometry, given by the offshell Kerr–NUT–(A)dS metric. The purpose of this section is to introduce the principal tensor, a ‘superhero’ of higherdimensional black hole physics, and discuss its basic properties.
2.8.1 Definition
2.8.2 Darboux and null frames
The condition that the principal tensor is nondegenerate requires that there are exactly \({{{n}}}\) nonvanishing eigenvalues \({x_\mu }\), which, in a suitable neighborhood, give \({{{n}}}\) functionally independent (nonconstant and with linearly independent gradients) functions.
2.8.3 Metric signature
All the formulae so far were adjusted to the Euclidean signature. For other signatures of the metric, most of the formulae can be written in the same way, only the reality of various quantities is different. In particular, for the Lorentzian signature one of the 1forms in the Darboux frame is imaginary and two of the null eigenvectors are real (and not complex conjugate anymore). One can also perform a suitable ‘Wick rotation’ and define real and properly normalized canonical frames. This will be done, for example, in the next chapter when discussing the canonical Darboux basis for the Kerr spacetime in four dimensions. In higher dimension, on other hand, we will use mostly the formal Euclidean definitions even in the case of Lorentzian signature (Chap. 5) and we will perform the Wick rotation only for the coordinate form of the metric, see Sect. 4.4.
However, the nonEuclidean signatures allow also other possibilities. The Darboux frame can take an exceptional ‘null form’ when some of the vectors in (2.110) are null, cf. Milson (2004). It can also happen that some of the eigenvalues \(x_\mu \) have a null gradient \({{\varvec{d}}}x_\mu \) which complicates the choice of the special Darboux frame discussed below, see (2.122). We do not consider such exceptional cases in our review. We assume that the principal tensor allows the choice of the Darboux frame in the form (2.122) and that eigenvalues \(x_\mu \) are not globally null (although they can become null on special surfaces as, for example, at the horizon). A study and the classification of the exceptional null cases is an interesting open problem.
2.8.4 Special Darboux frame
The two properties (2.133) and (2.134) play a crucial role in the construction of the canonical metric admitting the principal tensor, see the discussion in Chap. 5 and original papers Houri et al. (2007), Krtouš et al. (2008), Houri et al. (2009).
2.8.5 Killing tower
The special Darboux frame is only the first consequence of the existence of the principal tensor. One of the keystone properties of the principal tensor is that it can be used to generate a rich symmetry structure which we call the Killing tower. It is a sequence of various symmetry objects which, in turn, guarantee many important properties of the physical systems in spacetimes with the principal tensor. Here we only shortly sketch how the Killing tower is build to get an impression of this symmetry structure. We return to the Killing tower in Chap. 5, where we explore its definitions and properties in much more detail, and in Chap. 6, where we review its main physical consequences.
 (i)Closed conformal Killing–Yano forms \({{\varvec{h}}^{(j)}}\) of rank 2j:$$\begin{aligned} {\varvec{h}}^{(j)} = \frac{1}{j!}\,{\varvec{h}}^{\wedge j}. \end{aligned}$$(2.135)
 (ii)Killing–Yano forms \({{\varvec{f}}^{(j)}}\) of rank \(({D2j})\):$$\begin{aligned} {\varvec{f}}^{(j)} = * {\varvec{h}}^{(j)}. \end{aligned}$$(2.136)
 (iii)Rank2 Killing tensors \({{\varvec{k}}_{(j)}}\),$$\begin{aligned} k_{(j)}^{ab} = \frac{1}{(D{}2j{}1)!}\, f^{(j)}{}^{a}{}_{c_1\dots c_{D{}2j{}1}}\,f^{(j)}{}^{bc_1\dots c_{D{}2j{}1}}. \end{aligned}$$(2.137)
 (iv)Rank2 conformal Killing tensors \({{\varvec{Q}}_{(j)}}\):$$\begin{aligned} Q_{(j)}^{ab} = \frac{1}{(2j{}1)!} \, h^{(j)}{}^{a}{}_{c_1\dots c_{2j{}1}}h^{(j)}{}^{bc_1\dots c_{2j{}1}}. \end{aligned}$$(2.138)
 (v)Killing vectors \({{\varvec{l}}_{(j)}}\):$$\begin{aligned} {\varvec{l}}_{(j)} = {\varvec{k}}_{(j)}\cdot {\varvec{\xi }}. \end{aligned}$$(2.139)
Remark
The objects in the Killing tower encode symmetry properties of the geometry. Killing vectors characterize its explicit symmetries, while Killing tensors describe the hidden symmetries. Together they generate a sufficient set of conserved quantities for a free particle motion, yielding such a motion completely integrable. They also define symmetry operators for the wave operator. The objects in the Killing–Yano tower enable one to separate the Dirac equation. We will discuss all these consequences in Chap. 6.
2.8.6 Geometry admitting the principal tensor
As can be expected, the existence of the principal tensor imposes very restrictive conditions on the geometry. In fact, it determines the geometry: the most general geometry consistent with the existence of the principal tensor is the offshell Kerr–NUT–(A)dS geometry. This geometry is the main object of our study in the following sections. Since it contains, as a special subcase, the metric for a general multiplyspinning black hole, it represents a generalization of the Kerr solution to an arbitrary dimension. For this reason, we start in the next section with a review of the properties of the fourdimensional Kerr solution.
In Chap. 4 we introduce the general higher dimensional offshell Kerr–NUT–(A)dS geometry. We define canonical coordinates in which the metric acquires a manageable form. With this machinery we shall return back to the discussion of the principal tensor in Chap. 5.
The Killing tower can be build directly from the principal tensor, without referring to a particular form of the metric. This construction, sketched above, is discussed in detail in Chap. 5. However, it is also useful to present these objects in an explicit coordinate form. This is the reason why we are postponing the further discussion of the Killing tower till Chap. 5, only after we introduce the metric itself. Since the metric is determined by the existence of the principal tensor, the utilization of the metric in the discussion of the principal tensor does not mean a loss of generality.
3 Kerr metric and its hidden symmetries
The main goal of this review is to describe properties of Kerr–NUT–(A)dS family of higherdimensional black holes related to hidden symmetries. As we shall see many of these properties are similar to those of the Kerr metric. A deep reason for this is the existence of the principal tensor. In order to prepare a reader for ‘a travel’ to higher dimensions, where all the formulas and relations look more complicated and the calculations are more technically involved, we summarize the results concerning the properties of the Kerr metric and its fourdimensional generalization described by the Kerr–NUT–(A)dS spacetime in this chapter. We also briefly discuss a related family of Plebański–Demiański spacetimes which share with the Kerr metric some of its hidden symmetries.
3.1 Kerr metric
The Kerr metric describes a rotating black hole. Found by Kerr (1963), it is the most general stationary vacuum solution of Einstein’s equations in an asymptotically flat spacetime with a regular event horizon. The general properties of the Kerr metric are well known and can be found in many textbooks, see, e.g., Misner et al. (1973), Hawking and Ellis (1973), Wald (1984), Chandrasekhar (1983), Frolov and Novikov (2012), Frolov and Zelnikov (2011). In this section, we discuss the Kerr solution from a perspective of its hidden symmetries. As we shall demonstrate later, many of the remarkable properties of the Kerr geometry, that stem from these symmetries, are naturally generalized to black holes of higherdimensional gravity.
3.2 Carter’s canonical metric
3.2.1 Offshell canonical metric
3.2.2 Going onshell: Kerr–NUT–(A)dS metric
If one requires that the offshell metric satisfies the Einstein equations, the functions \(\varDelta _r(r)\) and \(\varDelta _y(y)\) take a special form. We call the metric (3.9) with such functions \(\varDelta _r(r)\) and \(\varDelta _y(y)\) an onshell metric.
For example, the onshell metric with functions \(\varDelta _r(r)\) and \(\varDelta _y(y)\) given by (3.8) reproduces the Kerr solution. However, one can easily check that this is not the most general vacuum onshell metric. For example, one can add a linear in y term, 2Ny, to the function \(\varDelta _y\). Such a generalization of the Kerr metric is known as the Kerr–NUT solution, and the parameter N is called the NUT (Newmann–Tamburino–Unti) parameter (Newman et al. 1963).
Remark
There are many publications which discuss the physical meaning and interpretation of the NUT parameter. In the presence of NUT parameters the spacetime is not regular and possesses a bad causal behavior, see, e.g., Griffiths et al. (2006), Griffiths and Podolský (2006a), Griffiths and Podolský (2006b), Griffiths and Podolský (2007), see also Clément et al. (2015) for more recent developments. \(\square \)
Remark
The general form of the Kerr–NUT–(A)dS metric in four dimensions was first obtained by Carter (1968b), and independently rediscovered by Frolov (1974) by using the Boyer–Lindquisttype coordinates. The charged generalization of the Kerr–NUT–(A)dS metric, which still takes the canonical form (3.9), was studied in Carter (1968c), Plebański (1975). In 1976 Plebanski and Demianski considered a metric that is conformal to the Kerr–NUT–(A)dS one and demonstrated that such a class of metrics includes also the accelerating solutions, known as the Cmetrics (Plebański and Demiański 1976) (see Sect. 3.9). \(\square \)
3.2.3 Hidden symmetries
The offshell metric (3.9) possesses the following property:
Theorem
The fact that \({\varvec{h}}\) obeys the closed conformal Killing–Yano equation (2.108) can be verified by a straightforward (but rather long) calculation, or perhaps more efficiently, by using the computer programs for analytic manipulations. The condition of nondegeneracy follows from the discussion of the Darboux frame below, proving that \({\varvec{h}}\) is a principal tensor. We may therefore apply the results of Sect. 2.8 and in particular construct the Killing tower associated with \({\varvec{h}}\).
The constructed Killing vectors \({\varvec{\xi }}_{(\tau )}\) and \({\varvec{\xi }}_{(\psi )}\), together with the Killing tensor \({\varvec{k}}\) and the metric \({\varvec{g}}\), are all independent and mutually (Nijenhuis–Schouten) commute. This means that the corresponding four integrals of motion for the geodesics are all independent and in involution, making the geodesic motion completely integrable.
3.2.4 Darboux basis and canonical coordinates

The eigenvalues of the principal tensor, r and y, determined by relations (3.31), are used as two of the canonical coordinates.

Since the principal tensor obeys \(\pounds _{{\varvec{\xi }}} {\varvec{h}}=0\) for both, the primary and secondary Killing vectors \({\varvec{\xi }}_{(\tau )}\) and \({\varvec{\xi }}_{(\psi )}\), its eigenvalues (r, y) are invariant under the action of \(\tau \) and \(\psi \) translations.

Since the Killing vectors \({\varvec{\xi }}_{(\tau )}\) and \({\varvec{\xi }}_{(\psi )}\) commute, they spread twodimensional invariant surfaces; the values of r and y are constant on each such surface. One can hence use the Killing parameters \(\tau \) and \(\psi \) as coordinates on the invariant surfaces. This completes the construction of the canonical coordinates \((\tau ,r,y,\psi )\).
3.2.5 Principal tensor: immediate consequences
 1.
The principal tensor \({\varvec{h}}\) exists for any offshell metric (3.9). It generates the Killing ‘turret’ of symmetries: Killing–Yano tensor \({\varvec{f}}\), conformal Killing tensor \({\varvec{Q}}\), Killing tensor \({\varvec{k}}\), and both generators of the isometries \({\varvec{\xi }}_{(\tau )}\) and \({\varvec{\xi }}_{(\psi )}\).
 2.
The integrability condition for \({\varvec{h}}\) implies, generalizing the result of Collinson (1974), that the spacetime is necessary of the special algebraic type D. See Mason and TaghaviChabert (2010) for a higherdimensional version of this statement.
 3.
The set \(\{{\varvec{g}},{\varvec{k}},{\varvec{\xi }}_{(\tau )},{\varvec{\xi }}_{(\psi )}\}\) forms a complete set of independent mutually (Nijenhuis–Schouten) commuting symmetries that guarantee complete integrability of geodesic motion, see Sect. 3.4.
 4.
The principal tensor also determines the preferred Darboux frame and the canonical coordinates \((\tau ,r,y,\psi )\). Such geometrically defined coordinates are convenient for separating the Hamilton–Jacobi and the wave equation, while the Darboux frame is the one where the Dirac equation separates, see Sect. 3.5.
 5.
The canonical metric (3.9) is the most general spacetime admitting the principal tensor, see also Sect. 3.3.
3.3 Uniqueness of the Kerr metric
The Kerr metric was originally obtained by Kerr (1963) as ‘one of many’ special algebraic type solutions (see Teukolsky 2015 for a historical account). A few years later the solution was rediscovered by Carter (1968b) by imposing a special metric ansatz (assuming two commuting Killing vectors) and by requiring that both the Hamilton–Jacobi and wave equations should be solvable by a method of separation of variables (see also Debever 1971). This not only allowed Carter to rederive the Kerr metric but to generalize it and to include the cosmological constant and the NUT parameter.
Remark
It is well known that any stationary and asymptotically flat black hole solution of the Einstein–Maxwell equations (with nondegenerate horizon) is the Kerr–Newman metric. The extended discussion of this uniqueness theorem and references can be found, e.g., in Mazur (2000), Hollands and Ishibashi (2012). It is interesting that another version of the uniqueness theorem can be formulated:
Theorem
The most general vacuum with \(\varLambda \) solution of the Einstein equations that admits a principal tensor is the Kerr–NUT–(A)dS geometry.
It is a special case of the higherdimensional uniqueness theorem (Houri et al. 2007; Krtouš et al. 2008), which will be discussed in Chap. 5. See Dietz and Rudiger (1981), Taxiarchis (1985) for earlier studies of this issue in four dimensions, where also exceptional metrics corresponding to the null forms of the principal tensor are discussed.
The proof of this statement proceeds in two steps. First, it can be shown that the most general offshell metric that admits the principal tensor has to admit two commuting Killing vectors and takes the form (3.9). Second, by imposing the Einstein equations, the remaining metric functions are uniquely determined and depend on 5 independent constants related to the mass, angular momentum, NUT charge, and the cosmological constant, yielding the Kerr–NUT–(A)dS spacetime. Note that if in addition we require regularity outside the horizon and in particular the absence of cosmic strings (see Sect. 3.8), the NUT charge has to vanish, and the Kerr–(A)dS geometry is recovered.
3.4 Geodesics
3.4.1 Integrals of motion
3.4.2 Firstorder form of geodesic equations
3.4.3 Actionangle variables
Instead of studying the details of particle’s orbits, one might be interested in such ‘global’ characteristics as, for example, the motion frequencies. A useful tool for this is provided by an actionangle formalism. This formalism is also useful for studying the adiabatic invariants and for the development of the perturbation theory when a system slightly differs from a completely integrable one. For the comprehensive discussion of this subject, we refer the reader to the remarkable books by Goldstein et al. (2002) and Arnol’d (1989). Here we just briefly discuss a construction of the actionangle variables for a free particle moving in the metric (3.9). See Appendix B for a general introduction to this subject.
For our dynamical system the coordinate \(\phi \) is cyclic while the value of the coordinate y is bounded and changes in the interval \((y^,y^+)\). The system admits different types of trajectories, depending on the concrete value of the integrals of motion \(\{m^2, K, E,L_\phi \}\) so that the range of the coordinate r may be unbounded. Let us here focus on the case of bounded trajectories for which the radial coordinate changes in the interval \((r^,r^+)\). In such a case the corresponding level set for \((r,y,\phi )\) sector is a compact threedimensional Lagrangian submanifold which, according to the general theorem, is a threedimensional torus. One can choose three independent cycles on this torus as follows. Let us fix y and \(\phi \) and consider a closed path, which propagates from the minimal radius \(r^\) to the maximal radius \(r^+\), and after this returns back to \(r^\) with opposite sign of the momentum. Another path is defined similarly for the ymotion. The third pass \(r=\)const, \(y=\)const is for the \(\phi \)motion.
We will return to the discussion of the actionangle variables in more details later, when discussing geodesics in the higherdimensional Kerr–NUT–(A)dS spacetimes.
3.4.4 Parallel transport
There are many problems with interesting astrophysical applications that require solving the parallel transport equations in the Kerr metric. One of them is a study of a star disruption during its close encounter with a massive black hole, see, e.g., Frolov et al. (1994) and references therein.
The principal tensor also allows one to solve an equation for a propagation of polarization of electromagnetic waves in the spacetime with the metric (3.9). In the leading order of the geometric optics approximation the Maxwell equations reduce to the equations for null geodesics. A vector of a linear polarization \({\varvec{q}}\) is orthogonal to null geodesics and parallelpropagated along them.
3.5 Separation of variables in the canonical metric
In this section we show that the fundamental physical equations do separate in the (offshell) canonical spacetime (3.9). We also discuss the intrinsic characterization of such separability, linked to the existence of the principal tensor. In particular, we concentrate on the Hamilton–Jacobi, Klein–Gordon, and Dirac equations, and do not discuss the electromagnetic and gravitational perturbations. Whereas for the Maxwell equations the link between separability and the principal tensor still can be found, e.g. Benn et al. (1997), Araneda (2016), this is not obvious for the gravitational perturbations.
3.5.1 Hamilton–Jacobi equation
Equations (3.46)–(3.48) allow one to find trajectories of massive particles in the Kerr spacetime. This problem can be alternatively studied by using the Hamilton–Jacobi equation, following Carter’s original paper (Carter 1968a).
Remark
The separability of the Hamilton–Jacobi equation (3.61) is intrinsically characterized by the existence of the separability structure, see Sect. 2.3. Namely, the Killing tensors \({\varvec{g}}\) and \({\varvec{k}}\), together with the Killing vectors \({\varvec{\xi }}_{(\tau )}\) and \({\varvec{\xi }}_{(\psi )}\) satisfy (2.36). Moreover, the Killing tensors have in common the following eigenvectors: \({\varvec{\partial }}_r\) and \({\varvec{\partial }}_y\) that together with \({\varvec{\xi }}_{(\tau )}\) and \({\varvec{\xi }}_{(\psi )}\) obey (2.37). Hence all the requirements of the theorem in Sect. 2.3 are satisfied and the separability of the Hamilton–Jacobi equation is justified. \(\square \)
3.5.2 Separability of the Klein–Gordon equation
Both equations (3.75) have a similar form—they can be written as the secondorder ordinary differential equations with polynomial coefficients. However, there is an essential difference between them. The coordinate y is restricted to the interval \(y\in [a,a]\) and the endpoints of this interval, \(y=\pm a\), are singular points of the yequation. Regularity of Y at these points cannot be satisfied for an arbitrary value of the parameter K, therefore, for regular solutions K has a discrete spectrum. In other words, one needs to solve the Sturm–Liouville boundary value problem. The solutions of this problem for the scalar field are called spheroidal wave functions. They were studied in detail by Flammer (1957). Similar spherical harmonics for fields of higher spin are called spinweighted spheroidal harmonics, see, e.g. Fackerell and Crossman (1977).
3.5.3 Separability of the Dirac equation
As we already mentioned the equations for massless fields with nonzero spin in the Kerr metric allow complete separation of variables. This was discovered by Teukolsky (1972, 1973). Namely, he demonstrated that these equations can be decoupled and reduced to one scalar (master) equation, which in its turn allows a complete separation of variables. Later Wald showed that the solution of the master equation allows one to reconstruct a solution of the original manycomponent equation (Wald 1978).
To separate variables in the massive Dirac equation in the Kerr metric, Chandrasekhar (1976, 1983) used another approach. Namely, he used a special ansatz for the spinor solution, and demonstrated that this allows one to obtain the separated equations for the functions which enter this ansatz. It turns out that the separability of the massive Dirac equation in the Kerr spacetime is also connected with its hidden symmetry, and, as a result, it also takes place in the canonical metric (3.9) for an arbitrary choice of the metric functions \(\varDelta _r(r)\) and \(\varDelta _y(y)\). Let us now demonstrate this result.
Remark
3.6 Special limits of the Kerr metric
3.6.1 Flat spacetime limit: \({M=0}\)
Let us finally note that the primary Killing vector is \({\varvec{\xi }}_{(\tau )}=\frac{1}{3}{\varvec{\nabla }}\cdot {\varvec{h}}={\varvec{P}}_T\), while the secondary Killing vector reads \({\varvec{\xi }}_{(\psi )}={\varvec{k}}\cdot {\varvec{\xi }}_{(\tau )}=a^2{\varvec{P}}_T+a{\varvec{L}}_Z\).
3.6.2 Extremal black hole: \(M=a\)
In the limit \(a=M\), the event and inner horizons have the same radius \(r_+=r_=M\). Such a rotating black hole is called extremal. The spatial distance to the horizon in the limit \(a\rightarrow M\) infinitely grows. It is interesting that some of the hidden symmetries in the vicinity of the horizon of extremal black holes become explicit. Two connected effects take place: the eigenvalues of the principal tensor become functionally dependent, and, besides \({\varvec{\partial }}_t\) and \({\varvec{\partial }}_{\phi }\), two new additional Killing vectors arise. Let us discuss the case of the extremal black hole in more detail.
3.6.3 Nonrotating black hole: \(a=0\)
3.7 Kerr–Schild form of the Kerr metric
It is a remarkable property of the Kerr metric that it can be written in the Kerr–Schild form, that is, as a linear in M deformation of flat spacetime (Kerr and Schild 1965; Debney et al. 1969). This property is intrinsically related to the special algebraic type of the Weyl tensor and the existence of hidden symmetries.
3.8 Remarks on the choice of angle variable
In the next chapter, we shall discuss higherdimensional metrics that generalize the fourdimensional Kerr metric (3.1). We shall see that there exists a natural canonical form for such metrics, where the coordinates are determined by the principal tensor. Part of these coordinates are Killing parameters associated with the corresponding primary and secondary Killing vectors. These angle coordinates are similar to the angle \(\psi \), used in (3.9). A natural question is how these angles are related to the other set of angle variables, similar to \(\phi \) in (3.1). In order to clarify this point, let us make here a few remarks, which will be useful later.
3.8.1 Axis of rotational symmetry
In a general case, one says that a Ddimensional manifold is cyclicly symmetric (or just cyclic) if it is invariant under an action of the oneparametric cyclic group SO(2). It requires that the Killing vector generating this symmetry has closed orbits.
As an example, consider a flat fourdimensional spacetime equipped with cylindrical coordinates \((T,Z,\rho ,\phi )\). In order to have a regular metric at \(\rho =0\), the coordinate \(\varphi \) must be periodic, with period \(2\pi \). The point \(\rho =0\) is a fixed point of the Killing vector \({\varvec{\xi }}_{(\varphi )}\) in the plane \(T,Z=\text {const}\). In a general case, however, a Killing vector field may not have fixed points. For example, consider a Killing vector \({\varvec{\eta }}={\varvec{\xi }}_{(\varphi )}+ \alpha {\varvec{\xi }}_{(Z)}\). One finds \({\varvec{\eta }}^2=\rho ^2 +\alpha ^2>0\) for a nonvanishing value of \(\alpha \). Thus, the Killing vector \({\varvec{\eta }}\) neither has fixed points nor it is cyclic. However, fixed points exist for \({\varvec{\xi }}_{(\varphi )}\). These two cases are illustrated in Fig. 1. The left figure shows a symmetry in a threedimensional flat space generated by the Killing vector \({\varvec{\eta }}\). The orbits are not closed and this vector field does not vanish anywhere. The case when the symmetry is cyclic group, orbits are closed, and there is an axis of symmetry, is shown in the right figure. The symmetry is generated by the Killing vector \({\varvec{\xi }}_{(\varphi )}\), which vanishes at the axis of symmetry, \(\rho =0\).
3.8.2 Twisting construction
In this spacetime the Killing vector \({\varvec{\eta }}\) is cyclic but (as in the previous case) it does not have fixed points. On other hand, the Killing vector \({\varvec{\xi }}_{(\varphi )}\) has still fixed points but it is not cyclic anymore. Its orbits are not closed and the corresponding symmetry group is not SO(2) but \({\mathbb {R}}\). The twisted spacetime thus has only a generalized axis of the symmetry at \(\rho =0\). This axis does not form a regular submanifold of the full twisted spacetime.
3.8.3 Rotating string and conical singularity
Because of the timelike nature of the Killing vector \({\varvec{\xi }}_{(T)}\), a new phenomenon occurs in this case. The Killing vector \({\varvec{\eta }}\) is spacelike far from the axis, for \(\rho >\alpha \), and timelike near the axis, for \(\rho <\alpha \). The surface \(\rho =\alpha \), where \({\varvec{\eta }}^2=0\), is an ergosurface of the Killing vector \({\varvec{\eta }}\). Let us emphasize that, although \({\varvec{\eta }}^2=0\), these are not fixed points of the Killing vector \({\varvec{\eta }}\) since \({\varvec{\eta }}\) is not vanishing here. The ergosurface contains orbits of the symmetry which are null closed curves, they correspond to light rays orbiting the axis in closed trajectories. Inside the ergosurface, where the Killing vector is timelike, the orbits of the symmetry are closed timelike curves. Clearly, such a behavior is not very physical. However, it seems that in a generic case it may not be escaped.
There is yet another aspect related to the identification of the axis of the symmetry and its regularity. Let us consider an axisymmetric spacetime with coordinate \(\varphi \in (0,2\pi )\) which parameterizes orbits of the cyclic symmetry. In general, the metric may not be regular on the axis—it can contain a conical singularity. Such a singularity can be eliminated choosing a different range of periodicity for coordinate \(\varphi \). It can be achieved by introducing a rescaled coordinate \(\phi =\beta \varphi \) which is required to be periodic on the interval \((0,2\pi )\). Physically, the conical singularity corresponds to a static thin string on the axis (Vilenkin and Shellard 2000; Griffiths and Podolský 2009).
3.8.4 Kerr geometry
For the canonical metric (3.9) of the rotating black hole spacetime, the coordinates \(\tau \) and \(\psi \) are directly connected with the principal tensor of this spacetime. Namely, they are the proper Killing coordinates for the primary \({\varvec{\xi }}_{(\tau )}\) and secondary \({\varvec{\xi }}_{(\psi )}\) Killing vectors. If one makes the coordinate \(\psi \) to be cyclic by identifying \(\psi =0\) and \(\psi =2\pi \), the corresponding spacetime is not axisymmetric, since the Killing vector \({\varvec{\xi }}_{(\psi )}\) does not have fixed points.
We can then ask if one can find the correct axisymmetric coordinate. For that we need to find a Killing vector which has fixed points. Let us consider a vector \({\varvec{\eta }}={\varvec{\xi }}_{(\psi )}+ \alpha {\varvec{\xi }}_{(\tau )}\) and require that it vanishes at some points. It can be shown that this happens only if the following two conditions are met: \(\varDelta _y=0\) and \(\alpha =a^2\). The vector \({\varvec{\eta }}={\varvec{\xi }}_{(\psi )}a^2{\varvec{\xi }}_{(\tau )}\) thus has fixed points at roots of \(\varDelta _y\). The coordinates adapted to this Killing vector are \(t=\tau +a^2\psi \), \({\varphi =\psi }\). However, if one makes \(\varphi \) periodic on the interval \((0,2\pi )\), there would be a conical singularity on the axis. One has to make an additional rescaling \({\phi =a\varphi }\) leading to the Boyer–Lindquist coordinates \((t,r,\theta ,\phi )\) given by (3.6). If the coordinate \(\phi \) is made periodic on interval \((0,2\pi )\), the axis is regular: the spacetime contains a cyclic Killing vector \({\varvec{\xi }}_{(\phi )}=\frac{1}{a}{\varvec{\xi }}_{(\psi )}a{\varvec{\xi }}_{(\tau )}\) with fixed points identifying the axis and there is no conical singularity on this axis.
3.8.5 Effect of NUT charges
Let us briefly comment on a more complicated metric with nontrivial NUT parameters for which the metric functions \({\varDelta _r}\) and \({\varDelta _y}\) are given by (3.18) with \({\varLambda =0}\). The polynomial \({\varDelta _y}\) has now two nontrivially different roots \({{}^\pm y}\) and the coordinate y runs between these roots, \({y\in ({}^y,{}^+y)}\). In this case one can find two candidates for the Killing vector with fixed points: \({{\varvec{\eta }}_+}\) and \({{\varvec{\eta }}_}\), with fixed points at \({y={}^+y}\) and \({y={}^y}\), respectively. One can choose one of the properly rescaled corresponding coordinates, say \({\phi _+}\), to be periodic with period \({2\pi }\). With such a choice, the submanifold \({y={}^+y}\) becomes the regular axis. Physically, it corresponds only to a semiaxis of the spacetime. The other semiaxis \({y={}^y}\) is not regular, the cyclic Killing vector \({{\varvec{\eta }}_+}\) does not have fixed points here. Of course, one can assume periodicity of the other coordinate \({\phi _}\), making thus the semiaxis \({y={}^y}\) regular. However, the semiaxis \({y={}^+y}\) becomes now nonregular. So, it is not a priori guaranteed that one can chose a unique Killing vector which makes the spacetime globally axisymmetric.
3.9 Hidden symmetries of the Plebański–Demiański metric
The Plebański–Demiański metric (Plebański and Demiański 1976) is the most general fourdimensional electrovacuum solution of Einstein’s equations that is stationary, axisymmetric, and whose Weyl tensor is of the special algebraic type D. It describes a wide family of spacetimes that generalize the Kerr–NUT–(A)dS family described in previous sections. Besides the cosmological constant, mass, rotation, and NUT parameter it also admits electric and magnetic charges and the acceleration parameter. As we shall discuss now, the Plebański–Demiański metric admits a ‘weaker’ (conformal) form of hidden symmetries of the Kerr geometry.
3.9.1 Solution
3.9.2 Hidden symmetries
For a discussion of the integrability of a charged particle motion in the Plebański–Demiański metric see Duval and Valent (2005). As a consequence of the existence of the conformal Killing–Yano 2form \({\varvec{h}}\), also the massless Hamilton–Jacobi, Klein–Gordon, and Dirac equations separate in the Plebański–Demiański backgrounds. We do not review here the corresponding calculations. The first two are easy to perform and we refer to the original papers (Kamran and McLenaghan 1983, 1984a) for the separability of massless Dirac equation; see also Torres del Castillo (1988), SilvaOrtigoza (1995) for a discussion of electromagnetic and Rarita–Swinger perturbations.
3.9.3 Higherdimensional generalizations
As we shall see in the next chapter, the fourdimensional Kerr–NUT–(A)dS metrics can be generalized to higher dimensions. However, similar attempts for the Plebański–Demiański metric have failed so far. In particular, people have tried to obtain a higherdimensional generalization of an accelerated black hole described by the so called Cmetric, which is a special case of the Plebański–Demiański class.
Remark
The Cmetric typically describes a pair of black holes moving in the opposite direction with constant acceleration caused either by a cosmic string of negative energy density between them or by two positiveenergy strings pulling the black holes from infinity. As the string is present, the corresponding solution does not represent, strictly speaking, a regular isolated black hole. \(\square \)
A straightforward method of multiplying the higherdimensional Kerr–NUT–(A)dS spacetime (4.1) with a properly chosen conformal factor \(\varOmega \), accompanied by a proper adjustment of metric functions \(X_\mu \), turned out to be very naive and does not work, e.g., Kubizňák and Krtouš (2007). However, a partial success has been achieved in five dimensions, where two different factors, rescaling various parts of the Kerr–NUT–(A)dS spacetime, have been used to construct a new metric whose limits lead to the black holes of spherical horizon topology on one side and to the black rings with toroidal horizon topology on the other side (Lü et al. 2009, 2010; Lü and VázquezPoritz 2014).
4 Higherdimensional Kerr–NUT–(A)dS metrics
Higherdimensional Kerr–NUT–(A)dS metrics (Chen et al. 2006a) describe a large family of geometries of various types and signatures that solve the vacuum Einstein equations with and without the cosmological constant. Parameterized by a set of free parameters that can be related to mass, rotations, and NUT parameters, they directly generalize the fourdimensional Carter’s canonical metric (3.9) studied in the previous chapter. The general rotating black holes of Myers and Perry (E.7) (Myers and Perry 1986), their cosmological constant generalizations due to Gibbons et al. (2004, 2005), the higherdimensional TaubNUT spaces (Mann and Stelea 2004, 2006; Clarkson and Mann 2006; Chen et al. 2007), or the recently constructed deformed and twisted black holes (Krtouš et al. 2016a), all emerge as certain limits or subcases of the Kerr–NUT–(A)dS spacetimes. All such geometries inherit hidden symmetries of the Kerr–NUT–(A)dS metrics.
In this chapter, we perform a basic analysis of the Kerr–NUT–(A)dS metrics, discussing their signature, coordinate ranges, scaling properties, and meaning of free metric parameters. We also identify their several special subcases, namely, the sphere, the Euclidean instanton, and various black hole solutions. The discussion of hidden symmetries is postponed to the next chapter.
4.1 Canonical form of the metric
4.1.1 Metric
Remark
As we shall see in the next chapter, both types of canonical coordinates are uniquely determined by the principal tensor \({\varvec{h}}\). Namely, \(x_{\mu }\)’s are the eigenvalues of the principal tensor and \(\psi _j\)’s are the Killing coordinates associated with the primary (\({j=0}\)) and secondary (\({j>0}\)) Killing vectors generated by this tensor. Such a choice of coordinates, internally connected with the principal tensor, makes the canonical form of the metric (4.1) quite simple. It is also directly ‘linked to’ the separability properties of the geometry. \(\square \)
The metric (4.1) is written in the most symmetric form adjusted to the Euclidean signature and is very convenient for the analysis of explicit and hidden symmetries. This most symmetric form is naturally broken when one describes the black hole case: in order to guarantee the Lorentzian signature, one needs to assume that some of the coordinates and parameters take imaginary values. In what follows we shall call this procedure a ‘Wick rotation’. We should also mention that coordinates \(\psi _j\) are different from the ‘standard azimuthal’ angles \(\phi _\mu \), used in the Boyer–Lindquist form of the Myers–Perry metric (see next section).
4.1.2 Special Darboux frame
4.1.3 Curvature
4.1.4 Onshell metric
Remark
It is interesting to note that, similar to four dimensions, a single equation corresponding to the trace of the Einstein equations, \({R=\frac{2D}{D2}\varLambda }\), almost fully determines relations (4.16). Once this equation is valid, all other Einstein’s equations require just equality of the absolute terms in all polynomials \({X_\mu }\) and otherwise they are identically satisfied (Houri et al. 2007). \(\square \)
4.2 Parameters and alternative form of the metric
Before we proceed to discussing various special cases of the onshell Kerr–NUT–(A)dS spacetimes, let us comment on a different, more convenient for its interpretation, form of the metric, and the parameters of the solution. For simplicity, in the rest of this section we restrict our discussion to even dimensions \({D=2{{n}}}\), that is \({\varepsilon =0}\), analysis in odd dimensions would proceed analogously.
4.2.1 Parametrization of metric functions
4.2.2 Two types of angular variables
As we already mentioned, the canonical ‘angles’ \(\psi _k\) in the metric (4.18) are the Killing parameters for the primary and secondary Killing vectors constructed from the principal tensor. In a general case, such Killing vectors do not have fixed points and the angles do not correspond to azimuthal angles in independent rotation 2planes. However, there may exist other angular variables such that the corresponding Killing vectors have fixed points and, hence, they define axes of symmetry and planes of rotation.
Remark
The same thing happens with the Kerr metric written in the canonical form (3.9). As explained in Sect. 3.8, the axisymmetry of the Kerr metric implies that, aside the Killing coordinate \(\psi \), there exists another angular variable \(\phi \), such that the Killing vector \({{\varvec{\partial }}}_{\phi }\) has fixed points and corresponds to the azimuthal angle in the 2plane of rotation. \(\square \)
4.2.3 Parameters of the solution
Taking into account this freedom, we find that for a fixed cosmological constant the onshell Kerr–NUT–(A)dS metric in \({D=2{{n}}}\) dimensions contains \(2{{n}}1\) independent parameters. In the black hole case they are connected with mass, \(({{n}}1)\) rotations parameters, and \(({{n}}1)\) NUT charges.
Similar counting would proceed in odd dimensions, where the analogous scaling freedom reduces the number of independent free parameters in \(D=2n+1\) dimensions to \(2{{n}}1\), giving mass, \({{n}}\) rotations parameters, and \(({{n}}2)\) NUT parameters for the black hole case, see Chen et al. (2006a).
4.3 Euclidean signature: instantons
The Kerr–NUT–(A)dS metric can describe various geometries. Depending on a choice of coordinate ranges and values of parameters it can have both Euclidean and Lorentzian signatures. We will see in the next chapter that common feature of the solution independent of a particular interpretation of the geometry is the presence of a rich symmetry structure. If one is interested mainly in the symmetries of the Kerr–NUT–(A)dS geometry and its integrability and separability properties, the general form of the metric presented above is sufficient to proceed directly to Chaps. 5 and 6.
In the rest of this chapter we make a short overview of several important special cases of the Kerr–NUT–(A)dS metric. In this section we explain appropriate coordinate ranges for Euclidean version of the geometry, in the next section we discuss the Wick rotations of coordinates appropriate for the Lorentzian signature.
4.3.1 Sphere
It is interesting to observe that this metric describes the maximally symmetric geometry of the sphere of the same radius \(1/\sqrt{\lambda }\) for any choice of parameters \({a_\mu }\). Going in the opposite direction, from the spherical geometry (4.32), expressed in coordinates \(({\rho _0,\,\rho _\mu ,\,\phi _\alpha })\), to the Kerr–NUT–(A)dS metric (4.28), and then to (4.22), expressed in the coordinates \(({x_\mu ,\,\phi _\alpha })\), it turns out that the parameters \({a_\mu }\) characterize a freedom in implicit definitions (4.29) of variables \({x_\mu }\) obeying the constrain (4.30). Jacobi coordinates \({x_\mu }\) are sortof elliptic coordinates (the surfaces of given \({x_\mu }\) being elliptical or hyperbolic surfaces) with an exact shape governed by parameters \({a_\mu }\).
For nonvanishing parameters \({b_\mu }\) one cannot use the orthogonality relation (D.24) and transform the Kerr–NUT–(A)dS metric (4.22) to the form (4.28). However, we have at least learned that coordinates \({x_\mu }\) take values between the roots of metric functions \({X_\mu }\), and these roots represents the axes of the Killing symmetry. This property survives in the generic case.
Let us finally note that the metric (4.32) or the corresponding Kerr–NUT–(A)dS form (4.22) can also describe a pseudosphere of various signatures, obtainable by a suitable Wick rotation of coordinates. We will discuss this below after we introduce the black hole solutions.
4.3.2 Euclidean instantons
Let us now describe the choice of coordinate ranges and parameters for which the Kerr–NUT–(A)dS metric describes a nontrivial geometry of the Euclidean signature.
Remark
For briefness we call such metrics Euclidean instantons or simply instantons. In fact, in order to be a ‘proper instanton’, the space must be regular and the corresponding gravitational action finite. These properties can impose additional restrictions on the parameters of the solution, which we do not study here and refer the interested reader to a vast literature on the subject of gravitational instantons, e.g. Hawking (1977), Page (1978a), Page (1978b), Gibbons and Hawking (1979), Eguchi et al. (1980), Hunter (1998), Mann (1999), Chamblin et al. (1999), Mann and Stelea (2004), Mann and Stelea (2006), Clarkson and Mann (2006), Chen et al. (2007), Yasui and Houri (2011). \(\square \)
For such a choice the metric (4.18) represents the Euclidean instanton of signature \({(++\cdots +)}\). Parameters \(b_\mu \) encode deformations of the geometry, namely how it deviates from the geometry of the sphere. For nonvanishing \(b_\mu \), parameters \(a_\mu \) become essential. That is they do not just label a choice of coordinates, as in the maximally symmetric case, but their change results in the change of the geometry (e.g. its curvature).
The global definition and regularity of the geometry described by metrics (4.18) or (4.22) has to be established by specifying which Killing angles should be cyclic and what are the periods of these cyclic angles. In the maximally symmetric case, which we discussed above, there was a natural choice of cyclic coordinates \({\phi _\alpha \in (\pi ,\pi )}\) with their natural identification at \({\phi _\alpha =\pm \pi }\). However, in general, any linear combination of Killing coordinates (with constant coefficients) forms again a Killing coordinate and it is not a priory clear which of the Killing coordinates should be periodic. Learning a lesson from the maximally symmetric case, the angles \(\psi _k\) are typically not those which should be periodic. Since Killing coordinates are nontrivially coupled in the metric (the metric is not diagonal in these directions), a particular choice of the periodicity of Killing coordinates can introduce a nontrivial twisting of the geometry, as well as possible irregularities on the axes. We will not discuss these characteristics in more detail as this is still an open problem awaiting its complete solution. For our purposes it is sufficient to simply remember that the Euclidean instanton describes a deformed and twisted sphericallike geometry. Other examples of compact Riemannian manifolds that can be obtained as special limits of the Kerr–NUT–(A)dS metrics include the most general explicitly known Einstein–Kähler and Einstein–Sasaki metrics, see e.g. Yasui and Houri (2011) and references therein.
4.4 Lorentzian signature: black holes
Let us now discuss the Kerr–NUT–(A)dS metrics with the Lorentzian signature. For vanishing NUT parameters such metrics describe an isolated rotating higherdimensional black hole in either asymptotically flat or asymptotically (anti)de Sitter spacetime. We start our discussion with the case of nonvanishing NUT parameters and proceed to the Kerr(A)dS and Myers–Perry black holes in the the next step.
4.4.1 General multiplyspinning black holes with NUTs
4.4.2 Vacuum rotating black holes with NUTs
4.4.3 Kerr–(A)dS, Myers–Perry, and Tangherlini metrics
4.5 MultiKerr–Schild form
In Sect. 4.5, we have seen that the Myers–Perry metric can be cast as a linear in mass deformation of the flat space, that is in the Kerr–Schild form (E.29). The same remains true for the higherdimensional Kerr(A)dS solutions (4.62) of Gibbons et al. (2004, 2005), replacing the flat space with the corresponding maximally symmetric geometry. Remarkably, in the presence of NUT charges, the onshell metric (4.1) can be written in the multiKerr–Schild form (Chen and Lu 2008), that is as a multilinear deformation of the maximally symmetric space, with deformation terms proportional to generalized masses, see (4.82) below. The modified construction goes as follows.
The coordinates \(\hat{\psi }_j\) need more attention. One has to modify their definition in such a way that in (4.79) the sum runs only over Wick rotated coordinates.
The case when only one coordinate, \(x_{{n}}=ir\), is Wick rotated covers the Lorentzian signature. It demonstrates, that the black hole solution (with vanishing NUT charges) can be written in the standard Kerr–Schild form. The fourdimensional case discussed in Sect. 3.7 is an example of this case, as well as the Kerr–Schild form of the Myers–Perry solution discussed in Appendix E (after some additional effort of identifying Myers–Perry and canonical coordinates).
The opposite case of the multiKerr–Schild form of the metric, when all \(x_\nu \) coordinates are Wickrotated, can be related to an analogous discussion in Chen and Lu (2008), where all coordinates \(\psi _j\) have been Wickrotated and the multiKerr–Schild form has been obtained.
5 Hidden symmetries of Kerr–NUT–(A)dS spacetimes
In the previous chapter the Kerr–NUT–(A)dS metric and its interpretation were discussed. Let us consider now the symmetries of this geometry in more detail. Namely, we shall show that, similar to its fourdimensional counterpart, this metric admits the principal tensor. The latter generates the whole tower of explicit and hidden symmetries. In fact, the geometry itself is uniquely determined by the principal tensor. Let us begin exploring this remarkable geometric construction.
5.1 Principal tensor
5.2 Killing tower
As we have already revealed in Chap. 2, from the principal tensor \({\varvec{h}}\) one can generate the whole tower of explicit and hidden symmetries of the Kerr–NUT–(A)dS geometry. We call this set a Killing tower. In what follows we shall review two methods for generating such a tower: a direct method of construction (based on theorems of Chap. 2) and the method of a generating function.
5.2.1 Direct method of construction
 (i)By employing the theorem (2.86) of Chap. 2, one can construct a tower of closed conformal Killing–Yano tensors by taking various wedge products of the principal tensor \({\varvec{h}}\) with itself (Krtouš et al. 2007a). Since \({\varvec{h}}\) is a nondegenerate 2form, this gives the following \({{{n}}+1}\) closed conformal Killing–Yano forms \({{\varvec{h}}^{(j)}}\) of increasing rank 2j \(({j=0,\dots ,{{n}}}\)):$$\begin{aligned} {\varvec{h}}^{(j)} = \frac{1}{j!}\,{\varvec{h}}^{\wedge j}. \end{aligned}$$(5.6)
Note that for \({j=0}\) we have a trivial 0form \({{\varvec{h}}^{(0)}=1}\). We also have \({{\varvec{h}}^{({{n}})}=\sqrt{A^{({{n}})}}\,{\varvec{\varepsilon }}}\) and \({{\varvec{h}}^{({{n}})} = \sqrt{A^{({{n}})}}\,{\varvec{\varepsilon }}\cdot {\hat{{\varvec{e}}}_{0}}}\) for even and odd dimensions, respectively. Here, as earlier, \({\varvec{\varepsilon }}\) is the LeviCivita tensor.
 (ii)As discussed after (2.65), the Hodge dual of a closed conformal Killing–Yano 2jform \({{\varvec{h}}^{(j)}}\) is a Killing–Yano \(({D{}2j})\)form, which we call \({{\varvec{f}}^{(j)}}\):$$\begin{aligned} {\varvec{f}}^{(j)} = * {\varvec{h}}^{(j)}. \end{aligned}$$(5.7)In particular, this gives the LeviCivita tensor \({{\varvec{f}}^{(0)}={\varvec{\varepsilon }}}\) for \(j=0\). In even dimensions one has \({{\varvec{f}}^{({{n}})} = \sqrt{A^{({{n}})}}}\). In odd dimensions, \({{\varvec{f}}^{({{n}})}=\sqrt{A^{({{n}})}}\,{\hat{{\varvec{e}}}^{0}}}\). The vector version of \({{\varvec{f}}^{({{n}})}}\) has to be a Killing vector. Namely, we get$$\begin{aligned} {\varvec{f}}^{({{n}})}=\frac{1}{\sqrt{c}}{{\varvec{\partial }}}_{\psi _{{n}}}. \end{aligned}$$(5.8)
 (iii)Partial contractions of squares of Killing–Yano forms \({{\varvec{f}}^{(j)}}\) define the following rank2 Killing tensors \({{\varvec{k}}_{(j)}}\), cf. (2.76),$$\begin{aligned} k_{(j)}^{ab} = \frac{1}{(D{}2j{}1)!}\, f^{(j)}{}^{a}{}_{c_1\dots c_{D{}2j{}1}}\,f^{(j)}{}^{bc_1\dots c_{D{}2j{}1}}. \end{aligned}$$(5.9)For \({j=0}\), the Killing tensor reduces to the metricFor odd dimensions the top Killing tensor is reducible, \({{\varvec{k}}_{({{n}})}=A^{({{n}})}\,{\hat{{\varvec{e}}}_{0}}\,{\hat{{\varvec{e}}}_{0}}}={c^{1}{{\varvec{\partial }}}_{\psi _{{n}}}{{\varvec{\partial }}}_{\psi _{{n}}}}\), whereas in even dimensions we define \({{\varvec{k}}_{({{n}})}={\varvec{0}}}\).$$\begin{aligned} k_{(0)}^{ab}= g^{ab}. \end{aligned}$$(5.10)
 (iv)Similarly, partial contractions of closed conformal Killing–Yano forms \({{\varvec{h}}^{(j)}}\) give rank2 conformal Killing tensors \({{\varvec{Q}}_{(j)}}\):$$\begin{aligned} Q_{(j)}^{ab} = \frac{1}{(2j{}1)!} \, h^{(j)}{}^{a}{}_{c_1\dots c_{2j{}1}}h^{(j)}{}^{bc_1\dots c_{2j{}1}}. \end{aligned}$$(5.11)We define \({{\varvec{Q}}_{(0)}={\varvec{0}}}\), and introduce a simpler notation \({{\varvec{Q}}}\) for the first conformal Killing tensor:$$\begin{aligned} Q^{ab} \equiv Q_{(1)}^{ab} = h^{a}{}_{c}\,h^{bc}. \end{aligned}$$(5.12)The conformal Killing tensors \({{\varvec{Q}}_{(j)}}\) contain essentially the same information as the Killing tensors \({{\varvec{k}}_{(j)}}\). Namely, for all \({j=1,\dots ,{{n}}}\) it holdswhere the scalar function \({A^{(j)}}\) can be expressed as$$\begin{aligned} {\varvec{k}}_{(j)}+{\varvec{Q}}_{(j)} = A^{(j)}\,{\varvec{g}}. \end{aligned}$$(5.13)Here we used the scalar product (A.5). It turns out that functions \({A^{(j)}}\) are exactly the symmetric polynomials (4.2) introduced earlier. The conformal Killing tensors and the Killing tensors are also related by$$\begin{aligned} A^{(j)} = {\varvec{h}}^{(j)}\bullet {\varvec{h}}^{(j)} = {\varvec{f}}^{(j)}\bullet {\varvec{f}}^{(j)} = \frac{1}{2j}\,Q_{(j)}{}^{n}_{n} = \frac{1}{D{}2j}\,k_{(j)}{}^{n}_{n}. \end{aligned}$$(5.14)$$\begin{aligned} Q_{(j)}^{ac} = h^{a}{}_{b}\, h^{c}{}_{d}\, k_{(j{}1)}^{bd} = Q^{a}{}_{b}\,k_{(j{}1)}^{bc}. \end{aligned}$$(5.15)
 (v)We conclude our construction by defining the following vectors:They turn out to be Killing vectors related to coordinates \({\psi _j}\), namely \({\varvec{l}}_{(j)}=\partial _{\psi _j}\). Since \({\varvec{l}}_{(j)}\) are constructed using the primary Killing vector \({{\varvec{\xi }}}\), we call them secondary Killing vectors. For \({j=0}\) we have \({{\varvec{l}}_{(0)}={\varvec{\xi }}}\). Since in even dimensions the top Killing tensor \({{\varvec{k}}_{({{n}})}}\) vanishes by definition, we have \({{\varvec{l}}_{({{n}})}=0}\). On the other hand in odd dimensions the top Killing vector is nontrivial and reads$$\begin{aligned} {\varvec{l}}_{(j)} = {\varvec{k}}_{(j)}\cdot {\varvec{\xi }}. \end{aligned}$$(5.16)$$\begin{aligned} {\varvec{l}}_{({{n}})} = \sqrt{cA^{({{n}})}}\,{\hat{{\varvec{e}}}_{0}}= {{\varvec{\partial }}}_{\psi _{{n}}}. \end{aligned}$$(5.17)These Killing vectors can be generated from the Killing copotentials \({{\varvec{\omega }}^{(j)}}\), e.g., Kastor et al. (2009), Cvetic et al. (2011),where$$\begin{aligned} {\varvec{l}}_{(j)}={{\varvec{\nabla }}}\cdot {\varvec{\omega }}^{(j)}, \end{aligned}$$(5.18)for \(j=0,\dots , {{n}}1\), and \({\varvec{\omega }}^{(n)}=\frac{1}{n!}\sqrt{c}\,{\varvec{*}}({\varvec{b}}\wedge {\varvec{h}}^{\wedge (n1)})\) in odd dimensions. Note that, apart from \({\varvec{\omega }}^{(0)}\), the Killing copotentials are not closed, \({\varvec{d\omega }}^{(j)}\ne 0\).$$\begin{aligned} \omega ^{(j)}_{ab}=\frac{1}{D{}2j{}1}\,k_{(j)}{}_{a}{}^n \,h_{nb}\, \end{aligned}$$(5.19)Let us also mention the following useful relation:which implies—through the Cartan identity and closeness of \({{\varvec{h}}}\)—that the principal tensor \({{\varvec{h}}}\) is conserved along the vector fields \({{\varvec{l}}_{(j)}}\),$$\begin{aligned} {\varvec{h}}\cdot {\varvec{l}}_{(j)} = \frac{1}{2}{{\varvec{d}}}A^{(j+1)}, \end{aligned}$$(5.20)$$\begin{aligned} \pounds _{{\varvec{l}}_{(j)}}{\varvec{h}} = 0. \end{aligned}$$(5.21)
Remark
Some properties of the above constructed Killing tower are simpler to prove than others. Namely, by theorems of Chap. 2 we know that \({{\varvec{h}}^{(j)}}\), \({{\varvec{f}}^{(j)}}\), \({{\varvec{k}}_{(j)}}\) and \({{\varvec{Q}}_{(j)}}\) are closed conformal Killing–Yano forms, Killing–Yano forms, Killing tensors, and conformal Killing tensors, respectively. However, to show that \({{\varvec{l}}_{(j)}}\) are Killing vectors (and in particular that \({{\varvec{\xi }}}\) given by (5.2) is indeed a primary Killing vector) and to demonstrate the commutation relations (5.22) poses a more difficult task. Of course, one way to show these is a ‘brute force’ calculation, employing the explicit form of the Kerr–NUT–(A)dS metric and the induced covariant derivative. However, it turns out that it is possible to prove all these relations directly from the integrability conditions of the principal tensor \({{\varvec{h}}}\), without referring to a particular form of the metric (Krtouš 2017). We will sketch the corresponding line of reasoning in Sect. 5.4.
5.2.2 Method of generating functions
There exists another (more compact) way for constructing the Killing tower (Krtouš 2017). Namely, it is possible to define a \(\beta \)dependent Killing tensor \({{\varvec{k}}(\beta )}\) and a \(\beta \)dependent Killing vector \({{\varvec{l}}(\beta )}\), both functions of a real parameter \({\beta }\), such that the Killing tensors \({\varvec{k}}_{(j)}\) and the Killing vectors \({\varvec{l}}_{(j)}\) in the Killing tower above emerge as coefficients of the \({\beta }\)expansion of \({{\varvec{k}}(\beta )}\) and \({{\varvec{l}}(\beta )}\), respectively. This procedure is related to Krtouš et al. (2007a), Houri et al. (2008a), where generating function for conserved observables is studied.
5.2.3 Killing tower in a Darboux frame
The link between definitions (5.26)–(5.29) and expansions (5.30)–(5.32) can be established by writing down all the quantities in the Darboux frame determined by the principal tensor \({\varvec{h}}\). For that, it is sufficient to specify the Darboux frame just in terms of the principal tensor, without refereing to its explicit coordinate form (4.7). However, if one seeks the expressions in terms of canonical coordinates, one can easily substitute relations (4.7) and (4.8).
5.3 Uniqueness theorem
It is obvious from the above construction of the Killing tower that the principal tensor \({\varvec{h}}\) determines uniquely a set of canonical coordinates. Namely, the set of n functionally independent eigenvalues \(x_\mu \) is supplemented by a set of \({{n}}+\varepsilon \) Killing coordinates \(\psi _j\) associated with the Killing vectors \({{\varvec{l}}_{(j)}}\). It is then no such a wonder that the principal tensor uniquely defines the corresponding geometry.^{16} This geometry has a local form of the offshell Kerr–NUT–(A)dS metric and is determined up to \({{n}}\) arbitrary metric functions of a single variable, \(X_\mu =X_\mu (x_\mu )\). It also possess a number of remarkable geometric properties. Namely the following central theorem has been formulated in Krtouš et al. (2008), culminating the previous results from Houri et al. (2007):
Uniqueness theorem
The most general geometry which admits a principal tensor can be locally written in the offshell Kerr–NUT–(A)dS form (4.1). When the Einstein equations are imposed, the geometry is given by the onshell Kerr–NUT–(A)dS metric described by the metric functions (4.16).
Moreover, this metric possesses the following properties:
Theorem
The offshell Kerr–NUT–(A)dS metric is of the special type D of higherdimensional algebraic classification. The geodesic motion in this spacetime is completely integrable, and the Hamilton–Jacobi, Klein–Gordon, and Dirac equations allow a separation of variables.
We refer to the literature (Houri et al. 2008a, 2007; Krtouš et al. 2008; Yasui 2008; Houri et al. 2009; Yasui and Houri 2011; Krtouš 2017) for various versions of the proof of the uniqueness theorem. The fact that the metric is of the type D (Hamamoto et al. 2007) of higherdimensional algebraic classification (Coley et al. 2004; Ortaggio et al. 2013; Pravda et al. 2007) follows directly from studying the integrability conditions of a nondegenerate conformal Killing–Yano 2form (Mason and TaghaviChabert 2010). The separability and integrability properties of the Kerr–NUT–(A)dS geometry will be demonstrated in the next chapter.
Remark
Perhaps the ‘shortest route’ to the uniqueness theorem and the Kerr–NUT–(A)dS metric is through the separability structure theory for the Hamilton–Jacobi and Klein–Gordon equations, see Sect. 2.3. Namely, the existence of the principal tensor implies the existence of a Killing tower of symmetries, which in its turn implies the separability of the Hamilton–Jacobi and Klein–Gordon equations. It then follows that one can use the canonical metric constructed in Benenti and Francaviglia (1979) admitting such separability structure. In the spirit of Carter’s derivation of the fourdimensional Kerr–NUT–(A)dS metric (Carter 1968b), this then directly leads to the higherdimensional Kerr–NUT–(A)dS geometry (Houri et al. 2008a, 2007; Yasui and Houri 2011), see also Kolář and Krtouš (2016). \(\square \)
Since the existence of a principal tensor \({\varvec{h}}\) uniquely determines the offshell Kerr–NUT–(A)dS geometry, when discussing the Killing tower one does not need to strictly distinguish among the properties that follow from general considerations with Killing–Yano tensors, the properties that follow from the existence of a (general) Darboux basis, and the properties that use the explicit form of the Darboux basis of the Kerr–NUT–(A)dS geometry. However, all the properties of the Killing tower can be derived directly from the properties of the principal tensor, without referring to the explicit form of the metric.
Let us finally note that when the nondegeneracy condition on the principal tensor is relaxed, one obtains a broader class of geometries that has been named the generalized Kerr–NUT–(A)dS geometry (Houri et al. 2008b, 2009; Oota and Yasui 2010). This class will be briefly reviewed in Sect. 7.4.
5.4 Proof of commutation relations
By now we have established most of the properties of objects in the Killing tower. However, we have not yet proved the commutation relations (5.22) or (5.33). Since \({{\varvec{k}}_{(0)}={\varvec{g}}}\), these relations in particular imply that \({{\varvec{k}}_{(j)}}\) and \({{\varvec{l}}_{(j)}}\) are Killing tensors and Killing vectors, respectively. The fact that \({{\varvec{k}}_{(j)}}\) are Killing tensors follows directly from their construction. However, that \({{\varvec{l}}_{(j)}}\) are Killing vectors we observed only using the identity (5.5) and its consequences (5.57). In other words, we have used the explicit form (4.1) of the Kerr–NUT–(A)dS metric. However, as mentioned above, it is possible to demonstrate the commutativity (5.33) directly from the existence of the principal tensor and without any reference to canonical coordinates, proving in particular that \({{\varvec{l}}_{(j)}}\) are Killing vectors. We give here a brief overview of such a procedure, for details see Krtouš (2017).
5.4.1 Commutation relations
To summarize, the proof of the Nijenhuis–Schouten commutativity (5.22) reduces to proving the properties (5.73) and (5.74) for the primary Killing vector \({{\varvec{\xi }}}\). It turns out, that both these conditions follow in a complicated way from the integrability conditions for the principal tensor. We discus this in more details in the next section.
5.4.2 Structure of the curvature
To complete the proof of the commutativity (5.22) we need first to discuss the integrability conditions for the principal tensor and establish their implications for the structure of the curvature tensor.
In particular, the commutativity (5.81) tells us that \([\,{\mathbf {Ric}},{\varvec{h}}\,] = 0\), which guarantees that the right hand side of (5.76) vanishes, proving thus that \({{\varvec{\xi }}}\) is a Killing vector. Moreover, (5.81) also implies that all the tensors \({\mathbf {Rh}^{(p)}}\) and \({\mathbf {Rich}^{(p)}}\) are diagonal in the Darboux frame. Indeed, the vectors of the Darboux frame are eigenvectors of \({\varvec{h}}^p\), which guarantees that the Ricci tensor has to have the structure (4.11).
Both conditions (5.73) and (5.74) thus follow from the integrability condition (5.77) for the principal tensor. That concludes the proof of the Nijenhuis–Schouten commutativity (5.33), respectively (5.22).
In Sect. 6.6 we will see that \({{\varvec{\xi }}}\) can be used as a vector potential for a special electromagnetic field which leads to an integrable motion of charged particles. The result (5.83) thus shows, that its Maxwell tensor \({{\varvec{F}}={{\varvec{d}}}{\varvec{\xi }}=2{{\varvec{\nabla }}}{\varvec{\xi }}}\) commutes with the principal tensor \({{\varvec{h}}}\) and can also be skewdiagonalized in the Darboux basis.
5.5 Principal tensor as a symplectic structure
5.5.1 Motivation
In the construction of the Killing tower from the principal tensor we have defined the Killing vectors \({{\varvec{l}}_{(j)}}\) by (5.16), or in terms of a generating function by (5.29). We then claimed that we can associate Killing coordinates with these Killing vectors and it turns out that those are exactly coordinates \({\psi _j}\) in the canonical metric (4.1), namely \({{\varvec{l}}_{(j)}={{\varvec{\partial }}}_{\psi _j}}\). However, we also mentioned that this last equality is not obvious and appears only after one reconstructs the full form of the metric, employing the uniqueness theorem in Sect. 5.3.
The Lie commutativity (5.85) has been shown (in the terms of Nijenhuis–Schouten brackets) in the previous section. However, this result can also be established by a slightly different argument which possesses a beauty on its own and to this argument we devote this section.
5.5.2 Symplectic structure on the spacetime
The principal tensor \({\varvec{h}}\) is a closed nondegenerate 2form on the configuration space M. As such it defines a symplectic structure on this space for M evendimensional, and a contact structure on M in the case of odd dimensions. To explore this idea, and since we have not introduced contact manifolds, let us restrict to the case of even number of dimensions. (The discussion in odd dimensions would proceed analogously.)
Remark
Let us stress that these operations are not related to analogous operations on the relativistic particle phase space, which is realized as the cotangent space \({\mathbf {T}^*M}\), see Sect. B.4. The Hamiltonian vector \({{\varvec{\varXi }}_{f}}\) is an ordinary vector field on M. The bracket (5.88) expects as arguments ordinary functions depending just on the position in M. The dynamics of a relativistic particle is governed by the Hamiltonian on the phase space \({\mathbf {T}^*M}\) and cannot be translated in a straightforward way to the language of the symplectic geometry generated by the principal tensor \({{\varvec{h}}}\). \(\square \)
6 Particles and fields: Integrability and separability
In this chapter, we study particles and fields in the vicinity of higherdimensional rotating black holes. As can be expected their behavior reflects the rich structure of hidden symmetries discussed in the previous chapter: the motion of particles and light is completely integrable and the fundamental physical equations allow separation of variables. Let us start our discussion with a brief overview of the discovery of these unexpected properties.
The Kerr–NUT–(A)dS metric in four spacetime dimensions possesses a number of remarkable properties related to hidden symmetries. In particular, those discovered by Carter (1968a), Carter (1968b), Carter (1968c) include the complete integrability of geodesic equations and the separability of the Hamilton–Jacobi and Klein–Gordon equations. A natural question is whether and if so how far these results can be extended to higher dimensions.
A first successful attempt on such a generalization, employing nontrivial hidden symmetries, was made by Frolov and Stojković (2003b, 2003a). In these papers the authors generalized Carter’s approach to fivedimensional Myers–Perry metrics with two rotation parameters, and demonstrated that the corresponding Hamilton–Jacobi equation in the Myers–Perry coordinates allows a complete separation of variables. This enabled to obtain an explicit expression for the secondrank irreducible Killing tensor present in these spacetimes.^{18}
These results were later generalized by Kunduri and Lucietti (2005) to the case of a fivedimensional Kerr–(A)dS metric. Fields and quasinormal modes in fivedimensional black holes are studied in Frolov and Stojković (2003b), Cho et al. (2012b), Cho et al. (2011). Page and collaborators (Vasudevan et al. 2005a, b; Vasudevan and Stevens 2005) discovered that particle equations are completely integrable and the Hamilton–Jacobi and Klein–Gordon equation are separable in the higherdimensional Kerr–(A)dS spacetime, provided it has a special property: its spin is restricted to two sets of equal rotation parameters. A similar result was obtained slightly later for the higherdimensional Kerr–NUT–(A)dS spacetimes subject to the same restriction on rotation parameters (Davis 2006; Chen et al. 2006b). With this restriction the Kerr–NUT–(A)dS metric becomes of cohomogeneitytwo and possesses an enhanced symmetry which ensures the corresponding integrability and separability properties.
Attempts to apply Carter’s method for general rotating black holes in six and higher dimensions have met two obstacles. First, the explicit symmetries of the Kerr–NUT–(A)dS metrics are, roughly speaking, sufficient to provide only half of the required integrals of motion. This means, that already in six dimensions one needs not one, but two independent Killing tensors, and the number of required independent Killing tensors grows with the increasing of number of spacetime dimensions. Second, more serious problem is that the separation of variables in the Hamilton–Jacobi equation may exist only in a very special coordinate system. However, how to choose the convenient coordinates was of course unknown. In particular, the widely used Myers–Perry coordinates have an unpleasant property of having a constraint (E.9), which makes them inconvenient for separation of variables in more than five dimensions.
The discovery of the principal tensor for the most general higherdimensional Kerr–NUT–(A)dS (Frolov and Kubizňák 2007; Kubizňák and Frolov 2007) spacetimes made it possible to solve both these problems. Namely, the associated Killing tower contains a sufficient number of hidden symmetries complementing the isometries to make the geodesic motion integrable. Moreover, the eigenvalues of the principal tensor together with the additional Killing coordinates, give the geometrically preferred canonical coordinates in the Kerr–NUT–(A)dS spacetime. It turns out that exactly in these coordinates the Hamilton–Jacobi as well as the Klein–Gordon equations separate. The following sections are devoted to a detailed discussion of these results.
6.1 Complete integrability of geodesic motion
The geodesic motion describing the dynamics of particles and the propagation of light in the Kerr–NUT–(A)dS spacetimes is completely integrable. In this section we prove this result, discuss how to obtain particles’ trajectories, and how to introduce the action–angle variables for the corresponding dynamical system.
6.1.1 Complete set of integrals of motion
Remark
It is interesting to note that the relations among the quadratic conserved quantities \(K_{j}\) are highly symmetric. One could actually study a space with the (inverse) metric given by the Killing tensor \({\varvec{k}}_{(i)}\), and all the tensors \({\varvec{k}}_{(j)}\) would remain Killing tensors with respect to this new metric, e.g., Rietdijk and van Holten (1996). This fact is precisely expressed by the first condition (6.3), giving \([{\varvec{k}}_{(i)}, {\varvec{k}}_{(j)}]_{\scriptscriptstyle \mathrm {NS}}=0\) for the Nijenhuis–Schouten brackets among these tensors. Similarly, all the vectors \({\varvec{l}}_{(j)}\) remain to be Killing vectors with respect to the new metric. The geodesic motion in any of the spaces with the metric given by \({\varvec{k}}_{(i)}\) is thus also complete integrable. However, in this context one should emphasize that only the space with \({\varvec{g}}={\varvec{k}}_{(0)}\) is the Kerr–NUT–(A)dS spacetime. Spaces with the metric given by \({\varvec{k}}_{(i)}\), \(i>0\), neither possess the principal tensor and the associated towers of Killing–Yano tensors, nor are solutions of the vacuum Einstein equations. Moreover, although the geodesic motion is integrable in these spaces, this is no longer true for the corresponding fields; the symmetry among Killing tensors does not elevate to the symmetry of the corresponding symmetry operators for the test fields in these spaces, see Sect. 6.3. \(\square \)
6.1.2 Particle trajectories
It is remarkable that the expression (6.7) for \(p_{x_\mu }\) depends only on one variable \(x_\mu \). This property stands behind the separability of the Hamilton–Jacobi equation discussed in the next section. Signs ± in equations (6.7) are independent for different \(\mu \) and indicate that for a given value of \(x_\mu \) there exist two possible values of momentum \(p_{x_\mu }\). We will return to this point below when discussing a global structure of the level set \(\mathcal {L}_{({K},{L})}\).
However, Eqs. (6.13) for \({\dot{x}_\mu }\) are not decoupled since the factor \({U_\mu }\) mixes the equations. In four dimensions, \({{{n}}=2}\), these factors are, up to a sign, the same, \({U_1=U_2=\varSigma }\), cf. (3.8), and they can be eliminated by the time reparamatrization, cf. (3.50). For general \({{{n}}}\) such a trick is not possible. However, the system can still be solved by an integration and algebraic operations. In four dimensions such a procedure was demonstrated by Carter (1968a), in the following we generalize it to an arbitrary dimension.
The procedure (6.15)–(6.19) may seem as an ad hoc manipulation. However, as we shall see below it is closely related to the Liouville construction for complete integrable systems.
We demonstrated that as a result of the complete integrability of geodesic equations in the higherdimensional Kerr–NUT–(A)dS spacetimes, finding solutions of these equations reduces to the calculation of special integrals. This integral representation of the solution is useful for the study of general properties of particle and light motion in these metrics. However, it should be emphasized that only in some special cases these integrals can be expressed in terms of known elementary and special functions. Let us remind that in four dimensions a similar problem can be solved in terms of elliptic integrals, the properties of which are well known. The integrals describing particle and light motion in higher dimensions contain square roots of the polynomials of the order higher than four, and this power grows with the increasing number of spacetime dimensions. Another complication is that the higherdimensional problem depends on a larger number of parameters. At present, the problem of classification of higherdimensional geodesics in Kerr–NUT–(A)dS metrics is far from its complete solution. Here we give some references on the publications connected with this subject. The particle motion in fivedimensional Kerr–(A)dS metrics was considered in Frolov and Stojković (2003a), Kagramanova and Reimers (2012), Diemer et al. (2014), Delsate et al. (2015). The papers Gooding and Frolov (2008), Papnoi et al. (2014) discuss the shadow effect for fivedimensional rotating black holes. Different aspects of geodesic motion in the higherdimensional black hole spacetimes were discussed in Hackmann et al. (2009), Enolski et al. (2011).
6.1.3 Conjugate coordinates on the level set
Having proved that the geodesic motion is completely integrable, let us now discuss the corresponding level sets (here) and the construction of the action–angle variables (below). For simplicity, in this exposition (till end of Sect. 6.1) we restrict ourselves to the case of even dimensions, \({D=2{{n}}}\).
6.1.4 Action–angle variables
Let us remind that for a completely integrable system with D degrees of freedom there exists D independent integrals of motion in involution \(P_i\). We called a level set a Ddimensional submanifold of the phase space \(\mathcal {L}_{P}\), where these integrals have fixed values. According to general theory (Arnol’d 1989; Goldstein et al. 2002) if this level set is compact, it has a structure of multidimensional torus with an affine structure, and one can introduce the so called action–angle variables.
Remark
The affine structure of the level set \(\mathcal {L}_{P}\) refers to the fact that coordinates Q, on the level set, which are conjugate to P, are given uniquely up to a linear transformation. In other words, if one uses a different combination of integrals of motions \(\bar{P}\), the corresponding conjugate coordinates \(\bar{Q}\) are related to Q on the given level set by a linear transformation. Action of the Hamilton flow associated with any of the conserved quantities is linear in the sense of affine structure—all conserved quantities generate Abelian group of translations on the level set. The torus structure of the level set must be compatible with the affine structure. However, its existence can be understood only after taking into account interpretation of the involved variables. \(\square \)
In the Kerr–NUT–(A)dS spacetime the conserved quantities are \(P=({K},{L})\) and the corresponding level set is \({\mathcal {L}_{({K},{L})}}\). As in the general case, this set is a Lagrangian submanifold, where the momenta \(p_a\) can be found as functions of the coordinates \((x_{\mu },\psi _j)\) and conserved quantities, cf. equations (6.7) and (6.8). We see that relations (6.7) are independent for each plane \({x_\mu }\)–\({p_{x_\mu }}\). In each of these planes the condition (6.7) defines a closed curve which spans the range \({(x_\mu ^,x_\mu ^+)}\) between turning points \({x_\mu ^\pm }\) for which \({{\mathcal {X}}_\mu =0}\). The curve has two branches over this interval, one with \({p_{x_\mu }>0}\), another with \({p_{x_\mu }<0}\).
The turning points \({x_\mu ^\pm }\) should exist for angular coordinates \(x_\mu \) since the ranges of these coordinates are bound. Situation can be different for the radial coordinate r (Wick rotated \({x_{{n}}}\)). Depending on the values of conserved quantities, one can have two turning points (bounded orbits), one turning point (scattering trajectories), or no turning points (fall into a black hole). For simplicity, here we discuss only the case where there are two turning points for r. Thus, one has a full torus structure in the xsector of the level set. The torus structure in Killing coordinates \({\psi _j}\) is also present. Condition (6.8) just fixes the momenta to be constant, but leaves the angles unrestricted. However, some linear combination of Killing angles \(\psi _j\) defines angular coordinates \({\varphi _\mu }\), which are periodic. In the maximally symmetric case or for the Myers–Perry solution these coordinates are simply \({\phi _\mu }\) discussed previously. In the periodic coordinates we get the explicit torus structure. The only exception is the time direction, for which one has an infinite range with a translation symmetry.
To summarize, the action variables \({({I},{A})}\) are defined by (6.25). The conjugate angle variables \({({\alpha },{\varPhi })}\) are related to \({({X},{\varPsi })}\) by linear relations (6.27), and \({({X},{\varPsi })}\) are defined in (6.22). It should be mentioned that although in the definition of the action variable \({I_\mu }\) we used the loop circling the torus just in \({x_\mu }\)–\({p_{x_\mu }}\) plane, the conjugate angle variable \({\alpha ^\mu }\) is not a function of just one coordinate \({x_\mu }\), it depends on all coordinates \({(x_1,\dots ,x_{{n}})}\). Similarly, in the inverse relations, \({x_\mu }\) depends on all angles \({\alpha ^\nu }\). However, the coordinates \({x_\mu }\) are multiplyperiodic functions of angle variables. When any angle \({\alpha _\nu }\) changes by period \({2\pi }\), all \({x_\mu }\) return to their original values, cf. general discussion in Chapter 10 of Goldstein et al. (2002).
6.2 Separation of variables in the Hamilton–Jacobi equation
Remark
In Sect. 2.3 we have mentioned that the separability of the Hamilton–Jacobi equation can be characterized by the corresponding separability structure (Benenti and Francaviglia 1979, 1980; Demianski and Francaviglia 1980; Kalnins and Miller 1981). The offshell Kerr–NUT–(A)dS geometry possesses \(({{n}}+\varepsilon )\)separability structure. Indeed, we can identify the ingredients of the first theorem of Sect. 2.3 as follows: in \(D=2{{n}}+\varepsilon \) dimensions, the Kerr–NUT–(A)dS geometry has \({{n}}+\varepsilon \) Killing vectors \({\varvec{l}}_{(j)}\), \(j=0,\dots ,{{n}}1+\varepsilon \), and \({{n}}\) Killing tensors \({\varvec{k}}_{(j)}\), \(j=0,\dots ,{{n}}1\). (i) All these objects commute in the sense of Nijenhuis–Schouten bracket, (5.22), and (ii) the Killing tensors have common eigenvectors \({{\varvec{\partial }}}_{x_\mu }\) which obviously Liebracket commute with the Killing vectors \({\varvec{l}}_{(j)}={{\varvec{\partial }}}_{\psi _j}\) and which are orthogonal to the Killing vectors. All the requirements of the theorem are thus satisfied and the result follows. \(\square \)
6.3 Separation of variables in the wave equation
Remark
The boxoperator in (6.40) is, in a sense, a first quantized version of the Hamiltonian (6.2). Similarly, one could define a secondorder operator \({\mathcal {K}=\nabla _{a} k^{ab}\nabla _{\!b}}\) for any symmetric secondrank tensor \({{\varvec{k}}}\). This can be understood as a ‘heuristic first quantization’ of a classical observable \({K=k^{ab} p_a p_b }\), using the rule \({{\varvec{p}}\rightarrow i{{\varvec{\nabla }}}}\). Of course, when applying this rule one has to chose a particular operator ordering. In our example we have chosen the symmetric ordering. In principle, one could also use a different (from the LeviCivita) covariant derivative. However, all these alternative choices would lead to operators that differ in lower order of derivatives, which could be studied separately. \(\square \)
Remark
The separability of the wave equation is again in an agreement with the theory of separability structures mentioned in Sect. 2.3. In the previous section we have already shown that the Kerr–NUT–(A)dS geometry possesses \(({{n}}+\varepsilon )\)separability structure. To fulfill the second theorem of Sect. 2.3, which guarantees the separability of the wave equation, one has to show that the eigenvectors \({{\varvec{\partial }}}_{x_\mu }\) are eigenvectors of the Ricci tensor. However, the Ricci tensor is diagonal in the special Darboux frame, (4.11), and vectors \({{\varvec{\partial }}}_{x_\mu }\) are just rescaled vectors \({{\varvec{e}}_{\mu }}\), cf. (4.8). This justifies the separability of the Klein–Gordon equation in offshell Kerr–NUT–(A)dS spacetimes. \(\square \)
Remark
When discussing the complete integrability of geodesic motion in the previous section, we mentioned that the geodesic motion in spaces with the metric given by any of the Killing tensors \({{\varvec{k}}_{(i)}}\) is also complete integrable. This property, however, does not elevate to the corresponding wave equations. Namely, the operators (6.41) given by the covariant derivative associated with the metric \({{\varvec{k}}_{(i)}}\), \({i\ne 0}\), no longer mutually commute; the anomalous conditions needed for the operator commutativity are satisfied only for the LeviCivita derivative associated with the Kerr–NUT–(A)dS metric. In particular, the Ricci tensors associated with the metric given by higher (\(i>0\)) Killing tensors are not diagonal in the common frame of eigenvectors of all Killing tensors and the separability structure does not obey the extra condition needed for the separation of the Klein–Gordon field equation, see Kolář (2014), Kolář and Krtouš (2015) for more details. \(\square \)
6.4 Dirac equation
The solution of the massive Dirac equation in the Kerr–NUT–(A)dS spacetimes can be found in a special (prefactor) separated form and the problem is transformed to a set of ordinary differential equations. Similar to the massive scalar equation, this solution is obtained as a common eigenfunction of a set of mutually commuting operators, one of which is the Dirac operator.
6.4.1 Overview of results
The study of Dirac fields in a curved spacetime has a long history. In 1973 Teukolsky rephrased the Dirac massless equation in Kerr spacetime in terms of a scalar ‘fundamental equation’ which could be solved by a separation of variables. However, such an approach does not work for the massive Dirac equation and it is difficult to generalize it to higher dimensions. There is yet another method which goes along the lines we used for the scalar wave equation: one can postulate the multiplicative ansatz for the solution of the Dirac equation and obtain independent (but coupled) differential equations for each component in this ansatz.
This approach dates back to the seminal paper of Chandrasekhar who in 1976 separated and decoupled the Dirac equation in the Kerr background (Chandrasekhar 1976), see Sect. 3.5. A few years later, Carter and McLenaghan (1979) demonstrated that behind such a separability stands a firstorder operator commuting with the Dirac operator which is constructed from the Killing–Yano 2form of Penrose (1973). This discovery stimulated subsequent developments in the study of symmetry operators of the Dirac equation in curved spacetime.
In particular, the most general firstorder operator commuting with the Dirac operator in four dimension was constructed by McLenaghan and Spindel (1979). This work was later extended by Kamran and McLenaghan (1984b) to Rcommuting symmetry operators. Such operators map solutions of the massless Dirac equation to other solutions and correspond to symmetries which are conformal generalizations of Killing vectors and Killing–Yano tensors.
With recent developments in higherdimensional gravity, the symmetry operators of the Dirac operator started to be studied in spacetimes of an arbitrary dimension and signature. The firstorder symmetry operators of the Dirac operator in a general curved spacetime has been identified by Benn and Charlton (1997) and Benn and Kress (2004). The restriction to the operators commuting with the Dirac operator has been studied in Cariglia et al. (2011a).
The higherdimensional Dirac equation has been also studied in specific spacetimes. In the remarkable paper Oota and Yasui (2008) separated the Dirac equation in the general offshell Kerr–NUT–(A)dS spacetime, generalizing the results (Chandrasekhar 1976; Carter and McLenaghan 1979) in four dimensions. The result of Oota and Yasui (2008) has been reformulated in the language of a tensorial separability and related to the existence of the commuting set of operators in Wu (2008, 2009c), Cariglia et al. (2011b), and generalized to the presence of a weak electromagnetic field in Cariglia et al. (2013a). Even more generally, separability of the torsion modified Dirac equation was demonstrated in the presence of U(1) and torsion fluxes of the Kerr–Sen geometry and its higherdimensional generalizations (Houri et al. 2010b) as well as in the most general spherical black hole spacetime of minimal gauged supergravity (Wu 2009a, b), see also Kubizňák et al. (2009), Houri et al. (2010a). The Dirac symmetry operators in the presence of arbitrary fluxes were studied in Acik et al. (2009), Kubizňák et al. (2011).
Before we review the results for Kerr–NUT–(A)dS spacetimes, let us make one more remark. Although the firstorder symmetry operators are sufficient to justify separability of the massless Dirac equation in the whole Plebanski–Demianski class of metrics in four dimensions or separability of the massive Dirac equation in Kerr–NUT–(A)dS spacetimes in all dimensions, they are not enough to completely characterize all Dirac separable systems and one has to consider higherorder symmetry operators, e.g., McLenaghan et al. (2000). In particular, there are known examples (Fels and Kamran 1990) where the Dirac equation separates but the separability is related to an operator of the secondorder. It means that the theory of separability of the Dirac equation must reach outside the realms of the so called factorizable systems (Miller 1988), as such systems are fully characterized by firstorder symmetry operators.
In the following we review the separability results for the Dirac equation in the Kerr–NUT–(A)dS spacetime (Oota and Yasui 2008; Cariglia et al. 2011a, b). A short overview of Dirac spinors in a curved spacetime of an arbitrary dimension can be found in Sect. F.1. In the same appendix, in Sect. F.2, one can also find a characterization of the first order operators that commute with the Dirac operator. Using these general results, we show below that the general offshell Kerr–NUT–(A)dS spacetime admits a set of mutually commuting firstorder operators including the Dirac operator, whose common eigenfunctions can be found in a tensorial Rseparable form.
6.4.2 Representation of Dirac spinors in Kerr–NUT–(A)dS spacetimes
A generic 2dimensional spinor can thus be written as \({\varvec{\chi }} = \chi ^\epsilon {\varvec{\vartheta }}_\epsilon = \chi ^+ {\varvec{\vartheta }}_+ + \chi ^ {\varvec{\vartheta }}_\), with components being two complex numbers \({\left( {\begin{matrix} \chi ^+ \\ \chi ^ \end{matrix}} \right) }\). Similarly, the Dirac spinors \({{\varvec{\psi }}\in \mathbf {D} M}\) can be written as \({\varvec{\psi }} = \psi ^{\epsilon _1\dots \epsilon _{{n}}} {\varvec{\vartheta }}_{\epsilon _1\dots \epsilon _{{n}}}\) with \({2^{{n}}}\) components \({\psi ^{\epsilon _1\dots \epsilon _{{n}}}}\).
6.4.3 Dirac symmetry operators in Kerr–NUT–(A)dS spacetimes
The Kerr–NUT–(A)dS spacetime is equipped with the full tower of Killing–Yano symmetry objects. As discussed in Sect. F.2, such objects allow one to define firstorder operators that commute with the Dirac operator. In fact, as we now demonstrate, it is possible to choose such a subset of Killing–Yano symmetries that yields a full set of D firstorder operators, one of which is the Dirac operator \({\mathcal {D}}\), that all mutually commute.
6.4.4 Tensorial Rseparability of common eigenfunctions
6.5 Tensor perturbations
The demonstrated separability of the Hamilton–Jacobi, Klein–Gordon, and Dirac equations in the general higherdimensional Kerr–NUT–(A)dS spacetime created hopes that higher spin equations might also possess this property. In particular, there were hopes that the electromagnetic and gravitational perturbations can be solved by either a direct separation of the corresponding field equations, or by their reduction to a master equation, which, in its turn, is separable. In spite of many attempts, only partial results were obtained. In this section we briefly discuss the tensor perturbations and return to the electromagnetic fields in the next section.
The study of gravitational perturbations of black holes is key for understanding their stability, and is especially important in higher dimensions where many black holes are expected to be unstable and may (as indicated in recent numerical studies) branch to other black hole families, e.g., Choptuik et al. (2003), Lehner and Pretorius (2010), Dias et al. (2009), Dias et al. (2010a), Dias et al. (2010b), Dias et al. (2014), Figueras et al. (2016), or even result in a formation of naked singularities (Figueras et al. 2017). The separability and decoupling of gravitational perturbations would also significantly simplify the study of quasinormal modes of these black holes or the study of Hawking radiation.
The gravitational perturbations have been analytically studied for higherdimensional black holes with no rotation, e.g., Gibbons and Hartnoll (2002), Kodama and Ishibashi (2003), Ishibashi and Kodama (2003), or for black holes subject to restrictions on their rotation parameters, e.g., Kunduri et al. (2006), Kodama (2009), Murata and Soda (2008a), Murata and Soda (2008b), Kodama et al. (2009), Kodama et al. (2010), Oota and Yasui (2010), Murata (2011), Murata (2013). Such black holes possess enhanced symmetries and are of a smaller cohomogeneity than the general Kerr–NUT–(A)dS spacetime. This allows one to decompose the corresponding perturbations into ‘tensor, vector and scalar’ parts that can be treated separately, yielding the corresponding master equations, e.g., Kunduri et al. (2006).
By the time this review is written it is unknown whether there exists a method which would allow one to separate and decouple gravitational perturbations of the general Kerr—NUT–(A)dS spacetimes. For example, as shown in papers by Durkee and Reall (2011a), Durkee and Reall (2011b) this goal cannot be achieved by following the ‘Teukolsky path’, employing the higherdimensional generalization of Newman–Penrose or Geroch’s formalisms (Pravdová and Pravda 2008; Durkee et al. 2010) building on Coley et al. (2004), Pravda et al. (2004), Ortaggio et al. (2007).
To conclude this section, let us briefly comment on a partial success by Oota and Yasui (2010), who demonstrated the separability of certain type of tensor perturbations in generalized Kerr–NUT–(A)dS spacetimes. In our discussion of the Kerr–NUT–(A)dS spacetimes we assumed that the principal tensor is nondegenerate, that is, it has n functionally independent eigenvalues, that were used as canonical coordinates. One obtains a more general class of metrics once this assumption is violated. The corresponding metrics, called the generalized Kerr–NUT–(A)dS solutions, were obtained in Houri et al. (2009, 2008b), and we will discuss them in more detail in Chap. 7. Here we just describe some of their properties that are required for the formulation of the results of Oota and Yasui (2010). The generalized metric has N essential coordinates which are nonconstant eigenvalues of the principal tensor, m parameters which are nonzero constant eigenvalues, and the degeneracy of a subspace responsible for the vanishing eigenvalue is \(m_0\). The total number of spacetime dimensions is thus \(D=2N+2m+m_0\). This space has a bundle structure. Its fiber is a 2ndimensional Kerr–NUT–(A)dS metric. All other dimensions form the base space. The tensor perturbations, analyzed in Oota and Yasui (2010), are those, that do not perturb the fiber metric and keep the bundle structure. These tensor perturbations admit the separation of variables and the corresponding field equations reduce to a set of ordinary secondorder differential equations.
6.6 Maxwell equations
The study of electromagnetic fields in general Kerr–NUT–(A)dS spacetimes is a complicated task. In particular, the procedure leading to the Teukolsky equation in four dimension (Teukolsky 1972, 1973) does not work in higher dimensions (Durkee and Reall 2011a, b). However, recently there was an important breakthrough in the study of possible separability of higherdimensional Maxwell equations in the Kerr–NUT–(A)dS spacetimes. Namely, Lunin (2017) succeeded to separate variables for some specially chosen polarization states of the electromagnetic field in the Myers–Perry metrics with a cosmological constant. In a general Ddimensional case, the number of polarizations of such a field is \(D2\). Lunin proposed a special ansatz for the field describing two special polarizations and demonstrated that it admits separation of variables. He also demonstrated how such a solution relates to the solution obtained in the Teukolsky formalism in four dimension. However, if one can obtain other components by a similar ansatz is still under investigation.
In the rest of this section we discuss yet other interesting test electromagnetic fields, namely fields aligned with the principal tensor. They include, for example, the field of weakly charged and magnetized black holes. It turns out, that they constitute the most general test electromagnetic field that preserves the integrability properties of the Kerr–NUT–(A)dS geometry.
6.6.1 Wald’s trick: electromagnetic fields from isometries
The study of electromagnetic fields in the vicinity of (rotating) black holes in four dimensions has interesting astrophysical applications and has been investigated by many authors, see e.g., Wald (1974), King et al. (1975), Bičák and Dvořák (1977), Bičák and Dvořák (1976), Bičák and Dvořák (1980), Bičák and Janiš (1985), Aliev and Galtsov (1989), Penna (2014). There is also a number of exact solutions of the Einstein–Maxwell system, ranging from the Kerr–Newman solution for the charged black hole (Newman and Janis 1965; Newman et al. 1965) to magnetized black holes of Ernst (Ernst 1968, 1976). However, it is often possible to restrict the description to a test field approximation assuming that the electromagnetic field obeys the Maxwell equations but does not backreact on the geometry.
Of course, the same trick also works in higher dimensions. This fact was used in Aliev and Frolov (2004) for a study of the gyromagnetic ratio of a weakly charged fivedimensional rotating black hole in an external magnetic field. This was later generalized to the Myers–Perry spacetimes (Aliev 2006).
As we mentioned earlier, in the Kerr–NUT–(A)dS spacetime in the canonical coordinates, the components of the principal tensor \(h_{ab}\) do not depend on the metric parameters. Thus, the operation (6.80) can be interpreted as a subtraction from \({{\varvec{d}}}{\varvec{\xi }}\) a similar quantity, calculated for the corresponding (anti)de Sitter background metric. This prescription was used by Aliev (2007b, 2007a) for obtaining a weakly charged version of the Kerr–(A)dS black holes in all dimensions. The weakly charged and magnetized black rings were studied in Ortaggio and Pravda (2006); Ortaggio (2005).
6.6.2 Aligned electromagnetic fields
6.6.3 Motion of charged particles
Let us now investigate the motion of charged particles in the ‘weakly charged’ Kerr–NUT–(A)dS spacetimes penetrated by the aligned electromagnetic field (6.82). A special case of the field (6.87) has been investigated in Frolov and Krtouš (2011) and Cariglia et al. (2013a).
6.6.4 Weakly charged operators
Similarly, one can also study test scalar and Dirac fields in the weakly charged Kerr–NUT–(A)dS spacetimes.
6.6.5 On a backreaction of the aligned fields
As demonstrated above, the aligned electromagnetic field (6.82) extends naturally most of the properties of Kerr–NUT–(A)dS spacetimes based on their high symmetry to the charged case, albeit this electromagnetic field is only a test field and does not modify the geometry itself. A natural question arises: is it possible to backreact this electromagnetic field to obtain the full solution of the Einstein–Maxwell system?
To answer this question, it is interesting to note that the expressions (6.81) for the Maxwell tensor, (6.82) for the vector potential, and (6.84) for the current do not contain a reference to the metric functions \({X_\mu }\). Indeed, the square roots of \({X_\mu }\) exactly compensate normalization factors included in the frame elements. It gives a hope that the metric functions could be chosen such that the geometry represents the gravitational back reaction of the aligned electromagnetic field. Even the stressenergy tensor of the electromagnetic field is diagonal in the Darboux frame (Krtouš 2007), and corresponds thus to the structure of the Ricci tensor (4.11). Unfortunately, except for the case of four dimensions, the diagonal elements of the Einstein equations do not match and, therefore, the Einstein equations with the electromagnetic field as a source cannot be satisfied (Krtouš 2007) (see also Aliev and Frolov 2004 for similar attempts).
Only in four dimensions the metric functions can be chosen so that the Einstein equations are fulfilled. The geometry then describes the charged Kerr–NUT–(A)dS spacetime (Carter 1968c; Plebański 1975). In higher dimensions, though, this is no longer possible within the realms of pure Einstein–Maxwell theory, see, however, Chong et al. (2005), and additional fields have to be introduced, e.g., Chow (2010). In other words, the exact higherdimensional analogue of the Kerr–Newman solution (without additional fields) remains elusive.
7 Further developments
In this chapter, we review several scattered results in the literature that are related to the existence of the principal tensor and its generalizations. Namely, we discuss the construction of parallel–transported frames along timelike and null geodesics, motion of classical spinning particles, and stationary configurations of strings and branes in the Kerr–NUT–(A)dS spacetimes. We then move beyond the Kerr–NUT–(A)dS spacetimes. Namely, we discuss what happens when some of the eigenvalues of the principal tensor become degenerate, which leads us to the generalized Kerr–NUT–AdS spacetimes. Some of these new spacetimes can be obtained by taking certain singular limits of the Kerr–NUT–(A)dS metric. The limiting procedure may preserve or even enhance the symmetries of the original metric. Hidden symmetries of warped spaces and the corresponding ‘lifting theorems’ are discussed next. We conclude this chapter by studying the generalizations of Killing–Yano objects to spacetimes with torsion and their applications to various supergravity backgrounds where the torsion can be naturally identified with the 3form flux present in the theory.
7.1 Parallel transport
In the previous chapters we have learned that the geodesic motion in general Kerr–NUT–(A)dS spacetimes is completely integrable. In this section we show that the existence of the principle tensor \({\varvec{h}}\) even allows one to construct a whole paralleltransported frame along these geodesics.
Such a frame provides a useful tool for studying the behavior of extended objects in this geometry. For example, in the fourdimensional case it was employed for the study of tidal forces acting on a moving body, for example a star, in the background of a massive black hole, e.g., Luminet and Marck (1985), Laguna et al. (1993), Diener et al. (1997), Ishii et al. (2005). In quantum physics the parallel transport of frames is an important technical element of the point splitting method which is used for calculating the renormalized values of local observables (such as vacuum expectation values of currents, stressenergy tensor etc.) in a curved spacetime. Solving the parallel transport equations is also useful when particles and fields with spin are considered, e.g., Christensen (1978).
7.1.1 Paralleltransported frame along timelike geodesics
Remark
One can do even more. Namely, it is possible to use the 2form \({\varvec{F}}\) to explicitly construct a frame which is paralleltransported along the timelike geodesic. To construct such a frame we use a method similar to the one developed by Marck for the fourdimensional Kerr metric (Marck 1983b). For more details concerning the solution of the parallel transport equations in the higherdimensional Kerr–NUT–(A)dS spacetime see Connell et al. (2008).

\({\varvec{F}}^2\cdot {\varvec{u}}=0\).

If \({\varvec{v}}\) obeys (7.3) then the vector \(\bar{{\varvec{v}}}={\varvec{F}}\cdot {\varvec{v}}\) obeys the same equation.

One also has \({\varvec{F}}\cdot \bar{{\varvec{v}}}=\lambda ^2\,{\varvec{v}}\).

Eigenvectors \({\varvec{v}}_\mu \) and \({\varvec{v}}_\nu \) of the operator \({\varvec{F}}^2\) with different eigenvalues \(\lambda _\mu \) and \(\lambda _\nu \) are orthogonal.
 Denote \(\dot{{\varvec{v}}}={{\varvec{\nabla }}}_{{\varvec{u}}} {\varvec{v}}\). Then since \({\varvec{F}}\) is paralleltransported, one has$$\begin{aligned} {\varvec{F}}^2\cdot \dot{{\varvec{v}}} ={{\varvec{\nabla }}}_{{\varvec{u}}} ({\varvec{F}}^2\cdot {\varvec{v}}) ={{\varvec{\nabla }}}_{{\varvec{u}}}(\lambda ^2\,{\varvec{v}})=\lambda ^2\, \dot{{\varvec{v}}}. \end{aligned}$$(7.4)
The whole construction of the paralleltransported frame in Kerr–NUT–(A)dS spacetimes is schematically illustrated in Fig. 5. The procedure is algorithmic and the actual calculation can be technically simplified by using the so called velocity adapted basis. We refer the interested reader to Connell et al. (2008) for more details.
7.1.2 Parallel transport along null geodesics
The described construction of paralleltransported frame does not straightforwardly apply to null geodesics. In this section we show how to modify this construction and to obtain a paralleltransported frame along null geodesics in Kerr–NUT–(A)dS spacetimes, generalizing the results obtained by Marck (1983a) for the fourdimensional Kerr metric. The section is based on Kubizňák et al. (2009) to where we refer the reader for more details.
The paralleltransported frame along null geodesics has applications in many physical situations. For example, it can be used for studying the polarized radiation of photons and gravitons in the geometric optics approximation, see, e.g., Stark and Connors (1977), Connors and Stark (1977), Connors et al. (1980) and references therein. It provides a technical tool for the derivation of the equations for optical scalars (Pirani 1965; Frolov 1977) and plays the role in the proof of the ‘peelingoff property’ of the gravitational radiation (Sachs 1961, 1962; Newman and Penrose 1962; Penrose 1965; Krtouš and Podolský 2004).
The other Darboux subspaces (with nonzero eigenvalues) are generically 2dimensional. To construct the paralleltransported vectors that span them one can proceed as in the timelike case. Explicit expressions for the parallel transported frame along a null geodesic in the Kerr–NUT–(A)dS spacetimes can be found in Kubizňák et al. (2009).
Let us finally mention that the above construction does not work for the special null geodesics that are the eigenvectors of the principal tensor. It turns out that such directions describe the principal null directions, or WANDs (Weyl aligned null directions). Such directions play an important role in many physical situations, e.g., Coley et al. (2004), Milson et al. (2005), Coley (2008), Ortaggio et al. (2013). In Kerr–NUT–(A)dS spacetimes, these directions can be explicitly written down and the paralleltransported frame can be obtained by a set of local Lorentz transformations of the principal Darboux basis (Kubizňák et al. 2009).
Remark
It was shown in Mason and TaghaviChabert (2010) that the eigenvectors of a nondegenerate (not necessarily closed) conformal Killing–Yano 2form are principal null directions and the corresponding spacetime is of the special algebraic type D. As we discussed in Sect. 4.5, these special directions may also play a role in the Kerr–Schild construction of solutions of the Einstein equations. \(\square \)
7.2 Classical spinning particle
So far we have discussed the geodesic motion of pointlike test particles as well as the propagation of test fields (possibly with spin) in the curved Kerr–NUT–(A)dS background. An interesting problem is to consider the motion of particles with spin. There exist several proposals for describing spinning particles in general relativity, ranging from the traditional approach due to Papapetrou (1951), Corinaldesi and Papapetrou (1951), accompanied by a variety of supplementary conditions, e.g., Semerák and Šrámek (2015); Semerák (2015), to some more recent proposals e.g., Rempel and Freidel (2016).
In this section we concentrate on the spinning particle described by the worldline supersymmetric extension of the ordinary relativistic pointparticle (Berezin and Marinov 1977; Casalbuoni 1976; Barrducci et al. 1976; Brink et al. 1976, 1977; Rietdijk and Holten 1990; Gibbons et al. 1993; Tanimoto 1995; Ahmedov and Aliev 2009b; Ngome et al. 2010), where the spin degrees of freedom are described by Grassmann (anticommuting) variables. Such a model is physically very interesting as it provides a bridge between the semiclassical Dirac’s theory of spin \(\frac{1}{2}\) fermions and the classical Papapetrou’s theory. Our aim is to show that the existence of the principal tensor provides enough symmetry to upgrade the integrals of geodesic motion to new bosonic integrals of spinning particle motion that are functionally independent and in involution. This opens a question of integrability of spinning particle motion in the Kerr–NUT–(A)dS spacetimes.
7.2.1 Theory of classical spinning particles
Let us start by briefly describing our model of a classical spinning particle. To describe a motion of the particle in D dimensions, we specify its worldline by giving the coordinates dependence on the proper time \(\tau \): \(x^a(\tau )\) (\(a=1,\dots , D\)). The particle’s spin is given by the Lorentz vector of Grassmannodd coordinates \(\theta ^A(\tau )\) (\(A = 1, \dots , D\)). We denote by A the vielbein index labeling an orthonormal vielbein \(\{{\varvec{e}}_{\!A}\}\) with components \(e_{\!A}^a\). These components are used to change coordinate indices to vielbain ones and viceversa, \(v^a=v^A e_{\!A}^a\).
7.2.2 Bosonic integrals of motion
7.2.3 Concluding remarks
Let us stress that the above results regard the bosonic sector and have not dealt with the fermionic part of the motion, whose integrability would require a separate analysis. For this reason, the question of complete integrability of the whole (bosonic and fermionic) system of equations of motion of the spinning particle remains open. However, there are reasons to expect that this system might be fully integrable. Perhaps the most suggestive one is the observation that the Dirac equation, that corresponds to the quantized system and can be formally recovered by replacing \(\theta \)’s with \(\gamma \) matrices and \(\varPi \)’s with the spinorial derivative, admits a separation of variables in these spacetimes, see Sect. 6.4. To achieve such separation it is enough to use a set of D mutuallycommuting operators, as many as the Poisson commuting functions that have been found for the motion of spinning particle.
7.3 Stationary strings and branes
7.3.1 Dirac–Nambu–Goto action for extended objects
There are interesting cases when the principal tensor allows one to integrate equations for some extended test objects, such as strings and branes. Such objects play a fundamental role in string theory. At the same time cosmic strings and domain walls are topological defects, which can be naturally created during phase transitions in the early Universe, e.g., Vilenkin and Shellard (2000), Polchinski (2004), Davis and Kibble (2005), and their interaction with astrophysical black holes may result in interesting observational effects, e.g., Gregory et al. (2013). Another motivation for studying these objects is connected with the braneworld models. For example, the interaction of a bulk black hole with a brane representing our world (Emparan et al. 2000; Frolov et al. 2003, 2004b, a; Majumdar and Mukherjee 2005) can be used as a toy model for the study of (Euclidean) topology change transitions (Frolov 2006), see also Kobayashi et al. (2007), Albash et al. (2008), HoyosBadajoz et al. (2007) for the holographic interpretation of this phenomenon. This model demonstrates interesting scaling and selfsimilarity properties during the phase transition that are similar to what happens in the Choptuik critical collapse (Choptuik 1993).
7.3.2 Killing reduction of action for a stationary string
Following Kubizňák and Frolov (2008), we will first discuss stationary strings. Consider a stationary spacetime and denote by \({\varvec{\xi }}\) its Killing vector. A stationary string is a string whose worldsheet \(\varSigma _\xi \) is aligned with this vector. In other words, the surface \(\varSigma _{\xi }\) is generated by a 1parameter family of the Killing trajectories (the integral lines of \({\varvec{\xi }}\)).
7.3.3 Solving stationary string equations in Kerr–NUT–(A)dS spacetimes
Stationary strings in the fourdimensional Kerr spacetime were studied in Frolov et al. (1989); Carter and Frolov (1989). It was demonstrated that the effective metric \({\tilde{{\varvec{q}}}}\) inherits symmetry properties of the Kerr metric and the stationary string equations are completely integrable. The same was found true in Frolov and Stevens (2004) for the fivedimensional Myers–Perry spacetime. It was shown in Kubizňák and Frolov (2008) that these results can be extended to all higher dimensions. Namely, the equations for a stationary string in the Kerr–NUT–(A)dS spacetime, that is a string aligned along the primary Killing vector \({\varvec{\xi }}={\varvec{l}}_{(0)}\), are completely integrable in all dimensions.
The Killing tensors \({\varvec{\tilde{k}}}_{(k)}\), together with the metric \({\varvec{\tilde{q}}}\) and the Killing vectors \({\varvec{l}}_{(j)}\) all mutually Nijenhuis–Schouten commute. Their existence therefore implies a complete set of mutually commuting constants of geodesic trajectories in the geometry \({\varvec{\tilde{q}}}\). Hence, the stationary string configurations in the Kerr–NUT–(A)dS spacetimes are completely integrable.
Let us conclude with the following remarks: (i) Although stationary string configurations in Kerr–NUT–(A)dS spacetimes are completely integrable, this is not true for strings aligned along other (rotational) Killing directions; the primary Killing vector is very special in this respect; (ii) A stationary string near a fivedimensional charged Kerr(A)dS black hole was discussed in Ahmedov and Aliev (2008; iii) The presented formalism of Killing reduction of the Dirac–Nambu–Goto action has been generalized to the case of spinning strings in Ahmedov and Aliev (2009a; iv) More recently, the notion of a stationary string has been generalized to the so called selfsimilar strings in Igata et al. (2016).
7.3.4 \(\xi \)branes
The notion of a stationary string readily generalizes to that of \(\xi \)branes (Kubizňák and Frolov 2008) which are pbranes formed by a 1parametric family of Killing surfaces. Suppose a Ddimensional spacetime admits p mutually commuting Killing vectors \({\varvec{\xi }}_{(M)}\) (\(M=1,\ldots ,p\)). According to the Frobenius theorem the set of p commuting vectors defines a pdimensional submanifold, which has the property that vectors \({\varvec{\xi }}_{(M)}\) are tangent to it. We call such a submanifold a Killing surface.
7.4 Generalized Kerr–NUT–(A)dS spacetimes
So far our discussion was mostly concentrated on Kerr–NUT–(A)dS spacetimes. Such spacetimes represent a unique geometry admitting the principal tensor which is a closed conformal Killing–Yano 2form whose characteristic feature is that it is nondegenerate. However, it is very constructive to relax the last requirement and consider more general geometries that admit a possibly degenerate closed conformal Killing–Yano 2form. Such geometries are now well understood and are referred to as the generalized Kerr–NUT–(A)dS spacetimes (Houri et al. 2008b, 2009; Oota and Yasui 2010; Yasui and Houri 2011). These metrics describe a wide family of geometries, ranging from the Kähler metrics, Sasaki–Einstein geometries, generalized TaubNUT metrics, or rotating black holes with some equal and/or some vanishing rotation parameters.
7.4.1 General form of the metric
7.4.2 Concrete examples
The onshell generalized Kerr–NUT–(A)dS metrics (7.50) describe a large family of vacuum (with cosmological constant) geometries of mathematical and physical interest. To obtain concrete examples one may simply specify the base metrics and the parameters of the solution.
To illustrate, a subfamily of solutions with vanishing NUT charges, describing the Kerr(A)dS black holes (Gibbons et al. 2004, 2005) with partially equal and some vanishing angular momenta, has been identified in Oota and Yasui (2010). Namely, in odd dimensions the generalrotating Kerr(A)dS spacetime (Gibbons et al. 2004, 2005) has an isometry \(\mathbb {R}\times U(1)^{n}\) and corresponds to identifying the base space with a product of the 2dimensional Fubini–Study metrics, \(B=\mathbb {CP}^{1}\times \cdots \times \mathbb {CP}^{1}\). When some of the rotation parameters become equal, the symmetry is enhanced and the dimension of the corresponding Fubini–Study metric enlarges. In particular, equal spinning Kerr(A)dS black hole has \(B=\mathbb {CP}^{n1}\) and its symmetry is \(\mathbb {R}\times U(n)\), see Oota and Yasui (2010), Yasui and Houri (2011) for more details.
Another example is that of ‘NUTty spacetimes’ describing twisted and/or deformed black holes has been studied more recently in Krtouš et al. (2016a). Such black holes correspond to the evendimensional ‘warped structure’ where all the Kähler metrics \({\varvec{g}}^i\) identically vanish and the metric \({\varvec{g}}^0\) becomes again the Kerr–NUT–(A)dS spacetime. As discussed in Krtouš et al. (2016a), these solutions have a full Killing tower of symmetries.
7.4.3 Special Riemannian manifolds
There is yet another, very effective, method for obtaining concrete examples of generalized Kerr–NUT–(A)dS metrics: the method of taking special limits of the original (possibly offshell) Kerr–NUT–(A)dS spacetimes (4.1). Especially interesting are the ‘singular limits’ where some of the originally functionally independent eigenvalues of the principal tensor become equal, constant, or vanish, or some of the original parameters of the Kerr–NUT–(A)dS metrics take special values/coincide. In what follows we shall give several examples of such limits that lead to interesting geometries.
As shown by Geroch (1969), limiting procedures of this kind are generally nonunique. This is related to a well known ambiguity in constructing the limiting spaces when some of the parameters limit to zero: there is always a possibility to make a coordinate transformation depending on the chosen parameters, before taking the limit. As we shall see on concrete examples below, to escape the pathology and to achieve a well defined limit, one should properly rescale both the metric parameters and the coordinates.
7.4.4 Partially rotating deformed black holes
Another class of generalized Kerr–NUT–(A)dS metrics is obtained when one tries to ‘switch off’ some of the rotation parameters of the canonical metric (4.1). In the Lorentzian signature this yields partially rotating black holes that are deformed by the presence of NUT charges. Similar to the special Riemannian manifolds above, these metrics possess enough explicit and hidden symmetries, inherited from the original Kerr–NUT–(A)dS spacetime, to guarantee the complete integrability of geodesic equations. In the following we sketch the idea of the corresponding limit, generalizing the procedure performed in Oota and Yasui (2010) for the case of vanishing NUT parameters. The details of the construction can be found in Krtouš et al. (2016a).
It is obvious from this ordering that in the limit when the first p rotation parameters \(\{ a_1,\ldots , a_p\}\) tend to zero, the first p angle coordinates, grasped between them, must tend to zero as well. In other words, to preserve the regularity of the metric one needs to, besides rescaling the rotation parameters, also properly rescale the first p angle coordinates. As shown in Krtouš et al. (2016a) this can be consistently done.
As a result, the principal tensor \({\varvec{h}}\) becomes degenerate and its matrix rank becomes \(2(np)\). The number of the rank2 Killing tensors, generated from \({\varvec{h}}\) is reduced to \(np\). At the same time, the limiting procedure generates new additional hidden symmetries, which provide one with additional p quadratic in momenta integrals of geodesic motion. The number of the first order in momenta integrals of motion, associated with Killing vectors, remains the same: n. Thus the total number of the integrals of motion, 2n, is sufficient to guarantee complete integrability of geodesics in the limiting spacetime.
The resulting metric has one less parameter and is a special case of the generalized Kerr–NUT–(A)dS metric with \(N=np\), \(m=0\), and \(m_0=2 p\). It has a warped structure: both components in the warped product are lowerdimensional Kerr–NUT–(A)dS metrics. We refer to Krtouš et al. (2016a) for more details and explicit formulas. A symmetry structure of warp product metrics has been studied in Krtouš et al. (2016b) and we will return to it in the next section.
7.4.5 NUTty spacetimes and near horizon geometries
Final interesting limiting cases of the Kerr–NUT–(A)dS metric that we are going to discuss in this section are those of NUTty spacetimes and near horizon geometries. They can be obtained as follows. Consider a coordinate \(x_{\mu }\). It belongs to an interval given by the roots of the metric function, see Sects. 4.3 and 4.4. Now, we want to study a ‘doubleroot’ limit of this metric function. In such a limit, the end points of the interval tend one to the other and the value of the coordinate \(x_{\mu }\), which is grasped between them, becomes in general a nonvanishing constant. This implies the degeneracy of the principal tensor. Such doubleroot limits generalize two interesting cases known from four dimensions: the Taub–NUT limit and the nearhorizon limit of the extremal Kerr black hole. As earlier, the corresponding limiting procedure has to be accompanied by a proper rescaling of coordinates.
It was shown in Kolář and Krtouš (2017), that when the doubleroot limit is taken for all angular coordinates \(x_{\mu }\), it leads to the ‘multiplyNUTty spacetime’, obtained by Mann and Stelea (2006, 2004).
If the doubleroot limit is taken for the metric function governing the position of the horizons, it leads to the nearhorizon limit of the extremal black hole metrics, which is similar to the extreme Kerr throat geometry in four dimensions (Bardeen and Horowitz 1999). This higherdimensional limiting spacetime geometry has enhanced symmetry, while some of the hidden symmetries of the original spacetime encoded by Killing tensors become reducible.
Remark
In the near horizon limit of an extremal Myers–Perry black hole in an arbitrary dimension the isometry group of the metric is enhanced to include the conformal factor SO(2, 1). In particular, when all n parameters of the rotation are equal this group is \(SO(2,1)\times U(n)\) (Galajinsky 2013). For the near horizon extremal Myers–Perry metric one of the rank 2 Killing tensors decomposes into a quadratic combination of the Killing vectors corresponding to the conformal group, while the remaining ones are functionally independent (Chernyavsky 2014). Similar result is valid for the Kerr–NUT–(A)dS metric. Namely, for the near horizon extremal Kerr–NUT–(A)dS geometry only one rank2 Killing tensor decomposes into a quadratic combination of the Killing vectors, which are generators of conformal group, while the others are functionally independent (Xu and Yue 2015). \(\square \)
Additional details and the discussion of various limiting geometries corresponding to double root limits can be found in Kolář and Krtouš (2017).
7.5 Lifting theorems: hidden symmetries on a warped space
As we have seen in the previous chapters, the existence of hidden symmetries imposes strong restrictions on the background geometry.
Consequently, not every geometry admits such symmetries. Even if the symmetries are present, finding their explicit form, by solving the corresponding differential equations, is a formidable task. For this reason, it is of extreme value to seek alternative ways for finding such symmetries. In this section we proceed in this direction. Namely, we study hidden symmetries on a warped space, formulating various criteria under which the Killing–Yano and Killing tensors on the base space can be lifted to symmetries of the full warped geometry. This decomposes a task of finding such symmetries to a simpler problem (that of finding hidden symmetries for a smaller seed metric) and opens a way towards extending the applicability of hidden symmetries to more complicated spacetimes.
Theorem
Theorem
Theorem
Theorem
There exist a number of examples, e.g., Krtouš et al. (2016b), where these theorems can be applied and exploited for finding hidden symmetries of complicated metrics. For example, a very nontrivial application happens for the NUTty spacetimes (Krtouš et al. 2016a, b) which inherit the full tower of hidden symmetries lifted from their two offshell Kerr–NUT–(A)dS bases \({\varvec{\tilde{g}}}\) and \({\varvec{\bar{g}}}\).
Let us finally mention that the lifting theorems presented in this section are not the only possibility for lifting hidden symmetries to higherdimensional geometries. For example, a completely different approach, the so called Eisenhart lift, (Eisenhart 1928) was recently used to construct spacetimes with higherrank Killing tensors (Gibbons et al. 2011) and subsequently applied to more complicated situations, e.g., Cariglia (2012), Galajinsky (2012), Cariglia (2012), Cariglia and Gibbons (2014), Cariglia et al. (2014a), Cariglia et al. (2014b), Cariglia (2014), Cariglia and Galajinsky (2015), Galajinsky and Masterov (2016).
7.6 Generalized Killing–Yano tensors
7.6.1 Motivation
Till now we have discussed mainly vacuum solutions of the higherdimensional Einstein equations with or without the cosmological constant. However, we already mentioned that, for example, in the fourdimensional case there exist the charged versions of the Kerr–NUT–(A)dS metric which are solutions of the Einstein–Maxwell equations and which also admit the Killing–Yano tensor (see e.g., Keeler and Larsen 2012). A natural question is how far can one generalize the presented in this review theory of hidden symmetries to nonvacuum solutions of the Einstein equations. For example, there are known solutions, describing black holes with nontrivial gauge fields, such as those of various supergravity theories which arise in low energy limits of string theory compactifications. It is also well known that some of these solutions, that can be thought of as generalizations of Kerr–NUT–(A)dS metrics, possess Killing tensors (see, e.g., Emparan and Reall 2008 and references therein) and allow separability of the Hamilton–Jacobi and Klein–Gordon equations (Chow 2010, 2016). In fact this is how some of these solutions were ‘constructed’.
In this section we demonstrate that the properties of such nonvacuum black holes can be explained by the existence of a deeper structure associated with the generalized Killing–Yano tensors.
7.6.2 Systematic derivation
In general, the generalized Killing–Yano system couples various homogeneous parts of inhomogeneous form \({\varvec{\omega }}\), and these only decouple for a special form of the flux \({\varvec{B}}\). In particular, this happens for \({\varvec{B}}=i{\varvec{A}}\frac{1}{4}{\varvec{T}}\), with a 1form \({\varvec{A}}\) and a 3form \({\varvec{T}}\), in which case the Killing–Yano system reduces to the torsion generalization of the conformal Killing–Yano equation (7.79) below. We refer to Kubizňák et al. (2011) for more details.
7.6.3 Killing–Yano tensors in a spacetime with torsion
Remark
Interestingly, the GKY tensors were first discussed from a mathematical point of view in Yano and Bochner (1953) many years ago, and rediscovered more recently in Rietdijk and van Holten (1996), Kubizňák et al. (2009) in the framework of black hole physics. The GCKY generalization (7.79) has been first discussed in Kubizňák et al. (2009). \(\square \)
 1.
A GCKY 1form is identical to a conformal Killing 1form.
 2.
The Hodge star \({\varvec{*}}\) maps GCKY pforms to GCKY \((Dp)\)forms. In particular, the Hodge star of a GCCKY pform is a GKY \((Dp)\)form and vice versa.
 3.
GCCKY tensors form a (graded) algebra with respect to a wedge product, i.e., when \({\varvec{h}}_1\) and \({\varvec{h}}_2\) is a GCCKY pform and qform, respectively, then \({\varvec{h}}_3={\varvec{h}}_1 \wedge {\varvec{h}}_2\) is a GCCKY \((p+q)\)form.
 4.
Let \({\varvec{k}}\) be a GCKY pform for a metric \({\varvec{g}}\) and a torsion 3form \({\varvec{T}}\). Then, \({\varvec{\tilde{k}}}=\varOmega ^{p+1} {\varvec{k}}\) is a GCKY pform for the metric \({\varvec{\tilde{g}}}=\varOmega ^2 {\varvec{g}}\) and the torsion \({\varvec{\tilde{T}}}=\varOmega ^2 {\varvec{T}}\).
 5.
Let \({\varvec{\xi }}\) be a conformal Killing vector, \(\pounds _{{\varvec{\xi }}} {\varvec{g}}=2f{\varvec{g}}\), for some function f, and \({\varvec{k}}\) a GCKY pform with torsion \({\varvec{T}}\), obeying \(\pounds _{{\varvec{\xi }}} {\varvec{T}}=2f{\varvec{T}}\). Then \({\varvec{\tilde{k}}}=\pounds _{{\varvec{\xi }}} {\varvec{k}} (p+1)f{\varvec{k}}\) is a GCKY pform with \({\varvec{T}}\).
 6.Let \({\varvec{h}}\) and \({\varvec{k}}\) be two generalized (conformal) Killing–Yano tensors of rank p. Thenis a (conformal) Killing tensor of rank 2.$$\begin{aligned} K_{ab}=h_{(a c_1\ldots c_{p1}}k_{b)}{}^{c_1\ldots c_{p1}} \end{aligned}$$(7.80)
The metrics admitting a nondegenerate GCCKY 2form have been locally classified in Houri et al. (2012). In general such metrics admit a tower of Killing tensors but no additional explicit symmetries. A subfamily of these metrics provided a new class of Calabi–Yau with torsion metrics (Houri et al. 2012), see also Houri et al. (2013) for the generalized Sasaki–Einstein metrics, and Hinoue et al. (2014) for a generalization of the Wahlquist metric. Further developments on the GKY tensors can be found in Chow (2015, 2016). We also refer to the wonderful review on applications of Killing–Yano tensors to string theory by Chervonyi and Lunin (2015).
7.7 Final remarks
This Living Review was mainly devoted to two subjects: hidden symmetries and higherdimensional black holes. Black holes in higher dimensions find applications in many physical situations. They naturally appear in low energy approximations of string theory, play an important role in braneworld scenarios, as well as provide a window to the nature of gravitational theory in four and higherdimensions. As we explained in this review, all higherdimensional Kerr–NUT–(A)dS black holes possess a set of explicit and hidden symmetries, which is sufficient to guarantee complete integrability of geodesic equations and separation of variables in physical field equations. The origin and seed of all these symmetries is a single very special object, called the principal tensor. This is a nondegenerate closed conformal Killing–Yano 2form. The existence of this object makes properties of higherdimensional black holes very similar to the properties of the fourdimensional Kerr metric.
During ten years that have passed since the discovery of the principal tensor, there have been published many papers devoted to hidden symmetries of higherdimensional black holes. In the present review, we collected the obtained results and provided the references to the main publications on this subject. It should be mentioned that during the work on the review we also obtained a number of new, yet unpublished, results that fill some loopholes in the literature. For example, we discussed in detail the solution of geodesic equations in terms of the action–angle variables, provided a direct proof of the commutation relations of the objects in the Killing tower without using the explicit form of the metric, studied a possibility of understanding the principal tensor as a symplectic form on the spacetime, or systematically discussed the meaning of coordinates and special cases of the Kerr–NUT–(A)dS metrics.
Let us mention several open problems that are immediately connected to the results presented in this review. For example, we showed that the geodesic equations in rotating black hole spacetimes are completely integrable in all dimensions. This provides a highly nontrivial infinite set of completely integrable dynamical systems. This might be of interest to researchers who study (finitedimensional) dynamical systems. In particular, we demonstrated how the actionangle variables approach can be developed for studying the particle and light motion. This opens an interesting possibility of applying the fundamental theorem of Kolmogorov–Arnold–Moses (Arnol’d 1989) to develop a perturbation theory for slightly distorted geodesics in such spacetimes. Another interesting mathematical problem, waiting for its solution, is the study of properties of the solutions of the ordinary differential equations which arise in the separation of variables of the Klein–Gordon and other field equations in the background of higherdimensional black holes. In particular, it is important to describe properties of higherdimensional spinweighted spheroidal harmonics (Berti et al. 2006; Kanti and Pappas 2010; Cho et al. 2012a; Brito et al. 2012; Kanti and Pappas 2012; Kanti and Winstanley 2015). These functions are defined as solutions of the Sturm–Liouville eigenvalue problem for the secondorder ordinary differential equation with polynomial coefficients, see Sect. 6.3.
There is a number of interesting possible extensions of the presented in this review subjects, which are still waiting for their study. These problems include, for example, the classification and complete study of metrics obtained from the Kerr–NUT–(A)dS metrics by different limiting procedures and, more generally, a thorough study of the generalized Kerr–NUT–(A)dS solutions. It would also be interesting to extend the applicability of hidden symmetries to nonempty and supersymmetric generalizations of the higherdimensional Einstein equations. More generally, the subject of hidden symmetries has many interesting applications that go well beyond the realms of black hole physics. It casts a new light on (integrable) dynamical systems, advances mathematical techniques, provides new tools for constructing solutions of Einstein’s equations, is related to special Riemannian manifolds, symmetry operators, and the Dirac theory. We refer to a beautiful review on hidden symmetries in classical and quantum physics (Cariglia 2014).
We would like to conclude this review by the following remark. The principal tensor, which exists in higherdimensional black holes, provides us with powerful tools that allow us to study these spacetimes. Why at all the Nature ‘decided’ to give us such a gift?
Footnotes
 1.
The sum over configuration space indices \(a=1,\dots ,D\) is assumed.
 2.
In the Lorentzian signature we additionally assume certain generality of eigenvectors and eigenvalues, see the discussion below.
 3.
We will use this notation just in this section since we want to be more explicit here. Elsewhere we raise indices implicitly, without the sharp symbol.
 4.
The eigenvectors can be chosen orthonormal with respect to the hermitian scalar product on the complexification of the tangent space, \({\langle {\varvec{a}},{\varvec{b}}\rangle ={\varvec{g}}(\bar{{\varvec{a}}},{\varvec{b}})}\). Such orthonormality relations translate into nullorthonormality conditions (2.114) written in terms of the metric.
 5.
Note that in the Lorentzian signature, the corresponding first eigenvalue is negative. This results in a slightly modified Darboux form of the principal tensor and the metric, see equations (3.34) and (3.35) below. Let us also notice that although the coordinate r is spacelike or timelike in a generic point, it becomes null at the horizon in the Kerr–NUT–(A)dS spacetime. In a more general case, one of the eigenvalues of the tensor \({\varvec{h}}\) might become null not only on a surface but in some domain (Dietz and Rudiger 1981; Taxiarchis 1985). In what follows we do not consider this case.
 6.
In fact, for massless particles one could instead of the Killing tensor \({\varvec{k}}\) use the conformal Killing tensor \({\varvec{Q}}\), defining \(Q^{a b}l_a l_b=K\) in (3.40) and similarly in (3.45). As can be expected, in the massless limit \(m\rightarrow 0\), the resultant equations (3.46)–(3.48) remain the same.
 7.
As we shall see, a similar trick does not work in higher dimensions. However, proceeding differently, the velocity equations can still be decoupled and in principle solved by integration, see Sect. 6.1.
 8.
This 2form is no longer closed. In consequence the structure of hidden symmetries is weaker than that of the Kerr–NUT–(A)dS spacetime. For example, only null but not timelike geodesics are integrable in the Plebański–Demiański background.
 9.
The Greek indices always take values \({\mu ,\nu ,{\dots }=1,\dots ,{{n}}}\) and, in even dimensions, the Latin indices from the middle of alphabet take values \({j,k,l,{\dots }=0,\dots ,{{n}}1}\). We do not use the Einstein summation convention for them but also we do not indicate limits in sums and products explicitly, i.e., \({\sum _\mu \equiv \sum _{\mu =1}^{{n}}}\), \({\sum _k\equiv \sum _{k=0}^{{{n}}1}}\).
 10.
 11.
As opposed to the Myers–Perry coordinates \((t,r,\mu _k, \phi _k)\), the coordinates \((t,r, x_{\bar{\mu }}, \phi _{\bar{\mu }})\) are already all independent, c.f. constraint (E.9). For this reason the metric (4.43) is ‘closer’ to the Boyer–Lindquist form of the Kerr geometry in four dimensions than the Myers–Perry form (E.7).
 12.
 13.Using the integrability relation (C.27), it is easy to show thatThus for the onshell metric, when \( R_{ac}=\frac{2}{D2}\varLambda g_{ac}\), the vector \({\varvec{\xi }}\) obeys the Killing equation. The same conclusion remains also true for any offshell Kerr–NUT–(A)dS metric. One way to demonstrate this is to use the explicit form of the offshell metric. However, it is also possible to prove this result without referring to the metric, using only the properties of the principal tensor. This proof is involved and we present it later, at the end of this chapter.$$\begin{aligned} 2(D2)\nabla _{\!(a}\xi _{b)} = h_{ac}R^c{}_b  R_{ac}\,h^{c}{}_{b}.\nonumber \end{aligned}$$
 14.
The inverse \({{\varvec{q}}^{1}}\) of a nondegenerate symmetric rank2 tensor \({{\varvec{q}}}\) is defined in a standard way: \({\varvec{q}}\cdot {\varvec{q}}^{1}={\varvec{1}}\), or in components, \({q_{ac} (q^{1}){}^{bc}=\delta ^b_a}\).
 15.
When speaking about eigenvectors of a rank2 tensor \({\varvec{h}}\), we always assume a proper adjustment of its indices (by using the metric) to form a linear operator \(h^{a}{}_{b}\).
 16.
As we mentioned in Sect. 2.8, we do not consider the exceptional null form of the principal tensor which is possible for the Lorentzian signature. Such a case would lead, in general, to its own special geometry.
 17.
Let us mention that relations (5.20) follow from the relation (5.64), together with the definition (5.29) and expansions (5.32) and (5.31). They can thus be deduced directly from the properties of the principal tensor, without referring to canonical coordinates. Also, here and below we use the relation \({{{\varvec{d}}}A^{(j+1)}=2\sum _\mu x_\mu A^{(j)}_\mu {{\varvec{d}}}x_\mu }\), which follows from (D.12), for example.
 18.
Let us note here that Carter’s method had been used to study higherdimensional black hole spacetimes prior to the works (Frolov and Stojković 2003b, a). For example, in Gibbons and Herdeiro (1999), Herdeiro (2000) the authors demonstrated the separability of the Hamilton–Jacobi and Klein–Gordon equations for the fivedimensional so called BMPV black hole (Breckenridge et al. 1997). In this case, however, the corresponding Killing tensor is reducible and the explicit symmetries of the spacetime are enough to guarantee the obtained results.
 19.
The factor 1 / 2 in these expressions is chosen for convenience, in order the final formulae can be directly translated to the action–angle expressions below. In the following we also assume the existence of turning points for the trajectories. This is automatically satisfied for \(x_\mu \) describing the angle variables, and restricts the discussion to the bounded trajectories regarding the radial coordinate.
 20.
A similar construction does not work for the secondary Killing vectors. One could try to use the Killing vector copotentials (5.19) instead of the principal tensor as a correction term. However, they are not closed, \({\varvec{d\omega }}^{(j)}\ne 0\) for \(j>0\), and cannot thus play a role of the correction to the electromagnetic field based on \({\varvec{d}}{\varvec{l}}_{(j)}\).
 21.
Various expressions below contain rescaled 1forms \({ \sqrt{\frac{X_\mu }{U_\mu }}{{\varvec{e}}^{\mu }}= {{\varvec{d}}}x_\mu }\) and \({ \sqrt{\frac{U_\mu }{X_\mu }}{\hat{{\varvec{e}}}^{\mu }}= \sum _k A^{(k)}_\mu {{\varvec{d}}}\psi _k}\). They have even simpler form than 1forms \({{{\varvec{e}}^{\mu }}}\) and \({{\hat{{\varvec{e}}}^{\mu }}}\) themselves and they could be used as a natural frame. However, such a frame is not normalized and we do not introduce it here explicitly, although, in some expressions we keep these terms together.
 22.
Notice that in Krtouš (2007), the relations (2.20) and (2.21) for the source in even dimensions have a wrong sign.
 23.
In this section we assume a Lorentzian signature of the metric. Moreover, to simplify the formulas in this section we simply put the mass m of a particle equal one. With this choice, \(m=1\), the inner time \(\sigma \) coincides with the proper time \(\tau \), and the momentum of the particle \(p_a\) is related to its velocity \(u^a\) as follows \(p_a=g_{ab}u^b\). We denote by dot the covariant derivative \(\nabla _{{\varvec{u}}}\).
 24.Let us remind that, if not indicated otherwise, the sums (and products) run over ‘standard’ ranges of indices:$$\begin{aligned} \sum _\mu \equiv \sum _{\mu =1}^{{n}},\qquad \sum _k \equiv \sum _{k=0}^{{{n}}1}. \end{aligned}$$
 25.
 26.
 27.
The cohomogeneity of a Ddimensional spacetime is p if there exist a group of symmetry acting on the spacetime with orbits having dimensionality \(Dp\).
 28.
We follow a common practice of omitting spinor indices even when writing components. A spinor \({\varvec{\psi }}\) has components \(\psi ^A\) which we collect to the ‘column’ \(\psi \). Similarly, a Clifford object \(\slash {\varvec{\omega }}\) has components \(\slash \omega ^A{}_B\), which we collect to the matrix \(\slash \omega \), (F.2). In other words, \(\slash {\varvec{\omega }}\in \mathbf {D}^1_1\,M\) is a tensorial object, and \(\slash \omega \) is a shorthand for its components. The gamma matrices \(\gamma ^a\) are shorthands for \(\gamma ^a{}^A{}_B\), components of the abstract generators of the Clifford algebra \({\varvec{\gamma }}\in \mathbf {T}^1_0\otimes \mathbf {D}^1_1\,M\). We also assume the implicit matrix multiplication denoted by juxtaposition of spinor matrices.
 29.
This choice is usually done by choosing an orthonormal frame \({\varvec{e}}_a\) in tangent space (but in the context of Lorentzian geometry a choice of a null frame is also common) and by specifying a particular form of the components of the gamma matrices \(\gamma ^a{}^A{}_B\). Different realizations of the gamma matrices which can be found in the literature can thus be understood as a different choice of the frame \({\varvec{\vartheta }}_A\) associated with the spacetime frame \({\varvec{e}}_a\). See (6.52) for a particular choice of the frame \({\varvec{\vartheta }}_A\) in the Kerr–NUT–(A)dS spacetime.
Notes
Acknowledgements
The authors thank Andrei Zelnikov for his help with the preparation of figures. V.F. thanks the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Killam Trust for their financial support, and thanks Charles University for hospitality during a stay and work on the review. P.K. was supported by successive Czech Science Foundation Grants P203120118 and 1701625S, and thanks University of Alberta and Perimeter Institute for hospitality in various stages of work on the review. D.K. acknowledges the Perimeter Institute for Theoretical Physics and the NSERC for their support. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.