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Introduction to Black Hole Physics$

Valeri P. Frolov and Andrei Zelnikov

Print publication date: 2011

Print ISBN-13: 9780199692293

Published to Oxford Scholarship Online: January 2012

DOI: 10.1093/acprof:oso/9780199692293.001.0001

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(p.414) Appendix D Hidden Symmetries

(p.414) Appendix D Hidden Symmetries

Source:
Introduction to Black Hole Physics
Publisher:
Oxford University Press

(p.414) Appendix D

Hidden Symmetries

D.1 Conformal Killing Tensor

The Kerr metric is an example of a spacetime with hidden symmetries. This property is discussed in Section 8.7. In this appendix we collect additional information, concerning spacetimes with hidden symmetries. We discuss first hidden symmetries in the 4D spacetime. Brief remarks on the hidden symmetries in the higher dimensions are given in Sections D.9 and D.10. Additional material can be found in the review article (Frolov and Kubizňák 2008).

By definition, a spacetime has a hidden symmetry if the geodesic equations possess conserved quantities of higher than the first order in momentum. The geometric structure responsible for such a conservation law is the Killing tensor. We define first a conformal Killing tensor. This is a symmetric tensor K μ 1 μ p of rank p that obeys the equation

K ( μ 1 μ p ; ν ) = g ν ( μ 1 K ˜ μ 2 μ p ) ,
(D.1.1)

where K ˜ μ 2 μ p is a symmetric tensor of rank p − 1. For null geodesics x μ(λ), where λ is the affine parameter, in a spacetime with the conformal Killing tensor the following quantity is conserved

K μ 1 μ p u μ 1 u μ p , u μ = d x μ d λ .
(D.1.2)

A Killing tensor is a special type of a conformal Killing tensor for which the right‐hand side of Eq. (D.1.1) vanishes. For the Killing tensor the quantity Eq. (D.1.2) is conserved for any (not necessary null) geodesics. For non‐null geodesics u μ = dx μ/, where τ is a proper time parameter for the time‐like geodesics and the proper distance for the space‐like one.

D.2 Killing–Yano Tensors

D.2.1 Conformal Killing–Yano tensor

A Killing tensor is a natural symmetric generalization of the Killing vector. There exists also an antisymmetric generalization, known as a Killing–Yano tensor. We introduce first a conformal Killing–Yano tensor. This is an anti–symmetric tensor h μ 1 μ 2 μ p of the rank p (p‐form) that obeys the equation

μ h μ 1 μ 2 μ p = [ μ h μ 1 μ 2 μ p ] + p g μ [ μ 1 h ˜ μ 2 μ p ] .
(D.2.1)

(p.415) The tensor h ˜ μ 2 μ p can be obtained by tracing both sides of Eq. (D.2.1) with respect to the indices μ and μ 1, and in 4D it has the form

h ˜ μ 2 μ p = 1 5 p μ h μ μ 2 μ p .
(D.2.2)

Another equivalent form of Eq. (D.2.1) is

( μ 1 h μ 2 ) μ 3 μ p + 1 = g μ 1 μ 2 h ˜ μ 3 μ p + 1 ( p 1 ) g [ μ 3 ( μ 1 h ˜ μ 2 ) μ p + 1 ] .
(D.2.3)

When h̃ = 0, and hence the right‐hand side of Eq. (D.2.3) vanishes, the corresponding antisymmetric tensor is called a Killing–Yano tensor. If a spacetime has a Killing–Yano tensor k μ 1 μ 2 μ p then

  • k μ 1 μ 2 μ p u μ p is parallelly propagated along a geodesic with a tangent vector u μ p .

  • K μ ν = ( k k ) μ ν k μ μ 2 μ p k ν μ 2 μ p is a Killing tensor.

The latter property means that in a spacetime with a Killing–Yano tensor there always exists a Killing tensor that is a ‘square’ of the Killing–Yano tensor. Note, that in the general case, the ‘square root’ of the Killing tensor does not exist, that is one cannot present the Killing tensor as a ‘square’ of some Killing–Yano tensor.

D.2.2 Closed conformal Killing–Yano tensor

When the first term on the right‐hand side of Eq. (D.2.1) vanishes

[ μ h μ 1 μ 2 μ p ] = 0 ,
(D.2.4)

the conformal Killing–Yano tensor is closed. Such a p‐form h can be written (at least locally) as

h = d b .
(D.2.5)

The (p − 1)‐form b is called a potential‐generating closed conformal Killing–Yano tensor.

Consider a rank‐2 skew‐symmetric tensor h μν in a four‐dimensional spacetime obeying the equation

h μ ν ; λ = g λ ν ξ μ g λ μ ξ ν .
(D.2.6)

The contraction of this relation gives

ξ μ = 1 3 h μ ; λ λ .
(D.2.7)

This tensor obeys the relations

( λ h μ ) ν = g ν ( λ ξ μ ) g λ μ ξ ν , [ λ h μ ν ] = 0 ,
(D.2.8)

and, hence, it is a rank‐2 closed conformal Killing–Yano tensor. This tensor can be written in the form (p.416)

h μ ν = μ b ν ν b μ = 2 b [ μ ; ν ] ,
(D.2.9)

where b μ is its potential.

D.2.3 Duality relations

There exists a duality relation between a closed conformal Killing–Yano tensor and a Killing– Yano tensor. To obtain this relation we use the following properties of the totally skew‐ symmetric tensor

e μ 1 μ 2 μ 3 μ 4 e ν 1 ν 2 ν 3 ν 4 = 24 δ μ 1 [ ν 1 δ μ 2 ν 2 δ μ 3 ν 3 δ μ 4 ν 4 ] , e μ 1 μ 2 μ 3 λ e ν 1 ν 2 ν 3 λ = 6 δ μ 1 [ ν 1 δ μ 2 ν 2 δ μ 3 ν 3 ] , e μ 1 μ 2 ϵ λ e ν 1 ν 2 ϵ λ = 4 δ μ 1 [ ν 1 δ μ 2 ν 2 ] , e μ 1 π ϵ λ e ν 1 π ϵ λ = 6 δ μ 1 ν 1 , e σ π ϵ λ e σ π ϵ λ = 24.
(D.2.10)

A Hodge dual tensor of an antisymmetric tensor h μν is defined as

k α β = 1 2 h μ ν e μ ν α β .
(D.2.11)

Lemma 3: If h μν is a closed conformal Killing–Yano tensor, then its Hodge dual k μν is a Killing–Yano tensor.

Proof: One has

k α β ; γ = 1 2 ( δ γ ν ξ μ δ γ μ ξ ν ) e μ ν α β = ξ μ e μ α β γ .
(D.2.12)

Hence, k α(β;γ) = 0 and k is a Killing–Yano tensor.

Lemma 4: If f μν is a Killing‐Yano tensor, then its Hodge dual h μν is a closed conformal Killing–Yano tensor.

Proof: According to the definition of the Killing‐Yano tensor f μν;λ = −f μλ;ν. Since f μν is antisymmetric with respect to its indices, f μν;λ is also an antisymmetric tensor. Let us denote

ξ ν = 1 6 f μ 1 μ 2 ; μ 3 e μ 1 μ 2 μ 3 ν .
(D.2.13)

Multiplying both sides of this relation by e ν 1 ν 2 ν 3 ν and using Eq. (D.2.10) we get

f μ 1 μ 2 ; μ 3 = ξ ν e ν μ 1 μ 2 μ 3 .
(D.2.14)

Consider now a Hodge dual tensor of f μν

h μ ν = 1 2 f α β e α β μ ν .
(D.2.15)

(p.417) for its covariant derivative we have

h μ ν ; λ = 1 2 f α β ; λ e α β μ ν .
(D.2.16)

Substituting the representation Eq. (D.2.14) in this relation and using Eq. (D.2.10) we get

h μ ν ; λ = g ν λ ξ μ g μ λ ξ ν .
(D.2.17)

Hence, h μν is a closed conformal Killing–Yano tensor.

Note that, because ∗∗h = ‐h, we have the relation f μν = −k μν.

D.3 Primary Killing Vector

Solutions of the Einstein equations with the cosmological constant Λ

R μ ν = Λ g μ ν ,
(D.3.1)

are called Einstein spaces. We prove now the following statement:

Lemma 5: Define a vector ξ in the Einstein space with a closed conformal Killing–Yano tensor h by the relation

ξ μ = 1 3 h μ λ ; λ .
(D.3.2)

If ξ ≠ 0 it is a Killing vector.

Proof: Using the formula for the commutator of the covariant derivatives we write

h μ ϵ ; λ ; ν h μ ϵ ; ν ; λ = h α ϵ R α μ λ ν + h μ α R α ϵ λ ν .
(D.3.3)

By multiplying this relation by g ϵλ we get

h μ λ ; λ ; ν h μ λ ; ν ; λ = h α λ R α μ λ ν h μ α R α ν .
(D.3.4)

Consider a vector ξ μ defined by Eq. (D.2.7). For this vector

ξ μ ; ν = 1 3 h μ λ ; λ ; ν ,
(D.3.5)

and Eq. (D.3.4) implies

2 ξ μ ; ν + g μ ν ξ λ ; λ = h α λ R α μ λ ν h μ α R α ν .
(D.3.6)

Contraction of this equation gives

ξ λ ; λ = 0.
(D.3.7)

After symmetrization of the relation Eq. (D.3.6) we get

ξ ( μ ; ν ) = 1 2 R ( ν α h μ ) α .
(D.3.8)

(p.418) for the Einstein space Eq. (D.3.1) the right‐hand side of this relation vanishes. It implies that ξ μ is a Killing vector. We call this vector a primary Killing vector.

D.4 Properties of the Primary Killing Vector

Lemma 6: The following relation is valid:

ξ α h μ ν ; α = 0.
(D.4.1)

Proof: Using Eq. (D.2.17) we get

ξ α h μ ν ; α = 2 ξ [ μ ξ ν ] = 0 .
(D.4.2)

We denote ζ α = −h αβ ξ β. This vector obeys the property ζ μ ξ μ = 0.

Lemma 7: The following relation is valid:

ζ μ = 1 4 ( h α β h α β ) ; μ .
(D.4.3)

Proof: One has

( h α β h α β ) ; μ = 2 h α β h α β ; μ = 2 h α β ( g μ β ξ α g μ α ξ β ) = 4 ζ μ .
(D.4.4)

Lemma 8: Let ξ be a primary Killing vector for the closed conformal Killing–Yano tensor h. Then, the following relation is valid:

ξ h μ ν = 0.
(D.4.5)

Proof: Using the definition of the Lie derivative we have

ξ h μ ν = ξ α h μ ν ; α + ξ μ ; α h α ν + ξ ν ; α h μ α .
(D.4.6)

We also have

ξ μ ; α h α ν = ξ α ; μ h α ν = ( ξ α h α ν ) ; μ + ξ α h α ν ; μ = g μ ν ξ 2 ζ ν ; μ ξ μ ξ ν .
(D.4.7)

Using Eq. (D.4.1) we obtain

ξ h μ ν = 2 ζ [ μ ; ν ] .
(D.4.8)

Finally, we use the relation Eq. (D.4.3) to prove Eq. (D.4.5).

D.5 Secondary Killing Vector

Let h μν be a closed conformal Killing–Yano tensor and k μν is its Hodge dual Killing–Yano tensor. Then, the Killing tensor associated with k μν

K μ ν = k μ λ k ν λ
(D.5.1)

(p.419) can be written as follows:

K μ ν = H μ ν 1 2 g μ ν H λ λ , H μ ν = h μ λ h ν λ .
(D.5.2)

The relation Eq. (D.5.2) can be checked directly by using the expression Eq. (D.2.11) and the relation Eq. (D.2.10).

Denote ζ μ = −h μν ξ ν, then direct calculations give

H μ ν ; λ = 2 [ h λ ( μ ξ ν ) g λ ( μ ζ ν ) ] , K μ ν ; λ = 2 [ h λ ( μ ξ ν ) g λ ( μ ζ ν ) + g μ ν ζ λ ] , K μ λ ; λ = 2 ζ μ .
(D.5.3)

Lemma 9: If K μν is a Killing tensor associated with the closed conformal Killing–Yano tensor h μν and ξ μ is a primary Killing vector then

η μ = K μ λ ξ λ
(D.5.4)

is a Killing vector. We call this Killing vector a secondary Killing vector.

Proof: One has

η μ ; ν = K μ λ ; ν ξ λ + K μ λ ξ λ ; ν .
(D.5.5)

Thus,

η ( μ ; ν ) = K λ ( μ ; ν ) ξ λ + 1 2 [ K μ λ ξ λ ; ν + K ν λ ξ λ ; μ ] .
(D.5.6)

Using the definition of the Killing tensor we get K λ ( μ ; ν ) = 1 2 K μ ν ; λ . The expression in the squared brackets in Eq. (D.5.6) can be rewritten as

K μ λ ξ ; ν λ + K ν λ ξ ; μ λ = ξ K μ ν ξ λ K μ ν ; λ .
(D.5.7)

Combining these results we obtain

η ( μ ; ν ) = 1 2 ξ K μ ν ξ λ K μ ν ; λ .
(D.5.8)

The first term vanishes because K μν is constructed in terms of h μν and g μν, for which ℒξ h μν = 0 and ℒξ g μν = 0. Using Eq. (D.5.3) one can check that the second term also vanishes. Thus, we proved that

η ( μ ; ν ) = 0 ,
(D.5.9)

and η μ is a Killing vector.

Lemma 10: Killing vector fields ξ μ and η μ commute with one another.

Proof: The following relations are valid:

[ ξ , η ] μ = ξ η μ = ( ξ K ν μ ) ξ ν + K ν μ L ξ ξ ν = 0.
(D.5.10)

(p.420) D.6 Darboux Basis

Consider a symmetric operator

H μ ν = h μ λ h ν λ ,
(D.6.1)

constructed from a closed conformal Killing–Yano tensor h. Consider the following eigenvalue problem:

H μ ν n ν = A n μ .
(D.6.2)

We assume that a vector n μ is normalized

( n , n )= ε n ,
(D.6.3)

where ε n = 1 for a space‐like vector and ε n = −1 for a time‐like one.

Lemma 11: If n ν is a normalized eigenvector of H μν with A ≠ 0, then

n ¯ μ = 1 | A | h λ μ n λ
(D.6.4)

is also a normalized eigenvector of the same operator with the same eigenvalue. One also has

n μ = s i g n ( A ) | A | h λ μ n ¯ λ .
(D.6.5)

Proof: Denote n ¯ μ = β h λ μ n λ . It is easy to see that (n, n¯) = 0, and one has

H ν μ n ¯ ν = β h ϵ μ h ν ϵ h λ ν n λ = β h ϵ μ H λ ϵ n λ = A n ¯ μ .
(D.6.6)

Thus, n¯μ is an eigenvector of H ν μ and its eigenvalue is A. For β = 1 / | A | one also has

( n ¯ , n ¯ ) = ε ¯ n , ε ¯ n = sign ( A ) ε n .
(D.6.7)

The relation Eq. (D.6.5) can be easily checked by substituting Eq. (D.6.4) into this relation and using Eq. (D.6.2). We call the vector n¯ conjugated to n. Two mutually conjugated orthonormal vectors {n,n¯} span a two‐dimensional eigenspace of H μ ν .

We call the closed conformal Killing–Yano tensor h μν in the 4D spacetimenon‐degenerate if the eigenvalue problem Eq. (D.6.2) has two different eigenvalues A 1 and A 2. In this case one has two linearly independent eigenspaces. Let us show that these spaces are orthogonal.

Lemma 12: Any two eigenvectors n 1 and n 2 of H with different eigenvalues are orthogonal.

Proof: Denote by A 1 and A 2 eigenvalues for the eigenvectors n 1 and n 2, respectively. Then, one has

A 2 ( n 1 , n 2 ) = ( n 1 , H n 2 ) = n 1 μ h μ λ h λ ν n 2 ν = ( H n 1 , n 2 ) = A 1 ( n 2 , n 1 ) .
(D.6.8)

(p.421) Since (n 1, n 2) = (n 2, n 1), for A 1A 2 one has

( n 1 , n 2 )=0 .
(D.6.9)

Thus, at each point of a 4D spacetime, a non‐degenerate closed conformal Killing–Yano tensor defines 2 mutually orthogonal planes. We call a plane space‐like if all its vectors are space‐like, and call it time‐like if it contains a time‐like vector. We assume that at any spacetime point the 4D tangent space is spanned by one time‐like and one space‐like 2D eigenspace.

Consider a space‐like eigenspace and denote the conjugated normalized vectors that span it by {n 2, n¯2}. For these vectors ε n ( 2 ) = ε ¯ n ( 2 ) = 1 , and the relation Eq. (D.6.7) implies that A 2 〈 0. Similar arguments for the vectors {n 1,n¯1}, which span the timelike 2D eigen‐subspace, show that A 1 〉 0. In what follows we choose n 1 to be a time‐like vector, so that n¯1 as well as n 2 and n¯2, are space‐like vectors. We also denote

A 1 = r 2 , A 2 = y 2 , r 0 , y 0.
(D.6.10)

Thus, we have the following result:

Lemma 13: In each point of a spacetime with a non‐degenerate closed conformal Killing– Yano tensor there exists a canonical basis of normalized vectors {n 1,n¯1,n 2, n¯2} in which

g μ ν = n 1 μ n 1 ν + n ¯ 1 μ n ¯ 1 ν + n 2 μ n 2 ν + n ¯ 2 μ n ¯ 2 ν , h μ ν = 2 r n 1 [ μ n ¯ 1 ν ] 2 y n 2 [ μ n ¯ 2 ν ] .
(D.6.11)

We call such a basis a Darboux basis.

The quantities r and y depend on the choice of a point. In the general case, they change when we move from one point to another. In other words, r and y are scalar functions on the spacetime manifold. We extend our assumption of non‐degeneracy of the closed conformal Killing–Yano tensor to include into it a requirement that r and y are functionally independent (at least in some region U of the spacetime). This allows one to use r and y as coordinates in U. We call these coordinates Darboux coordinates.

We assume that the associated Killing vectors ξ μ and η μ do not vanish and are linearly independent of U. The integral lines x μ(τ) and y μ(ψ) for these vector fields are defined by equations

d x μ ( τ ) d τ = ξ μ , d y μ ( ψ ) d ψ = η μ .
(D.6.12)

We can use τ and ψ as two coordinates. We call them Killing coordinates.

To summarize, in the presence of a non‐degenerate closed conformal Killing–Yano tensor there exist, at least locally, coordinates (τ, r, y, ψ), which we call canonical coordinates, such that (r,y) are the Darboux coordinates and τ, ψ are the Killing coordinates.

(p.422) D.7 Canonical Form of Metric

D.7.1 Off‐shell canonical metric

Let us consider the following metric

d s 2 = 1 r 2 + y 2 [ Δ r ( d τ + y 2 d ψ ) 2 + Δ y ( d τ r 2 d ψ ) 2 ] + ( r 2 + y 2 ) [ d r 2 Δ r + d y 2 Δ y ] ,
(D.7.1)

where Δr = Δr(r) and Δy = Δy(y) are arbitrary functions.

Theorem: The metric Eq. (D.7.1) has a closed conformal Killing–Yano tensor h = d b. The corresponding potential b is

b = 1 2 [ ( r 2 y 2 ) d τ + r 2 y 2 d ψ ] .
(D.7.2)

This statement can be checked by direct calculations, which are straightforward but rather long. Another, much more efficient way is to use a computer for analytical calculations. For example, the GRTensor package can be used to make the required calculations in a very short time.

The form of the metric Eq. (D.7.1) was introduced by Debever (1971). Carter (1968b) demonstrated separation of variables in the Hamilton–Jacobi and Schrodinger equations in such a spacetime. For the discussion of the closed conformal Killing–Yano tensor for the metric Eq. (D.7.1) (see (Frolov and Kubizňák 2008; Kubiznak 2008) and references therein.)

We collect below some formulas for geometrical objects calculated for the metric Eq. (D.7.1). The required calculations are straightforward, but some of them are really long. The formulas below are obtained by using GRTensor package developed for the analytical calculations of the geometrical objects in a curved spacetime (see page 244). In particular, the determinant g of the metric Eq. (D.7.1) is

g = ( r 2 + y 2 ) 2 .
(D.7.3)

The closed conformal Killing–Yano tensor h and its Hodge dual Killing tensor k = ∗h are

h = h μ ν d x μ d x ν = y d y ( d τ r 2 d ψ ) + r d r ( d τ + y 2 d ψ ) , k = k μ ν d x μ d x ν = r d y ( d τ r 2 d ψ ) y d r ( d τ + y 2 d ψ ) .
(D.7.4)

The tensor H and the Killing tensor K are

H μ ν = ( r 2 y 2 0 0 r 2 y 2 0 r 2 0 0 0 0 y 2 0 1 0 0 0 ) ,
(D.7.5)
(p.423)
K μ ν = ( 0 0 0 r 2 y 2 0 y 2 0 0 0 0 r 2 ( r 2 y 2 ) 1 0 0 0 ) .
(D.7.6)

Here, tensor indices are ordered in accordance with (τ,r,y,ψ) choice of the coordinates. The One also has H μ μ = 2 ( r 2 y 2 ) .

D.7.2 Darboux basis for canonical metric

The characteristic equation for the eigenvalues of H is

det ( H λ I ) = 0.
(D.7.7)

Here, I is a unit matrix. The equation Eq. (D.7.7) has the form

( λ r 2 ) 2 ( λ + y 2 ) 2 = 0.
(D.7.8)

Thus, the eigenvalues of H are r 2 and −y 2, and (r, y) are the Darboux coordinates.

A corresponding normalized Darboux basis is

n 1 μ d x μ = Δ r r 2 + y 2 ( d τ + y 2 d ψ ) , n ¯ 1 μ d x μ = r 2 + y 2 Δ r d r , n 2 μ d x μ = Δ y r 2 + y 2 ( d τ r 2 d ψ ) , n ¯ 2 μ d x μ = r 2 + y 2 Δ y d y .
(D.7.9)

The primary and secondary Killing vectors are

ξ μ μ = τ , η μ μ = ψ .
(D.7.10)

The coordinates τ and ψ are corresponding Killing coordinates for these vectors.

Note that Eqs (D.7.2)–(D.7.6) do not contain the functions Δr(r) and Δy(y). That is, these equations are valid even if Δr(r) and Δy(y) describe the metrics that do not fulfill the Einstein equations. We call the metric Eq. (D.7.1) with arbitrary functions Δy and Δy the off‐shell metric, to distinguish it from an on‐shell metric satisfying the Einstein equations.

D.7.3 On‐shell canonical metric

We discuss now conditions on the metric functions Δr(r) and Δy(y) imposed by the vacuum Einstein equations with the cosmological constant

R μ ν = Λ g μ ν .
(D.7.11)

We use the GRTensor program to calculate the components of the Ricci tensor for the metric Eq. (D.7.1). We consider first the trace equation

R = 4 Λ ,
(D.7.12)

(p.424) which takes a very simple form

r 2 Δ r + y 2 Δ y = 4 Λ ( r 2 + y 2 ) .
(D.7.13)

This equation allows a separation of variables

r 2 Δ r + 4 Λ r 2 = C , y 2 Δ y + 4 Λ y 2 = C .
(D.7.14)

A solution of each of these two equations contains 2 independent integration constants. Thus, together with C one has 5 integration constants. Note that the metric Eq. (D.7.1) remains invariant under the following rescaling

r p r , y p y , τ p 1 τ , ψ p 3 ψ , Δ r p 4 Δ r , Δ y p 4 Δ y .
(D.7.15)

This means that one of the 5 integration constants can be excluded by means of these transformations. We write the answer in the following standard form

Δ r = ( r 2 + a 2 ) ( 1 Λ r 2 / 3 ) 2 M r , Δ y = ( a 2 y 2 ) ( 1 + Λ y 2 / 3 ) + 2 N y .
(D.7.16)

The four parameters in these functions are Λ, M, N, and a. For Λ = 0 and N = 0 this metric coincides with the Kerr metric, M and a being the mass and the rotation parameter, respectively. In addition to these two parameters, a general solution Eq. (D.7.16) contains the cosmological constant Λ, and a so‐called NUT parameter N. Solutions with a non‐trivial N contain singularities in the black hole exterior. For this reason we do not consider them here. In the general case, a solution with parameters M, a, and Λ describes a rotating black hole in the asymptotically de Sitter (for Λ 〉 0), anti‐de Sitter (for Λ 〈 0), or flat (for λ = 0) spacetime. A similar solution containing the NUT parameter N is known as Kerr‐NUT‐(A)dS spacetime. Direct calculation shows that for the choice Eq. (D.7.16) of the metric functions the Einstein equations are satisfied.

The metric for the Kerr‐NUT‐(A)dS spacetime can be written in a more symmetric form. Let us denote r = iz, N = N y, M = iN z,

Δ z = ( a 2 z 2 ) ( 1 + Λ z 2 / 3 ) + 2 N z z , Δ y = ( a 2 y 2 ) ( 1 + Λ y 2 / 3 ) + 2 N y y .
(D.7.17)

Then, the Kerr‐NUT‐(A)dS metric Eq. (D.7.1) takes the form

d s 2 = 1 y 2 z 2 [ Δ y ( d τ + z 2 d ψ ) 2 Δ z ( d τ + y 2 d ψ ) 2 ] + ( y 2 z 2 ) [ d y 2 Δ y d z 2 Δ z ] .
(D.7.18)

This metric is symmetric with respect to the formal substitution zy.

(p.425) D.8 Separation of Variables in Canonical Coordinates

A massive Klein–Gordon equation and the Hamilton–Jacobi equation allow a separation of variables in the off‐shell canonical metric Eq. (D.7.1). To demonstrate this we calculate the inverse metric to Eq. (D.7.1). Using this metric one can write

g g μ ν μ ν = Δ r ( r ) 2 + Δ y ( y ) 2 + Δ y 1 ( y 2 τ ψ ) 2 Δ r 1 ( r 2 τ + ψ ) 2 .
(D.8.1)

The Klein‐Gordon equation

Φ μ 2 Φ = 0
(D.8.2)

in the canonical coordinates takes the form

g ( μ 2 ) Φ = r ( Δ r r Φ ) + y ( Δ y y Φ ) 1 Δ r ( r 2 τ + ψ ) 2 Φ + 1 Δ y ( y 2 τ ψ ) 2 Φ μ 2 ( r 2 + y 2 ) Φ = 0.
(D.8.3)

To demonstrate that this equation allows a separation of variables we write

Φ = e i ω τ e i Ψ ψ R ( r ) Y ( y ) .
(D.8.4)

Substitution of this expression into Eq. (D.8.3) gives

r ( Δ r r R ) + 1 Δ r ( r 2 ω Ψ ) 2 R μ 2 r 2 R + λ R = 0 , y ( Δ y y Y ) 1 Δ y ( y 2 ω + Ψ ) 2 Y μ 2 y 2 Y λ Y = 0.
(D.8.5)

The parameter λ is a separation constant.

Similarly, the Hamilton–Jacobi equation

g μ ν μ S ν S + μ 2 = 0
(D.8.6)

can be separated. Indeed, let us write

S = ω τ + Ψ ψ + R ( r ) + Y ( y ) .
(D.8.7)

After multiplying Eq. (D.8.6) by g and substituting Eq. (D.8.7) one obtains

Δ r ( r ) 2 Δ r 1 ( r 2 ω Ψ ) 2 + μ 2 r 2 + λ = 0 , Δ y ( y Y ) 2 + Δ y 1 ( y 2 ω + Ψ ) 2 + μ 2 r 2 λ = 0.
(D.8.8)

D.9 Higher‐Dimensional Generalizations

D.9.1 Principal conformal Killing‐Yano tensor

In the models with large extra dimensions there exist a variety of black objects, which are generalizations of the four‐dimensional black holes. We discuss them in Section 10.6.2. These black objects differ by the topology of the event horizon. As in four dimensions, the horizon (p.426)

Appendix D Hidden Symmetries

Fig. D.1 Schematical illustration of the properties of conformal Killing‐Yano tensors. Points inside a large oval correspond to conformal Killing‐Yano tensors. Horizontal lines connect two conformal Killing‐Yano tensors, related by the Hodge–duality transformation. Two triangles represent closed conformal Killing–Yano tensors and Killing‐Yano tensors. The Hodge, duality transformation gives a map between these objects. An exterior product maps two closed conformal Killing‐Yano tensors to another closed conformal Killing–Yano tensor. A ‘square’ of a Killing‐Yano tensor gives a rank‐2 Killing tensor.

of a higher‐dimensional black object is a null surface. A black hole surface, that is a spatial slice of the horizon, is a compact space‐like surface of the dimension D − 2, where D is the dimension of the spacetime. We focus here on the black objects with the topology of the horizon surface S D−2. We call them black holes. The most general known solution of the higher‐dimensional Einstein equations for rotating higher‐dimensional black holes was obtained in (Chen et al. 2007). This, so‐called, higher‐dimensional Kerr‐NUT‐(A)dS metric is quite complicated. The remarkable fact is that this solution in many aspects is quite similar to its four‐dimensional ‘cousin’. The main reason for this is that it possesses a non‐degenerate closed conformal Killing–Yano tensor (Frolov and Kubizňák 2007; Kubiznak and Frolov 2007). Here, we briefly summarize important properties of these solutions. More detailed information and references can be found, e.g., in (Frolov and Kubizňák 2008).

We denote the number of dimensions of the spacetime by D = 2n + ε, where ε = 0 for even dimensions, and ε = 1 for odd dimensions. The basic objects responsible for the hidden symmetry are, Killing–Yano tensor and conformal Killing‐Yano tensor. They are antisymmetric tensors, or forms. To describe their properties it is very convenient to use the ‘language’ of differential forms (see Section 3.1.4). The exterior (or wedge) product of p‐form α p and q‐form β q is a (p + q)‐form α pβ q. The Hodge dual of a p‐form α p is a (Dp)‐form (∗α)Dp. An exterior derivative of the p‐form α p is a (p + 1)‐form p.

A conformal Killing–Yano tensor in D dimensions is defined by the same equation Eq. (D.2.1) as in, 4‐dimensional case. For the closed conformal Killing‐Yano tensor the first term in the right‐hand side of this equation vanishes. The conformal Killing–Yano tensor in the D‐dimensional spacetime possesses the following properties:

  1. (p.427) 1. The Hodge dual of a conformal Killing–Yano tensor is a conformal Killing‐Yano tensor.

  2. 2. The Hodge dual of a closed conformal Killing–Yano tensor is a Killing‐Yano tensor.

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

  4. 4. An exterior product of two closed conformal Killing–Yano tensors is a closed conformal Killing–Yano tensor.

Consider a 2‐form h obeying the equation

h μ ν ; λ = g λ ν ξ μ g λ μ ξ ν .
(D.9.1)

This equation implies

h [ μ ν ; λ ] = 0 , ξ μ = 1 D 1 h μ λ ; λ .
(D.9.2)

This object is a closed conformal Killing–Yano tensor. We assume that it is non‐degenerate, that is it matrix rank is 2n. Since this object plays a fundamental role in study of the hidden symmetries, we call it a principal conformal Killing–Yano tensor.

D.9.2 Killing‐Yano tower

Starting with a principal conformal Killing–Yano tensor h one can construct a set of new closed conformal Killing–Yano tensors

h j = h h h j times .
(D.9.3)

The index j enumerates how many hs are in the exterior product, so that h ∧1 = h. For j = n the object h n is either proportional to the totally antisymmetric tensor (for ε = 0), or its dual is a vector (for ε = 1). Excluding these ‘trivial’ cases, one has (n − 1) non‐trivial closed conformal Killing–Yano tensors. The Hodge dual of these tensors

k j = * h j ,
(D.9.4)

are (n − 1) Killing–Yano tensors, which can be used to construct (n − 1) Killing tensors

K j = k j k j .
(D.9.5)

The latter notation means that we use the operation described on page 415. Since the metric g is a (trivial) Killing tensor, this construction shows that in a spacetime with a principal conformal Killing–Yano tensor there exist n integrals of motion for geodesic equations, that are quadratic in momentum.

Let us show that in a general case this spacetime also has (n + ε) Killing vectors. The first of this set is a primary Killing vector ξ defined in Eq. (D.9.2). This vector obeys the relation

ξ ( μ ; ν ) = 1 D 2 R α ( ν h μ ) α .
(D.9.6)

(p.428) In an Einstein space, which is a solution of the equation

R μ ν = 2 D 2 Λ g μ ν ,
(D.9.7)

the relation Eq. (D.9.6) implies that ξ (μ;ν) = 0 is a Killing vector.1

It is also possible to show that the following objects constructed from the primary Killing vector

ξ μ j = K μ ν j ξ ν
(D.9.8)

are, in fact, again Killing vectors. In the odd‐dimensional spacetime there exists one more Killing vector that can be obtained as a Hodge dual of h n. We call these objects secondary Killing vectors. The total number of the (primary plus secondary) Killing vectors is n + ε. This gives (n + ε) additional integrals for geodesic motion, that are of the first order in momentum.

Thus, in a spacetime with a principal conformal Killing–Yano tensor geodesic equations have

n +( n + ε )=2 n + ε = D
(D.9.9)

integrals of motion. It is possible to show that they are in involution and (generically) they are functionally independent. This provides the complete integrability of geodesic equations in a spacetime with a principal conformal Killing‐Yano tensor. This is a direct generalization of the similar result for the four‐dimensional Kerr metric. It should be emphasized that this gives a new important example of a physically interesting dynamical system, which allows the complete integrability.

D.10 Higher‐Dimensional Kerr‐NUT‐(A)dS Metric

In a spacetime with symmetry generated by Killing vectors one can find special coordinates, in which the form of the metric is simplified. In such coordinates the metric coefficients are functions of fewer variables and their number is less than in the general case. Spacetimes with a hidden symmetry have a similar property. In particular, the existence of a principal conformal Killing–Yano tensor h imposes constraints on the geometry.

To describe a canonical form of the metric, in which the hidden symmetries generated by the Killing–Yano tower induced by h become most transparent, we define first a Darboux basis. Consider an orthonormal basis (e a, e ā, e 0). The last vector, e 0, is present only when ε = 1, that is the number of spacetime dimensions is odd. The indices a and ā, which enumerate the vectors, take values 1,2, …, n. Denote by (ω a, ω ā, ω 0) a dual basis of 1‐forms. It is possible to choose the basis in such a way, that

g μ ν = a = 1 n ( ω μ a ω ν a + ω μ a ¯ ω ν a ¯ ) + ε ω μ 0 ω ν 0 , h μ ν = 2 a = 1 n x a ω [ μ a ω ν ] a ¯ .
(D.10.1)

(p.429) This is a special choice of the Darboux basis, which possesses an additonal property: in this basis the metric is also diagonal. We define H μ ν = h μ λ h λ ν . For fixed a two vectors e a and e ā span a two‐dimensional eigenspace of H, and the corresponding coordinate x a is determined by the eigenvalue of H. Since h is non‐degenerate, its eigenvalues x a are functionally independent and can be used as Darboux coordinates on the spacetime manifold. By adding (n + ε) Killing coordinates ψ b one obtains the canonical coordinates. It is possible to show (Krtous et al. 2008; Houri et al. 2009), that in such (canonical) coordinates the metric takes the form

d s 2 = a = 1 n [ d x a 2 Q a + Q a ( b = 0 n 1 A a ( b ) d ψ b ) 2 ] ε c A ( n ) ( b = 0 n A ( b ) d ψ b ) 2 .
(D.10.2)

Here, A (b) and A a ( b ) are polynomials in the Darboux coordinates x a, defined by the following relations

a = 1 n ( 1 + λ x a 2 ) = b = 0 n λ b A ( b ) , ( 1 + λ x a 2 ) 1 a = 1 n ( 1 + λ x a 2 ) = b = 0 n 1 λ b A a ( b ) , Q a = X a U a , U a = a a ( x a 2 x a 2 ) .
(D.10.3)

Quantities X a are functions of one variable, X a = X a(x a). Equations (D.10.2) and (D.10.3) give the most general form of the metric that allows a principal conformal Killing–Yano tensor.

In the case when the canonical metric is a solution of the D‐dimensional Einstein equations, the functions X a take the form

X a = k = ε n c k x a 2 k 2 b a x a 1 ε + ε c x a 2 .
(D.10.4)

With this choice of X a the metric obeys Eq. (D.9.7) with

Λ = ( D 1 ) ( D 2 ) 2 ( 1 ) n c n .
(D.10.5)

The asymptotically flat spacetime is recovered when c n = 0

Time is denoted by ψ 0, azimuthal (Killing) coordinates by ψ k, k = 1,…, n − 1 + ε, and x a, a = 1,…, n, stand for Darboux coordinates. The metric with proper Lorentzian signature is recovered from Eq. (D.10.2) after the analytical continuation of the radial coordinate r = −ix n and the redefinition of the mass parameter M = (−i)1+ϵ b n.

The total number of constants that enter the solution is 2n + 1: ε constants c, n + 1 − ε constants c k and n constants b a. However, the form of the metric is invariant under a 1‐parameter scaling coordinate transformations, thus the total number of independent parame ters is 2n. These parameters are related to the cosmological constant, mass, angular momenta, (p.430) and NUT parameters. One of them, say Λ, may be used to define a scale, while the other 2n − 1 parameters can be made dimensionless (Chen et al. 2007). Similar to the 4D case, the signature of the symmetric form of the metric depends on the domain of x as and the signs of X as.

In the four‐dimensional spacetime the number of independent parameters in the solution is 4. They are the cosmological constant, mass, angular momentum and NUT parameter. When the cosmological constant and the NUT parameter vanish, after a proper analytical continuation one recovers the Kerr metric.

Notes:

(1) This property is also valid ‘off‐shell’, that is even if the Einstein equations are not satisfied. The proof of this is, however, much more complicated (see, e.g, (Krtous et al. 2008; Houri et al. 2009)).