Separation of Maxwell equations in Kerr-NUT-(A)dS spacetimes
aa r X i v : . [ h e p - t h ] J u l Separation of Maxwell equations in Kerr–NUT–(A)dSspacetimes
Pavel Krtouˇs a Valeri P. Frolov b David Kubizˇn´ak c a Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University,V Holeˇsoviˇck´ach 2, Prague, Czech Republic b Theoretical Physics Institute, Department of Physics, University of Alberta,Edmonton, Alberta, T6G 2G7, Canada c Perimeter Institute, 31 Caroline St. N. Waterloo Ontario, N2L 2Y5, Canada
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[email protected] , [email protected] , [email protected] Abstract:
In this paper we explicitly demonstrate separability of the Maxwell equationsin a wide class of higher-dimensional metrics which include the Kerr–NUT–(A)dS solutionas a special case. Namely, we prove such separability for the most general metric admittingthe principal tensor (a non-degenerate closed conformal Killing–Yano 2-form). To thispurpose we use a special ansatz for the electromagnetic potential, which we representas a product of a (rank 2) polarization tensor with the gradient of a potential function,generalizing the ansatz recently proposed by Lunin. We show that for a special choice ofthe polarization tensor written in terms of the principal tensor, both the Lorenz gaugecondition and the Maxwell equations reduce to a composition of mutually commutingoperators acting on the potential function. A solution to both these equations can bewritten in terms of an eigenfunction of these commuting operators. When incorporatinga multiplicative separation ansatz, it turns out that the eigenvalue equations reduce toa set of separated ordinary differential equations with the eigenvalues playing a role ofseparability constants. The remaining ambiguity in the separated equations is related toan identification of D − Keywords:
Electromagnetic Field, Proca Field, Separability, Black Holes, Higher Dimen-sions, Hidden Symmetries
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28D Technicalities & Proofs 30 – 1 –
Introduction
A method of separation of variables plays an important role in the theory of partial dif-ferential equations (PDEs). It allows one to reduce these equations to a set of ordinarydifferential equations (ODEs). The latter are simpler and can be solved either analyticallyor by simple numerical methods. In particular, the separation of variables in the equationsfor physical fields in a curved space of a stationary black hole allowed one to study manyphysical processes in the vicinity of these black holes such as propagation, scattering andcapture of waves. Separated equations for quasinormal modes were used to study the blackhole stability and its ringing radiation. The method of separation of variables is also usedto study the quantum Hawking effect.Separation of variables in the physical field equations in the rotating black hole space-time described by the Kerr geometry has a long history. It started in 1968, when Carterdemonstrated that a scalar field equation can be solved by a method of separation of vari-ables [1]. In 1972, Teukolsky [2, 3] decoupled equations for the electromagnetic and grav-itational perturbations and demonstrated that decoupled equations can be solved by theseparation of variables. The massless neutrino equations were separated by Teukolsky [3]and Unruh [4] in 1973, and the massive Dirac equations were separated by Chandrasekhar[5] and Page [6] in 1976.More recently, the development of brane-world models and the discussion of the possi-bility of mini black-hole creation in colliders attracted a lot of attention to the problem ofseparation of variables in higher-dimensional black hole spacetimes. This problem is ratherstraightforward for the (spherically symmetric) Tangherlini metric, which is a simple gen-eralization of the Schwarzschild geometry. However, in the presence of rotation and NUTparameters it becomes quite complicated. One of the reasons is that even if the equationsare separable in a given geometry, the separation occurs only in a very special coordinatesystem which is a priori not known. Separation of variables in the Klein–Gordon equa-tion in the five-dimensional Myers–Perry metric was first demonstrated in [7], see also [8]for the 5-dimensional Kerr-(A)dS generalization. Page and collaborators [9–11] discoveredthat the Klein–Gordon equation is separable in a special case of the higher-dimensionalKerr-(A)dS spacetime, provided the black hole spin is restricted to two sets of equal rota-tion parameters. Upon this restriction, the explicit symmetry of the spacetime is enhancedand makes the separation of variables possible. Similar results, exploiting the enhancedsymmetry of black holes arising from a restriction on rotation parameters, were obtainedin [12, 13].The discovery [14] of the principal tensor in the most general higher-dimensional Kerr–NUT–(A)dS spacetime [15] made it possible to solve the problem of separability for theKlein–Gordon equation without any restriction on rotation parameters [16]. The principaltensor is a non-degenerate rank-2 closed conformal Killing–Yano tensor. The discussion ofits remarkable properties can be found in a comprehensive review [17]. This tensor gen-erates a complete set (tower) of symmetries, which consists of Killing vectors and rank-2Killing tensors. Moreover, the eigenvalues of the principal tensor, together with the appro-priate choice of the Killing coordinates, define special, the so called canonical coordinates.– 2 –t is in these coordinates the separability property is valid for the Klein–Gordon equation.Later, it was shown that using the Killing vectors and Killing tensors in the Killing towerone can construct a full set of the corresponding first-order and second-order covariantdifferential operators, all mutually commuting, such that their eigenvalues coincide withthe corresponding separation constants of the Klein–Gordon equation [18–21]. This resultdemonstrated a close relationship between the separability structure of the spacetime andthe existence of the principal tensor.Let us write D = 2 N + ε for the number of spacetime dimensions, with ε = 1 forodd dimensions and ε = 0 for even ones. As shown in [22, 23], the most general metricthat possesses the principal tensor admits N arbitrary metric functions of one variable.We call such metrics off-shell. For the on-shell metric, when the Einstein equations areimposed, these metric functions reduce to polynomials, and, in the Lorentzian signature, werecover the Kerr–NUT–(A)dS solution [15]. Interestingly, the separation of variables in theKlein–Gordon equation remains valid for a general higher-dimensional off-shell geometry.The separability of the massive Dirac equation in the higher-dimensional off-shell Kerr–NUT–(A)dS spacetimes was proved in [24], see also [25–28] for the intrinsic characterizationof this separability in terms of the commuting operators. A partial success regarding theseparation of variables for the special type of gravitational perturbations in these spacetimeswas achieved in [29, 30].The question of separability of Maxwell equations in higher-dimensional rotating blackhole spacetimes remained open for a long time. In four dimensions both electromagneticstrength field F and its Hodge dual ∗ F are 2-forms. The complex self-dual and anti-self-dual2-forms F ± i ∗ F describe independent right- and left-polarization states of propagatingelectromagnetic waves. This property was essentially used in various schemes of reductionof the Maxwell equations to a set of complex scalar equations and their further separationin the 4D Kerr–NUT–(A)dS metrics. Unfortunately, such a method cannot be generalizedto higher dimensions.A breakthrough in the problem of separability of the Maxwell equations in higher-dimensional rotating black hole spacetimes came in recent Lunin’s paper [31]. Luninhas proposed a special ansatz for the vector potential, which can be reformulated as A a = B ab ∇ b Z , where Z is a complex scalar function and B is a special tensor, whichwe call the polarization tensor. In his work Lunin has written down the ansatz for thevector potential in a special frame, effectively specifying the polarization tensor. He usedspecial coordinates, different from the Myers–Perry coordinates, which are closely relatedto the canonical coordinated connected with the principal tensor [32, 33]. In this settingLunin demonstrated [31] that the Maxwell equations in the higher-dimensional Kerr–(A)dSspacetimes imply separable equations for the function Z .In this paper we propose an essential development of Lunin’s approach. Our analysisis performed for general off-shell metrics which admit a non-degenerate principal tensor h .We first find a covariant (coordinate-independent) expression for the polarization tensor B in terms of the principal tensor. Next we show that the Lorenz condition for the vectorpotential, which becomes a second-order wave-like operator acting on the function Z , canbe understood as a composition of N second-order commuting operators. These, sup-– 3 –lemented with derivatives along the explicit spacetime symmetries, form a system of D mutually commuting operators which possess a common system of eigenfunctions. Amongthese eigenfunctions one can find those solving the Lorenz condition. It turns out thatthese solutions are labeled by D eigenvalues and by a discrete choice of N − N second-order ordinarydifferential equations. Such solutions are labeled by D − D − Off-shell Kerr–NUT–(A)dS geometry
This section contains a brief summary of properties of spacetimes admitting the principaltensor. A thorough discussion of these spacetimes, the principal tensor, the associatedKilling tower, and the on-shell and off-shell Kerr–NUT–(A)dS geometries can be found inthe recent review [17].
Principal tensor and metric
In what follows we denote by D = 2 N + ε the number of dimensions. We are interested inspaces which possess the principal tensor: a non-degenerate closed conformal Killing–Yano2-form. The principal tensor h satisfies the following equation: ∇ c h ab = g ca ξ b − g cb ξ a , (2.1)where ξ is a primary Killing vector, ξ a = 1 D − ∇ b h ba . (2.2)The non-degeneracy of the principal tensor essentially means that the principal tensor hasnon-degenerate imaginary eigenvalues ± ix µ , ( µ = 1 , . . . , N ), and that x µ are independentfunctions which, when supplemented with an appropriate set of Killing angles, can be usedas canonical coordinates . The metric can be written in a formally Euclidian Darboux framein which the principal tensor has a semi-diagonal form: g = X µ (cid:0) e µ e µ + ˆ e µ ˆ e µ (cid:1) + ε ˆ e ˆ e , (2.3) h = X µ x µ e µ ∧ ˆ e µ . (2.4)In the canonical coordinates, the metric reads g = N X µ =1 (cid:20) U µ X µ d x µ + X µ U µ (cid:16) N − X j =0 A ( j ) µ d ψ j (cid:17) (cid:21) + ε cA ( N ) (cid:16) N X k =0 A ( k ) d ψ k (cid:17) , (2.5)where A ( k ) , A ( j ) µ , and U µ are explicit polynomial functions of coordinates x ν , A ( k ) = X ν ,...,ν k ν < ··· <ν k x ν . . . x ν k , A ( j ) µ = X ν ,...,ν j ν < ··· <ν j ν i = µ x ν . . . x ν j ,U µ = Y νν = µ ( x ν − x µ ) , (2.6)and each metric function X µ is a function of a single coordinate x µ : X µ = X µ ( x µ ) . (2.7) We write sums over coordinate indices µ, ν, . . . and k, l, . . . explicitly, but we usually do not indicatetheir ranges. If they are not indicated, we assume P µ = P Nµ =1 and P k = P N − k =0 . – 5 –f these functions are chosen arbitrary, the geometry in general does not satisfy the Einsteinequations and we call it off-shell. If the Einstein equations are imposed, X µ must take aform of specific polynomials [15, 22]: X µ = − b µ x µ + N X k =0 c k x kµ for D even , − cx µ − b µ + N X k =1 c k x kµ for D odd . (2.8)Here the parameter c N gives the cosmological constant, while other parameters are relatedto the mass, NUT parameters, and rotations, see [17] for more details. In particular,in the Lorentzian signature we would recover the on-shell Kerr–NUT–(A)dS spacetimes[15]. However, in what follows we do not assume this specific choice and the subsequentdiscussion is valid for the full off-shell family of spacetimes.The Darboux frame of 1-forms e µ , ˆ e µ ( µ = 1 , . . . , N ), and ˆ e (in odd dimensions) read e µ = (cid:16) U µ X µ (cid:17) d x µ , ˆ e µ = (cid:16) X µ U µ (cid:17) N − X j =0 A ( j ) µ d ψ j , ˆ e = (cid:16) cA ( N ) (cid:17) N X k =0 A ( k ) d ψ k , (2.9)with the dual vector frame e µ , ˆ e µ and ˆ e given by e µ = (cid:16) X µ U µ (cid:17) ∂ x µ , ˆ e µ = (cid:16) U µ X µ (cid:17) N − ε X k =0 ( − x µ ) N − − k U µ ∂ ψ k , ˆ e = (cid:0) cA ( N ) (cid:1) − ∂ ψ N . (2.10)The primary Killing vector in the canonical coordinates and the Darboux frame is ξ = ∂ ψ = X µ (cid:16) X µ U µ (cid:17) ˆ e µ + ε (cid:16) cA ( N ) (cid:17) ˆ e . (2.11)The square of the principal tensor is a conformal Killing tensor, Q ab = h ac h bd g cd , (2.12)which identifies e µ and ˆ e µ as its eigenvectors with the eigenvalue x µ : Q = X µ x µ (cid:0) e µ e µ + ˆ e µ ˆ e µ (cid:1) . (2.13)The metric (2.5) describes a wide class of geometries, both Riemannian and Lorentzian,subject to possible Wick rotations of coordinates and a choice of signs of metric functions.We will not attempt to classify this family of geometries here, see [17] for a discussion. Wejust recall that the family contains the on-shell Kerr–NUT–(A)dS black holes, which, whenthe NUT parameters are turned off, are equivalent to the Myers–Perry spacetimes [36] withpossibly a cosmological constant [37, 38]. The coordinates used here generalize Carter’scoordinates known in four dimension, with x N being the Wick rotated radial coordinate andother x ν corresponding to (cosine of) latitudinal angular coordinates. Killing coordinates ψ k correspond to explicit symmetries of the space: time and longitudinal angles. However,this relation is not direct, see appendix A for more details.– 6 – illing tower The principal tensor guarantees the existence of a rich symmetry structure, the so calledKilling tower of Killing and Killing–Yano objects [39]. Here we are going to introduce onlythe Killing tensors and Killing vectors since they are directly related to the symmetries ofvarious fields in the studied spaces. The Killing tower can be defined in terms of generatingfunctions. First we define a β -dependent conformal Killing tensor q ( β ), q ( β ) = g + β Q , (2.14)and scalar functions A ( β ) and A µ ( β ), A ( β ) = s Det q ( β )Det g = Y ν (1 + β x ν ) , (2.15) A µ ( β ) = A ( β )1 + β x µ = Y νν = µ (1 + β x ν ) . (2.16)In the following, we usually skip the argument β to keep the expressions more compact.The generating Killing tensor k ( β ) and the generating Killing vector l ( β ) are definedas k = A q − , (2.17) l = k · ξ . (2.18)The Killing tower of Killing tensors k ( j ) and Killing vectors l ( j ) is given by an expansionin β k ( β ) = X j k ( j ) β j , (2.19) l ( β ) = X j l ( j ) β j . (2.20)Note that only terms for j = 0 , , . . . , N − ε are nonvanishing. One also has A ( β ) = N X j =0 A ( j ) β j , A µ ( β ) = N − X j =0 A ( j ) µ β j , (2.21)with A ( j ) , A ( j ) µ being the standard symmetric polynomials introduced in (2.6).In the Darboux frame, the generating Killing tensor and Killing vector are k = X µ A µ (cid:0) e µ e µ + ˆ e µ ˆ e µ (cid:1) + εA ˆ e ˆ e , (2.22) l = X µ A µ (cid:16) X µ U µ (cid:17) ˆ e µ + εA (cid:16) cA ( N ) (cid:17) ˆ e , (2.23)– 7 –hile in coordinates they read k = X µ A µ U µ (cid:20) X µ ∂ x µ + 1 X µ (cid:16) N − ε X k =0 ( − x µ ) N − − k ∂ ψ k (cid:17) (cid:21) + ε AA ( N ) ∂ ψ n , (2.24) l = N − ε X j =0 β j ∂ ψ j . (2.25)Similar expressions for individual Killing tensors and Killing vectors from the tower areobtained by a simple β -expansion. We emphasize only l ( j ) = ∂ ψ j . (2.26)A trace k aa ( β ) of the generating Killing tensor is k aa = 2 X µ A µ + εA = − β ddβ ( β − D A ) , (2.27)since the traces of individual Killing tensors are k ( j ) aa = 2 X µ A ( j ) µ + εA ( j ) = ( D − j ) A ( j ) , (2.28)and k aa ( β ) = N − X j =0 k ( j ) aa β j . (2.29)The covariant derivative of the generating Killing tensor is [17] ∇ c k ab = 2 β A (cid:0) k ab k cn h nm + h mn k n ( a k b ) c + k m ( a k b ) n h nc (cid:1) ξ m . (2.30)The contraction gives ∇ n k na = β A (2 k am k mn h nl ξ l − k cc k am h mn ξ n ) = (cid:16) k an − k cc g an (cid:17) A ∇ n A , (2.31)where we used another useful relation:12 ∇ a A = β h an l n . (2.32)Finally, the generating Killing tensor commutes with the principal tensor in the senseof matrix multiplication h an k nb = k an h nb . (2.33)All these definitions and relations have been discussed in the literature and are reviewedin [17]. – 8 – Field ansatz
We want to study a test electromagnetic field in the background of the off-shell Kerr–NUT–(A)dS spacetime. We are looking for a field which solves the Maxwell equations ina separable form. However, we have to face the fact that the electromagnetic field hasseveral components and that these components are coupled together. The long-standingproblem of decoupling the Maxwell equations in higher dimension was successfully attackedby Lunin [31] in the case of the field in the background of the Myers–Perry and Kerr–(A)dSblack holes.In four dimension we have demonstrated [40] that Lunin’s ansatz for the field can bereformulated covariantly in terms of the principal tensor. Similarly, in higher dimensionswe assume that the electromagnetic vector potential A has the form A a = B ab ∇ b Z . (3.1)Here, Z is an auxiliary complex scalar function that plays a role of a kind of scalar potentialfor the vector potential A . This function will be searched for and found in a multiplicativeseparated form.Let us first concentrate on the polarization tensor B in the ansatz (3.1). It is definedin terms of the principal tensor h as B ac ( g cb − βh cb ) = δ ab . (3.2) B ( β ) thus depends on a parameter β , which is in general complex.Since (3.2) means that B = ( g − β h ) − , the ‘symmetric square’ of B is closely relatedto the generating Killing tensor k B · g · B T = ( g − β h ) − · g · ( g + β h ) − = ( g + β Q ) − = 1 A k , (3.3)or in indices, B ak B bl g kl = 1 A k ab . (3.4)Inverting B T , we find B ab = 1 A g am ( g mn + βh mn ) k nb . (3.5)From here we can read off the symmetric and antisymmetric parts of B : B ( ab ) = 1 A k ab ,B [ ab ] = βA h an k nb = βA k an h nb . (3.6) Unfortunately, the letter A is heavily used for various alternatives of metric functions, namely, A , A µ , A ( k ) , A ( k ) µ . Therefore we use Serif font for the vector potential A a to avoid a confusion. Consistently, weuse F ab for the field strength. Here and later we use a dot to denote a contraction of two subsequent tensors with respect to their twoneighbor indices. For example, for two tensors with components X ab and Y cd , X · Y means a tensor withcomponents X ac Y cb . – 9 –hanks to this, the trace of B is B nn = k aa A . (3.7)Taking a covariant derivative of definition (3.2) and employing relation (2.1) and (3.2),one finds ∇ c B ab = β ( B ac ξ n B nb − B an ξ n B cb ) . (3.8)Contractions yield ∇ n B nb = βA ( k aa ξ n B nb − ξ n k nb ) , ∇ n B an = βA ( ξ n k na − k bb B an ξ n ) . (3.9) We use the ansatz (3.1) to obtain solutions of the Maxwell equations in the higher di-mensional off-shell Kerr-NUT-(A)dS spacetimes. We proceed as follows. First, we imposethe Lorenz condition on the potential A and demonstrate that the obtained second orderequation for the potential Z allows the separation of variables in the canonical coordinates.After this we show that the Maxwell field equations are satisfied provided (i) the Lorenzequation is valid and (ii) an additional equation for Z is valid. And finally, we show thatthis additional equation is also satisfied provided Z obeys the separable equation, obtainedfrom the Lorenz condition.For simplicity, starting with this section we restrict ourselves to even dimensions.Thanks to that the coordinate expressions for differential operators are slightly shorter.The full expression for scalar operators in odd dimensions can be found in [20]. Similarexpressions could be written for the electromagnetic case. Covariant form of the Lorenz condition
Let us start investigating the Lorenz condition ∇ a A a = 0 . (4.1)In the appendix (see (D.4), (D.5)) we show that the divergence of the vector potential (3.1)reads ∇ m A m = ∇ m (cid:0) B mn ∇ n Z (cid:1) = ∇ m (cid:16) A k mn ∇ n Z (cid:17) + βA (cid:16) k aa A − (cid:17) l n ∇ n Z . (4.2)Taking the factor 1 /A out in the first term, one can also write ∇ m (cid:0) B mn ∇ n Z (cid:1) = 1 A ∇ m (cid:0) k mn ∇ n Z (cid:1) + 1 A (cid:18) − A ( ∇ m A ) k mn + β (cid:16) k aa A − (cid:17) l n (cid:19) ∇ n Z . (4.3)
Coordinate form of the Lorenz condition
The first term in (4.3) is, up to a prefactor 1 /A , the scalar wave operator associated withthe Killing tensor k . Such operators have been studied in [20] and we can use its coordinate– 10 –orm (B.3) reviewed in the appendix B. Using (2.15) and (2.24), we find that the first termin the brackets in (4.3), which is linear in ∇ Z , has the form − A ( ∇ m A ) k mn ∇ n Z = − X ν A ν U ν X ν β x ν β x ν ∂∂x ν Z . (4.4)In the appendix we prove the identity (D.8), k aa A − β − D X ν A ν U ν − β x ν β x ν , (4.5)which allows us to express the second term in the brackets linear in ∇ Z in (4.3), β (cid:16) k aa A − (cid:17) l n ∇ n Z = β X ν A ν U ν − β x ν β x ν β − N ) X j β j ∂∂ψ j Z . (4.6)Putting these together, the coordinate expression for the divergence of the vector potential(4.3) reads ∇ m (cid:0) B mn ∇ n Z (cid:1) = 1 A X ν A ν U ν ˜ C ν Z , (4.7)where ˜ C ν =(1+ β x ν ) ∂∂x ν h X ν β x ν ∂∂x ν i + 1 X ν hX j ( − x ν ) N − − j ∂∂ψ j i + β − β x ν β x ν β − N ) X j β j ∂∂ψ j . (4.8) Covariant form of the Maxwell equations
The left-hand side of the Maxwell equations written in terms of the vector potential reads ∇ n F an = − (cid:3) A a + R an A n + ∇ a (cid:0) ∇ n A n (cid:1) , (4.9)with (cid:3) ≡ ∇ m ∇ m . Inserting ansatz (3.1), we get ∇ n F an = −∇ m ∇ m ( B an ∇ n Z ) + R am B mn ∇ n Z + ∇ a (cid:0) ∇ m ( B mn ∇ n Z ) (cid:1) . (4.10)In appendix D we derive a nontrivial identity (D.11) for the first two terms, which gives us ∇ n F an = − B am ∇ m (cid:0) (cid:3) Z + 2 βξ k B kn ∇ n Z (cid:1) + 2 βB ak ξ k ∇ m ( B mn ∇ n Z ) + ∇ a (cid:0) ∇ m ( B mn ∇ n Z ) (cid:1) . (4.11)Clearly, if the Lorenz condition is satisfied, ∇ m (cid:0) B mn ∇ n Z (cid:1) = 0, then the last two termsvanish and the vacuum Maxwell equations read B am ∇ m (cid:16) (cid:3) Z + 2 βξ k B kn ∇ n Z (cid:17) = 0 . (4.12)– 11 – oordinate form of the Maxwell equations We already know the coordinate form of the Lorenz condition, so we concentrate on theoperator (cid:0) (cid:3) + 2 βξ k B kn ∇ n (cid:1) Z . (4.13)The box operator is given by expression for K in (B.2). The second term in the bracket,using (3.5), (2.18), and (2.32), yields2 βξ k B kn ∇ n Z = β A l n ∇ n Z − A ( ∇ n A ) ∇ n Z . (4.14)Employing identity (D.10) and the coordinate form (2.25) in the first term and A ∇ A = ∇ log A = P ν ∇ log(1+ β x ν ) with the index raised using the coordinate metric component g νν = X ν U ν in the second term, we obtain2 βξ k B kn ∇ n Z = − X ν X ν U ν β x ν β x ν ∂∂x ν Z + β X ν U ν − β x ν β x ν β − N ) X k β k ∂∂ψ k Z . (4.15)The first term nicely combines with the coordinate expression for the box, yielding (cid:2) (cid:3) + 2 βξ k B kn ∇ n (cid:3) Z = X ν U ν ˜ C ν Z (4.16)for the operator (4.13), with ˜ C ν defined by (4.8). Massive vector field equations
Although it is not the main topic of this paper, let us briefly comment on a generalizationof the Maxwell field to the massive case. The vector Proca field A satisfies the followingfield equations [41–44]: ∇ n F an + m A a = 0 . (4.17)As a direct consequence, the Lorentz condition (4.1) must be satisfied. Employing (4.11),(4.17) for our ansatz gives B am ∇ m (cid:16) (cid:3) Z + 2 βξ k B kn ∇ n Z (cid:17) = m B am ∇ m Z . (4.18)Clearly, it is satisfied if h (cid:3) + 2 βξ k B kn ∇ n i Z = m Z . (4.19)The sufficient conditions for the Proca equations are thus the Lorentz condition (4.1) andthe eigenfunction equation (4.19) of the operator (4.13).In coordinates, the previous results (4.7) and (4.16) require1 A X ν A ν U ν ˜ C ν Z = 0 , (4.20) X ν U ν ˜ C ν Z = m Z . (4.21)The electromagnetic case is recovered upon switching off the mass, m = 0.– 12 – Structure of the equations
In this section we are going to discuss the general structure of the obtained equations, theassociated system of commuting operators, and the corresponding eigenvalue problem. Westart with the following observation. k – ν transform Let O k , k = 0 , . . . , N − N ‘objects’. Define the following ‘polynomials’:˜ O ν = X k ( − x ν ) N − − k O k . (5.1)˜ O ν are thus polynomials in variable x ν with the same coefficients O k . Applying the algebraicrelation (D.1) we can write O k = X ν A ( k ) ν U ν ˜ O ν . (5.2)Moreover, we can define the following ‘generating’ polynomial O , depending on an auxiliaryvariable β : O ≡ X k O k β k = X ν A ν U ν ˜ O ν . (5.3)We can think of the above relations as ( k – ν )-transform between O k and ˜ O ν objects. Inparticular, if O k are ordinary numbers, ˜ C ν are normal polynomials. In what follows,however, we will use this transform also for the differential operators. In such a case, ˜ O ν will typically be an operator in variable x ν only. System of commuting operators
The Lorenz condition (4.7) and the modified box operator (4.16) are constructed using thesame operators ˜ C ν (4.8). Introducing the Killing-vector operators L k , L k = − i ∂∂ψ k , (5.4)together (by employing the ( k – ν )-transform) with the associated operators ˜ L ν and L ˜ L ν = X k ( − x ν ) N − − k L k , L k = X ν A ( k ) ν U ν ˜ L ν , (5.5) L = X k L k β k = X ν A ν U ν ˜ L ν , (5.6)the operators ˜ C ν take the following form:˜ C ν = (1+ β x ν ) ∂∂x ν h X ν β x ν ∂∂x ν i − X ν ˜ L ν + iβ − β x ν β x ν β − N ) L . (5.7)– 13 –n a similar manner, starting from ˜ C ν , we introduce C k and C ,˜ C ν = X k ( − x ν ) N − − k C k , C k = X ν A ( k ) ν U ν ˜ C ν , (5.8) C = X k C k β k = X ν A ν U ν ˜ C ν . (5.9)Using these definitions, we can present the operators (4.7) and (4.16) in the following form: ∇ m (cid:2) B mn ∇ n (cid:3) = 1 A X ν A ν U ν ˜ C ν = 1 A C , (5.10) (cid:2) (cid:3) + 2 βξ k B kn ∇ n (cid:3) = X ν U ν ˜ C ν = C . (5.11)It is important to notice that unlike operators L k and ˜ L µ , operators C k and ˜ C µ are β -dependent. We do not write this dependence explicitly but we should remember it. Incase of the Killing-vector operator L the expansion in β gives directly L k . Since C k dependon β , the same is not true for β -expansion of C , although the relation (5.9) still holds true.Let us observe here that the operator ∇ m (cid:2) B mn ∇ n (cid:3) is symmetric for β imaginary. Thisfollows from the fact that β enters the definition of B in a combination with antisymmetric h , cf. (3.2). This might suggest we set β = − iµ , (5.12)assuming µ to be real; this notation has been used in [40]. However, such a choice wouldpossibly restrict the ability to describe a sufficient set of independent polarizations as canbe seen from discussion in [45]. For this reason in what follows we continue working witha general complex β .An important property of the operators C k and L k is that, for a fixed value of β , theseoperators mutually commute[ C k , C l ] = 0 , [ C k , L l ] = 0 , [ L k , L l ] = 0 . (5.13)Beware, however, that for different values of β , this is no longer true, and [ C k ( β ) , C l ( β )] = 0.The commutation of the Killing-vector operators L k is obvious. The commutation of C k follows from the commutation of operators ˜ C ν . Each ˜ C ν contains just one x -variable x ν ,derivatives with respect to x ν , and derivatives with respect to all ψ k . Therefore, for κ = λ operators ˜ C κ and ˜ C λ trivially commute. For κ = λ they commute only for the same valueof β , when they are identical. For fixed β , we thus have ˜ C κ ˜ C λ = ˜ C λ ˜ C κ . Expanding the rightoperators using (5.8) we get X l (cid:16) ( − x λ ) N − − l ˜ C κ C l + δ κλ H lκ C l (cid:17) = X k (cid:16) ( − x κ ) N − − k ˜ C λ C k + δ λκ H kλ C k (cid:17) , (5.14)where H jν = ˜ C ν ( − x ν ) N − − j . Sums of terms with H ’s on both sides are the same. Applyingrelation (5.8) once more we get X k,l ( − x κ ) N − − k ( − x λ ) N − − l C k C l = X k,l ( − x κ ) N − − k ( − x κ ) N − − l C l C k . (5.15)– 14 –ince the matrix ( − x ν ) N − − j (indexed by ν and j ) is nonsingular, the commutativity[ C k , C l ] = 0 follows. System of eigenfunctions
For fixed β we have commuting operators C k and L k . We can thus introduce a systemof common eigenfunctions Z ≡ Z ( β ; C , . . . , C N − , L , . . . , L N − ) labeled by eigenvalues C k and L k , C k Z = C k Z , L k Z = L k Z . (5.16)The eigenvalues L k are related to the explicit symmetries corresponding to coordinates ψ k .For periodic angular coordinates, such operators would acquire discrete values. We referto the appendix A for the corresponding discussion.On the other hand, at the moment we do not have a covariant form for the operators C k which would connect the eigenvalues C k with some physical quantities. We expect that suchoperators are related to the hidden symmetries encoded by Killing tensors. Unfortunately,some obvious guesses for C l as for example ∇ · k ( l ) · ∇ − iβ l ( l ) · B · ∇ do not quite work.Although the operators C k depend on β , we understand eigenvalues C k , as well as L k , tobe β -independent. In what follows, we shall use the eigenfunctions Z to generate solutions of the Maxwellequations.Let us finish this section by introducing some auxiliary notation. Starting with con-stants C k and L k , and using the ( k – ν )-transform as we did for operators, we define poly-nomials ˜ C ν ≡ ˜ C ν ( x ν ) and C ≡ C ( β ), and ˜ L ν ≡ ˜ L ν ( x ν ) and L ≡ L ( β ) as follows˜ C ν = X k ( − x ν ) N − − k C k , C k = X ν A ( k ) ν U ν ˜ C ν , (5.17) C = X k C k β k = X µ A ν U ν ˜ C ν , (5.18)˜ L ν = X k ( − x ν ) N − − k L k , L k = X ν A ( k ) ν U ν ˜ L ν , (5.19) L = X k L k β k = X µ A ν U ν ˜ L ν . (5.20) This is just a convention how to parameterize eigenfunctions Z for various values of β . Such aparametrization is possible if we assume that for different values of β the spectrum of operators remainsthe same. This assumption may be too strong for the detailed study of the spectrum following from e.g.regularity of the eigenfunctions. Since we do not do such a study here, we can ignore potential problemsand assume that, at least in some range of β , the spectrum remains the same, and that we can use thesame eigenvalues for different β to parameterize these eigenfunctions. – 15 – Solutions: magnetic polarizations
Parametrization using polarizations
As it was shown above, the solution of the Maxwell equations can be generated throughthe ansatz (3.1) by function Z satisfying the Lorenz condition and the condition (4.12),which, using (5.10) and (5.11), are C Z = 0 , C Z = 0 . (6.1)The second condition can be easily satisfied by our eigenfunctions Z with C = 0.Note that the trivial C is no longer required for the massive vector field discussed in theprevious section, where we effectively require C = m , cf. (4.21); see [45] for a discussionof interesting consequences. The first condition, when applied to Z , requires C ( β ) ≡ X k C k β k = 0 . (6.2)This could be trivially satisfied by setting all C k = 0, but it would reduce our system ofeigenfunctions too much. Fortunately, we can utilize here the freedom in parameter β bysetting it to one of N − β , . . . , β N − of the polynomial C ( β ). We thus define N − Z P mg , P = 0 , . . . , N −
2, each labeled by 2 N − Z P mg ( C , . . . , C N − , L , . . . , L N − ) = Z ( β P ; 0 , C , . . . , C N − , L , . . . , L N − ) . (6.3)Setting C = 0 implies that one of the roots, say β , of the polynomial C ( β ) is zero, β = 0, cf. definition (5.18). However, for β = 0 our ansatz (3.1) gives a pure gauge field.We thus have only N − P = 1 , . . . , N − D = 2 N dimensions one should have D − N − N − N − Alternative parametrization
In the picture described above we use eigenvalues C k , L k to parameterize solutions and foreach choice of them we set β to one of the roots β P . It gives us a discrete choice of thepolarization for given eigenvalues. Clearly, changing constants C k , L k varies the roots β P and these roots can mix their values. The nature of independence of the polarizations isthus not completely clear.One can therefore prefer a different parametrization of functions satisfying the Lorenzcondition C Z = 0. Instead of constants C , . . . , C N − , which define a root β ∗ , one can use– 16 –he root β ∗ and constants C , . . . , C N − as independent and find a value of C so that C ( β ∗ ) = 0. Clearly, C must be given by¯ C = − β ∗ N − X k =0 C k +1 β k ∗ . (6.4)The Maxwell equations then require ¯ C = 0. It can be achieved by setting β ∗ = 0, whichleads to a pure gauge as before, or by imposing a linear constraint N − X k =0 C k +1 β k ∗ = 0 (6.5)on the remaining constants C , . . . , C N − . It can be solved, for example, by evaluating C in terms of other constants, ¯ C = − β ∗ N − X k =0 C k +2 β k ∗ . (6.6)The “magnetic” solutions of the Maxwell equations can thus be generated through theansatz (3.1) using functions Z mg ( β ∗ ; C . . . , C N − , L , . . . , L N − ) = Z ( β ∗ ; 0 , ¯ C , C , . . . , C N − , L , . . . , L N − ) . (6.7)In this parametrization we do not distinguish a discrete choice of the polarization, insteadwe have a direct control over the root β ∗ . However, changing β ∗ and C , . . . , C N − , L , . . . , L N − freely should cover the same set of function as Z P mg introduced above.This parametrization corresponds more to Lunin’s approach, as far as we are able tocompare. Yet another parametrization
Another way how to solve the Lorenz condition, i.e., to enforce that β ∗ is a root of C ( β ),is to require that the polynomial C has the following form: C = ( β − β ∗ ) Q , Q = N − X k =0 Q k β N − − k ) . (6.8)It gives C k in terms of Q , . . . , Q N − and β ∗ , namely for C , C = − β ∗ Q N − . (6.9)Setting thus Q N − = 0 guarantees the Maxwell equations. Hence, the solution is generatedby function Z mg ′ Z mg ′ ( β ∗ ; Q . . . , Q N − , L , . . . , L N − ) = Z ( β ∗ ; 0 , C , C , . . . , C N − , L , . . . , L N − ) , (6.10)with C k evaluated from β ∗ and Q , . . . , Q N − using (6.8).– 17 –t will be useful in a discussion of the separation of variables to evaluate polynomials˜ C ν in this parametrization. Employing (6.8) and (5.17), one easily gets˜ C ν = (1 + β ∗ x ν ) ¯ Q ν , (6.11)where we have introduced polynomials ¯ Q ν ≡ ¯ Q ν ( x ν )¯ Q ν = N − X k =0 Q k ( − x ν ) k , (6.12)with the highest power missing when the Maxwell equations are imposed. D=4
The last parametrization is suitable for a discussion of the four-dimensional spacetimes.Namely, for N = 2, both polynomials Q and ¯ Q ν reduce to a constant and the Maxwellequations require this constant to be zero. The solution is generated by the function Z mg ′ ( β ∗ , L , L ) parameterized just by the root β ∗ and Killing-vector constants L , L .Clearly, C k = 0, as well as ˜ C ν = 0, and β ∗ is unconstrained. In the discussion of magnetic polarizations we have lost one solution, since β = 0 leadsto a pure gauge potential A = ∇ Z . In this section we attempt to recover the missingpolarization by investigating the behavior of our system of eigenfunctions (5.16) in thelimit β →
0, see appendix C. The obtained results inspire the following new ansatz for thevector potential: A a = h an ∇ n Z . (7.1)The Lorenz condition and the Maxwell equations then read ∇ n A n = ( D − ξ n ∇ n Z = 0 , (7.2) ∇ n F an = − h an ∇ n (cid:3) Z + 2 ξ a (cid:3) Z + ( D − ∇ a ( ξ n ∇ n Z ) = 0 . (7.3)Both these equations can be satisfied by requiring (cid:3) Z = 0 , (7.4) ξ n ∇ n Z = 0 . (7.5)The solutions of the wave operator have been studied before [16, 20]. In appendix B werecall that they are given by the eigenfunctions ˜ Z ( K , . . . , K N − , L , . . . , L N − ) of the oper-ators K k and L k , labeled by their eigenvalues. The first condition (7.4) sets the eigenvalueof the wave operator itself to zero, K = 0. The second condition (7.5) requires L = 0.We can thus generate solutions to the Maxwell equations via ansatz (7.1) using func-tions Z el ( K , . . . , K N − , L , . . . , L N − ) = ˜ Z (0 , K , . . . , K N − , , L , . . . , L N − ) . (7.6)We call these solutions the “electric polarization”. This family of solutions is degeneratesince it is parameterized just by 2 N − Separation of variables
Until now our discussion of the solutions of the Maxwell equations has been rather abstract,based on the eigenfunctions of the operators C k and L k . Here we demonstrate that theseeigenfunctions can be found using the method of separation of variables. This reducesthe problem to solving ordinary differential equations instead of having to deal with thecomplicated partial differential operators.We proceed as follows. First, we show that the eigenvalue problem for the operators C k and L k , and common eigenfunction Z , can be solved by employing the separation ansatz(8.2) below. Next, we discuss a refined method of separation of variables that is applicableto test fields in the higher-dimensional Kerr–NUT–(A)dS spacetimes and show that boththe Lorenz condition and the remaining Maxwell equations can be solved by this method.By comparing the obtained separated equations with the equations for the eigenfunction Z we conclude that the separation constants are precisely the eigenvalues of the operators C k and L k . Multiplicative separation ansatz
A possibility to use the method of separation of variables is based on the fact that theoperators C k and L k have a special form C k = X ν A ( k ) ν U ν ˜ C ν , L k = X ν A ( k ) ν U ν ˜ L ν , (8.1)where each ˜ C ν and ˜ L ν are operators in just one x -variable x ν (and Killing variables ψ k ). Wecan take an advantage of this special coordinate dependence and of the additive structureby imposing the multiplicative separation ansatz for a function on which the operators act.Namely, we set Z = (cid:16)Y ν R ν (cid:17) exp (cid:16) i X j L j ψ j (cid:17) , (8.2)where functions R ν are functions of just one variable, R ν = R ν ( x ν ). Note that in terms ofperiodic angular coordinates ϕ ν (A.1) and constants m ν (A.7), the exponent reads X k L k ψ k = X ν m ν ϕ ν . (8.3) Eigenvalue problem
Let us consider the eigenvalue problem (5.16), with the eigenfunction ansatz (8.2). Thesecond set of equations, L k Z = L k Z , is automatically satisfied. The first set of equationsreads C k Z = C k Z . (8.4) The same structure has been already recognized in the discussion of the scalar field in [20] with operators K k and ˜ K ν , see appendix B for a short review. Naturally, we follow this case. – 19 –riting the l.h.s. explicitly for the ansatz (8.2), we find C k Z = Z X ν A ( k ) ν U ν R ν (cid:18) (1+ β x ν ) (cid:16) X ν β x ν R ′ ν (cid:17) ′ − ˜ L ν X ν R ν + iβ − β x ν β x ν β − N ) LR ν (cid:19) , (8.5)where the prime denotes the derivative with respect to x ν . Applying the k – ν transformationto (8.4), the sum in (8.5) disappears on the r.h.s. and polynomials ˜ C ν defined in (5.17)appear on l.h.s., yields the following ordinary differential equations for functions R ν :(1+ β x ν ) (cid:16) X ν β x ν R ′ ν (cid:17) ′ − ˜ L ν X ν R ν + iβ − β x ν β x ν β − N ) LR ν − ˜ C ν R ν = 0 . (8.6)Functions R ν , each of one variable x ν , satisfying equations (8.6) thus give eigenfunc-tions Z ( β, C , . . . , L , . . . ) via multiplicative ansatz (8.2). Refined separation of variables
We now want to demonstrate that the eigenvalues, which label our eigenfunctions, canbe interpreted as separation constants. We start by describing the refined method ofseparation of variables that is applicable in our case.An elementary formulation of the method of separation of variables states that if onehas N functions f ν , each of which depends on one variable only, f ν = f ν ( x ν ), and if theyadd to zero, P ν f ν = 0, then each f ν has to be a constant and these constants have to sumto zero, f ν = q ν , X ν q ν = 0 . (8.7) q ν are called separation constants and only N − Separation lemma.
Let f ν are N functions of one variable only, f ν = f ν ( x ν ) . If theycomposite to a zero according to X ν U ν f ν = 0 , (8.8) then they have to be given by the same polynomial of degree N − : f ν = N − X k =0 Q k ( − x ν ) k ≡ ¯ Q ν . (8.9) We call the coefficients of these polynomials, Q , . . . , Q N − , the separation constants. Thereare N − of them, one less than the number of independent variables. The implication (8.9) to (8.8) follows directly from identity (D.1). The proof of theopposite implication has been sketched in [34].The lemma encodes the greatest freedom in functions f ν which compose to zero throughthe sum of type (8.8). It can be easily generalized to a non-trivial right hand side if oneknows at least one particular solution f ν for that right hand side. Namely, using again the– 20 –dentity (D.1), we find: Generalized separation lemma.
Functions f ν of one variable satisfying X ν U ν f ν = C , (8.10) with C = const, must be given by a polynomial of degree N − , f ν = N − X k =0 C k ( − x ν ) N − − k ≡ ˜ C ν , (8.11) where the constant C specifies the highest order term. Separation constants
Using this insight, we can revisit the Lorenz condition A C Z = 0, cf. (5.10). Employing theseparation ansatz (8.2), it yields1 A X ν A ν U ν R ν (cid:18) (1+ β x ν ) (cid:16) X ν β x ν R ′ ν (cid:17) ′ − ˜ L ν X ν R ν + iβ − β x ν β x ν β − N ) LR ν (cid:19) = 0 . (8.12)At first sight this equation does have the form (8.8) useful for the separation lemma since A ν is a function of all variables { x κ } except x ν and we need the exact opposite. Fortunately,the definition (2.16) of A ν shows that A ν /A = (1 + β x ν ) − is function of just x ν . TheLorenz condition thus takes the form (8.8) where X ν U ν R ν (cid:18)(cid:16) X ν β x ν R ′ ν (cid:17) ′ − ˜ L ν (1+ β x ν ) X ν R ν + iβ − β x ν (1+ β x ν ) β − N ) LR ν (cid:19) = 0 , (8.13)and the separation lemma gives(1+ β x ν ) (cid:16) X ν β x ν R ′ ν (cid:17) ′ − ˜ L ν X ν R ν + iβ − β x ν β x ν β − N ) LR ν − (1+ β x ν ) ¯ Q ν R ν = 0 . (8.14)On the other hand, the remaining Maxwell equations reduce to C Z = 0, cf. (5.11).Slightly more generally, we can consider the eigenfunction equation C Z = C Z , (8.15)setting C = 0 later. Substituting the multiplicative separation ansatz (8.2), we obtain X ν U ν (cid:18) (1 + β x ν ) (cid:16) X ν β x ν R ′ ν (cid:17) ′ − ˜ L ν X ν R ν + iβ − β x ν β x ν β − N ) LR ν (cid:19) = C . (8.16)The generalized separation lemma (8.10) above then implies that the brackets must beequal to the same polynomial ˜ C ν in the respective variable x ν ,(1 + β x ν ) (cid:16) X ν β x ν R ′ ν (cid:17) ′ − ˜ L ν X ν R ν + iβ − β x ν β x ν β − N ) LR ν − ˜ C ν R ν = 0 . (8.17)– 21 –he eigenvalue C determine the highest order of the polynomials ˜ C ν and, as we said, thesource-free Maxwell equations require C = 0. We also realize that the separated equations(8.17) are identical to the conditions (8.6) obtained from the eigenfunction equations foroperators C k . It means that the separability constants C k (coefficients of the polynomials˜ C µ from the generalized separation lemma) are exactly the eigenvalues of operators C k .Moreover, by comparing (8.14) and (8.17), we recover the relation (6.11),˜ C ν = (1 + β x ν ) ¯ Q ν , (8.18)which we derived originally from a completely different perspective. However, the basisfor this relation remains the same. It reflects the requirement that the Lorenz condition C Z = 0 holds for given β . Let us return to the electric polarization discussed in section 7. One can apply the multi-plicative separation ansatz as in the previous section and recover the separable structureof the eigenfunctions ˜ Z , see appendix B.However, we will look at this case from a different perspective, restricting to the specialcase L k = 0 , (9.1)i.e., to the field independent of ψ k .In section 7 we mentioned that the ansatz (7.1) for the vector potential and the fieldequations (7.4), (7.5) can be motivated by the limiting procedure β → R ν in the multiplicative separationansatz (8.2) also expand as R ν = 1 + βS ν + O ( β ) , (9.2)with functions S ν depending just on one variable, S ν = S ν ( x ν ). The multiplicative separa-tion ansatz thus reduces to Z = 1 + β X ν S ν + O ( β ) . (9.3)It motivates us to search for the electric polarization (7.1), A a = − h an ∇ n S , (9.4)in the form of an additive separation ansatz S = X ν S ν . (9.5)The vector potential yields A a = − h an X ν S ′ ν ∇ n x ν = X ν x ν X ν S ′ ν U ν X k A ( k ) ν d a ψ k . (9.6)– 22 –he Lorenz condition (7.5) is satisfied automatically since L = 0. The Maxwell con-dition (7.4) is the massless scalar wave equation K S = 0, and upon inserting the additiveseparation ansatz, we get X ν U ν (cid:0) X ν S ′ ν (cid:1) ′ = 0 . (9.7)The separability lemma gives us that S ν must satisfy the following differential equation: (cid:0) X ν S ′ ν (cid:1) ′ = ¯ Q ν , (9.8)where ¯ Q ν are polynomials (8.9) of degree N − x ν . Eq. (9.8) can be integrated once,leading to x ν X ν S ′ ν = q ν x ν + ˜ P ν , (9.9)where q ν is an integration constant and ˜ P ν is a polynomial of degree N − x ν withoutan absolute term, say ˜ P ν = N − X l =0 P l ( − x ν ) N − − l . (9.10)Substituting to the vector potential (9.6), and using (D.1), we obtain A = X ν q ν x ν U ν N − X k =0 A ( k ) ν d ψ k + N − X k =0 P k d ψ k . (9.11)The second term is a pure gauge can be ignored. The first term reproduces exactly theelectromagnetic fields aligned with the principal tensor found in [34] and discussed in [19].In other words, we have just demonstrated that the aligned fields can be understood as aspecial case of the electric solutions for which the dependence on all Killing coordinatesvanishes.
10 Summary
In this paper we have demonstrated the separability of the Maxwell equations in the back-ground of the most general higher-dimensional spacetime admitting the principal tensor—the off-shell Kerr–NUT–(A)dS geometry. This goal was achieved by adopting a specialansatz (3.1) for the vector potential of the electromagnetic field. We demonstrated thatthis ansatz solves the Maxwell equations if the corresponding potential function Z hasthe form (8.2) provided mode functions R ν are solutions of the second-order ODEs (8.6).These equations contain the metric functions X ν . For a general off-shell metric, theseare arbitrary functions of one variable x ν . For the on-shell metric these functions becomepolynomials, so that the coefficients that enter the equations are rational functions of thecorresponding variables.In order to adapt the obtained separated equations to the physical Kerr–NUT–(A)dSblack hole spacetimes (in Lorentzian signature), one needs to apply additionally the Wickrotation to the radial coordinate and the mass parameter [17]. Consequently, one of theseparated equations will be in the radial sector, while the other equations become the– 23 –atitudinal angle equations. The requirement that the solutions of these angular equationsare regular fixes the spectrum of some of the separation constants. We did not discuss theseimportant details in the present paper, but we would like to emphasize that the proof ofthe completeness of the set of the solutions, obtained by in this paper described method isan important open problem.We also demonstrated that for the constructed potential function the electromagneticfield potential satisfies the Lorenz gauge condition.Let us emphasize that the approach used in this paper is in its spirit similar to the oneproposed by Lunin [31]. However, there are number of differences. First, we consideredthe off-shell Kerr–NUT–(A)dS metrics, and in this sense, we obtained a non-trivial andfar-reaching generalization of Lunin’s results. Second, contrary to Lunin’s paper our con-struction is totally covariant and entirely based on the principal tensor. Third, the proofof the separability of the Maxwell equations proposed in our paper is carried out in ananalytic form.The key role in this proof is played by the rich geometrical structure generated from theprincipal tensor. Using this tensor we defined a covariant form of the polarization tensorwhich modifies the gradient of a generating scalar function in the ansatz for the vectorpotential. The rich symmetry structure has a consequence that both the Lorenz conditionand the Maxwell equations can be written as a composition of operators separated inlatitudinal variables. This allowed us to construct the system of commuting operators,which define a set of common eigenfunctions. In terms of these eigenfunctions we havebeen able to identify the separable solutions. These are labeled, in general, by the correctnumber of D − Z can be identified with the eigenfunctionsof a complete set of the first-order and second-order differential operators. For the Klein–Gordon field, the covariant form of these operators is well known—they are constructedfrom the Killing vectors and Killing tensors present in the Killing tower. A covariant formfor the operators acting on Z is currently unknown and finding it poses an interestingproblem for future studies. Acknowledgments
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 hospital-ity. P.K. was supported by Czech Science Foundation Grant 17-01625S. D.K. acknowledgesthe Perimeter Institute for Theoretical Physics and the NSERC for their support. Researchat Perimeter Institute is supported by the Government of Canada through the Departmentof Innovation, Science and Economic Development Canada and by the Province of Ontariothrough the Ontario Ministry of Research, Innovation and Science.– 24 –
Angular coordinates
Periodic angular coordinates
In this appendix, let us return to the metric (2.5) and discuss the meaning of the Killingcoordinates ψ k . Such coordinates correspond to explicit symmetries and represent timeand longitudinal angles. However, this relation is not direct. Even for vanishing NUT pa-rameters these coordinates cannot be directly identified with the standard periodic angularcoordinates around axes of rotation but instead are their linear combination [17, 38, 46].With non-vanishing NUTs, the situation is even more complicated because it is not clear,what are the “correct” periodic angular coordinates [17, 47–49].The reason is that the metric (2.5) itself does not specify a global geometry. It has to beaccompanied by a specification of what the ranges of coordinates are and which coordinatesare periodic. In some cases (as vanishing NUTs, i.e., the Myers–Perry geometry) there isa natural choice of such angular coordinates which guarantees the regularity of axes ofrotation. With non-vanishing NUTs or even for the off-shell geometries described by (2.5),the axes cannot be, in general, made regular. Physically it means that there are somelinear sources along the axes and such sources cannot be eliminated. The specification ofthese sources is hidden exactly in an identification of the periodic longitudinal coordinates.In any case, the coordinates ψ k are not typically the periodic coordinates. ψ is a timecoordinate, but the periodic angular coordinates ϕ ν are given by a liner combination of ψ ’s. In even 2 N dimensions, which we mainly considered in the main text, it is useful towrite such transformation in the form ϕ ν = N − X k =0 ˚ A ( k ) ν ˚ U ν ψ k , ψ k = X ν ( − ˚ x ν ) N − − k ϕ ν , (A.1)where ˚ x ν are constants and all other quantitative as ˚ A ( k ) ν , ˚ U ν are build from ˚ x ν in the sameway as A ( k ) ν , U ν from x ν . A specification of the constants ˚ x ν thus defines the correct periodiccoordinates ϕ ν . It has to be accompanied by setting correct ranges of this periodicity. Allthese choices identify what singular sources are present on the axes.For vanishing NUT parameters, the relation to the Myers–Perry metric includes setting˚ x ν to values of rotational parameters a ν , see [17, 47, 48].We will not discuss these issues in more detail. The only thing we need is that theperiodic angular coordinates ϕ k are related to ψ k by transformation (A.1). Thus, whenstudying the spectra of operators related to Killing vectors, we can expect that operators ∂∂ϕ µ have a simple discrete spectrum and spectra of ∂∂ψ k must be derived from them using(A.1). For that it is useful to write down relations of the coordinate Killing vectors: ∂ ψ k = X ν ˚ A ( k ) ν ˚ U ν ∂ ϕ ν , ∂ ϕ ν = N − X k =0 ( − ˚ x ν ) N − − k ∂ ψ k , (A.2) Even in the regular case of the Myers–Perry black hole one can superimpose rotating strings along theaxes which is effectively done exactly be changing ranges of angular coordinates and rewinding angular andtime coordinates among themselves. This causes irregularities on the axes. The regularity of Myers–Perrysolution means that such irregularities can be eliminated by a proper choice of time and angular coordinates. – 25 –here we used the important identity (D.1).
Operators J ν and L k Operators L k , ˜ L ν , and L can be also expressed in terms of period coordinates ϕ ν introducedin (A.1). If we define J ν = − i ∂∂ϕ ν = X k ( − ˚ x ν ) N − − k L k , (A.3) L ’s operators are L k = X ν ˚ A ( k ) ν ˚ U ν J ν , (A.4)˜ L ν = X ν U ν Y κκ = ν (˚ x κ − x µ ) J ν , (A.5) L = X ν ˚ A ν ˚ U ν J ν . (A.6)Operators J ν commute with all operators C k and L k , since they are related to L k just by a linear transformation (A.3) with constant coefficients. We can thus introduceeigenvalues of operators J ν , J ν Z = m ν Z , (A.7)which we expect to have a simple discrete spectrum. The eigenvalues L k and polynomials˜ L ν and L are then related as L k = X ν ˚ A ( k ) ν ˚ U ν m ν , (A.8)˜ L ν = X ν U ν Y κκ = ν (˚ x κ − x µ ) m ν , (A.9) L = X ν ˚ A ν ˚ U ν m ν . (A.10)In the main text, we continue to use eigenvalues L k as basic ones. Transformation (A.8)can be always carried out at the end. B Separability of the scalar wave equation
On several places in the main text we refer to the separability of the scalar wave equation inthe off-shell Kerr–NUT–(A)dS spacetime. For convenience of the reader, in this appendixwe give a short overview of this result. The separability has been first demonstrated in[16] and later elaborated on in [20] where one can find also the results for odd dimensions.Here we restrict to even dimensions. – 26 –sing the Killing tensors k ( j ) one can construct the tower of symmetric second orderoperators K j = ∇ a (cid:2) k abj ∇ b (cid:3) . (B.1)In the canonical coordinates these operators read [20] K j Z = X ν A ( j ) ν U ν (cid:20) ∂∂x ν h X ν ∂∂x ν i + 1 X ν hX i ( − x ν ) N − − i ∂∂ψ i i (cid:21) Z . (B.2)Similarly, the generating Killing tensor k defined in (2.19), defines the operator K Z ≡ ∇ m (cid:0) k mn ∇ n Z (cid:1) = X ν A ν U ν (cid:20) ∂∂x ν h X ν ∂∂x ν i + 1 X ν hX j ( − x ν ) N − − j ∂∂ψ j i (cid:21) Z . (B.3)By a similar argument as for the operators C j in section 5, one can show that theseoperators, together with the operators L j , mutually commute,[ K k , K l ] = 0 , [ K k , L l ] = 0 , [ L k , L l ] = 0 . (B.4)Therefore, they have common eigenfunctions ¯ Z labeled by eigenvalues K j and L j , K j ¯ Z = K k ¯ Z , L j ¯ Z = L k ¯ Z . (B.5)Let us concentrate on the eigenfunction equation of the zeroth operator K ≡ (cid:3) , K Z = K Z . (B.6)Substituting the multiplicative separation ansatz (8.2), it boils to X ν U ν (cid:18) R ν (cid:16) X ν R ′ ν (cid:17) ′ − ˜ L ν X ν (cid:19) = K . (B.7)The generalized separation lemma (8.10) then implies that the brackets must be equal tothe same polynomial ˜ K ν in the respective variable x ν with K governing the highest orderterm, i.e., (cid:16) X ν R ′ ν (cid:17) ′ − ˜ L ν X ν R ν − ˜ K ν R ν = 0 , (B.8)where ˜ K ν ≡ N − X k =0 K k ( − x ν ) N − − k . (B.9)Plugging this most general solution of (B.6) to the operators K j we find that such Z is the eigenfunction ¯ Z with eigenvalues K j . Using the system of eigenfunctions of theoperators K j is thus equivalent to solving equation (B.6) by the separation of variables andthe eigenvalues correspond to the separation constants.Note also that by plugging ¯ Z into operator (B.3), we get K ¯ Z = K ¯ Z , (B.10)with K = X j K j β j = X µ A ν U ν ˜ K ν . (B.11)– 27 – Limiting procedure β → In the discussion of magnetic polarizations we have lost one of the solutions, since β = 0leads to a pure gauge potential A = ∇ Z . In the hope to recover the missing polarization,let us investigate the behavior of our system of eigenfunctions (5.16) for β →
0. As we shallsee, this will naturally lead to the definition of the electric polarization (7.1) in the maintext. We shall also use this to recover the special solutions of Maxwell equations known asthe aligned fields, see Sec. 9.
Behavior of eigenfunctions for β → C ν given by (5.7), we see a potential problem with the last termwhich is proportional to β − N ) . To investigate its behavior, we expand also the fractionalfactor in β ,1 − β x ν β x ν β − N ) = − β x ν β − N ) = − ∞ X j =0 β j +1 − N ) ( − x ν ) j . (C.1)We see that the sum contains plenty of terms with negative powers of β . However, operators C k are given as a sum (5.8) of operators ˜ C ν . Keeping just terms with non-positive powersof β and changing j → N − − j , the contributions to C k are X ν A ( k ) ν U ν − β x ν β x ν β − N ) = − X ν A ( k ) ν U ν + 2 N − X j =0 β − j X ν A ( k ) ν U ν ( − x ν ) N − − j + O ( β ) . (C.2)Sums over ν are easily evaluated using the identity (D.1), giving X ν A ( k ) ν U ν − β x ν β x ν β − N ) = − δ N − k β − N − + 2 β − k + O ( β ) . (C.3)Substituting it to (5.7), we get C k = K k + 2 iβ k X l =0 β − k − l ) L l + O ( β ) (C.4)for k = 0 , . . . , N − C N − = K N − + iβ N − X l =0 β − N − − l ) L l + O ( β ) (C.5)for k = N −
1. Here we used operators K k given in (B.2).We see that the operators C k are mostly divergent for β →
0. Therefore, one cannotexpect the eigenfunctions Z to behave reasonably in this limit. However, one could avoidthis problem for a subclass of eigenfunctions, namely for those with L k = 0, i.e., for thoseindependent of ψ k . Operators C k acting on such eigenfunctions reduce just to K k (with thelast term vanishing).We can thus hope that by expanding the eigenfunctions Z with vanishing L k we couldfind a subfamily of solutions of the Maxwell equations corresponding to the missing polar-ization. – 28 – ehavior of Z for β → Z . We just want to solve (6.1): C Z = 0, and C Z = 0. Fortunately, C is regular for small β , C = K + 2 iβ L + O ( β ) , (C.6)as well as operator C , C = X k β k C k = K + (2 N − iβ L + O ( β ) . (C.7)Unfortunately, they differ in the first order of β by a term proportional to L . Therefore,if we want satisfy both conditions (6.1) up to the first order, the function Z must satisfy K Z = 0 , (C.8) L Z = 0 . (C.9)The first condition is actually the scalar wave equation for Z . The second condition ismilder than setting L k = 0 for all k but it is still a non-trivial condition. The coordinate ψ represents time, so we are obtaining the condition of stationarity. Behavior of field equations for β → Z = Z + βZ + O ( β ) , (C.10)and B = g + β h + O ( β ) following from (3.2), the ansatz (3.1) for the vector potentialreads A = ∇ Z + β (cid:0) h · ∇ Z + ∇ Z (cid:1) + O ( β ) . (C.11)The leading term is a pure gauge, but the first order term is not. The nontrivial contributioncomes from the zeroth-order function Z .The Lorenz condition reads ∇ · A = (cid:3) Z + β (cid:16) ( D − ξ · ∇ Z + (cid:3) Z (cid:17) + O ( β ) , (C.12)where we have used (2.2) and the antisymmetry of the principal tensor h . The Maxwelltensor F is sensitive only to the gauge non-trivial part of the vector potential and thereforeit is of the first-order, F = ∇ ∧ A = β ∇ ∧ (cid:0) h · ∇ Z (cid:1) + O ( β ) . (C.13)Calculating the Maxwell equations is more involved, but it essentially follows the samesteps as in the case of arbitrary β discussed in section 4. Alternatively, one can just expand(4.11). It yields ∇ · F = β (cid:16) h · ∇ (cid:3) Z − ξ (cid:3) Z − ( D − ∇ ( ξ · ∇ Z ) (cid:17) + O ( β ) . (C.14)– 29 –e see that it can be set equal to zero up to the first-order in β provided that (cid:3) Z ≡ K Z = 0 , (C.15) ξ · ∇ Z ≡ i L Z = 0 . (C.16)We recovered that the function Z must satisfy the scalar wave equation and the stationaritycondition, the results (C.8) and (C.9) above.Let us note that the same result can be obtained if one assumed function Z in theform Z = 1 + βZ + β Z + O ( β ) . (C.17)The expansions of the Lorenz condition and of the Maxwell equations look exactly thesame as in (C.12) and (C.14), respectively, just with higher power of β . D Technicalities & Proofs
In this appendix we gather some important technical results and present proofs that arereferred to in the main text.First we list some important identities for the symmetric polynomials: X µ A ( k ) µ ( − x µ ) N − − l U µ = δ kl , (D.1) X µ A ( k ) µ x µ U µ = A ( k ) A ( N ) , (D.2) A ( N − = X µ A ( N − µ . (D.3) Proof of (4.3)Employing expression (3.8) for the derivative of B , symmetric part (3.6) of B , relation(3.5), and definition (2.18), we can write ∇ m (cid:0) B mn ∇ n Z (cid:1) = B mn ∇ ( m ∇ n ) Z + ( ∇ m B mn ) ∇ n Z = 1 A k mn ∇ m ∇ n Z + βA (cid:0) k aa ξ m B mn − ξ m k mn (cid:1) ∇ n Z = 1 A ∇ m (cid:0) k mn ∇ n Z (cid:1) − A ( ∇ m k mn ) ∇ n Z + βA (cid:16) k aa A l n + β k aa A l m h mn − l n (cid:17) ∇ n Z . (D.4)Substituting (2.31), and using (2.32) we obtain ∇ m (cid:0) B mn ∇ n Z (cid:1) = 1 A ∇ m (cid:0) k mn ∇ n Z (cid:1) − A ( ∇ m A ) (cid:0) k mn − kg mn (cid:1) ∇ n Z + 1 A (cid:16) − k aa A ( ∇ n A ) + β (cid:0) k aa A − (cid:1) l n (cid:17) ∇ n Z = 1 A ∇ m (cid:0) k mn ∇ n Z (cid:1) + 1 A (cid:16) − A ( ∇ m A ) k mn + β (cid:0) k aa A − (cid:1) l n (cid:17) ∇ n Z , = ∇ m (cid:0) A k mn ∇ n Z (cid:1) + βA (cid:0) k aa A − (cid:1) l n ∇ n Z , (D.5)which is what we wanted to show. – 30 – roof of (4.5)Using identities (D.1), (D.2), and (D.3) for k, l = N −
1, one has X µ A ( N − µ U µ (cid:16) x µ − (cid:17) = 2 A ( N − A ( N ) − P µ A ( N − µ A ( N ) − . (D.6)Under substitution x ν → β x ν functions U µ , A ( N − µ , and A ( N ) behave as U µ → β N − U µ , A ( N − µ → A µ , and A ( N ) → A . It gives us the relation β − N X µ A µ U µ (cid:16)
21 + β x µ − (cid:17) = 1 A X µ A µ − . (D.7)On the right-hand side we identify expression (2.27) for the trace k aa of the generatingKilling tensor and we obtain k aa A − β − N X µ A µ U µ − β x µ β x µ . (D.8) Proof of (D.9)We have β N − A = X ν U ν
11 + β x ν . (D.9)Indeed, it is just (D.2) with k = 0, in which we substitute x ν → β x ν . Subtractingzero 0 = P ν U ν (cf. (D.1) with k = 0 and l = N − A = β − N ) X ν U ν − β x ν β x ν . (D.10) Proof of (D.11)We want to prove − ∇ m ∇ m ( B an ∇ n Z ) + R am B mn ∇ n Z = − B an ∇ n (cid:3) Z + 2 βB ak ξ k ∇ m ( B mn ∇ n Z ) − βB am ∇ m (cid:0) ξ k B kn ∇ n Z (cid:1) . (D.11)We start by pulling B an from the covariant derivatives, −∇ m ∇ m ( B an ∇ n Z ) + R am B mn ∇ n Z = − B an ∇ m ∇ n ∇ m Z − ∇ m B an ) ∇ m ∇ n Z − ( ∇ m ∇ m B an ) ∇ n Z + R am B mn ∇ n Z = − B an ∇ n (cid:3) Z − β (cid:0) B am ξ k B kn − B ak ξ k B mn (cid:1) ∇ m ∇ n Z − β (cid:0) ∇ m ( B am ξ k B kn − B ak ξ k B mn ) (cid:1) ∇ n Z . (D.12)– 31 –n the last step we used the Ricci identities to interchange covariant derivatives, producinga curvature term which canceled the term with Ricci tensor. Here we also used that B ab commutes with R ab as matrices, since h ab commutes with R ab . Next we used twice theexpression (3.8). Pushing ξ k B kn and B mn in the second term in the last expression backunder the derivative gives −∇ m ∇ m ( B an ∇ n Z ) + R am B mn ∇ n Z = − B an ∇ n (cid:3) Z + 2 βB ak ξ k ∇ m ( B mn ∇ n Z ) − βB am ∇ m (cid:0) ξ k B kn ∇ n Z (cid:1) + β (cid:16) B am ∇ m ( ξ k B kn ) − ξ k B kn ∇ m B am + B mn ∇ m ( B ak ξ k ) − B ak ξ k ∇ m B mn (cid:17) ∇ n Z = − B an ∇ n (cid:3) Z + 2 βB ak ξ k ∇ m ( B mn ∇ n Z ) − βB am ∇ m (cid:0) ξ k B kn ∇ n Z (cid:1) + β (cid:16) B am ( ∇ m ξ k ) B kn + B am ξ k ( ∇ m B kn ) βA (cid:0) k am ξ m − k cc B am ξ m (cid:1) ξ k B kn + B am ( ∇ k ξ m ) B kn + ( ∇ m B ak ) ξ k B mn − βA B ak ξ k (cid:0) k cc ξ m B mn − ξ m k mn (cid:1)(cid:17) ∇ n Z , (D.13)where we used relations (3.9). Because ξ is a Killing vector, terms with ∇ ξ cancel eachother, as well as terms proportional to k . Using once more (3.8), we obtain −∇ m ∇ m ( B an ∇ n Z ) + R am B mn ∇ n Z = − B an ∇ n (cid:3) Z + 2 βB ak ξ k ∇ m ( B mn ∇ n Z ) − βB am ∇ m (cid:0) ξ k B kn ∇ n Z (cid:1) + β (cid:16) − A (cid:0) k am ξ m ξ k B kn − B ak ξ k ξ m k mn (cid:1) + B am ξ k (cid:0) B km ξ l B ln − B kl ξ l B mn (cid:1) + (cid:0) B am ξ l B lk − B al ξ l B mk (cid:1) ξ k B mn (cid:17) ∇ n Z = − B an ∇ n (cid:3) Z + 2 βB ak ξ k ∇ m ( B mn ∇ n Z ) − βB am ∇ m (cid:0) ξ k B kn ∇ n Z (cid:1) + β A (cid:16) − k al ξ l ξ k B kn + B ak ξ k ξ l k ln + k ak ξ k ξ l B ln − B al ξ l ξ k k kn (cid:17) ∇ n Z (D.14)= − B an ∇ n (cid:3) Z + 2 βB ak ξ k ∇ m ( B mn ∇ n Z ) − βB am ∇ m (cid:0) ξ k B kn ∇ n Z (cid:1) , where we canceled the terms proportional to ξ k ξ l B kl and used relation (3.4). References [1] B. Carter,
Hamilton-Jacobi and Schrodinger separable solutions of Einstein’s equations , Commun. Math. Phys. (1968) 280–310.[2] S. A. Teukolsky, Rotating black holes - separable wave equations for gravitational andelectromagnetic perturbations , Phys. Rev. Lett. (1972) 1114–1118.[3] S. A. Teukolsky, Perturbations of a rotating black hole. I. Fundamental equations forgravitational electromagnetic and neutrino field perturbations , Astrophys. J. (1973) 635–647.[4] W. G. Unruh,
Separability of the neutrino equations in a Kerr background , Phys. Rev. Lett. (1973) 1265.[5] S. Chandrasekhar, The solution of Dirac’s equation in Kerr geometry , Proc. R. Soc. Lond.,Ser A (1976) 571–575. – 32 –
6] D. N. Page,
Dirac equation around a charged, rotating black hole , Phys. Rev. D (1976) 1509–1510.[7] V. P. Frolov and D. Stojkovi´c, Quantum radiation from a 5-dimensional rotating black hole , Phys. Rev. D (2003) 084004, [ gr-qc/0211055 ].[8] H. K. Kunduri and J. Lucietti, Integrability and the Kerr-(A)dS black hole in fivedimensions , Phys. Rev. D (2005) 104021, [ hep-th/0502124 ].[9] M. Vasudevan and K. A. Stevens, Integrability of particle motion and scalar field propagationin Kerr-(anti) de Sitter black hole spacetimes in all dimensions , Phys. Rev. D (2005) 124008, [ gr-qc/0507096 ].[10] M. Vasudevan, K. A. Stevens and D. N. Page, Particle motion and scalar field propagation inMyers-Perry black hole spacetimes in all dimensions , Class. Quantum Grav. (2005) 1469–1482, [ gr-qc/0407030 ].[11] M. Vasudevan, K. A. Stevens and D. N. Page, Separability of the Hamilton-Jacobi andKlein-Gordon equations in Kerr-de Sitter metrics , Class. Quantum Grav. (2005) 339–352,[ gr-qc/0405125 ].[12] P. Davis, A killing tensor for higher dimensional Kerr-AdS black holes with NUT charge , Class. Quantum Grav. (2006) 3607–3618, [ hep-th/0602118 ].[13] W. Chen, H. Lu and C. N. Pope, Separability in cohomogeneity-2 Kerr-NUT-AdS metrics , JHEP (2006) 008, [ hep-th/0602084 ].[14] D. Kubizˇn´ak and V. P. Frolov,
Hidden symmetry of higher dimensional Kerr-NUT-AdSspacetimes , Class. Quantum Grav. (2007) F1–F6, [ gr-qc/0610144 ].[15] W. Chen, H. Lu and C. N. Pope, General Kerr-NUT-AdS metrics in all dimensions , Class. Quantum Grav. (2006) 5323–5340, [ hep-th/0604125 ].[16] V. P. Frolov, P. Krtouˇs and D. Kubizˇn´ak, Separability of Hamilton-Jacobi and Klein-Gordonequations in general Kerr-NUT-AdS spacetimes , JHEP (2007) 005, [ hep-th/0611245 ].[17] V. P. Frolov, P. Krtouˇs and D. Kubizˇn´ak,
Black holes, hidden symmetries, and completeintegrability , Living Rev. Rel. (2017) 6, [ ].[18] B. Carter, Separability of the Killing–Maxwell system underlying the generalized angularmomentum constant in the Kerr–Newman black hole metrics , J. Math. Phys. (1987)1535–1538.[19] I. Kol´aˇr and P. Krtouˇs, Weak electromagnetic field admitting integrability inKerr-NUT-(A)ds spacetimes , Phys. Rev. D (2015) 124045, [ ].[20] A. Sergyeyev and P. Krtouˇs, Complete set of commuting symmetry operators forKlein-Gordon equation in generalized higher-dimensional Kerr-NUT-(A)dS spacetimes , Phys. Rev. D (2008) 044033, [ ].[21] V. P. Frolov and P. Krtouˇs, Charged particle in higher dimensional weakly charged rotatingblack hole spacetime , Phys. Rev. D (2011) 024016, [ ].[22] T. Houri, T. Oota and Y. Yasui, Closed conformal Killing-Yano tensor and Kerr-NUT-deSitter spacetime uniqueness , Phys. Lett. B (2007) 214–216, [ ].[23] P. Krtouˇs, V. P. Frolov and D. Kubizˇn´ak,
Hidden symmetries of higher dimensional blackholes and uniqueness of the Kerr-NUT-(A)dS spacetime , Phys. Rev. D (2008) 064022,[ ]. – 33 –
24] T. Oota and Y. Yasui,
Separability of Dirac equation in higher dimensional Kerr-NUT-deSitter spacetime , Phys. Lett. B (2008) 688–693, [ ].[25] M. Cariglia, P. Krtouˇs and D. Kubizˇn´ak,
Dirac equation in Kerr-NUT-(A)dS spacetimes:Intrinsic characterization of separability in all dimensions , Phys. Rev. D (2011) 024008,[ ].[26] M. Cariglia, P. Krtouˇs and D. Kubizˇn´ak, Commuting symmetry operators of the Diracequation, Killing-Yano and Schouten-Nijenhuis brackets , Phys. Rev. D (2011) 024004,[ ].[27] M. Cariglia, V. P. Frolov, P. Krtouˇs and D. Kubizˇn´ak, Electron in higher-dimensional weaklycharged rotating black hole spacetimes , Phys. Rev. D (2013) 064003, [ ].[28] D. Kubizˇn´ak, C. M. Warnick and P. Krtouˇs, Hidden symmetry in the presence of fluxes , Nucl. Phys. B (2011) 185–198, [ ].[29] H. K. Kunduri, J. Lucietti and H. S. Reall,
Gravitational perturbations of higher dimensionalrotating black holes: Tensor perturbations , Phys. Rev. D (2006) 084021,[ hep-th/0606076 ].[30] T. Oota and Y. Yasui, Separability of gravitational perturbation in generalized Kerr-NUT-deSitter spacetime , Int. J. Mod. Phys.
A25 (2010) 3055–3094, [ ].[31] O. Lunin,
Maxwell’s equations in the Myers-Perry geometry , JHEP (2017) 138,[ ].[32] V. P. Frolov and D. Kubizˇn´ak,
Higher-dimensional black holes: Hidden symmetries andseparation of variables , Class. Quantum Grav. (2008) 154005, [ ].[33] Y. Chervonyi and O. Lunin, Killing(-Yano) tensors in string theory , JHEP (2015) 182,[ ].[34] P. Krtouˇs,
Electromagnetic field in higher-dimensional black-hole spacetimes , Phys. Rev. D (2007) 084035, [ ].[35] W. Chen and H. Lu, Kerr-Schild structure and harmonic 2-forms on (A)dS-Kerr-NUTmetrics , Phys. Lett. B (2008) 158–163, [ ].[36] R. C. Myers and M. J. Perry,
Black holes in higher dimensional space-times , Ann. Phys. (N.Y.) (1986) 304–347.[37] G. W. Gibbons, H. Lu, D. N. Page and C. N. Pope,
The general Kerr-de Sitter metrics in alldimensions , J. Geom. Phys. (2005) 49–73, [ hep-th/0404008 ].[38] G. W. Gibbons, H. Lu, D. N. Page and C. N. Pope, Rotating black holes in higher dimensionswith a cosmological constant , Phys. Rev. Lett. (2004) 171102, [ hep-th/0409155 ].[39] P. Krtouˇs, D. Kubizˇn´ak, D. N. Page and V. P. Frolov, Killing-Yano tensors, rank-2 Killingtensors, and conserved quantities in higher dimensions , JHEP (2007) 004,[ hep-th/0612029 ].[40] V. P. Frolov, P. Krtouˇs and D. Kubizˇn´ak,
Separation variables in Maxwell equations inPleba´nski–Demia´nski metric , Phys. Rev. D (2018) 101701(R), [ ].[41] A. Proca, Sur la th´eorie ondulatoire des ´electrons positifs et n´egatifs , Journal de Physique etle Radium (1936) 347–353.[42] F. J. Belinfante, The interaction representation of the Proca field , Phys. Rev. (1949) 66–80. – 34 –
43] N. Rosen,
A classical Proca particle , Found. Phys. (12, 1994) 1689–1695.[44] M. Seitz, Proca field in a space-time with curvature and torsion , Class. Quantum Grav. (1986) 1265–1273.[45] V. P. Frolov, P. Krtouˇs, D. Kubizˇn´ak and J. E. Santos, Massive vector fields in rotatingblack-hole spacetimes: Separability and quasinormal modes , Phys. Rev. Lett. (2018) 231103, [ ].[46] Z. W. Chong, G. W. Gibbons, H. Lu and C. N. Pope,
Separability and Killing tensors inKerr-Taub-NUT-de Sitter metrics in higher dimensions , Phys. Lett. B (2005) 124–132,[ hep-th/0405061 ].[47] I. Kol´aˇr and P. Krtouˇs,
NUT-like and near-horizon limits of Kerr-NUT-(A)dS spacetimes , Phys. Rev. D
D95 (2017) 124044, [ ].[48] P. Krtouˇs, D. Kubizˇn´ak, V. P. Frolov and I. Kol´aˇr,
Deformed and twisted black holes withNUTs , Class. Quantum Grav. (2016) 115016, [ ].[49] I. Kol´aˇr and P. Krtouˇs, “Fixed points of isometries in Kerr-NUT-(A)dS spacetimes.” 2018.].[49] I. Kol´aˇr and P. Krtouˇs, “Fixed points of isometries in Kerr-NUT-(A)dS spacetimes.” 2018.