Integrability of the Dirac equation on backgrounds that

PHYSICAL REVIEW D 95, 084022 (2017)

Integrability of the Dirac equation on backgrounds that are the direct product of bidimensional spaces Joás Venâncio* and Carlos Batista† Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50740-560, Brazil (Received 2 February 2017; revised manuscript received 15 March 2017; published 12 April 2017) The field equation for a spin 1=2 massive charged particle propagating in spacetimes that are the direct product of two-dimensional spaces is separated. Moreover, we use this result to attain the separability of the Dirac equation in some specific static black hole solutions whose horizons have topology R × S2 × × S2 . DOI: 10.1103/PhysRevD.95.084022

I. INTRODUCTION Besides the detection of gravitational radiation and observation of the direct interaction between objects via gravitation, the most natural and simple way to probe the gravitational field permeating our spacetime is by letting other fields interact with it. This is the main reason why the study of scalar fields, spin 1=2 fields, and gauge fields (Abelian and non-Abelian) propagating in curved spacetimes plays a central role on the study of general relativity and any other theory of gravity. Moreover, through the investigation of these interactions one can test the stability of certain gravitational configurations, such as black holes for example. Nevertheless, in a general spacetime it is quite difficult to integrate and even separate the equation of motion for these fields. Luckily, some of the most important spacetimes are endowed with geometrical structures that allow the integration of these field equations. For instance, this is the case for the Schwarzschild spacetime, which possesses four Killing vector fields, and the Kerr spacetime, which has two Killing vectors and one Killing-Yano tensor [1,2]. Indeed, in Refs. [3,4] the separation and asymptotic integration of some of the mentioned field equations have been attained in the Schwarzschild background, while in Refs. [5–8] the problem is tackled in the Kerr spacetime. Many of these results are summarized by the celebrated Teukolsky master equation, which is the radial part of the equation of motion for a field of generic spin propagating in the four-dimensional Kerr background [9–11]. In the past forty years, a great amount of research has been headed toward the investigation of higherdimensional spacetimes, especially due to their interest for string theory [12], which requires the spacetime to have 10 dimensions, and because of applications in field theory by means of the AdS=CFT correspondence [13]. Moreover, there are many other theories that tries to explain our current understanding of the Universe by means of higherdimensional theories [14]. Aiming applications on some of * †

[email protected] [email protected]

2470-0010=2017=95(8)=084022(14)

these fields, in the present article we shall work out the separation of the Dirac equation for a massive spin 1=2 charged particle propagating in the black hole background described in Ref. [15], which is a static black hole whose horizon topology is R × S2 × × S2 . One interesting feature of this black hole is that, in addition to the electric charge, it has a magnetic charge, differently from the higherdimensional generalization of the Reissner-Nordstrom solution [16], which only has electric charge. Thus, in spite of the static character of the black hole considered here, the physics involved can be quite rich. Spaces that are the direct product of two-dimensional spaces can also be of relevance to model internal spaces in string theory compactifications [17]. One of the applications of the calculations performed here is on the study of the quasinormal modes of the Dirac field. The notion of quasinormal modes and their spectrum are of great physical relevance, inasmuch as these are the modes that survive for a longer time when a background is perturbed and, therefore, these are the configurations that are generally measured by experiments [18–20]. Therefore, this theme acquired even greater importance after the recent measurement of gravitational radiation [21]. In addition, the quasinormal modes are of great relevance for studying the stability of certain solutions [22]. Another interesting application of the results presented in the sequel is on the investigation of superradiance phenomena for the spin 1=2 field. Although bosonic fields like scalar, electromagnetic, and gravitational fields can exhibit superradiant behavior in four-dimensional Kerr spacetime [23], curiously, this is not the case for the Dirac field [24]. Thus, it would be interesting to investigate whether an analogous thing happens in the background considered here. The outline of the paper is the following. Section II sets the notation and the conventions used throughout the article. Then, in Sec. III we work out how the Dirac operator behaves under a conformal transformation of the metric, a result that will be of great value for the separation of the Dirac equation in the black hole background. In Sec. IV we show that a generalized Dirac equation can be separated in a space that is

084022-1

© 2017 American Physical Society

JOÁS VENÂNCIO and CARLOS BATISTA


the direct product of two-dimensional spaces. Section Vuses the results obtained in the previous sections to attain the separation of the Dirac equation in a black hole background. Then, in Sec. VI we sum up what have been done and digress about future applications of this research. As an addendum, in Appendix A the results of Sec. IV are applied to a higher-dimensional version of the charged Nariai spacetime, whereas in Appendix B we generalize the separation of the Dirac equation to odd-dimensional spaces that are the direct product of bidimensional spaces with a real line or a circle.

In what follows, we shall deal with an even-dimensional manifold M endowed with a metric g and a torsion-free connection ∇ that is compatible with the metric. Regarding the signature, in general, we shall not restrict ourselves to a specific choice of signature, although our main motivation are the Lorentzian spacetimes. The dimension will be denoted by d ¼ 2n. Our indices conventions are the greek letters from the middle of the alphabet (μ, ν), are coordinate indices and range from 1 to 2n. The greek letters from the beginning of the alphabet (α, β, ε) run from 1 to 2n and label the vector fields of an orthonormal frame feα g. Lowercase ~ range from 1 ~ b) Latin indices with and without tildes (a; b; a; to n and are also used to label the vector fields of an orthonormal frame, but in a pairwise form fea ; ea~ g, which will be quite suitable to our intent, as will be clear in the ~ run from 2 to n and serve to label sequel. The indices ðl; lÞ the angular directions of the black hole spacetime considered here. Finally, the indices ðs; s1 ; s2 ; Þ can take the values 1 and label spinorial degrees of freedom. In what follows, Einstein’s summation convention is adopted for pairs of equal indices that are in opposite places, one up and the other down. But equal indices that are in the same position, both up or both down, should not be summed, in principle, unless an explicit symbol of sum is included. By an orthonormal frame we mean that 8 > < gðea ; eb Þ ¼ δab ↔ gðea ; eb~ Þ ¼ 0 ; > : gðea~ ; eb~ Þ ¼ δa~ b~

where the fact that the indices a and b can be thought as labeling the first n vector fields of the orthonormal frame feα g has been used, while a~ and b~ label the remaining n vectors of the frame feα g. If the signature is not Euclidean, some of the vector fields eα might be imaginary in order for the frame be orthonormal. The derivatives of the frame vector fields determine the spin connection according to the following relation: ∇α eβ ¼ ωαβ ε eε :

1 γ ðα γ βÞ ¼ ðγ α γ β þ γ β γ α Þ ¼ δαβ 1; 2

ð1Þ

with 1 standing for the 2n × 2n identity matrix. The covariant derivative of a spinorial field ψ is, then, given by

II. NOTATION AND CONVENTIONS

gðeα ; eβ Þ ¼ δαβ

Indices of the spin connection are raised and lowered with δαβ and δαβ , respectively, so that frame indices can be raised and lowered unpunished. Since the metric is covariantly constant, it follows that ωαβε ¼ ωα½βε , where indices inside the square brackets are antisymmetrized. Analogously, indices enclosed by round brackets are assumed to be symmetrized. The Dirac matrices γ α are 2n × 2n matrices obeying the Clifford algebra,

1 ∇α ψ ¼ ∂ α ψ − ωα βε γ β γ ε ψ; 4 with ∂ α denoting the partial derivative along the vector field eα . Given an orthonormal frame feα g, we can define the dual frame of 1-forms fEα g, which is defined to be such that Eα ðeβ Þ ¼ δα β : Thus, if fxμ g is a local coordinate system in our manifold M, the line element can be written as ds2 ¼ gμν dxμ dxν ¼ δαβ Eα Eβ ¼ Eα Eα ; where gμν ¼ gð∂ μ ; ∂ ν Þ are the components of the metric in the coordinate frame. III. CONFORMAL TRANSFORMATION AND THE DIRAC OPERATOR In this section we shall obtain the conformal transformation of the Dirac operator, which will be of future relevance in order to simplify the Dirac equation in the spacetime of our interest. Let gˆ be a metric that is conformally related to our initial metric, gˆ ¼ Ω2 g, with Ω being a positive definite function throughout the manifold. Then, if fêα g is an orthonormal frame with respect to the metric gˆ we have Eˆ α ¼ ΩEα

and eˆ α ¼ Ω−1 eα :

ð2Þ

Since we are dealing with metric-compatible connections, a change of metric leads to a different spin connection, which is defined by the following relation: ˆ α eˆ β ¼ ωˆ αβ ε eˆ ε : ∇

ð3Þ

Using the identity gˆ μν ¼ Ω2 gμν to find the relation between the Levi-Civita symbols of the two metrics and using Eqs. (2) and (3), we eventually arrive at the following expression:

084022-2

INTEGRABILITY OF THE DIRAC EQUATION ON …

ωˆ α βε ¼ Ω−1 ωα βε þ 2Ω−2 ∂ ½β Ωδε α :


ð4Þ

Now, let us obtain how the Dirac operator, defined by D ¼ γ α ∇α , behaves under conformal transformations. If ψ is a spinorial field, let us define ψˆ ¼ Ωp ψ; with p being a constant parameter that will be conveniently chosen in the sequel. Then, using Eq. (4) we eventually obtain the identity below: ˆ α ðΩp ψÞ ˆ ψˆ ¼ γ α ∇ D 1 ¼ Ωp−1 Dψ þ p þ n − Ωp−2 ð∂ α ΩÞγ α ψ: 2 Thus, choosing p to be 12 − n, it follows that ˆ ψ; ˆ Dψ ¼ Ωðnþ2Þ D 1

1

where ψˆ ¼ Ωð2−nÞ ψ:

ˆ aaa ωˆ aaa~ ¼ −ω ~

ð5Þ

σ1 ¼

0 1 1 0

σ2 ¼

;

The goal of the present section is to show that the Dirac equation minimally coupled to an electromagnetic field is separable in spaces that are the direct product of bidimensional spaces. Let ðM; gˆ Þ be a 2n-dimensional space that is the direct product of n bidimensional spaces, namely, the space can be covered by coordinates fx1 ; y1 ; x2 ; y2 ; …; xn ; yn g such that the line element is given by dˆs2 ¼

n X a¼1

dˆs2a ¼

n X ðEˆ a Eˆ a þ Eˆ a~ Eˆ a~ Þ;

ð7Þ

0

−i

i

0

;

σ3 ¼

1

0

0

−1

:

γ a ¼ σ 3 ⊗ ⊗ σ 3 ⊗ σ 1 ⊗ I ⊗ ⊗ I; |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} ðn−aÞ times

ða−1Þ times

γ a~ ¼ σ 3 ⊗ ⊗ σ 3 ⊗ σ 2 ⊗ I ⊗ ⊗ I: |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

ð9Þ

ðn−aÞ times

ða−1Þ times

Indeed, we can easily check that the Clifford algebra given in Eq. (1) is properly satisfied by the above matrices. Regarding the spinors, it is useful to define the following column vectors: ξþ

IV. DIRECT PRODUCT SPACES AND THE SEPARABILITY OF THE DIRAC EQUATION

ð8Þ

Using this notation, a convenient representation of the Dirac matrices in 2n dimensions is the following:

ð6Þ

Since, generally, Ω is a nonconstant function, it follows that the massive Dirac equation is not conformally invariant, whereas the massless Dirac equation is invariant under conformal transformations. In spite of the lack of conformal invariance of the Dirac equation with mass, Eq. (6) will be of great help for the separation of the Dirac equation in the black hole spacetime considered in this work.

ˆ a~ a~ a : and ωˆ aa ~ a~ ¼ −ω

Thus, for example, ωˆ aab~ ¼ 0 and ωˆ aab ¼ 0 whenever a ≠ b. Furthermore, the nonzero components of the spin connection associated to the index a depend just on the ˆ 111~ is a function that coordinates xa and ya . For instance, ω 1 1 depends just on x and y . In order to accomplish the separability of the Dirac equation, it is necessary to use a suitable representation for the Dirac matrices. In what follows, the 2 × 2 identity matrix will be denoted by I, while the usual notation for the Pauli matrices is going to be adopted:

In particular, this relation enables us to investigate the conformal invariance of the Dirac equation. Indeed, if ψ is a spinorial field of mass m that obeys the Dirac equation in the spacetime with metric g then ˆ ψˆ ¼ ðΩ−1 mÞψ: ˆ Dψ ¼ mψ ⇒ D

and theirs components depend just on the coordinates, x1 and y1 . In such a case, the only components of the spin connection that can be nonvanishing are

1 ¼ 0

and

ξ−

0 ¼ : 1

ð10Þ

If we assume that the index s can take the values “þ1” and “−1,” the action of the Pauli matrices on the above column vectors can be summarized quite concisely as σ 1 ξs ¼ ξ−s ;

σ 2 ξs ¼ isξ−s ;

σ 3 ξs ¼ sξs :

ð11Þ

The spinor space, in which the Dirac matrices act, can be spanned by the direct product of the elements ξs n times. More precisely, a general spinor field can be written as X ψˆ ¼ ð12Þ ψˆ s1 s2 sn ξs1 ⊗ ξs2 ⊗ ⊗ ξsn ; fsg

a¼1

where the two-dimensional line elements dˆs2a and the 1-forms Eˆ a and Eˆ a~ depend just on the two coordinates corresponding to their bidimensional spaces. For instance, ~ dˆs21 , Eˆ 1 and Eˆ 1 depend just on the differentials dx1 and dy1

in which ψˆ s1 s2 sn stands for the components of the spinor and the sum over fsg means the sum over all possible values of the set fs1 ; s2 ; …; sn g. Since every sa can take two values, it follows that this sum comprises 2n terms, which is the number of components of a spinor in d ¼ 2n

084022-3



dimensions. Using this basis, we can easily compute the action of the Dirac matrices on the spinor field. Indeed, using Eqs. (9), (11), and (12) we have X ðs1 s2 sa−1 Þψˆ s1 s2 sn ξs1 ⊗ ξs2 ⊗ γ a ψˆ ¼ fsg

respectively. In order for the latter equation to be separable in blocks depending only on the coordinates fx1 ; y1 g, fx2 ; y2 g and so on, the functions Aˆ a and Aˆ a~ must depend ˆ only on the two coordinates fxa ; ya g and the function m must be a sum of functions depending on these pairs of coordinates:

⊗ ξsa−1 ⊗ ξ−sa ⊗ ξsaþ1 ⊗ ⊗ ξsn X ¼ ðs1 s2 sa Þsa ψˆ s1 s2 sa−1 ð−sa Þsaþ1 sn ξs1

Aˆ a ¼ Aˆ a ðxa ; ya Þ; Aˆ a~ ¼ Aˆ a~ ðxa ; ya Þ; n X ˆ a ðxa ; ya Þ: ˆ ¼ m m

fsg

⊗ ξs2 ⊗ ⊗ ξsa−1 ⊗ ξsa ⊗ ξsaþ1 ⊗ ⊗ ξsn ; where from the first to the second line we have changed the index sa to −sa , which does not change the final result, since we are summing over all values of sa , which comprise the same list of the values of −sa . Moreover, we have used that ðsa Þ2 ¼ 1. Analogously, we have X ðs1 s2 sa−1 Þðisa Þψˆ s1 s2 sn ξs1 ⊗ ξs2 ⊗ γ a~ ψˆ ¼ fsg

With these necessary assumptions for attaining separability, we are left with the following equation: n X a¼1

1 sa ð−sa Þ ˆ a ¼ 0; ðs1 s2 sa Þ sa Da ψˆ a − im ψˆ a

Dsaa

fsg

1 1 ˆ ˆ ˆ ˆ ˆ ~ a~ − Aa þ ∂ a~ þ ω ˆ ~ − Aa~ : ¼ isa ∂ a þ ω 2 aa 2 aaa ð17Þ

⊗ ξs2 ⊗ ⊗ ξsa−1 ⊗ ξsa ⊗ ξsaþ1 ⊗ ⊗ ξsn : Now, we have the tools to try to separate the general equation ð13Þ

ˆ is the Dirac in its two-dimensional blocks, where D operator of the space with metric (7) while Aˆ a , Aˆ a~ , and ˆ are arbitrary functions. In order to accomplish this goal, m we shall assume that the components of the spinor field (12) take the separable form ψˆ s1 s2 sn ¼ ψˆ s11 ðx1 ; y1 Þψˆ s22 ðx2 ; y2 Þ ψˆ snn ðxn ; yn Þ:

ð14Þ

Using this hypothesis and noting that the Dirac operator is ˆ a þ γ a~ ∇ ˆ a~ , it follows that Eq. (13) is given by ˆ ¼ γa ∇ D n X X a¼1 fsg

ð16Þ

where the operator Dsaa used above is defined by

⊗ ξsa−1 ⊗ ξ−sa ⊗ ξsaþ1 ⊗ ⊗ ξsn X ¼ −i ðs1 s2 sa Þψˆ s1 s2 sa−1 ð−sa Þsaþ1 sn ξs1

ˆ − ðAˆ a γ a þ Aˆ a~ γ a~ Þψˆ ¼ m ˆ ψˆ ½D

ð15Þ

a¼1

Since each term in the sum over a in Eq. (16) depends just on the two coordinates fxa ; ya g, it follows that each of these terms in the sum must be a constant, otherwise they could not sum to zero. Let us denote these P separation constants by iηa. Thus, Eq. (16) requires that a ηa ¼ 0. However, it is worth noting that Eq. (16) provides not only one equation but rather a total of 2n independent equations, since for each choice of fsg ≡ fs1 ; s2 ; …; sn g we have one equation. For each of these equations we can have different separation constants. Therefore, ηa can depend on the choice of fsg, so that it is appropriate to write these fsg separation constants as iηa . Then, we arrive at the following equations: ð−sa Þ

ðs1 s2 sa ÞDsaa ψˆ a

fsg

ˆ a þ ηa Þψˆ saa : ¼ iðm

ð18Þ

s

aþ1 a−1 ðs1 s2 sa Þψˆ s11 ψˆ sa−1 ψˆ snn ψˆ aþ1

1 1 ˆ ˆ ˆ ˆ × isa ∂ a þ ωˆ aa − Aa þ ∂ a~ þ ωˆ aaa − Aa~ 2 ~ a~ 2 ~ ð−sa Þ s1

× ψˆ a

ξ ⊗ ξs 2 ⊗ ⊗ ξs n X s s ˆ ¼ im ψˆ 11 ψˆ 22 ψˆ snn ξs1 ⊗ ξs2 ⊗ ⊗ ξsn ; fsg

where by ∂ˆ a and ∂ˆ a~ we mean the derivatives along the vector fields eˆ a and eˆ a~ , namely, ðêa Þμ ∂ μ and ðêa~ Þμ ∂ μ ,

These equations enable us to integrate the fields ψˆ saa and, therefore, find the solutions for the generalized Dirac equation (13). Although these equations are first order differential equations, they are coupled in pairs, namely, the ˆ −a as source and vice equations involving the field ψˆ þ a have ψ versa. Therefore, after unraveling this system, we are left with a decoupled second order differential equation for each component ψˆ saa , thus achieving the separability that we were looking for. Note that, in accordance with Eq. (16), the separation constants must obey

084022-4

INTEGRABILITY OF THE DIRAC EQUATION ON … n X

fsg

ηa ¼ 0:


ð19Þ

a¼1

Since the collective “index” fsg can take 2n values, the latter equation comprise 2n constraints. Let us unravel these constraints. Note that, since the left-hand side of Eq. (18) is fsg independent of saþ1 ; saþ2 ; …; sn , it follows that ηa cannot fsg depend on these indices. In particular, η1 depends just on s1 , so that we can write fsg

η1 ¼ s1 κs11 ;

ð20Þ

Note that this equation must hold for all possible choices of fsg. Now, let us manipulate this equation in order to solve such constraint. Isolating κ s11 , we have κs11 ¼ −

a¼2

Note that the left-hand side depends just on s1 . Therefore, the sum on the right-hand side can depend just on s1 . However, since none of the terms in the sum depend on s1 we conclude that this sum is a constant, namely, κ s11 ¼ −c1

κ s11

is a pair of constants that depends just on s1 . where Since ðsa Þ2 ¼ 1, it follows from Eq. (18) that we can write 1 sa ð−sa Þ fsg ˆ a þ ηa Þ: Da ψˆ a ¼ iðs1 s2 sa Þðm ψˆ saa

ð21Þ

Inasmuch as the left-hand side of this equation depends just on sa , it follows that the right-hand side of this equation should, likewise, depend just on sa . Therefore, we have fsg

ˆ a þ ηa ¼ ðs1 s2 sa Þκ saa m

if a ≥ 2;

ð22Þ

where, for each a, κ saa is a pair of parameters that depends just on sa . Thus, for instance, κ saa does not depend on sa−1 ˆ a is nonconstant, we and on saþ1 . But, since, in principle, m cannot say that the parameters κsaa are constant for a ≥ 2. Nevertheless, taking the derivative of both sides of the latter equation, we have ˆ a ¼ ðs1 s2 sa Þ∂ μ κ saa ∂ μm

if a ≥ 2:

Since the left-hand side of the latter equation does not depend on fsg, it follows that ∂ μ κ saa must vanish, which, in ˆ a should be constants for a ≥ 2. But, its turn, implies that m ˆ 2 ðx2 ; y2 Þ; m ˆ 3 ðx3 ; y3 Þ; …; m ˆ n ðxn ; yn Þ are constants we if m can, without loss of generality, make all of them zero and ˆ 1 ðx1 ; y1 Þ. Therefore, we can say absorb these constants in m that a consistent separability process requires that ˆ2 ¼m ˆ 3 ¼ ¼ m ˆ n ¼ 0: m

fsg

ηa ¼ ðs1 s2 sa Þκsaa ;

ð24Þ

where, for each a, κ saa is a pair of constants. Thus, the constraint (19) now writes as n X ðs1 s2 sa Þκ saa ¼ 0: a¼1

ð25Þ

and

n X ðs2 s3 sa Þκsaa ¼ c1 ; a¼2

where c1 is a constant that does not depend on fsg. Now, the latter equation can be written as κ s22 − s2 c1 ¼ −

n X ðs3 sa Þκ saa : a¼3

Following the same reasoning that we have just used, the left-hand side of the latter equation depends just on s2 while the terms on the right-hand side clearly do not depend on s2 , we can thus conclude that κ s22 ¼ s2 c1 − c2

and

n X ðs3 sa Þκsaa ¼ c2 ; a¼3

where c2 is a constant that does not depend on fsg. Following the same procedure until reaching the term κsnn , we end up with the following final result that solves the constraint (25): κ saa ¼ sa ca−1 − ca

with c0 ¼ cn ¼ 0:

ð26Þ

Concerning the constants c1 ; c2 ; …; cn−1 , they are arbitrary. Thus, in this problem we have (n − 1) separation constants. Finally, inserting Eqs. (23), (24), and (26) into Eq. (18) we arrive at the following equations: ð−s1 Þ

ˆ 1 − c1 Þψˆ s11 ¼ iðs1 m

ð−sa Þ

¼ iðsa ca−1 − ca Þψˆ saa ;

Ds11 ψˆ 1

Dsaa ψˆ a

ð23Þ

Assuming this requirement to hold, Eqs. (20) and (22) immediately lead to

n X ðs2 s3 sa Þκ saa :

if a ≥ 2;

ð27Þ

where the operators Dsaa were defined in Eq. (17). The above set of equations provides two equations for each a, one for sa ¼ þ1 and the other for sa ¼ −1. Manipulating these two equations one easily obtains second order differential equations for ψˆ saa , achieving the separability of the generalized Dirac equation that we are looking for. For appropriate boundary conditions, the constants fc1 ; …; cn−1 g can only take discrete values. The general solution of Eq. (13) is, then, a linear combination of the particular solutions for each of the possible “eigenvalues” fc1 ; …; cn−1 g.

084022-5



As a closing comment for this section, it is well known that every two-dimensional line element can be locally brought to a conformally flat form by means of a change of coordinates. For instance, passing from the coordinates fxa ; ya g to harmonic coordinates fˇxa ; yˇ a g, the line element dˆs2a can be written as dˆs2a ¼ Φ2a ðˇxa ; yˇ a Þ½ðdˇxa Þ2 þ ðdˇya Þ2 :

ds2 ¼ −fðrÞ2 dt2 þ

ð28Þ

Then, it is natural to wonder whether the separation process would be easier using such coordinates. The quick answer is no. The reason being that the metric components enter into the Dirac equation only by means of the spin ˆ aaa coefficients ωˆ aa ~ a~ and ω ~ , which appear at the operator Dsaa ; see Eq. (17). In spite of its simple form, the line element (28) has nonvanishing spin coefficients, so that no practical simplification would be in order. Nevertheless, it is interesting noting that, adopting the frame Ea ¼ Φa dˇxa and Ea~ ¼ Φa dˇya , the spin coefficients for the bidimensional line element (28) are given by 1 ∂ a logðΦa Þ ¼ ∂ a logðΦa Þ; Φa xˇ 1 ¼ ∂ a logðΦa Þ ¼ ∂ a~ logðΦa Þ: Φa yˇ

ωˆ aa ~ a~ ¼ ωˆ aaa ~

spin 1=2 on the background of black holes whose horizons have topology R × S2 × × S2. These black hole solutions, which possess electric and magnetic charge, have been obtained in Refs. [15,25,26] and are given by

ð31Þ where f ¼ fðrÞ is the following function of the coordinate r: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2M Q2e ðd − 3Þ Q2m Λr2 fðrÞ ¼ − d−3 þ − − : d−3 r 2ðd − 2Þr2ðd−3Þ 4ðd − 5Þr2 d − 1 ð32Þ In the latter expression, d ¼ 2n is the dimension of the spacetime, M, Qe , and Qm are the mass, the electric charge, and the magnetic charge of the black hole, respectively, while Λ is the cosmological constant; see Ref. [15] for details. This spacetime is the solution of Einstein-Maxwell equations with a cosmological constant Λ and electromagnetic field F ¼ dA, where the gauge field A is given by

Using these relations, we immediately see that the operator Dsaa given in Eq. (17) can be nicely written as Dsaa ¼ isa ð∂ˆ a − Bˆ a Þ þ ð∂ˆ a~ − Bˆ a~ Þ;

ð29Þ

1 ˆ Bˆ μ ¼ Aμ þ ∂ μ − logðΦ1 Φ2 Φn Þ : 2

ð30Þ

where

A¼

V. BLACK HOLE SPACETIMES In this section we shall separate the Dirac equation corresponding to a massive and electrically charged field of

n X Qe dt þ Q cos θl dϕl : m rd−3 l¼2

A suitable orthonormal frame for such a spacetime is given by

Comparing Eqs. (17) and (29), we see that using the conformally flat form of the bidimensional line elements, Eq. (28), we can pretend that our space is flat, i.e., with vanishing spin coefficients, as long as we perform the gauge transformation (30) in the field Aˆ μ . In other words, a change in the conformal factors Φa is equivalent to a gauge transformation on the field Aˆ μ . As a direct application of the results obtained in the present section, in Appendix A we study the Dirac equation in a higher-dimensional version of the charged Nariai spacetime. A more challenging example of the usefulness of the results obtained here is worked out in the next section, where we separate the Dirac equation in some black hole spacetimes. Additionally, in Appendix B we generalize the results of this section to odd-dimensional spaces.

n X dr2 2 þ r ðdθ2l þ sin2 θl dϕ2l Þ; fðrÞ2 l¼2

E1 ¼ ifdt;

~

E1 ¼ f −1 dr; ~

El ¼ r sin θl dϕl ; El ¼ rdθl ; where, as explained earlier, the index l ranges from 2 to n. Using this frame, the line element is given by ds2 ¼

n X ðEa Ea þ Ea~ Ea~ Þ: a¼1

In its turn, the gauge field can be written as A ¼ Aa Ea þ Aa~ Ea~ ; where A1 ¼

−iQe ; frd−3

Al ¼

Qm cot θl ; r

Aa~ ¼ 0:

ð33Þ

A field of spin 1=2 with electric charge q and mass m minimally coupled to the electromagnetic field and propagating in this spacetime obeys the following version of the Dirac equation:

084022-6

INTEGRABILITY OF THE DIRAC EQUATION ON … α

γ ð∇α − iqAα Þψ ¼ mψ:


ð34Þ

Using the definition of the Dirac operator, D ¼ γ α ∇α along with the fact that Aa~ ¼ 0, it follows that the above equation is written as

operator, obtained in Sec. III, we can write the field equation (35) in terms of an equation in the space with line element dˆs2 , so that the separability results of Sec. IV can be fully used. Indeed, defining 1

Dψ ¼ ðm þ iqAa γ a Þψ:

The aim of the present section is to integrate this equation. In Sec. IV, we have been able to separate an analogous equation for spaces that are the direct product of bidimensional spaces. However, the black hole line element (31) is not of this special type, due to the warping factor r2 in front of the angular part of the metric. Nevertheless, the conformal transformation 2

2

2

2

ds → dˆs ¼ Ω ds

with

Ω¼r

dˆs2 ¼ −

n X f2 2 dr2 dt þ þ ðdθ2 þ sin2 θl dϕ2l Þ: ð36Þ r2 ðrfÞ2 l¼2 l

A suitable orthonormal frame for this space is given by (

Eˆ 1 ¼ ifr−1 dt;

~ Eˆ 1 ¼ ðrfÞ−1 dr;

~ Eˆ l ¼ sin θl dϕl ; Eˆ l ¼ dθl :

ð37Þ

ˆ ψˆ ¼ Ω−1 ðm þ iqAa γ a Þψ: ˆ D

ˆ 111~ ¼ rf0 − f; ωˆ 111 ~ ¼ −ω ˆ lll~ ¼ cot θl ; ωˆ lll ~ ¼ −ω

ˆ ¼m ˆ 1 ¼ rm; m

Aˆ a ¼ iqrAa ;

Aˆ a~ ¼ 0;

ð41Þ

it follows that Eq. (40) takes exactly the form of the equation studied in Sec. IV; namely, we obtain Eq. (13). Moreover, and foremost, defining the coordinates x1 ¼ t;

y1 ¼ r;

x l ¼ ϕl ;

yl ¼ θl ;

ð42Þ

ˆ and the gauge field Aˆ α are it follows that the function m exactly of the form necessary to attain separability; namely, the constraints (15) and (23) are obeyed. Therefore, due to Eqs. (12), (14), and (39), it follows that a solution for Eq. (34) in the black hole background is provided by ψ ¼ rð2−nÞ

X s ψ 11 ðt; rÞψ s22 ðϕ2 ; θ2 Þ ψ snn ðϕn ; θn Þξs1 fsg

⊗

ð38Þ

where f 0 stands for the derivative of f with respect to its variable r. Using the conformal transformation of the Dirac

ð40Þ

Then, defining

1

The nonvanishing components of the spin connection are

ð39Þ

and using Eq. (5), it follows that the field equation (35) can be written as

−1

leads us to the following line element that is the direct product of bidimensional spaces:

1

ψˆ ¼ Ωð2−nÞ ψ ¼ rðn−2Þ ψ;

ð35Þ

ξs2

⊗ ⊗ ξsn :

ð43Þ

From Eqs. (17), (27), (33), and (37)–(41), it follows that the functions ψ saa must be solutions of the following differential equations:

r qQe 1 ð−s Þ 0 is1 ∂ − þ rf∂ r þ ðrf − fÞ ψ 1 1 ¼ iðs1 rm − c1 Þψ s11 if t frd−4 2 1 1 ð−s Þ isl ∂ ϕl − iqQm cot θl þ ∂ θl þ cot θl ψ l l ¼ iðsl cl−1 − cl Þψ sll : sin θl 2 As explained in Sec. IV, the constants c1 ; c2 ; …; cn−1 are separation constants that generally take discrete values once boundary conditions and regularity requirements are imposed. The constant cn , on the other hand, is zero. Note that the coefficients in the above equations do not depend on the coordinates t and ϕl , which stems from the fact that these are cyclic coordinates of the metric, so that ∂ t and ∂ ϕl are killing vector fields of both metrics g and gˆ . Therefore, it is convenient to decompose the dependence of the fields ψ saa on these coordinates in the Fourier basis, namely,

ψ s11 ðt; rÞ ¼ eiωt Ψs11 ðrÞ;

ð44Þ

ψ sll ðϕl ; θl Þ ¼ eiωl ϕl Ψsll ðθl Þ: ð45Þ

The final general solution for the field ψ must, then, include a “sum” over all values of the Fourier frequencies ω and ωl with arbitrary Fourier coefficients. While ω can be interpreted as related to the energy of the field, ωl are related to angular momentum. Note that in order to avoid conical singularities in the spacetime, the coordinates ϕl must have

084022-7



period 2π, namely ϕl and ϕl þ 2π should be identified. As it is well known, a spin 1=2 field changes its sign after a 2π rotation, which implies that the angular frequencies ωl must be half-integers [27,28]: 1 3 5 ωl ¼ ; ; ; …: 2 2 2

ð46Þ

Finally, inserting the decomposition (45) into Eq. (44), we end up with the following pairwise coupled system of differential equations: d 1 ωr qQe ð−s Þ 0 rf þ ðrf − fÞ þ is1 − Ψ1 1 dr 2 f frd−4

it turns out that the angular part of Eq. (47) can be written in the following simpler way in terms of the fields Φsll ðθÞ:

d 1 ωl ð−s Þ þ cot θl − sl − qQm cot θl Φl l dθl 2 sin θl ¼ isl λl Φsll :

Although it may seem that we did not achieve much simplification by the redefinition of the fields and separation constants, it turns out that in the case in which the black hole has vanishing magnetic charge, Qm ¼ 0, these equations reduce to

d 1 s ω ð−s Þ þ cot θl − l l Φl l ¼ isl λl Φsll : dθl 2 sin θl

¼ iðs1 mr − c1 ÞΨs11 d 1 ωl ð−s Þ þ cot θl − sl − qQm cot θl Ψl l dθl 2 sin θl

¼ iðsl cl−1 − cl ÞΨsll :

ð47Þ

A. The angular part of Dirac’s Equation Now, we shall investigate a little further the above equations. Let us start with the angular part of the equations, namely, the equations for Ψsll . One can make a simplification on these equations by performing a field redefinition along with a redefinition of the separation constants, as we show in the sequel. Instead of using the n − 1 separation constants c1 ; c2 ; …; cn−1 , we shall use the constants λ2 ; λ3 ; …; λn , defined by λl ≡

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2l−1 − c2l ;

ð48Þ

where it is worth recalling that cn ¼ 0, by definition. Inverting these relations, we find that the old constants can be written in terms of the new constants as follows: cl−1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ λ2l þ λ2lþ1 þ þ λ2n :

ð49Þ

Then, defining the parameter ζl ¼ arctanhðcl =cl−1 Þ; we find that cl−1 ¼ λl cosh ζl

and cl ¼ λl sinh ζ l ;

so that the following relation holds: sl cl−1 − cl ¼ sl λl e−sl ζl : Thus, performing the field redefinition given by Ψsll ðθÞ ¼ esl ζl =2 Φsll ðθÞ;

ð51Þ

ð50Þ

In the latter form, the angular equations are identical to the equation DS2 Φ ¼ iλΦ, where DS2 is the Dirac operator in the two-dimensional unit sphere. To check this claim, one should use the frame e1 ¼ sin θdϕ and e2 ¼ dθ along with the Dirac matrices γ 1 ¼ σ 1 and γ 2 ¼ σ 2 . The solutions of the eigenvalue equation DS2 Φ ¼ iλΦ are well known; the components of the two-component spinor Φ are written in terms of Jacobi polynomials [27,29]. From a geometrical point of view, these solutions can be understood in terms of the Wigner elements of the group SpinðR3 Þ, that give rise to the so-called spin weighted spherical harmonics [30,31], which are tensorial generalizations of the spherical harmonics. Moreover, the allowed eigenvalues λl are also known; they must be nonzero integers [27,32]: λl ¼ 1; 2; 3; …: Solutions with noninteger eigenvalues are not well defined on the whole sphere, while a vanishing eigenvalue is forbidden by the Lichnerowicz theorem [33], since the sphere is a compact manifold with positive curvature. Regarding the general case in which the black hole magnetic charge is nonvanishing, Qm ≠ 0, we have tried to make a redefinition of the fields Φsll by means of a general − linear combination of the fields Φþ l and Φl , with nonconstant coefficients, in order to convert Eq. (51) into the eigenvalue equation DS2 Φ ¼ iλΦ. However, it turns out that the coefficients of the linear combination must obey fourth-order differential equations, whose solutions seem to be quite difficult to attain analytically. In spite of this, we can make an important progress regarding the system of − equations (51) by decoupling the fields Φþ l and Φl , which, after all, is our goal at this paper. The final result is that the fields Φsll satisfy the following second order differential equation:

084022-8



dΦsll 1 d ð1 þ 2qQm Þωl cos θl 1 þ 2qQm þ 2ω2l ð1 − 4q2 Q2m Þcos2 θl 2 sin θl þ − þ − λl Φsll ¼ 0: sin θl dθl dθl sin2 θl 2sin2 θl 4sin2 θl It is worth stressing that the latter equation must be supplemented by the requirement of regularity of the fields Φsll at the points θl ¼ 0 and θl ¼ π, where our coordinate system breaks down. These regularity conditions transform the task of solving the latter equation in a Sturm-Liouville problem, so that the possible values assumed by the separation constants λl form a discrete set. Since the case Qm ¼ 0 in Eq. (52) has a known solution, as described above, it follows that we can look for solutions for the case Qm ≠ 0 by means of perturbation methods, with Qm being the perturbation parameter. Indeed, in the celebrated paper [10], a similar path has been taken by Press and Teukolsky in order find the solutions and their eigenvalues for the angular part of the equations of motion for fields with arbitrary spin on Kerr spacetime, in which case the angular momentum of the black hole was the order parameter. In this respect, see also Ref. [34]. B. The radial part of Dirac’s Equation In order to solve the pair of radial equations in (47), we − should first decouple the fields Ψþ 1 and Ψ1 . This can be easily attained by defining 1 1 ωr qQe 0 Bs1 ðrÞ ¼ ðrf − fÞ − is1 − ; rf 2 f frd−4 i Cs1 ðrÞ ¼ − ðs1 mr þ c1 Þ; rf in terms of which the radial equation in (47) can be written as d s1 1 Ψ ¼ −Bs1 Ψs11 þ Cs1 Ψ−s 1 : dr 1

ð53Þ

Then, deriving this equation with respect to r and using 1 in terms of Ψs11 , we are Eq. (53) to substitute Ψ−s 1 eventually led to the following decoupled second order differential equation: s1 d2 Ψs11 1 dCs1 dΨ1 s1 þ B þ B − Ψ þ B s1 −s1 s1 1 Cs1 dr dr dr2 dBs1 þ ð54Þ − B2s1 − Cs1 C−s1 Ψs11 ¼ 0: dr An analytical exact solution of the latter differential equation is, probably, out of reach. Nevertheless, we can use Eq. (54) to infer the asymptotic forms of the solution near the infinity, r → ∞, as well as near the horizon

ð52Þ

r → r⋆ , where r⋆ is a root of the function f, namely, fðr⋆ Þ ¼ 0. Particularly, we shall prove in the sequel that, in the case of vanishing cosmological constant, the wellknown case d ¼ 4 is qualitatively different from the higherdimensional cases d ≥ 6. In order to do this analysis, we shall write Eq. (54) as d2 Ψs11 dΨs11 s1 þ hs01 ðrÞΨs11 ¼ 0; þ h ðrÞ 1 dr dr2

ð55Þ

where hs11 and hs01 are defined by comparing Eqs. (54) and (55). Then, we can work out the asymptotic forms of the coefficients hs11 and hs01 in the region of interest. In particular, for a consistent investigation of the asymptotic form of the solutions of Eq. (55) in the limit r → ∞, if we want to know Ψs11 up to order r−p we need to know hs11 up to order r−ðpþ1Þ and consider hs01 up to order r−ðpþ2Þ. Looking at the function fðrÞ in Eq. (32), we see that the term that multiplies Λ becomes the dominant one as we approach the infinity, r → ∞. Therefore, it is intuitive to guess that the cases of vanishing and nonvanishing Λ should be qualitatively different. Thus, let us separate the analysis of these two cases. First, let us consider the case Λ ≠ 0. Collecting the dΨ

s1

coefficients that multiply dr1 and Ψs11 in Eq. (54) and then expanding them in powers of r−1 , we can find, after some algebra, the following asymptotic forms: ðd − 1Þm2 iðd − 1Þs1 ω 1 hs01 ðrÞ ¼ ; þ þ O Λr3 r4 Λr2 1 s1 c1 2ðd − 1Þ c21 1 1 s1 h1 ðrÞ ¼ − 2 þ − 2 3þO 4 : r mr ðd − 3ÞΛ m r r In particular, considering the expansion of hs01 up to order r−2 and the expansion of hs11 up to order r−1, we are led to the following asymptotic form: Ψs11 ðrÞ ∼ C0 sin

pffiffiffiffiffiffiffiffiffiffiffi m d−1 1 pffiffiffiffi logðrÞ þ φ0 þ O ; r Λ

where C0 and φ0 are arbitrary integration constants. Note, however, that the field that is “the solution” of the Dirac equation, ψ, has a further decaying multiplicative factor rð1−dÞ=2 , in accordance with Eq. (43). Now, let us consider the context of vanishing cosmological constant, Λ ¼ 0. In such a case, one can see, after some algebra, that the asymptotic forms of the functions h0 and h1 are the following when d ≥ 6:

084022-9


hs01 ðrÞ


¼ ½ðd − 3Þ2 ω2 − ðd − 3Þm2 3 iωc1 þ þ ðd − 3Þ c21 − 4 m

ðd − 3Þ2 Q2m m2 ðd − 3Þ3 Q2m ω2 1 1 þO 3 ; − þ 2 4ðd − 5Þ 2ðd − 5Þ r r 1 sc 1 hs11 ðrÞ ¼ − − 1 21 þ O 3 : r mr r However, these formulas do not apply to the well-studied situation d ¼ 4, in which case hs01 has a term of order r−1 depending on M and Qe and the coefficient of order r−2 has additional terms also depending on M and Qe . Analogously, for d ¼ 4, the function hs11 also has additional contributions of order r−2 stemming from the mass and the electric charge of the black hole. Thus, we conclude that, for Λ ¼ 0, the spinor field that represents a charged particle of spin 1=2 moving in the black hole (31) has qualitatively different fall off properties in the asymptotic infinity depending on whether d ¼ 4 or d ≥ 6. A similar asymptotic analysis can be performed near the horizons, namely, near the values of r for which the function f vanishes. In such a case the coordinate r ceases to be reliable and we should change the radial coordinate to tortoise-like coordinates; see [10] for instance. Besides such asymptotic behaviors, one can also look for approximate solutions valid in a broader domain by means of other approximation methods. For instance, in Ref. [35] an approximate solution for the Dirac field on the Kerr spacetime has been obtained using the WKB method after transforming the radial second order differential equation into a Schrödinger equation; see also [36,37]. VI. CONCLUSIONS AND PERSPECTIVES In this article we have shown that the Dirac equation coupled to a gauge field can be decoupled in evendimensional manifolds that are the direct product of bidimensional spaces, provided that the gauge field is also “separated” in accordance with the bidimensional blocks, as shown in Eq. (15). Then, we have used this fact along with the conformal transformation of the Dirac operator to decouple the equation of motion of a charged test field of spin 1=2 propagating in the background of the black hole solution (31). We have shown that the latter problem reduces to solving a second order radial differential equation, whose asymptotic behavior has been worked out, along with a second order angular differential equation with regularity conditions. In particular, we have argued that if the black hole has a vanishing magnetic charge then the angular equation reduces to the eigenvalue problem for the Dirac operator on the sphere, whose solutions are known. The separation attained in the present work paves the way to analyze the quasinormal modes associated to a field of spin 1=2 on the background of the black hole considered

here. Moreover, in four dimensions, considering the case of positive cosmological constant and taking the limit of equal temperatures for the black hole horizon and the cosmological horizon we end up with the so-called Nariai spacetime [15], whose geometric structure is much simpler than the black hole solution. Therefore, hopefully, one can use the tools presented here to find, analytically, the quasinormal modes of a spin 1=2 field on higher-dimensional versions of the Nariai spacetime. This line of research is of physical relevance both from the theoretical and experimental points of view. Indeed, quasinormal modes are related to the analysis of stability of black holes and their knowledge are of relevance on applications of the AdS/CFT correspondence. Furthermore, these modes play a central role on the measurements of gravitational radiation as well as on the characterization of astronomical objects [18–20]. Therefore, we intend to continue our work addressing these points in the near future. In addition, we aim to investigate the possibility of a superradiance phenomenon in these spacetimes in forthcoming works. ACKNOWLEDGMENTS C. B. would like to thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), for their partial financial support, and to Universidade Federal de Pernambuco (UFPE), for a partial financial support through Qualis A grant. J. V. thanks CNPq for their financial support. APPENDIX A: GENERALIZED CHARGED NARIAI SPACETIMES As a direct application of the results presented in Sec. IV, let us work out the separation of the Dirac equation in a generalized version of the Nariai spacetime. In four dimensions the Nariai spacetime is the direct product of the twodimensional de Sitter space with the two-dimensional sphere. Thus, a natural higher-dimensional generalization of this spacetime is provided by the direct product of the two-dimensional de Sitter space with several spheres [15], it is this spacetime that we shall consider in the sequel. The line element of the higher-dimensional version of the Nariai spacetime is given by ds2 ¼ R21 ðdx2 − sin2 xdt2 Þ þ R22

n X l¼2

dθ2l þ sin2 θl dϕ2l ;

where the constant parameters R1 and R2 are given by

084022-10

8 ðd − 3ÞQ2 −1=2 > > ; > < R1 ¼ Λ − 2ðd − 2Þ −1=2 > Q2 > > ; : R2 ¼ Λ þ 2ðd − 2Þ


(

in which case this metric is a solution of Einstein’s equation with cosmological constant Λ and in the presence of an electric field F ¼ dA whose gauge field is given by A ¼ QR21 cos xdt:

E1 ¼ iR1 sin xdt;

ðA2Þ

The 1-form gauge field is then written as

ðA1Þ

Our aim is to separate the latter equation. In order to accomplish this, we first should define a frame. A suitable choice is

1 ω111 ~ ¼ −ω111~ ¼ R1 cot x; 1 ωlll ~ ¼ −ωlll~ ¼ R2 cot θl :

A spin 1=2 particle of mass m and electric charge q propagating in such a background is represented by a spinorial field ψˆ obeying Dirac’s equation ˆ ðD − iqAα γ α Þψˆ ¼ mψ:


A ¼ −iQR1 cot xE1 : Taking this into account, in order to put Eq. (A1) in the form of the generalized Dirac equation considered in Sec. IV, Eq. (13), we should set ˆ A1 ¼ qQR1 cot x; Aˆ 1~ ¼ 0; Aˆ l ¼ 0; Aˆ l~ ¼ 0;

~

m1 ¼ m; ml ¼ 0:

E1 ¼ R1 dx; ~

El ¼ R2 sin θl dϕl ; El ¼ R2 dθl ; where the index l ranges from 2 to n. In such a frame, the spin coefficients are given by

Inserting these fields along with the spin coefficients (A2) into Eq. (27), we eventually arrive at the desired decoupled equations:

s1 1 1 ð−s Þ cot x ψˆ 1 1 ¼ iðs1 m − c1 Þψˆ s11 ; ∂ t − is1 qQR1 cot x þ ∂ x þ R1 2R1 R1 sin x isl 1 1 ð−s Þ ∂ ϕl þ ∂ θl þ cot θl ψˆ l l ¼ iðsl cl−1 − cl Þψˆ sll : R2 2R2 R2 sin θl

ðA3Þ

Since ∂ t and ∂ ϕl are Killing vector fields, we shall expand the spinorial fields in the Fourier basis, ψˆ sll ðϕl ; θl Þ ¼ eiωl ϕl Ψsll ðθl Þ;

ψˆ s11 ðt; xÞ ¼ eiωt Ψs11 ðxÞ;

in which case Eq. (A3) becomes d 1 ω ð−s Þ 2 þ cot x þ is1 − qQR1 cot x Ψ1 1 ¼ iR1 ðs1 m − c1 ÞΨs11 ; dx 2 sin x d 1 ω s ð−s Þ þ cot θl − l l Ψl l ¼ iR2 ðsl cl−1 − cl ÞΨsll : dθl 2 sin θl

ðA4Þ

As explained in Sec. V, the angular frequencies ωl must be half-integers. In addition, note that the angular part, Eq. (A4), has the same form of Eq. (51) as long as we set Qm ¼ 0. Therefore, the solutions of the angular differential equations in Eq. (A4) are related to the eigenmodes of the Dirac operator in the sphere, as explained in Sec. V. Concerning the 1 differential equation for the fields Ψs11 , differentiating the first relation in Eq. (A4) and eliminating the field Ψ−s 1 , we eventually arrive at the following decoupled second order differential equation: d2 Ψs11 dΨs11 1 1 1 2 2 2 2 2 4 2 2 cos x 2 2 Ψs1 þ cot x þ R1 ðc1 − m Þ þ q Q R1 þ cot x þ ωðis1 − 2qQR1 Þ 2 þ ω − iqQR1 − 4 2 sin2 x 1 dx sin x dx2 ¼ 0:

ðA5Þ

APPENDIX B: ODD-DIMENSIONAL CASE In this appendix we shall consider the task of separating the Dirac equation for the case in which the space is the direct product of bidimensional spaces and a

one-dimensional line (or circle). More precisely, here we will assume that the dimension of the space is d ¼ 2n þ 1, with n being an integer, and that the line element is given by

084022-11


dˆs2 ¼

n X


Now, we shall decompose the components of the spinor ψˆ obeying Eq. (B2) as the following product of functions:

dˆs2a þ hðzÞ2 dz2 ;

a¼1

where dˆs2a are bidimensional line elements that are independent from each other, just as explained below Eq. (7), whereas hðzÞ is an arbitrary function of the extra coordinate z. This is the natural odd-dimensional generalization of the spaces considered throughout this article. Now, defining the coordinate z to be such that dz ¼ hðzÞdz, we end up with the following simpler line element: dˆs2 ¼

n X

dˆs2a þ dz2 :

ðB1Þ

a¼1

ψˆ s1 s2 sn ¼ ψˆ s11 ðx1 ; y1 Þ ψˆ snn ðxn ; yn Þψˆ nþ1 ðzÞ: Then, following steps completely analogous to the ones adopted in Sec. IV, one can see that in order to achieve the ˆ should have separability of the Eq. (B2) the fields Aˆ μ and m the following form: Aˆ a ¼ Aˆ a ðxa ;ya Þ; Aˆ a~ ¼ Aˆ a~ ðxa ;ya Þ; n X ˆ nþ1 ðzÞ: ˆ a ðxa ;ya Þ þ m ˆ¼ m m a¼1

Our goal is to separate the equation, ˆ − Aˆ a γ a − Aˆ a~ γ a~ − Aˆ nþ1 γ nþ1 Þψˆ ¼ m ˆ ˆ ψ; ðD

ðB2Þ

ˆ standing for the in the odd-dimensional space (B1), with D Dirac operator in such space and γ nþ1 denotes the extra Dirac matrix associated to the added dimension. As usual, the indices a are assumed to range from 1 to n. Before heading to separate such equations, let us first recall that in d ¼ 2n þ 1 dimensions spinors can be represented by column vectors with 2n components, just as in d ¼ 2n dimensions, and that, besides the 2n Dirac matrices γ a and γ a~ , we need to add one further matrix, which will be denoted by γ nþ1. The latter matrix should obey the following algebra: 8 nþ1 a a nþ1 > ¼ 0; > > : γ nþ1 γ nþ1 ¼ 1:

Thus, assuming the latter relations, one eventually obtains that Eq. (B2) leads to the following partial differential equation: X n 1 ð−s Þ â ðs1 s2 sa Þ sa Dsaa ψˆ a a − im ψˆ a a¼1 1 ˆ ˆ nþ1 ¼ 0; ð∂ − Anþ1 Þψˆ nþ1 − im þ ðs1 s2 sn Þ ψˆ nþ1 z where the differential operator Dsaa has been defined in Eq. (17). Since the coordinates in each term of the latter differential equation are independent, we conclude that the following identities must hold: ð−sa Þ

ðs1 sa ÞDsaa ψˆ a

fsg

ˆ a þ ηa Þψˆ saa ; ¼ iðm

fsg ˆ nþ1 þ ηnþ1 Þψˆ nþ1 ; ðB4Þ ðs1 sn Þð∂ z − Aˆ nþ1 Þψˆ nþ1 ¼ iðm fsg

Adopting the representation showed on Eq. (9) for the matrices γ a and γ a~ , it follows that the latter algebra is satisfied if γ nþ1 is written as

fsg

where ηa and ηnþ1 are separation constants that must obey the constraint X n

γ nþ1 ¼ σ 3 ⊗ σ 3 ⊗ ⊗ σ 3 : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

fsg ηa

fsg

þ ηnþ1 ¼ 0:

ðB5Þ

a¼1

n times

In particular, writing a general spinor field as X ψˆ ¼ ψˆ s1 s2 sn ξs1 ⊗ ξs2 ⊗ ⊗ ξsn ;

Aˆ nþ1 ¼ Aˆ nþ1 ðzÞ

ðB3Þ

fsg

it follows that the action of γ nþ1 on such a field is given by X ðs1 sn Þψˆ s1 sn ξs1 ⊗ ⊗ ξsn ; γ nþ1 ψˆ ¼

Now, our final and important task is to solve the latter constraint. Since the multi-index fsg can assume 2n values, the latter equation comprises a set of 2n relations, rather than just one. Following procedures analogous to the ones shown in Eqs. (20)–(26), we conclude that, since ma and mnþ1 are independent of the spinorial indices fsg, one must have that m2 ; m3 ; …; mnþ1 must be constants. Absorbing these constants into m1 , one can, equivalently, assume that m2 ¼ m3 ¼ ¼ mn ¼ mnþ1 ¼ 0:

fsg

whereas the action of the matrices γ a and γ a~ are the same given previously in Sec. IV.

ðB6Þ fsg

In addition, one find that the separation constants ηa and fsg ηnþ1 must assume the following form:

084022-12

INTEGRABILITY OF THE DIRAC EQUATION ON … fsg

ηa ¼ ðs1 s2 sa Þðsa ca−1 − ca Þ; fsg

ηnþ1 ¼ ðs1 s2 sn Þcn ;

with

PHYSICAL REVIEW D 95, 084022 (2017) ð−s1 Þ

ˆ 1 − c1 Þψˆ s11 ; ¼ iðs1 m

ð−sa Þ

¼ iðsa ca−1 − ca Þψˆ saa ;

Ds11 ψˆ 1

c0 ¼ 0;

Dsaa ψˆ a

ðB7Þ

where in the above equation the constants ca are arbitrary integration constants. Thus, we end up with a total of n integration constants, which is one more than what is necessary in the even-dimensional case d ¼ 2n. As a check, one can easily verify that Eq. (B7) provides a solution to the constraint (B5). Finally, inserting Eqs. (B6) and (B7) into Eq. (B4), we are left with the separated equations that we were looking for:

[1] B. Carter, Global structure of the Kerr family of gravitational fields, Phys. Rev. 174, 1559 (1968). [2] M. Walker and R. Penrose, On quadratic first integrals of the geodesic equations for type f22g spacetimes, Commun. Math. Phys. 18, 265 (1970). [3] R. H. Price, Nonspherical perturbations of relativistic gravitational collapse. 1. Scalar and gravitational perturbations, Phys. Rev. D 5, 2419 (1972). [4] R. H. Price, Nonspherical perturbations of relativistic gravitational collapse. II. Integer-spin, zero-rest-mass fields, Phys. Rev. D 5, 2439 (1972). [5] S. A. Teukolsky, Rotating Black Holes-Separable Wave Equations for Gravitational and Electromagnetic Perturbations, Phys. Rev. Lett. 29, 1114 (1972). [6] W. Unruh, Separability of the Neutrino Equations in a Kerr Background, Phys. Rev. Lett. 31, 1265 (1973). [7] S. Chandrasekhar, The solution of Dirac’s equation in Kerr geometry, Proc. R. Soc. A 349, 571 (1976). [8] D. N. Page, Dirac equation around a charged, rotating black hole, Phys. Rev. D 14, 1509 (1976). [9] S. A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J. 185, 635 (1973). [10] W. H. Press and S. A. Teukolsky, Perturbations of a rotating black hole. II. Dynamical stability of the Kerr metric, Astrophys. J. 185, 649 (1973). [11] S. A. Teukolsky and W. H. Press, Perturbations of a rotating black hole. III. Interaction of the hole with gravitational and electromagnet ic radiation, Astrophys. J. 193, 443 (1974). [12] S. Mukhi, String theory: A perspective over the last 25 years, Classical Quantum Gravity 28, 153001 (2011). [13] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38, 1113 (1999); Adv. Theor. Math. Phys. 2, 231 (1998). [14] C. Csáki, in From Fields to Strings, edited by M. Shifman et al. (World Scientific, Singapore, 2005), Vol. 2, pp. 967–1060.

ð∂ z − Aˆ nþ1 Þψˆ nþ1 ¼ icn ψˆ nþ1 :

if a ≥ 2; ðB8Þ

In particular, the solution for the first order differential equation for ψˆ nþ1 ðzÞ is given by Z ψˆ nþ1 ðzÞ ¼ exp ½Aˆ nþ1 ðzÞ þ icn dz :

[15] C. Batista, Generalized charged Nariai solutions in arbitrary even dimensions with multiple magnetic charges, Gen. Relativ. Gravit. 48, 160 (2016). [16] F. R. Tangherlini, Schwarzschild field in n dimensions and the dimensionality of space problem, Nuovo Cimento 27, 636 (1963). [17] A. R. Brown and A. Dahlen, Spectrum and stability of compactifications on product manifolds, Phys. Rev. D 90, 044047 (2014). [18] K. D. Kokkotas and B. G. Schmidt, Quasinormal modes of stars and black holes, Living Rev. Relativ. 2, 2 (1999). [19] H. P. Nollert, Quasinormal modes: The characteristic sound of black holes and neutron stars, Classical Quantum Gravity 16, R159 (1999). [20] E. Berti, V. Cardoso, and A. O. Starinets, Quasinormal modes of black holes and black branes, Classical Quantum Gravity 26, 163001 (2009). [21] B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016). [22] A. Zhidenko, Linear perturbations of black holes: Stability, quasi-normal modes and tails, arXiv:0903.3555. [23] J. G. Rosa, Superradiance in the sky, Phys. Rev. D 95, 064017 (2017). [24] R. Gueven, Wave mechanics of electrons in Kerr geometry, Phys. Rev. D 16, 1706 (1977). [25] H. Maeda, M. Hassaine, and C. Martinez, Magnetic black holes with higher-order curvature and gauge corrections in even dimensions, J. High Energy Phys. 08 (2010) 123. [26] M. Ortaggio, J. Podolsky, and M. Zofka, RobinsonTrautman spacetimes with an electromagnetic field in higher dimensions, Classical Quantum Gravity 25, 025006 (2008). [27] R. Camporesi and A. Higuchi, On the Eigen functions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20, 1 (1996). [28] A. A. Abrikosov, Jr., Fermion states on the sphere S2 , Int. J. Mod. Phys. A 17, 885 (2002). [29] A. A. Abrikosov, Jr., Dirac operator on the Riemann sphere, arXiv:hep-th/0212134.

084022-13



[30] J. N. Goldberg, A. J. MacFarlane, E. T. Newman, F. Rohrlich, and E. C. G. Sudarshan, Spin s spherical harmonics and edth, J. Math. Phys. (N.Y.) 8, 2155 (1967). [31] M. Boyle, How should spin-weighted spherical functions be defined?, J. Math. Phys. (N.Y.) 57, 092504 (2016). [32] A. Trautman, Spin Structures on Hypersurfaces and the Spectrum of the Dirac Operator on Spheres, Spinors, Twistors, Cli ord Algebras and Quantum Deformations (Kluwer Academic Publishers, Dordrecht, 1993). [33] A Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris Ser. A. B. 257, 7 (1963).

[34] S. K. Chakrabarti, On mass-dependent spheroidal harmonics of spin one-half, Proc. R. Soc. A 391, 27 (1984). [35] B. Mukhopadhyay and S. K. Chakrabarti, Solution of Dirac equation around a spinning black hole, Nucl. Phys. B582, 627 (2000). [36] B. F. Schutz and C. M. Will, Black hole normal modes: A semianalytic approach, Astrophys. J. 291, L33 (1985). [37] E. Berti, A black-hole primer: Particles, waves, critical phenomena and superradiant instabilities, arXiv:1410 .4481.

084022-14