Integrability of the Dirac Equation on Backgrounds that are the Direct Product of Bidimensional Spaces

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by  Joás Venâncio

Integrability of the Dirac Equation on Backgrounds that are the Direct Product
of Bidimensional Spaces
Jo´as Venˆancio and Carlos Batista∗
Departamento de F´ısica, Universidade Federal de Pernambuco, Recife, Pernambuco 50740-560, Brazil
The field equation for a spin 1/2 massive charged particle propagating in spacetimes that are
the direct product of 2-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
7 topology R×S2×···×S2.
1
0 Keywords: DiracEquation,Separability,Spinors
2

n
I. INTRODUCTION versebymeanshigher-dimensionaltheories[14]. Aim-
a
J ingapplicationsonsomeofthesefields,inthepresent
7 Besidesthedetectionofgravitationalradiationand article we shall workout the separation of the Dirac
2 observation of the direct interaction between objects equationforamassivespin1/2chargedparticleprop-
via gravitation, the most natural and simple way to agatingintheblackholebackgrounddescribedinRef.
] probe the gravitational field permeating our space- [15], which is a static black hole whose horizontopol-
c
q time is by letting other fields interact with it. This is ogyisR S2 S2. Oneinterestingfeatureofthis
× ×···×
- the main reason why the study of scalar fields, spin black hole is that, in addition to the electric charge,
gr 1/2 fields and gauge fields (abelian and non-abelian) it has a magnetic charge, differently from the higher-
[ propagating in curved spacetimes plays a central role dimensionalgeneralizationofthe Reissner-Nordstrom
on the study of general relativity and any other the- solution[16], which only has electric charge. Thus, in
1
ory of gravity. Moreover, through the investigation spite of the static character of the black hole consid-
v
3 of these interactions one can test the stability of cer- ered here, the physics involved can be quite rich.
6 tain gravitational configurations, such as black holes One of the applications of the calculations per-
0 for example. Nevertheless, in a general spacetime it formedhereisonthestudyofthequasi-normalmodes
8 is quite difficult to integrate and even separate the of the Dirac field. The notion of quasi-normal modes
0
equation of motion for these fields. Luckily, some of and their spectrum are of great physical relevance,
.
1 themostimportantspacetimesareendowedwithgeo- inasmuch as these are the modes that survive for
0 metricalstructuresthatallowthe integrationofthese a longer time when a background is perturbed and,
7 field equations. For instance, this is the case for the therefore,these are the configurationsthat are gener-
1
Schwarzschild spacetime, that possesses four Killing allymeasuredbyexperiments[17–19]. Therefore,this
:
v vector fields, and the Kerr spacetime, which has two theme acquired even greater importance after the re-
i Killing vectors and one Killing-Yano tensor [1, 2]. In- cent measurement of gravitational radiation [20]. In
X
deed, in Refs. [3, 4] the separation and asymptotic addition, the quasi-normal modes are of great rele-
r
a integration of some of the mentioned field equations vance for studying the stability of certain solutions
have been attained in the Schwarzschild background, [21]. Another interesting application of the results
whileinRefs. [5–8]theproblemistackledintheKerr presentedinthesequelisontheinvestigationofsuper-
spacetime. Many of these results are summarized by radiance phenomena for the spin 1/2 field. Although
the celebrated Teukolsky master equation, which is bosonic fields like scalar, electromagnetic and gravi-
the radialpartof the equationofmotion for a field of tational fields can exhibit superradiant behaviour in
genericspinpropagatinginthefour-dimensionalKerr four-dimensionalKerrspacetime[22],curiously,thisis
background [9–11]. notthecasefortheDiracfield[23]. Thus,itwouldbe
In the past forty years, a great amount of research interesting to investigate whether an analogous thing
has been headed toward the investigation of higher- happens in the background considered here.
dimensionalspacetimes,speciallyduetotheirinterest Theoutlineofthepaperisthefollowing. SectionII
for String theory [12], which requires the spacetime sets the notation and the conventions used through-
to have 10 dimensions, andbecause of applications in outthearticle. Then,inSec. IIIweworkouthowthe
fieldtheorybymeansoftheAdS/CFTcorrespondence Diracoperatorbehavesunderaconformaltransforma-
[13]. Moreover, there are many other theories that tion of the metric, a result that will be of great value
tries to explainour currentunderstandingof the Uni- for the separation of the Dirac equation in the black
hole background. In Sec. IV we show that a general-
ized Dirac equation can be separated in a space that
is the direct product of 2-dimensional spaces. Sec. V
∗ [email protected],[email protected] uses the results obtained in the previous sections to
2
attain the separationof the Dirac equation in a black ω = ω , where indices inside square brackets
αβε α[βε]
hole background. Finally, inSec. VI we sumup what are anti-symmetrized. Analogously, indices enclosed
have been done and digress about future applications by round brackets are assumed to be symmetrized.
of this research. The Dirac matrices γ are 2n 2n matrices obeying
α
×
the Clifford algebra,
1
II. NOTATION AND CONVENTIONS γ γ = (γ γ + γ γ ) = δ 1, (1)
(α β) 2 α β β α αβ
In what follows, we shall deal with an even- with 1 standing for the 2n 2n identity matrix. The
dimensionalmanifoldM endowedwithametricgand covariant derivative of a s×pinorial field ψ is, then,
a torsion-free connection that is compatible with given by
∇
the metric. Regarding the signature, in general, we
shallnotrestrictourselvestoaspecificchoiceofsigna- 1
ψ = ∂ ψ ω βεγ γ ψ,
ture,althoughourmainmotivationaretheLorentzian ∇α α − 4 α β ε
spacetimes. Thedimensionwillbedenotedbyd=2n.
with ∂ denoting the partialderivative along the vec-
Our indices conventions are: the Greek letters from α
tor field e . Given an orthonormal frame e , we
the middle of the alphabet (µ, ν) are coordinate in- α α
can define the dual frame of 1-forms Eα ,{wh}ich is
dices and range from 1 to 2n; the Greek letters from
{ }
defined to be such that
the beginning of the alphabet (α, β, ε) run from 1 to
2nandlabelthevectorfieldsofanorthonormalframe
Eα(e ) = δα .
e ; lowercaseLatin indices with and without tildes β β
α
{ }
(a, b, a˜, ˜b) range from 1 to n and are also used to la- Thus,if xµ isalocalcoordinatesysteminourman-
bel the vector fields of an orthonormal frame, but in { }
ifold M, the line element can be written as
a pairwise form e ,e , which will be quite suitable
a a˜
{ }
toourintent,aswillbeclearinthesequel;theindices ds2 = g dxµdxν = δ EαEβ = EαE ,
µν αβ α
(ℓ, ℓ˜) run from 2 to n and serve to label the angular
directionsoftheblackholespacetimeconsideredhere; whereg = g(∂ ,∂ )arethecomponentsofthemet-
µν µ ν
finally, the indices (s, s , s , ) can take the values ric in the coordinate frame.
1 2
···
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, III. CONFORMAL TRANSFORMATION
oneupandtheotherdown. Butequalindicesthatare AND THE DIRAC OPERATOR
in the same position, both up or both down, should
notbe summedinprinciple,unlessanexplicitsymbol In this section we shall obtain the conformaltrans-
of sum is included. formation of the Dirac operator, which will be of fu-
By an orthonormalframe we mean that ture relevance in order to simplify the Dirac equation
in the spacetime of our interest.
g(ea,eb) = δab Letgˆ be ametricthatis conformallyrelatedto our
g(eα,eβ) = δαβ ↔ g(ea,e˜b) = 0 , initial metric, gˆ = Ω2g, with Ω being a positive defi-
g(ea˜,e˜b) = δa˜˜b nite function throughout the manifold. Then, if eˆα
{ }
is an orthonormalframe with respect to the metric gˆ
whereithas beenusedthe factthatthe indices aand we have
b can be thought as labeling the first n vector fields
of the orthonormal frame {eα}, while a˜ and ˜b label Eˆα = ΩEα and eˆα = Ω−1eα. (2)
the remaining n vectors of the frame e . If the
α
{ }
signature is not Euclidean some of the vector fields Since we are dealing with metric-compatible connec-
e might be imaginary in order for the frame be or- tions, a change of metric lead to a different spin con-
α
thonormal. The derivatives of the frame vector fields nection, which is defined by the following relation
determinethespinconnectionaccordingtothefollow-
ing relation ˆ eˆ = ωˆ εeˆ . (3)
∇α β αβ ε
e = ω εe . Using the identity gˆ = Ω2g to find the relation
∇α β αβ ε µν µν
between the Levi-Civita symbols of the two metrics
Indices of the spin connection are raised and low- and using Eqs. (2) and (3), we eventually arrive at
ered with δαβ and δαβ respectively, so that frame in- the following expression:
dices can be raised and lowered unpunished. Since
the metric is covariantly constant, it follows that ωˆ βε = Ω−1ω βε + 2Ω−2∂[βΩ δε] . (4)
α α α
3
Now, let us obtain how the Dirac operator, defined Thus, for example, ωˆ = 0 and ωˆ = 0 whenever
aa˜b aab
by D = γα , behaves under conformal transforma- a = b. Furthermore, the non-zero components of the
α
∇ 6
tions. If ψ is a spinorial field, let us define spinconnectionassociatedto the index a depend just
on the coordinates xa and ya. For instance, ωˆ is a
ψˆ = Ωpψ, 11˜1
function that depends just on x1 and y1.
withpbeing aconstantparameterthatwillbe conve- Inordertoaccomplishthe separabilityofthe Dirac
niently chosen in the sequel. Then, using Eq. (4) we equation, it is necessary to use a suitable representa-
eventually obtain the identity below: tionforthe Dirac matrices. Inwhatfollows,the 2 2
identity matrix will be denoted by I, while the us×ual
Dˆψˆ = γα ˆ (Ωpψ)
∇α notationforthePaulimatricesisgoingtobeadopted:
1
= Ωp−1Dψ + p+n Ωp−2(∂ Ω)γαψ.
α
− 2 0 1 0 i 1 0
(cid:18) (cid:19) σ1 = 1 0 , σ2 = i −0 , σ3 = 0 1 .
Thus, choosing p to be 1 n, it follows that (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) − (cid:21)
2 −
Dψ = Ω(n+21)Dˆψˆ, where ψˆ = Ω(12−n)ψ. (5) Usingthisnotation,aconvenientrepresentationofthe
Dirac matrices in 2n dimensions is the following:
In particular, this relation enables to investigate the
conformal invariance of the Dirac equation. Indeed, γ = σ σ σ I I,
if ψ is a spinorial field of mass m that obeys Dirac a 3⊗···⊗ 3⊗ 1⊗ ⊗···⊗
equation in the spacetime with metric g then (a−1)times (n−a)times
Dψ = mψ Dˆψˆ = (Ω−1m)ψˆ. (6) | {z } | {z } (9)
⇒ γ = σ σ σ I I .
a˜ 3 3 2
Since, generally, Ω is a non-constant function, it fol- ⊗···⊗ ⊗ ⊗ ⊗···⊗
lows that the massive Dirac equation is not confor- (a−1)times (n−a)times
mally invariant, whereas the massless Dirac equation | {z } | {z }
Indeed, we can easily check that the Clifford algebra
isinvariantunderconformaltransformations. Inspite
given in Eq. (1) is properly satisfied by the above
of the lack of conformal invariance of the Dirac equa-
matrices. Regarding the spinors, it is useful to define
tion with mass, Eq. (6) will be of great help for the
the following column vectors:
separation of the Dirac equation in black hole the
spacetime considered in this work.
1 0
ξ+ = and ξ− = . (10)
0 1
(cid:20) (cid:21) (cid:20) (cid:21)
IV. DIRECT PRODUCT SPACES AND THE
SEPARABILITY OF THE DIRAC EQUATION If we assume that the index s can take the values
“+1” and “ 1”, the action of the Pauli matrices on
−
The goal of the present section is to show that the the above column vectors can be summarized quite
Dirac equation minimally coupled to an electromag- concisely as
netic field is separable in spaces that are the direct
product of bidimensional spaces. σ1ξs = ξ−s , σ2ξs = isξ−s , σ3ξs = sξs. (11)
Let (M,gˆ) be a 2n-dimensional space that is
the direct product of n bidimensional spaces, Thespinorspace,inwhichtheDiracmatricesact,can
namely the space can be covered by coordinates be spanned by the direct product of the elements ξs
x1,y1,x2,y2, ,xn,yn such that the line element n times. More precisely, a general spinor field can be
{ ··· }
is given by written as
n n
dsˆ2 = dsˆ2 = (EˆaEˆa + Eˆa˜Eˆa˜), (7) ψˆ = ψˆs1s2···sn ξs1 ξs2 ξsn, (12)
a ⊗ ⊗···⊗
Xa=1 Xa=1 X{s}
where the 2-dimensional line elements dsˆ2 and the 1-
forms Eˆa andEˆa˜ depend just on the twoacoordinates in which ψˆs1s2···sn stands for the components of the
spinor and the sum over s means the sum over all
corresponding to their bidimensional spaces. For in- { }
possiblevaluesoftheset s ,s , ,s . Since,every
stance, dsˆ2, Eˆ1 and Eˆ˜1 depend just on the differen- { 1 2 ··· n}
1 s can take two values, it follows that this sum com-
tials dx1 and dy1 and theirs components depend just a
prises 2n terms, which is the number of components
on the coordinates x1 and y1. In such a case, the
ofaspinorind=2ndimensions. Using this basis,we
only components of the spin connection that can be
can easily compute the action of the Dirac matrices
non-vanishing are
on the spinor field. Indeed, using Eqs. (9), (11) and
ωˆ = ωˆ and ωˆ = ωˆ . (8) (12) we have:
aaa˜ aa˜a a˜aa˜ a˜a˜a
− −
4
γ ψˆ = (s s s )ψˆs1s2···sn ξs1 ξs2 ξsa−1 ξ−sa ξsa+1 ξsn
a 1 2 a−1
··· ⊗ ⊗···⊗ ⊗ ⊗ ⊗···⊗
X{s}
= (s s s )s ψˆs1s2···sa−1(−sa)sa+1···sn ξs1 ξs2 ξsa−1 ξsa ξsa+1 ξsn.
1 2 a a
··· ⊗ ⊗···⊗ ⊗ ⊗ ⊗···⊗
X{s}
Where from the first to the second line we have changed the index s to s , which does not change the final
a a
−
result, since we are summing overallvalues ofs , which comprisethe same list ofthe values of s . Moreover,
a a
we have used that (s )2 =1. Analogously, we have: −
a
γ ψˆ = (s s s )(is )ψˆs1s2···sn ξs1 ξs2 ξsa−1 ξ−sa ξsa+1 ξsn
a˜ 1 2 a−1 a
··· ⊗ ⊗···⊗ ⊗ ⊗ ⊗···⊗
X{s}
= i (s s s )ψˆs1s2···sa−1(−sa)sa+1···sn ξs1 ξs2 ξsa−1 ξsa ξsa+1 ξsn
1 2 a
− ··· ⊗ ⊗···⊗ ⊗ ⊗ ⊗···⊗
X{s}
Now, we have the tools to try to separate the general equation
Dˆ (Aˆ γa+Aˆ γa˜) ψˆ = mˆ ψˆ (13)
a a˜
−
h i
in its 2-dimensionalblocks,where Dˆ is the Dirac operatorof the space with metric (7) while Aˆ , Aˆ and mˆ are
a a˜
arbitrary functions. In order to accomplish this goal, we shall assume that the components of the spinor field
(12) take the separable form
ψˆs1s2···sn = ψˆs1(x1,y1)ψˆs2(x2,y2) ψˆsn(xn,yn). (14)
1 2 ··· n
UsingthishypothesisandnotingthattheDiracoperatorisDˆ =γaˆ + γa˜ˆ ,itfollowsthatEq. (13)isgiven
a a˜
∇ ∇
by:
n
(s s s )ψˆs1 ψˆsa−1ψˆsa+1 ψˆsn
1 2··· a 1 ··· a−1 a+1 ··· n
Xa=1X{s}
1 1
is ∂ˆ + ωˆ Aˆ + ∂ˆ + ωˆ Aˆ ψˆ(−sa)ξs1 ξs2 ξsn =
a a 2 a˜aa˜− a a˜ 2 aa˜a− a˜ a ⊗ ⊗···⊗
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
imˆ ψˆs1ψˆs2 ψˆsnξs1 ξs2 ξsn,
1 2 ··· n ⊗ ⊗···⊗
X{s}
where by ∂ˆ and ∂ˆ we mean the derivatives along the vector fields eˆ and eˆ , namely (eˆ )µ∂ and (eˆ )µ∂
a a˜ a a˜ a µ a˜ µ
respectively. In order for the latter equation to be separable in blocks depending only on the coordinates
x1, y1 , x2, y2 and so on, the functions Aˆa and Aˆa˜ must depend only on the two coordinates xa, ya and
{ } { } { }
the function mˆ must be a sum of functions depending on these pairs of coordinates:
n
Aˆ = Aˆ (xa,ya), Aˆ = Aˆ (xa,ya), mˆ = mˆ (xa,ya). (15)
a a a˜ a˜ a
a=1
X
With these necessary assumptions for attaining separability, we are left with the following equation:
n
1
(s s s ) D/saψˆ(−sa) imˆ = 0, (16)
1 2··· a ψˆsa a a − a
a=1 (cid:20) a (cid:21)
X
where the operator D/sa used above is defined by
a
1 1
D/sa =is ∂ˆ + ωˆ Aˆ + ∂ˆ + ωˆ Aˆ . (17)
a a a 2 a˜aa˜− a a˜ 2 aa˜a− a˜
(cid:18) (cid:19) (cid:18) (cid:19)
Since each term in the sum over a in Eq. (16) de- pends just on the two coordinates xa, ya , it follows
{ }
5
that each of these terms in the sum must be a con- depend on s and on s . But, since, in principle,
a−1 a+1
stant, otherwise they could not sum to zero. Let us mˆ is non-constant, we cannot say that the parame-
a
denote these separation constants by iη . Thus, Eq. ters κsa are constant for a 2. Nevertheless, taking
a a ≥
(16) requires that η = 0. However, it is worth the derivative of both sides of the latter equation, we
a a
noting that Eq. (16) provides not only one equation have
but rather a total oPf 2n independent equations, since
∂ mˆ = (s s s )∂ κsa if a 2.
for each choice of s s1,s2, ,sn we have one µ a 1 2··· a µ a ≥
{ } ≡ { ··· }
equation. Foreachoftheseequationswecanhavedif-
Since the left hand side of the latter equation does
ferentseparationconstants. Therefore,ηa candepend notdepend on s ,it followsthat ∂ κsa must vanish,
onthe choice of s ,so that it is appropriateto write { } µ a
which, in its turn, implies that mˆ should be con-
{ } a
these separation constants as iηa{s}. Then, we arrive stants for a 2. But, if mˆ2(x2,y2), mˆ3(x3,y3), ,
≥ ···
at the following equations: mˆ (xn,yn) are constants we can, without loss of gen-
n
erality, make all of them zero and absorb these con-
(s1s2···sa)D/saaψˆa(−sa) = i(mˆa + ηa{s})ψˆasa. (18) stants in mˆ1(x1,y1). Therefore, we can say that a
consistent separability process requires that
These equations enable us to integrate the fields ψˆsa
a
and, therefore, find the solutions for the generalized mˆ2 = mˆ3 = = mˆn = 0. (23)
···
Dirac equation (13). Although these equations are
Assumingthisrequirementtohold,Eqs. (20)and(22)
first order differential equations, they are coupled in
immediately lead to
pairs, namely the equations involving the field ψˆ+
a
have ψˆa− as source and vice-versa. Therefore, after ηa{s} = (s1s2···sa)κsaa, (24)
unraveling this system, we are left with a decoupled
where, for each a, κsa is a pair of constants. Thus,
secondorderdifferentialequationfor eachcomponent a
the constraint (19) now writes as
ψˆsa,thusachievingtheseparabilitythatwewerelook-
a
ing for. n
Note that, in accordance with Eq. (16), the sepa- (s1s2···sa)κsaa = 0. (25)
ration constants must obey a=1
X
Note that this equation must hold for all possible
n
η{s} = 0. (19) choices of s . Now, let us manipulate this equation
a in order to{s}olve such constraint. Isolating κs1, we
a=1 1
X have
Since the collective “index” s can take 2n values,
n
{ }
the latter equation comprise 2n constraints. Let us κs1 = (s s s )κsa.
unravel these constraints. Note that, since the left 1 − 2 3··· a a
a=2
X
hand side of Eq. (18) is independent of s , s ,
a+1 a+2
Note that the left hand side depends just on s .
, s , it follows that η{s} cannot depend on these 1
··· n a Therefore, the sum on right hand can depend just on
{s}
indices. Inparticular,η1 dependsjustons1,sothat s1. However, since none of the terms in the sum de-
we can write pend on s we conclude that this sum is a constant,
1
namely:
η{s} = s κs1, (20)
1 1 1
n
where κs11 is a pair of constants that depends just on κs11 = −c1 and (s2s3···sa)κsaa = c1,
s1. Since (sa)2 =1,itfollowsthatfromEq. (18)that Xa=2
we can write
where c is a constant that does not depend on s .
1
{ }
1 Now, the latter equation can be written as
D/saψˆ(−sa) = i(s s s )(mˆ + η{s}). (21)
ψˆsa a a 1 2··· a a a n
a
κs2 s c = (s s )κsa.
Inasmuch as the left hand side of this equation de- 2 − 2 1 − 3··· a a
a=3
X
pends just on s , it follows that the right hand side
a
Following the same reasoning that we have just used,
of this equation should, likewise, depend just on s .
a
the left hand side of the latter equation depends just
Therefore, we have
on s while the terms on the right hand side clearly
2
mˆa + ηa{s} = (s1s2···sa)κsaa if a≥2, (22) do not depend on s2, we can thus conclude that
n
wdehpeerne,dsfojrusetaochn sa,.κTsaahuiss,afopraiinrstoafnpcea,raκmsaetdeorsesthnaott κs22 = s2c1 − c2 and (s3···sa)κsaa = c2,
a a a=3
X
6
where c is a constant that does not depend on s . Manipulating these two equations one easily obtain
2
Followingthe same procedureuntil reachingthe t{erm} second order differential equations for ψˆsa, achieving
a
κsn, we end up with the following final result that theseparabilityofthegeneralizedDiracequationthat
n
solves the constraint (25): we are looking for. For appropriate boundary condi-
tions, the constants c , ,c can only take dis-
1 n−1
κsa = s c c with c = c = 0. (26) cretevalues. Thegen{eral·s·o·lution}ofEq. (13)is,then,
a a a−1 − a 0 n
a linear combination of the particular solutions for
Concerning the constants c1, c2, ···, cn−1, they are each of the possible “eigenvalues” {c1,··· ,cn−1}. In
arbitrary. Thus, in this problem we have (n 1) sep- thenextsection,weshallusetheseresultstoseparate
−
aration constants. Finally, inserting Eqs. (23), (24) the Dirac equation in some black hole spacetimes.
and(26)intoEq. (18)wearriveatthefollowingequa-
tions:
V. BLACK HOLE SPACETIMES
D/s1ψˆ(−s1) = i(s mˆ c )ψˆs1
1 1 1 1− 1 1
Inthis sectionwe shallseparate the Dirac equation
(27)
corresponding to a massive and electrically charged
D/saaψˆa(−sa) = i(saca−1−ca)ψˆasa , if a≥2, field of spin 1/2 on the background of black holes
whosehorizonshavetopologyR S2 S2. These
where the operators D/sa were defined in Eq. (17). black hole solutions, that posse×ss e×lec·t·r·i×c and mag-
a
The abovesetofequationsprovidestwoequationsfor netic charge, have been obtained in Refs. [15, 24, 25]
each a, one for s = +1 and the other for s = 1. and are given by
a a
−
dr2 n
ds2 = f(r)2dt2+ +r2 (dθ2+sin2θ dφ2), (28)
− f(r)2 ℓ ℓ ℓ
ℓ=2
X
where f =f(r) is the following function of the coordinate r:
1 2M Q2(d 3) Q2 Λr2
f(r)= + e − m . (29)
sd 3 − rd−3 2(d 2)r2(d−3) − 4(d 5)r2 − d 1
− − − −
In the latter expression, d = 2n is the dimension of where, as explained earlier, the index ℓ ranges from 2
the spacetime, M, Q and Q are the mass,the elec- to n. Using this frame, the line element is given by
e m
tric charge and the magnetic charge of the black hole
n
respectively, while Λ is the cosmologicalconstant, see
ds2 = (EaEa + Ea˜Ea˜).
Ref. [15] for details. This spacetime is the solution of
a=1
Einstein-Maxwell equations with a cosmological con- X
stant Λ and electromagnetic field F =dA, where the In its turn, the gauge field can be written as
gauge field A is given by
A = A Ea + A Ea˜,
a a˜
where
n
Q
A= rd−e3 dt + Qm cosθℓdφℓ. A = −iQe , A = Qm cotθ , A =0. (30)
Xℓ=2 1 frd−3 ℓ r ℓ a˜
A field of spin 1/2 with electric charge q and mass
A suitable orthonormal frame for such a spacetime is m minimally coupledto the electromagneticfield and
given by propagatinginthisspacetimeobeysthefollowingver-
sion of the Dirac equation:
γα( iqA )ψ = mψ. (31)
E1 = ifdt, E˜1 = f−1dr, ∇α− α
UsingthedefinitionoftheDiracoperator,D =γα
Eℓ = r sinθℓdφℓ , Eℓ˜ = rdθℓ, along with the fact that Aa˜ = 0, it follows that ∇thαe

7
above equation is written as Thenon-vanishingcomponentsofthespinconnection
are
ωˆ = ωˆ =rf′ f,
Dψ = (m + iqA γa)ψ. (32) 1˜11 − 11˜1 −
a (35)

ωˆ = ωˆ =cotθ ,
 ℓℓ˜ℓ − ℓℓℓ˜ ℓ
The aim of the present section is to integrate this 
wheref′ standsforthederivativeoff withrespectto
equation.
its variable r. Using the conformal transformation of
In Sec. IV, we have been able so separate an anal- the Dirac operator,obtained in Sec. III,we canwrite
ogous equation for spaces that are the direct product the field equation (32) in terms of an equation in the
of bidimensional spaces. However, the black hole line space with line element dsˆ2, so that the separability
element (28) is not of this special type, due to the results of Sec. IV can be fully used. Indeed, defining
warping factor r2 in front of the angular part of the
metric. Nevertheless, the conformal transformation ψˆ = Ω(12−n)ψ = r(n−12)ψ, (36)
and using Eq. (5), it follows that the field equation
(32) can be written as
ds2 dsˆ2 = Ω2ds2 with Ω=r−1
−→
Dˆψˆ = Ω−1(m + iqA γa)ψˆ. (37)
a
leadus to the followingline elementthat is the direct Then, defining
product of bidimensional spaces:
mˆ =mˆ =rm , Aˆ =iqrA , Aˆ =0, (38)
1 a a a˜
it follows that Eq. (37) takes exactly the form of the
f2 dr2 n
dsˆ2 = dt2+ + (dθ2+sin2θ dφ2). (33) equation studied in Sec. IV, namely we obtain Eq.
−r2 (rf)2 ℓ ℓ ℓ (13). Moreover, and foremost, defining the coordi-
ℓ=2
X nates
x1 =t, y1 =r , xℓ =φ , yℓ =θ , (39)
A suitable orthonormal frame for this space is given it follows that the function mˆ andℓ the gaugℓe field Aˆ
by α
are exactly of the form necessary to attain separabil-
ity, namely the constraints (15) and (23) are obeyed.
Therefore, due to Eqs. (12), (14) and (36), it follows
Eˆ1 = ifr−1dt, Eˆ˜1 = (rf)−1dr, that a solution for Eq. (31) in the black hole back-
(34) ground is provided by

Eˆℓ = sinθℓdφℓ , Eˆℓ˜ = dθℓ.

ψ = r(21−n) ψ1s1(t,r)ψ2s2(φ2,θ2)···ψnsn(φn,θn)ξs1 ⊗ξs2 ⊗···⊗ξsn. (40)
X{s}
From Eqs. (17), (27), (30) and (34)-(38), it follows that the functions ψsa must be solutions of the following
a
differential equations:
r qQ 1
is ∂ e + rf∂ + (rf′ f) ψ(−s1) = i(s rm c )ψs1
1 if t− frd−4 r 2 − 1 1 − 1 1
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
(41)
1 1
is ∂ iqQ cotθ + ∂ + cotθ ψ(−sℓ) = i(s c c )ψsℓ.
ℓ sinθ φℓ − m ℓ θℓ 2 ℓ ℓ ℓ ℓ−1− ℓ ℓ
(cid:20) (cid:18) ℓ (cid:19) (cid:18) (cid:19)(cid:21)
As explained in Sec. IV, the constants c , c , , c are separation constants that generally take discrete
1 2 n−1
···
values onceboundaryconditionsandregularityrequirementsareimposed. The constantc , onthe other hand,
n
is zero. Note that the coefficients in the above equations do not depend on the coordinates t and φ , which
ℓ
stems from the fact that these are cyclic coordinates of the metric, so that ∂ and ∂ are killing vector fields
t φℓ
of both metrics g and gˆ. Therefore, it is convenient to decompose the dependence of the fields ψsa on these
a
8
coordinates in the Fourier basis, namely,
ψs1(t,r) = eiωtΨs1(r) , ψsℓ(φ ,θ ) = eiωℓφℓΨsℓ(θ ). (42)
1 1 ℓ ℓ ℓ ℓ ℓ
The final general solution for the field ψ must, then, include a “sum” over all values of the Fourier frequencies
ω and ω with arbitrary Fourier coefficients. While ω can be interpreted as related to the energy of the field,
ℓ
ω are related to angular momentum. Note that in order to avoid conical singularities in the spacetime, the
ℓ
coordinatesφ musthaveperiod2π,namelyφ andφ +2π shouldbeidentified. Asitiswell-known,aspin1/2
ℓ ℓ ℓ
field changes its sign after a 2π rotation, which implies that the angular frequencies ω must be half-integers
ℓ
[26, 27]:
1 3 5
ω = , , , . (43)
ℓ
±2 ±2 ±2 ···
Finally, inserting the decomposition (42) into Eq. (41), we end up with the following pairwise coupled system
of differential equations:
d 1 ωr qQ
rf + (rf′ f)+is e Ψ(−s1) = i(s mr c )Ψs1
dr 2 − 1 f − frd−4 1 1 − 1 1
(cid:20) (cid:18) (cid:19)(cid:21)
(44)
d 1 ω
+ cotθ s ℓ qQ cotθ Ψ(−sℓ) = i(s c c )Ψsℓ.
dθ 2 ℓ− ℓ sinθ − m ℓ ℓ ℓ ℓ−1− ℓ ℓ
(cid:20) ℓ (cid:18) ℓ (cid:19)(cid:21)
A. The angular part of Dirac’s Equation as follows:
c = λ2+λ2 + +λ2 . (46)
ℓ−1 ℓ ℓ+1 ··· n
Now, we shall investigate a little further the above
q
equations. Let us start with the angular part of the Then, defining the parameter
equations, namely the equations for Ψsℓ. One can
ℓ
make a simplification on these equations by perform- ζ = arctanh(c /c ),
ℓ ℓ ℓ−1
ing a field redefinitionalongwith aredefinition ofthe
separation constants, as we show in the sequel. In- we find that
stead of using the n 1 separation constants c , c ,
1 2
, cn−1, we shall us−e the constants λ2, λ3, , λn, cℓ−1 = λℓcoshζℓ and cℓ = λℓsinhζℓ,
··· ···
defined by
so that the following relation holds:
s c c = s λ e−sℓζℓ.
ℓ ℓ−1 ℓ ℓ ℓ
−
λ c2 c2, (45) Thus, performing the field redefinition given by
ℓ ≡ ℓ−1 − ℓ
q Ψsℓ(θ) = esℓζℓ/2Φsℓ(θ), (47)
ℓ ℓ
where it is worth recalling that c = 0, by definition. it turns out that the angular part of Eq. (44) can be
n
Inverting these relations, we find that the old con- written in the following simpler way in terms of the
stants can be written in terms of the new constants fields Φsℓ(θ):
ℓ
d 1 ω
+ cotθ s ℓ qQ cotθ Φ(−sℓ) = is λ Φsℓ. (48)
dθ 2 ℓ− ℓ sinθ − m ℓ ℓ ℓ ℓ ℓ
(cid:20) ℓ (cid:18) ℓ (cid:19)(cid:21)
Although it may seem that we did not achieve much theblackholehasvanishingmagneticcharge,Q =0,
m
simplificationbytheredefinitionofthefieldsandsepa-
rationconstants,itturnsoutthatinthecaseinwhich
9
these equations reduce to [26, 31]:
λ = 1, 2, 3, .
ℓ
± ± ± ···
d 1 s ω Solutions with non-integer eigenvalues are not well-
+ cotθ ℓ ℓ Φ(−sℓ) = is λ Φsℓ.
dθ 2 ℓ− sinθ ℓ ℓ ℓ ℓ defined on the whole sphere, while a vanishing eigen-
(cid:20) ℓ ℓ(cid:21) value is forbidden by the Lichnerowicz theorem [32],
since the sphere is a compact manifold with positive
curvature.
Inthe latter form,the angularequationsareidentical Regarding the general case in which the black hole
otopetrhaeteoqruiantitohneD2-Sd2iΦme=nsiioλnΦal,wunhietrespDhSe2rei.stThoeDchiercakc tmriaegdnteoticmcahkaeragereidsenfionnit-ivoannoisfhtihneg,fiQeldms6=Φsℓ0,bywemheaanves
ℓ
this claim, one should use the frame e1 = sinθdφ of a general linear combination of the fields Φ+ and
ℓ
and e2 = dθ along with the Dirac matrices γ1 = σ Φ−,withnon-constantcoefficients,inordertoconvert
1 ℓ
and γ2 = σ . The solutions of the eigenvalue equa- Eq. (48) into the eigenvalue equation D Φ = iλΦ.
2 S2
tion D Φ = iλΦ are well-known, the components However, it turns out that the coefficients of the lin-
S2
of the 2-component spinor Φ are written in terms of ear combination must obey fourth-order differential
Jacobipolynomials[26,28]. Fromageometricalpoint equations, whose solutions seem to be quite difficult
ofview,thesesolutionscanbe understoodintermsof to attain analytically. In spite of this, we can make
the Wigner elements ofthe groupSpin(R3), thatgive an important progress regarding the system of equa-
rise to the so-called spin weighted spherical harmon- tions(48)bydecouplingthefieldsΦ+ andΦ−,which,
ℓ ℓ
ics [29, 30], which are tensorial generalizations of the after all, is our goalat this paper. The final results is
spherical harmonics. Moreover, the allowed eigenval- that the fields Φsℓ satisfy the following second order
ℓ
uesλ arealsoknown,they mustbenon-zerointegers differential equation:
ℓ
1 d dΦsℓ (1+2qQ )ω cosθ 1+2qQ +2ω2 (1 4q2Q2 ) cos2θ
sinθ ℓ + m ℓ ℓ m ℓ + − m ℓ λ2 Φsℓ = 0.
sinθℓdθℓ (cid:18) ℓ dθℓ (cid:19) (cid:20) sin2θℓ − 2 sin2θℓ 4sin2θℓ − ℓ(cid:21) ℓ
(49)
It is worth stressing that the latter equation must be B. The radial part of Dirac’s Equation
supplemented by the requirement of regularity of the
fields Φsℓℓ at the points θℓ =0 and θℓ = π, where our Inordertosolvethepairofradialequationsin(44),
coordinatesystembreaksdown. Theseregularitycon- we should first decouple the fields Ψ+ and Ψ−. This
1 1
ditions transform the task of solving the latter equa- can be easily attained by defining
tioninaSturm-Liouvilleproblem,sothatthepossible
values assumedby the separationconstantsλ forma 1 1 ωr qQ
ℓ B (r) = (rf′ f) is e ,
discrete set. Since the case Qm =0 in Eq. (49) has a s1 rf 2 − − 1 f − frd−4
knownsolution,asdescribedabove,itfollowsthatwe (cid:20) (cid:18) (cid:19)(cid:21)
i
canlookforsolutionsforthecaseQ =0bymeansof C (r) = (s mr+c ),
m 6 s1 −rf 1 1
perturbation methods, with Q being the perturba-
m
tion parameter. Indeed, in the celebrated paper [10],
in terms of which the radial equation in (44) can be
asimilarpathhasbeentakenbyPressandTeukolsky
written as
in order find the solutions and their eigenvalues for
the angular part of the equations of motion for fields d
Ψs1 = B Ψs1 + C Ψ−s1. (50)
with arbitrary spin on Kerr spacetime, in which case dr 1 − s1 1 s1 1
theangularmomentumoftheblackholewastheorder
Then, deriving this equation with respect to r and
parameter. In this respect, see also Ref. [33].
usingEq. (50) to substitute Ψ−s1 in terms ofΨs1, we
1 1
are eventually led to the following decoupled second
order differential equation:
d2Ψs1 1 dC dΨs1 dB
1 + B +B s1 1 +B Ψs1 + s1 B2 C C Ψs1 = 0. (51)
dr2 s1 −s1 − C dr dr s1 1 dr − s1 − s1 −s1 1
(cid:18) s1 (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
10
Ananalyticalexactsolutionofthelatterdifferential the functions h andh arethe followingwhend 6:
0 1
≥
equation is, probably, out of reach. Nevertheless, we
can use Eq. (51) to infer the asymptotic forms of the hs1(r)= (d 3)2ω2 (d 3)m2 +
0 − − −
solution near the infinity, r , as well as near the
3 iωc
horizon r r , where r →is a∞root of the function (cid:2)+(d 3) c2 1 (cid:3)
→ ⋆ ⋆ 4 − 1− m −
f, namely f(r ) = 0. Particularly, we shall prove in (cid:20) (cid:18) (cid:19)
⋆
the sequel that, in the case of vanishing cosmological (d−3)2Q2mm2 + (d−3)3Q2mω2 1 +O 1 ,
constant, the well-known case d = 4 is qualitatively 4(d 5) 2(d 5) r2 r3
− − (cid:21) (cid:18) (cid:19)
different from the higher-dimensional cases d 6. In
≥
order to do this analysis, we shall write Eq. (51) as
1 s c 1
hs1(r)= 1 1 +O .
1 − r − mr2 r3
d2Ψs1 dΨs1 (cid:18) (cid:19)
1 + hs1(r) 1 + hs1(r)Ψs1 = 0, (52)
dr2 1 dr 0 1 However, these formulas do not apply to the well-
studiedsituationd=4,inwhichcasehs1 hasatermof
wherehs11 andhs01 aredefinedbycomparingEqs. (51) orderr−1 depending onM andQe and0the coefficient
and(52). Then,wecanworkouttheasymptoticforms of order r−2 has additional terms also depending on
of the coefficients hs11 and hs01 in the region of inter- M and Qe . Analogously, for d = 4, the function hs11
est. Inparticular,fora consistentinvestigationofthe also has additional contributions of order r−2 stem-
asymptotic form of the solutions of Eq. (52) in the ming from the mass and the electric charge of the
limit r →∞, if we wantto know Ψs11 up to order r−p black hole. Thus, we conclude that, for Λ = 0, the
weneedtoknowhs11 uptoorderr−(p+1) andconsider spinor field that represents a charged particle of spin
hs1 up to order r−(p+2). 1/2 moving in the black hole (28) has qualitatively
0
Looking at the function f(r) in Eq. (29), we see different fall off properties in the asymptotic infinity
that the term that multiplies Λ becomes the domi- depending on whether d=4 or d 6.
≥
nant one as we approach the infinity, r . There- A similar asymptotic analysis can be performed
→∞
fore,itisintuitivetoguessthatthe casesofvanishing near the horizons, namely near the values of r for
andnon-vanishingΛshouldbe qualitativelydifferent. which the function f vanishes. In such a case the co-
Thus, let us separate the analysis of these two cases. ordinate r ceases to be reliable and we should change
First, let us consider the case Λ = 0. Collecting the radial coordinate to tortoise-like coordinates, see
the coefficients that multiply dΨs11 6and Ψs1 in Eq. [10]forinstance. Besidessuchasymptoticbehaviours,
dr 1 one can also look for approximate solutions valid in
(51) and then expanding them in powers of r−1, we
a broader domain by means of other approximation
canfind, aftersome algebra,the followingasymptotic
methods. Forinstance,inRef.[34]anapproximateso-
forms:
lution for the Dirac field on the Kerr spacetime has
been obtained using the WKB method after trans-
(d 1)m2 i(d 1)s ω 1
hs1(r)= − + − 1 +O , forming the radial second order differential equation
0 Λr2 Λr3 r4
(cid:18) (cid:19) into a Schr¨odinger equation, see also [35, 36].
1 s c 2(d 1) c2 1 1
hs11(r)= r − m1r21 + (d −3)Λ − m12 r3 +O r4 . VI. CONCLUSIONS AND PERSPECTIVES
(cid:20) − (cid:21) (cid:18) (cid:19)
In particular, considering the expansion of hs1 up to In this article we have shown that the Dirac Equa-
0
order r−2 and the expansion of hs1 up to order r−1, tioncoupledtoagaugefieldcanbedecoupledineven-
1
dimensional manifolds that are the direct product of
we are led to the following asymptotic form:
bidimensional spaces, provided that the gauge field
is also “separated” in accordance with the bidimen-
m√d 1 1
Ψs1(r) C sin − log(r) + ϕ +O , sional blocks, as shown in Eq. (15). Then, we have
1 ∼ 0 (cid:20) √Λ 0(cid:21) (cid:18)r(cid:19) used this fact along with the conformal transforma-
tionofthe Diracoperatortodecouple the equationof
where C0 and ϕ0 are arbitrary integration constants. motionof a chargedtest field of spin 1/2propagating
Note, however,that the field that is “the solution” of inthe backgroundofthe blackhole solution(28). We
the Dirac equation, ψ, has a further decaying multi- have shown that the latter problem reduces to solv-
plicative factor r(1−d)/2, in accordancewith Eq. (40). ing a second order radial differential equation, whose
Now, let us consider the context of vanishing cos- asymptoticbehaviourhasbeenworkedout,alongwith
mological constant, Λ = 0. In such a case, one can asecondorderangulardifferentialequationwithregu-
see, after some algebra, that the asymptotic forms of larityconditions. Inparticular,wehavearguedthatif