Solution of Dirac equation in the near horizon geometry of an extreme Kerr black hole

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Solution of the Dirac equation in the near horizon geometry of an extreme Kerr black hole

I. Sakalli*and M. Halilsoy†

Physics Department, EMU G.Magosa, Mersin 10, Turkey

共Received 15 January 2004; published 14 June 2004兲

The Dirac equation is solved in the near horizon limit geometry of an extreme Kerr black hole. We decouple equations first, as usual, into an axial and angular part. The axial equation turns out to be independent of the mass and is solved exactly. The angular equation reduces, in the massless case, to a confluent Heun equation. In general for a nonzero mass, the angular equation is expressible, at best, as a set of coupled first order differential equations apt for numerical investigation. The axial potentials corresponding to the associated Schro¨dinger-type equations and their conserved currents are found. Finally, based on our solution, we verify in a similar way the absence of superradiance for Dirac particles in the near horizon, a result which is well known within the context of a general Kerr background.

DOI: 10.1103/PhysRevD.69.124012 PACS number共s兲: 04.20.Jb I. INTRODUCTION

During the last three decades the study of spin-1

2 particles

on type-D spacetimes has attracted much interest and by now accumulated results are already available in the literature 关1–3兴共and references cited therein兲. The main reason for this is that all well-known black holes共BHs兲 are in this category and their better understanding involves a detailed analysis of various physical fields in their vicinity. Dirac particles with 共and without兲 mass constitute one such potential candidate whose interaction and behavior around BHs may reveal in-formation of much significance. Tests of the Dirac equation in spacetimes other than BHs have also started to arouse interest for various reasons. From this token we cite the Robertson-Walker and Bertotti-Robinson 共BR兲 spacetimes 关4–6兴. The latter in particular has already gained much rec-ognition in connection with extremal BHs, higher dimen-sions, and the brane world.

In this paper we consider the Dirac equation in the near horizon geometry of an extreme Kerr BH. This constitutes the most important region of the outerworld prior to the ho-rizon of a Kerr BH enhanced with the extremality condition. Extremal BHs are believed to have connection with the ground states of quantum gravity. This alone justifies, in spite of the absence of backreaction effects, the importance of spin-12 particles on such backgrounds. The throat

geom-etry is a completely regular vacuum solution with an en-hanced symmetry group SL(2,R)⫻U(1). In many aspects, this solution shares common features with the AdS2⫻S2 ge-ometry arising in the near horizon limit of extreme Reissner-Nordstro¨m BHs. The behavior of massless scalar fields in the extreme Kerr throat has been considered, and it is found that certain modes with large azimuthal quantum number exhibit superradiance 关7兴. This implies that the geodesics near the horizon can escape to infinity carrying energy-momentum more than the amount that infalls. Our solution enables us to investigate a similar phenomenon with Dirac fields which turns out to be negative as far as superradiance is conserved.

This result is in accordance with the treatment of the Dirac equation in the general Kerr background in the absence of an exact solution 关1兴. The analysis of fermions in the Kerr-Newman background and the absence of superradiance was shown first by Lee关3兴. Our aim is to reexamine this item— not only by separating equations and deducing results on general grounds—but rather obtaining exact solutions and employing them. The advantage of confining ourselves to the near horizon alone bears fruits by allowing solutions ex-pressible in terms of known polynomials. In the general background of the Kerr family of BHs even this much re-mains a problem beyond technical reach. Meanwhile, it is important to note that in contrast to the general Kerr metric, the throat metric is not asymptotically flat. This difference naturally shows itself in the potentials, too. In other words, the extreme Kerr throat metric represents such a local region that the geometry of interest is different than the geometry of the general Kerr metric. The steeply rising potential prevents any particle共or field兲 flow to infinity to make superradiance. Hence, we can mentally say that the behavior of particles in two geometries must be considered separately.

In order to separate equations we employ the well-known method due to Chandrasekhar so that we prefer to label the set of equations as Chandrasekhar-Dirac共CD兲 equations. We separate the y 共axial兲 and␪ 共angular兲 dependence in such a way that the resulting axial equation remains independent of mass. This leads us to an exact solution irrespective of mass. The angular equation on the other hand depends strictly on the mass. For the massless case 共which we refer to as a neutrino equation兲 the angular equation reduces to a conflu-ent Heun equation 关8兴. When the mass is nonzero, however, we cannot identify our equations but instead we express them as a set of linear equations suitable for numerical analy-sis.

The organization of the paper is as follows: in Sec. II we review the near horizon geometry of an extreme Kerr BH and separation of variables of the CD equation. Solution of the axial equation follows in Sec. III. The massless and mas-sive cases are discussed in Secs. IV and V, respectively. The reduction of our equations to one-dimensional Schro¨dinger-type equations with their conserved currents and superradi-*Email address: izzet.sakalli@emu.edu.tr

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ance is included in Sec. VI. The paper ends with a conclusion in Sec. VII.

II. EXTREME KERR THROAT GEOMETRY AND SEPARATION OF DIRAC EQUATION ON IT The extreme Kerr metric in the Boyer-Lindquist coordi-nates is given by ds2⫽e2␯d t˜2⫺e2␺共d␾˜⫹␻d t˜兲2⫺␳2

冉

dr ˜2 ⌬˜ ⫹d␪ 2

冊

_, _共1兲 where e2␯⫽ ⌬˜␳ 2 共r˜2_⫹M2_兲2_⫺⌬˜M2_sin2_␪, e2(␯⫹␺)⫽⌬˜ sin2␪, ⌬˜⫽共r˜⫺M 兲2_, ␻⫽2 M 2_˜e_r 2␯ ⌬˜␳2 , ␳2_⫽r˜2_⫹M2_cos2_␪_. _共2兲

In the extreme case, both the total mass M and the rotation parameter a become identical so that the angular momentum

J⫽M2 and the extremal horizon corresponds to r˜⫽M. The area of the horizon is A⫽8␲J.

To describe the near horizon 共or throat兲 limit of the ex-treme Kerr metric, due originally to Bardeen and Horowitz 关7兴, one can set

r ˜_⫽M⫹␭r, t˜⫽t

⬘

␭, ␾˜_⫽_␾_⫺ t

⬘

2␭M, 共3兲

and take the limit␭→0. In these new coordinates, the throat metric is obtained as ds2⫽F

冋

r 2 r0 2dt

⬘

2_⫺r0 2 r2dr 2_⫺r 0 2 d␪2

册

⫺r0 2_sin2_␪ F

冉

d␾⫹ r r₀2dt

⬘

冊

2 , 共4兲 where F⫽1⫹cos 2_␪ 2 , r₀2⫽2M2_. _共5兲

We set further, for simplicity, r₀2⫽1. This throat spacetime no longer has asymptotic flatness.

Finally, passing to more general coordinates,

y⫽ 1 2r关r 2_共1⫹t

_⬘

2_兲⫺1兴, cot t⫽ 1 2t

⬘

r关r 2_共1⫺t

_⬘

2_兲⫹1兴,

␾⫽␸⫹ln

冏

cos t⫹y sin t

1⫹t

⬘

r

冏

, 共6兲

we can write the throat metric 共4兲 as

ds2⫽F

冋

共1⫹y2兲dt2⫺ d y 2 1⫹y2⫺d␪ 2

册

_⫺sin 2_␪ F 共d␸⫹ydt兲 2_. 共7兲 The metric functions in Eq. 共7兲 depend only on the vari-able␪ and thus as expected in the search for a solution of the Dirac equation the angular equation forms the crux of the problem. The coordinates⫺⬁⬍t⬍⬁, ⫺⬁⬍y⬍⬁ cover the entire, singularity free spacetime. The Killing vector⳵/⳵t is

not timelike everywhere; it admits a region 共for sin2␪ ⬎0.536) in which it becomes spacelike. Therefore by a co-ordinate transformation this particular region is transform-able into the spacetime of colliding plane waves 关9兴. Re-cently, it has also been shown that the metric 共7兲 can be obtained as a solution to dilaton-axion gravity which is simi-lar to the rotating BR spacetime关10兴.

The singularity free character can best be seen by check-ing the Weyl scalar ⌿2 and the Kretschmann scalar:

⌿2⫽ 2

共1⫹cos2_␪_兲3关3 cos

2_␪_{⫺1⫹i cos}_␪_共cos2_␪_⫺3兲兴, 共8兲 R_␮␯␳␴R␮␯␳␴⫽ 192 sin 2_␪ 共1⫹cos2_␪_兲6关共1⫹cos 2_␪_兲2_{⫺16 cos}2_␪_兴. 共9兲 When the backreaction of the spin-12 test particles on the

background geometry is neglected, the Dirac field equation is given by the CD equations关1兴 on a fixed spacetime 共7兲.

We choose a complex null tetrad兵l,n,m,m¯其 that satisfies the orthogonality conditions l•n⫽⫺m•m¯⫽1. We note that, throughout the paper, an overbar denotes complex conjuga-tion. Thus the covariant one-forms can be written as

F sin␪⳵␸. 共11兲

One can determine the nonzero Newman-Penrose 共NP兲 complex spin coefficients 关11兴 in the above null tetrad as

␲⫽⫺␶⫽sin␪共cos␪⫺i兲 共2F兲3/2 ,

␧⫽␥⫽ y

2

冑

2F共1⫹y2兲 ,

␣⫽⫺␤⫽2 cot␪⫺i sin␪

2共2F兲3/2 . 共12兲 The CD equations in the NP formalism are then 关1兴

共D⫹␧⫺␳兲F1⫹共¯␦⫹␲⫺␣兲F2⫽i␮pG1, 共⌬⫹␮⫺␥兲F2⫹共␦⫹␤⫺␶兲F1⫽i␮pG2, 共D⫹␧¯⫺¯␳_兲G₂_⫺共_␦_⫹_␲¯_⫺_␣¯_兲G₁_⫽i_␮_p_F₂_,

共⌬⫹␮¯_⫺_␥¯_兲G

1⫺共¯␦⫹␤¯⫺¯␶兲G2⫽i␮pF1, 共13兲 where␮*⫽

冑

2␮p is the mass of the Dirac particle.

The form of the CD equations suggests that we introduce 关1,6兴

F1⫽ f1共y兲A1共␪兲ei(␴t⫹m␸),

G₁⫽g₁共y兲A₂共␪兲ei(␴t⫹m␸)_,

F2⫽ f2共y兲A3共␪兲ei(␴t⫹m␸),

G2⫽g2共y兲A4共␪兲ei(␴t⫹m␸), 共14兲 where ␴ is the frequency of the corresponding Compton wave of the Dirac particle and m is the azimuthal quantum number of the wave. Our convention is that ␴ is always positive.

¯ g1⫽⫺i␭g2, 共18兲

Z

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and

LA3⫹i␮*A2

冑

F⫽i␭A1, 共20兲

L⫹A1⫹i␮*A4

冑

F⫽i␭A3. 共21兲 It is clear from Eqs. 共18兲, 共19兲 that g₁⫽g¯₂.

III. SOLUTION OF THE AXIAL EQUATION If we decouple the axial equations共18兲, 共19兲 in Eq. 共18兲 to get g1, we obtain

Z ˜_共Z˜¯g

1兲⫽␭2g1. 共22兲

Similarly one can decouple the axial equations in Eq.共18兲 for g2. The explicit form of Eq.共22兲 can be obtained as

共1⫹y2_兲g 1

⬙

共y兲⫹2yg1

⬘

共y兲 ⫹ 1 1⫹y2

冉

1 2⫹␴ 2_⫹y 2 4 ⫺␭ 2_共1⫹y2_兲⫹i_␴_y

冊

_g 1共y兲⫽0. 共23兲共Throughout the paper, a prime denotes a derivative with respect to its argument.兲

Thus the solutions of the decoupled equations for g1, Eq. 共23兲, and g2 共not given here兲 can be found in terms of the associated Legendre functions as follows:

g1共y兲⫽c1P_␭⫺1/2␤ˆ 共iy兲, g2共y兲⫽c2P_␭⫺1/2␤ˆ 共⫺iy兲, 共24兲 where ␤ˆ_⫽

冑

_␴2_⫹␴ 2⫹ 1 4 共25兲

and c₁, c₂ are complex constants.

Here, as a result of the physical necessities, we considered only the first kind of the associated Legendre functions. Al-though solutions共24兲 seem like complex solutions, it is pos-sible to draw real functions from the above associated Leg-endre functions. We may define

␭⫽m˜_⫹1 2 with m˜⫽1,2,3, . . . 共26兲 and ␴⫽1 4共

冑

16n ˜2_⫺3⫺1兲 _共27兲 so that ␤ˆ_{⫽n˜ with n˜⫽⫺m}_{˜ ,}_⫺m˜_{⫹1, . . . ,⫺1,1, . . . ,m}˜_⫺1,m_{˜ .} 共28兲

In order to get the real functions for solutions 共22兲, the re-quired condition is m˜⫺兩n˜兩⫽even number.

It is worth also drawing attention to the following re-marks.

共i兲 In the case of ␭⫽0, Eqs. 共18兲, 共19兲 reduce to simple first order differential equations which admit the solutions

g1共y兲⫽c3共1⫹y2兲⫺1/4ei␴ tan

⫺1_{(y )}

,

g2共y兲⫽c4共1⫹y2兲⫺1/4e⫺i␴ tan

⫺1(y )_, _共29兲 with c3,c4 complex constants.

These two solutions can be interpreted as representing ingoing and outgoing waves.

共ii兲 In the case of ␭⫽1

2, we obtain the following complex

solutions from Eqs. 共18兲, 共19兲:

⫺␤ˆ/2 , 共30兲

where again c_j with j⫽5,6,7,8 are complex constants. IV. REDUCTION OF THE ANGULAR EQUATION TO THE

HEUN EQUATION: THE MASSLESS CASE In this section, we shall show that the angular equations 共20兲, 共21兲 for the neutrino particles can be decoupled to the confluent Heun equation. To the end that let us reconsider Eqs. 共20兲, 共21兲 in an explicit form for␮*⫽0:

A₃

⬘

共␪兲⫹共K⫹G兲A3共␪兲⫽i␭A1共␪兲, 共31兲 A₁

⬘

共␪兲⫹共K⫺G兲A1共␪兲⫽i␭A3共␪兲, 共32兲 where K⫽ 1 2F

冉

冊

, 共33兲 G⫽ mF sin␪. 共34兲

By introducing the scalings

A1共␪兲⫽H1共␪兲exp

冉

⫺

冕

共K⫺G兲d␪

冊

, 共35兲

A3共␪兲⫽H3共␪兲exp

冉

⫺

冕

共K⫹G兲d␪

冊

, 共36兲 one gets

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H₃

⬘

共␪兲⫽i␭H₁共␪兲exp

冉

冕

2Gd␪

冊

. 共38兲 If we decouple Eqs.共37兲, 共38兲 in Eq. 共37兲 for H₁(␪), we get

H₁

⬙

共␪兲⫹2GH₁

⬘

冊

H3

⬘

共z兲 ⫺ ␭ 2 z共z⫺1兲H3共z兲⫽0. 共42兲

Let us recall the general confluent form of Heun equation 关8兴 H

⬙

共z兲⫹

冉

A⫹B z⫹ C z⫺1

冊

H

⬘

共z兲⫺ DBz⫺h z共z⫺1兲H共z兲⫽0. 共43兲 Drawing the similarities between Eq. 共43兲 and Eqs. 共41兲, 共42兲, we observe the following correspondences.

共a兲 For Eq. 共41兲,

D⫽0, h⫽␭2, A⫽⫺2m, B⫽1 2⫹m, C⫽ 1 2⫺m. 共44兲共b兲 For Eq. 共42兲, D⫽0, h⫽␭2, A⫽⫺2m, B⫽m⫺1 2, C⫽m⫹ 1 2. 共45兲 The confluent Heun equation 共43兲, with its accessory pa-rameter h, has two regular singular points at z⫽0,1 with exponents (0,1⫺B) and (0,1⫺C), respectively, as well as an irregular singularity at infinity. In the vicinity of the point z ⫽0, its power series can be written as

H共D,A,B,C,h;z兲⫽

兺

j⫽0

⬁

Wjzj 共46兲

and the coefficient Wj satisfies a three-term recurrence

rela-tion 关8兴 W0⫽1, W1⫽ ⫺h B , 共 j⫹1兲共 j⫹B兲Wj⫹1⫺A共 j⫺1⫹D兲Wj⫺1 ⫽关 j共 j⫺1⫺A⫹B⫹C兲⫺h兴Wj. 共47兲

It is also possible to obtain the power series solution in the vicinity of the point z⫽1 by a linear transformation inter-changing the regular singular points z⫽0 and z⫽1. Namely,

z→1⫺z.

Expansion of solutions to the confluent Heun equation in terms of the hypergeometric and confluent hypergeometric functions can be seen in关8兴. In Ref. 关8兴, it is also shown that the confluent Heun equation can be normalized to constitute a group of orthogonal complete functions and the confluent Heun equation also admits quasipolynomial solutions for particular values of the parameters.

Since D⫽0 in our case, it follows from the three-term recurrence relation that H(D,A,B,C,h;z) is a polynomial solution if W1(h)⫽0, where W1 stands for a polynomial of degree 1 in h. Namely , there is only one eigenvalue hifor h

such that W1(hi)⫽0 共i.e., ␭⫽0).

V. REDUCTION OF THE ANGULAR EQUATION INTO A SET OF LINEAR FIRST ORDER DIFFERENTIAL

EQUATIONS: THE CASE WITH MASS

To complete our analysis of the angular equation, we need to discuss the angular equation for the Dirac particles with mass.

The angular equations 共20兲, 共21兲 can be rewritten in the forms

LA3⫹i␮p

冑

1⫹cos2␪ A1⫽i␭A1, 共48兲

L⫹A1⫹i␮p

冑

1⫹cos2␪ A3⫽i␭A3. 共49兲 With substitutions

A1共␪兲⫽关A0共␪兲⫹iB0共␪兲兴exp

冉

冕

cos3␪ 2 sin␪Fd␪

冊

, 共50兲 A3共␪兲⫽

冋

共M0共␪兲⫹iN0共␪兲兴exp

冉

冕

cos3␪ 2 sin␪Fd␪

冊

, 共51兲 we can transform Eqs. 共48兲, 共49兲 into a set of first order differential equations

Decoupling is attained by introducing

P1共u兲⫽

冑

E_⫹T共u兲, 共66兲

P2共u兲⫽

冑

E⫺S共u兲, 共67兲 where we obtain a pair of one-dimensional Schro¨dinger-type equations

T

⬙

共u兲⫹V₁T共u兲⫽0, 共68兲 S

⬙

共u兲⫹V2S共u兲⫽0, 共69兲 with the potentials

⫺gJy_d_␪_d_␸

冊

冏

horizon ⬍0, 共72兲 where g is the determinant of the spacetime metric and Jy _is

the axial component of the neutrino particle current. We re-call from metric共7兲 that we have

冑

⫺g⫽F sin␪. 共73兲

It is clear from transformations 共3兲, 共6兲 that the horizon of metric共7兲 corresponds to y→(⫺⬁). In other words, integral 共72兲 is taken over y→(⫺⬁).

In the more standard spinor formalism, Jyis introduced as 关1兴 1

冑

冑

⫺gJy_d_␪_d_␸_⫽3.246_␲_共兩Z

2兩2⫺兩Z1兩2兲. 共78兲 From Eqs.共60兲, 共61兲, 共62兲, and 共63兲, we successively find

兩Z2兩2⫺兩Z1兩2⫽ i 2共P1P¯2⫺P2P¯1兲 ⫽_2E⫺i ⫹关P1, P ¯ 1兴u, 共79兲

where关P1, P¯1兴u is the Wronskian.

Therefore, in order to check the existence of superradi-ance, it will suffice to seek a solution for P1 at the horizon. The reality that the potentials V1 and V2 become infinite at both the horizon and y→⬁ leads us to think of the prob-lem as a probprob-lem of particles in an infinite potential well. Since the particles are bound inside the well, the principal physical fact requires that the solutions of the wave equa-tions 共68兲, 共69兲 must be identically zero at the walls 共the horizon and y→⬁). Clearly, the Wronskian vanishes at the horizon and it follows that the number of particles exiting the horizon per unit time is zero. Consequently, similar to the general Kerr background关1,3兴, there is also no superradiance in the extreme Kerr throat geometry.

VII. CONCLUSION

Our aim in this paper was to do more than separating the Dirac equation in a sector of Kerr—namely, the extremal Kerr throat geometry and obtain exact solutions if possible. This premise has mostly been accomplished and it definitely will contribute to the wave mechanical aspects of spin-1

2

par-ticles prior infalling into the extreme Kerr BH.

In the general Kerr background the radial Dirac equation was the harder part to be tackled compared with the angular part 关1兴. In the present problem of the extremal Kerr throat we have the opposite case: the axial part poses no more difficulty than the angular part does. For the massless case, we overcome the difficulty and attain an exact solution in terms of Heun polynomials. Inclusion of mass prevents this reduction and as a result we are unable to express the angular equation in terms of a set of known equations. This part of the problem can be handled numerically. Alternatively, the angular equation is cast into a pair of Schro¨dinger-type equa-tions. Unlike the scalar field case Dirac fields exhibit no superradiance. The charge coupling of a Dirac particle to an extremal Kerr-Newman BH in its near horizon limit may reveal more information compared to the present case. This is the next stage of study that interests us.

关1兴 S. Chandrasekhar, The Mathematical Theory of Black Holes 共Clarendon Press, London, 1983兲.

关2兴 E.G. Kalnins and W. Miller, Jr., J. Math. Phys. 33, 286 共1992兲.

关3兴 C.H. Lee, Phys. Lett. 68B, 152 共1977兲. 关4兴 A. Zecca, J. Math. Phys. 37, 874 共1996兲.

关5兴 S.O. Gilberto, Gen. Relativ. Gravit. 33, 395 共2001兲. 关6兴 I. Sakalli, Gen. Relativ. Gravit. 35, 1321 共2003兲.

关7兴 J. Bardeen and G.T. Horowitz, Phys. Rev. D 60, 104030 共1999兲.

关8兴 A. Ronveaux, Heun’s Differential Equations 共Oxford Science,

Oxford, 1995兲.

关9兴 M. Halilsoy, ‘‘Near horizon geometry of extreme black holes

and colliding waves,’’ gr-qc/0210055.