Twisted spinors on Schwarzschild and Reissner-Nordström black holes

1 July 1999

Physics Letters B 458 Ž1999. 29–35

Twisted spinors on Schwarzschild and Reissner-Nordstrom ¨ black holes Yu.P. Goncharov Theoretical Group, Experimental Physics Department, State Technical UniÕersity, Sankt-Petersburg 195251, Russia Received 31 March 1999; received in revised form 6 May 1999 Editor: P.V. Landshoff

Abstract We describe twisted configurations of spinor field on the Schwarzschild and Reissner-Nordstrom ¨ black holes that arise due to existence of the twisted spinor bundles over the standard black hole topology R 2 = S 2. From a physical point of view the appearance of spinor twisted configurations is linked with the natural presence of Dirac monopoles that play the role of connections in the complex line bundles corresponding to the twisted spinor bundles. Possible application to the Hawking radiation is also outlined. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 04.20.Jb; 04.70.Dy; 14.80.Hv Keywords: Black holes; Topological properties; Twisted fields; Dirac monopoles

1. Introductory remarks Recently it has appeared an interest in studying topologically inequivalent configurations ŽTICs. of various fields on the 4D black holes w1–4x since TICs might give marked additional contributions to the quantum effects in the 4D black hole physics, for instance, such as the Hawking radiation w2x and also might help to solve the problem of statistical substantiation of the black hole entropy w3x. So far, however, only TICs of complex scalar field have been studied more or less on the Schwarzschild ŽSW., Reissner-Nordstrom ¨ ŽRN. and Kerr black holes. The next physically important case is the one of spinor fields. The present paper will be, therefore,

devoted just to a description of twisted TICs of spinor field in the form convenient to physical applications within the framework of the SW and RN black hole geometry. As was discussed in Refs. w1–4x, TICs exist owing to high nontriviality of the standard topology of the 4D black hole spacetimes which is of the R 2 = S 2-form. High nontriviality of the given topology consists in the fact that over it there exist a huge Žcountable. number of nontrivial real and complex vector bundles of any rank N ) 1 Žfor complex ones for N s 1 too.. In particular, TICs of complex scalar field on the 4D black holes are conditioned by the availability of countable number of complex line bundles over the R 2 = S 2-topology underlying the

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 5 7 2 - 9

Yu.P. GoncharoÕr Physics Letters B 458 (1999) 29–35

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4D black hole physics. In turn, TICs of spinor field can be tied with the twisted spinor bundles on the given topology. We write down the black hole metrics under discussion Žusing the ordinary set of local coordinates t,r,q , w . in the form ds 2 s gmn dx m m dx n ' adt 2 y ay1 dr 2 y r 2 Ž dq 2 q sin2q d w 2 . 2

2

Ž 1.

2

with a s 1 y 2 Mrr q a M rr , a s QrM, where M, Q are, respectively, a black hole mass and a charge. Besides we have < g < s
complex line bundle with Chern number n, we can construct tensorial product SŽ M . m j . As is known w7x, over any noncompact spacetime the bundle SŽ M . is trivial and, accordingly, the Chern number of 4-dimensional vector bundle SŽ M . m j is equal to n as well. Under the circumstances we obtain the twisted Dirac operator D :SŽ M . m j ™ SŽ M . m j , so the wave equation for corresponding spinors C Žwith a mass m 0 . as sections of the bundle SŽ M . m j may look as follows DC s m 0C ,

Ž 2.

and we can call Žstandard. spinors corresponding to n s 0 Žtrivial complex line bundle j . untwisted while the rest of the spinors with n / 0 should be referred to as twisted. From general considerations w5,6,8x the explicit form of the operator D in local coordinates x m on a 2 k-dimensional Žpseudo.riemannian manifold can be written as follows D s i=m ' ig c Ecm Ž Em y 12 vm a bg ag b y ieAm . ,

a - b,

Ž 3. 2. Description of TICs Mathematical grounds for the existence of spinor field TICs on black holes lie in the fact that over the standard black hole topology R 2 = S 2 there exists only one Spin-structure wconforming to the group SpinŽ1,3. s SLŽ2,C.x. Referring for the exact definition of Spin-structure to Refs. w5,6x, we here only note that the number of inequivalent Spin-structures for manifold M is equal to the one of elements in H 1 Ž M,Z 2 ., the first cohomology group of M with coefficients in Z 2 . In our case H 1 ŽR 2 = S 2 ,Z 2 . s H 1 ŽS 2 ,Z 2 . which is equal to 0 and thus there exists the only Žtrivial. Spin-structure. On the other hand, the nonisomorphic complex line bundles over the R 2 = S 2-topology are classified by the elements in H 2 Ž M,Z., the second cohomology group of M with coefficients in Z w1x, and in our case this group is equal to H 2 ŽS 2 ,Z. s Z and, consequently, the number of complex line bundles is countable. As a result, each complex line bundle can be characterized by an integer n g Z which in what follows will be called its Chern number. Under this situation, if denoting SŽ M . the only standard spinor bundle over M s R 2 = S 2 and j the

where A s Am dx m is a connection in the bundle j and the forms v a b s vm a b dx m obey the Cartan structure equations de a s v ab n e b with exterior derivative d, while the orthonormal basis e a s ema dx m in cotangent bundle and dual basis Ea s EamEm in tangent bundle are connected by the relations e a Ž Eb . s d ba. At last, matrices g a represent the Clifford algek bra of the corresponding quadratic form in C 2 . Below we shall deal only with 2D euclidean case Žquadratic form Q2 s x 02 q x 12 . or with 4D lorentzian case Žquadratic form Q1,3 s x 02 y x 12 y x 22 y x 32 .. For the latter case we take the following choice for g a g 0s

ž 10

0 , y1

/

g bs

ž

sb , 0

0 y sb

/

bs1,2,3 ,

Ž 4.

where s b denote the ordinary Pauli matrices s1 s

ž 01 10 / ,

s2 s

ž 0i

yi , 0

/

s3 s

ž 01

0 . y1

/

Ž 5.

It should be noted that further in lorentzian case, Greek indices m , n , . . . will be raised and lowered with gmn of Ž1. or its inverse g mn and Latin indices a,b, . . . will be raised and lowered by ha b s h a b s diagŽ1,y 1,y 1,y 1., so that ema enb g mn s h a b , Eam Ebn gmn s ha b and so on.

Yu.P. GoncharoÕr Physics Letters B 458 (1999) 29–35

Using the fact that all the mentioned bundles SŽ M . m j can be trivialized over the chart of local coordinates Ž t,r,q , w . covering almost the whole manifold R 2 = S 2 , we can concretize the wave Eq. Ž2. on the given chart for TIC C with the Chern number n g Z in the case of metric Ž1.. Namely, we can put e 0 s 'a dt, e 1 s drr 'a , e 2 s rdq , e 3 s r sinq d w and, accordingly, E0 s E tr 'a , E1 s 'a Er , E2 s Eq rr, E3 s EwrŽ r sinq .. This entails

v 01 s y

1 da 2 dr

v 13 s y'a sinq d w ,

v 23 s ycosq d w .

Ž 6.

As for the connection Am in bundle j then the suitable one was found in Refs. w1x and is A s Am dx m s y

Q r

dt y

n e

cosq d w .

Ž 7.

Under the circumstances, as was shown in Refs. w1x, integrating F s dA over the surface t s const, r s const with topology S 2 gives rise to the Dirac charge quantization condition n

HS F s 4p e s 4p q 2

Ž 8.

with magnetic charge q, so we can identify the coupling constant e with electric charge. Besides, the Maxwell equations dF s 0, d) F s 0 are fulfilled w1x with the exterior differential d s E t dt q Er dr q Eq dq q Ew d w in coordinates t,r,q , w , where ) means the Hodge dual form. We come to the same conclusion that in the case of TICs of complex scalar field w1–4x: the Dirac magnetic UŽ1.-monopoles naturally live on the black holes as connections in complex line bundles and hence physically the appearance of TICs for spinor field should be obliged to the natural presence of Dirac monopoles on black hole and due to the interaction with them the given field splits into TICs. Also it should be emphasized that the total Žinternal. magnetic charge Q m of black hole which should be considered as the one summed up over all the monopoles remains equal to zero because Qm s

1 e

Ý ngZ

Returning to Eq. Ž2., we can see that with taking into account all the above it takes the form ig 0

ns0 .

Ž 9.

1

'a

qig 2 qig 3

v 12 s y'a dq ,

dt ,

31

ž 1 r

E t y 12 v t 01 g 0g 1 q

ieQ

/

r

q ig 1'a Er

Ž Eq y 12 vq 12 g 1g 2 . 1

r sinq

Ž Ew y

1 2

vw 13 g 1g 3

y 12 vw 23 g 2g 3 q incosq . C s m 0C .

Ž 10 .

After a simple matrix algebra computation with using Ž4., Ž6. and the ansatz

C s e i v t ry1

ž

F1 Ž r . F Ž q , w . F2 Ž r . s 1F Ž q , w .

/

F

with a 2D spinor F s F 1 , we can from Ž10. obtain 2 the system

ž /

'a Er F1 q 'a Er F2 q

ž ž

1 d'a

l q

2 dr

r

1 d'a

l y

2 dr

r

/ /

F1 s i Ž m 0 y c . F2 ,

F2 s yi Ž m 0 q c . F1

Ž 11 . with an eigenvalue l for the eigenspinor F of the operator Dn s yi s 1w i s 2 Eq q i s 3 sin1q Ž Ew y 12 s 2 s 3 = cosq q incosq .x, so that s 1 Dn s yDn s 1 , while c . s 1 Ž v q eQ r . We need, therefore, explore the 'a

equation DnF s lF .

3. Spectrum of the operator Dn As is not complicated to see, the operator Dn has the form Ž3. with g 0 s yi s 1 s 2 , g 1 s yi s 1 s 3 , e 0 s dq , e 1 s sinq d w , v 01 s cosq d w , Am dx m s y ne cosq d w , i.e., it corresponds to the abovementioned quadratic form Q2 and this is just twisted Žeuclidean. Dirac operator on the unit sphere and the conforming complex line bundle is the restriction of

Yu.P. GoncharoÕr Physics Letters B 458 (1999) 29–35

32

bundle j on the unit sphere. Again simple matrix algebra computation results in Dn s

ž

D1 n yD 2 n

D2 n yD1 n

/

with D 1 n s i Ž Eq q 12 cotq ., D 2 n s y sin1q Ž Ew q incotq .. Then it is easy to see that the equation DnF s lF can be transformed into the one

ž

0 Dnq

Dny Fq F 0 s lF 0 ,F 0 s , Fy 0

/

ž /

methods Žsee, e.g. Refs. w10x.. The corresponding eigenfunctions of the above Dq-operator can be chosen by miscellaneous ways, for instance, as follows Žsee, e.g., Ref. w9x. PmkX nX Ž cosq . X

si

Ž 12 .

where Dn" s D 1 n " D 2 n s i w Eq q Ž 12 . n .cotq x . sin1q Ew , F "s F 1 " F 2 . From here it follows that 2 q y 2 DnyDq n F qs l Fq, Dn Dn F ys l FyX or, with employing the ansatz F "s P " Žq . eyi m w , in explicit form

=

ž

=

sin2q

/

=P " Ž q . s Ž 14 y n2 y l2 . P " Ž q . . It is known w9x that differential operator

Dq s Eq2 q cotqEq y

mX 2 q nX 2 y 2 mX nX cosq

Ž 13 .

Ž 14 . sin2q has eigenfunctions in the interval 0 F q F p , which are finite at q s 0,p , only for eigenvalues yk Ž k q 1., where k is positive integer or half-integer simultaneously with mX ,nX while the multiplicity of such an eigenvalue is equal to 2 k q 1. In our case of Eq. Ž13. we have that nX s n " 1r2 is half-integer because the Chern number n g Z. As a result, we should put mX s m y 1r2 with an integer m and then < mX < F k s l q 1r2 with a positive integer l and, accordingly, 1r4 y n2 y l2 s yk Ž k q 1. which en2 tails Ždenoting l s Ž l q 1 . y n2 . that spectrum of Dn consists of the numbers "l with multiplicity 2 k q 1 s 2Ž l q 1. of each one. Besides, it is clear that under this yl F m F l q 1,l G < n <. This just reflects the fact that from general considerations w5,6,8x the spectrum of twisted euclidean Dirac operator on even-dimensional manifold is symmetric with respect the origin. At n s 0 we get the spectrum of just Dirac operator on S 2 , i.e., "l s "Ž l q 1. g Z _  04 which may be obtained by purely algebraic

(

1 y cosq k

Ý

X X jsmax Ž m , n .

ž

1 y cosq 2

/

mXqnX 2

Ž k q j . !i 2 j Ž k y j . ! Ž j y mX . ! Ž j y nX . ! j

/

Ž 15 .

with the orthogonality relation at nX fixed

2

y

ž

(

Ž k y mX . ! Ž k y nX . ! Ž k q mX . ! Ž k q nX . !

1 q cosq

=

Eq2 q cotqEq mX 2 q Ž n . 1r2 . y 2 mX Ž n . 1r2 . cosq

X

ym yn

p

H0

X

Pm)X nkX Ž cosq . PmkXX nX Ž cosq . sinq dq 2

d X d X XX , Ž 16 . 2 kq1 kk m m where ŽU . signifies complex conjugation. As a consequence, we come to the conclusion that spinor F 0 of Ž12. can be chosen in the form s

F0s

ž

C1 PmkX ny1r2 C2 PmkX nq1r2

/

X

eyi m w

with some constants C1,2 . Now we can employ the relations w9x mX yEq PmkX nX " nX cotq y P kX X sinq m n

ž

/

s yi k Ž k q 1 . y nX Ž nX " 1 . PmkX nX " 1

(

Ž 17 .

holding true for functions PmkX nX to establish that C1 s C2 s C corresponds to eigenvalue l while C1 s yC2 s C conforms to yl. Thus, the eigenF

spinors F s F 1 of the operator Dn can be written 2 as follows

ž /

F"ls

C 2

ž

PmkX ny1r2 " PmkX nq1r2 PmkX ny1r2 . PmkX nq1r2

/

X

eyi m w ,

Ž 18 .

Yu.P. GoncharoÕr Physics Letters B 458 (1999) 29–35

where the coefficient C may be defined from the normalization condition p

2p

H0 H0 Ž
1

< 2 q
Ž 19 .

(

can get the second order equations for them in the form

ž /

q 4

s

4. Increase of Hawking radiation for spinor particles As follows from the above, when quantizing twisted TICs of spinors we can take the set of spinors 1

'2pv

e i v t ry1

ž

F1 Ž r ," l . F " l F2 Ž r ," l . s 1F " l

/

Ž 20 .

L42 ŽR 2 = S 2 .

as a basis in and realize the procedure of quantizing, as usual, by expanding in the modes Ž20. `

Cs

lq1

Ý Ý Ý

H

"l ls < n < msyl m 0 `

Cs

lq1

Ý Ý Ý

dv Ž

avyn l mCl q bvqn l mCy l

.,

H

"l ls < n < msyl m 0

2 dr

r

/

l2 y

eQ

vq

F1,2

r2

r

2

/

F1,2 .

Ž 22 .

2 rq

ln

rqy ry

r y rq

y

2M

2 ry

rqy ry

ln

r y ry 2M Ž 23 .

and by going to the dimensionless quantities x s r ) rM, y s rrM, k s v M Eqs. Ž22. can be rewritten in the Schrodinger-like equation form ¨ d2 dx 2

E1,2 q k 2 y Ž m 0 M .

2

E1,2

s V1,2 Ž x ,k , a , l . E1,2

Ž 24 .

with E1,2 s E1,2 Ž x,k, a , l. s F" Ž Mx ., F" Ž r ) . s F1,2 w r Ž r ) .x while the potentials V1,2 are given by V1,2 Ž x ,k , a , l . s

q y

y6 Ž x .

l .

2

yŽ x. ya 2

4

Ž 21 . where C s g 0C † is the adjoint spinor and Ž†. stands for hermitian conjugation. As a result, the operators av"nl m , bv"n l m of Ž21. should be interpreted as the creation and annihilation ones for spinor particle in the gravitational field of the black hole, in the field of monopole with Chern number n and in the external electric field of black hole. As to the functions F1,2 Ž r . of Ž20. then in accordance with Eqs. Ž11. we

ž

l "

By replacing

`

d v Ž avqn l mCl q bvyn l mCy l . ,

ž

dr

a m20 y

1

`

1 d'a

2

d'a

1

r ) srq

C" l s

'a Er

aEr aEr F1,2 q a

lq 1 p

with using the relation Ž16. that yields C s . These spinors form an orthonormal basis in L22 ŽS 2 .. Finally, it should be noted that the given spinors can be expressed through the monopole spherical harmonics Yml nŽq , w . s Pml nŽcosq . eyi m w which naturally arise when considering twisted TICs of complex scalar field w1–4x but we shall not need it here. In general, for physical applications the condition Ž19. seems to be quite enough rather than explicit form of F " l .

33

y2 Ž x .

(

1y

1y

y2 Ž x .

a2 q

q yŽ x.

e2 Q2 y

y2 Ž x .

q yŽ x.

yŽ x. ,

y2 Ž x .

a2

2

2

y Ž x.

a2

2

y

2

y4 Ž x .

2

l2 q

1 3a 2 y 2 y Ž x .

y2 Ž x .

2 Ž m0 M . y

2 keQ yŽ x.

Ž 25 .

Yu.P. GoncharoÕr Physics Letters B 458 (1999) 29–35

34

where y Ž x . is a function reverse to the following one xŽ y. syq y

2 yq

yqy yy 2 yy

yqy yy

y y yq

ln

L Ž n . s lim

r™`

2

y E1,2

x ,k . ;

½

ln

sA Ý

,

2

Ž 1 ,2 . yi k x e i k x q s12 e q

1 64 k 2

x ™y`,

,

Ž 27 .

x ™q`,

²0 < Tt r <0: d s `

Ý 2 Ž l q 1. H

0

Ž1. < s11 Ž k , l. < 2

e 8p k q 1

dk

Ž 32 .

with the vacuum expectation value ²0 < Tt r <0: and the surface element d s s r 2 sinq dq n d w while A s c5 c" 1r2 f 0.125728 P 10 55 erg P sy1 P My1 Ž M in GM Ž G . g.. We can interpret LŽ n. as an additional contribution to the Hawking radiation due to the additional spinor particles leaving black hole because of the interaction with monopoles and the conforming radiation should be called the monopole Hawking radiation. Under this situation, for the total luminosity L of black hole with respect to the Hawking radiation concerning the spinor field to be obtained, one should sum up over all n, i.e. `

Ž 1,2 . yi k x e q 22 yi k x

2

" l ls < n <

Ž 1 ,2 . i k x s11 e ,

°s ~ Ž x ,k . ; ¢e

HS

`

y y yy

y "s 1 " '1 y a 2 , Ž 26 . so y Ž x . is the one-to-one correspondence between Žy`,`. and Ž yq,`.. Further we shall for the sake of simplicity restrict ourselves to the SW black hole case Ž Q s 0. and massless spinors Ž m 0 s 0.. Then, as can be seen, when x ™ q`, V1,2 ™ 0 and at x ™ y`, V1,2 ™ 1r64. This allows us to pose the scattering problem on the whole x-axis for Eq. Ž24. at k ) 0 E1q,2 Ž

tion for TIC with the Chern number n Žin usual units.

1

64 k Ž 1,2 . i k x q s21 e ,

2

Ls

x ™ y`,

,

ngZ

x ™ q`

Ž 28 .  siŽ1,2. Ž k, l.4 . s siŽ1,2. j j

with S-matrices of Ž11. one can obtain the equality

Then by virtue

Ž1. Ž2. s11 Ž k , l . s ys11 Ž k , l. .

Ž 29 .

Having obtained these relations, one can speak about the Hawking radiation process for any TIC of spinor field on black holes. Actually, one can notice that Eq. Ž2. corresponds to the lagrangian i L s < g < 1r2 Cg m=mC y =mC g mC y m 0CC , 2 Ž 30 .

ž

/

and one can use the energy-momentum tensor for TIC with the Chern number n conforming to the lagrangian Ž29. i Tmn s Cgm=nC q Cgn=mC y =mC gnC 4

ž

y Ž =nC . gmC ,

Ý

L Ž n . s L Ž 0. q 2

/

Ž 31 .

to get, according to the standard prescription Žsee, e.g., Ref. w11x. with employing Ž19. and Ž28., the luminosity LŽ n. with respect to the Hawking radia-

Ý LŽ n. ,

Ž 33 .

ns1

since LŽyn. s LŽ n.. As a result, we can expect marked increase of Hawking radiation from black holes for spinor particles. But for to get an exact value of this increase one should apply numerical methods, so long as the scattering problem for general Eq. Ž24. does not admit any exact solution and is complicated enough for consideration – the potentials V1,2 Ž x,k, a , l. of Ž25. are given in an implicit form. One can remark that, for instance, in the Schwarzschild black hole case the similar increase for complex scalar field can amount to 17 % of total Žsummed up over all the TICs. luminosity w2x. We hope to obtain numerical results elsewhere. 5. Concluding remarks It is clear that the next important case is the Kerr black hole one or, more generally, the Kerr-Newman black hole one but the equations here will be more complicated. Besides, as was mentioned above, the corresponding scattering problems require serious study since contribution of TICs Že.g., to Hawking radiation. can be computed only numerically and when doing so it is very important to know whether

Yu.P. GoncharoÕr Physics Letters B 458 (1999) 29–35

the elements of the corresponding S-matrices exist in the strict mathematical sense which enables one to get exact, for example, integral equations for their numerical computation. It is difficult task and it has still been studied only in a number of cases for complex scalar field w2,12x. Acknowledgements The work was supported in part by the Russian Foundation for Basic Research Žgrant No. 98-0218380-a. and by GRACENAS Žgrant No. 6-18-1997.. References w1x Yu.P. Goncharov, J.V. Yarevskaya, Mod. Phys. Lett. A 9 Ž1994. 3175; A 10 Ž1995. 1813; Yu.P. Goncharov, Nucl. Phys. B 460 Ž1996. 167.

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w2x Yu.P. Goncharov, N.E. Firsova, Int. J. Mod. Phys. D 5 Ž1996. 419; Nucl. Phys. B 486 Ž1997. 371. w3x Yu.P. Goncharov, Int. J. Mod. Phys. A 12 Ž1997. 3347; Pis’ma v ZhETF 67 Ž1998. 1021; Mod. Phys. Lett. A 13 Ž1998. 1495. w4x Yu.P. Goncharov, Phys. Lett. B 398 Ž1997. 32; Int. J. Mod. Phys. D 8 Ž1999. 123. w5x Geometrie Riemannian en Dimension 4. Seminaire Arthur ´ ´ Besse, CedicrFernand Nathan, Paris, 1981. w6x H.B. Lawson Jr., M.-L. Michelsohn, Spin Geometry, Princeton U.P., Princeton, 1989. w7x R. Geroch, J. Math. Phys. 9 Ž1968. 1739; C.J. Isham, Proc. R. Soc. ŽLondon. A 364 Ž1978. 591. w8x A.L. Besse, Einstein Manifolds, Springer, Berlin, 1987. w9x N.Ya. Vilenkin, Special Functions and Theory of Group Representations, Nauka, Moscow, 1991. w10x M. Cahen, S. Gutt, Simon Stevin 62 Ž1988. 209; M. Cahen, A. Franc, S. Gutt, Lett. Math. Phys. 18 Ž1989. 165. w11x D.V. Gal’tsov, Particles and Fields in the Vicinity of Black Holes, Moscow University Press, Moscow, 1986. w12x N.E. Firsova, Int. J. Mod. Phys. D 7 Ž1998. 509.