Volume 688, number 2 PHYSICS LETTERS 23 May 1977

MASSIVE SPIN-; WAVE AROUND A KERR-NEWMAN BLACK HOLE

C.H. LEE

International Centre for Theoretical Physics, Trieste, Italy

Received 21 February 1977

Using the separation of variables of the Dirac equations for the electron in the Kerr-Newman geometry, it is ex- plicitly shown that the super-radiant scattering is not allowed for the electron wave on a Kerr-Newman black hole.

Problems concerning the perturbations of the electromagnetic, the gravitational and the neutrino fields around

a Kerr black hole have been extensively investigated by various authors. The separability of the equations governing

the perturbations of these massless fields in the Kerr geometry was a major factor that made the analyses possible.

Recently, Chandrasekhar [l] has succeeded in separating the Dirac equations for massive fields in the Kerr geome- try using the Newman-Penrose formalism, and Page [2] has extended the result to the Kerr-Newman geometry. This separation of variables opens many possibilities for the treatment of the various problems which have been analysed for massless fields.

The derivation of the equations given by Chandrasekhar and Page is summarized for later references in this paper. Then the asymptotic solutions both at infinity and at the event horizon are obtained. We give the expression for the stress-energy tensor in terms of the quantities relevant in the Newman-Penrose formalism, and finally show ex- plicitly the nonexistence of super-radiant scattering for spin- $ fields with mass and charge on a Kerr-Newman black hole. This result is the same as Unruh’s [3] for the neutrino field on a Kerr black hole. Also, without using the separation of variables but with the restrictions of a <M and a < w- I, Maeda [4] showed the same result as

ours. As Hawking [.5] and Damour [6] explained, the non-existence of super-radiance should be a common proper- ty of all fermions regardless of particular background geometry.

The Dirac equations for the electron in the electromagnetic field can be derived from the Lagrangian

L = ?iyp(Vcc + ie-4,)$ - pG$, (1)

where A is the electromagnetic potential, p is the mass of the electron and yp are the Dirac matrices satisfying

y’Iyv t yvyp = 2gM”. (2)

With

the two-spinors satisfy

&a’J_&(V, tieA,)xB = -@A, JZopAB(Vi, tieAJ&j = -ipxA. (3)

By taking the hermitian conjugate of eq. (3) and comparing it with the equation for 6 one can see that $ =(gA, xi). In the Newman-Penrose formalism with a null tetrad system (I, IZ, m, m*) and the spinor components defined by

XA =x0@ t+, [A = $OA t $A (4)

(OA and & are basis spinors), eq. (3) becomes

152

Volume 68B, number 2 PHYSICS LETTERS 23 May 1977

&[(D + e - p + ielccA,)xu +(6* t 7r - 0L t iem*ccA,)xl] = i#;,

&[(A + P - 7 + ienfi,4Jx1 + (6 + /3 - r + iem~~&-,] = -in&,

4]@ + f* - P* + ielPA,)[i + (6 + 7r* - o* t ie&A,)$z] = -ipxl,

@[(A + P* - y* + ierPA,)[l t (6* + fl* - r* t iem*pA,)lE] = ipxu.

Here D E ZW/W), A E nj+j/W), 6 = mp(a/?W) and e, p, etc. are the spin coefficients [7]. Eq. (S), written out explicitly for the Kerr-Newman geometry using the Kinnersley tetrad

(5)

( ( ICC = v,l,O,z), r+=(*,-$,0,&J, mu=\/2(rt~ncos8)(i.Sin8,0,1,~);

Zrr2 +a2 c0$e, A=r2-2Mrta2tQ2 , )

and A, = -Qr/Z (1, 0, 0, -a sin2e), can be separated by writing

xu = e-iwt eirn@ ‘&)Y(r) ET, = -e-iwt eim@ %\e)k-'ce>

fi(r-iacose)’ fi(r t ia cos e) ’

XI = e-iwteim~A-l/2’SI(e)‘fi)(r), l; = e-iwt eim$A-1/2(j)(e)(g(r).

The resulting equations for s(e) and R(r) are.

(6)

( 5 cos e

--awsinets (?=(--Xtpacos.8) 1

(-

de 2sin8 s’,

d COSe -t-taawsine- de 2 sine

$? = (h t pa cos e) '?,

AlI2 $ t i {w(r2 +a2) - ma t eQr} I

‘2 = (h - ipr) (2,

AlI2 - i{w(r2 t a2) - ma t eQr} 1 ‘ii’=(htipr)(S.

(7)

o-9

The decoupled second-order differential equation for !?’ can be obtained by eliminating (5’ from eq. (7),

[~~(sinB$)th~~~~se~t(fta~cosB)2-(m~i~~e~-;i2a~m-a2w2

+ ap($cos e + aw sin28 - m)

htapcose - a2p2 c0S2e tX2 1 'i\ej= 0.

(9)

y- (n 6) satisfies the same equation. The condition that S be regular over 0 < 8 < n makes it an eigenvalue equa- tion for X. Similarly, the decoupled equation for (&’ is

ctK2ti(r-MYC_2iwr_ieQ_ PK A YXj?

- p2r2 - X2 1 P(r) = 0, (10) where K z (r2 4 a2)w - ma t eQr. !?( r )* a so satisfies&q. (10). The general solution can be expressed as a linear 1

superposition of the solutions corresponding to all possible values of X. For example,

153

Volume 68B, number 2 PHYSICS LETTERS 23 May 1977

e-iwt eim@ C (?(h; 0) (i’(X; r), etc.

X0 = fi (r - if2 cos 0) h (11)

The asymptotic solutions at infinity (r + m), with the connections between the coefficients determined through

eq. (8) are

(-) R(X;r)=-; (1 -fL$)Zin(h) epior +Z,,t(h)t+, (12)

where (yZ&J ,u2M- ewQ In,. dr’ r2 +a2

,2 -p2 ’ -&=a,

and 2, and Zout are constant coefficients. As for the asymptotic solutions at the event horizon, the boundary condition that the physically well-behaved observer near the horizon see non-singular fields (or that the radial group velocity of a wave packet be negative, i.e. going into the hole) eliminates half of the possible solutions. The physically acceptable asymptotic solutions at the horizon (r + r+ =‘M t &I2 - a2 - Q2) are

V(A) = A(h) eS@, Y(h) (h t ipr+)A(A) A1j2 e-@

z-p (r+ - M) - 2i [w(rj + a2) - ma + eQr1 ’ (13)

where m - eQr+

)

,

r? +a2 r.

The stress-energy tensor of the electron given by Brill and Wheeler [8] (extended to the case where the electro- magnetic field is present) is

(14)

where the parentheses (p V) indicate symmetrization. To express it in terms of the quantities relevant in the Newman-Penrose formalism, the following relationships for the tetrad:

/cc = o~A;@o~ .

n’ = op.&j tA iB , mP = awAi OA $ (15)

and the corresponding relationships for the spin coefficients [7] are used. The final result after a rather lengthy derivation is

TP,, = tolvj + c.c., (16)

with

+(lto12 - Ixo12)(zp(Y*~, + e*nv tarn, t pm:) t mE(r1, t w2, -pm, - urn;))

+(ltl I2 - lx1 12)(n,(rZv ten, tP*mv ta*m:) tm,(-vl, - nn, thv tpm:))

t ($$l - x~~~)(Z,(ul, t 7*n, - yrn: - r*rnt) +n,(nZ, + K*n, -em; -e*m,*) tm,(-Al, -a*nv tcvn~ tP*mJ)

tm&d,-p*n,tcr*m,*t/3m~)) -P.I(l~ol2+Ix0l2)Z~A~+(l1;1l2+Ix~l2)~~~,+~~~~~~+X~X~~m,~,~~ I 4

154

Volume 68B, number 2 PHYSICS LETTERS 23 May 1977

If one puts the solution in the form of eq. (11) into eq. (16) and evaluates the limiting values at infinity with the use of eq. (12), one obtains the following expressions for the energy density and flux in the asymptotic flat region:

d2E -=,.2Ttt dr dS1

= r2 T,,

l -

d2E - = $Ttr dt da

= -r2 T,,. t-00 r+-

(W

At first sight the double sum over X and X’ looks troublesome, especially since the orthogonality of the angular functions for different values of h is not guaranteed because of the peculiar involvement of X in eq. (9). However,

from eq. (7), one can easily show that

Hence, the integral on the left-Qand side*of eq. (19) is zero unless X = h’. Eq. (19) can also be used to show that X is real by putting h’ = h* and (S)(x*) = !,S)(x>* (then (h* - h)J [I( + I ‘$o()12] dQ = 0). Integrating eqs. (17) and (18) over the solid angle one obtains for the ener y density per unit radial length and the energy flux per unit time, with the normalization such that J[‘?(?I)~ t (-(X)2] da = 1, syp

(20)

To guarantee a positive definite energy density in the asymptotic flat region we restrict o to be positive. Also, since the energy of an electron is always

From the fact that both (ii)(r) and eater than the rest mass, w > /.L

(&)* satisfy eq. ( ) 10 , we derive a Wronskian which is a conserved quantity (i.e. independent of r). After a simple manipulation,

g s$ Al/‘$j t ipr)- = 0.

(22) Therefore W is independent of r and its values at infinity and at the horizon are the same,

WI,=, = WI,=,+.

Or, explicitly, putting the asymptotic forms (eqs. (12) and (13)) into eq. (22), eq. (23) becomes

(23)

(24)

Eq. (24) indicates that

155

Volume 68B, number 2 PHYSICS LETTERS 23 May 1977

and proves the non-existence of super-radiance.

The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. He has benefitted from discussions with Professor R. Ruffini.

References

[l] S. Chandrasekhar, Proc. Roy. Sot. (London) A349 (1976) 571. [2] D. Page, Phys. Rev. D14 (1976) 1509. [3] W. Unruh, Phys. Rev. Letters 31 (1973) 1265. [4] K. Maeda, Progr.Theoret. Phys. (Kyoto) 55 (1976) 1677. [5] S. Hawking, Commun. Math. Phys. 43 (1975) 199. [6] T. Damow, invited talk at the M. Grossmann Meeting, held at the ICTP, Trieste, 1976 (to appear in the proceedings, ed. R.

Ruffini, North-Holland). [7] E. Newman and R. Penrose, J. Math. Phys. 3 (1962) 566. [S] D. Brill and J. Wheeler, Rev. Mod. Phys. 29 (1957) 465.

156