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Dirac Equation in Kerr Space-Time

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Dirac Equation in Kerr Space-Time Ptami~ta, Vol. 8, No. 6, 1977, pp. 500-511. @ Printed in India. Dirac equation in Kerr space-time B R IYER and ARVIND KUMAR Department of Physics, University of Bombay, Bombay 400098 MS received 20 October 1976; in revised form 7 February 1977 Abstract. The weak, field-low velocity approximation of Dirac equation in Kerr space-time is investigated. The interaction terms admit of an interpretation in terms of a ' dipole-dipole' interaction in addition to coupling of spin with the angu- lar momentum of the rotating source. The gravitational gyro-faetor for spin is identified. The charged case (Kerr-Newman) is studied using minimal pre- scription for electromagnetic coupling in the locally inertial frame and to the leading order the standard electromagnetic gyro-factor is retrieved. A first order perturba- tion calculation of the shift of the Schwarzschild energy level yields the main interesting result of this worl~: the anomalous Zeeman splitting of the energy level of a Dirac particle in Kerr metric. Keywords. Kerr space-time; Dirac equation ; anomalous Zeeman effect. 1. Introduction The general formulation of Dime equation in curved space-time has been known for a long time (Brill and Wheeler 1957). In tiffs paper we take up the investi- gation of Dirac equation in the specific case of Kerr geometry which is supposed to represent the gravitational field of a rotating black hole or approximately the field outside of a rotating body. The corresponding problem of the scalar particle in Kerr metric has been recently studied by Ford (1975) and our purpose is to extend this work and obtain new results characteristic of the spin haft case. In addition, using minimal prescription for electromagnetic coupling in the local Lorentzian frame we extend the results to Kerr-Newman geometry (charged rotat- ing source). The spin half case is complicated by the fact that the transformation properties of spinors (unlike tensors) are defined only in local Lorentzian frame. This necessi- tates the use of vierbein (tetrad) formalism (Boulware 1975). In the next section we evaluate the necessary vierbein components and ' spinor atfmity ' for the Kerr metric and take a systematic weak field-low velocity limit. The interaction terms admit of interpretation in terms of a 'dipole-dipole' interaction in addition to a direct coupling of spin with the angular momentum of the rotating source. Tiffs leads to the identification of the gravitational gyro-factor for spin. Extension to the charged case in section 3 retrieves, in the leading order, the standard electro- magnetic gyro-factor. In sec. 4, we calculate in first-order perturbation theory, 500 Dirac equation in Kerr space-tlme 501 the splitting of the Schwarzschild energy level and, in contrast to the scalar case which gives normal ' Zeeman pattern', we obtain anomalous ' gravitational Zevman pattern' for a Dirac particle in Kerr geometry. 2. Spin half equation in Kerr metric (Notation: In what follows, the Latin indices a, b, c, etc., run over 0, 1, 2, 3 and refer to local Lorentzian co-ordinates. The Greek indices ,t, fl, t~, v, etc.. run over the four general co-ordinates t, r, 0, ~b.) 2.1. The Kerr metric The Kerr metric in terms of Boyer-Lindqnist co-ordinates is given by ds 2 ~ dr 2 _1_ p~d) 2 + (r 2 + a~) sin2 0 + 2Mr a2 sin4 0 d~ 2 4Mr ( 2 Mr '~ p---~- a sin~O de dt -- 1 p2 ] dt2; A = r 2-q- a 2-2Mr, ps= r 2q_ a 2cos20. (1) The contravatiant g~t~ can be read from gafl~x~b bx/~b _p-z A~+~+(sin -20-a zA -1) b¢ 4Mar ~ aO s a s ~2 ] -- & beb~--{A-'(r 2+ -- sinS0}~ . (2) The coefficients of affine connection ga~ (bg[~ a bgT~ bgfl,~ can be obtained directly. The expressions for the thirty-two non-vanishing /'fl.~ are lengthy and will not be given here. The Kerr geometry is stationary and axially-symmetric with horizon at r -- re----- M + (M s- a~)v~. For a = 0 it reduces to the Schwarzschild geometry. It represems the geometry of a rotating black hole (of mass M and angular momentum K-: Ma) in the region at and outside the horizon well after the collapse when gravitational radia- tion has died out. It also approximately describes the external gravitation field of a rotating star. 2.2. Dirac equation in curved space-time The generalization of ordinary flat space Dirac equation to curved space-time is more complicated than the corresponding problem of scalar or vector equations because Dirac spinors are defined relative to Minkowski co-ordinates. One way 502 B R/yer and Arvind Kumar to handle this situation is to introduce local Lorentzian frames with respect to which the transformation properties of Dirac spinor are known (Boulwaro op. cit). Lot #o (x) be a set of orthonormal basis vectors defined at every point of space-time : ~° • eb = ~ob; ~oo = ~Txx = ~7z2 = ~ss = 1. (3) The Loreatz transformation of the local frame is given by £'a (X) = A° ~ (x) e~ (x) with Ao ~ ha "~t "~b)' = ~a° °- Under infinitesimal Lorontz transformations (x) ~ ~' (x) = ~ (x) + ½i a,o.~s'~ (x) wh~e s "~ = -~ [r °, rq. The transformation between the co-ordinate basis vectors ~a and the local vectors e, is given by e, (x) = e? (x) % (4) where e, a (x) are called the vierbein components. It is easily seen that e, ~ = ~,b #'~ ~-# (5) whore ~ are the locally inertial co-ordinates. In terms of these vierbein components it is possible to generalize the ordinary derivative of a spinet to one that is covariant under transformation of the local Lorentzian frame. The details of this procedure are available in the literature (Boulware op. cit)and will not be repeated here. The covariant derivative of a spinor ~ is of the form: v, ¢ = O, + ½i s °' o,.~,) ¢ (6) where the 'spinet atKaity' is given by eit,;, = ~p eat, -- Fx, ebx. (7) In terms of the covariant derivative, Dime equation in curved space takes the form 19irae equation in Kerr space-time 503 where Vo = eo~ V/~. (9) To write eq. (8) explicitly for Kerr space-time we need the viorbein components and spinet affinities for this case. After a straightforward computation, we obtain the results given in tables 1 and 2. Using these, the Dirac equation in Kerr space-time is given by [ 1 2MarsinO 3 ~ 1 ~.~a m+ rw~o ~,_ ~V/S w~ ~,+__ ~1~, +~ M--r(1 q- ~) + i~/~ sin 0 2ipv'A --Mar~v/-Ac°sOTrp5W~ y5 + cotO(~ 1 -- a~ sin'V~p, O) sinO( ] +-2pag r, l--p3/ Y5 ~=0; 2Mr~ v~ W = (1 / • (10) It is easily verified that for a = 0 this equation reduces to the known Dirac equa- tion in Schwarzschild space-time. 2.3. The weak-field low-velocity limit Consider the solutions of eq. (10): Table 1. Viorboin components eo# for Kerr-Newman metric. Here a stands for the Lorentz index and ~ for the co-ordinate index. The axes of the local frames are oriented parallel to the co-ordinate axes except for the third direction. For Kerr metric put Q = 0 and for Schwarzschild metric Q = a = O, i a~ t r 0 o ~,I o o o 1 o ~A 0 0 P 1 2 0 0 p- 0 3 asin0 ( 1) H," ~A ~" -- ~. 0 0 v'A- sin 0 504 B R Iyer and Arvincl l~umar Table 2. Spinor affinity toa~ for Kerr-Newman metric given as matrix elements (to/~)a~. The unlisted ones may be obtained from the antisymmetry of t%~/~: o,a~/z (00~)ox ---- ~ [M + r (W~ -- 1)] [~ o _ asinS0 ~/s] a sin 20 (m~).~ = ~ (W----~)[(r ~ +a')8~ a-a~ °] a [Msin 0 ~/1 ..~ (W g _ 1) (r sin 0 ~/£1 _ /~ cos 0 ~2)] (°t~)°a = ~a W ~ ~/A (o'~M V'A~,~ [~[a ssin 20 ~1 + rS/S] (o~)x. -- sinoiv0 I.p[a~(M" -- ,r + rW'Z) (8/~° -- asin s 0 8~ s) -- rWs b s ] (~'~),3= ~~/~cos 0 [~( W~-l)(asin~0~ a_ S 0)_ 8 8 ] The two-component spinors u and v are found to satisfy the equations: E 2MaEr sin 0 o3 v + .-=---~'A ~1 b,v mu -- ~ u + p~ ~/ ~ W ~p M--r 1-~-~ 1 2 ~0 "13 "JI- W (y3 ~b "0 ~ cr1 + ~p g i %/~ sin O 2ip V~A v +2--~pC°t0 (1__ a s sins_~ 0 1~ ~s v-- Marp5B~'A cos 0 ~1 u Ma sin 0 (1 2r~'~ a s u = 0 (12) + 2p8 Wz -- ps] E 2MaEr sin 0 ~s u -- V'A ,1 b,u mv + ~ v -- pz A/ A W ip 1 s W M--r(1 + ~) "--tpcr ~0 u -- i~/~ sine ~s ~,u + 2ip ~ A __ ~z u cot0 ( a s sin s 0)~2 Mar ~/-~ cos0 ol 2ip 1 -- ps u + p~ WS v Ma sin 0 ( 2r2'~ as ---2p s W~ 1-- ps} v=0. (13) In the extreme weak-field low-velocity limit r --~ 0% E --~ m the spinet v -+ 0. Thus in the far-away region v is the smaller component of ~b as expected. To proceed further, notice that oq. (13) has no derivative term in v. We invert the matrix coofiiciont of v up to order 1Jrs and eq. (13) then gives v in terms of u J~irac equation in Kerr space-tlme 505 and its derivatives up to this order. This solution for v is substituted in eq. (12) and retaining terms up to the leading,order, the equation for u is obtained. The actual calculation based on the above procedure is considerably lengthy but straightforward.
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