Separation of variables and symmetry operators for the neutrino and Dirac equations inthe spacetimes admitting a twoparameter abelian orthogonally transitive isometrygroup and a pair of shearfree geodesic null congruencesN. Kamran and R. G. McLenaghan Citation: Journal of Mathematical Physics 25, 1019 (1984); doi: 10.1063/1.526269 View online: http://dx.doi.org/10.1063/1.526269 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/25/4?ver=pdfcov Published by the AIP Publishing

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Separation of variables and symmetry operators for the neutrino and Dirac equations in the space-times admitting a two-parameter abelian orthogonally transitive isometry group and a pair of shearfree geodesic null congruences8

)

N. Kamran and R. G. McLenaghan Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3Gl

(Received 28 June 1983; accepted for publication 7 October 1983)

We show that there exist a coordinate system and null tetrad for the space-times admitting a two­parameter abelian orthogonally transitive isometry group and a pair of shearfree geodesic null congruences.in which the neutrino equation is solvable by separation of variables ifand only if the Weyl tensor IS Petrov type D. The massive Dirac equation is separable if in addition the conformal ~actor satisfi~s a certai~ functional equation. As a corollary, we deduce that the neutrino equation IS separable In a canomcal system of coordinates and tetrad for the solution of Einstein's type D vacuum or electrovac field equations with cosmological constant admitting a nonsingular aligned Max~ell field and that the Dirac equation is separable only in the subclass of Carter's [A] solutIOns and. the Debever-McLenaghan null orbit solutionAo. We also compute the symmetry operators WhICh arise from the above separability properties.

PACS numbers: 04.20.Jb, 03.65.Ge, 02.20.Km

1. INTRODUCTION

Central to the analytic solution of the relativistic hydro­gen atom problem 1.2 stands the fact that in Minkowski space-time, the Dirac equation for a central potential is solv­able by separation of variables in spherical coordinates, the symmetry operator3 which underlies this separability prop­erty being the total angular momentum operator well known to Dirac himself. 4

In curved space-time, although the separability proper­ties of scalar equations such as the Hamilton-Jacobi and Klein-Gordon equations have been studied in detail by Carte..s in the physically important situation of a charged rotating black hole described by the Kerr-Newman solu­tion, it is only in the last decade that significant progress has been achieved in understanding the separability of higher spin wave equations. In particular, the separability of the Weyl neutrino equation (spin one-halt), Maxwell's equations (spin one), and the perturbed Einstein gravitational field equations (spin two) has been established in the Kerr back­ground by Teukolsky6 and, using an analogous method, in the seven-parameter Plebafiski-Demiafiski background by Dudley and Finley.7 All attempts to extend Teukolsky's se­paration method, where previously decoupled single-com­ponent equations are separated, to the Dirac equation for a massive charged spin one-half particle were unsuccessful un­til Chandrasekhar8 ingeniously performed separation prior to decoupling, establishing the separability of the Dirac equation in the Kerr background. Chandrasekhar's result, which was immediately extended to the Kerr-Newman background by Page9 and ToopiO and to the Kinnersley case II vacuum solutions by Guven, II was subsequently analyzed by Carter and McLenaghan I2

•13 who constructed an opera­

tor commuting with the Dirac operator in the Kerr-New-

"This work was supported in part by a grant from the National Sciences and Engineering Research Council of Canada.

man background, admitting the separated solutions as eigen­functions with the separation constant as eigenvalue, which is the symmetry operator associated to this separability, gen­eralizing the total angular momentum operator existing in flat space-time. The symmetry operators for the Dirac oper­ator on an aribtrary curved background were given a tensor­ial interpretation by Carter and McLenaghan 14 and a tensor­ial characterization by McLenaghan and Spindel. 15 The symmetry operators for the neutrino operator have recently been characterized tensorially by Kamran and McLen­aghan. 16

The purpose of this paper is to perform a systematic study of the separability properties of the neutrino and mas­sive charged Dirac equations and an explicit computation of the associated symmetry operators in the class of Lorentzian spaces characterized by the existence of a two-dimensional abelian group of local isometries acting orthogonally transi­tively and the existence of two geodesic and shearfree real null congruences. The motivation behind this choice of background lies in a theorem of Debever, Kamran, and McLenaghan 17. 18 which gives a canonical form for the met­ric and Maxwell field for the class :D of solutions of Einstein's vacuum and electrovac field equations with cosmological constant, for Petrov type D, with a nonsingular aligned Maxwell field. This canonical form, in which the Einstein­Maxwell equations have been integrated and a single expres­sion for the general solution has been given, 19 is the starting point of our study, our aim being to impose only those condi­tions that are required for separability rather than work in the explicit form of the integrated solutions. Section 2 is de­voted to the explicit statement of our hypotheses and results. Sections 3, 4, 5, and 6 contain the proofs of our theorems.

2. HYPOTHESIS AND STATEMENT OF RESULTS

Weare investigating the separability properties of the neutrino equation

1019 J. Math. Phys. 25 (4), April 1984 0022-2488/84/041019-09$02.50 @ 1984 American Institute of PhYSics 1019

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HN¢: = iY'Vk ¢ = 0,

and the Dirac equation

(2.Ia)

H D ¢: = [iY'(Vk - ieA k ) - V~J]¢ = 0, (2.Ib)

where! Yk J is a set of Dirac matrices associated to a Lorent­zian metricgij'V k denotes the covariant differentiation oper­ator20 on four-spinors corresponding to the choice of Y' 's and to the Levi-Civita connection of gij' and where A = Ak dxkis a I-form field, in the class of Lorentzian spaces admitting a metric of the form

ds2 = 2(8 18 2 - 8 3

(4), (2.2a)

where

8 1= (vLT(w,X))-IIZ(w,x)II/2[JW(W)Z(W,X)-1

X (tldu + m(x)dv) +g-2W(W)-ldw], (2.2b)

8 2 = (vLT(w,X))-IIZ(w,x)II/2[ W(w)Z(W,X)-1

X(tldu + m(x)dv) - /g-2W(W)-ldw], (2.2c)

8 3 = (vLT(w,X))-IIZ(w,x)II/2[X(X)Z(W,X)-1

X (t2du + p(w)dv) + lX(X)-ldx], (2.2d)

8 4 = 7P, (2.2e)

with

Ak dxk = T(w,x)(2Z(w,X))-1I2[g-2H(w)W(W)-1

XU8 1+ ( 2) + G(x)X(x)-1(8 3 + ( 4)],(2.2f)

where

Z(w,x) = tIP(W) - t2m(x), g = ((1 + F)/2)1/2, (2.3)

and where tp t2' and/are real constants satisfying ci + t~ =I-° and all functions are real valued.

The Lorentzian spaces whose metric is given by (2.2) and (2.3) are characterized21 by the following properties:

(i) There exists a two-dimensional abelian group oflocal isometries acting orthogonally transitively.

(ii) There exist two geodesic and shearfree real null con­gruences.

The motivation for the study of the separability proper­ties of Eqs. (2.1) in the Lorentzian spaces whose metric is given by (2.2) and (2.3) with the I-form field A = Ak dx k by (2.3) lies in the following result: Let ~ denote the class of solutions of Einstein's vacuum and electrovac field equa­tions with cosmological constant, which may be written as

Rij - !Rgij + Agij = FikF} k - };.gijFrsF rs , (2.4a)

Fik; k = 0, Flij;k I = 0; Fij = 2A lj;i!' (2.4b)

where we permit the cosmological constant A and the elec­tromagnetic field tensor Fij to vanish, that satisfy the follow­ing conditions:

HI. The Weyl tensor is everywhere of Petro v type D. H2. If the Maxwell field tensor Fij is nonzero, it is nonsingu­lar with its principal null directions aligned with the repeat­ed principal null directions of the Weyl tensor. H3. The hypothesis of the generalized Goldberg-Sachs theorem is satisfied, insuring that the null congruences asso­ciated to the principal directions of the Weyl tensor are geo­desic and shearfree.

1020 J. Math. Phys., Vol. 25, No.4, April 1984

Then, we have the following result, proved in Debever, Kamran, and McLenaghan (DKM).22

Theorem 1: For every solution in ~, there exists a sys­tem oflocal coordinates (u,v,w,x) and a null basis of I-forms ! ()" J such that the metric and self-dual Maxwell field take the form

+ F =B(w,x)(8 11\8 2

- 8 3 1\( 4),

(2.5a)

(2.5b)

where B is a complex valued function, with the I-forms ()" and a real vector potential A given by (2.2) and (2.3).

The field equations (2.4) have been integrated for the class ~ and a single expression has been given in DKM for the general solution in the tetrad and coordinates of Theorem 1.

We shall work in the general context of the metric and I-form (2.2) rather than in the explicit form of the integrated ~ solutions, imposing those conditions that are necessary and sufficient for separability and subsequently determining those subclasses of ~ satisfying the separability require­ments.

The main results of this paper are given in the following theorems:

Theorem 2: In the Lorentzian space of Theorem 1, the neutrino equation (2.Ia) admits, in the Weyl representation, an R- separable23 solution of the form

(PO) ( e

icrJ

R I(x)S2(W) ) ,I. = !I = ei(au +f3vIT 3 / 2z -1/4 e

idJ R 2(x)SI(W)

'f' QO' e-l:1IIR I(x)SI(W) ,

Q I. e - i1ll R2(x)S2(W) (2.6a)

where a and f3 are arbitrary real constants and where

dYJ = (4Z)-I(tlm'(x)dw + t2p'(w)dx), (2.6b)

if and only if the Petrov type D condition HI, which insures the existence of YJ 24 as given in (2. 6b), is satisfied.

Corollary: In all space-times in the class ~, the neutrino equation (2.Ia) admits, in the Weyl representation, an R­separable solution of the form (2.6a).

Special cases of our corollary have been obtained using a different approach, where starting from the Weyl neutrino equation in terms of two-spinors, previously decoupled sec­ond-order equations are separated, by Teukolsky25 for the Kerr solution and Dudley and Finley26 for the seven-param­eter Plebanski-Demianski solutions. It should be noted that this decoupling prior to separation procedure seems to re­quire either the vacuum ~ equations or all but two of the electrovac ~ equations27 rather than the weaker conformal­ly invariant Petrov type D equation appearing in Theorem 2. Although Theorem 2 and its corollary deal with the separa­bility of the neutrino equation in ~ and hence could be en­tirely formulated in terms of a single two-spinor, we employ a four-spinor formalism to be able to compare the above results with the following separability theorems for the mas­sive charged Dirac equation, which may be formulated ei­ther in terms of four-spinors or a pair of two-spinors.

N. Kamran and R. G. McLenaghan 1020

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Theorem 3: In the Lorentzian space of Theorem 1, the Dirac equation (2.1b) admits, in the Weyl representation, an R-separable solution of the form given in (2.6a) and (2.6b) if and only if:

(i) The Petrov type D condition (HI)' which insures the existence of f!iJ as given in (2.6b), is satisfied.

(ii) There exist real valued functions h (w) and g(x) such that

(2.7)

Theorem 3 enables us to characterize those space-times in :tI admitting a separable Dirac equation through the following corollary.

Corollary: Within the class :tI of space-times, the Dirac equation (2.1b) admits, in the Weyl representation, an R­separable solution of the form given in Eqs. (2.6a) and (2.6b) only in Carter's [A] family of solutions and in the Ao null orbit solution of Debever and McLenaghan.28

Our corollary becomes an immediate consequence of Theorem 3 once the values of Z, T, and f!iJ as given in DKM, which contains the result of the integration of the field equa­tion (2.2) in the canonical (u,v,w,x) class of coordinates of Theorem 1, are substituted in Eq. (2.7). Special cases of our corollary have been obtained for the Kerr solution by Chan­drasekhar,29 whose approach was subsequently applied by Page30 and TOOp31 to show separability of the Dirac equa­tion in the Kerr-Newman solution and Giiven32 in the Kin­nersley case II vacuum solutions.

There is a well-known connection between separation of variables and constants of the motion. In the case of the Hamilton-Jacobi equation for the massive charged particle orbits, it has been established in DKM that Carter's [A] fam­ily of solutions and theAo null orbit solution are those space­times in :tI where one has separation. This separation ofvar­iables gives rise to a quadratic first integral of the equations of motion which implies the existence ofa (O,2)-Killing ten­sor and whose Poisson bracket with the Hamiltonian vanish­es. In the case of the Hamilton-Jacobi equation for the zero rest mass particles (null geodesics) it has been shown by Deb­ever and McLenaghan33 (in the electrovac case) and Czapor and McLenaghan34 (in the vacuum case) that a separable coordinate system exists for every solution in:tl. This separa­tion of variables gives in turn rise to a function on the cotan­gent bundle which is also quadratic in the momenta, whose existence yields a (O,2)-conformal Killing tensor, and whose Poisson bracket with the Hamiltonian will beproportional to this Hamiltonian.

When studying the motion of test particles in the first­quantized limit rather than in the classical description pro­vided by the Hamilton-Jacobi equation, the connection between separation of variables and constants of motion is reflected in commutation relations rather than Poisson bracket relations (for a general discussion of this point, see Carter35). In the case of the Klein-Gordon equation for charged (massive or massless) spin zero particles, the separa­bility properties in the class :tl have been studied in detail in DKM. The constants of the motion appear there as symme­try operators admitting the separated solutions as eigenfunc­tions with the separation constant as eigenvalue. Explicitly,

1021 J. Math. Phys., Vol. 25, No.4, April 1984

denoting by HK the massless conjormally invariant Klein­Gordon operator, for which R-separability was established in DKM for the entire:tl class, we have a symmetry operator Kk defined by

KK = TUKI(UKxWKW - UKwWKx)T-I, (2.8a)

where

UKx = - E2mT-2, UKw = E!pT - 2, UK = UKw + UKx '

W Kx = X 2J; + (X2)'Jx + X -2[( - mau + E1aV)2

+ ieG( - mau + EIJv ))

(2.8b)

+ GX -2(ie( - mau + Etav ) - e2G) + i(X 2)", (2.8c)

W Kw =jW2a; + (f(W2)' - g-2(1 - j2)(pau - E2av ))aw + g-4W- 2 [ - j(pJu - E

2av )2

- gZieH(pJu - E2av )]

- 2- lg- 2 (1 - j2)p'au

+ HW- 2( - ieg- 2(pJu - e2av )

+ e2g-'iH) + U(W 2)".

If the wave function takes the R-separable form

¢(u,v,w,x) = ei(Uu HV)¢I(W)¢2(X)T(w,x),

(2.8d)

(2.8e)

the symmetry operator Kk satisfies the eigenvalue equation

(2.8t)

where A K denotes the separation constant arising from the separation of variables, and the commutation relation

[KK,HK ) = rKHK,

where

(2.8g)

rK =TUK1 ![UKx,WKw ] - [UKw,WKw]JT-t,

(2.8h)

and where use has been made in the expressions of those :tI field equations required for separability. It can be checked that in the massive case, one obtains in the subclass of:tl where one has separability, that is the Carter [A] and theAo null orbit solutions, a symmetry operator which reduces to a commuting operator and still satisfies the eigenvalue equa­tion (2.8t).

Central to the Poisson bracket discussion of the con­stants of the motion for the charged particle orbits in :tI and the commutator discussion of the constants of the motion for the first-quantized spin zero test particles in :tl, as they arise from the separability of the Hamilton-Jacobi and Klein­Gordon equations, stands a lemma, first stated by Carter36 in the Poisson bracket case and subsequently employed by Carter and McLenaghan37 in the commutator case, that will also be applied to the derivation of the symmetry operators associated to the separability of the neutrino and Dirac equa­tions in :tI.

Lemma: Let M and N be matrix differential operators and P, PI' P2 be nonsingular matrices not involving any dif­ferentiation operators. Then

N. Kamran and A. G. McLenaghan 1021

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[P -I(M + N),P -I(PIN - P2M)]

=P-I[M,P-l(P1 +P2)]N

- P -I [N,P -l(P1 + P2)]M

+P- 2(P1 +P2 )[M,N] +P-I([N,PI ] - [M,P2 ]

-NP-I[PI,P] -MP- l [P2,P]jP- l(M +N). (2.9)

The commutation relation (2.8g) can be deduced from this lemma by letting

P=Uk , PI=UKx ' P2 =UKw ' M=WKx ' N=WKw (2.10)

in Eq. (2.9). The separability Theorems 2 and 3 and their corollaries

give rise to matrix symmetry operators for the neutrino and Dirac equations, which we will compute using the above Lemma. The expressions of these symmetry operators are given in the following theorems.

Theorem 4: In the single expression given in DKM for the class ::3) solutions, in the tetrad and coordinates of Theorem 1, let KN be the operator defined by

KN =SNU Ii I(UNx WNw - UNwWNx)SIi I,

where

(2.lla)

SN

= T 3/2 diag{ [b 2(ew cos r + k) - i(ex sin r + I)] -1/2, [b 2(ew cos r + k) - i(ex sin r + I)] -1/2, [b 2(ew cos r + k) + i(ex sin r + I)] -1/2, [b 2(ew cos r + k) + i(ex sin r + I)] -1/2l, (2.IIb)

UNx = T-li(ex sin r + I)diag(I, - 1,1, - 1), (2.llc)

UNw = T-Ib 2(ew cos r + k )diag(I, - 1, - 1,1), (2.lld)

UN = UNx + UNw , (2.IIe)

a a a

LD W N, ~ ( ~ a -L/

L- a x

-Lx+ a a

a a - f.-le(ex sin r + I)

a W -

(

a

Nw - _ ~;:;

a a a

L+ w

L+ w

a a a

(2.llt)

L x- = 2- 1/2( - iXJx + X-l(E1Jv - mJu) - i2- IX'), (2.IIh)

L:; = 2- l/2(WJw + jg-ZW-1(pJu - E2J v ) + 2- 1 W'), (2.lli)

L;:; = 2- l/2( - fWJw + g-2W- l(pJu - E2J v ) - 2- lfW'). (2.1Ij)

The operator KN has the following properties: (i) The operator KN is a symmetry operator for the neu­

trino operator HN , that is

[KN,HN] = rNHN'

where

(2.12a)

r N = S N U Ii 1 { [ U Nx , WNw] - [ U Nw' W Nx Jl S Ii I. (2.12b)

(ii) The R-separable solution (2.6) is an eigenspinor of KN with the separation constant 4 N arising from the separa­tion of variables as eigenvalue:

(2.13)

Now, for the Dirac equation, we have: Theorem 5: In the expression given for the Carter [.4 ]

solutions and theAo null orbit solution in DKM, which uses the tetrad and coordinates of Theorem 1, let KD be the oper­ator defined by

KD =SDU;;I(UDxWDW - UDwWDxlS;;I,

where

SD = T- 3IZSN, UDx = TUNx '

UDw = TUNw ' UD = UDx + UDw '

(2.14a)

(2.14b)

WDx =

(

f.-le(ex si~n r + I)

-Dx+

D-x

a f.-le (ex sin r + I)

a

_ (- if.-leb 2(e~ cos r + k)

W Dx - -D-w

a

D + =L + -i2- 1/2eX- I G x x '

D x- = L x- - i2- 1/2eX -IG,

D:; =L:; _i2-1/2jg-2eW-IH,

D;:; = L;:; - i2-l/2g-2eW-IH.

a if.-leb 2(ew cos r + k)

a D+

w

(2.14d)

(2.14e)

1022 J. Math. Phys., Vol. 25, No.4, April 1984

D+ w

a if.-leb 2(ew cos r + k)

a

-~;:; ) a '

- if.-leb 2(ew cos r + k)

The operator KD has the following properties:

(2.14c)

(i) The operator KD commutes with the Dirac operator

HD :

(2.15)

N. Kamran and R. G. McLenaghan 1022

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(ii) The R-separable solution (2.6) is an eigenspinor of Ko with the separation constantA D arising from the separa­tion of variables as eigenvalue:

(2.16)

Operators commuting with the Dirac operator have previously been considered, the first example of which is given in fiat space-time by the total angular momentum op­erator 7 38 which is notably associated to the separability in spherical coordinates of the Dirac equation for a central po­tential and whose eigenvalues yield quantum numbers.

First-order operators which commute with the Dirac opera­tor on a curved background were obtained in the Kerr-New­man solution by Carter and McLenaghan,39 using the above Lemma in their analysis of Chandrasekhar's separation of variables procedure. A tensorial characterization of the op­erators commuting with the Dirac operator on an arbitrary curved background was given by Carter and McLenaghan40

and McLenaghan and Spindel.41 The symmetry operators for the neutrino operator on an arbitrary curved space-time have been characterized tensorially by Kamran and McLen­aghan.42

3. PROOF OF THEOREM 2

The essence of our proof lies in the analysis performed by Carter and McLenaghan43 of Chandrasekhar's separation of variables procedure for the Dirac equation in the Kerr space-time.

In the Newman-Penrose spin coefficient formalism, the neutrino equation (2.1a) reads,44 using the Weyl representation for¢,

(

a a

A+/-l-Y - (8" + 1T - a)

a a

- (0 +p'-r)

D+,€-p "8+p-r

a a

In the tetrad and coordinates of Theorem 1, we have, letting

(J 1 = ni dxj, (J2 = Ii dxi , (J3 = - mj dx

j, (J4 = - mj dxi ,

the following:

D + E - P = T2- 1 / 2Z -1/2[ waw + jg-2W- 1(pau - E2av )

-lWZ-'E,( -p' + im') + !W' - ~T,wT-IW], A - Y + /-l = T2- 1I2Z -1/2[ - fWaw + g-2W- 1(pau - E2av )

+ lfWZ-'EI( -p' +im') - !fW' +~fT'wT-'W],

o +P' - r = T2- 1/2Z -1/2 [ - iXa" +X -1(Elav - maul - ~z -IE2(p' - im') - !iX' + ~iT,,, T-1X],

"8 - a + 1T = T2-1/2Z -1/2[iXax +x-1(Elav - maul + lXZ -IE2(p' - im') + ~iX' - ~iT,x T-IX].

We first perform the four-spinor transformation defined by

¢=S¢',

with

where

dYJ = (4Z)-I(E}m' dw + E2P' dx),

the integrability condition of which is satisfied thanks to the Petrov type D condition (Hl).45 The resulting operator acting on the transformed spinor ¢' is

S-'H S~ TZ-"'( . ~ a e- 2j:5iJL .: eU'L<-) 0 e- 2j:5iJ L x+ e - 2j:5iJ L w-

N 2.:5iJ L - - e2j:5iJL,,- a a ' e w

_ e2j:5iJ L / e2j:5iJ L': a a r

where the operators L ,,+ , L ,,- , L .: , and L ,;; are defined by Eqs. (2.11). We next mUltiply the operator which acts on the transformed spinor ¢' by the nonsingular separating matrix U defined by

( 0 -I a W N = US HNS=

-L';;

U = Z 1I2T -I diag(e2j:5iJ, _ e2i:5iJ, _ e - 2j:5iJ ,e - 2i:5iJ), (3.5) -Lx+

to obtain an operator W N given by

a L+ w

a -L,,+

L-"

a L+ w a

1023 J. Math. Phys., Vol. 25, No.4, April 1984 N. Kamran and R. G. McLenaghan

(3.1a)

(3.lb)

(3.2a)

(3.2b)

(3.2c)

(3.2d)

(3.3a)

(3.3b)

(3.3c)

(3.4)

L,,- ) -L,;;

a . a (3.6)

1023

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Now, taking into account the existence for all solutions in ::tl of a two-parameter abelian group of local isometries,46 which manifests itself in the form given in Theorem 1 for the met rices of the ::tl class by u and v being ignorable coordi­nates, we may write

(

Fl(W'X)) .li(U v W x) = ei(au + f3v) F2(w,x) 'i" " , Gl(w,x) ,

G2(w,x)

(3.7)

where a and f3 are arbitrary constants.

When the form (3.7) for ¢' is substituted into the trans­formed equation W N¢' = 0, we obtain the following system of first-order partial differential equations:

L :: Gl + L x- G2 = 0, - L :;; Fl + L x- F2 = 0, (3.8a)

L / Gl + L :;; G2 = 0, - L / Fl + L :: F2 = 0, (3.8b)

where L / , L x- , L :: , and L:;; denote the operators given in Eqs. (2.11) where the derivatives with respect to ignorable coordinates a / au and a/au have been, respectively, replaced by ia and if3. Following Chandrasekhar,47 we seek separable solutions of the form

Fl(w,x) = R l(x)S2(W), Gl(w,x) = Rl(x)SI(W),

F2(w,x) = R2(x)Sj(w), G2(w,x) = R2(x)S2(W),

which, when substituted into the system (3.8), yield

Rl(x)i,:: SI(W) + S2(w)L x- R2(x) = 0,

SI(W)i, / Rl(X) + R2(x)i,:;; S2(W) = 0,

- R I (x)L :;; S2(W) + SI(W)i, x- R2(x) = 0,

- S2(W)i, x+ Rj(x) + R2(x)i, :: SI(W) = o.

4. PROOF OF THEOREM 3

(3.9a)

(3.9b)

(3. lOa)

(3. lOb)

(3.lOc)

(3.lOd)

These relations give then immediately

L :: SI(W) = AlS2(W), L:;; S2(W) = A3SI(W), (3.11a)

L x-R2(x) = -AlRl(x), L x-R2(x)=A3R l(x), (3.11b)

L:;; S2(W) = A2SI(W), L:: SI(W) = A4S2(W), (3.11c)

L / Rl(x) = - A2R2(x), L / Rl(X) = A4R2(x), (3.lId)

where AI> A2, A3, A4 are the separation constants, which for consistency must satisfy

(3.12)

Equations (3.11) are thus equivalent to the system of coupled first-order ordinary differential equations

L :: SI(W) = ANS2(W), L:;; S2(W) = - ANSl(W), (3.13a)

L x-R2(x) = -ANR1(x), L x+R l(x)=ANR2(x),(3.13b)

and the proof of Theorem 2 is complete. Note that the first­order system (3.13) can be written as a system of decoupled second-order ordinary differential equations which reads as follows:

L :: L :;; S2(W) = - A ;,S2(W),

L :;; L :: Sj(w) = - A ;,Sj(w), L / L x- Rz(x) = - A ;'R2(x). (3.14b)

It may be noted that one recovers as special cases of Eqs. (3.14) TeukolskY's48 decoupled equations in the Kerr back­ground and Dudley and FinleY's49 decoupled equations in the Plebaiiski-Demiaiiski background when the metric functions appearing in Theorem 1 are appropriately special­ized.

The first steps of our proof are similar to those of Theorem 2. In the Newman-Penrose formalism. the Dirac equation (2.1b) reads,50 using the Weyl representation for ¢,

- ill e

( 0

0 D+"E-p-eAj 8H-a_ieA')C) 0 - ille {} + 1J - T - ieA4 .:i + Jl - r - ieA2 !.I = 0 (4.1)

.:i + Il - r - ieA z - (8 + f3 - 1') + ieA3 - ille o QO' 0 '

- ({) + 1T - a) + ieA4 D+€-p-ieA l 0 - ille (F 0

where Aa are the components of the vector potential I-form in the basis lea J. Having written Eq. (4.1) with the tetrad, coordinates, and vector potential given in Eqs. (2.3), (2.4), (2.5), and (3.1b) we perform, as we did for the neutrino equation, the four-spinor transformation defined by Eq. (3.3) and we multiply on the left the Dirac operator S -IHDS acting on the transformed spinor ¢' by the separating matrix U given in (3,5) to obtain an operator W D given by

o D+ w

liJ.e Z IIzT -le2idJ

D;;

-Dx-

illeZ l/ZT -Ie - Zid! - D:;; (4.2)

Dx

- )

. ZI/~T-l -21d! ' D::

where the operators D x+ , D x- , D :: , and D:;; are those giv­en in Eqs. (2.14).

A necessary condition for the separability of the trans­formed equation W D ¢' = 0 is that each diagonal entry in the expression given in Eq. (4.2) for the operator W D split into the sum of a function of x and a function of w, in other words

1024 J. Math. Phys., Vol. 25, NO.4, April 1984

o - Ille e i

that there exist real valued functions/(x), g(x), h (w), and k (w) such that

ZI/ZT- 1e2i&J = [f(x)+ig(x)] + [h(w)+ik(w)]. (4.3)

However, this condition is not sufficient for separability with ¢' given (following Chandrasekhar) by

N. Kamran and R. G. McLenaghan 1024

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@t(X)K2(W))

t/J(U,V,W,x) = ei1au + /:Iv) H 2(x)Kt(W) . Ht(x)Kt(w)

2(X)K2(W)

(4.4)

Indeed, substituting Eqs. (4.3) and (4.4) into W D t/J' = 0 with W D as given by (4.2), we obtain, in the same way in which we obtained Eqs. (3.11), the following equations:

D': Kt(w) - if-te(h (w) + ik (w))Kz(w) = f-ttK2(W), (4.Sa)

D x- Hz(x) - if-te(f(x) + ig(x))Ht(x) = - f-ttHt(x), (4.Sb)

D;;; Kz(w) - if-te(h (w) + ik (w))Kt(w) = f-t2Kt(W), (4.Sc)

D x+ Ht(x) - if-te(f(x) + ig(x))H2(X) = - f-t2Hz(x), (4.Sd)

D;;; Kz(w) - if-te(h (w) - ik (w))Kt(w) = f-t3Kt(W), (4.Se)

- D x- H 2(x) - if-te(f(x) - ig(x))Ht(x) = - f-t3H t(X), (4.Sf)

D': Kt(w) - if-te(h (w) - ik (w))Kz(w) = f-t4Kz(w), (4.Sg)

- D x+ Ht(x) - if-te(f(x) - ig(x))H2(X) = - f-t4H 2(X), (4.Sh)

where D x+ , D x- , D .: , and D;;; denote the operators given in Eqs. (2.14) where the derivatives with respect to the ignor­able coordinates a lau and a lav have been, respectively, re­placed by ia and if3.

Subtracting Eq. (4.Sg) from Eq. (4.Sa), we obtain

k (w)=c t , (4.6a)

wherec t is a real constant and adding Eq. (4.Sh) to Eq. (4.Sd) we obtain

(4.6b)

where C2 is a real constant. Defining now new functions g(x) and h (w) by

g(x) = g(x) + Ct, h (w) = h (w) + Cz, (4.7)

the condition (4.3) reduces, dropping tildes, to the following:

Z tl2T -te2i3iJ = h (w) + ig(x), (4.8)

which along with the Petrov type D condition HI constitutes a necessary and sufficient condition for the existence of a separable solution of the form given in Eqs. (2.6); indeed, substituting Eq. (4.8) into Eq. (4.2) yields, by a consistency argument similar to that used for the neutrino equation, the following system of coupled first-order ordinary differential equations:

D': Kt(w) - if-teh (W)K2(W) = ADK 2(w),

D;;; K 2(w) - if-teh (w)Kt(w) = - ADKt(w), (4.9a)

D x- H 2(x) + f-teg(x)Ht(x) = - ADHt(x),

D x+ Ht(x) + f-teg(x)H2(X) = ADH2(x). (4.9b)

This completes the proof of Theorem 3. It should be noted that one recovers as special cases of

Eqs. (4.9) the separated equations ofChandrasekha~t in the Kerr solution, of Carter and McLenaghan52 in the Kerr­Newman solution and of Giiven53 in the Kinnersley case II vacuum solutions. One also notes that the first-order system (4.9) can be written as a system of decoupled second-order ordinary differential equations which reads as follows:

1025 J. Math. Phys., Vol. 25, No.4, April 1984

o 0 if-te (D ;;; h (w)) 0 + D;;; D': Kt(w) - D w Kt(w)

AD + if-teh (w)

+ (A t + f-t;h (w)Z)Kt(w) = 0,

o 0 iJJ (D + h (w)) 0

D + D - K (w) - r'e w D - K (w) w w 2 '+' h() W 2 - ""-D If-te W

+ (A t + f-t;h (W)Z)K2(W) = 0,

D x- D x+ Ht(x) - f-te(D x- g(x)) D x+ Ht(x) - AD + f-teg(x)

+ (A t - f-t;g(x)2)Ht(x) = 0,

D x+ D x- H 2(x) - f-te(D x+ g(x)) D x- Hz(x) AD + f-teg(X)

+ (A t - f-t;g(x)Z)Hz(x) = O.

(4. lOa)

(4. lOb)

(4.lOc)

(4.lOd)

We again remark that one recovers as special cases of Eqs. (4.10) the decoupled second-order equations ofChandrasek­har54 in the Kerr solution, of Page 55 and Carter and McLen­aghan56 in the Kerr-Newman solution, and of Giiven57 in the Kinnersley case II vacuum solutions.

5. PROOF OF THEOREM 4

The main tool in our proof is the identity (2.9). Ifwe go back to Eq. (3.6), we see that we have a splitting

W N = WNw + W Nx' (S.l)

where WNw and W Nx are given in Eqs. (2.11). It should be noted that except for derivatives with respect to the ignora­ble coordinates a 1 au and a 1 av, WNw depends only on wand W Nx depends only on x, moreover it may be readily checked that

[W Nw' W Nx] = O. (S.2)

Now, if we refer ourselves to the single expression presented in Ref. 18 for the general solution of the class Sl) field equa­tions, we have as solution to Eqs. (2.6c)

f!lJ = (iI4) [In(b 2(CW cos r + k ) - i(cx sin r + I)) -In(b 2(CW cos r + k) + i(cx sin r + I))], (S.3)

which implies that the matrix UN' obtained upon substitu­tion of f!lJ's expression (4.3) into the definition (3.S) ofthe matrix U, splits as follows

UN = UNw + UNx , (S.4a)

with U Nw and U Nx given by Eqs. (2.11). Also, one checks immediately that

[UNw,UNx ] = O. (S.4b)

Thus, if we set in the identity (2.9)

P= UN' Pt = UNx ' P2 = UNw ' M = WNx ' NNw'(S,S)

we obtain part (i) of Theorem 4, thanks to Eqs. (5.2) and (S.4). To prove part (ii) of Theorem 4, we proceed as follows. Equa­tions (3.13) are equivalent, using Eqs. (3.9), to the following relations:

i .: F2(w,x) = AN Gz(w,x), i x+ F\(w,x) = ANG2(w,x), (S.6a)

i.: Gt(w,x) = ANFt(w,x), i x+ Gt(w,x) = ANFz(w,x), (S.6b)

N. Kamran and R. G. McLenaghan 1025

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i x- Gz(w,x) = - ANFI(w,x), i;; Gz(W,x) = - ANFz(w,x), (5.6c)

i x- Fz(w,x) = - ANGI(W,X), i;; FI(w,x) = - ANGI(w,X). (5.6d)

Now, on account of Eqs. (5.1b) and (3.7), it is easily seen that Eqs. (5.6) are equivalent to

(5.7)

Equations (5.7) enable us to show as follows that tf is an eigenspinor of KN with the separation constant AN as eigen­value. Let us first note that S N as defined in (2.11 b) is ob­tained by replacing §J with its expression (5.3) for the class:D solutions in the definition (3.3b) of S. We have then from (2.11a), (3.3a), (5.4a), and (5.7):

KNtf=SU ii I(UNxWNwtf' - UNwWNxtf')

=ANSU iil(UNx + UNw )¢' =ANtf·

6. PROOF OF THEOREM 5

(5.8)

Ifwe consider the separability condition (4.8) and we refer ourselves to the single expression presented in DKM for the general solution of the class :D field equations, we see that in the tetrad and class of coordinates of Theorem 1, we have separability for those solutions with

T(w,x) = 1, (6.1)

in the above-mentioned single expression, that is, the Carter [A] solutions and theAo null orbit solution listed in the Cor­ollary to Theorem 3.

Using now Eq. (6.1) along with the expression (5.3) for §J and the fact58 that

Z (w,x) = b 4(ew cos r + k)Z + (ex sin r + I)z, (6.2)

in the above-mentioned single expression, we see from (4.2) that there is a splitting

W D = W Dw + W Dx' (6.3)

where W Dw and W Dx are as given in Eqs. (2.14). It should be noted that in analogy with the neutrino

case, W Dw and W Dx depend, except for derivatives with re­spect to the ignorable coordinates a/au and a/au, only on w and x, respectively. We also have

(6.4)

Now, the matrix UD' obtained upon substitution of §J's expression (5.3) and the separability condition (6.1) into the definition (3.5) of the matrix U, splits as follows:

(6.5)

with UDw and UDx given by Eqs. (2.14). It may then be veri­fied by direct calculation that

[ U Dw ,U Dx ] = 0, [ U Dw' W Dx ] = 0, [ U Dx ,W Dw] = 0. (6.6)

It is then straightforward to obtain part (i) of Theorem 5 by applying the Identity (2.9) with

P= UD' PI = UDx , Pz = UDw ,

M=WDx , N=WDw , (6.7)

and using Eqs. (6.4), (6.5), and (6.6).

1026 J. Math. Phys., Vol. 25, No.4, April 1984

To prove part (ii) of Theorem 5, we note first that Eqs. (6.2), (6.1), (5.3), and (4.8) y~ld

h (w) = b Z(ew cos r + k), g(x) = ex sin r + I, (6.8)

so that Eqs. (4.9) imply, using Eqs. (3.7), (4.4), and (6.8) that

h.: Fz(w,x) - ilteb Z(ew cos r + k )Gz(w,x) = ADG2(W,X), (6.9a)

h x+ FI(w,x) + Ite(ex sin r + I)G2(w,x) = AD Gz(w,x), (6.9b)

h.: G1(w,x) - ilteb Z(ew cos r + k )Fl(W,X) = ADFI(w,x), (6.9c)

h / GI(w,x) + Ite(ex sin r + 1 )Fz(w,x) = ADFz(w,x), (6.9d)

h x- Gz(w,x) + Ite(ex sin r + 1 )FI(W,X) = - ADFI(w,x), (6.ge)

h;; G2(w,x) - liJ,eb 2(ew cos r + k )Fz(w,x) = - ADFz(w,x), (6.9f)

D x- F2(w,x) +lte(ex sin r + I)GI(w,x) = -ADG1(W,x), (6.9g)

h;; FI(w,x) - ilteb 2(ew cos r + k )GI(w,x) = - AD GI(w,x). (6.9h)

Now, Eqs. (6.9), using the expressions (2. 14c) for W Dw and W Dx and Eq. (3.7), are equivalent to

WDwtf' =ADtf', WDxtf' = -ADtf'· (6.10)

We then have from Eqs. (2. 14a), (3.3a), (6.5), and (6.10)

KDtf = SU;; l(UDx WDwtf' - UDw WDxtf')

=ADSU;;l(UDX + UDw)¢'=ADtf. (6.11)

Let us finally note that the expression of the operators W Dw and WDx verifying the property (6.10) was computed in the Kerr-Newman background by Carter and McLenaghan59

in their analysis of Chandrasekhar's separation of variables procedure.

Ip. A. M. Dirac, The Principles o/Quantum Mechanics (Oxford. U. P., London, 1958), 4th ed., p. 269.

2L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955), 2nd ed., p. 485.

3W. Miller, Jr., Symmetry and Separation 0/ Variables (Addison-Wesley, Reading, MA, 1977), p. 2.

'Reference I, p. 267. 5B. Carter, Phys. Rev. 174,1559 (1968); Commun. Math. Phys. 10, 280 (1968).

"S. A. Teukolsky, Astrophys. J. 185,635 (1973). 7 A. L. Dudley and J. D. Finley, III, J. Math. Phys. 20,311 (1979). 8S. Chandrasekhar, Proc. R. Soc. London Ser. A 349,571 (1976). 9D. N. Page, Phys. Rev. D 14, 1509 (1976). ION. Toop, preprint, D. A. M. T. P., Cambridge (1976). II R. Giiven, Proc. R. Soc. London Ser. A 356, 465 (1977). 12B. Carter and R. G.McLenaghan, Phys. Rev. D 19, 1093(1979). I3B. Carter and R. G.McLenaghan, in Proceedings o/the Second Marcel

Grossmann Meeting on Recent Developments o/General Relativity Trieste, 1979), edited by Ruffinni (North-Holland, Amsterdam, 1982).

I'Reference 12. I5R. G. McLenaghan and Ph. Spindel, Phys. Rev. D 20, 409 (1979); Bull.

Soc. Math. Belg. XXXI, 65 (1979). 16N. Kamran and R. G. McLenaghan, Lett. Math. Phys. 7, 381 (1983). I7R. Debever, N. Kamran, and R. G. McLenaghan, J. Math. Phys. (in

press). 18The results of Ref. 17 were announced by the same authors in Phys. Lett.

93A, 399 (1983) and in Bull. Cl. Sci. Acad. Roy. Belg. LXVIII, 592 (1982). 19Reference 17. 20 A. Lichnerowicz, in Relativity, Groups and Topology, edited by B. S.

N. Kamran and R. G. McLenaghan 1026

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DeWitt and C. DeWitt (Gordon and Breach, New York, 1963), p. 823. 2'R. Debever, Bull. Cl. Sci. Acad. Roy. Belg. LV, 8 (1969). 22Reference 17. 23p. Moon and D. E. Spencer, Field Theory Handbook (Springer-Verlag,

New York, 1971), 2nd ed., p. 96. 24Reference 17, Eq. (4.2). 25Reference 6. 26Reference 7. 27J. M. Stewart and M. Walker, Proc. R. Soc. London Ser. A 341, 49 (1974). 28R. Debever and R. G. McLenaghan, J. Math. Phys. 22,1711 (1981). 29Reference 8. 30Reference 9. 31 Reference 10. 32Reference 11. "Reference 28. 34S. R. Czapor and R. G.McLenaghan, J. Math. Phys. 23, 2159 (1982). "B. Carter, Phys. Rev. D 16, 3395 (1977). 36B. Carter, in Black Holes, edited by B. DeWitt and C. DeWitt (Gordon

and Breach, New York, 1973), p. 118. 37Reference 12. 3"Reference 4. 39References 12 and 13.

1027 J. Math. Phys., Vol. 25, No.4, April 1984

4°Reference 12. 4'Reference 15. 42Reference 16. "Reference 12. "Reference 8, Eqs. (9) to (13). 45Reference 17, Eqs. (4.1) and (4.2). '''This was demonstrated in the electrovac case in Ref. 28 and in the vacuum

case in Ref. 34. 47Reference 8, Eqs. (30). 48Reference 6, Eqs. (4.9) and (4.10) with lsi = \. 49Reference 7, Eqs. (5.18a) and (5.18b) with lsi = \. 50Reference 9, Eqs. (3H6). "Reference 8, Eqs. (40) and (41). 52Reference 13, Eqs. (4.9) and (4.10). "Reference II, Eqs. (47H50). ;4Reference 8, Eqs. (44) and (45). "Reference 9, Eqs. (19) and (20). 56Reference 13, Eqs. (4.13) and (4.14). 57Reference II, Eqs. (54) and (55). 58Reference 17, Eqs. (2.6aH2.6c). 59Reference 13, Eq. (4.5).

N. Kamran and R. G. McLenaghan 1027

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