PHYSICAL HEVIE% 0 VOLUME 10, NUMB ER 6 15 SEPTEMBER 1974

Teukoh)ky equation and Penrose wave equation*

Michael P. Ryan, Jr.Center for Relativity Theory, Univeruty of Texas at Austin, Austin, Texas 7S712

(Received 28 March 1974)

It is shown that Teukolsky's equation can be derived from a secondwrder wave equation for theRiemann tensor, The derivation is done in a way that emphasizes the modern tensor-analysis content ofthe Newman-Penrose formalism.

I. INTRODUCTION

Recently Teukolsky" published a separableequation for gravitational-wave perturbations ina Kerr black-hole background. This wave equationwas derived for certain of the Newman-Penrosequantities' and the derivation was done entirelyin the Newman-Penrose formalism. In that for-malism the equation seems to emerge mysteriouslyfrom the arcane-looking Newman-Penrose Bianchiidentities. We shall show that Teukolsky's equa-tion can be derived simply from a second-orderequation for the Riemann tensor that goes back toPenrose. "We shall also do the derivation in moreconventional notation to clarify the relationshipbetween the tmo equations. At first we will restrictourselves to vacuum perturbations, and then showhow to extend the derivation to perturbations wherea small amount of stress-energy is allowed.

Rif ua8 ~ b+Rif „ba.s +R)f us b. n =0. (2.1)

Covariantly differentiate them once, and subtracttwo covariantly differentiated Bianchi identitiesto arrive at

II. THE PENROSE W'AVE EQUATION

The idea of a second-order equation for theRiemann tensor has had a peripatetic career. Itseems to have been written down first by Pen-rose." Such an equation could have been givenbefore this; in fact it was possible any time afterthe discovery of the Bianchi identities in 1902,'but a diligent search of the literature indicates thatit was not. Since it appeared it has surfaced sev-eral times' ' with no reference to its antecedents,and has been rediscovered at least once."

Since the derivation is simple, we will. give ithere. Begin with the Bianchi identities

if uns; be+ )f uba 8 e+ p vsb ae Rp vbne8 if uebas Rif vne b 8 p vs be n )f ue8; bn )1 vbe8n —0

(2.2)

Use the Ricci identities to replace the commutators [second, and fourth, third, and seventh terms of (2.2)]by products of Riemann tensors; next write the sixth and eighth terms as R,„,. b. s and R,s„„.b. and usethe differential Bianchi identities to replace them by -R,b„.„.s-R«ub. „.s and -R,sb„, n-Res vb. „.a.Transvect the resulting equation by g ' to arrive at

Rif vas b e 2RifysbRv a +2R)1ynb&v 8 +R)fubYR nsbe y b y b

R„+„)„R g R„„qgR ~--R~~. „.g+R„„.„g+R„g „.~ .R„g „~=0.-(2.3)y Y

Notice that only the first four terms on the left-hand side of (2.3) survive in a, vacuum.

III. THE TEUKOLSKY EQUATION IN A VACUUM

We want to consider Eq. (2.3) in a vacuum in the following frame:

8eo= l )f

~x(3.1)

~'= -n„dx", ~' = —l„dx", cv' = m „dx",

where for the Kerr" metric in Boyer-I indquist coordinates" we have"

f" =((r' +a), 1, Oa)//b, , n" =((t'+a'), -n„0, a)/2&, ~" =(-&a stnt), 0, -1, -&/sfnt))P /v2,

where p = -(r —iacos 6) . The Kerr metric in Boyer-Lindquist coordinates has the form

(3 2)

10

10 TEUKOLSKY EQUATION AND PENROSE WA VE EQUATION

ds'= — 1 — dt2 —4Marsin'8 Z dtdp+ Z ~ dr'+Zd8'+sin'8 r'+a' -2Ma'rsin'8 Z d, 3.32Mr

with

~ =r '+ a' cos'8

and

g =r' —2Mr+a2.

In the frame (3.1) the metric becomes

0 -1 00-1 0 000 0 0 1

0 0 1 0

(3.4)

~op» ~2) ~ok» ~3) ~ok»

Io —I3 I0„-I2 IS —Z'2

(3.'I)

Before we write down the nonzero coefficientsfor the Kerr metric in our frame, notice that be-cause m" is complex, any component of a geo-metrical object obeys

since g„„is constant,

I'"..='( c"-.+g. g"'&'..+g, .g""c'..) (3 8)

The equation dg„„=&„„+&„„=0implies the fol-lowing relations among the connection coefficients:

We can now calculate the connection coefficientsby means of the first Cartan" equation:

pp 0 ~ ~ 2 0 ~ ~ 3) pp t ~ 0 30 ~ ~ 2~ ~ ~ 3o ~ ~ 2 j so ~ 2I ~ e3 ~

d~jf (djf ~ (d& pjf ~CF (dV Cjl ~Of ~1/P Il (X av

(3.5)

This coupled with (3.7) means that we need onlywrite down certain of the I'"„~ for Kerr. We findthat

I",, = fa sin8/&2Z,

I",, = iap' sin8/V2,

I '„= rn. /Z'+-(r —I)/Z,I",, = fap*sin8/v2,

I'„, = pb, /2Z,

I"o3 = -»I",, = iap' sin8/v 2 + p cot8/v2,

I'~» = ia cos 8 6/Z',

I",, = i ap*' sin 8/v 2 —p*cot8/v'2,

I'~» = iap'sin8/v2,

(3.8)

and all the connection coefficients that are not obtainable from these by means of (3.7) or complex conjuga-tion are zero.

Ony more useful set of relations is provided by R„„=g R~», =0 in a vacuum. They are

R2030 R2131 =Ro212 =R0313

R Ioo2 =R2o32 1003 3023» 0112 2132» 0113 3123 (3 9)

1001 3021 2031 0213 1203 3223

In the Kerr spacetime the only nonzero R„„.N have p.veP any permutation of 0123, or 0101, or 3232.Equation (2.3) in a vacuum reads

y y h SyRj.ne;~ .—2Rjys~R. a +2Rt ye~Re 8 +Rj usyR as=0 (3.10)

Let us consider perturbations of this equation. We shall require that our perturbations maintain the metric(3.4), that is, the perturbed metric is to be described in a perturbed frame where (3.4) remains unchanged.Since we are requiring the metric components to be constant in our frame, we must construct a waveequation for some other object. The most natural candidate is the Riemann tensor.

Let us let I'"„„=I'„""+c&i'"„„,R„„s=R~„' 8+&SR„„„8and g„„=g'„'„' (we preserve our frame). Insertingthese into the 0202 term of (3.10) and retaining terms to first order in e we find

(CI, &Riemann)„„—8(R,",', +R,'P, +R,'P, )(-51' I',","+5l' I',",")=4(R",', +R ~0„', —R,'P, )5R„ (3.11)

where Ob, 6Riemann is ~R„„~8.z. ,g ', where the covariant derivatives are with respect to the backgroundgeometry "We hav.e used the fact that the 51'"„„obey (3.7) and the 6R„„„S(3.9). We have

MIC HAH L P. RYAN, JR. 10

(o 5II )„„=g"[5R„„„„-45R„„,„1,„»-25R„„,r,„»„25R,„„I,„»„

&'202 so + 3&'02 ao + oa&a oo +2 320& a)& } o

+2( RD~DRI"5)„'"+ Roooylaa'")I 2„' — Roao, 31"2„' +2(5Ryaoal'03' +5ROq»I"33' )I'„„' ),(3.12)

where, as usual, A, „=-e„A,. Using the symmetries of &R„„„swe find that

(i:jb25Rtema&m}oa» =g" 15R»oa. a. 5Roaoa. 3~I) 3 Rsaoa. ~o)&

R»02~02 ~ 0 4OROR 0( 02 22 } 0202( 0)1» 0 22

[2P(0&RP(0) & + 2(P (0)0 + P(0&2)(T(0)0 ~ P(0)2) ~ (P(0)0 + P(0&2)T(0) 3]

(5p(0&ap(0)s + 51"(0)01'(0&3+2I'(0) sp(0) 3)j3202 2P 0~ + 0[1 0I + 0&

%e can now eliminate the terms containing &R»02, „and 1,"„',3, . Since

R&0)[& p(0)o p(0&p I'(0) p&(0)o p&0»'@&0)o p(0)o(p[0&1' p(o)&'}2

and R~,"and R0»' are zero, we can eliminate I',"„"„in favor of products of the I'",„. The differentialBianchi identities B„„&„8.z, =0 imply that

I) o&oo: (ba&o& )'o&&32 'I I&I (t&+ 2 &'&0&o I ol 33+ 2 0'&&2 oo)+ I) a&os ) I Its& 0 l o2&.3(P& 3 ){0) )' (0) (0) )' {0) (0)

(3.13)

where 2~2&„=—(5eo)A. They can be used to eliminate 5R»„„in favor of products of 5RD„,and5R»» and theI'&oo&", 5R»» 0, and products ofR„D„„Dand51', 3(the derivatives of R„'„' 3 all vanish). Two beautiful cancellationsnow occur. The coefficient of 5R»»becomes zero and theR(0 5I'terms exactly cancel those in Eq. (3.11). Wenow arrive at an equation for 5R»„only. Notice that the first two terms of the right-hand side of (3.13}arethe scalar d' Alembertian of 5B~~ and form a quantity that can be written as

ax" . ax"

for g " in the coordinate frame of (3.3). We finally find that Eq. (3.11)becomes

Z sin(9 Bx8 2 g 0 ( 0202}»02 1 03 0202 3 01Bx

020R ~ 0( 01 21 } 0202 2( 03 23 } 0202, 3( 02 23

+ 25R [4I &o&oI (o&s 4I (o)aI (o&3+ 8(I &o&o+ I (o)a)I (o&s 3(1 (o&o+I (o&a}I (0)30202 30 01 12 03 01 21 03 03 23 01

+go 0[ (+&0)0+P(0)2) ~ 2(P(0)0+ P(0)2)(P(0)D+ P(0)2) +2+(0)DP(0)3+ (P(0)0 P(0&2)P(0) 6]0P 2P»' 0P 2P OII 2L& 3P 0II + 06 + 26 P I&

(3.14)

Inserting our values for I "„,we arrive at

1 825R„sin2 g

&3(r —M) i cos 8 55R M(3'2 —aa)sin'8 8 p

adR„—r —iacos8 " +(4cot'8 —2)5R»»=0. (3.15)

This is Teukolsky's equation for s =+2.Suppose we now construct a new basis,

01 001 000

eojd .00010010

Under this change of bases the metric is preservedand the l „" may be treated as tensors, and theonly change consists of exchanging the indices 0and 1, and 2 and 3. If we make this, exchange onthe nonzero I',"of (3.8), we find that in the newframe the same connection coefficients are nonzeroexcept for I 00 and Z,0 whi. ch become nonzero while

Fpl and F» become zero. In this new frame, then,

TEUKOLSKY EQUATION AND PENROSE %AVE EQUATION

the reduction of Eq. (3.11) for 5R»» is very nearly the same as that in the original frame. Returning tothe original frame, we arrive at an equation for 5R»» that is almost the same as Eq. (3.15), except thatthe last three derivative terms on the left-hand side change sign, the coefficient of I»» becomes

cot'8+ 2 —4 p + 12p'6 + 4 fa(cos 8 —sin 8) p —12a' cos 8 sin 8p'

and two new derivative terms appear: +Spas5R»»/ar and +Bia cos8ps5R»»/s 8. This new equation reducesto an equation for p 5R»» that is just Eq. (3.15) with the sign of the last three derivative terms changedand the coefficient of p '5R»» equal to (4 cot'8+2). This is Teukolsky's equation for s =-2.

IV. TEST MATTER

(4.1)

we arrive at a wave equation for C„v„8.he y hy 6y

Cjl un8; 6e 2C)1 ) 8 hCu n +2CII ynhCv 8 Cjl v6)'C n8 g)18 Rb&C u n gunC)I p8 6 R

6y Fj 1 6e+gusCv pns R +gvnRs pCu s + CupvnRs Cv vps +ng2vRns v esg 2gvs Rneu; s g

1 m6 be be h n~fn2gun~~8)I ~ o. e g + &gv8 ~njf ~ e; 6 & + 6tp8 +un~~ ~ 6 ~ eS + egjf ng8 v~~: 6;eg 2 &gj18 gvn Bjf ngv8&~~ ~( y

If the gravitational waves in Eq. (2.2) are not in a vacuum, but are driven by stress-energy that is notsufficient to affect the background we must allow a 5R„„in perturbing Eq. (2.2), while maintaining therequirement that R„",' =0. It is simplest to do this by means of the %'eyl tensor. Inserting the R„v 8 into(2.2) in terms of the Weyl tensor C„„s,

1 I 1 1R)1 vn8 =Cq vn8 + zgjfnR8 v 2gp8Rnv —2gvnR8& + 2gu8 Rn& + eg&8 gnuR —eg&ng8 uR,

+2(Z sR"n-g nR's)R ~ +~(g. R~s-g. sR"n)R + (R nRsu RnuR-sv)

u (Zvs Rnv avnRsu)R 6 (g Runs vgvsRun)RE@2+ 6(g)18 gun gIIngv8~ ' "p. n v. 8 + "vn p. 8+ ")18 ~ v n "u8 ~ )I ~ n (4.2)

Opv2t2 20 0;2 0212;0 22'0 0 (4.3)

(the covariant derivations are with respect to the

If we now write C =C"'+eDC and R =e6R andkeep only terms to first order, we find that in theframe of Eq. (3.11) only the first four and lastfour terms of the left-hand side of (4.2) contribute,and we find that 5C„„obeys (3.10) with R replacedby C and with new terms in 6R„, on the right-handside. Because g" vC„v8 =0, the analysis of Sec.III carries over directly, and we find that &C0202obeys (3.15) with a driving term:

background). 5C»» obeys the equation mentionedat the end of Sec. III, and the driving term in thiscase is

II'3'3 31;i;3 13;3;1 33;1' j. ' (4.4)

Because

1 1(R» + —,g» R),„s =R». n. s + —,Z» R.„.s,

we find that 5R„, in (4.3) and (4.4) can be replacedby 8n4T„„ the test stress-energy driving thegravitational perturbation.

*Work supported in part by NSF Grant No. GP 34639X-2.~S. A. Teukolsky, Phys. Rev. Lett. 29, 1114 (1972).2S. A. Teukolsky, Astrophys. J. 185, 635 (1973).3E. Newman and R. Penrose, J. Math. Phys. 3, 566

(1962).4R. Penrose, Ann. Phys. (N.Y.) 10, 171 {1960).~R. Penrose, in Theories Relativistes de la Gravitation

(CNRS, Paris, 1962).L. Bianchi, Atti Acad. Naz. Lincei C1.. Sci. Fis. Mat.Nat. Bend. (V) 11, 3 (1902).

B. DeWitt, in Gravitation: An Entroduction to Current

Research, edited by L. Witten (Wiley, New York, 1962).8A. Petrov, in Contemporary Problems of Gravitation,

edited by D. Ivanenko (Tblissi Univ. Press, Tblissi, 1967).SC. Misner, K. Thorne, and J. Wheeler, Gravitation

(Freeman, San Francisco, 1973).~OC. Lopez, Nuovo Cimento Lett. 6, 608 (1973).

R. Kerr, Phys. Rev. Lett. 11, 237 (1963).~2R. Boyer and R. Lindquist, J. Math. Phys. 8, 265

{1967).~3W. Kinnersley, J. Math. Phys. 10, 1195 (1969).~4K. Cartan, Lemons sur la geometric des espaces de

MICHAEL P. RYAN, JR. 10

Biemann (Gauthier-ViQars, Paris, 1951).L. Hughston (private comznunication) has pointed outthat the structure of the right-hand side of {3.11) isnatural for (2,2} spacettmes. In fact. he has shown

that a necessary and sufficient condition for a spacetimeto be f2, 2} ts that the Inst three terms of (8.10) belinear in R pvag