separation of variables

Communications in Commun, math. Phys.59,285—302 (1978) Mathematical Physics ©by Springer-Verlag 1978 Separable Coordinates forFour-Dimensional Riemannian Spaces C.P.Boyer IIMAS, Universidad Nacional Autonoma deMexico, Mexico 20,D.F., Mexico E.G.Kalnins Mathematics Department, University ofWaikato, Hamilton, New Zealand Willard Miller, Jr. School ofMathematics, University ofMinnesota, Minneapolis, Minnesota 55455, USA Abstract. Wepresent acomplete listofallseparable coordinate systems for 4 4 theequations }g~"6(g'?g'0,)=E®and ¥g'/0,wo,W =Ewithspe- ij=1 ij=t cial emphasis?onnonorthogonal coordinates. Applications togeneral relativity theory areindicated. 1.Introduction Westudy theproblem ofseparation ofvariables fortheequations a A " a)A,@=¥7gae)=E®iJ=1/g te (1.1) b)¥g0,Wo,W=E. ij=t Here,ds?=Ygjax'dx! isacomplex Riemannian metric, g=det(g,,), La"9ik=&, 9i;=9)»andEisanonzero complex constant. (Furthermore, wehaveadopted ; ow , ‘ thenotation0,W=W,-%)Thus(1.1a)istheHelmholtzequationonafour dimensional complex Riemannian space and (1.1b) istheassociated Hamilton- Jacobi (HJ) equation. Inthispaper weclassify allmetrics andcoordinate systems forwhich Equations (1.1) admit solutions viaseparation ofvariables. For (1.1a) theseparation isin terms ofaproduct whereas for(1.1b) iticinterms ofasum 4 Hx)=[[O%), WK)=YWX’). (1.2) i=1 i=1 0010-36 16/78/0059 /0285/$03.60 286 C.P.Boyer etal. Byapplying appropriate reality conditions onthemetric tensor wecanuseour results toobtain theseparable andpartially separable systems forcorresponding realequations such astheEuclidean space Helmholtz equation, theMinkowski space Klein-Gordon equation [1]andvarious equations ofgeneral relativity theory. Indeed theinterest insolving relativistic equations viaseparation of variables methods, seee.g.[2-5], ismotivation forourwork. Inaforthcoming paper weshall make these applications explicit byclassifying allseparable systems for(1.1a) and (1.1b) inRicci-flat spaces. There isadeep relationship between thesymmetry groups ofEquations (1.1), thecoordinate systems inwhich these equations admit solutions viaseparation ofvariables andtheproperties oftheseparated (special function) solutions so obtained. See[6]foranexamination ofthiscorrespondence inthecase ofsome ofthemost common partial differential equations ofmathematical physics. We mention inparticular that themyriad addition theorems, generating functions andexpansion formulas forthespecial functions ofmathematical physics canbe derived systematically interms ofthisrelationship. Thus themethod ofseparation ofvariables assumes animportance farbeyond thefactthatitpermits thecons- truction ofexplicit solutions forpartial differential equations. The classical Stackel method forseparating variables inEquations (1.1) is wellknown, [7],[8].Webriefly review themain ideas fortheHJequation (1.1b). Let{x'}beaprospective separable system fortheHJequation withseparated ordinary differential equations 4 Pe &,=f(x')(W) +g(x)W +h) +YS{x)=0 i=1,...,4. (13) j=1 Here c,=—Eandc,,c,,c, aretheseparation constants. Itisassumed thatthe Stackel determinant S=det(S;,) isnonzero. Furthermore, iff,#0wecanrequire thatf,=1.Torelate (1.3)with (1.1b) onelooks forfunctions @,(x’,...,x*) such that 4 4 ¥66,= ¥g0,Wo,w —-E (1.4) i=1 ij=1 identically intheseparation constants E,c,,c,,c,. Stackel [8]showed thatthe only solution isf,=1,g,=h,=0,0;=M,,/S where M,,isthe(j1)cofactor of S.Inparticular themetric must beorthogonal. The classification procedure utilized here ismore general than that ofStackel andisbased onamethod introduced in[9]forthree-dimensional Riemannian manifolds. Here theseparable systems areclassified interms ofthenumber of ignorable coordinates they contain. (Avariable x!inaseparable system istermed ignorable if69,=0for1Sj,k<4,ie.themetric tensor isindependent ofie Otherwise thevariable x'isessential. Iftheseparated ordinary differential equation intheessential variable x!isfirstorder inW®then x!isoftype J,ifsecond order then x!isoftype2. Ifx*isignorable then theHJequation admits solutions oftheform W= Separable Coordinates 287 W(x1, x2,x*)+c,x*where c,isaconstant, and(1.1b) reduces to 3 3 Yg°0,W0,W +2c,»940,W +cig** =E, (1.5) ij=1 j=1 anequation inthreevariables. Ifallremaining variables arerequired tobeessential, wecanthen apply theStackel method to(1.5) andfind anequivalent system of three ordinary differential Equations (1.3) where now i=1,2,3,jissummed from 1to3andf;need notbenonzero. If,however, two, three orfour variables are ignorable wecanfirst split offthese variables, introducing aseparation constant c¢,foreach ignorable variable, and apply theStackel method tothereduced equation intheremaining essential variables. The orthogonal separable systems weobtain areexactly those which one would find byemploying theclassical Stickel method. However, forsystems with atleast one ignorable variable wefind truly nonorthogonal coordinates which donotseem toappear intheliterature. Our lists arealso simpler and more explicit than those given heretofore. This isprimarily because, aspointed outin[9],onecanidentify allcoordinate systems which leadtothesame families ofseparable solutions forEquations (1.1). (For example, if{x'}isaseparable system withx*ignorable andx’,x?,x*essential thenitiseasytoseethat{X‘} with XJ=x/,j=1,2,3,X* =x*+a,(x') +a,(x?) +a,(x°),a, arbitrary, isalso aseparable system with X*ignorable and X',X?, X°essential. Furthermore, thetwosystems have thesame separable solutions. Weregard allsuch systems as equivalent and merely give one representative from each equivalence class in ourlists. The functions a,arechosen such that thisrepresentative isassimple aspossible. Inparticular, ifpossible wechoose thea,such that theresulting metric isorthogonal. Ifthiscannot bedone wesaythat theseparable system is truly nonorthogonal. Allother separable systems areequivalent toorthogonal separable systems. Similar remarks hold fortwo, three and four ignorable co- ordinates.) Inthispaper welistallpossible ways variables canlocally separate ona four-dimensional Riemannian manifold. Arelated problem notsolved here is: Given aparticular manifold Mlistallseparable systems forM.Tosolve this problem wemust determine which ofthemetrics listed here canbeinterpreted asametric forM.Inparticular, forflatspace wemust require that thecurvature tensor corresponding toeach separable metric vanish identically. Fororthogonal separable systems wehave solved thisproblem inflatspaces E,,[1]andinthe space ofconstant curvature S,,[10]. Fornonorthogonal separable systems in E,,andS,thesolutions willbegiven inforthcoming articles. Themost notable paper concerning separation ofvariables forEquations (1.1) isundoubtedly that byEisenhart [11]. Itfollows easily from Stackel’s original paper [8]thatanyorthogonal separable system for(1.1a) also separates (1.1b). Robertson [12]found anecessary andsufficient condition thataseparable system for(1.1b) also separate (1.1a). Eisenhart’s great contribution was toshow that theRobertson condition amounted totherequirement R,,;=0 fori#jwhere R,istheRicci tensor oftheRiemann manifold. Itfollows immediately from thisresult thatforallEinstein spaces (inparticular forflatspace andspaces of 288 C.P.Boyer etal. constant curvature) theHelmholtz and HJequations separate inexactly the same orthogonal coordinate systems. Furthermore, this result makes feasible thecomputation ofallseparable orthogonal systems foragiven Riemann manifold provided thecurvature tensor ofthemanifold canbecharacterized inareasonably simple manner. Among recent contributors, Havas hasinfluenced usthemost. Inhispapers [13]and[14]hegave auseful summary oftheclassical work relating Equations (1.1), listed both orthogonal andnonorthogonal separable (and partially separable) systems forthese equations innvariables and emphasized therelevance ofthe condition R,;=0,i#j, even fortheHelmholtz separability ofnonorthogonal systems. Our work differs from hisprimarily intheexplicit nature ofourresults andinthenewnonorthogonal systems wefind. Most ofthenonorthogonal systems obtained byHavas andearlier workers appear tobeequivalent toorthogonal separable systems’ inthesense discussed in[9].Inthispaper weshow that the condition R,=0,i1# j,forallnon-ignorable variables x‘,x/isnecessary and sufficient forHelmholtz separability ofeven nonorthogonal HJseparable systems with twoexceptions, thenonorthogonal E1andE2systems. These special systems, particularly E1,areimportant counterexamples forseveral conjectures concerning separable coordinates. There isanother series ofpapers [3-5], based onsome results ofWoodhouse, concerning separability ofEquations (1.1) andexplicitly pointing outtherelevance ofvariable separation tothesymmetry properties of(1.1) andtomodern relativity theory. However, theauthors ofthese papers adopt avery special definition of separation andpartial separation ofvariables which rules outmany oftheclassical separable systems. They consider only thetype ofvariable separation inwhich one variable canbeexplicitly separated from theremaining variables in(1.1). This definition omits such wellknown separable systems astheLamé ellipsoidal coordinates and paraboloidal coordinates inflatspace forwhich allvariables must beseparated simultaneously (i.c.,thefullStackel matrix machinery must be used [11]). Thevariable separation treated inthese papers corresponds toour types IandJ. Asafirststepintheapplication ofourresults torelativity theory weestablish inthispaper that aRicci-flat space admitting aseparable coordinate system must alsoadmit aKilling vector. Furthermore aseparable system foranon-flat Ricci- flatspace must contain anignorable coordinate. Forcorresponding treatments ofEquations (1.1) inthree andfour variables with E=0see[15-17]. ; 2.Equations with Ignorable Coordinates Wenow enumerate thepossible separable coordinate systems fortheHJand Helmholtz equations which contain atleast oneignorable variable. A.AllVariables Ignorable Byapplying alinear transformation x!=ax!totheignorable coordinates {x} Separable Coordinates 289 wecanobtain anequivalent setofignorable coordinates {x'}forwhich g,,=4,;. The HJequation becomes [A] W2+W7+W;+W,=E, (2.1) andthecorresponding Helmholtz equation isalso separable inthese flatspace variables. B.Three Ignorable Variables Ifx*istheessential variable theng,,=G,{x’). Byre-defining theignorable4 variables {%}according todx!=dx'+h(x')dx' where g,,=— ))9,h,.j= k=2 2,3,4 wecanobtain theHJequation 4 [B] Wi+ ¥gix)WW,=E. (2.2) i,j=2 Thecorresponding Helmholtz equation also separates inthese coordinates. C.Two Ignorable Variables with Two Essential Variables ofType 2 We will treat this case insome detail toindicate our methods ofderivation. If theessential variables arex’,x?then forseparation oftheHJequation thecontra- variant metric must have the form oeaGe=Q,g'° =Qa(x'), g?*BisQb(x?) (2.3) g'*=Q(x"), g?*=Od(x?), 9°=Ofe,(x*) +e,(x?)] g*=OLA) +07)],9°* =Ofh,(x*) +h(x")],g"?=0 where Q=1/[K,(x')— K,(x’)]. Bydefining equivalent ignorable coordinates x3,x*totheoriginal coordinates x,x*by dx3=dx?—adx' —bdx?, dx*=dx*—cdx! —ddx? (2.4) wecanassume a=b=c=d=0 in(2.3). Thus theHJequation becomes 1 [Cc]Kw[W2+W2+(e,+¢,)W2+2h,+h,)W,W, +(f, +f,)Wi]=E. (2.5) Thecorresponding Helmholtz equation separates ifandonly if (KLiK,) | é,,8]————_ 2 |=9 (2.6) a2le+e,)(f,+4)—(h,+h)? andthiscondition isequivalent toR,,=0where R,,istheRicci tensor. 1 290 ; C.P.Boyer etal. D.Two Ignorable Variables withOneEssential Variable ofEach Type There aretwocases todistinguish, thefirstbeing 1 [D1]agli +2a,WaW,+2b,WAW,+dW (2.7) +2(f,+f)WW,+e,We]=E, where thesubscript iona,b,d,e, f,Kdenotes theessential variable x!onwhich thefunction depends. Here a,b, +0.Thenecessary andsufficient condition for separation ofthecorresponding Helmholtz equation is (Kim) 01)sp =0; 2.8 12°"[2aybalf, +)—a3e1—bed, Be which isequivalent toR,,=0. The second case corresponds to 1 +2f,W,W,+e,W2Z]=E. (2.9) Thenecessary andsufficient condition forseparation oftheHelmholtz equation is (KtogK,)a ayain| a,Hd,=i) (2.10) andthisisequivalent toR,,=0. E.Two Ignorable Variables with Two Essential Variables ofType 1 Here there arethree cases todistinguish. Thefirstpossibility is 1 [Et]Baa Ns Le Mate +(c, +c,)W7]=E,a,a, #0. (2.11) Necessary andsufficient conditions forseparation oftheHelmholtz equation are 6,,(4,0,W +4,0,W) =0,,(0,W +0,~) =0, (2.12) aie Inthiscasethecondition R,,=0where it,= a"—a" Ry,=0,0—40,0,y —4!226,9+2,)—!:2 (2.13)aja; a,—a, neither implies norisaconsequence of(2.12). Toshow thiswefirstfindallsolutions ofEquations (2.12). Thegeneral solution ofthesecond equation is W=f(x—x?)+g(x*) +h(x?) Separable Coordinates 291 andsubstituting thisresult into thefirst equation weobtain Lemma 1.Acoordinate system oftype E1permits separation oftheHelmholtz equation ifandonly ifitcorresponds tooneofthefollowing three types: i)a,=cosh x',a,=cosh x”,wy=—In(e*—*' —1) ii)a,=e", a,=e",K,=0 ili)0,,~=0. Itiseasy tocheck thatR,,#0forthesystems oftypes i)andii).Furthermore thesystem B=(34:4) 5)x1,(a,2x7, Ke K=O satisfies R,,=0butnotEquations (2.12). Webriefly investigate thepossible E1systems which canoccur inflatspace. The only nonvanishing elements ofthecurvature tensor foranElsystem are Ry444>Ri223>R2413 andR,,,,- Adirect computation shows thatifthefirstthree ofthese elements arerequired tobezero then thesystem satisfies 0,,\/=0,hence permits separation oftheHelmholtz equation. Lemma 2.Atype Elsystem inflat space permits separation oftheflat space Helmholtz equation. The second possibility fortype Esystems is 1 [E2]K,-K, [2W,W,+2W,W,+2b,W,W,+(c,+¢,)WZ]=E (2.14)1 with thecondition forHelmholtz separability, 0,,n(K, —K,)=0. (2.15) Here KK! BER! Pi eo Ap ee122(K,—K,)? 222K,—K,) sothecondition R,,=0implies (2.15) although (2.15) doesn’t imply R,,=0. The third possibility is 1 [E3] eA, +2W,W,+c,Wy+dW]=E (2.16) ae) with thecondition forHelmholtz separability 6,,In(K, —K,)=0 (2.17) which isequivalent toR,,=0. F.One Ignorable Variable with Three Essential Variables ofType 2 The HJequation is 1 [F] glG2—43)We+(43—4)We+(4,—92)We +[71@2 =45)+1243 —41)+73(41 —42)We]=E (2.18) S=ACEi93)a5(95=q;)+33(4;paq2)- 292 C.P.Boyer etal. Since these coordinates areorthogonal theconditions forHelmholtz separability areR,;=0 foralli#j.However, thecondition R,,=0,!=1,2,3, issatisfied automatically. G.OneIgnorable Variable withOneEssential Coordinate ofType 1 andTwo Coordinates ofType 2 There aretwo cases toconsider, thefirst being 1 [G1]gue+W?+2(l,—1)W,W,+(m,—m,)W2]=E, (2.19) Q=k, —k,+9,(l, —1,). The conditions forHelmholtz separability areequivalent to 6,,1nee=0, 1si<js3, (2.20)2h,=FL, which areprecisely R,;=0. The second case is 1 [G2]GlasWi+lW2 +20, +uals+93f,)We]=E, (2.21) Q=k, +0,1,+937,- The conditions forHelmholtz separability are 6,,nQ=0, 1si<js3 (2.22) which areprecisely R,,=0. H.NoIgnorable Variables This isthemost complicated case andthemetric isnecessarily orthogonal. The HJEquation reads <Mji2: [H]pyWise (2.23) i= where SisaStackel determinant andM,,isthe(j1)cofactor ofS.Eisenhart [11] hasshown quite generally thatthecorresponding Helmholtz equation separates ifandonlyifR,,=0 foralli#j.Adetailed classification ofthese coordinates willbegiven inSection 4. Theorem 1.Aseparable coordinate system fortheHJequation inafour dimen- sional Riemannian space isequivalent toexactly oneofthesystems {x'}oftypes A-H.If {x'}isnotoftypeElandR,=forallpairsofdistinct essential variables x',x/, then {x'}alsoseparates thecorresponding Helmholtz equation. Except for thecoordinates oftypes E1andE2thiscondition isalsonecessary forHelmholtz separation. Separable Coordinates 293 Since theRicci tensor vanishes inflatspace thefollowing result isanimmediate consequence ofTheorem |and Lemma 2. Corollary 1,Acoordinate system infour dimensional flatspace provides asepa- ration ofvariables fortheHJequation 4 Ow 2$(Mej1 dz! ifandonly ifitprovides separation fortheHelmholtz equation 4owYa =Ev.voy This result iswellknown fororthogonal separable systems [11], butwehave extended ittononorthogonal systems. Defining theEinstein tensor G,,=R,,—7Rg;; itfollows immediately from Theorem 1that Corollary 2.Let{x'}beasystem, notoftype El,which separates variables in theHJEquation andsuchthatthevacuum Einstein equations G,j=0aresatisfied. Then {x'}separates variables intheHelmholtz equation. 3.Equations with Partial Separation Wenextenumerate thesystems {x'}fortheHJandHelmholtz Equations such that one ortwo variables can beseparated from therest butthevariables are nottotally separable. Weshould emphasize that while these systems canbe considered asgeneralizations ofthecoordinates oftypesA—Glisted inSection 2, theyarenottruegeneralizations oftype Hcoordinates. Indeed themost interesting andcomplicated ofthetype Hcoordinates (such asellipsoidal coordinates) do notpermit thesplitting ofoneortwovariables from theremaining variables. For these coordinates allvariables must beseparated simultaneously. The seven types ofpartially separable systems willnow beclassified interms ofessential and ignorable variables. I.One Essential Variable ofType2 The HJequation canbewritten as 1 4 NN ie A,(x?, x3,x*)W, =E. 3.1 Oyeee" +2NyeidiMittd Thecondition forpartial Helmholtz separation isthateither K;orKisconstant, which isequivalent toR,j=0,j=2,3,4. J.One Ignorable Variable The HJequation hastheform 4 ey YG'WW,=E (3.2) jl=i "i ' 294 C.P.Boyer etal. where 8G/"/ax! =0.Thecorresponding Helmholtz equation alsoadmits partial separation inx'. K.Two Essential Variables ofType 2 TheHJequation hastheform 1 4alove.me+H(x,x*)W?+Eas.xtngm| =E (3.3) Q=K,(x!)G +K,(x?)H +K(x*,x*) andthecondition forpartial Helmholtz separability isthattwoofthefunctions K,,K,,K areconstant, which isequivalent toR,,=0,R,,=0, t=j= 3,4 L.Two Ignorable Variables The HJequation is 4 YG3, x*)W,W, =E (3.4) jl=1 andthecorresponding Helmholtz equation isalsopartially separable. M.OneEssential Variable ofType 2andOneIgnorable Variable TheHJequation hastheform 1 4M —______.___| W? (x3, x*)W,W, |= 330,RaEaeral74DAeeae|= Ge andpartial Helmholtz separability isachieved ifeither K,orKisconstant, which isequivalent toR,;=0,j=3,4. N.OneEssential Variable ofType 1andOneIgnorable Variable TheHJequation hastheform 1 4 N ——_—— |2 (X?,xX)W,W, |=E 6 IN]arya WatDAdeme;| ae andpartial Helmholtz separability isachieved ifeitherK,orKisconstant. In thiscaseHelmholtz separability appears unrelated toanyRiccitensor condition. 0.Variable Splitting Here theHJequation becomes 1 2 4 (@) So Ton A,(x!,x2)W,W, B.(x3,x* [1]memes 2,ae:™itd,abe4 wm,|=E (3.7) Separable Coordinates 295 andthecorresponding Helmholtz equation partially separates ifeither KorL isconstant, which isequivalent toR;,=0,i=1,2,j=3,4. 4.Equations with NoIgnorable Variables Wenow present adetailed classification oftheHelmholtz separable systems of typeH.These areorthogonal systems {x”}forwhich themetric canbewritten 4 ds*=Y)H3(dx/)?, H,=S/M,,, (4.1) j=1 SisaStaéckel determinant andMyisthe(j,1)cofactor ofS.Furthermore, none ofthevariables x'isignorable andtheRobertson condition Rin=OJFKis satisfied. Asiswell-known [11] theStackel form condition (4.1) isequivalent tothesystem ofequations 6,inH?—0,InH?+0,inH70,InHF +0,InH?0,in Hy=0, G#h, (4.2) and the Robertson condition reads Ry=30,,1n(H?H?)=0, (j,i,#). (4.3) From (4.2)wehave0,In(H7/H) =0.Combining thisresult with(4.3)wefind Hj=99n>Hy=Pjx 8,0, =9,Pxj=Wie =Oi =9; Oo,aa01;WineWirj- Furthermore, OyInH?=0,;1nHZ=0,,;InH?=—Oy;InHF=f; (4.5) where 0,f;=0.Integrating (4.5)andmaking useof(4.4)weseethatInH?isasum offunctions, each function depending onatmost twoofthefour variables x',x/,x*,x!.Substituting thisformforInH?into(4.4)weobtain Lemma 3.Ifthemetric ds?isinStdckel formandsatisfies theRobertson condition, thenthereexistnonzero functions X,=X(x'),€,;=¢,{x',/) =5.=mx’) = n,,suchthat HT=X1F545o3S 24MMagia Hy=X054F136MaasaMor (4.6) HZ=X302614 CoaMaiMaoMaa Hy=KaGracaatlantaatiaee 296 C.P.Boyer etal. Here AelEnEnuitp)=0, (ssi, 1). (4.7) Property (4.7)follows from substitution of(4.6)into(4.2)fori=j.Thenonzero functions X,arearbitrary andcanbemodified atwillbythetrivial change of variable x”=x"(x"),m=1,...,4, which does notaffect variable separation. Tocomplete ourclassification weneed only satisfy theEquations (4.2) for i,j,kdistinct. Note thatthe¢,,andn,,arenotuniquely determined by(4.6).Indeed these expressions areinvariant under thereplacements S12>4400512 Noy>4b2bC21 £13>Biesh13 Nyy>43b34,C; N31 S14>16a14 Nay>44bgay"by‘Nas (48) Sa3>42b3025 Nz>43*C3Cqb3 ‘N35 bog>Onb4oag Naa7IqCqz“C2Nap S34>4344534 Nhs>Cgbg*b5*cx"Nas where a,,b,,c, arenonzero functions ofthesingle variable x‘.Inparticular, if éikIngi=0forsomej,kthenwithout lossofgenerality wecanassume €iesBe Denote eachofthetwelve Equations (4.2)fori,j,kdistinct by[i;j,k](=[isk./]). Differentiating each equation [i;j,k] withrespect tox'andx!where j,k,i,! are distinct, weobtain twelve equations which areequivalent totheeight conditions A,,(A34 +2B34) +Ay3(424 +2B,4) =9 A34(Ay2 —2B,,) +A24(A,3 —2B,3) =0 A,3(Ay4— 2B,4)—A34(A12+2B,,)=0 | Aj(A534+2B34)—A,4(423 +2B,3)=0 A,4(A23 —2B23) +Az4(A,3 +2By3) =9 (4.9) Ay3(Ay4 —2B,4)+Ay3(424 —2B24)=0 A,3(A34 —2B34) —4o3(414 +2By4) =9 A,2(A34 —2B34) +Az4(Ai3 +2B,,)=9 A,,=0,, In¢,,,By=0,0n,;- Ouranalysis ofthepossible Helmholtz separable systems depends strongly on which ofthevarious factors vanish inexpressions (4.9). Weexamine allpossible cases. Case 1.A,,+2B,,=0 foralli,j. Then A;,=0 andwecanassume that¢,,=1. Case2.A;(Ay,+2B,,)#0forsome choice ofi,j,k,|. Tobedefinite weassume (i,j,k, |)=(1,3,2,4). Then itfollows from (4.9) that Separable Coordinates 297 A,,,433,434 andA,,arenonzero, aswell asA,,+2B3,,A,4—2By4,41.+ 2B,,,A>, +2B,,. Furthermore, relations (4.9) imply that0,,In A,,=0,, In A,,=4,, InA,,=0,,In A,,=0.Integrating these equations andsubstituting into 8s9(51361452362412) =9, aspecial case of(4.7), weseethat thislastcondition cannot besatisfied. Thus Case 2doesn’t occur. Case 3.Case 2doesn’t holdbutforsome choice ofdistinct i,j,k,1,A;{A,, +2B,) # 0,A,;—2B,,=0. Without lossofgenerality wecanassume (i,j,k,!)=(1,3,2,4). Itfollows from (4.9)thatA,,,4,,+2B,,,4,4,4,3 +2B,, arenonzero. Since Case 2doesn’t hold,wehaveA,,=2B,,#0,A,,=2B,,#0,A,,= —2B,,and A,,+2B,,=0. This lastpairofequalities implies A,,=0,which isimpossible. Case 4,Case 2doesn’t hold butforsome choice ofdistinct i,j,k, 1,A;(Ay — 2B,,) +0,A,,+2B,=9. Byacomputation analogous toCase 3wecanshow thatthisisimpossible. Itfollows from the above that A,{Au+2B,,)=0 (4.10) foralldistinct i,j,k,l. The possibility that thesecond factor isalways zero has been treated inCase 1.Wenow treat theremaining possibilities. Case5.A;i=0foralldistinct i,j. Here wecanassume that¢;j=l. Case6.Aj,Ax,A;,=0,i,j,k,|distinct. Without loss ofgenerality wecan assume that (i,j,k, 1)=(1,2,3,4). Then Ay>,A,3,444#0 andfrom (4.10) 4,=A,4= 4,=B34,=B,4=B,, =0.We canuse(4.8) and assume without loss ofgenerality that 7,,=1.4="23=1 €,,=1,6,, =,4(x*). Differentiating equation [3;2,1], (4.2), with respect to x*and[4;2, 1]withrespect tox?weobtain A,,(20, Iné,,—@, In¢,, —0,Iny,,)=0 A,3(— 20,In¢,,—30,In¢,, —0,In,,)=0 which implies A,,=B,,=0, acontradiction. Case7.A;;,Ay,Ay; #9. Without lossofgenerality wecanassume that(i,j,k) =(1,2,4). Then 4,,,A,,, A,,#0and A,,=A,,=A,,=B,,=B;,=B,,=0. Wecanuse(4.8) and assume without lossofgenerality that7,4=M3, =1,54 =€54(%°)s 31=310%"), €5=652°), 232="32(x°). Writing thesimultaneous equations ¢@,[4;2,3], 6,[2;1,3],@,[4;1,3],6,[1;2, 3],@,[1;3,4],,[2;3, 4]weseethattheseequations areconsistent onlyif€,,,€,,,¢3, and,,areconstants. Thus, x°isanignorable variable, which isimpossible. 298 C.P.Boyer etal. Case8.Aj,Aj+0,allother A,,=0.Without lossofgenerality wecanassume (i,j,k) =(1,2,4). ThenA,,,424 aretheonlynonzero A,andB,,=B,,=0. Employing (4.8)wecanrequire Na=M13=S34=Cra=1,01€13 =92623=0-Then equations 0,[4;2,3],0,[4;1, 3],0,[1;2,3] areconsistent onlyif0,6,, =03623 =03%23 =0.Itfollows that x?isignorable andthisisacontradiction. Case 9,Only onenonzero function A,,. Without lossofgenerality wecanassume thatA,,istheonly nonzero A. ThusB,,=Oandbyemploying (4.8)wecanrequire 7,4=654=C4=10,654=0. From equations @,[3;2,4] and@,[3;1,4] wefindthat0,Inn,,= —20,In¢,,, 6,nny,=—30,Inoo4>By4 =B,,=0 andby(48)wecanrequire 0,7,4=854=0.Theadditional condition 0,a(613603614%24"34) =0from(4.7)implies 84£54=080€54,N,4 and1,areconstants. Itfollows thatx*isignorable, which isacontradiction. Wehave strengthened Lemma 3to Lemma 4.Ifthemetric ds?satisfies thehypotheses ofLemma 3andadmits no ignorable coordinates, then H?=X(x')tlngsx).ng =Mp1SiS4 (4.11) Fi Aut =0,1Sk<IS4. Inconsequence oftheStackel conditions (4.2),thecomponents oftheRiemann curvature tensor fori,j,kdistinct maybewritten [11] Rie=3H76;,InH?. (4.12) Animmediate consequence ofLemma 4is Lemma 5.Ifds?satisfies thehypotheses ofLemma 3andadmits noignorable coordinates, thenRj=.Itisremarkable thatthestrong condition R;,;,=0isautomatically satisfied byaHelmholtz separable system withnoignorable variables. InRef.[18]twooftheauthors computed allHelmholtz separable systems infourvariables such thatthetechnical condition Rj,=wassatisfied. Theresults werealsoreportedinRefs.[1],[17]andwereobtained bysubstituting expressions (4.11)intothe twelve equations [i;j,k] andsolving forthen,,.Wenowseethattheanswer to ourpresent problem canbeobtained byeliminating theseparable metrics with ignorable variables fromourprevious list.Thuswehave 4 Theorem 2.Themetric ds?=),H?(dx')? defines aHelmholtz separable system 1 withnoignorable variables ifandonlyifthemetric coefficients takeoneofthe following forms: [Ha] Hi=/X,(a— 5), H?=X,(¢, —¢,) HZ=X,(0; —94),Hi=X,(03 —%4) o,=0{x'), 6,#0, ——E—&x=— Separable Coordinates 299 [Hb] H{=X,(¢,—¢,),H}=X,(o, —¢,) H3=X,0,0,(0, —0,), Hj=X,0,0,(0; —0,) 0;£0, [Hc] H?=X,o,—2¢)(o,—4,)(,— 4) with i,j,k,|distinct, [Hd] Hi=X,,HZ=X,0,(023 +052)(Oo4 +F42) H3=X54(052 +Fy3)(0g4 +43)HG=X491042 +Fy4)(F43 +534), 9,79. Here¢;;isafunction ofx‘alone. Wewill now useourresults toprove atheorem about Ricci-flat spaces, i.e. complex Riemannian spaces which satisfy thevacuum Einstein equations. (At thispoint itisconvenient towrite x,inplace ofx'.) Theorem 3.IfinaRicci-flat (R;,=0)complex Riemannian space theHJequation admits aseparation ofvariables with noignorable coordinates, then thespace is flat. Before proving thisweprove asimple butuseful lemma. Lemma 6.Suppose themetric ds?isinStéckel form, isRicciflat(i.e.R,,=9), andadmits noignorable coordinates. Furthermore, suppose thatforafixed value ofi R;,;,=0fortwovalues ofj#i.Then thespace isflat. Proof. Since R;,=0 implies theRobertson condition, itfollows from Lemma 5 thatR,j;,,=0.Now without lossofgenerality wetakeasourtwovanishing compo- nents ofRj,;,Ry22;=R,33, =9.Itfollows immediately from R,,=0 thatRy44,=0.Theremaining Ricciequations are R R Ro,Sone+=0 R R R..= 2332ae3443 =0 R R R,,=~2442433443_9, aa+P Itiseasytoseethattheonlysolution tothese equations isR,,,,=R 44.= Roagg=0. QED. Inorder toprove Theorem 3wewillneed thefollowing expression ([19], p.44). =H?{6,In 0,In Hd,InHiRy=Hi0,0;+0,inHe;inJ H, 2+1(¢,In,+10Hin) H?H?aH>7p0,InH0,InH,. (4.13)AFji k 300 C.P.Boyer etal. Proof ofTheorem 3.Since R,,=0 HJseparation implies Helmholtz separation| andbyTheorem 2weneed onlyprove theresult forthefourtypes ofmetrics [Ha-d]. Weproceed bycases: (Ha)Itiseasytoseefrom(4.13) thatR,33,=Ry44, =0andbyLemma 6R;,,,=0. (Hb)Ifeither o,oro,isconstant, wecanredefine variables toobtain anignorable coordinate. Thus, ¢,,0, arenotconstant andweredefine x,,x, suchthatHa= X,(x,—x,), H}=X,(x, —x,), Hj=X5x,x,(03 —04),HZ=X,x,x,(65,—¢,). From (4.13) wefind wore talaan) (z)-&)1221—____} _____{—444]}+3{{— ]—|—_]]x HZ 2%=x). xp \Xa BNA hy ae and Ry331RygaraI111x1=|—1331—i441__1)[4___4——i+——______2. |. H3 Hy 4]xtxy(ey— XQ) XX, X2(%—2)X2 Forming theRicci component R R R R,.= 1221Je1331+1441 (4.14)bee te wefind Femi aeSSIIRE SIO aMAE6i (x,—x,)? x64, —x,) x}x,(%,—%,)X, 4X, Xx ‘aN 1.\+stealle)-(é)°2%,=x,)L\Xy X, Consider theoperator (x,—x,)?0,= A.Computing A*R,,=0 weobtain an expression independent ofX,.Equating, then, coefficients ofpowers ofx,to 1(3)zero,wefind(z)=0.Bysymmetryofthecoordinates x,andx,,weconclude 2 1\®)(x)=0.Pluggingthisinformation backintoR,,=0,wefind 1 2 1 2eo eex,"—(cx}+dx,) fromwhich R,33;=R444; =0-Thusthedesired result follows byLemma 6. (Hc)Thisisthemostdifficult case.Wecanassume thatnoa;isconstant; forifa o,wereconstant, wecould findanignorable coordinate. Thuswecantakeo;=x;, andfrom (4.13) wefind 7-1 1 cents aeakeSPSSS HH; (x,—x)—xO-x)%—%)? 20; x),—%) %s1 1a 1 (1)r Ax,—x)-—x))\ Xi) 205- x(x, —x)O;-%) \X; ll Separable Coordinates 301 ax) 2Sa EeEe eares Ax,=x))"(x; —x); -—x)X; 1 1 1+eeeserst-—+—— (x;—x); —x); —x) (x,—x)? 2x,-—x); —x) 1 1 1 1 Tape (ae aao, eae =Set AX.x;-—x)(x%j—%)\ Xj) 2—x)"—*)" 0—XD)X, RFE)2(j— x)? =x) —x4)21 fori,j,k, 1different. Computing theRicci component R,,from (4.14) wefind eoHy) —x2); —x5)(0—x4) X, =x)? &i—x);—%), Ha)+5) fry.1 EAN,Gex,—x\X +(x) i} DONA ASPNET aNSes 3&)—x),=We,=*) Multiplying thisexpression by(x,—x,)°(x, —x3)(x, —x,) anddifferentiating (5)fivetimeswithrespecttox,,weobtain(+)=0.Bysymmetryweconclude 2 1\©)=) Plugging thisinformation back intoR,,, weseebyequating powers thatthe 1 coefficientsin(=)ofthepowersofx,areindependent ofi,i.e. I 1(3)=axt+bx?+cx}+dx;+e;. Astraightforward buttedious computation then shows that R,,,, =0,andour result follows from Lemma 6. (Hd) Notice thatif~,isconstant wecanmake x,anignorable coordinate. Assuming 7,#constant weredefine x,sothatg,=x,.Aneasycomputation using (4.13) shows that Hi(ieee.) Ry=—-|=+—> }- uyAUNSt5 Itfollows from (4.14) that 1 1X’ =—3(—54+——}. Rulat) ThusR,,=0implies R,,,,=0andinvoking Lemma 6onceagain weobtain the theorem. Wehave thefollowing simple result: 302 C.P.Boyeretal. Corollary 3.IfinaRicci-flat complex Riemannian space theHamilton-Jacobi equation admits aseparation ofvariables, thenthespace admits aKilling vector (i.e.aninfinitesimal isometry). References 1.Kalnins, E.G.,Miller, W.Jr.:Lietheory andthewaveequation inspace time.IV.TheKlein- Gordon equation andthePoincaré group. J.Math. Phys. (toappear) 2.Carter, B.:Hamilton-Jacobi andSchrédinger separable solutions ofHinstein’s equations. Commun. math. Phys. 10,280-310 (1968) 3.Woodhouse, N.M.J.:Killing tensors andtheseparation oftheHamilton-Jacobi equations. Commun. math. 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Princeton: Princeton University Press1949(2ndprinting) Communicated byJ.Glimm Received November 14,1977