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Vacuum Einstein metrics with bidimensional Killing leaves, ∗ I-Local aspects. 3 0 0 G. Sparano #,§, G. Vilasi #,§§, A. M. Vinogradov #,§ 2 # Istituto Nazionale diFisica Nucleare, GruppoCollegato diSalerno, Italy. n § Dipartimento di Matematica e Informatica, Università di Salerno, Italy. a J §§ Dipartimento di ScienzeFisiche E.R.Caianiello,Università diSalerno, Italy. 7 (e-mail: [email protected], [email protected], [email protected]) 1 v February 7, 2008 0 2 0 1 0 3 0 Abstract / c q ThesolutionsofvacuumEinstein’sfieldequations,fortheclassofRiemannianmetricsad- - mittinganonAbelianbidimensionalLiealgebraofKillingfields,areexplicitlydescribed. They r g areparametrizedeitherbysolutionsofatranscendentalequation(thetortoiseequation), orby : solutions of a linear second order differential equation in two independentvariables. Metrics, v corresponding to solutions of the tortoise equation, are characterized as those that admit a i X 3-dimensionalLiealgebraofKillingfieldswithbidimensionalleaves. Subj. Class.: Differential Geometry, General Relativity. Keywords: Einstein metrics, Killing vectors. MSclassification: r a 53C25 ∗PublishedinDiff. Geom.Appl. 16(2002)95-120 ResearchsupportedinpartbytheItalianMinisterodell’UniversitaèdellaRicercaScientificaeTecnologica. 1 1 Introduction Inthispaperwedescribeinanexactformlocalsolutions(metrics)ofthevacuumEinsteinequations assuming that they admit a Lie algebra of Killing vector fields such that: G I. the distribution , generated by the vector fields belonging to , is bidimensional, D G II. the distribution , orthogonal to , is completely integrable and transversalto . ⊥ D D D Global, in a sense, solutions of the Einstein equations constructed on the basis of the local solutionsfoundinthispaperarediscussedinthesubsequentone. Therecanoccurtwoqualitatively different cases according to whether the dimension of is 2 or 3. Both of them, however, have G an important feature in common, which makes reasonable to study them together. Namely, all manifolds satisfying the assumptions I and II are in a sense fibered over ζ-complex curves (see section 7 and [10]). dim =2 Recall that, up to isomorphisms, there are two bidimensional Lie algebras: Abelian and G non-Abelian, which in what follows will be denoted by and respectively. A metric g 2 2 A G satisfyingtheassumptionsIandII,with = or ,willbecalled -integrable. Thestudy 2 2 G A G G of -integrablemetricswerestartedbyBelinsky,Geroch,Khalatnikov,Zakharovandothers 2 A [3],[4],[7]. Someremarkablepropertiesofthereduced,accordinglywiththeabovesymmetry assumptions, vacuum Einstein equations were discovered in 1978. In particular, a suitable generalization of the Inverse Scattering Transform, allowed to integrate the equations and to obtainsolitary wavesolutions[4]. Some physicalconsequencesof these reducedequations were analyzed in a number of works (see for instance [5, 2]). This paper will be devoted to theanalysisof -integrablesolutions,forwhichsomepartialresultscanbefoundin[8,1,6]. 2 G In this case, the Killing fields ”interact” non-trivially one another (for instance, [X,Y]=Y, for a suitable choice of the basis vectors in ), while in the Abelian case these fields are G absolutely free (i.e., [X,Y]=0). Hence, it is natural to expect that the former case is more rigid, with respect to the latter, and, as such, it allows a more complete analysis. It occurs tobethe case,namely,metricsinquestionareparametrizedbysolutionsofalinearequation intwoindependentvariables,which,initsturn,dependslinearlyonachoiceofaζ-harmonic function. Thus, this class of solutions has a ”bilinear structure” and, hence, is subjected to two superposition laws. 2 dim =3 In this case, assumption II follows automatically from I and the local structure of this class G ofEinsteinmetricscanbe explicitly described. Somewellknownexactsolutions[9], suchas, for instance, that of Schwarzschild, belong to this class. Geometrical properties of solutions described in the paper will be discussed with more details separately. In the paper, as it is usual, everything is assumed to be of C class and the following ∞ terminological and notational convention are adopted. manifolds are assumed to be connected and C , ∞ • metric refers to a non-degenerate symmetric (0,2) tensor field, • k-metric refers to a metric on a k-dimensional manifold, • the Lie algebra of all Killing fields of a metric g is denoted by il(g) while the term Killing • K algebra refers to a subalgebra of il(g), K integral submanifolds of the distribution, generated by vector fields of a Killing algebra , • G are called Killing leaves, stands for a bidimensional Abelian Lie algebra, while for a non-Abelian one, 2 2 • A G a -integrable metric is a metric satisfying the assumptions I and II, with = or . 2 2 • G G A G theelementsofamatrixwillbedenotedwiththecorrespondinglowercaseletter,forinstance • A=(a ). ij 2 Metrics admitting a bidimensional Lie algebra of Killing 2 G fields. For a given s R,s=0, we fix a basis e,ε in such that [e,ε]=sε. It is defined uniquely up 2 ∈ 6 { } G to transformations of the form e λe+µε,ε λ 1ε, λ,µ R,λ=0. − 7→ 7→ ∈ 6 The parameter s is introduced in order to include, into our subsequent analysis, the Abelian case (s=0 ) as well. In what follows, it will be useful the following general fact. 3 Lemma 1. Let g be a metric on a differential manifold M. If X =0 and fX, f C (M), are ∞ 6 ∈ two of its Killing fields, then f is constant. Proof. The proof results from the formula L (g)=fL (g)+i (g)df, (1) fX X X where the secondterm in the right hand side is the symmetric product of two differential 1-forms, and i (g) the natural insertion of X in g. Indeed, L (g)=0 and L (g)=0 imply, in view of X X fX relation (1), i (g)df = 0. This shows that df vanishes at those points where i (g) = 0. Since X X 6 g is non-degenerate, i (g) vanishes exactly at the same points where X does. Therefore, df =0, X on suppX = a M X =0 . On the other hand, if a Killing field vanishes on an open subset of a { ∈ | 6 } M, then, obviously, it vanishes everywhere on M. For this reason suppX is dense on M and, so, df =0 on M. Let g be a metric on a manifold M admitting as a Killing algebra. Then, for the Killing 2 G vector fields X and Y corresponding, respectively, to e and ε, one has [X,Y]=sY. (2) Denote by the Frobenius distribution, possibly with singularities, generated by X and Y. D Proposition2. Thedistribution is bidimensional andinaneighborhood ofanon-singularpoint D of there exists a local chart (x ) in M such that α D X =∂ , Y =esxn−1∂ . n 1 n − Proof. First of all, show that dim = 2. Indeed, in view of the above lemma if locally X = φY, D then φ is constant and X and Y commute, in contradiction with Eq.(2). Thus, the vector Y and a X are independent for almost all points a M, i.e. in an everywhere dense open subset M of a 0 ∈ M. Choose now a function φ such that the fields X and φY commute. In view of Eq. (2), this is equivalenttoX(φ)+sφ=0.This equationadmits,obviously,asolutioninaneighborhoodofany point a M . In a local chart (y ) in which X = ∂ , φY = ∂ , the equality X(φ)+sφ = 0 ∈ 0 µ ∂yn−1 ∂yn looks as ∂φ +sφ=0 and hence, φ=e syn−1+λ where the function λ does not depend ony . ∂yn−1 − n−1 By passing now to coordinates (x ) with x = y , α < n, and x = β(y ,s,y ,y ) one finds α α α n 1 n 2 n − the desired result with β such that ∂β =e λ. Indeed, since λ does not depend on y , the last ∂yn − n−1 equation admits a solution not depending on y . n 1 − 4 Definition1. Achartofthekindintroducedintheabovepropositionwillbecalled semi-adapted (with respect to X,Y). All metrics g admitting the X,Y Killing algebra, i.e. such that L (g) = L (g) = 0, are Y X { } characterizedby the following proposition. Proposition 3. An n-metric g admits the vector fields X and Y as Killing fields iff in a semi- adapted chart it has the following block matrix form (g ) (sm x +l ) ( m ) ij i n i i − M (g)= (sm x +l )T s2λx2 2sµx +ν sλx +µ C  i n i n− n − n  (−mi)T −sλxn+µ λ   where C = dx , and g ,m ,l ,λ,µ,ν, are functions of x , 1 l n 2. µ ij i i l { } ≤ ≤ − Proof. Indeed, the invariance with respect to X shows that the components of the metric do not depend on x while the invariance with respect to Y is equivalent to n 1 − ∂ g =0, i,j n 2 (3) n ij ∀ ≤ − ∂ g +sg =0 (4) n n 1n 1 nn 1 − − − ∂ g +sg =0 (5) n n 1n nn − ∂ g =0 (6) n nn ∂ g +sg =0 (7) n in 1 in − ∂ g =0 (8) n in Eq. (3) tells that, for i,j < n 1, the components g do not depend also on x , while Eqs. ij n − (4), (5) and (6), imply that, for a,b=n 1,n − s2λx2 2sµx +ν sλx +µ (gab)= ns−λx +µn − λn , (9) n (cid:18) − (cid:19) where λ, µ and ν depend only on the coordinates x . Eqs. (7) and (8) have the solution i g , g = sm x +l (x ), m (x ) . in 1 in i n i j i j − − (cid:0) (cid:1) (cid:0) (cid:1) where l and m are arbitrary functions. i i For further computations it is more convenient to work with a basis, say e , of vector fields i { } invariant with respect to the Killing algebra. It is easy to see that all such fields are linear 5 combinations of e =∂ , e =∂ +sx ∂ e = ∂ . (10) i i n 1 n 1 n n, n n − − − whose coefficients are -invariantfunctions, i.e. not depending onx ,x . So, the set (10) can 2 n 1 n G − be taken as such a basis. Obviously, the basis of differential 1-forms Θ= ϑi dual to e i { } (cid:8) (cid:9) ϑi =dx , ϑn 1 =dx , ϑn =sx dx dx . (11) i − n 1 n n 1 n − − − is also -invariant. The bases (10), (11) are ”slightly” non- holonomic because in the relations 2 G 1 [e ,e ]=Cα e , dϑα = Cα ϑµ ϑν, µ ν µν α −2 µν ∧ allthe structure constants Cα are vanishing, except Cn , which equals s. They will be called µν n−1n − non-holonomic semi-adapted. The expression of the metric of proposition 3 in terms of the basis (11) is g =g ϑiϑj +λϑnϑn+νϑn 1ϑn 1 2µϑn 1ϑn+2l ϑiϑn 1+2m ϑiϑn. ij − − − − i − i Corollary 4. An n-metric g admits the vector fields X and Y as Killing fields iff its components, in a semi-adapted non-holonomic basis Θ, do not depend on x and x . The matrix of g with n 1 n − respect to the basis Θ is (g ) (l ) (m ) ij i i M (g)= (l )T ν µ . Θ  i −  (m )T µ λ i −   3 Killing leaves The assumption II of the Introduction imposed on the metrics g considered in this paper allows, obviously, to construct semi-adapted charts, x , such that the fields e = ∂ , i = 1,..,n 2, { i} i ∂xi − belong to . In such a chart, called from now on, adapted, the components l ’s and m ’s vanish. ⊥ i i D The corresponding non-holonomic semi-adapted bases will be called non-holonomic adapted. Wewillcallorthogonalleaf anintegral(bidimensional)submanifoldof . Since isassumed ⊥ ⊥ D D to be transversal to , the restriction of g to any Killing leaf, say S, is non-degenerate. So, D (S,g )isa homogeneousbidimensionalRiemannianmanifold. Inparticular,the Gausscurvature |S K =K(S)ofthe Killingleavesisconstant. Itcanbe easilycomputedby noticingthatthe matrix of the components of g with respect to the chart x= x , y = x is |S n−1|S n|S s2λy2 2sµy+ν sλy+µ M(dx,dy)(g|S)= s−λye+µ − eλ !, − e e ee ee e ee e ee e e 6 where the symbol ”tilde” refers to the restriction to S and λ, µ, and ν are constants according to proposition 3. The result is e e e λs2 K(S)= , λν µ2 =M (g ). µ2 λν − (dx,dy) |S e− This shows that the following cases can occurefoer (Se,g ). e e e ee |S 1. λ > 0, λν µ2 > 0: (S,g ) is a non-Euclidean plane, i.e. a bidimensional Riemannian − |S manifold of negative constant Gauss curvature. e ee e 2. λ < 0, λν µ2 > 0: (S,g ) is an ”anti” non-Euclidean plane, i.e. is endowed with the − |S metric of the previous case multiplied by 1. e ee e − 3. λν µ2 <0: (S,g ) is any indefinite bidimensional metric of constant Gauss curvature. − |S ee e Since the Killing leaves are parametrized by x ,x , the function 1 2 λs2 K =K(x ,..,x )= 1 n−2 µ2 λν − describes the behavior of the Gauss curvature when passing from one Killing leave to another. It is worth to note that the Killing algebra is a subalgebra of the algebra il(g ), g being 2 0 0 G K a bidimensional metric of constant curvature (for instance, g = g ). If g is positive (respec- 0 |S 0 tively, negative) definite and of positive (respectively, negative) Gauss curvature, then il(g ) is 0 K isomorphic to so(3). But so(3) does not admit bidimensional subalgebras at all. This explains why g cannotbe apositively(respectively,negative)curvedmetricinthecase(1)(respectively, |S (2)). Similarly,ifg is apositiveornegativedefinite flatmetric,then il(g )admits onlyAbelian 0 0 K bidimensional subalgebras. This explains why both positive and negative definite flat metrics are absent in the above list for g . In all other cases, the algebra il(g ) admits bidimensional non- |S K 0 Abelian subalgebras. More exactly, if g is not flat, then il(g ) is isomorphic to so(2,1). Let g 0 0 K be the Killing form of so(2,1). Then, the tangent planes to the isotropic cone of g exhaust the bidimensional non-Abelian Lie subalgebras of so(2,1). If g is flat and, thus, indefinite, then any 0 bidimensional subspace of the algebra il(g ) different from its commutator, which is Abelian, is 0 K a non-Abelian subalgebra. It is not difficult to describe the algebra il(g ) in the semi-adapted K |S coordinates (x,y). A direct computation shows that il(g ) has the following basis: 0 K Xe=e∂ , Y =esx∂ , Z =e sx 2 sλy µ ∂ + s2λy2 2sµy+ν ∂ , x y − x y − − e eh (cid:16) (cid:17) (cid:16) (cid:17) i e e e e e ee e e ee ee e e 7 X,Y =sY, X,Z = sZ, Y,Z =2sλX, − h i h i h i e e e e e e e e ee Inthecaseλ=0,themetric g isflatindefiniteanditisconvenienttoidentify(S,g )withthe |S |S standardplane R2,dξ2 dη2 ,R2 = (ξ,η) . Todothatitisnecessarytochooseabidimensional − { } non-commutativ(cid:0)esubalgebrai(cid:1)n il dξ2 dη2 (theyareallequivalent). Forinstance,bychoosing K − Y = ∂ +∂ , X = η∂ ξ∂ , w(cid:0)e have [X (cid:1),Y ] = Y , X ,Y il dξ2 dη2 and, for s = 0, 0 ξ η 0 ξ η 0 0 0 0 0 − − ∈ K − 6 one can identify the quadruple S, 2 dxdy ydx2 , X , Y with(cid:0)R2, dξ2 (cid:1)dη2, X , Y . − |S |S − 0 0 (cid:0) (cid:0) (cid:1) (cid:1) (cid:0) (cid:1) The simply connected Lie group G corresponding to is isomorphic to the group of affine e e e e G transformationsofR. Then,bothS andR2 arediffeomorphictoGashomogeneousG spacesand − the above identification of them is an equivalence of G spaces. The Killing form of determines − G naturally a symmetric covariant tensor field on the G space G which is identified with dx2 on S − 2 andwith dξξ−−ηdη onR2. WewillcontinuetocallitKillingform. Thus,intheaboveidentiefication the metri(cid:16)c g fo(cid:17)r λ=0 and s=0 corresponds to |S dξ dη 2 µ dξ2 dη2 +ν − . (12) − ξ η (cid:18) − (cid:19) (cid:0) (cid:1) e e This representation of the metric g will be used to describe global solutions of the Einstein |S equations in section 5. 4 The Ricci tensor field Inthe following we will consider4-dimensionalmanifolds andwill use the following conventionfor the indices: Greek letters take values from 1 to 4; the first Latin letters take values from 3 to 4, while i,j from1 to2. Letg be a -integrable4-metric. The resultsofthe previoussectionsallow 2 G to choose a non-holonomic adapted basis Θ such that the matrix M (g) associated to g is of the Θ form F 0 M (g)= (13) Θ 0 H (cid:18) (cid:19) where F and H are 22 matrices whose elements depend only on x and x . We will distinguish 1 2 two cases according to whether F, i.e., the matrix associated to the metric restricted to , has ⊥ D negative or positive determinant. detF<0. In this case, owing to the bidimensionality of , and the independence of F on ⊥ • D x andx ,thecoordinatesx andx ,canbefurtherspecifiedtobecharacteristiccoordinates 3 4 1 2 8 on any integral submanifold of , so that, without changing the properties of M (g) in ⊥ Θ D (13), F takes the following form 0 f F= . f 0 (cid:18) (cid:19) detF>0. Similarly,inthiscase,insomeisothermalcoordinates,thematrixFgetstheform • f 0 F= 0 f (cid:18) (cid:19) Thus, we have: Proposition 5. A 4-metric g, is -integrable iff there exists a non-holonomic adapted basis Θ 2 G such that the matrix M (g) of g takes one of the following block forms, according to whether Θ detF<0 or detF>0 . 0 f f 0 0 0 M (g)= f 0 , M (g)= 0 f Θ Θ     0 H 0 H     ν µ H= − µ λ (cid:18) − (cid:19) λ, µ, ν being arbitrary functions of x . In the corresponding adapted holonomic basis C = dxµ i { } we have 0 f f 0 0 0 M (g)= f 0 , M (g)= 0 f C C     0 H 0 H     where s2λx2 2sµx +ν sλx +µ H= 4− 4 − 4 . sλx +µ λ 4 (cid:18) − (cid:19) It is worth to observe that detH=detH=λν µ2 is a functions of x ’s only. i − Inthefollowingsectionsthe explicitexpressionsofthe componentsR oftheRiccitensorfield µν in terms of the function f and of the elements h of the matrix H in the adapted non-holonomic ab basis of corollary 5 are found. Recall that R =Rβ =e (γβ )+γβ γρ Cρ γβ . µν µνβ [ν β]µ [νρ β]µ− νβ ρµ with the Christoffel symbols 1 γα = gασ( e (g )+e (g )+e (g )) µν 2 − σ µν µ σν ν σµ 1 Cα +gραg Cσ +gραg Cσ . − 2 νµ σµ νρ σν µρ (cid:0) (cid:1) 9 It is easy too see that the γα ’s and R ’s are first order polynomials in s and it is convenient to µν µν single out their constant terms Γα and S , respectively. More exactly, one has: µν µν 1 γ =Γ +Λ = g 1G +Λ µ µ µ − µ µ 2 where γ ,Γ ,Λ ,G are matrices whose elements γα , Γα ,Λα ,G , are defined by µ µ µ µ µν µν µν µαν 1 Γα = gασ( e (g )+e (g )+e (g )) µν 2 − σ µν µ σν ν σµ 1 Λα = Cα +gραg Cσ +gραg Cσ (14) µν −2 νµ σµ νρ σν µρ G = e (cid:0)(g )+e (g )+e (g ). (cid:1) µσν σ µν µ σν ν σν − R =S +T µν µν µν where S =e Γβ +Γβ Γρ µν [ν β]µ [νρ β]µ T =e Λβ + Γ Λ β + Λ Γ β + Λ Λ β Cρ γβ . µν [ν β]µ [ν β] µ [ν β] µ [ν β] µ− νβ ρµ (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Now we pass to the calculation of the Ricci tensor. 4.1 The Ricci tensor in the case det F < 0 Note that for s = 0 the adapted non-holonomic basis becomes holonomic and coincides with the one used in [4]. This is why the expressions for S given below coincide with the expressions for µν the components of the Ricci tensor found in [4]. Observe also that only the fields e and e give 1 2 nontrivialcontributions to expressions (14) for the Γα ’s and all components Λα , except possibly µν µν Λc , vanish. ba Components R : ij • Let us note that T =Λβ Γρ Cρ γβ =0, ij βρ ji− jβ ρi thisisduetothefactthatCρ =0,thecomponentsΛα withanindexequalto1or2vanish jβ µν and Γa =0. So, ij R =S =e Γβ +Γβ Γρ . ij ij [j β]i [jρ β]i (cid:16) (cid:17) 10