Skew-symmetric endomorphisms in \mathbb{M}^{1,n}: A unified canonical form with applications to conformal geometry
aa r X i v : . [ g r- q c ] D ec Skew-symmetric endomorphisms in M ,n : A unified canonical formwith applications to conformal geometry Marc Mars and Carlos Pe´on-NietoInstituto de F´ısica Fundamental y Matem´aticas, Universidad de SalamancaPlaza de la Merced s/n 37008, Salamanca, SpainDecember 23, 2020
Abstract
We show the existence of families of orthonormal, future directed bases which allow to cast every skew-symmetric endomorphism of M ,n (SkewEnd (cid:0) M ,n (cid:1) ) in a single canonical form depending on a minimal num-ber of parameters. This canonical form is shared by every pair of elements in SkewEnd (cid:0) M ,n (cid:1) differing by anorthochronous Lorentz transformation, i.e. it defines the orbits of the orthochronous Lorentz group under theadjoint action on its algebra. Using this form, we obtain the quotient topology of SkewEnd (cid:0) M ,n (cid:1) /O + (1 , n ).From known relations between SkewEnd (cid:0) M ,n (cid:1) and the conformal Killing vector fields (CKVFs) of thesphere S n , a canonical form for CKVFs follows immediately. This form is used to find adapted coordinatesto an arbitrary CKVF that covers all cases at the same time. We do the calculation for even n and obtainthe case of odd n as a consequence. Finally, we employ the adapted coordinates to obtain a wide classof TT-tensors for n = 3, which provide Cauchy data at conformally flat null infinity I . Specifically, thisclass of data is characterized for generating Λ > I . The class of data is infinite dimensional, depending on two arbitraryfunctions of one variable as well as a number of constants. Moreover, it contains the data for the Kerr-deSitter spacetime, which we explicitly identify within. Contents O + (1 , d − -classes 17 V ) /O + (1 , d −
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 TT-Tensors 38
Having a Lie group G acting on a space X respresenting a set of physical quantities is always a desirable featurein a physical problem, as Lie groups represent symmetries (either global or gauge) and their presence is oftentranslated into a simplification of the formal aspects of the problem. Roughly speaking, in this situation the“relevant” part for the physics effectively happens in the quotient space X/G . For the study of these quotientspaces, one may be interested in obtaining a unified form to give a representative for every orbit in
X/G , i.e. a canonical form (also known as normal form). A particularly relevant case is when X is a Lie algebra g and G itsLie group acting by the adjoint action, in which case the orbits are also called conjugacy classes of G (see e.g.[5]). In the first part of this paper, we study the conjugacy classes of the pseudo-orthogonal group O (1 , n ) (orLorentz group), for which we will obtain a canonical form. Our main interest on this, addressed in the secondpart of the paper, lies in its relation with the Cauchy problem of general relativity (GR) (cf. [11] and e.g. [7],[14]) and more precisely, its formulation at null infinity I for the case of positive cosmological constant Λ (cf.[12], [13]). In the remainder of this introduction we summarize our ideas and results, and we will also brieflyreview some results on conjugacy classes of Lie groups related to our case, as well as the Cauchy problem ofGR with positive Λ.A typical example of a canonical form in the context described above, is the well-known Jordan form,which represents the conjugacy classes of GL ( n, K ) (where K is usually R , C or the quaternions H ). Besidesthis example, the problem of finding a canonical representative for the conjugacy classes of a Lie group hasbeen adressed numerous times in the literature. The reader may find a list of canonical forms for algebraswhose groups leave invariant a non-degenerate bilinear form in [9] (this includes symmetric, skew-symmetricand simplectic algebras over R , C and H ) as well as the study of the affine orthogonal group (or Poincar´e group)in [8] or [19]. Notice that these works deal, either directly or indirectly, with our case of interest O (1 , n ), whosealgebra o (1 , n ) will be represented in this paper as skew-symmetric endomorphisms of Minkowski spacetime M ,n . When giving a canonical form, it is usual to base it on criteria of irreducibility rather than uniformity(e.g. [8], [9], [19]). This is similar to what is done when the Darboux decomposition is applied to two-forms (i.e.elements of o (1 , n )), for example in [23] or for the low dimensional case n = 3 (e.g. [18], [29]). As a consequence,all canonical forms found for the case of o (1 , n ) require two different types of matrices to represent all orbits, oneand only one fitting a given element. Our first aim in this paper is to give a unique matrix form which representseach element F ∈ o (1 , n ), depending on a minimal number of parameters that allows one to easily determineits orbit under the adjoint action of O (1 , n ). This is obviously achieved by loosing explicit irreducibility in theform. However, this canonical form will be proven to be fruitful by giving several applications. This same issuehas also been adressed in [24] for the case of low dimensions, i.e. O (1 ,
2) and O (1 , O + (1 , n ) ⊂ O (1 , n ) is homomorphic to the group ofconformal transformations of the n -sphere, Conf ( S n ). The conformal structure of I happens to be fundamentalfor the Cauchy problem at null infinity of GR for spacetimes with positive Λ , as it is the gauge group for theset of initial data. Such a set consists of a manifold Σ endowed with a (riemannian) conformal structure [ γ ],2epresenting the geometry of null infinity I := (Σ , [ γ ]), together with the conformal class [ D ] of a transverse(i.e. zero divergence), traceless, symmetric tensor D (TT-tensor) of I . If the spacetime generated by the datais to have a Killing vector field, the TT-tensor must satisfy a conformally covariant equation depending on aconformal Killing vector field (CKVF) of I , the so-called Killing Initial Data (KID) equation [25].The class of data in which [ γ ] contains a constant curvature metric (or alternatively the locally conformallyflat case) includes the family of Kerr-de Sitter black holes and its study could be a possible route towards acharacterization result for this family of spacetimes. Even in this particular (conformally flat) case, it is difficultto give a complete list of TT-tensors. An example can be found in [3], where the author gives a class of solutionswith a direct and elegant method, but the solution is restricted in the sense that global topological conditionsare imposed on I . Namely, the solutions obtained by this method must be globally regular on S and hencecannot contain the family of Kerr-de Sitter, which is known (see e.g. [23]) to have I with topology S minus twopoints, which correspond with the loci where the Killing horizons “touch” I . The local problem for TT tensorsis much more difficult to solve with generality, so our idea is to simplify it by imposing two KID equationsto the data, so that the corresponding spacetimes have at least two symmetries. Using the homomorphismbetween O + (1 , n + 1) and Conf ( S n ), we induce a canonical form for CKVFs from the canonical form obtainedfor o (1 , n + 1). Since this form covers all orbits of CKVFs under the adjoint action of Conf ( S n ), our adaptedcoordinates fit every CKVF and in addition, since the KID equation is conformally covariant, we can choosea conformal gauge where this CKVF is a Killing vector field, which makes the KID equation trivial. Hence, aremarkable feature from our method is that by solving one simple equation, we are solving many cases at once.This has already been done in the case of S in [24] and here we extend it to the more interesting and difficultcase of (open domains of) S .Specifically, we obtain the most general class of TT-tensors on a conformally flat I such that the Λ > I . This provides a potentially interesting ”sandbox“ to trythe consistence of possible definitions of (global) mass and angular momentum (see [30] for a review on thestate of the art). Recall that symmetries are well-known to be related to conserved quantities, in particular,axial symmetry is related to conservation of angular momentum and time symmetry to conservation of energy.Moreover, for a spacetime to have constant mass, one may require no radiation escaping from or coming withinthe spacetime, a condition which, following the criterion of [10], is guaranteed by conformal the flatness of I .Finally, the presence of the Kerr-de Sitter data within the set of TT-tensors contributes to its physical relevanceand furnishes the possibility of looking for new characterization results for this family of spacetimes.As an additional sidenote concerning our results, notice that both the canonical form of CKVFs as well asthe adapted coordinates are obtained in arbitrary dimensions, so similar applications may be worked out inarbitrary dimensions which, needless to say, is a considerable harder problem. On the possible extension tomore dimensions of this type of TT-tensors, one should mention that the Cauchy problem at I for positivecosmological constant is known to be well-posed in arbitrary even dimensions [2]. However the KID equationsare only known to be a necessary consequence of having symmetries, but sufficiency is an open problem inspacetime dimensions higher than four.The paper is organized as follows. In order to properly define the canonical form, in Section 2 we rederivea classification result for skew-symmetric endomorphisms (cf. Theorem 2.6), employing only elementary linearalgebra methods. The results of this section are known (see e.g. [16], [17], [20]), but the method is original andwe believe more direct than other approaches in the literature. We also include the derivations in order to makethe paper self-contained. Section 2 leads to the definition of canonical form in Section 3. Section 4 deals witha particular type of skew-symmetric endomorphisms (the so-called simple , i.e. of minimal matrix rank), whichwill be useful for future sections. In Section 5 we work out some applications of our canonical form: identifying3nvariants which characterize the conjugacy classes of the orthochronous Lorentz group (cf. Theorem 5.1) andobtaining the topological structure of this quotient space (cf. Section 5.1).In Section 6 we use the homomorphism between O + (1 , n + 1) and Conf ( S n ) and apply the canonical formobtained for skew-symmetric endomorphisms to give a canonical form for CKVFs, together with a decomposedform (cf. Proposition 6.1) which analogous to the one given for skew-symmetric endomorphisms. In Section7, we adapt coordinates to CKVFs in canonical form, first in the even dimensional case, from which the odddimensional case is obtained as a consequence. Finally, in Section 8 we employ the adapted coordinates to findthe most general class of data at I corresponding to spacetime dimension four, such that I is conformallyflat and the (Λ > In this section we derive a classification result for skew-symmetric endomorphisms of Lorentzian vector spaces.Let V be a d -dimensional vector space endowed with a pseudo-Riemannian metric g . If g is of signature( − , + , · · · , +), then ( V, g ) is said to be Lorentzian. Scalar product with g is denoted by h , i . An endomorphism F : V −→ V is skew-symmetric when it satisfies h x, F ( y ) i = −h F ( x ) , y i ∀ x, y ∈ V. We denote this set by SkewEnd ( V ) ⊂ End ( V ). We take, by definition, that eigenvectors of an endomorphism arealways non-zero. We use the standard notation for spacelike and timelike vectors as well as for spacelike, timelikeand degenerate vector subspaces. In our convention all vectors with vanishing norm are null (in particular, thezero vector is null). We denote ker F and Im F , respectively, to the kernel and image of F ∈ End ( V ). Lemma 2.1. [Basic facts about skew-symmetric endomorphisms] Let F be a skew-symmetric endomorphism ina pseudo-riemannian vector space V . Thena) ∀ w ∈ V , F ( w ) is perpendicular to w , i.e. h F ( w ) , w i = 0 .b) Im F ⊂ (ker F ) ⊥ and ker F ⊂ (Im F ) ⊥ .c) If w ∈ ker F ∩ Im F then w is null.d) If w ∈ V is a non-null eigenvector of F , then its eigenvalue is zero.e) If w is an eigenvector of F with zero eigenvalue, then all vectors in Im F are orthogonal to w , i.e. Im F ⊂ w ⊥ .f ) If F restricts to a subspace U ⊂ V (i.e. F ( U ) ⊂ U ), then it also restricts to U ⊥ .Proof. a) is immediate from h w, F ( w ) i = −h F ( w ) , w i . For b ), let v ∈ ker F and w be of the form w = F ( u ) forsome u ∈ V , then h w, v i = h F ( u ) , v i = −h u, F ( v ) i = 0the last equality following because F ( v ) = 0. c ) is a consequence of b ) because w belongs both to ker F and toits orthogonal, so in particular it must be orthogonal to itself, hence null. d ) is immediate from0 = h w, F ( w ) i = λ h w, w i
4o if w is non-null, its eigenvalue λ must be zero. e ) is a corollary of b ) because by hypothesis w ∈ ker F soIm F ⊂ (ker F ) ⊥ ⊂ w ⊥ the last inclusion being a consequence of the general fact U ⊂ U ⇒ U ⊥ ⊂ U ⊥ . Finally, f ) is true because forany u in a F -invariant subspace U and w ∈ U ⊥ h F ( u ) , w i = − h u, F ( w ) i . Another well-known property of skew-symmetric endomorphisms that we will use is that dim Im F is alwayseven. Equivalently, dim ker F has the same parity than dim V . To see this, consider the 2-form F assigned toevery F ∈ SkewEnd ( V ) by the standard relation F ( e, e ′ ) = h e, F ( e ′ ) i , ∀ e, e ′ ∈ V. The matrix representing F is skew in the usual sense, hence the dimension of Im F ⊂ V ⋆ (the dual of V ) is therank of that matrix, which is known to be even (see e.g. [15]) and clearly dim Im F = dim Im F .The strategy that we will follow to classify skew-symmetric endomorphisms of V with g Lorentzian is via F -invariant spacelike planes. Conditions for F -invariance of spacelike planes are stated in the following lemma: Lemma 2.2.
Let F ∈ SkewEnd ( V ) . Then F has a F -invariant spacelike plane Π s if and only if F ( u ) = µv, F ( v ) = − µu, (1) for Π s = span { u, v } with u, v ∈ V spacelike, orthogonal, unit and µ ∈ R . Moreover, (1) is satisfied for µ = 0 ifand only if ± iµ are eigenvalues of F with (null) eigenvectors u ± iv , for u, v ∈ V spacelike, orthogonal with thesame square norm.Proof. If (1) is satisfied for u, v ∈ V spacelike, orthogonal, unit, then Π s = span { u, v } is obviously F -invariantspacelike. On the other hand, if Π s is F -invariant, then it must hold that F ( u ) = a u + a v, F ( v ) = b u + b v, a , a , b , b ∈ R , for a pair of orthogonal, unit, spacelike vectors u, v spanning Π s . Using skew-symmetry and the orthogonalityand unitarity of u, v , the constants are readily determined: a = b = 0 and a = − b =: µ , which implies (1).This proves the first part of the lemma.For the second part, it is immediate that if (1) holds with µ = 0, then ± iµ are eigenvalues of F with respectiveeigenvectors u ± iv . The orthogonality of u, v follows from h F ( u ) , u i = 0 = µ h v, u i and the equality of normfrom skew-symmetry h F ( u ) , v i = − h u, F ( v ) i ⇒ µ h v, v i = µ h u, u i . Assume now that F has an eigenvalue iµ = 0with (necessarily null) eigenvector w = u + iv , for u, v ∈ V . Since F is real, neither u nor v can be zero. Fromthe nullity property h w, w i = 0, it follows that h u, u i − h v, v i = 0 and h u, v i = 0. Hence, u, v are orthogonal withthe same norm, so they are either null and proportional, which can be discarded because it would imply that u (and v ) is a real eigenvector with complex eigenvalue; or otherwise u, v are spacelike, thus the lemma follows.There is an analogous result for F -invariant timelike planes:5 emma 2.3. Let F ∈ SkewEnd ( V ) . Then F has a F -invariant timelike plane Π t if and only if F ( e ) = µv, F ( v ) = µe, (2) for Π t = span { e, v } with e, v ∈ V for e timelike unit orthogonal to v spacelike, unit and µ ∈ R . Moreover, (1) issatisfied for µ = 0 if and only if ± µ are eigenvalues of F with (null) eigenvectors e ± v , for e, v ∈ V orthogonal,timelike and spacelike respectively with opposite square norm.Proof. For the first claim, repeat the first part of the proof of Lemma 2.2 assuming u = e timelike.For the second claim, assume (1) is satisfied with µ = 0. Then it is immediate that h F ( e ) , e i = 0 = µ h v, e i ,hence e, v are orthogonal and by skew-symmetry h F ( e ) , v i = − h e, F ( v ) i ⇒ µ h v, v i = − µ h e, e i , i.e. must haveopposite square norm. Conversely, let ± µ = 0 be a pair of eigenvalues with respective null eigenvectors q ± , thatw.l.o.g can be chosen future directed. Then e := q + + q − and v := q + − q − are orthogonal, with opposite squarenorm h e, e i = 2 h q + , q − i = − h v, v i <
0, and they satisfy (2).The F -invariant spacelike or timelike planes will be often be refered to as “eigenplanes” and µ will be denotedas the “eigenvalues” of Π. Notice that a simple change of order in the vectors switches the sign of the eigenvalue µ . Thus, unless otherwise stated, we will consider the eigenvalues of eigenplanes (both spacelike and timelike)non-negative by default.The first question we address here is under which conditions such a plane exists (cf. Proposition 2.1). Butbefore doing so, we need to prove some results first. Lemma 2.4.
Let V be a Lorentzian vector space F ∈ SkewEnd ( V ) . Then there exist two vectors x, y ∈ V ,with x = 0 , such that one of the three following exclusive possibilities hold(i) x is a null eigenvector of F .(ii) x is a non-null eigenvector (with zero eigenvalue).(iii) x, y are orthogonal, spacelike and with the same norm, and define an eigenplane of F with non-zeroeigenvalue, i.e. F ( x ) = µy, F ( y ) = − µx, µ ∈ R \ { } . If, instead, V is riemannian, only cases (ii) and (iii) can arise.Proof. From the Jordan block decomposition theorem we know that there is at least one, possibly complex,eigenvalue s + is with eigenvector x + iy , that is, F ( x + iy ) = ( s + is )( x + iy ), or equivalently: F ( x ) = s x − s y, (3) F ( y ) = s x + s y. (4)This system is invariant under the interchange ( x, y ) → ( − y, x ), so without loss of generality we may assume x = 0. The respective scalar products of (3) and (4) with x , y yield s h x, x i − s h x, y i = 0 s h y, y i + s h x, y i = 0 (cid:27) ⇐⇒ (cid:18) h x, x i − h x, y ih y, y i h x, y i (cid:19) (cid:18) s s (cid:19) = (cid:18) (cid:19) . (5)Observe that if s + is = 0 the determinant of the matrix must vanish. i.e. h x, y i ( h x, x i + h y, y i ) = 0. Hence,we can distinguish the following possibilities: 6a) s = s = 0. Then x is an eigenvector of F with vanishing eigenvalue so we fall into cases ( i ) or ( ii ).(b) s + is = 0. From h x, y i ( h x, x i + h y, y i ) = 0 we distinguish two cases:(b.1) h x, y i = 0. If s = 0 then (5) forces x and y to be both null and, being also orthogonal to each other,there is a ∈ R such that y = ax and we fall into case ( i ). So, we can assume s = 0 (and then s = 0). Let µ := − s , thus ( iii ) follows from equations (3), (4) and Lemma 2.2.(b.2) h x, y i 6 = 0. Then h x, x i = −h y, y i and the matrix problem (5) reduces to s h x, x i − s h x, y i = 0 . In addition, (3) and (4) imply h F ( x ) , y i = s h x, y i − s h y, y i = s h x, y i + s h x, x i = h F ( y ) , x i . But skew-symmetry requires h F ( x ) , y i = − h F ( y ) , x i , so h F ( y ) , x i = 0 and we conclude s h x, y i + s h x, x i = 0 . Combining with (2) yields (cid:18) h x, x i − h x, y ih x, y i h x, x i (cid:19) (cid:18) s s (cid:19) = (cid:18) (cid:19) . The determinant of this matrix is non-zero which yields a contradiction with s + is = 0. So this case isempty.To conclude the proof, we must consider the case when the vector space V is riemannian. The proof isidentical except from the fact that all cases involving null vectors are imposible from the start. Remark 2.1.
One may wonder why the lemma includes the possibility of having a spacelike eigenplane (case(iii)), but not a timelike eigenplane. The reason is that invariant timelike planes, which are indeed possible, fallinto case (i) by Lemma 2.3, because e ± v are null eigenvectors. In the case of Riemmanian signature, Lemma 2.4 can be reduced to the following single statement:
Corollary 2.4.1.
Let V be Riemannian of dimension d and F ∈ SkewEnd ( V ) . If d = 1 then F = 0 and if d ≥ then there exist two orthogonal and unit vectors u, v satisfying F ( u ) = µv, F ( v ) = − µu, µ ∈ R (6) Proof.
The case d = 1 is trivial, so let us assume d ≥
2. By the last statement of Lemma 2.4 either there existsan eigenvector x with zero eigenvalue or the pair { u, v } claimed in the corollary exists. In the former case,we consider the vector subspace x ⊥ . Its dimension is at least one and F restricts to this space so again eitherthe pair { u, v } exists or there is y ∈ x ⊥ satisfying F ( y ) = 0. But then { x, y } are orthogonal and non-zero.Normalizing we find a pair { u, v } that satisfies (6) with µ = 0,Lemma 2.4 lists a set of cases, one of which must always occur. However, we now show that, if the dimensionis sufficiently high, case ( i ) of that lemma implies one of the other two:7 emma 2.5. Let F ∈ SkewEnd ( V ) , with V Lorentzian of dimension at least four. If F has a null eigenvector,then it also has either a spacelike eigenvector or a spacelike eigenplane.Proof. Let k ∈ V be a null eigenvector of F . The space A := k ⊥ ⊂ V is a null hyperplane and F restricts to A .On this space we define the standard equivalence relation y ∼ y iff y − y = ak , a ∈ R . The quotient A/ ∼ (which has dimension at least two) inherits a positive definite metric g and F also descends to the quotient.More precisely, if we denote the equivalence class of any y ∈ A by y , then for any y ∈ A/ ∼ and any y ∈ y theexpression F ( y ) = F ( y ) is well-defined (i.e. independent of the choice of representative y ) and hence defines anendomorphism F of A/ ∼ which, moreover, satisfies h F ( y ) , y i g = −h y , F ( y ) i g . In other words F is a skew-symmetric endomorphism in the riemannian vector space A/ ∼ . By Corollary 2.4.1(here we use that the dimension of A/ ∼ is at least two) there exists a pair of orthogonal and g -unit vectors { e , e } satisfying F ( e ) = a e , F ( e ) = − a e , a ∈ R . Select representatives e ∈ e and e ∈ e . In terms of F , the condition (2) and the fact that k is eigenvectorrequire the existence of constants σ, a, λ and λ such that F ( k ) = σk, F ( e ) = ae + λ k, F ( e ) = − ae + λ k. Whenever a + σ = 0 the vectors u := e − a + σ ( aλ + σλ ) k, v := e + 1 a + σ ( aλ − σλ ) k satisfy F ( u ) = av and F ( v ) = − au . Since u and v are spacelike, unit and orthogonal to each other the claimof the proposition follows (with µ = a ). If σ = a = 0, then either λ = λ = 0 and then { e , e } are directlythe vectors { u, v } claimed in the proposition (with µ = 0), or at least one of the λ s (say λ ) is not zero. Then e := e − λ λ e is a spacelike eigenvector of F .Now we have all the ingredients to show one of the main results of this section, that will eventually allow usto classify skew-symmetric endomorphisms of Lorentzian vector spaces. Proposition 2.1.
Let V be a Lorentzian vector space of dimension at least five and F ∈ SkewEnd ( V ) . Then,there exists a spacelike eigenplane.Proof. We examine each one of the three possibilities described in Lemma 2.4. Case ( iii ) yields the resulttrivially, so we can assume that F has an eigenvector x .If we are in case ( ii ), the vector x is either spacelike or timelike. If it is timelike we consider the riemannianspace x ⊥ where F restricts. We may apply Corollary 2.4.1 (note that x ⊥ has dimension at least four) andconclude that the vectors { u, v } exist. So it remains to consider the case when x is spacelike and F admits notimelike eigenvectors. We restrict to x ⊥ which is Lorentzian and of dimension at least four. Applying againLemma 2.4, either there exists a spacelike eigenplane, or a second eigenvector y ∈ x ⊥ , which can only bespacelike or null. If y is spacelike, { u := x, v := y } span a spacelike eigenplane with µ = 0. If y is null, wemay apply Lemma 2.5 to F | x ⊥ to conclude that either a spacelike eigenplane exists, or there is a spacelikeeigenvector e ∈ x ⊥ , so the pair { u := e, v := x } satisfies (1) with µ = 0. This concludes the proof of case ( ii ).8n case ( i ), i.e. when there is a null eigenvector x we can apply Lemma 2.5 and conclude that either { u, v } exist, or there is a spacelike eigenvector e ∈ V , in which case we are into case ( ii ), already solved. This completesthe proof.Proposition 2.1 provides the basic tool to classify systematically skew-symmetric endomorphisms if thedimension d is at least five. The idea is to start looking for a first spacelike eigenplane Π. Then, we restrict toΠ ⊥ , that is Lorentzian of dimension d −
2. If d − ≥
5, Proposition 2.1 applies again and we can keep on goinguntil we reach a subspace of dimension three if d odd or dimension four if d even. Therefore, for a completeclassification it only remains to solve the problem in three and four dimensions. This has already been donein [24], where a canonical form based on the classification of skew-symmetric endomorphisms is introduced.The results from [24] that we shall need are summarized in Proposition 2.2 and Corollary 2.5.1 and their mainconsequences in the present context are discussed in Remarks 2.2 and 2.3 below, where we also relate thecanonical form with the classification of skew-symmetric endomorphisms. For a proof and extended discussion,we refer the reader to [24]. In the remainder, when we explicitly write a matrix of entries F αβ , where α is therow and β the column, we refer to a linear transformation expressed in a vector basis { e α } d − α =0 acting on thevectors v = v α e α ∈ V by F ( v ) = F αβ v β e α . Proposition 2.2.
For every non-zero F ∈ SkewEnd ( V ) , with V Lorentzian four-dimensional, there exists anorthonormal basis B := { e , e , e , e } , with e timelike future directed, into which F is F = − σ τ − − σ − τ − σ σ τ τ , σ, τ ∈ R , (7) where σ := − Tr F and τ := − F , with τ ≥ . Moreover, if τ = 0 the vector e can be taken to be anyspacelike unit vector lying in the kernel of F . Corollary 2.5.1.
For every non-zero F ∈ SkewEnd ( V ) , with V Lorentzian three-dimensional, there exists anorthonormal basis B := { e , e , e } , with e timelike future directed, into which F is F = − σ − − σ − σ σ , σ := −
12 Tr (cid:0) F (cid:1) ∈ R . (8) Remark 2.2.
A classification result follows because only two exclusive possibilities arise:1. If either σ or τ do not vanish, F has a timelike eigenplane and an orthogonal spacelike eigenplane withrespective eigenvalues µ t := p ( − σ + ρ ) / and µ s := p ( σ + ρ ) / for ρ := p σ + τ ≥ . (9) The inverse relation between µ t , µ s and σ, τ is σ = µ s − µ t and τ = 2 µ t µ s .2. Otherwise, σ = τ = 0 if and only if ker F is degenerate two-dimensional. Equivalently, F has nulleigenvector orthogonal to a spacelike eigenvector both with vanishing eigenvalue. ne can easily check that when τ = 0 , the sign of σ determines the causal character of ker F , namely σ < if ker F is spacelike, σ = 0 if ker F is degenerate and σ > if ker F is timelike. Obviously, τ = 0 implies ker F = { } . The characteristic polynomial of F is directly calculated from (7) P F ( x ) = ( x − µ t )( x + µ s ) . Remark 2.3.
For a classification result in the three dimensional case, one can see by direct calculation that q := (1 + σ/ e + (1 − σ/ e generates ker F and furthermore h q, q i = − σ . Hence, the sign of σ determines thecausal character of ker F , namely it is spacelike if σ < , degenerate if σ = 0 and timelike if σ > . Moreover,when σ = 0 , F has an eigenplane with opposite causal character than q and eigenvalue p | σ | . The characteristicpolynomial of F reads P F ( x ) = x ( x + σ ) . We have now all the necessary ingredients to give a complete classification of skew-symemtric endomor-phisms of Lorentzian vector spaces. In what follows we identify Lorentzian (sub)spaces of d -dimension with theMinkowski space M ,d − . Also, for any real number x ∈ R , [ x ] ∈ Z denotes its integer part. Theorem 2.6 (Classification of skew-symmetric endomorphisms in Lorentzian spaces) . Let F ∈ SkewEnd ( V ) with V Lorentzian of dimension d > . Then V has a set of [ d − ] − mutually orthogonal spacelike eigenplanes { Π i } , i = 1 , · · · , [ d − ] − , so that V admits one of the following decompositions into direct sum of F -invariantsubspaces:a) If d even V = M , ⊕ Π d − ⊕ · · · ⊕ Π and either F | M , = 0 or otherwise one of the following cases holds:a.1) F | M , has a spacelike eigenvector e orthogonal to a null eigenvector with vanishing eigenvalue and then M , = M , ⊕ span { e } .a.2) F | M , has a spacelike eigenplane Π d − (as well as a timelike eigenplane M , orthogonal to Π d − ) andthen M , = M , ⊕ Π d − .b) If d odd V = M , ⊕ Π d − ⊕ · · · ⊕ Π and either F | M , = 0 or otherwise one of the following cases holds:b.1) F | M , has a spacelike eigenvector e and then M , = M , ⊕ span { e } .b.2) F | M , timelike eigenvector t and then M , = span { t } ⊕ Π d − .b.3) F | M , has a null eigenvector with vanishing eigenvalue.Proof. The proof is a simple combination of the previous results. First, if d ≥
5, we can apply Proposition 2.1to obtain the first spacelike eigenplane Π . Then Π ⊥ is Lorentzian of dimension d −
2. If d − ≥
5, we canapply again Proposition 2.1 to obtain a second eigenplane Π . Continuing with this process, depending on d ,two things can happen:a) If d even, we get d − (cid:0) = [ d − ] − (cid:1) spacelike eigenplanes, until we eventually reach a Lorentzian vectorsubspace of dimension four, M , , where Proposition 2.1 cannot be applied. In M , , either F | M , = 0 orotherwise cases a.
1) and a.
2) follow from Remark 2.2, cases 2 and 1 respectively.b) If d odd, we get d − (cid:0) = [ d − ] − (cid:1) spacelike eigenplanes, until we reach a Lorentzian vector subspace ofdimension three, M , . In M , , either F | M , = 0 or by Remark 2.3 there exists a unique eigenvector σ withvanishing eigenvalue. If σ null, case b.
3) follows. If it is spacelike e := σ , F restricts to e ⊥ = M , ⊂ M , and b.
1) follows. If σ timelike, the same argument applies with t := σ and t ⊥ ⊂ M , defines the remainingspacelike plane Π d − . 10 Canonical form for skew-symmetric endomorphisms
Our aim here is to extend the results in Proposition 2.2 and Corollary 2.5.1 to arbitrary dimensions. To dothat, we will employ the classification Theorem 2.6 derived in Section 2, from which it immediately follows adecomposition of any F ∈ SkewEnd ( V ) into direct sum of skew-symmetric endomorphisms of the subspacesthat F restricts to, namely F = F | M , [ d − ] − M i =1 F | Π i if d even , (10) F = F | M , [ d − ] − M i =1 F | Π i if d odd , (11)where Π i are spacelike eigenplanes. In what follows, we will denote p := [( d − / − . Notice that the blocks F | M , and F | M , may also admit different subdecompositions depending on the case,but our purpose is to remain as general as possible, so we leave this part unaltered. It will be convenient forthe rest of the paper to give a name to the decompositions (10) and (11): Definition 3.1.
Let F ∈ SkewEnd ( V ) non-zero for V Lorentzian d -dimensional. Then, a decomposition of theform (10) or (11) is called block form of F . A basis that realizes a block form is called block form basis . Writing F in block form form allows us to work with F as a sum of skew-symmetric endomorphisms ofriemmanian two-planes plus one skew-symmetric endomorphism of a three or four dimensional Lorentzianvector space. For the latter we will employ the canonical forms in Proposition 2.2 and Corollary 2.5.1, and forthe former, it is immediate that in every (suitably oriented) orthonormal basis of Π i F | Π i = (cid:18) − µ i µ i (cid:19) , ≤ µ i ∈ R . (12)Having defined a canonical form for four, three and two dimensional endomorphisms (i.e. matrices (7), (8)and (12) respectivley), the idea is to extend this result to arbitrary dimensions finding a systematic way toconstruct a block form (10), (11) such that each of the blocks are in canonical form. This is not immediate,firstly, because the block form does not require the blocks F | M , or F | M , to be non-zero and secondly, because,unlike in the four and three dimensional cases, the parameters σ, τ of the four and three dimensional blockscannot be invariantly defined as, for example, traces of F or determinant of F . The first of these concerns iseasily solved by suitably choosing a block form: Lemma 3.1.
Let F ∈ SkewEnd ( V ) be non-zero for V Lorentzian of dimension d . Then there exists a blockform (10) and (11) such that F | M , and F | M , are non-zero and they either contain no spacelike eigenplanesor they contain one with largest eigenvalue (among all spacelike eigenplanes of F ). In addition, the rest ofspacelike eigenplanes Π i are sorted by decreasing value of µ i , i.e. µ ≥ µ ≥ · · · ≥ µ p .Proof. If ker F is degenerate, it must correspond with cases a.
1) ( d even) or b.
3) ( d odd) of Theorem 2.6. Hence,in any block form the blocks F | M , and F | M , are non-zero and they do not contain any spacelike eigenplane,11s claimed in the lemma. So let us assume that ker F is non-degenerate or zero, which discards cases a.
1) and b.
3) of Theorem 2.6. In all possible cases, any block form admits the following splitting in F | M , = F | Π t ⊕ F | Π s , F | M , = F | span { v } ⊕ F | v ⊥ , (13)with Π s , Π t spacelike and timelike eigenplanes with (possibly zero) respective eigenvalues µ s and µ t , v a timelikeor spacelike eigenvector (in ker F ) and v ⊥ ⊂ M , an eigenplane with opposite causal character than v . If v isspacelike, then either F | v ⊥ is non-zero, in which case F | M , = 0 and clearly contains no spacelike eigenplanes(which is one of the possibilities in the lemma), or F | v ⊥ = 0 and then F | M , = 0, so we can rearrange thedecomposition (13) using some timelike vector v ′ ∈ v ⊥ instead of v , i.e. F | M , = F | span { v ′ } ⊕ F | v ′⊥ . Hence,in the case of d odd, we may assume that v is timelike and v ⊥ ⊂ M , is a spacelike eigenplane. Let Π µ be aspacelike eigenplane of F with largest eigenvalue µ among Π s ( d even) or v ⊥ ( d odd) and Π , · · · , Π p . Then,switching F | Π s or F | Π v ⊥ by F | Π µ we constructˆ F | M , := F | Π t ⊕ F | Π µ , ˆ F | M , := F | span { v } ⊕ F | Π µ . The resulting matrix is still in block form and has non-zero blocks ˆ F (cid:12)(cid:12)(cid:12) M , , ˆ F (cid:12)(cid:12)(cid:12) M , containing a spacelikeeigenplane with largest eigenvalue, which is the other possibility in the lemma. The last claim follows by simplyrearranging the remaining spacelike eigenplanes Π i by decreasing order of µ i .With a skew-symmetric endomorphism F in the block form given in Lemma 3.1 we can take each of theblocks to its respective canonical form. Let us denote F στ := F | M , (if d even), F σ := F | M , (if d odd) and F µ i := F | Π i when written in the canonical forms (7), (8) and (12) respectively. Consequently F = F στ p M i =1 F µ i (d even) , F = F σ p M i =1 F µ i (d odd) , (14)where, notice, each of the blocks is written in an orthonormal basis of the corresponding subspace, whichmoreover is future directed if the subspace is Lorentzian, i.e. M , or M , (c.f. Proposition 2.2 and Corollary2.5.1). Hence, the form given in (14) corresponds to a future directed, orthonormal basis of M ,d − .Our aim now is to give an invariant definition of σ, τ, µ i . A possible way to do this is through the eigenvaluesof F . One may wonder why not to use directly the eigenvalues of F . One reason is that since we are interestedin real Lorentzian vector spaces V (although, for practical reasons, we may rely on the complexification V C forsome proofs), it is more consistent to give our canonical form in terms of real quantities, while the eigenvaluesof F may be complex. In addition, the canonical form will require to sort them in some way, for which usingreal numbers is better suited.The characteristic polynomial of F is known (e.g. [23]) to possess the following parity: P F ( x ) = ( − d P F ( − x ) . (15)Thus, a simple calculation relates the characteristic polynomials of F and F P F ( x ) = det( xId d − F ) = det (cid:0) √ xId d − F (cid:1) det (cid:0) √ xId d + F (cid:1) = ( − d P F ( √ x ) P F ( −√ x ) = (cid:0) P F ( √ x ) (cid:1) , (16) √ x being any of the square roots of x in C and Id d the d × d identity matrix. We can extract some conclusionsfrom (16): 12 emma 3.2. Let F ∈ SkewEnd ( V ) for V Lorentzian of dimension d . Then the non-zero eigenvalues of F have even multiplicity m a and the zero eigenvalue has multilplicity m with the parity of d . In addition, F possesses p a (resp. exaclty one) spacelike (resp. timelike) eigenplanes with eigenvalue µ = 0 if and only if F has a negative (resp. positive) non-zero eigenvalue − µ (resp. µ ) with multiplicity m a := 2 p a (resp. exactlytwo).Proof. It is an immediate consequence of equation (16) that non-zero eigenvalues of F must have even multi-plicity m a . Moreover, since the sum of all multiplicites adds up to the dimension d , the multiplicity of the zero m has the parity of d .Combining Lemma 2.2 and equation (16), F has a spacelike eigenplane Π with non-zero eigenvalue µ if andonly if F has a negative double eigenvalue − µ . If d ≤
4, there cannot be any other spacelike eigenplanes inΠ ⊥ , so applying the same argument to F | Π ⊥ ∈ SkewEnd (cid:0) Π ⊥ (cid:1) , the multiplicity m a of − µ must be m a = 2.If d > m a ≥
4, then − µ is an eigenvalue of ( F | Π ⊥ ) with multiplicity m a −
2, thus F has a secondspacelike eigenplane with eigenvalue µ in Π ⊥ . Repeating this argument, F has a negative eigenvalue − µ withmultilplicity m a if and only if F has p a = m a / µ .Finally, by Lemma 2.3 and equation (16), F has a timelike eigenplane Π with non-zero eigenvalue µ if andonly if F has a positive double eigenvalue µ . Obviously, the maximum number of timelike eigenplanes that F can have is one. Thus, F | Π ⊥ cannot have timelike eigenplanes and hence ( F | Π ⊥ ) has no additional positiveeigenvalues. Consequently, the multiplicity of µ is exactly two.Taking into account Lemma 3.2, we will employ the eigenvalues of − F rather than those of F , so weassign positive eigenvalues of F with spacelike eigenplanes and negative eigenvalues to timelike eigenplanes.This amounts to employ the roots of the characteristic polynomial P F ( − x ).We now discuss how to invariantly define the parameters σ, τ, µ i for d even and σ, µ i for d odd. The resultof the argument is formalized below in Definition 3.2. Recall that the characteristic polynomial of a direct sumof two or more endomorphisms is the product of their individual characteristic polynomials, in particular, thecharacteristic polynomial of − F equals to the product of the characteristic polynomials of − F στ or − F σ timesthose of each − F µ i (c.f. equation (14)). Let us define: Q F ( x ) := ( P F ( − x )) / (d even) , Q F ( x ) := (cid:18) P F ( − x ) x (cid:19) / (d odd) , (17)Starting with d even, from formula (14) it is immediate that µ i are double roots of P F ( − x ), which by Lemma3.1 satisfy µ ≥ · · · ≥ µ p ≥
0. On the other hand, let µ t := p ( − σ + ρ ) / iµ s := i p ( σ + ρ ) / ρ := √ σ + τ ≥
0, that by Remark 2.2, are roots of P F στ ( x ), thus roots of P F ( x ) . By equation (16), − µ t , µ s are double roots of P F ( − x ). The set (cid:8) − µ t , µ s , µ , · · · , µ p (cid:9) are in total p + 2 = [( d − /
2] + 1 = d/ P F ( − x ). In other words, (cid:8) − µ t , µ s , µ , · · · , µ p (cid:9) is the set of all roots of thepolynomial Q F ( x ). If ker F is degenerate, then ker F στ is degenerate and by Remark 2.2 it must happen µ t = µ s = 0. Hence µ ≥ µ ≥ · · · µ p ≥ µ s = − µ t = 0. Otherwise, also by Remark 2.2, F στ contains aspacelike eigenplane with eigenvalue µ s (which by Lemma 3.1 is the largest) as well as a timelike eigenplanewith eigenvalue µ t . In this case µ s ≥ µ ≥ · · · µ p ≥ ≥ − µ t .We next discuss σ, µ i for d odd. Again, from (14) we have that µ i are double roots of P F ( − x ), which byLemma 3.1 also satisfy µ ≥ · · · ≥ µ p ≥
0. By Remark 2.3, √ σ is a root of P F σ ( x ), thus a root of P F ( x ), so by We adopt the convention that a root with multiplicity m ≥ Q F ( x ) is a polynomial because all the roots of P F ( − x ) are double. σ is a double root of P F ( − x ). Also, P F ( − x ) has at least one zero root and hence, P F ( − x ) /x is a polynomial with d − (cid:8) σ, µ , · · · , µ p (cid:9) are all double roots of P F ( − x ) /x , which are p + 1 = [( d − /
2] = ( d − / Q F as defined in (17) is also apolynomial and (cid:8) σ, µ , · · · , µ p (cid:9) is the set of all its roots. If ker F is timelike, then ker F σ is timelike, whichhappens if and only if σ > F σ has a spacelike eigenplane with eigenvalue p | σ | ,that by Lemma 3.1 is the largest eigenvalue among spacelike eigenplanes. Thus σ ≥ µ ≥ · · · ≥ µ p . In the caseker F not timelike, the inequalities become µ ≥ · · · ≥ µ p ≥ ≥ σ .Summarizing, the paramaters σ, τ, µ i correspond to the set of all roots of Q F sorted in a certain order fullydetermined by the causal character of ker F . This allows us to put forward the following definition: Definition 3.2.
Let
Roots ( Q F ) denote the set of roots of Q F ( x ) repeated as many times as their multiplicity.Thena) If d odd, (cid:8) σ ; µ , · · · , µ p (cid:9) := Roots ( Q F ) sorted by σ ≥ µ ≥ · · · ≥ µ p if ker F is timelike and µ ≥ · · · ≥ µ p ≥ ≥ σ otherwise.b) If d even, σ := µ s − µ t , τ := 2 | µ t µ s | with (cid:8) − µ t , µ s ; µ , · · · , µ p (cid:9) := Roots ( Q F ) sorted by µ ≥ · · · ≥ µ p ≥ µ s = − µ t = 0 if ker F is degenerate and µ s ≥ µ ≥ · · · ≥ µ p ≥ ≥ − µ t otherwise. In addition, we also summarize the results concerning the canonical form in the following Theorem:
Theorem 3.3.
Let F ∈ SkewEnd ( V ) non-zero, with V Lorentzian of dimension d ≥ and p := [( d − / − .Then there exists an orthonormal, future oriented basis such that F is given (14) where F στ := F | M , , F σ := F | M , , F µ i := F | Π i are given by (7) , (8) , (12) respectively and σ, τ, µ i are given in Definition 3.2. In particular, F στ , F σ are non-zero and they either do not contain a spacelike eigenplane or they contain one with maximaleigenvalue (among all spacelike eigenplanes of F ) and the eigenvalues µ i are sorted by µ ≥ µ ≥ · · · µ p . Definition 3.3.
For any F ∈ SkewEnd ( V ) , for V Lorentzian d -dimensional, the form of F given in Theorem3.3 is called canonical form and the basis realizing it is called canonical basis . The first and obvious reason why the canonical form is useful is that it allows one to work with all elements F ∈ SkewEnd ( V ) at once. The fact that we can give a canonical form for every element without splitting intocases is a great strenght, since we can perform a general analysis just in terms of the parameters that define thecanonical form. Moreover, as we will show in Section 5, this form is the same for all the elements in the orbitgenerated by the adjoint action of the orthochronous Lorentz group O + (1 , d − O + (1 , d −
1) invariance (or covariance) which, as discussed in Section 6, isdirectly related to certain conformally covariant problems in general relativity.We finish this section with two corollaries that will be useful later. The first one is trivial from the canonicalform (14)
Corollary 3.3.1.
The characteristic polynomial of F ∈ SkewEnd ( V ) is P F ( x ) = ( x − µ t )( x + µ s ) p Y i =1 ( x + µ i ) ( d even) , P F ( x ) = x ( x + σ ) p Y i =1 ( x + µ i ) ( d odd) , (18) where − µ t := σ − √ σ + τ , µ s := σ + √ σ + τ . The second gives a formula for the rank of F . We base our proof in the canonical form (14) because it isstraightforward. However, we remark that this corollary can also be regarded as a consequence of Theorem 2.6.14 orollary 3.3.2. Let F ∈ SkewEnd ( V ) , with V Lorentzian of dimension d and m the multiplicity of the zeroeigenvalue. Then, only of the following exclusive cases hold:a) ker F is non-degenerate or zero if and only if rank F = d − m .b) ker F is degenerate if and only if m > and rank F = d − m + 2 .Proof. Consider F in canonical form (14) and let k ∈ N be the number of parameters µ i that vanish. For d evenwe have dim ker F = 2 k + dim ker F στ . On the one hand, ker F degenerate implies ker F στ degenerate, whichby Remark 2.2 happens if and only if σ = τ = 0 and in addition dim ker F στ = 2. Therefore dim ker F = 2 k + 2and by (18), m = 2 k + 4 ( > F = d − dim ker F = d − m + 2. On the other hand, ker F non-degenerate if at most one of σ or τ vanish. If τ = 0 (so that µ s = 0 and µ t = 0), dim ker F στ = 0 and m = 2 k = dim ker F . Consequently rank F = d − m . If τ = 0 (and σ = 0, so that exactly one of µ s , µ t vanish),by Remark 2.2 dim ker F στ = 2 and by (18) m = 2 k + 2. Hence dim ker F = 2 k + 2 and rank F = d − m .For d odd, we have dim ker F = 2 k + dim ker F σ = 2 k + 1, because dim ker F σ = 1 (c.f. Remark 2.3). ker F is degenerate if and only if ker F σ is degenerate, which by Remark 2.3 occurs if and only if σ = 0. Hence, byequation (18), m = 2 k + 3 ( >
2) and rank F = d − dim ker F = d − m + 2. For the ker F non-degenerate case, σ = 0 and also by (18) m = 2 k + 1 = dim ker F . Therefore rank F = d − m . By simple skew-symmetric endomorphism we mean a G ∈ SkewEnd ( V ) satisfying rank G = 2. As usual e ♭ ≡ h e, ·i is the one-form obtained by lowering index to a vector e ∈ V . Then, a simple skew-symmetricendomorphism can be always written as G = e ⊗ v ♭ − v ⊗ e ♭ for two linearly independent vectors e, v ∈ V and its action on any vector w ∈ V is G ( w ) = h v, w i e − h e, w i v. Since the two-fom associated to a simple endomorphism is G = e ♭ ∧ v ♭ , it follows from elementary algebra thattwo simple skew-symmetric endomorphisms G = e ⊗ v ♭ − v ⊗ e ♭ and G ′ = e ′ ⊗ v ′ ♭ − v ′ ⊗ e ′ ♭ are proportional ifand only if span { e, v } = span { e ′ , v ′ } . This freedom in the pair { e, v } defining G can be used to choose themorthogonal. Lemma 4.1.
Let G ∈ SkewEnd ( V ) be simple. Then there exist two non-zero orthogonal vectors e, v ∈ V suchthat G = e ⊗ v ♭ − v ⊗ e ♭ with v spacelike.Proof. By definition G = ˜ e ⊗ ˜ v ♭ − ˜ v ⊗ ˜ e ♭ for two linearly indepedent vectors ˜ e, ˜ v ∈ V . If one of them is non-null,we set ˜ v := v and decompose V = span { v } ⊕ v ⊥ . Thus ˜ e = av + e with a ∈ R and e ∈ v ⊥ and G takes the form G = ( av + e ) ⊗ v ♭ − v ⊗ ( av + e ) ♭ = e ⊗ v ♭ − v ⊗ e ♭ , as claimed. If ˜ e and ˜ v are both null, consider V = span { ˜ e } e ⊕ (˜ e ) c (we use e ⊕ because this direct sum is not by orthogonal spaces) where (˜ e ) c is a spacelike complement of span { ˜ e } .Then we can write ˜ v = a ˜ e + v ′ , with a ∈ R and v ′ ∈ ˜ e c non-null. Thus G = ˜ e ⊗ v ′ ♭ − v ′ ⊗ ˜ e ♭ , with v ′ non-nulland we fall into the previous case. All in all, G = e ⊗ v ♭ − v ⊗ e ♭ with e, v orthogonal. Consequently, either oneof the vectors is spacelike or both are null and proportional which would imply G = 0, against our hypothesisrank G = 2. 15he decomposition G = e ⊗ v ♭ − v ⊗ e ♭ is not unique even with the restriction of v being spacelike unit andorthogonal to e . One can easily show that the remaining freedom is given by the transformation e ′ = ae − b h e, e i v , v ′ = be + av with a, b ∈ R restricted to a + b h e, e i = 1. Nevertheless, the square norm h e ′ , e ′ i is invariantunder this change, so the following definition makes sense: Definition 4.1.
Let G ∈ SkewEnd ( V ) be simple, with G = e ⊗ v ♭ − v ⊗ e ♭ , e, v ∈ V orthogonal with v spacelikeunit. Then G is said to be spacelike, timelike or null if the vector e is spacelike, timelike or null respectively. Inthe non-null case, G is called spacelike (resp. timelike) unit whenever h e, e i = +1 (resp. h e, e i = − ). By Lemma 4.1, it is immediate that Definition 4.1 comprises any possible simple endomorphism (up to amultiplicative factor).We next obtain the necessary and sufficient conditions for a simple endomorphism G to commute with agiven F ∈ SkewEnd ( V ). We first make the simple observation that the composition of a one-form e ♭ and askew-symmetric endomorphism F satisfies (simply apply for sides to any w ∈ V ) e ♭ ◦ F = − F ( e ) ♭ . An immediate consequence is that for any pair of vectors e, v ∈ V and F ∈ SkewEnd ( V ) it holds F ◦ ( e ⊗ v ♭ ) = F ( e ) ⊗ v ♭ , ( e ⊗ v ♭ ) ◦ F = − e ⊗ F ( v ) ♭ (19)The following commutation result will be used later. Lemma 4.2.
Let
F, G ∈ SkewEnd ( V ) with G = e ⊗ v ♭ − v ⊗ e ♭ simple and e, v ∈ V as in Definition 4.1. Then [ F, G ] = 0 if and only if there exist µ ∈ R such that: F ( e ) = h e, e i µv, F ( v ) = − µe. (20) Proof.
The commutator is [
F, G ] = F ◦ G − G ◦ F [ F, G ] = F ◦ G − G ◦ F = F ◦ ( e ⊗ v ♭ − v ⊗ e ♭ ) − ( e ⊗ v ♭ − v ⊗ e ♭ ) ◦ F = F ( e ) ⊗ v ♭ − F ( v ) ⊗ e ♭ + e ⊗ F ( v ) ♭ − v ⊗ F ( e ) ♭ , (21)where we have used (19). The “if” part is obtained by direct calculation inserting (20) in (21). To prove the “onlyif” part, the condition [ F, G ] = 0 requires the two endomorphisms F ( e ) ⊗ v ♭ − v ⊗ F ( e ) ♭ and F ( v ) ⊗ e ♭ − e ⊗ F ( v ) ♭ to be equal. One such endomorphism is either identically zero or simple. This implies that span { F ( e ) , v } andspan { e, F ( v ) } are either both one dimensional or both two-dimensional and equal. In the first case, F ( v ) = − µe and F ( e ) = αv for µ, α ∈ R , which are determined by skew-symmetry to satisfy α = µ h e, e i , so the lemmafollows. The second case is empty, for it is necessary that v = ae + bF ( v ) with a, b ∈ R , which implies h v, v i = h ae + bF ( v ) , v i = b h F ( v ) , v i = 0, against the hypothesis of v being spacelike. Corollary 4.2.1.
Let
G, G ′ ∈ SkewEnd ( V ) be simple, spacelike and linearly independent. Let { e, v } , { e ′ , v ′ } be orthogonal spacelike vectors such that G = e ⊗ v ♭ − v ⊗ e ♭ and G ′ = e ′ ⊗ v ′ ♭ − v ′ ⊗ e ′ ♭ . Then [ G, G ′ ] = 0 if andonly if { e, v, e ′ , v ′ } are mutually orthogonal.Proof. By the previous lemma [
G, G ′ ] = 0 if and only if there exist µ ∈ R such that G ( e ′ ) = h e ′ , v i e − h e ′ , e i v = µv ′ , G ( v ′ ) = h v ′ , v i e − h v ′ , e i v = − µe ′ . (22)If µ = 0, then span { e, v } = span { e ′ , v ′ } and G and G ′ are proportional, agains hypothesis. Thus, µ = 0 and by(22) the set { e, v, e ′ , v ′ } is mutually orthogonal. 16 O + (1 , d − -classes In this section we use the canonical form of Section 3 to characterize skew-symmetric endomorphisms of V underthe adjoint action of the orthochronous Lorentz group O + (1 , d − O (1 , d − F ] O + for a given element F ∈ SkewEnd ( V ).The charaterization of these orbits by a set of independent invariants is known and it can be found in [23] interms of two-forms and other references such as [5]. What we do here is, first, to give an alternative way tocharacterize the orbits [ F ] O + by a convenient set of invariants and second, to show that the canonical formis the same for every element in a given orbit. This makes the canonical form specially useful as a tool forproblems with O + (1 , d −
1) invariance.Although we restrict here to the orthochronous component O + (1 , d −
1) because of its relation with conformaltransformations of the sphere S d − (see Section 6), from this case, the orbits of the full group O (1 , d −
1) areeasy to determine. Recall that the time-reversing component O − (1 , d −
1) is one-to-one with O + (1 , d − − ∈ O − (1 , d −
1) to elements in Λ + ∈ O + (1 , d −
1) by e.g. Λ + := Λ − η ,where η = diag( − , + , · · · , +) is an element in O − (1 , d −
1) with the same matrix form as the metric. ThenΛ + F (Λ + ) − = Λ − ηF η (Λ − ) − = − Λ − F (Λ − ) − , where the last equality follows from skew-symmetry. Hence, the elements F and − F belong to the same orbitunder the action of O (1 , d − O + ] = SkewEnd ( V ) /O + (1 , d −
1) and [ O ] = SkewEnd ( V ) /O (1 , d − O ] = [ O + ] / Z . This fact is of course well-known and can be inferred from general references e.g. [18].A consequence of equation (15) is that the characteristic polynomial of F ∈ SkewEnd ( V ) must have theform P F ( x ) = x d + q X b =1 c b x d − b , (23)where we have introduced q := [ d ]. The coefficients c b can be obtained using the Fadeev-LeVerrier algorithm,summarized by the following matrix determinant [15]: c b = 1(2 b )! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Tr F b − · · · F Tr F b − · · · F b − Tr F b − · · · · · · F b Tr F b − · · · · · · Tr F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Since the traces of odd powers vanish by skew-symmetry, the coefficients c b depend on the entries of F onlythrough the traces of the squared powers of F : I b = 12 Tr ( F b ) , b = 1 , · · · , q. Recall that the adjoint respresentation Ad of a matrix Lie group G is a linear representation of G on its Lie17lgebra automorphisms Aut( g ) given byAd : G −→ Aut( g ) g −→ Ad( g ) := Ad g : g → g X → gXg − . The traces I b are obviously invariant under the adjoint action of O + (1 , d −
1) and so are the coefficients c b .Another invariant that plays an important role in the classification of conjugacy classes is the rank of F . Sincethis is always even, we denote it by rank F = 2 r, and clearly r ≤ q . From now we say rank parameter to refer to r . In the following proposition we show thatthis set of invariants actually identifies the canonical form. Proposition 5.1.
Let F, e F ∈ SkewEnd ( V ) , for V Lorentzian of dimension d . Then the invariants { c b , r } and { e c b , e r } of F and e F respectively are equal if and only if their canonical forms given by Theorem 3.3 are the same.Proof. The “if” part ( ⇐ ) is trivial, because the invariants c b , r are independent on the basis, so they can becalculated in a canonical basis. Hence, same canonical form implies same invariants. For the “only if” part ( ⇒ ),we notice that if the coefficients c b and e c b of P F and P e F are equal, so are their characterisic polynomials, themultiplicities of their zero eigenvalue and the polynomials Q F and Q e F (equation (17)). Since rank F = rank e F ,Corollary 3.3.2 implies that ker F and ker e F must have the same causal character. The canonical form onlydepends on the roots Q F and the causal character of ker F through Definition 3.2. Thus, F and e F must havethe same canonical form.We now characterize the classes [ F ] O + in terms of the same invariants given in Proposition 5.1. As mentionedabove, this result is known [23], but we give here an alternative and very simple proof based on our canonicalform: Theorem 5.1. [23] Let F, e F ∈ SkewEnd ( V ) , for V Lorentzian of dimension d . Then their invariants { c b , r } and { e c b , e r } are the same if and only if F and e F are O + (1 , d − -related.Proof. The if ( ⇐ ) part is immediate, since it is trivial from their definitions that the quantities { c b , r } areLorentz invariant. To prove the “only if” ( ⇒ ), by Proposition 5.1, F and e F have the same canonical form incanonical bases B and e B respectively. By definition (c.f. Theorem 3.3), these bases are unit, future orientedand orthonormal. Thus, the transformation taking B to e B transforms F into e F and both must be O + (1 , d − O + (1 , d − σ, µ i or σ, τ, µ i ) totally define the equivalence class of skew-symmetric endomorphisms up to O + (1 , d − V ), such as the one in [9], where they seekirreducibility of the blocks, so they must give two different forms to cover every case.Next, we discuss some facts about the coefficients of the characteristic polynomial, also stated in [23], wherethe proof is only indicated, and which can now be easily proven using the canonical form.18 emma 5.2. Let F ∈ SkewEnd ( V ) be non-zero and let r = rank F . Then c r > , c r = 0 , c r < if and onlyif ker F is timelike, null or spacelike (or zero) respectively. Moreover, if r < q , c q = c q − = · · · = c r +1 = 0 .Proof. Taking into account that the parities of d and m are equal (Lemma 3.2), q − [ m ] = [ d ] − [ m ] = d − m ,so equation (23) can be rewritten P F ( x ) = x m x d − m + q − [ m / X b =1 c b x d − m − b = x m x d − m + d − m X b =1 c b x d − m − b , (24)where we have explicitly substituted all zero coefficients by extracting the common factor x m , thus the remainingcoefficients c b = 0 for b = 1 , · · · , ( d − m ) /
2. By Corollary 3.3.2, ker F degenerate if and only if 2 r = d − m + 2and m >
2, so the sum in (24) runs up to ( d − m ) / r −
1, which means c r = c r +1 = · · · = c q = 0, as statedin the lemma. Also by Corollary 3.3.2, ker F non-degenerate if and only if 2 r = d − m . In this case, the sum in(24) runs up to ( d − m ) / r , hence c r = 0 and if r < q , the next coefficients vanish c r +1 = c r +2 = · · · = c q = 0.In addition c r is the independent term in the polynomial in parentheses. Let µ , · · · , µ λ be all the non-zeroparameters among the { µ i } of the canonical form of F given in (14). By equation (18), c r can be written for d odd: c r = σµ · · · µ λ . Then, the sign of σ determines the sign of c r and, by Remark 2.3, also the causal character of ker F σ , hence,the causal character of ker F in accordance with the stament of the lemma. For d even, also from (18) we have c r = − τ µ · · · µ λ < τ = 0) , c r = σµ · · · µ λ ( τ = 0) , where the expression for τ = 0 follows because in this case either µ t or µ s (or both) vanish, hence either c r = µ s µ · · · µ λ or c r = − µ t µ · · · µ λ and σ equals µ s in the first situation and − µ t in the second. By Remark2.2, when τ = 0 we have ker F στ = { } and hence ker F is always spacelike or zero and when τ = 0, the causalcharacter of ker F στ (and that of ker F ) is determined by the sign of σ in accordance with the statement of thelemma. Remark 5.1.
A converse version of Lemma 5.2 also holds, in the sense that the number ν of last vanishingcoefficients restricts the allowed rank parameters r . Let ν be defined by ν = 0 if c q = 0 and, otherwise, by thelargest natural number satisfying c q = c q − = · · · c q − ν +1 = 0 . By equation (24) it follows ν = [ m / , and sincethe dimension d and m have the same parity (cf. Lemma 3.2), d − m = 2[ d/ − m /
2] = 2( q − ν ) which inparticular shows that ν determines m uniquely. If m > , by Corollary 3.3.2 the rank parameter admits twopossibilities r = { q − ν, q − ν + 1 } , each of which determined by the causal character of ker F . If m ≤ , alsoby Corollary 3.3.2 the ker F degenerate case cannot occur and r = ( q − ν ) is uniquely determined. In particular,if d = 4 , r is always determined by c , c , because r = 2 happens if and only if ν = 0 and otherwise r = 1 (unless F is identically zero, in which case r = 0 ). SkewEnd ( V ) /O + (1 , d − By Theorem 5.1, the q -tuple ( c , · · · , c q ) corresponding to the coefficients of the characteristic polynomial of askew-symmetric endomorphism, does not suffice to determine a point in the quotient space SkewEnd ( V ) /O + (1 , d − ν of last vanishing coefficients c b , the allowed rank parameters are r ∈ { q − ν, q − ν + 1 } ,and r = q − ν + 1 is only possible provided m > c q = 0 then necessarily r = q ). Onesays that there is a degeneracy for the value of the rank at certain points in the space of coefficients c b . In thesubmanifold { c q = · · · = c q − ν +1 = 0 , c q − ν = 0 } , the possible rank parameters are r ∈ { q − ν, q − ν + 1 } . When aboundary point where the number of last vanishing coefficients increases by exactly one is approached, the rankparameter may remain equal to q − ν or jump to q − ν − c i are continuousfunctions of F , the rank is only lower semicontinuous, e.g. [21]). As we shall see in this section, this behaviourgives rise to special limit points which make the space of parameters defining the canonical form (i.e. the spaceof conjugacy classes) a non-Hausdorf topological space, when endowed with the natural quotient topology. Letus start by locating these limit points using the canonical form. Degeneracies can only occur in dimensions d = 5 or larger because in dimension three the rank is two for any non-trivial F and in dimension four the rankis uniquely determined by the invariants (c.f. Remark 5.1). We thus consider first the case d = 5 and thenextend to all values d ≥
5. In d = 5 the space of parameters A defining the [ F ] O + classes is (see fig. 1 ) A := (cid:8) ( σ, µ ) ∈ R × R + | σ ≥ µ if σ > (cid:9) . Consider a [ F ] O + in the region R + := (cid:8) σ ≥ µ > (cid:9) and let F be a representative of [ F ] O + in a canonical basis B = { e α } α =0 , ··· , , that is F = − σ − − σ − σ σ ⊕ (cid:18) − µµ (cid:19) . (25)Let us define the functions C ± ( x ) := x ± x . Then, the following change of basis to B ′ = { e ′ α } is well defined in R + : e ′ = C + ( µ ) ( C + ( √ σ ) e + C − ( √ σ ) e ) − C − ( µ ) e , e ′ = − e , e ′ = C − ( √ σ ) e + C + ( √ σ ) e e ′ = − C − ( µ ) ( C + ( √ σ ) e + C − ( √ σ ) e ) + C + ( µ ) e , e ′ = − e . (26)By direct calculation, F is written in basis B ′ as F = − µ − − µ − µ µ ⊕ (cid:18) −√ σ √ σ (cid:19) . (27)The basis B ′ is non-canonical because µ < σ . However, if we vary the parameters so that µ → σ unchanged), the matrix (27) becomes canonical (i.e. of the form (14)) in the limit and the class [lim µ → F ] O + is given by l = (0 , σ ). On the other hand, F in canonical form (25) also admits a limit µ →
0, which isalso canonical and whose representative [lim µ → F ] O + is given by l = ( σ, R + . However this sequence has twodifferent limit points. As a consequence, the space of canonical matrices, and therefore the quotient spaceSkewEnd ( V ) /O + (1 , d − R − := { σ < , µ > } . Let F be a representative in canonical formof a point [ F ] O + in this region. Then, F has a timelike eigenplane Π t with eigenvalue p | σ | (c.f. Remark 2.3),a spacelike eigenvector e as well as spacelike eigenplane Π s with eigenvalue µ . Thus V = Π t ⊕ span { e } ⊕ Π s and there exist a (non-canonical) basis B ′ adapted to this decomposition, into which F takes the form F = p | σ | p | σ | ⊕ (cid:18) − µµ (cid:19) . (28)20eeping µ unchanged, expression (28) has a limit σ →
0, which has a spacelike eigenplane Π s of eigenvalue µ and it is identically zero on Π ⊥ . Hence, ker F is timelike and using Definition 3.2, the canonical form of thislimit lim σ → F is given by σ ′ = µ and µ ′ = 0. Thus [lim σ → F ] O + is represented by the point l = ( µ , F also admits a limit σ →
0, whose class [lim σ → F ] O + is obviouslyrepresented by the point l = (0 , µ ).PSfrag replacements R − R R + l l µ σ Figure 1:
Representation of
SkewEnd ( V ) /O + (1 , d − in the region A ⊂ R . The shadowed region is not included. The same reasoning can be carried out to arbitrary odd dimension. First, define the regions R ( d, := (cid:8) σ ≥ µ ≥ · · · ≥ µ p > (cid:9) and R ( d, − := (cid:8) σ < , µ ≥ · · · ≥ µ p > (cid:9) and also the limit regions R ( d, := (cid:8) σ = 0 , µ ≥ · · · ≥ µ p > (cid:9) and R ( d, := (cid:8) σ ≥ µ ≥ · · · ≥ µ p − > µ p = 0 (cid:9) . Consider representatives F + and F − (in canonical form) of points ( σ + , ( µ +1 ) , · · · , ( µ + p ) ) and ( σ − , ( µ − ) , · · · , ( µ − p ) )in the regions R ( d, and R ( d, − respectively. Then F + has a spacelike eigenplane Π + s with eigenvalue µ + p aswell as timelike eigenvector e + and spacelike eigenplane Π + t with eigenvalue √ σ + . Restricting to the sub-space W + = span { e + } ⊕ Π + t ⊕ Π + s we can repeat the procedure followed for the five dimensional case andconclude that [lim µ + p → F + ] has simultaneously limits on the points ( σ + , ( µ +1 ) , · · · , ( µ + p − ) , ∈ R ( d, and(0 , ( µ +1 ) , · · · , ( µ + p ) ) ∈ R ( d, . Analogously F − has a spacelike eigenplane Π − s with eigenvalue µ − p as well asspacelike eigenvector e − and timelike eigenplane Π ′− s with eigenvalue p | σ − | . Restricting to the subspace W − =Π − s ⊕ span { e − }⊕ Π ′− s , the above arguments for the five dimensional case show that [lim σ − → F − ] has simultaneouslimits on the points (( µ − p ) , ( µ − ) , · · · , ( µ − p − ) , ∈ R ( d, and (0 , ( µ − ) , · · · , ( µ − p ) ) ∈ R ( d, . Thus the regions R ( d, and R ( d, − limit simultaneously with R ( d, and R ( d, as µ p and σ tend to zero respectively. Indeed, thesame ideas can be applied again to R ( d, and R ( d, − := (cid:8) σ < , µ ≥ · · · ≥ µ p − > µ p = 0 (cid:9) , so that they alsolimit simultaneously, as µ p − and σ go to zero respectively, with R ( d, := (cid:8) σ = 0 , µ ≥ · · · ≥ µ p − > µ p = 0 (cid:9) and R ( d, := (cid:8) σ > , µ ≥ · · · ≥ µ p − > µ p − = µ p = 0 (cid:9) . In general, the regions R ( d,i ) ± analogously defined,i.e. where i gives the number of vanishing parameters µ p = · · · = µ p − i = 0 and the subindex ± gives the signof σ , have simultaneous limits in R ( d,i )0 and R ( d,i +1)+ , where the subindex stands for vanishing σ .For the even dimensional case (with d ≥ τ = 0 is equivalent tothe odd dimensional case direct sum with a one dimensional zero endomorphism (of a Riemannian line). Hence,21he previous reasoning for odd dimensions also applies for even dimensions and τ = 0. For example, considerin d = 6 dimensions the regions R + = (cid:8) τ = 0 , σ ≥ µ > (cid:9) and R − = (cid:8) τ = 0 , σ < , µ > (cid:9) . Then they bothassume limit in R = (cid:8) τ = 0 , σ = 0 , µ > (cid:9) and R ( d, = (cid:8) τ = 0 , σ > µ = 0 (cid:9) . Notice that if we keep τ = 0no degenerate limits of this kind occur. This can be justified as follows. Let µ t , µ s be defined as in (9). Then,it can be readily checked that det F = − µ t µ s µ , so if σ, τ, µ = 0, then rank F = 6. If we keep τ = 0 (thus both µ s , µ t are different from zero), µ = 0 and make σ →
0, the limit must have always rank F = 6. Hence, it is notpossible that a limit σ → τ = 0, the limit µ → F = 4 and therefore, µ → τ = 0 is also straightforward from the odd dimensional case discussed above, which wenow summarize in the following remark: Remark 5.2.
In the case of d odd, consider the subset of R q given by A ( odd ) := { (cid:0) σ, µ , · · · , µ p (cid:1) ∈ R × (cid:0) R + (cid:1) p | µ ≥ · · · ≥ µ p − and if σ > , σ ≥ µ ≥ · · · ≥ µ p } . Define also the subsets of A ( odd ) given by R ( d,i )+ := n ( σ, µ , · · · , µ p ) ∈ A ( odd ) | σ ≥ µ ≥ · · · ≥ µ p − i > µ p − i +1 = · · · = µ p = 0 o , R ( d,i ) − := n ( σ, µ , · · · , µ p ) ∈ A ( odd ) | σ < , µ ≥ · · · ≥ µ p − i > µ p − i +1 = · · · = µ p = 0 o , R ( d,i )0 := n ( σ, µ , · · · , µ p ) ∈ A ( odd ) | σ = 0 , µ ≥ · · · ≥ µ p − i > µ p − i +1 = · · · = µ p = 0 o . Then in the quotient topology of
SkewEnd ( V ) /O + (1 , d − the sequences of R ( p − i ) ± with limit at R ( d,i )0 alsohave limit at R ( d,i +1)+ .In the case of d even, first define µ s as in (9) and let A ( even ) be the subspace of R q given by: A ( even ) := { (cid:0) σ, τ, µ , · · · , µ p (cid:1) ∈ R × (cid:0) R + (cid:1) p | µ ≥ · · · ≥ µ p and if τ = 0 or σ > , µ s ≥ µ ≥ · · · ≥ µ p } . Define also the following subsets of A ( even ) R ( d,i )+ := n ( σ, µ , · · · , µ p ) ∈ A ( even ) | τ = 0 , σ ≥ µ ≥ · · · ≥ µ p − i > µ p − i +1 = · · · = µ p = 0 o , R ( d,i ) − := n ( σ, µ , · · · , µ p ) ∈ A ( even ) | τ = 0 , σ < , µ ≥ · · · ≥ µ p − i > µ p − i +1 = · · · = µ p = 0 o , R ( d,i )0 := n ( σ, µ , · · · , µ p ) ∈ A ( even ) | τ = 0 , σ = 0 , µ ≥ · · · ≥ µ p − i > µ p − i +1 = · · · = µ p = 0 o . Then in the quotient topology of
SkewEnd ( V ) /O + (1 , d − the sequences of R ( d,i ) ± with limit at R ( d,i )0 also havelimit at R ( d,i +1)+ . One interesting applications of our previous results is based on the relation between skew-symmetric endomor-phisms and the set of conformal Killing vector fields (CKVFs) of the n -sphere, CKill( S n ). These vector fieldsare the generators of the conformal transformations of the n -sphere Conf ( S n ), i.e. the group of transformations22 Λ that scale the spherical metric γ , ψ ⋆ Λ ( γ ) = Ω γ , where Ω is a smooth positive function of S n . A standardtechnique to describe these transformations consists in viewing S n as the (real) projectivization of the null futurecone in M ,n +1 , in such a way that Conf ( S n ) is induced from the isometries of M ,n +1 . This is discussed indetail for the four dimensional case in [26] and in arbitrary dimensions in [23] and in [28] (the latter considersarbitrary signature and the projectivization of the null ”cone” in M p +1 ,q +1 , giving S p × S q ). This procedurestablishes a group homomorphism ψ : O (1 , n + 1) → Conf( S n ) , Λ ψ Λ , which is one-to-one when restrictedto the orthochronous component O + (1 , n + 1) ⊂ O (1 , n + 1).The Euclidean space E n = ( R n , g E ) and S n are well-known to be conformally related via the stereographicprojection St N : S n \{ N } → E n , where N denotes the point w.r.t. which the projection is taken. Observe thatthe stereographic projection depends not only on the point N but also on the (signed) distance between N and the plane onto which the projection is performed. We do not reflect this dependence in the notation forsimplicity.Hence, the composition of a transformation ψ Λ ∈ Conf ( S n ) with the stereographic projection yields St N ◦ ψ Λ ◦ St − N =: φ Λ ∈ Conf ( E n ), which is a conformal transformation of E n . Strictly speaking, these transformations arenot diffeomorphisms of E n , as they require to remove the two points p , p ∈ E n satisfying ψ Λ ◦ St − N ( p ) = N and ψ − ◦ St − N ( p ) = N , which are the “preimage” and the “image” of infinity under φ Λ respectively. Nevertheless,since ψ : O (1 , n + 1) → Conf( S n ) is a group homomorphism, so is φ : O (1 , n + 1) → Conf( E n ) , Λ φ Λ aswell as the map which assigns ψ Λ φ Λ . In that sense Conf ( S n ) and Conf ( E n ) are the same. These grouphomomorphisms, induce Lie algebra homomorphisms between SkewEnd (cid:0) M ,n +1 (cid:1) , CKill( S n ) and CKill( E n ) (thevector fields generating Conf ( E n )). The precise form of these maps depends, firstly, on the representative usedto describe the projective cone (i.e. S n ) and secondly on the point N as well as on the signed distance fromthis point to the plane. In [23], the morphism ξ := φ ⋆ : SkewEnd (cid:0) M ,n +1 (cid:1) −→ CKill( E n ) ,F ξ ( F ) =: ξ F , (29)is constructed related to each other using the representative with { x = 1 } ∩ { x α x α = 0 } for the projectivecone, where { x α } ( α, β = 0 , · · · , n + 1) are Minkowskian coordinates of M ,n +1 , N is the point with coordinates { x = − x = 1 , x A +1 = 0 } ( A, B = 1 , · · · , n ) and the image plane for the stereographic projection is { x = x = 1 } . The result is a representation of CKill( E n ) where the vector vector fields are expressed in Cartesiancoordinates { y A } induced from the Minkowskian coordinates by means of { x = x = 1 , x A +1 = y A } . Theorem 6.1. [[23]] Let M ,n +1 endowed with Minkowskian coordinates { x α } and consider any element F ∈ SkewEnd (cid:0) M ,n +1 (cid:1) written in the basis { ∂ x α } in the form F = − ν − a t + b t / − ν − a t − b t / − a + b/ a + b/ − ωωω , (30) where a, b ∈ R n are column vectors, t stands for the transpose and ωωω is a skew-symmetric n × n matrix ( ωωω = − ωωω t ).Then, in the Cartesian coordinates { y A } of E n defined by the embedding i : E n ֒ → M ,n +1 , i ( E n ) = { x = x =1 , x A +1 = y A } , the CKVFs of E n are ξ F = (cid:18) b A + νy A + ( a B y B ) y A −
12 ( y B y B ) a A − ω AB y B (cid:19) ∂ y A . (31) The method in [23] is based on the unit spacelike hyperboloid in Minkowski instead of on the null cone. However, the twomethods are easily seen to be equivalent to the one we describe oreover, ξ Ad Λ ( F ) = φ Λ ⋆ ( ξ F ) for every Λ ∈ O + (1 , n + 1) and ξ is a Lie algebra antihomomorphism, i.e. [ ξ F , ξ G ] = − ξ [ F,G ] . Remark 6.1.
For later use, we write explicitly the parameters of the vector field ν, a A , b A , ω AB in terms of theentries F αβ of the endomorphism F : ν = − F , a A = − (cid:0) F A +1 + F A +1 (cid:1) ,b A = 12 (cid:0) F A +1 − F A +1 (cid:1) , ω AB = − F A +1 B +1 . (32) where capital Latin indices are lowered with the Kronnecker δ AB . Unless otherwise stated, ξ without subindexrefers to the map ξ given in (29) while ξ F refers to the CKVF which is image under ξ of the skew-symmetricendomorphism F . The freedom of choosing a representative for S n (as well as the point N and the projection stereographicplane) can be also seen in a more “passive” picture. Consider two different sets of Minkowskian coordinates { x α } and { x ′ α } related by a O + (1 , n + 1) transformation Λ, x ′ α = Λ αβ x β . Using Theorem 6.1, we obtain twodifferent embeddings i, i ′ : E n ֒ → M ,n +1 associated to { x α } and { x ′ α } respectively, for which i ( E n ) = { x = x = 1 , x A +1 =: y A } and i ′ ( E n ) = { x ′ = x ′ = 1 , x ′ A +1 =: y ′ A } , as well as two associated maps ξ, ξ ′ . Let F ∈ SkewEnd (cid:0) M ,n +1 (cid:1) , defined by (30) with parameters { ν, a A , b A , ω AB } and { ν ′ , a ′ A , b ′ A , ω ′ AB } in the bases { ∂ x α } and { ∂ x ′ α } respectively. Then, F can be associated to two vector fields ξ F = (cid:0) b A + νy A + ( a B y B ) y A −
12 ( y B y B ) a A − ω AB y B (cid:1) ∂ y A ,ξ ′ F = (cid:0) b ′ A + ν ′ y ′ A + ( a ′ B y ′ B ) y ′ A −
12 ( y ′ B y ′ B ) a ′ A − ω ′ AB y ′ B (cid:1) ∂ y ′ A , which are equal in the following sense. If we transform the representative S ′ n = { x ′ = 1 } ∩ { x ′ α x ′ α = 0 } with Λ, we obtain a new representative of the projective cone which in coordinates x α is precisely S n = { x =1 } ∩ { x α x α = 0 } . Abusing the notation, the map χ Λ := St N ◦ Λ ◦ St − N ′ is such that χ Λ ⋆ ( ξ ′ F ) = ξ F . Then,considering i ( E n ) and i ′ ( E n ) as respresentations of the same space in two different global charts ( y A , R n ) and( y ′ A , R n ), χ Λ can be seen as a change of coordinates y A = ( χ Λ ( y ′ )) A , with the property that the Euclideanmetric in coordinates { y ′ A } transforms as g E = δ AB d y ′ A d y ′ B = Ω ( y ) δ AB d y A d y B for a smooth positive function Ω. In other words, changing to different Minkowskian coordinates in M ,n +1 induces a change of coordinates in E n in such a way that the form (31) of the map ξ is preserved. Notice that asimilar result holds if we change the point w.r.t. which we take the stereographic projection, because any two N, N ′ ∈ S n must be related by a SO ( n ) ⊂ O + (1 , n + 1) transformation.Therefore, for the rest of this section, we will often adapt our choice of Minkowskian coordinates { x α } of M ,n +1 to simplify the problem at hand. With this choice, it comes a corresponding set of cartesian coordinates (cid:8) y A (cid:9) of E n such that ξ F is given by equation (31) and the Euclidean metric is g E = Ω( y ) δ AB d y A d y B . Whichcoordinates are adequate obviously depends on the problem. For example, from the block form (10) and (11)of skew-symmetric endomorphisms, consider each of the blocks F | M , F | M , as endomorphisms of M ,n +1 ,extended as the zero map in ( M , ) ⊥ and ( M , ) ⊥ respectively, and similarly for each F | Π i . If we denote by ξ F | M , , ξ F | M , and ξ F | Π i the corresponding images by ξ , one readily gets following decomposition: ξ F = ξ F | M , + p X i =1 ξ F | Π i ( n even) , ξ F = ξ F | M , + p X i =1 ξ F | Π i ( n odd) , (33)24here in terms of n , p is given by p = (cid:20) n + 12 (cid:21) − F is defined is d = n + 2, cf. Theorem 6.1). The explicitform of each of the terms in (33) is direct from (32). Namely, the terms ξ F | M , and ξ F | M , are given by (31)with vanishing parameters a A , b A , ω AB for A, B ≥ A, B ≥ ξ F | Π i is proportionalto a vector field of the form η := y A ∂ y B − y B ∂ y A (35)with A , B ∈ { , · · · , n } such that A = B . More specifically, ξ F | Π i = µ i η i , where η i is given by equation (35)with B = A + 1 and A = 2 i if n even while A = 2 i + 1 if n odd. Vector fields of the form (35) will play animportant role in the following analysis. They have the form of axial Killing vector fields, although in generalthey are CKVFs because of the conformal factor in g E = Ω( y ) δ AB d y A d y B . From the previous discussion, itfollows that there exists a conformal transformation χ Λ ∈ Conf ( E n ) such that g ′ E := χ ⋆ Λ ( g E ) = δ AB d y A d y B .Then by the properties of the Lie derivative it is immediate0 = L η χ ⋆ Λ ( g E ) = L χ Λ ⋆ η g E . In other words, η is an axial Killing vector of g ′ E and χ Λ ⋆ η is an axial Killing vector of g E . Thus, we define: Definition 6.1.
A CKVF of an Euclidean metric g E , η , is said to be a conformally axial Killing vector field(CAKVF) if and only if the exist a χ Λ ∈ Conf ( E n ) such that χ Λ ⋆ ( η ) is an axial Killing vector field of g E .Equivalently, η is a CAKVF if and only if it is an axial Killing vector field of χ ⋆ Λ ( g E ) . Remark 6.2.
Using Theorem 6.1, it is immediate to verify that a CKVF is a CAKVF if and only if it is theimage under ξ of a simple unit spacelike endomorphism G . Notice that the terms in (33) form a commutative subset of CKill ( E n ). This is an immediate consequence ofthe fact that ξ is a Lie algebra antihomomorphism (c.f. Theorem 6.1) and the blocks F | M , (resp. F | M , ) and F | Π i are pairwise commuting. In addition, a straightforward calculation shows that they form an orthogonalset g E ( e ξ, η i ) = 0 , g E ( η i , η j ) = 0 ( i = j )where e ξ := ξ F | M , for n even and e ξ := ξ F | M , for n odd. In fact, as we show next, orthogonality of two CKVFsimplies commutativity provided one of them is a CAKVF. If both are CAKVF, then orthogonality turns out tobe equivalent to commutativity. Lemma 6.2.
Let η, η ′ be non-proportional CAKVFs and ξ F a CKVF. Then [ η, η ′ ] = 0 if and only if there existcartesian coordinates such that η = y n − ∂ y n − − y n − ∂ y n − and η ′ = y n − ∂ y n − y n ∂ y n − . Equivalently [ η, η ′ ] = 0 if and only if g E ( η, η ′ ) = 0 . In addition, [ ξ F , η ] = 0 if g E ( ξ F , η ) = 0 .Proof. Let
G, G ′ ∈ SkewEnd (cid:0) M ,n +1 (cid:1) be such that ξ ( G ) = η, ξ ( G ′ ) = η ′ . Since G and G ′ are simple, spacelikeand unit (cf. Remark 6.2), we can write G = e ⊗ v ♭ − v ⊗ e ♭ and G ′ = e ′ ⊗ v ′ ♭ − v ′ ⊗ e ′ ♭ for spacelike, unitvectors { e, e ′ , v, v ′ } , such that 0 = h e, v i = h e ′ , v ′ i . By Corollary 4.2.1, it follows that [ G, G ′ ] = 0 if and onlyif { e, e ′ , v, v ′ } are mutually orthogonal. Let us take cartesian coordinates of M ,n +1 such that e = ∂ x n − , v = ∂ x n − , e ′ = ∂ x n , v ′ = ∂ x n +1 . Then, in the associated coordinates (cid:8) y A (cid:9) of E n it follows η = y n − ∂ y n − − y n − ∂ y n − η ′ = y n − ∂ y n − y n ∂ y n − . This proves the first part of the lemma. From this result, it is trivial that [ η, η ′ ] = 0implies g E ( η, η ′ ) = 0.To prove that g E ( η, ξ F ) = 0 implies [ η, ξ F ] = 0 (which in particular establishes the converse g E ( η, η ′ ) =0 = ⇒ [ η, η ′ ] = 0 for CAKVFs), let us take coordinates (cid:8) y A (cid:9) such that η = y n − ∂ y n − y n ∂ y n − . Then, writing ξ F as a general CKVF (31), we obtain by direct calculation: g E ( η, ξ F ) = Ω (cid:18) y n b n − − y n − b n − y B y B a n y n − − a n − y n ) + ω n − B y B y n − ω nB y B y n − (cid:19) = 0 . Therefore a n , a n − , b n , b n − , ω nB , ω n − B must vanish. This implies that the associated endomorphisms G and F to η and ξ F adopt a block structure from which it easily follows that [ G, F ] = 0 and hence [ η, ξ F ] = 0. Definition 6.2.
Let ξ F ∈ CKill ( E n ) . Then a decomposed form of ξ F is ξ F = e ξ + P pi =1 µ i η i for an orthogonalsubset { e ξ, η i } , where η i are CAKVFs, µ i ∈ R for i = 1 , · · · , p . A set of cartesian coordinates (cid:8) y A (cid:9) such that η i = y A i ∂ y Ai +1 − y A i +1 ∂ y Ai , for A i = 2 i for n odd and A i = 2 i + 1 for n even, is called a set of decomposed coordinates. Remark 6.3.
Observe that the e ξ is a CKVF. By Lemma 6.2 and its proof, the parameters { ν, a, b, ω } defining e ξ in a set of decomposed coordinates must all vanish except possibly { ν, a , a , b , b , ω = − ω } when n iseven or { ν, a , b } when n is odd. This means that there is a skew-symmetric endomorphism e F with restrictsto M , ⊂ M ,n ( n even) or M , ⊂ M ,n ( n odd) and vanishes identically on their respective orthogonalcomplements such that e ξ = ξ e F . We will exploit this fact in an essential way below. With the definition of decomposed form of CKVFs, we can reformulate Theorem 2.6 in terms of CKVFs.
Proposition 6.1.
Let ξ F ∈ CKill ( E n ) . Then there exist an orthogonal set { η i } pi =1 of CAKVFs such that [ ξ F , η i ] = 0 . For every such a set { η j } pj =1 and i ∈ { , · · · , p } there exist µ i ∈ R such that g E ( η i , η i ) µ i = g E ( ξ F , η i ) . In addition,with the definition e ξ := ξ F − P µ i η i the expression ξ F = e ξ + P µ i η i provides a decomposedform of ξ F .Proof. The existence of p commuting CAKVFs is a direct consequence of decompositions (10) and (11) of theassociated skew-symmetric endomorphism F , for n even and odd respectively. Indeed, for each such decom-position of F , it follows a set of p CAKVFs commuting with ξ F . Let us denote { η i } any such set. Each η i isassociated to a simple, spacelike unit endomorphism G i that commutes with F . By Lemma 4.2, G i defines aspacelike eigenplane Π i of F . The orthogonality of any two such eigenplanes Π i , Π j , i = j is a consequence ofCorollary 4.2.1 because [ G i , G j ] = 0. In other words, given a set of p CAKVFs commuting with ξ F , we have ablock form of F , thus, defining e ξ := ξ F − P µ i η i , it is immediate that ξ F = e ξ + P µ i η i is a decomposed formwith g E ( η i , η i ) µ i = g E ( ξ F , η i ).The next step now is to give a definition of canonical form for CKVFs, which we induce from the canonicalform of the associated skew-symmetric endomorphism. Definition 6.3.
A CKVF ξ F is in canonical form if it is the image of a skew-symmetric endomorphism F in canonical form, i.e. ξ F = e ξ + P µ i η i such that e ξ is given, in a cartesian set of coordinates { y A } denoted canonical coordinates , by the parameters a = 1 , b = σ/ , a = 0 , b = τ / if n even and a = 1 , b = σ/ if n odd (the non-specified parameters all vanish) and η i are CAKVFs η i = y A i ∂ y Ai +1 − y A i +1 ∂ y Ai , for A i = 2 i for n odd and A i = 2 i + 1 for n even, and where σ, τ, µ i are given by Definition 3.2. ξ F , the existence of a canonical form and canonical coordinates are guaranteed by Theorem3.3. By Theorem 6.1, the conformal class [ ξ F ] Conf of a CKVF ξ F is equivalent to the equivalence class [ F ] O + of F under the adjoint action of O + (1 , n + 1), and this is determined by the canonical form of F (c.f. Theorem5.1). This argument together with the results of Section 5 yield the following statement. Theorem 6.3.
Let ξ F ∈ CKill ( S n ) be in canonical form. Then its conformal class [ ξ F ] Conf is determined by ( σ, τ, µ i ) if n even and ( σ, µ i ) if n odd. Moreover, the structure of CKill( S n ) / Conf ( S n ) corresponds with thatof Remark 5.2. Given a canonical form ξ F = e ξ + P µ i η i the set of vectors { e ξ, η i } are pairwise commuting and linearlyindependent. As we will next prove, in the case of odd dimension this set is a maximal (linearly indepen-dent) pairwise commuting set of CKVFs commuting with ξ (i.e. it is not contained in a larger set of lin-early independent vectors commuting with ξ ). In the case of even dimension it is not maximal. By Remark6.3, e ξ = e ξ ( ν, a , a , b , b , ω ), where the right-hand side denotes a CKVF of the form (35) whose parametersvanish, except possibly { ν, a , a , b , b , ω := ω } . As mentioned in the Remar, the corresponding skew-symmetric endomorphism e F satisfying ξ e F = e ξ can be understood as an element e F ∈ SkewEnd (cid:0) M , (cid:1) , with M , = span { e , e , e , e } , that is identically zero in (cid:0) M , (cid:1) ⊥ . Fix the orientation in M , so that the basis { e , e , e , e } is positively oriented. The Hodge star maps two-forms into two-forms. This defines a naturalmap ⋆ : SkewEnd (cid:0) M , (cid:1) −→ SkewEnd (cid:0) M , (cid:1) , e F e F ⋆ . From standard properties of two-forms, (see also [24]) it follows that e F ⋆ commutes with e F . We may extend e F ⋆ to an endomorphism on M ,n +1 that vanishes identically on ( M , ) ⊥ , just as e F . It is clear that the commutationproperty is preserved by this extension. The image of e F ⋆ under ξ is the vector field e ξ ⋆ := (cid:16)e ξ ( ν, a , a , b , b , ω ) (cid:17) ⋆ = e ξ ( − ω, a , − a , − b , b , ν ) , which by construction commutes with e ξ . In the case that e ξ is the first element in a decomposed form ξ F = e ξ + P µ i η i , it is immediately true that e ξ ⋆ also commutes with all of the CAKVFs η i . Hence, { e ξ, e ξ ⋆ , η i } is apairwise commuting set, all of them commuting with ξ . This set can be proven to be maximal: Proposition 6.2.
Let ξ F = e ξ + P µ i η i be a CKVF in canonical form. If n is odd, { e ξ, η i } is a maximal linearlyindependent pairwise commuting set of elements that commute with ξ F . If n is even, { e ξ, e ξ ⋆ , η i } is a maximallinearly independent pairwise commuting set of elements that commute with ξ F .Proof. Suppose that there is an additional CKVF ξ ′ commuting with each element in { e ξ, η i } if n odd or { e ξ, e ξ ⋆ , η i } if n even (in either case ξ ′ clearly commutes with ξ F also). Since it commutes with each η i , by Proposition6.1, it admits a decomposed form ξ ′ = e ξ ′ + P pi =1 µ ′ i η i , where e ξ ′ is a CKVF orthogonal to each η i and whichmust verify [ e ξ ′ , e ξ ] = 0. Equivalently, their associated endomorphisms satisfy e F ′ ∈ C ( e F ), where C ( e F ) denotes thecentralizer of F , i.e. the set of all skew-symmetric endomorphisms that commute with F . From the results in[24], C ( e F | M , ) = span { e F | M , } when n is odd and C ( e F | M , ) = span { e F | M , , e F ⋆ | M , } when n is even. Here, e F ⋆ is the skew-symmetric endomorphim associated with e ξ ⋆ and we restrict to M , because the action of theendomorphisms is identically zero in ( M , ) ⊥ . Thus e ξ ′ = a e ξ, a ∈ R , if n odd and e ξ ′ = b e ξ + c e ξ ⋆ , b, c ∈ R if n even. 27 Adapted coordinates
In the previous section we obtained a canonical form for each CKVF of euclidean space based on the canonicalform of skew-symmetric endomorphisms in Section 3. As an application, we consider in this section the problemof adapting coordinates in E n to a given CKVF ξ F . The use of the canonical form will allow us to solvethe problem for every possible ξ F essentially in one go. Actually it will suffice to consider the case of evendimension n and assume that at least one of the parameters σ, τ in the canonical form of ξ F is non-zero. Thecase where both σ and τ vanish will be obtained as a limit (and we will check that this limit does solve therequired equations). The case of odd dimension n wil be obtained from the even dimensional one by explitingthe property that E m +1 can be viewed as a hyperplane of E m +2 in such a way that the given CKVF ξ F in E m +1 extends conveniently to E m +2 . Restricting the adapted coordinates already obtained in the evendimensional case to the appropriate hyperplane we will be able to infer the odd dimensional case. As we willjustify the process of adapting coordinates is different for n = 2 and n ≥ n = 2 has beentreated in detail in [24], so it will suffice to consider even n ≥ E n endowed with a CKVF ξ F . First of all we adapt the Cartesian coordinates of E n so that ξ F takes its canonical form and we fix the metric of E n to take the explictly flat form in these coordinates. Wefurther assume (for the moment) that n is even. For notational reasons it is convenient to rename the canonicalcoordinates as z := y , z := y and x i := y i +1 , y i := y i +2 for i = 1 , · · · , p , where in the even casecase p = n/ − ξ F can be decomposed as a sum of CKVFs e ξ and η i and,additionally one can construct canonically yet another CKVF e ξ ⋆ . This collection of CKVFs defines a maximalcommutative set. Moreover, { η i } are all mutually orthogonal and perpendicular to e ξ and e ξ ⋆ . It is thereforemost natural to try and find coordinates adapted simultaneously to the whole family { e ξ, e ξ ⋆ , η i } . This will leada (collection of) coordinate systems where the components of ξ F are simply constants. From here one canimmediately find coordinates that rectify ξ F , if necessary. It is important to emphasize that selecting the wholeset { e ξ, e ξ ⋆ , η i } to adapt coordinates provides enough restrictions so that the coordinate change(s) can be fullydetermined. Imposing the (much weaker) condition that the system of coordinates rectifies only ξ F is just atoo poor condition to solve the problem. This is an interesting example where the structure of the canonicaldecomposition of ξ F (or of F ) is exploited in full.By Theorem 6.1, the explicit form of { e ξ, e ξ ⋆ , η i } in the canonical coordinates is e ξ = σ z − z − p X i =1 ( x i + y i ) !! ∂ z + (cid:16) τ z z (cid:17) ∂ z + z p X i =1 ( x i ∂ x i + y i ∂ y i ) (36) e ξ ⋆ = − (cid:16) τ z z (cid:17) ∂ z + σ − z − z − p X i =1 ( x i + y i ) !! ∂ z − z p X i =1 ( x i ∂ x i + y i ∂ y i ) η i = x i ∂ y i − y i ∂ x i . We are seeking coordinates { t , t , φ i , v i } adapted to these vector fields, i.e. such that ∂ t = e ξ, ∂ t = e ξ ⋆ , ∂ φ i = η i .It is clear that if { t , t , φ i , v i } is an adapted coordinate system, so it is { t − t , ( v ) , t − t , ( v ) , φ i − φ ,i ( v ) , v i } for arbitrary functions t , ( v ), t , ( v ) and φ ,i ( v ), where v = ( v , · · · , v p ). This will be used to simplify the The fact that we tag the coordinates { z , z , x i , y i } with lower indices has no particular meaning. It is simply to avoid anotational clash of upper indices and powers that will appear later ∂z ∂t = σ z − z − p X i =1 ( x i + y i ) ! , ∂z ∂t = τ z z , ∂x i ∂t = z x i , ∂y i ∂t = z y i , (37) ∂z ∂t = σ − z − z − p X i =1 ( x i + y i ) ! , ∂z ∂t = − τ − z z , ∂x i ∂t = − z x i , ∂y i ∂t = − z y i , (38) ∂z ∂φ i = 0 ∂z ∂φ i = 0 ∂x i ∂φ i = − y i ∂y i ∂φ i = x i (39)The additional p coordinates v i , will appear through functions of integration. It is clear that the structure ofthe equations is different for n = 2, where there are no { x i , y i } , which implies that the process of integrationfollows a different route. The case n = 2 has been treated in full detail in [24], where the the complex structureof S can be exploited to simplify the problem. Here we adress the problem for n ≥ z = z ( t , t , v ) , z = z ( t , t , v ), so that the secondpair becomes a harmonic oscillator in x i , y i , whose solution is x i = ρ i ( t , t , v ) cos( φ i − φ ,i ( t , t , v )) , y i = ρ i ( t , t , v ) sin( φ i − φ , ,i ( t , t , v )) , (40)where ρ i and φ ,i are arbitrary functions (depending only on the variables indicated) and ρ i is not identicallyzero.Inserting (40) in any of the two right-most equations of (37) and (38) and equating terms multiplyingsin( φ i + φ (0) i ) and cos( φ i + φ (0) i ) yields: z = 1 ρ i ∂ρ i ∂t , z = − ρ i ∂ρ i ∂t , ∂φ (0) i ∂t = 0 , ∂φ (0) i ∂t = 0 . Thus, φ ,i is a function only of v , which may be absorbed on the coordinate φ i as discussed above. The twofirst equations imply 1 ρ i ∂ρ i ∂t = 1 ρ j ∂ρ j ∂t , ρ i ∂ρ i ∂t = 1 ρ j ∂ρ j ∂t ⇐⇒ ρ i = ˆ α i ( v )ˆ ρ ( t , t , v ) , for arbitrary (non-zero) functions ˆ α i and ˆ ρ . Defining ρ := p P i =1 ρ i = (cid:18) p P i =1 ˆ α i (cid:19) ˆ ρ we can write ρ i = ˆ α i ˆ ρ = ˆ α i ǫ qP pj =1 ˆ α j ρ = α i ρ, where α i := ˆ α i ǫ/ qP pj =1 ˆ α j , with ǫ = 1, form a set of arbitrary (non-zero) functions of v such that p P i =1 α i = 1.The function ρ satisfies z = 1 ρ ∂ρ∂t , z = − ρ ∂ρ∂t . (41)Inserting (41) in the two left-most equations in (37) and (38), with the change of variable U = ρ − , we obtainafter some algebra the following covariant system of PDEs (indices a, b = 1 , { t , t } ) ∇ a ∇ b U = U A ab + 12 U (1 + ∇ c U ∇ c U ) g ab with A = 12 ( − σ d t + σ d t + 2 τ d t d t ) , g = d t + d t , (42)29nd where ∇ is the Levi-Civita covariant derivative of g . Lemma 7.1.
Up to shifts t → t − t , ( v ) and t → t − t , ( v ) , the general solution of (42) with either σ or τ non-zero is given by U = ǫµ t + µ s ( β cosh( t + ) − α cos( t − )) with β = q α + µ t + µ s (43) where α is a function of integration (depending on v ), ǫ = 1 and t + := µ t t + µ s t , t − := µ t t − µ s t , with µ s , µ t given by (9) . The solution (43) admits a limit σ = τ = 0 (i.e. µ t = µ s = 0 ) provided α > , which is lim µ s µ t → U = ǫ α t + t ) + ǫ α . (44) Up to shifts t → t − t , ( v ) and t → t − t , ( v ) , this function is the general solution of (42) for σ = τ = 0 .Proof. The coordinates t + , t − defined in the lemma diagonalize A and g simultaneously and yield A = 12 (d t − d t − ) , g = 1 µ s + µ t (d t + d t − ) . From this and equation (42) it follows that ∂ U/∂t + ∂t − = 0 or, equivalently, U ( t + , t − ) = U + ( t + ) + U − ( t − ).Substracting the { t + , t + } and { t − , t − } components of (42) one obtainsd U + d t − d U − d t − = U = U + + U − = ⇒ d U + d t − U + = d U − d t − + U − = ˆ a for an arbitrary separation function ˆ a ( v ). The general solution is clearly U + = − ˆ a + a cosh( t + ) + b sinh( t − ) U − = ˆ a + c cos( t − − δ ) , (45)where a, b, c, δ are also functions of v . Since ˆ a drops out in U = U + + U − we may set ˆ a = 0 w.l.o.g. Inserting(45) in (any of) the diagonal terms of (42) and one simply gets a − b = 1 µ s + µ t + c . Hence | a | > | b | and we may use the freedom of translating t + by a function of v to write U + = a cosh( t + )(i.e. b = 0). A similar translation in t − sets δ = 0. Rescaling the functions a, c as a = ( µ s + µ t ) − β and c = − ( µ s + µ t ) − α we get U = U − + U − = βµ s + µ t cosh( t + ) − αµ s + µ t cos( t − ) , β = µ s + µ t + α . (46)It is obvious that sign( U ) = sign( β ). Thus taking β as the positive root β = p α + µ s + µ s and addinga multiplicative sign ǫ in (46), we obtain (43). To evaluate the convergence as both σ, τ tend to zero, orequivalently µ s , µ t →
0, consider the series expansion β cosh( t + ) = (cid:18) | α | + µ s + µ t | α | + o (4) µ t ,µ s (cid:19) (cid:18) µ s t + µ t t ) o (4) µ t ,µ s (cid:19) ,α cos( t − ) = α − α ( µ t t − µ s t ) o (4) µ t ,µ s , o (4) µ t ,µ s denotes a sum of homogeneous polynomials in µ t , µ s starting at order four, whose coeficients maydepend on t , t and α . Then, the expansion of U is U = ǫµ s + µ t (cid:18) ( | α | − α )(1 + µ s µ t t t ) + | α | µ s + αµ t t + | α | µ t + αµ s t + µ s + µ t | α | + o (4) µ t ,µ s (cid:19) . It is clear that lim µ s ,µ t → o (4) µ t ,µ s / ( µ s + µ t ) = 0 and the rest of the equation converges if and only if α > t , t ) is the generalsolution of (42) when σ, τ = 0.Having the general general solution (43) of (42) we can give the expression of the adapted coordinates z = − U ∂U∂t = (cid:12)(cid:12)(cid:12)(cid:12) U (cid:12)(cid:12)(cid:12)(cid:12) αµ s sin( t − ) − βµ t sinh( t + ) µ s + µ t , (47) z = 1 U ∂U∂t = (cid:12)(cid:12)(cid:12)(cid:12) U (cid:12)(cid:12)(cid:12)(cid:12) αµ t sin( t − ) + βµ s sinh( t + ) µ s + µ t , (48) x i = α i U cos( φ i ) , y i = α i U sin( φ i ) , (49)where no sign of α is in principle assumed , except for the case µ s = µ t = 0, where U must be understood asthe limit (with α >
0) (44) and z = − U − ∂U/∂t , z = U − ∂U/∂t . This coincides with the limit of the RHSexpressions (47), (48), which is z = − α t α ( t + t ) , z = 2 α t α ( t + t ) . (50)From equations (47), (48) and (49) it is obvious that the sign ǫ is not relevant in the definition of the adaptedcoordinates. This is because the two branches ǫ = 1 and ǫ = − U >
U < π in the φ i angles. Hence, w.l.o.g. weconsider ǫ = 1, i.e. U >
0. Also notice that the dependence on the variables v i appears through the functions α i and α , with P pi =1 α i = 1. The set { α i , α } define p independent arbitrary functions of the variables v i , so itis natural to use as coordinates { α i , α } themselves, provided they are restricted to satisfy P pi =1 α i = 1.We now calculate the region of E n covered by the adapted coordinates. It is clear that in no case this regioncan include neither the zeros of the vector fields e ξ and e ξ ⋆ and η i nor the points where these p + 2 vectors arelinearly dependent. We therefore start by locating those points. Denoting the loci of the zeros of e ξ and e ξ ⋆ and η i by Z ( e ξ ), Z ( e ξ ) ⋆ and Z ( η i ) respectively, a simple calculation gives Z ( e ξ ) = (cid:16) { p \ j =1 { x j = y j = 0 }} ∩ { z = ± µ t , z = ∓ µ s } (cid:17) ∪ (cid:16) { z = 0 } ∩ { z + p X j =1 ( x j + y j ) = µ s − µ t } if µ s µ t = 0 (cid:17) , (51) Z ( e ξ ⋆ ) = (cid:16) { p \ j =1 { x j = y j = 0 }} ∩ { z = ± µ t , z = ∓ µ s } (cid:17) ∪ (cid:16) { z = 0 } ∩ { z + p X j =1 ( x j + y j ) = µ t − µ s } if µ s µ t = 0 (cid:17) , (52) Z ( η i ) = { x i = y i = 0 } . The domain of definition of α will be later restricted under the condition that the adapted coordinates define a one to one map. µ s , µ t and imply that in the case µ s = µ t = 0, Z ( e ξ ) = Z ( e ξ ⋆ ) = (cid:8) T pj =1 { x j = y j = 0 } (cid:9) ∩ { z = z = 0 } , which is contained in each Z ( η i ) = { x i = y i = 0 } .On the other hand, since { e ξ, η i } is an orthogonal set of CKVFs (cf. Lemma 6.2), they are pointwise linearlyindependent at all points where they do not vanish. Similarly, { e ξ ⋆ , η i } is also an orthogonal set, so linearindependence is guaranteed away from the zero set. Away from this set, the set of vectors { e ξ, e ξ ⋆ , η i } is linearlydependent only at points where e ξ and e ξ ⋆ are proportional to each other with a non-zero proportionality factor, e ξ = a e ξ ⋆ , a = 0. One easily checks that, away from Z ( e ξ ) and Z ( e ξ ⋆ ), the set of point where e ξ − a e ξ ⋆ vanishes isempty except when µ s = 0 , µ t = 0 and a = µ t µ s . It turns out to be useful to determine the set of points where µ s e ξ − µ t e ξ ⋆ = 0 when at least one of { µ s , µ t } is non-zero. We call this set Z ( µ s e ξ − µ t e ξ ⋆ ), and a straightforwardanalysis gives Z ( µ s e ξ − µ t e ξ ⋆ ) = { µ s z = − µ t z } ∩ { ( µ s + µ t ) z + µ s p P i =1 ( x i + y i ) = ( µ s + µ t ) µ s } if µ s = 0 { µ s z = − µ t z } ∩ { ( µ s + µ t ) z + µ t p P i =1 ( x i + y i ) = ( µ s + µ t ) µ t } if µ t = 0 (53)Obviously, the two expressions are equivalent when both µ s and µ t are non-zero. The interest of this set is thatit happens to always contain Z ( e ξ ) and Z ( e ξ ⋆ ). This, together with the fact that when µ s = µ t = 0 these setsare contained in the axes Z ( η i ) will allow us to ignore them altogether. Lemma 7.2.
Assume that at least one of { µ s , µ t } is non-zero. Then Z ( e ξ ) , Z ( e ξ ⋆ ) ⊂ Z ( µ s e ξ − µ t e ξ ⋆ ) .Proof. Consider first µ s , µ t = 0. Then at Z ( µ s e ξ − µ t e ξ ⋆ ) ∩ (cid:8) T pj =1 { x j = y j = 0 } (cid:9) we have that z = ± µ t and z = ∓ µ s which establishes Z ( e ξ ) , Z ( e ξ ⋆ ) ⊂ Z ( µ s e ξ − µ t e ξ ⋆ ) in this case. When µ t = 0 , µ s = 0, by definition ofthe respective sets we have Z ( e ξ ) = Z ( µ s e ξ − µ t e ξ ⋆ ). Moreover, directly from (52) one finds Z ( e ξ ⋆ ) = p \ j =1 { x j = y j = 0 } ∩ { z = 0 , z = ± µ s } , which (cf. the first expression in (53)) is clearly contained in Z ( µ s e ξ − µ t e ξ ⋆ ). An analogous argument applies inthe case µ t = 0 , µ s = 0.Let us define the following auxiliary coordinatesˆ z + := µ s z + µ t z pP pi =1 ( x i + y i ) , ˆ z − := µ s z − µ t z pP pi =1 ( x i + y i ) , ˆ x i := x i , ˆ y i := y i . Except for the case µ s = µ t = 0 (which will be analyzed later) the coordinates { ˆ z + , ˆ z − , ˆ x i , ˆ y i } obviously cover R n \ (cid:8) T pj =1 { x j = y j = 0 } (cid:9) . In terms of the adapted coordinates, they readˆ z + = α sin( t − ) , ˆ z − = β sinh( t + ) ˆ x i = α i U cos( φ i ) , ˆ y i = α i U sin( φ i ) . (54)Let us analyze the points where (54) fails to be a change of coordinates and hence restrict the domain of definitionof { α, t − , t + , α i , φ i } . The first thing to notice is that a change of sign in the coordinate α i is equivalent to arotation of angle π in the coordinate φ i . Moreover, at points where α i = 0, i.e. the axis of η i , the coordinate φ i is completely degenerate, which obviously excludes S pj =1 { x j = y j = 0 } from the region covered by the adapted32oordinates. To avoid duplications, we must restrict α i ∈ (0 ,
1) and φ ∈ [ − π, π ) or alternatively α i ∈ ( − , \{ } and φ i ∈ [0 , π ). We choose the former for definiteness.The hypersurface { α = const , t − = const , t + = const } is an n − U − , namely { ˆ z − = const , ˆ z + = const } ∩ { P pi =1 ( x i + y i ) = U − = const } . This gives a straightforward splitting of R n \{ n − } , with 0 n − := { T pj =1 { x j = y j = 0 } (cid:9) , into R × ( R n − \{ n − } ), where R n − \{ n − } is foliated by n − Z ( µ s e ξ − µ t e ξ ⋆ ) respects this foliation, so it descends to R × R + (the lastfactor is the radius of the n − Z ( µ s e ξ − µ t e ξ ⋆ ) to denote thisquotient set. We next show that the adapted coordinates actually cover the largest possible domain, namely R n \ {Z ( µ s e ξ − µ t e ξ ⋆ ) ∪ S pj =1 { x j = y j = 0 }} . From the previous discussion, this is a consequence of the followingresult. Lemma 7.3.
Assume that at least one of { µ s , µ t } is not zero. Then. the transformation (ˆ z + , ˆ z − , U ) : R × [ − π, π ) × R + −→ (cid:0) R × R + (cid:1) \Z ( µ s e ξ − µ t e ξ ⋆ )( t + , t − , α ) (ˆ z + , ˆ z − , U ) . (55) is a diffeomorphism.Proof. The determinant of the jacobian of (55) reads (cid:12)(cid:12)(cid:12)(cid:12) ∂ (ˆ z + , ˆ z − , U ) ∂ ( t + , t − , α ) (cid:12)(cid:12)(cid:12)(cid:12) = αU. Since U is strictly positive (cf. (43) and recall that we chose ǫ = 1 w.l.o.g.), the conflictive points are α = 0.To calculate the locus { α = 0 } we obtain the inverse transformation of α in terms of U, ˆ z + , ˆ z − by solving (43)and the first two in (54). The result is, after a straightforward computation, α = ± (cid:18) ˆ z + 14 U ( µ s + µ t ) (ˆ z + ˆ z − − U ( µ s + µ t ) + ( µ s + µ t )) (cid:19) / . (56)It follows that α = 0 is equivalent to ˆ z + = 0 and ˆ z − + µ s + µ t = U ( µ s + µ t ) . When translated into the originalcoordinates { z , z , x i , y i } this set is precisely Z ( µ s e ξ − µ t e ξ ⋆ ). Also, from (56) it is obvious that α is multivalued,which also implies that t − is multivalued after substituting α as a function of ˆ z + , ˆ z − , U in the first equation in(54) . We solve this issue by restricting α to be strictly positive and let t − take values in [ − π, π ).We have shown that the adapted coordinates cover all R n except S pj =1 Z ( η j ) ∪ Z ( µ s e ξ − µ t e ξ ⋆ ) . The domainof definition of the coordinates t , t depends on µ t and µ s , because − π ≤ t − = µ t t − µ s t < π . This definesa band B ( µ s , µ t ) := {− π ≤ t − = µ t t − µ s t < π } , whose width and tilt is determined by σ, τ through µ s , µ t (see figure 2). Nevertheless, the coordinate change is well defined for all values of t and t and involves onlyperiodic functions of t − . Thus, we can extend the domain of definition of t , t to all of R . This defines acovering of the original space R n \ ( S pj =1 Z ( η j ) ∪ Z ( µ s e ξ − µ t e ξ ⋆ )) which unwraps completely the orbits of e ξ and e ξ ⋆ . It is not the universal covering because it does not unwrap the orbits of the axial vectors. This result is ageneralization to higher dimensions of the covering dicussed in detail in [24].The limit case µ s = µ t = 0 (that is σ = τ = 0) corresponds with a band of infinite width, i.e. B ( µ s , µ t ) = R .In this case, the adapted coordinates also cover the largest possible set R n \ ( S pj =1 Z ( η j )). Recall that in thiscase the only points where { e ξ, e ξ ⋆ , η i } is not a linearly independent set is the union of Z ( e ξ ) , Z ( e ξ ⋆ ), and Z ( η i )and we have already seen that in this case Z ( e ξ ) = Z ( e ξ ⋆ ) ⊂ Z ( η i ), for i = 1 , · · · , p . This limit case is the sameresult that we would have obtained, had we performed a direct analysis using U as given by (44). This was already evident by observing that a change of sing in α is cancelled by a rotation of π in t − θw t t Figure 2: Band B ( µ s , µ t ) where the coordinates t , t are defined. The tilt is given by θ = arctan (cid:16) µ s µ t (cid:17) and thewidth w is 2 π/µ t if µ t = 0, 2 π/µ s if µ t = 0 , µ s = 0 and w → ∞ if µ s = µ t = 0.Once we have determined the adapted coordinates and the region they cover, we may proceed to calculatethe expression of the Euclidean metric g E = d z + d z + p X i =1 (cid:0) d x i + d y i (cid:1) . (57)in adapted coordinates. We start with the term p P i =1 (cid:0) d x i + d y i (cid:1) , which is straightforward p X i =1 (cid:0) d x i + d y i (cid:1) = d U U + 1 U p X i =1 (cid:0) d α i + α i d φ i (cid:1)(cid:12)(cid:12) P pi =1 α i =1 − UU p X i =1 α i d α i ! = d UU + 1 U γ S n − , (58)where in the last equality we used P pi =1 α i d α i = 0, which follows from P pi =1 α i = 1 and we have defined γ S n − := p X i =1 (cid:0) d α i + α i d φ i (cid:1)(cid:12)(cid:12) P pi =1 α i =1 . (59)The notation is justified because the right-hand side corresponds to the standard unit metric on S n − . Thisfollows because P pi =1 (cid:0) d α i + α i d φ i (cid:1) is obviously flat and the restriction P pi =1 α i = 1 defines a unit sphere.We emphasize, however that the notation γ S n − refers to the quadratic form above, not to the spherical metricin any other coordinate system. Observe also that d U in (58) should be understood as a short name for theexplicit differential of U in terms of d t , d t , d α . Using (57) and (58), we have g t t = (cid:18) ∂z ∂t (cid:19) + (cid:18) ∂z ∂t (cid:19) + 1 U (cid:18) ∂U∂t (cid:19) , g t t = α + µ t U . Notice that g t t = g E ( e ξ, e ξ ), g t t = g E ( e ξ ⋆ , e ξ ⋆ ) and g t ,t = g E ( e ξ, e ξ ⋆ ). From the expressions in cartesiancoordinates it is straightforward to show g E ( e ξ, e ξ ) = g E ( e ξ ⋆ , e ξ ⋆ ) − σ p X i =1 ( x i + y i ) = g E ( e ξ ⋆ , e ξ ⋆ ) − σU , g E ( e ξ, e ξ ⋆ ) = τ p X i =1 ( x i + y i ) = τ U where we have used U − = p P i =1 ( x i + y i ) (see (49)). Thus g t t = g t t + σU = α + µ s U , g t t = τ U = µ s µ t U . The remaining terms are rather long to calculate. With the aid of a computer algebra system one gets g αα = (cid:18) ∂z ∂α (cid:19) + (cid:18) ∂z ∂α (cid:19) + 1 U (cid:18) ∂U∂α (cid:19) = 1 β U g αt = ∂z ∂α ∂z ∂t + ∂z ∂α ∂z ∂t + 1 U ∂U∂α ∂U∂t = 0 ,g αt = ∂z ∂α ∂z ∂t + ∂z ∂α ∂z ∂t + 1 U ∂U∂α ∂U∂t = 0 . Notice that no terms in d α i , d φ i appear but those in γ S n − , since neither U nor z , z depend on α i , φ i . Puttingall these results together we obtain the following expression: Lemma 7.4.
In adapted coordinates { t , t , α, α i , φ i } , the Euclidean metric g E takes the form g E = 1 U (cid:18) ( α + µ t )d t + ( α + µ s )d t + 2 µ s µ t d t d t + d α α + µ s + µ t + γ S n − (cid:19) . (60)We would like to stress the simplicity of this result. Except in the a global conformal factor, the metric doesnot depend in t and t (so, both e ξ and e ξ ⋆ are Killing vectors of U g E ). The dependence in the coordinate α and the conformal class constants { µ s , µ t } is also extremely simple. Even more, the fact that all dependence in { α i , φ i } arises only in γ S n − allows us to use any other coordinate system on the unit S n − . Any such coordinatesystem is still adapted to e ξ and e ξ ⋆ but (in general) no longer to { η i } . This enlargement to partially adaptedcoordinates is an interesting consequence of the foliation of R n by ( n − n case. As already discussed, we will base the analysis on the even dimensionalcase by restricting to a suitable a hyperplane. The underlying reason why this is possible is given in the followinglemma. Lemma 7.5.
Fix n ≥ odd. Let ξ F be a CKVF of E n in canonical form and let { z , x i , y i } be canonicalcoordinates. Consider the embedding E n ֒ → E n +1 where E n is identified with the hyperplane { z = 0 } , for acartesian coordinate z of E n +1 . Then ξ F extends to a CKVF of E n +1 with the same value of σ, µ i and τ = 0 . roof. By Remark 6.3 and Theorem 6.1, the expression of ξ F in the canonical coordinates { z , x i , y i } is ξ F = σ z − p X i =1 ( x i + y i ) !! ∂ z + z p X i =1 ( x i ∂ x i + y i ∂ y i ) + p X i =1 µ i ( x i ∂ y i − y i ∂ x i ) := e ξ + p X i =1 µ i η i . Define ξ ′ F on E n +1 in cartesian coordinates { z , z , x i , y i } by ξ ′ F = e ξ ′ + p P i =1 µ i ( x i ∂ y i − y i ∂ x i ) where e ξ ′ is givenby (36) with τ = 0. It is clear that this vector is a CKVF of E n +1 written in canonical form, that it is tangentto the hyperplane z = 0 and that it agrees with ξ F on this submanifold.Consequently, introducing adapted coordinates for the extended CKVF and restricting to { z = 0 } willprovide adapted coordinates for ξ F . The restriction will obviously reduce the domain of definition of theadapted coordinates ( t , t , α, α i , φ i ) to a hypersurface. It is straightforward from equation (48) and the secondequation in (50) that for the three cases σ > σ = 0 or σ <
0, the hyperplane { z = 0 } corresponds to { t = 0 } . It follows that the remaining coordinates { t , α, α i , φ i } are adapted to e ξ and all η i . Their domain ofdefinition is t ∈ R , α ∈ R + , α i ∈ (0 , φ i ∈ [ − π, π ) and the coordinate change is given by (47) (or the first in(50)) together with (49) after setting τ = 0 and t = 0. Depending on the sign of σ one gets for z z = − | U + | α sin( √ σt ) √ σ , σ > − | U − | √ α + | σ | sinh( √ | σ | t ) √ | σ | , σ < − | U | αt , σ = 0 , (61)where U + := 1 σ ( p α + σ − α cos( √ σt )) , U − := 1 − σ ( p α − σ cosh( √− σt ) − α ) , U := 12 ( αt + 1 α ) , and for all three cases x i = α i U ǫ cos( φ i ) , y i = α i U ǫ sin( φ i ) , (62)where we write U ǫ for the function U + , U − or U according with sign of σ .The range of variation of { t , α, α i , φ i } was inferred before from the corresponding range of variation of { t , t , α, α i , φ i } in E n +1 . It may happen, however, that when we restrict to the hyperplane { z = 0 } , the rangegets enlarged and additional points get covered by the adapted coordinate system. The underlying reason isthat, in effect, we are no longer adapting coordinates to e ξ ′ ⋆ , so the points on z = 0 where this vector is linearlydependent to e ξ ′ (or zero) are no longer problematic. When τ = 0, one has( µ s = √ σ, µ t = 0) if σ ≥ , ( µ s = 0 , µ t = p | σ | ) if σ ≤ . We may ignore the case σ = 0 because Z ( e ξ ′ ) = Z ( e ξ ′ ⋆ ). It follows from (51) and (53) that Z ( e ξ ′ ) (cid:12)(cid:12)(cid:12) z =0 = { z = 0 } ∩ (cid:26) p P i =1 ( x i + y i ) = σ (cid:27) if σ > T pj =1 { x j = y j = 0 } ∩ n z = ± p | σ | o if σ < Z ( µ s e ξ ′ − µ t e ξ ′ ⋆ ) (cid:12)(cid:12)(cid:12) z =0 = { z = 0 } ∩ (cid:26) p P i =1 ( x i + y i ) = σ (cid:27) if σ > { z + p P i =1 ( x i + y i ) = | σ |} if σ < . σ >
0, the two sets are the same and no extension of the coordinates { t , α, α i , φ i } is possible. However,when σ <
0, the set Z ( µ s e ξ ′ − µ t e ξ ′ ⋆ ) | z =0 is strictly larger than Z ( e ξ ′ ) | z =0 . From expressions (61) and (62) onechecks that Z ( µ s e ξ ′ − µ t e ξ ′ ⋆ ) | z =0 \ Z ( e ξ ′ ) | z =0 corresponds exactly to the value α = 0 and that Z ( e ξ ) = Z ( e ξ ′ ) | z =0 is at the limit t → ±∞ . Thus, a priori there is the possibility that the adapted coordinates { t , α, α i , φ i } canbe extended regularly to α = 0 when σ <
0. It follows directly from (61) that this is indeed the case (observethat, to the contrary, the limit α → σ ≥
0, in agreement with the previous discussion).Thus, the range of definition of α is [0 , ∞ ) when σ <
0. The conclusion is that, irrespectively of the value of σ , the adapted coordinates { t , α, α i , φ i } cover the largest possible domain of E n , namely all points where e ξ isnon-zero away from the axes of { η i } .To obtain the Euclidean metric in E n for n odd in adapted coordiantes we simply restrict (60) (with n → n +1)to the hypersurface t = 0, and get g ǫE = 1( U ǫ ) (cid:18)(cid:18) α + (1 − ǫ ) | σ | (cid:19) dt + dα α + | σ | + γ S n − (cid:19) , (63)where ǫ = − , , σ < , σ = 0 , σ > Remark 7.1.
The three odd dimensional cases can be unified into one. The function U coincides with thelimits of U + and U − when σ → . However, the analytical continuation of U + to negative values of σ does notdirectly yield U − . To solve this we introduce the function W ( y ) = 1 σ (cid:16)p y + σ − y cos (cid:0) √ σt (cid:1)(cid:17) , which is analytic in σ and takes real values for real σ . We observe that U + ( α = y ) = W ( y ) for σ > , U ( α = y ) = W ( y ) ( σ = 0 ) and U − ( α = + p y + σ ) = W ( y ) ( σ < ). This suggests introducing thecoordinate change α = y for σ ≥ and α = + p y + σ for σ < . From the domain of α , it follows that y takesvalues in y > when σ ≥ and y ≥ √− σ when σ < . In terms of y , the three metrics metric g ǫ take theunified form g ǫE = 1 W ( y ) (cid:18) y d t + d y y + σ + γ S n − (cid:19) . The function W is the analytic continuation of U + to negative values of σ . We could have started with U − and continued analytically to positive values of σ . Instead of repeating the argument, we simply introduce a newvariable z defined by y = √ z − σ with range of variation z > √ σ for σ ≥ and z ≥ for σ < . The metrictakes the (also unified and even more symmetric) form g ǫE = 1 W ( z ) (cid:18) ( z − σ )d t + d z z − σ + γ S n − (cid:19) , W ( z ) := 1 σ (cid:16) z − p z − σ cos (cid:0) √ σt (cid:1)(cid:17) . The function W ( z ) is again analytic in σ , takes real values on the real line, and now it extends U − . Morespecifically, U − ( α = z ) = W ( z ) ( σ < ), U ( α = z ) = W ( z ) ( σ = 0 ) and U + ( α = √ z − σ ) = W ( z ) ( σ > ). Remark 7.1 allows us to work with all the odd dimensional cases at once, which will be useful for Section 8.However, this unified form does not arise naturally when the odd dimensional case is viewed as a consequenceof the n + 1 even dimensional case. So, leaving aside this remark for Section 8, we summarize the results of thissection in the following Theorem. 37 heorem 7.6. Given a CKVF ξ F of E n , with n ≥ even, in canonical form ξ F = e ξ + P pi =1 µ i η i , the coordinates t , t , φ i , α, α i , for i = 1 , · · · p and P pi =1 α i = 1 , defined by z = − U ∂U∂t , z = 1 U ∂U∂t x i = α i U cos( φ i ) , y i = α i U sin( φ i ) with U = p α + µ t + µ s cosh( µ t t + µ s t ) − α cos( µ t t − µ s t ) µ t + µ s , which admits a limit lim µ s µ t → U = α ( t + t ) + α , furnish adapted coordinates to e ξ = ∂ t e ξ ⋆ = ∂ t η i = ∂ φ i ,which cover the maximal possible domain, namely E n \ (cid:16)S pj =1 Z ( η j ) ∪ Z ( µ s e ξ − µ t e ξ ⋆ ) (cid:17) for t , t ∈ B ( µ s , µ t ) , φ i ∈ [ − π, π ) , α i ∈ (0 , and α ∈ R + . Moreover, the metric g E , which is flat in canonical cartesian coordinates,is given by g E = 1 U (cid:16) ( α + µ t )d t + ( α + µ s )d t + 2 µ s µ t d t d t + d α α + µ s + µ t + p X i =1 (cid:0) d α i + α i d φ i (cid:1)(cid:12)(cid:12) P pi =1 α i =1 (cid:17) . (64) If n ≥ is odd and ξ F is in canonical form, ξ F = e ξ + P pi =1 µ i η i , the coordinates { t , φ i , α, α i } adapted to e ξ = ∂ t η i = ∂ φ i are given by the case of n +1 (even) dimensions, for τ = 0 restricted to t = 0 (which defines the embed-ding E n = { z = 0 } ⊂ E n +1 ) and cover again the maximal possible domain, given by E n \ (cid:16)S pj =1 Z ( η j ) ∪ Z ( e ξ ) (cid:17) for t ∈ R , φ i ∈ [ − π, π ) , α i ∈ (0 , and α ∈ R + when σ ≥ and α ∈ R + ∪ { } when σ < . Moreover,the metric g E , which is flat in canonical cartesian coordinates, is given by the pull-back of (64) at t = 0 aftersetting τ = 0 . Explicitly g E is, depending on the sign of σ , given by (63) with γ S n − as in (59) . The adapted coordinates derived in Section 7 provide a useful tool to solve geometric equations involvingCKVFs. In this section we give an example of this in the context of Λ-vacuum spacetimes admitting a smoothnull conformal infinity.Recall that for such spacetimes the data at I is a conformal class [ g ] of riemannian metrics and a conformalclass of transverse and divergence-free tensors. More specifically, for a representative metric g in the conformalclass, there is associated a symmetric tensor D AB satisfying g AB D AB = 0 (divergence-free) and ∇ A D AB = 0(transverse). For any other metric ˜ g = Ω g in the conformal class, the associated tensor is Ω − ( n +2) D AB , whichis again a TT tensor with respect to ˜ g . In dimension n = 3, it has been shown in [25] that the spacetimegenerated by the Cauchy data at I admits a Killing vector if and only if the metric g admits a CKV ξ (whichis the restriction of the Killing vector to I ) and D satisfies the so-called Killing initial data (KID) equation.This equation admits a natural generalization to arbitrary dimension which is L ξ D AB + n + 2 n div g ξD AB = 0 , (65)where div g ξ is the divergence of ξ . Equation (65) reduces to the KID equation of Paetz in dimension n = 3 andit is conformally covariant, i.e. if { g AB , D AB , ξ A } is a solution, then so it is { Ω g AB , Ω − ( n +2) D AB , ξ A } . Weemphasize however, that in higher dimension ( n ≥
4) it is not known whether a spacetime admiting a smooth38 such that the corresponding data at null infinity solves the KID equation for some CKV ξ , must necessarilyadmit a Killing vector.A CKVF satisfying (65) will be called KID vector for short. An important property of KID vectors is thatthey form a Lie subalgebra of CKVFs, i.e. if ξ, ξ ′ are KIDs for a given T T tensor D , then [ ξ, ξ ′ ] is also a KIDfor D . The problem of obtaining all TT-tensors with generality for a given conformal structure is hard, evenin the conformally flat case (see e.g. [3]). In this section we exploit the results above to obtain the generalsolution of the KID equations in dimension n = 3 for spacetimes which possess two commuting symmetries, oneof which is axial. This case is specially relevant since n = 3 corresponds to the physical case of four spacetimedimensions and the class necessarily contains the Kerr-de Sitter family of spacetimes, which is a particularlyinteresting explicit familiy of spacetimes. Our strategy is to take an arbitrary CKVF ξ , derive its canonicalform ξ F = e ξ + µη , adapt coordinates to e ξ and η and impose the KID equations to e ξ and η . The problemsimplifies notably in the conformal gauge to g := ( U ǫ ) g ǫE because both e ξ and η become Killing vector fields.From Remark 7.1, we may treat all cases σ < , σ = 0 , σ > g = d z z − σ + ( z − σ )d t + d φ , e ξ = ∂ t , η = ∂ φ . (66)We remark that even though we solve the problem by fixing the coordinates and conformal gauge, we shallwrite the final result in fully covariant form (cf. Theorem 8.2 below).In the conformal gauge of g , the condition that a TT-tensor D satisfies KID equations for both e ξ and η (which is equivalent to imposing that ξ and η are KID vectors) is trivial in the adapted coordinates obtainedin the previous section: L e ξ D AB = ∂ t D AB = 0 , L η D AB = ∂ φ D AB = 0 . Thus, D AB are only functions of z . The transversality condition is also quite simple in adapted coordinates:d D zz d z − z (cid:18) D zz z − σ + ( z − σ ) D tt (cid:19) = 0 , (67)d D zt d z + 2 zz − σ D zt = 0 (68)d D zφ d z = 0 , (69)while the traceless condition imposes g AB D AB = D zz z − σ + ( z − σ ) D tt + D φφ = 0 . (70)There are no equations for D tφ so D φt = h ( z ) with h ( z ) an arbitrary function. The general solution of equations(68) and (69) is obtained at once and reads D zt = K z − σ , D zφ = K , K , K ∈ R . For equations (67) and (70), we let D zz =: f ( z ) be an arbitrary function and obtain the remaining components D φφ = − z d f d y , D tt = 1 z ( z − σ ) d f d z − f ( z − σ ) . Summarizing 39 emma 8.1.
In the three-dimensional conformally flat class [ g ] , let ξ F be a CKVF. Decompose ξ in canonicalform ξ = e ξ + µη and fix the conformal gauge so that g given by (66) . Then the most general symmetric TT-tensor D satifying the KID equations for ξ and η simultaneously is, in adapted coordinates { z, t, φ } , a combination(with constants) of the following tensors D f := f ∂ z ⊗ ∂ z + (cid:18) z ( z − σ ) d f d z − f ( z − σ ) (cid:19) ∂ t ⊗ ∂ t − z d f d z ∂ φ ⊗ ∂ φ ,D h := h ( ∂ t ⊗ ∂ φ + ∂ φ ⊗ ∂ t ) , D e ξ := 1 z − σ ( ∂ z ⊗ ∂ t + ∂ t ⊗ ∂ z ) , D η = ∂ z ⊗ ∂ φ + ∂ φ ⊗ ∂ z , where f and h are arbitrary functions of z . Having obtained the general solution in a particular gauge, our next aim is to give a (diffeomorphism andconformal) covariant form of the generators in Lemma 8.1. From [23], we know that, for any CKV ξ of any n -dimensional metric g (not necessarily conformally flat) the following tensors are TT w.r.t. to g and satisfythe KID equation with respect to ξ . D ( ξ ) = 1 | ξ | n +2 g ξ ⊗ ξ − | ξ | g n g ♯ ! , where | · | g denotes the norm w.r.t. g and g ♯ the contravariant form of g . Thus, we can rewrite D f as D f = (cid:18) − z − σ ) / f + ( z − σ ) / z d f d z (cid:19) D ( e ξ ) − (cid:18) fz − σ + 1 z d f d z (cid:19) D ( η ) . We now restore the conformal gauge freedom by considering the metric b g = Ω g and b D f = D f / Ω , for any(positive) conformal factor Ω. Since the tensors D ( e ξ ) , D ( η ) are already conformal and diffeomorphism covariant,we must impose their multiplicative factors in b D f to be conformal and diffeomorphism invariant. With the gaugefreedom restored, the norms of the CKVFs now are | e ξ | b g = Ω p z − σ, | η | b g = Ω . Then, considering f =: √ X b f (X) as function of the conformal invariant quantity X = | e ξ | b g / | η | b g = √ z − σ , onecan directly cast b D f in the following form: b D f = X ddX b f (X)X / ! D ( e ξ ) − ddX (cid:16) X / b f (X) (cid:17) D ( η ) , which is a conformal and diffeomorphism covariant expression. Notice that the expression is symmetric underthe interchange e ξ ↔ η because the coefficient of D ( η ) expressed in the variable Y = X − is identical in form tothe coefficient of D ( ξ ).For the tensor b D h := D h / Ω , redifining h =: b h | e ξ | − / , it is immediate to write b D h = b D b h := b h | η | / b g | e ξ | / b g ( e ξ ⊗ η + η ⊗ e ξ ) , (71)40hich is obviously conformal and diffemorphism covariant if and only if b h is conformal invariant, e.g. considering b h ≡ b h (X). We remark that the form (71) already appeared (with different powers due to the different dimension)in the classification [24] of TT tensors in dimension two satisfying the KID equation.For the remaining tensors b D e ξ := D e ξ / Ω and b D η := D η / Ω , we define a conformal class of vector fields χ ,which in the original gauge coincides with χ := ∂ z . This vector is divergence-free ∇ A χ A = 0, and this equationis conformally invariant provided the conformal weight of χ is − b g = Ω g , the corresponding vector is b χ = Ω − χ ). We therefore impose this conformal behaviour of χ . The direction of χ is fixed by orthogonalityto e ξ and η . The combination of norms that has this conformal weight and recovers the appropriate expressionin the gauge of Lemma 8.1 is | χ | b g := | e ξ | − b g | η | − b g (note that the orthogonality and norm conditions fix χ uniquelyup to an irrelevant sign in any gauge). Thus, we may write D e ξ = 1 | e ξ | b g ( χ ⊗ e ξ + e ξ ⊗ χ ) , D η = 1 | η | b g ( χ ⊗ η + η ⊗ χ ) , which are conformally covariant expressions. Therefore, we get to the final result: Theorem 8.2.
Let ξ be a CKVF of the class of three dimensional conformally flat metrics and let ξ = e ξ + µη acanonical form. For each conformal gauge, let us define a vector field χ with norm | χ | b g := | e ξ | − b g | η | − b g , orthogonalto e ξ and η . Then, any TT-tensor satisfying the KID equations (65) for e ξ and η is a combination (with constants)of the following tensors: b D b f = X ddX b f (X)X / ! D ( e ξ ) − ddX (cid:16) X / b f (X) (cid:17) D ( η ) , b D b h = b h | η | / b g | e ξ | / b g ( e ξ ⊗ η + η ⊗ e ξ ) ,D e ξ = 1 | e ξ | b g ( χ ⊗ e ξ + e ξ ⊗ χ ) , D η = 1 | η | b g ( χ ⊗ η + η ⊗ χ ) , for arbitrary functions b f and b h of X = | e ξ | b g / | η | b g . Remark 8.1.
The vector field χ defined in this Theorem is divergence-free. This property would have beendifficult to guess (and even to prove) in the original Cartesian coordinate system. Remark 8.2.
A corollary of this theorem is that the general solution of the Λ -vacuum Einstein field equationin four dimensions with a smooth conformally flat null infinity and admitting an axial symmetric and a secondcommuting Killing vecor can be parametrized by two functions of one variable and two constants. Recall thatin the Λ = 0 case, the general asympotically flat stationary and axially symmetric solution of the Einstein fieldequations can be parametrized (in a neighbourhood of spacelike infinity, by two numerable sets of mass andangular multipole moments (satisfying appropriate convergence properties), see [1], [4], [6] for details. Thereis an intriguing paralelism between the two situations, at least at the level of crude counting of degrees offreedom. This suggests that maybe in the Λ > case it is possible to define a set of multipole-type moments thatcharacterizes de data at null infinity (and hence the spacetime), at least in the case of a conformally flat nullinfinity. This is an interesting problem, but well beyond the scope of the present paper. This choice may appear somewhat ad hoc at this point. However, the condition of vanishing divergence appears naturaly whenstudying (for more general metrics) under which conditions a tensor ξ ⊗ W + W ⊗ ξ is a TT tensor satisfying the KID equation for ξ . We leave this general analysis for a future work. emark 8.3. It is natural to ask whether Theorem 8.2 is general for TT-tensors admitting two commutingKIDs, e ξ, η , without the condition of η being conformally axial. In Appendix C of [23] one can explicitly find,for an arbitrary CKVF ξ , the set C ( ξ ) of elements that commute with ξ . Then, from a case by case analysis,one concludes that except in one special situation, for any linearly independent pair ξ, ξ ′ , with ξ ′ ∈ C ( ξ ) it is thecase that there is a CAKVF η ∈ C ( ξ ) such that span { ξ, η } = span { ξ, ξ ′ } . Thus, all these cases are covered byTheorem 8.2. The exceptional case is when ξ, ξ ′ are conformal to translations. It is immediate to solve the TTand KID equations for such a case directly in Cartesian coordinates. The solution given in Theorem 8.2 provides a large class of initial data, which we know must contain theso-called Kerr-de Sitter-like class with conformally flat I (see [23] for precise definition and properties of thisclass), which in turn contains the Kerr-de Sitter family of spacetimes. It is interesting to identify this classwithin the general solution given in Theorem 8.2. The characterizing property of the Kerr-de Sitter-like classin the conformally flat case is D = D ( ξ ) for some CKVF ξ , where moreover, only the conformal class of ξ matters to determine the family associated to the data. Decomposing canonically ξ = e ξ + µη , a straightforwardcomputation yields D ( ξ ) = X (X + µ ) / D ( e ξ ) + µ (X + µ ) / D ( η ) + µ X / (X + µ ) / b D b h =1 , which comparing with Theorem 8.2 yields the following corollary: Corollary 8.2.1.
The Kerr-de Sitter-like class with conformally flat I is determined by the TT-tensor D KdS = b D f + b D b h with b f = −
13 X / (X + µ ) / , b h = µ X / (X + µ ) / . It is also of interest to identify the the Kerr-de Sitter family. To that aim we combine the results in [23]to those in the present paper to show that this family corresponds to σ <
0. The classification of conformalclasses of ξ in [23] is done in terms of the invariants b c = − c and b k = − c together with the rank parameter r , where c and c are the coefficients of the characteristic polynomial of the skew-symmetric endomorphism F associated to ξ . In terms of these objects, it is shown in [23] that the Kerr-de Sitter family corresponds to either S = { b k > , b c ∈ R and r = 2 } , or S = { b k = 0 , b c > r = 1 } , the latter defining the Schwarzschild-deSitter family. It is immediate to verify that, since (cf. Corollary 3.3.1) b k = − σµ < b c = − σ − µ , then S = { σ < , µ = 0 } and S = { σ < , µ = 0 } (the condition µ = 0 implies r = 2 and µ = 0 implies r = 1).Thus, in terms of the classification developed in this paper, the Kerr-de Sitter family corresponds to σ <
0. It isinteresting that in the present scheme we no longer need to specify the rank parameter to identify the Kerr-deSitter family (unlike in [23]) and that the whole family is represented by an open domain. We emphazise thatthe dependence in σ in the solutions given in Theorem 8.2 and Corollary 8.2.1 is implicit through the norm of e ξ . Acknowledgements
The authors acknowledge financial support under the projects PGC2018-096038-B-I00 (Spanish Ministerio deCiencia, Innovaci´on y Universidades and FEDER) and SA083P17 (JCyL). C. Pe´on-Nieto also acknowledges thePh.D. grant BES-2016-078094 (Spanish Ministerio de Ciencia, Innovaci´on y Universidades).42 eferences [1] A. E. Ace˜na. Convergent null data expansions at space-like infinity of stationary vacuum solutions.
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