Block diagonalisation of four-dimensional metrics
aa r X i v : . [ m a t h . DG ] N ov BLOCK DIAGONALISATION OF FOUR-DIMENSIONAL METRICS
JAMES D.E. GRANT AND J.A. VICKERS
Abstract.
It is shown that, in 4-dimensions, it is possible to introduce coordinates so thatan analytic metric locally takes block diagonal form. i.e. one can find coordinates such that g αβ = 0 for ( α, β ) ∈ S where S = { (1 , , (1 , , (2 , , (2 , } . We call a coordinate system inwhich the metric takes this form a ‘doubly biorthogonal coordinate system’. We show that allsuch coordinate systems are determined by a pair of coupled second-order partial differentialequations. Introduction
This paper is concerned with making coordinate choices to put general metrics into simplifiedor canonical forms. A metric in 2-dimensions depends upon × g , g and g . On the other hand, the diffeomorphism freedom f : R → R ( x, y ) ( f ( x, y ) , f ( x, y ))contains 2 arbitrary functions. Given any 2-dimensional metric, one would therefore expect to beable to introduce local coordinates such that the metric depended on only 3 − ds = Ω ( x, y ) (cid:0) dx + dy (cid:1) . The proof for analytic metrics goes back to Gauss [Gau22], while the proof for smooth metrics ismore recent (see for example [Spi75] for details).In 3-dimensions the metric depends upon × f : R → R involves 3 functions. One would therefore expect to be ableto introduce coordinates such that a 3-dimensional metric was specified by 6 − ds = A ( x, y, z ) dx + B ( x, y, z ) dy + C ( x, y, z ) dz . Again the proof of this result in the analytic case goes back a long way [Car45]. The proof in thesmooth case was, again, much more recent [DY84] and uses the theory of the characteristic varietyof an exterior differential system.In 4-dimensions the metric depends upon × f : R → R gives 4 functions. One would therefore expect to be ableto write a 4-dimensional metric in a canonical form that depended upon 10 − Mathematics Subject Classification.
Primary 58A15; Secondary 53B99.
Key words and phrases.
Exterior differential systems, special coordinate systems.This work was partially supported by START-project Y237–N13 of the Austrian Science Fund.
J.D.E. GRANT AND J.A. VICKERS form for 4-dimensional metrics was the ‘block diagonal’ form g αβ = A B
B C
D E
E F . In this paper we use Cartan’s theory of exterior differential systems to show that it is indeedpossible to write an analytic 4-dimensional metric in this form, at least locally. We show thatthe problem of finding local coordinates that block-diagonalise a metric may be reformulated asa condition on an orthonormal tetrad (see equation (3.4)). From this reformulation, we constructan exterior differential system on the orthonormal frame bundle of our manifold, the integralmanifolds of which give rise to solutions of our block-diagonalisation problem. This exteriordifferential system is not involutive, however, so we must go to the first prolongation. At thispoint, we discover a consistency condition for our system, (4.10), that must be satisfied. Imposingthis constraint on our exterior differential system gives rise to an involutive Pfaffian system, towhich the Cartan–K¨ahler theorem may be applied to show existence of solutions. Note that theconsistency condition mentioned above may be interpreted on our manifold as a relation betweena curvature component and various components of the connection (cf. equation (4.16) for theRiemannian version of this constraint and equation (4.19) for the Lorentzian version in Newman–Penrose formalism). At the level of our four-dimensional manifold, this constraint may be deduceddirectly as being a consequence of the conditions (3.4) imposed on the orthonormal tetrad. Theconstraint involves the extrinsic curvature of the two surfaces and does not impose any additionalgeometrical restrictions on our manifold. Indeed the fact that we have a Pfaffian system on thefirst prolongation which satisfies the conditions for the Cartan–K¨ahler theorem shows that theblock-diagonalisation of any four-dimensional metric may be carried out locally.Although our results for local canonical forms have assumed that the metric is Riemannian, theyremain true in the Lorentzian case (with obvious modifications). Similarly, we will assume thatour metric is Riemannian, although the proof may easily be adapted for metrics of Lorentzian or( − , − , + , +) signature. The Lorentzian version of the 4-dimensional result is, in particular, usefulin establishing certain results in general relativity. For example, it can be used to establish someresults concerning the geometry of generalised cosmic strings [Kin97] and can also be used to makea gauge choice within the 2 + 2 formalism [dS80] in which all the shifts β iα vanish.Given a local canonical form for a metric one can ask what transformations preserve that form.For the case of a metric in 2-dimensions a conformal (in the sense of complex analytic) transfor-mation of the flat metric will map isothermal coordinates into isothermal coordinates. Similarlyin the 3-dimensional case the problem is essentially the same as finding all ‘triply orthogonalcoordinate systems’ which are coordinates in which the flat metric is diagonal. The problem offinding all such coordinate systems was solved by Darboux [Dar98], who showed that it requiredthe solution of a certain third-order partial differential equation. Similarly in 4-dimensions theproblem is essentially the same as finding all ‘doubly biorthogonal coordinate systems’ which arecoordinates in which the flat metric is in block diagonal form. We will show in Section 5 that allsuch coordinates are determined by the solution of a pair of coupled second-order equations.The plan of this paper is as follows. In Section 2 we briefly review the proofs that 3-dimensionalmetrics may be diagonalised in both the analytic and smooth case. In Section 3 we explain whythese methods fail to give a direct proof of the block diagonalisation of a 4-dimensional metric.However, we reduce the problem of block diagonalising a metric to the problem of constructing anorthonormal tetrad that satisfies a particular set of identities (3.4). In Section 4 we show, usingthe theory of exterior differential systems, that, in the case where the metric is analytic, suchan orthonormal tetrad can always be constructed. As such, we deduce that a four-dimensionalanalytic metric can be block-diagonalised. In Section 5 we discuss triply orthogonal systemsin 3-dimensions to motivate the discussion of doubly biorthogonal systems of coordinates in 4-dimensions. In order to make the paper reasonably self-contained, we have collected together LOCK DIAGONALISATION OF FOUR-DIMENSIONAL METRICS 3 the main background material that we require from the theory of exterior differential systems inAppendix A.
Notation : In the earlier sections of this paper, we will often have cause to refer to a singlediagonal component g αα of a metric. Also, when working with exterior differential systems, it issometimes convenient to explicitly write out the terms in a sum individually, rather than use thesummation convention. Therefore, we will generally not use the Einstein summation conventionin this paper, with the exception of Section 5, where the above issues do not arise.Note also that we will use Greek letters for coordinate indices and Latin letters for frame indices.2. Diagonalising metrics in -dimensions In this section, we review the methods of proving that a 3-dimensional smooth metric can bediagonalised.In the analytic case, rather than working with the covariant metric g αβ it is more conve-nient to consider the equivalent problem of diagonalising the contravariant metric g αβ . Given g αβ ( x , x , x ) we wish to find new coordinates { x α ′ ( x , x , x ) : α = 1 , , } such that g α ′ β ′ = X γ,δ ∂x α ′ ∂x γ ∂x β ′ ∂x δ g γδ = 0 for α ′ = β ′ . This is a non-linear system of 3 equations (taking ( α ′ , β ′ ) to be (1 , ,
3) and (2 , x ′ , x ′ and x ′ . In the analytic case one can show that solutions to these equationsexist but the solutions are not unique (there are trivial transformations given by replacing x ′ , x ′ and x ′ with h ( x ′ ), h ( x ′ ) and h ( x ′ )) and the strongly non-linear nature of the equationsmakes it hard to utilise this method in the smooth case. Instead, DeTurck and Yang [DY84] seekan orthonormal coframe ǫ , ǫ , ǫ , and a coordinate system x , x , x such that ǫ i = f i dx i , i = 1 , , . (2.1)(Recall no summation .) Clearly such a frame would imply that g µν is diagonal in the coordinatesystem of the x µ .The advantage of condition (2.1) is that, by the Frobenius theorem, it is (locally) equivalent tothe existence of a coframe such that ǫ i ∧ d ǫ i = 0 , i = 1 , , ǫ i to be unique (up to relabelling) since the lack of uniquenessin the coordinates noted above is absorbed into the f i .Let { ǫ i } be some fixed orthonormal frame for g αβ in some open set. Then, since ǫ i and ǫ i areboth orthonormal, they are related by some SO(3) transformation a ij ǫ i ( x ) = X j a ij ( x ) ǫ j ( x ) . (2.3)We now substitute (2.3) into (2.2) to obtain X j,k a ij ǫ j ∧ d (cid:0) a ik ǫ k (cid:1) = 0 , i = 1 , , . (2.4)Note that this gives 3 equations for 3 unknowns (such as the Euler angles) which parameteriseelements of SO(3). To show that there exist solutions to (2.4), DeTurck and Yang write the secondterm as d (cid:0) a ik ǫ k (cid:1) = X l (cid:0) a ik | l ǫ l ∧ ǫ k + a ik d ǫ k (cid:1) (where f | i = e i ( f ) = P µ e iµ ∂f∂x µ , with e i the dual basis to ǫ i ). They then use Cartan’s firststructure equation to write d ǫ k = X l,m γ klm ǫ l ∧ ǫ m , J.D.E. GRANT AND J.A. VICKERS where γ kml are the connection coefficients with respect to the frame ǫ i . (Our conventions are that d ǫ i = − Γ ij ∧ ǫ j with Γ ij = P k γ ijk ǫ k .)Substituting in (2.4) gives X σ ∈ Σ X j,k,l,m (sign σ ) a iσ ( j ) (cid:0) a iσ ( k ) | σ ( l ) + a im γ mσ ( k ) σ ( l ) (cid:1) = 0 , i = 1 , , . One can then solve for a ik | l and show that the resulting system is diagonal hyperbolic (a specialcase of symmetric hyperbolic). In the smooth case one has existence and uniqueness theorems forsuch systems of equations (see e.g. [Tay81]), so that one can show the existence of a unique (up torelabelling) orthonormal frame satisfying (2.1) and hence a diagonal metric. Note however thatas remarked earlier the coordinate expression (2.1) is not unique, but one is free to replace x by h ( x ) etc, so the actual diagonal entries of the metric are not unique.3. Block diagonalisation of -dimensional metrics In this section and the next, we shall show that it is possible, in the analytic case, to intro-duce coordinates that block-diagonalise a 4-dimensional metric. The proof will eventually be byan application of the Cartan–K¨ahler theorem, a generalisation of the Cauchy–Kovalevskya theo-rem [BCG + g αβ ( x , x , x , x ), we want to find new coordinates { x α ′ ( x , x , x , x ) : α = 1 , . . . , } such that g α ′ β ′ = X γ,δ ∂x α ′ ∂x γ ∂x β ′ ∂x δ g γδ = 0 for ( α ′ , β ′ ) ∈ S, where S = { (1 , , (1 , , (2 , , (2 , } . This gives 4 equations for 4 unknowns.For ease of notation, we let x α ′ ( x , x , x , x ) = y α ( x , x , x , x ) = y α ( x β ). We now lineariseabout y αo ( x β ) and obtain X γ,δ (cid:16) y α,γ y βo,δ + y β,γ y αo,δ (cid:17) g γδ = − X γ,δ y αo,γ y βo,δ g γδ for ( α, β ) ∈ S. This is a system of the form P α ∂∂x α y = c , where P α = y ,αo y ,αo y ,αo y ,αo y ,αo y ,αo y ,αo y ,αo , c = − y o,α y o,β g αβ − y o,α y o,β g αβ − y o,α y o,β g αβ − y o,α y o,β g αβ , and y β,αo = y βo,γ g αγ .Unfortunately, when one attempts to find the characteristic surfaces, one findsdet( P α ξ α ) = 0 , ∀ ξ α ∈ R , so that there are no non-characteristic surfaces and the initial data must satisfy some constraint.As a result, one cannot directly apply the Cauchy–Kovalevskya theorem, unlike in the apparentlysimilar problem of diagonalising a metric in 3-dimensions. LOCK DIAGONALISATION OF FOUR-DIMENSIONAL METRICS 5
We therefore turn to the method of DeTurck and Yang. In this case, this involves finding acoframe { ǫ i } and a coordinate system such that ǫ ∧ ǫ = f dx ∧ dx , (3.1a) ǫ ∧ ǫ = gdx ∧ dx . (3.1b)Note that (3.1a) implies ǫ i = X µ =1 , ǫ iµ dx µ i = 1 , ǫ i = X µ =3 , ǫ iµ dx µ i = 3 , g µν = P i,j δ ij ǫ iµ ǫ jν is block diagonal. Conversely, if g µν is block diagonal, we cancertainly find a coframe that satisfies (3.2) and (3.3) and hence (3.1a) and (3.1b).This leads to the following characterisation of metrics that can be block-diagonalised: Proposition 3.1.
A Riemannian metric g can be block-diagonalised if and only if it admits anorthonormal coframe, { ǫ a : a = 1 , . . . , } , that satisfies the relations ǫ ∧ ǫ ∧ d ǫ = 0 , ǫ ∧ ǫ ∧ d ǫ = 0 , ǫ ∧ ǫ ∧ d ǫ = 0 , ǫ ∧ ǫ ∧ d ǫ = 0 . (3.4) Proof.
Given a coframe that obeys relations (3.4), the Frobenius theorem implies the existence oflocal coordinates ( t, x, y, z ) and functions α, . . . , θ such that ǫ = α dt + β dx, ǫ = γ dt + δ dx, ǫ = ǫ dy + ζ dz, ǫ = η dy + θ dz. (3.5)The metric g is then block-diagonal in this coordinate system. Conversely, if the metric g isblock-diagonal with respect to a coordinate system ( t, x, y, z ), then we can choose a coframe ofthe form (3.5), which then automatically satisfies (3.4). (cid:3) Remark . Although we have stated the block-diagonalisation problem in terms of Riemannianmanifolds, it is clear that the problem of block-diagonalising a metric is conformally invariant,In particular, a coordinate system that block-diagonalises a representative metric in a conformalequivalence class will block-diagonalise all representatives in that conformal equivalence class. Wewill pursue the Riemannian version of the problem for simplicity, although all of our calculationscan be reformulated in a conformally equivariant fashion.In the next section, the characterisation given in Proposition 3.1 will be used to show that allanalytic four-dimensional metrics can be block-diagonalised.4.
Exterior differential systems
In this section, we use the theory of exterior differential systems, in particular the Cartan–K¨ahlertheorem, to show that, for a given analytic metric g , we can find an orthonormal coframe thatsatisfies the conditions (3.4) of Proposition 3.1. Our notation, generally, follows that of [BCG + + X be an oriented four-manifold with a Riemannian metric g , and let π : F → X be thebundle of orthonormal coframes of ( X, g ). We will denote points in F by either p or, since weare working locally, we will assume a trivialisation π − ( X ) ∼ = X × SO(4) and denote points in
J.D.E. GRANT AND J.A. VICKERS F by ( x, g ) where x ∈ X and g ∈ SO(4). The bundle F comes equipped with a canonical basisof 1-forms consisting of the components, { ω a } a =1 ,..., , of the tautological 1-form on F and thecomponents, { ω ab } a,b =1 ,..., , of the Levi-Civita connection (see, e.g., [IL03]). These differentialforms have the following properties: • Reproducing property : An orthonormal coframe { ǫ a } a =1 on M defines a correspondingsection f : X → F . Pulling back the tautological 1-forms on F by this section reproducesthe coframe { ǫ a } i.e. f ∗ ω a = ǫ a . • Canonical coframing : A canonical coframing of F consists of the tautological 1-forms ω a , a = 1 , . . . , ω ab , where a, b = 1 , . . . , a < b . Notethat we will often write summations that involve terms of the form ω ab with a > b . Inthis case, we identify ω ab with − P c,d δ ac δ bd ω dc , consistent with the SO(4) nature of theconnection. We adopt similar conventions with quantities such as λ bca introduced later. • Cartan structure equations : The one-forms { ω a , ω ab } obey the Cartan structure equa-tions d ω a + X b ω ab ∧ ω b = 0 ,d ω ab + X c ω ac ∧ ω cb = Ω ab , where Ω ab = 12 X c,d R abcd ω c ∧ ω d ∈ Ω ( F , so (4))is the curvature form of the connection form ω ab . (Recall our convention mentioned abovefor ω ab with a > b .)Following Proposition 3.1, let I ⊂ Ω ∗ ( F ) be the exterior differential system on F generated bythe 4-forms Θ := ω ∧ ω ∧ d ω = ω ∧ ω ∧ ω ∧ ω + ω ∧ ω ∧ ω ∧ ω , Θ := ω ∧ ω ∧ d ω = ω ∧ ω ∧ ω ∧ ω + ω ∧ ω ∧ ω ∧ ω , Θ := ω ∧ ω ∧ d ω = − ω ∧ ω ∧ ω ∧ ω − ω ∧ ω ∧ ω ∧ ω , Θ := ω ∧ ω ∧ d ω = − ω ∧ ω ∧ ω ∧ ω − ω ∧ ω ∧ ω ∧ ω . (4.1)(Therefore, I is the ideal in Ω ∗ ( F ) generated, algebraically, by the 4-forms Θ i and the 5-forms d Θ i .) We consider the exterior differential system with independence condition ( I , Ω ) on theten-dimensional manifold F , where the independence condition is defined by the 4-form Ω := ω ∧ · · · ∧ ω ∈ Ω ( F ) . As a result of the previous discussion, we have the following:
Lemma 4.1.
Let U ⊆ X is an open set, and f : U → F a section of F that satisfies f ∗ ϕ = 0 ,for all ϕ ∈ I , and f ∗ Ω = 0 on U . Then the -forms ǫ a := f ∗ ω a ∈ Ω ( U ) define an orthonormalcoframe on U that satisfies (3.4) . Let E ⊂ T p F be a 4-dimensional integral element of ( I , Ω ) based at point p ∈ F (i.e. ϕ | E = 0,for all ϕ ∈ I and Ω | E = 0). The space of such integral elements is denoted by V ( I , Ω ), andis a subset of Gr ( T F , Ω ), which is the subset of the Grassmannian bundle Gr ( T F ) consistingof 4-planes, E , for which Ω | E = 0. Let ( v , . . . , v ) be a basis for E which, without loss ofgenerality, we may take to be of the form v a = ∂∂ ω a ( p ) + X b 4. Substituting (4.2)into (4.1), we find that this is equivalent to the conditions λ = λ , λ = λ ,λ = λ , λ = λ . (4.3)At each point p ∈ F , these equations impose 4 linear constraints on the coordinates λ bca . Ittherefore follows that V ( I , Ω ) is a smooth submanifold of Gr ( T F ) of codimension 4.We now consider an integral flag (0) p ⊂ E ⊂ E ⊂ E ⊂ E ⊂ T p ( F ), and wish to calculatethe integers c k , k = 0 , . . . , I contains no non-zero1-forms, 2-forms or 3-forms, it follows that c = c = c = 0and, from its definition, we have c = dim F − . Therefore, it only remains to calculate c . To do this, we first define the one-forms π ab := ω ab ( p ) − X c λ abc ω c ( p ) ∈ T ∗ p F . Note that the π ab , with a < b , span the subspace of T ∗ p ( F ) that annihilate the vectors v a . It thenfollows that E may be described as E = n v ∈ T p F : π ab ( v ) = 0 , for a, b = 1 , . . . , a < b o . We now note that, by (4.3), we may write Θ = ω ∧ ω ∧ ω ∧ π + ω ∧ ω ∧ ω ∧ π , Θ = ω ∧ ω ∧ ω ∧ π + ω ∧ ω ∧ ω ∧ π , Θ = − ω ∧ ω ∧ ω ∧ π − ω ∧ ω ∧ ω ∧ π , Θ = − ω ∧ ω ∧ ω ∧ π − ω ∧ ω ∧ ω ∧ π . We let E := span { e , e , e } ⊂ E , where e i = X a =1 e ai v a , i = 1 , , , and define the quantities A := (cid:0) ω ∧ ω ∧ ω (cid:1) ( e , e , e ) , B := (cid:0) ω ∧ ω ∧ ω (cid:1) ( e , e , e ) ,C := (cid:0) ω ∧ ω ∧ ω (cid:1) ( e , e , e ) , D := (cid:0) ω ∧ ω ∧ ω (cid:1) ( e , e , e ) . We then wish to consider the polar space H ( E ) := n v ∈ T p F : ϕ ( v , e , e , e ) = 0 , ∀ ϕ ∈ I o (see Definition A.3 in Appendix A). It follows that v ∈ T p F lies in H ( E ) if and only if Θ ( v , e , e , e ) = − A π ( v ) − B π ( v ) = 0 , Θ ( v , e , e , e ) = − A π ( v ) − B π ( v ) = 0 , Θ ( v , e , e , e ) = D π ( v ) + C π ( v ) = 0 , Θ ( v , e , e , e ) = D π ( v ) + C π ( v ) = 0 . (4.4)Since π , π , π , π are linearly-independent 1-forms on F , it follows that the number oflinearly-independent constraints imposed on a vector v ∈ T p F by equations (4.4) is equal to the J.D.E. GRANT AND J.A. VICKERS rank of the matrix α := − A − B − A − BD C D C . Since det α = 0, it follows that c ≤ 3. Any flag (0) p ⊂ E ⊂ E ⊂ E ⊂ E such that rank α = 3(e.g. A = C = 1, B = D = 0) will give rise to 3 linearly-independent 1-forms, ( π , π , π ), suchthat H ( E ) = (cid:8) v ∈ T p F : π ( v ) = π ( v ) = π ( v ) = 0 (cid:9) . Hence c = 3 for such an integral flag. Corollary 4.2. The exterior differential system with independence condition ( I , Ω ) contains nointegral elements of dimension that pass Cartan’s test.Proof. The codimension of V ( I , Ω ) at any integral element is equal to 4. Any four-dimensionalintegral flag has c = c = c = 0 and c ≤ 3. Therefore c + c + c + c ≤ = 4, so no suchintegral element passes Cartan’s test. (cid:3) Note that the non-maximality of the rank of α is essentially the same algebraic condition thatled to the non-existence of non-characteristic surfaces when we studied the linearisation of theblock-diagonalisation problem in Section 3.4.1. Prolongation. Since the system ( I , Ω ) is not involutive, we cannot directly apply theCartan–K¨ahler theorem. There is a standard technique for dealing with such non-involutive exte-rior differential systems, namely prolongation (see, e.g., [BCG + 91, IL03, Olv95]). In the currentcontext, the (first) prolongation of the system ( I , Ω ) is a Pfaffian system defined on the manifoldof four-dimensional integral elements, V ( I , Ω ), of the system ( I , Ω ). In particular, recall that( x, g, λ bca ) define a local coordinate system on the Grassmannian bundle Gr ( T F ) of four-planesin the tangent bundle of F . Moreover, the space M (1) := V ( I , Ω ) is a thirty-dimensional mani-fold of the form F × R , with the parameters λ bca subject to the symmetry conditions (4.3) ascoordinates in the R direction. (In particular, the conditions imposed by the exterior differentialsystem ( I , Ω ) have already been imposed.) As such M (1) may be viewed as a subspace of thebundle Gr ( T F ). The bundle Gr ( T F ) comes equipped with a natural set of contact forms, andthe Pfaffian system that we consider on M (1) is generated by the restriction of these differentialforms.More explicitly, we now consider the exterior differential system with independence condition,( I (1) , Ω ), on the space M (1) generated by the 1-forms θ ab := ω ab − X c λ abc ω c , a, b = 1 , . . . , a < b, (4.5)where ω ab and ω a now denote the pull-backs to M (1) of the corresponding forms on F , with theindependence condition defined by the 4-form Ω := ω ∧ ω ∧ ω ∧ ω .We now look for four-dimensional integral elements, E ∈ V ( I (1) , Ω ), of this system. The pointis that if U is an open subset of X and f : U → M (1) a local section of the bundle M (1) with theproperty that f ∗ θ ab = 0, f ∗ Ω = 0, then ǫ i := f ∗ ω i define an orthonormal coframe on U thatobeys (3.4). As such, integral manifolds of ( I (1) , Ω ) define solutions of our block-diagonalisationproblem. As a first step in showing the existence of such integral manifolds we show that the system( I (1) , Ω ) on M (1) is involutive. Applying the Cartan–K¨ahler theorem then gives the solution toour block-diagonalisation problem. Our method here follows that of [BCG + § d θ ab ≡ − X c dλ abc ∧ ω c + 12 X c,d T abcd ω c ∧ ω d mod θ , (4.6)where we have defined T abcd := R abcd + X e [ λ abe ( λ edc − λ ecd ) − λ aec λ ebd + λ aed λ ebc ] . LOCK DIAGONALISATION OF FOUR-DIMENSIONAL METRICS 9 The second term in Equation (4.6) implies that there is torsion in the Pfaffian system. We wouldlike to absorb the torsion terms by writing (4.6) in the form d θ ab ≡ − P c π abc ∧ ω c mod θ , where π abc ≡ dλ abc mod ω i and the 1-forms π abc , a, b, c = 1 , . . . , b < c obey symmetry relationsanalogous to (4.3) (e.g. π = π ). However, in the present case, there is an obstruction to theexistence of such 1-forms π bca , which lies in the quantity, T ( x, g, λ ), defined by the relation ω ∧ ω ∧ d θ + ω ∧ ω ∧ d θ + ω ∧ ω ∧ d θ + ω ∧ ω ∧ d θ ≡ − T ( x, g, λ ) ω ∧ ω ∧ ω ∧ ω mod θ . (4.7) T ( x, g, λ ) is then given in terms of the curvature by the expression T ( x, g, λ ) := R ( x, g ) + λ (cid:0) λ − λ (cid:1) + λ (cid:0) λ − λ (cid:1) + λ (cid:0) λ − λ (cid:1) + λ (cid:0) λ − λ (cid:1) . In particular, an explicit calculation (for details, see Appendix B.1) shows that it is possibleto absorb most of the torsion terms in (4.7) and there exist 1-forms, π bca , on M (1) satisfying π bca ≡ dλ bca mod ω i in terms of which equations (4.6) take the form d θ ≡ − X a π a ∧ ω a mod θ ,d θ ≡ − X a π a ∧ ω a mod θ ,d θ ≡ − X a π a ∧ ω a mod θ ,d θ ≡ − X a π a ∧ ω a mod θ ,d θ ≡ − X a π a ∧ ω a + 2 T ω ∧ ω mod θ ,d θ ≡ − X a π a ∧ ω a mod θ . (4.8)Equation (4.7) implies, however, that it is not possible to absorb the remaining torsion by aredefinition of the forms π bca . In particular, it implies that there is essential torsion in the systemcharacterised by the function T . The existence of such essential torsion implies that a necessarycondition for the existence of an integral element E ⊂ T p M (1) based at a point p ∈ M (1) is that p satisfies the compatibility condition T ( p ) = 0. We define the non-singular part of the subspacewhere this condition holds, S (1) := n p ∈ M (1) : T ( p ) = 0 , dT ( p ) = 0 o , which (by the implicit function theorem) is a codimension-one submanifold, i : S → M (1) , of M (1) . Remark . Note that an explicit calculation of dT shows that, given ( x, g ) ∈ F , for generic λ we have dT ( x, g, λ ) = 0.We define the 1-forms e θ ab := i ∗ θ ab , e ω a := i ∗ ω a on S , and consider the Pfaffian system ( e I , e Ω)on S generated by { e θ ab } with independence condition e Ω := i ∗ Ω = e ω ∧ e ω ∧ e ω ∧ e ω . We thenhave the following: Proposition 4.4. There exist -forms, e π bca ∈ Ω ( S ) , for a, b, c = 1 , . . . , with b < c , that satisfy (1) e π bca ≡ i ∗ (cid:0) dλ bca (cid:1) mod e ω i , (2) e π = e π , e π = e π , e π = e π , e π = e π ,with the property that d e θ ab ≡ − X c e π abc ∧ e ω c mod e θ . (4.9) Proof. Taking the pull-back of equations (4.8) to S , and using the fact that T ◦ i = 0, we deducethat the 1-forms e π bca := i ∗ (cid:0) π bca (cid:1) on S have the required properties. (cid:3) Rather than using λ bca as coordinates it will be useful to introduce new coordinates y , . . . , y and z , . . . , z on M (1) defined by y := λ , y := 12 (cid:0) λ − λ (cid:1) , y := λ , y := 12 (cid:0) λ − λ (cid:1) ,y := λ , y := 12 (cid:0) λ − λ (cid:1) , y := λ , y := 12 (cid:0) λ − λ (cid:1) and z := 12 (cid:0) λ + λ (cid:1) , z := 12 (cid:0) λ + λ (cid:1) ,z := 12 (cid:0) λ + λ (cid:1) , z := 12 (cid:0) λ + λ (cid:1) . In terms of these coordinates our constraint equation takes the form T ( x, g, y, z ) = R ( x, g ) + 2 (cid:0) y y − y y + y y − y y (cid:1) = 0 , (4.10)so that the constraint does not depend upon the z coordinates.We now write the structure equations (4.9) in the form d e θ e θ e θ e θ e θ e θ ≡ π ∧ e ω e ω e ω e ω mod e θ . (4.11)Here, the matrix of 1-forms π (which, modulo { e θ , e ω } , is the tableau matrix of ( e I , e Ω ) at x ) is givenby π = − e π e π e π e π e π e π e π e π e π e π e π e π e π e π e π e π e π e π e π e π e π e π e π e π (4.12)In order to simplify notation, we define the 1-forms e π α , α = 1 , . . . , 8, by e π := e π , e π := 12 (cid:16)e π − e π (cid:17) , e π := e π , e π := 12 (cid:16)e π − e π (cid:17) , e π := e π , e π := 12 (cid:16)e π − e π (cid:17) , e π := e π , e π := 12 (cid:16)e π − e π (cid:17) , which have the property that e π α ≡ i ∗ dy α mod e ω a for α = 1 , . . . , 8. We also define the 1-forms f ρ a , a = 1 , . . . , e ρ := 12 (cid:16)e π + e π (cid:17) , e ρ := 12 (cid:16)e π + e π (cid:17) , e ρ := 12 (cid:16)e π + e π (cid:17) , e ρ := 12 (cid:16)e π + e π (cid:17) , LOCK DIAGONALISATION OF FOUR-DIMENSIONAL METRICS 11 which have the property that e ρ a ≡ i ∗ dz a (mod e ω , . . . , e ω ) for a = 1 , . . . , 4. In addition, we define1-forms { e µ a } a =1 and { e ν a } a =1 by (cid:16)e µ , . . . , e µ (cid:17) := (cid:16)e π , e π , e π , e π (cid:17) , (cid:16)e ν , . . . , e ν (cid:17) := (cid:16)e π , e π , e π , e π (cid:17) . In this notation we have π = − e µ e µ e µ e µ e ρ − e π e π − e ρ + e π e π e ρ − e π e π e π − e ρ − e π e π e ρ + e π − e ρ + e π e π e π e ρ + e π e π − e ρ − e π e ν e ν e ν e ν . (4.13)We now note that, since the functions ( x, g, y ) obey equation (4.10) on S (1) , they will not befunctionally independent when pulled back to S . In particular, we require that i ∗ ( dT ) = 0, whichtranslates into the condition that2 (cid:0)e y d e y + e y d e y − e y d e y − e y d e y + e y d e y + e y d e y − e y d e y − e y d e y (cid:1) + X a Φ a e ω a ≡ e θ on S , whereΦ a = i ∗ (cid:18) ∂∂ ω a R (cid:19) + X b he λ b a e R b + e λ b a e R b + e λ b a e R b + e λ b a e R b i , and e y α := i ∗ y α = y α ◦ i , etc, denote the pull-backs to S of the corresponding functions on M (1) .In particular, since e π α ≡ i ∗ dy α mod e ω a for α = 1 , . . . , 8, there exist functions Ψ a on S such that i ∗ ( dT ) = e y e π + e y e π − e y e π − e y e π + e y e π + e y e π − e y e π − e y e π + X a Ψ a e ω a ≡ e θ . (4.14)Recall, however, that the 1-forms, e π bca , are not uniquely determined, and that we may add tothem any linear combination of the 1-forms { e ω a } consistent with equations (4.11) and (4.12). At ageneric point p ∈ S at which e y ( p ) , . . . , e y ( p ) are all non-zero, it is shown in Appendix B.2 that allthe functions Ψ a in equation (4.14) may be absorbed into a redefinition of the 1-forms e π , . . . , e π and e ρ , . . . , e ρ . Noting that the non-vanishing of e y , . . . , e y is an open condition, we deduce thatwe may take the 1-forms e π , . . . , e π to obey the linear-dependence condition e y e π + e y e π − e y e π − e y e π + e y e π + e y e π − e y e π − e y e π ≡ e θ (4.15)on an open neighbourhood, U , of the point p in S . This relationship implies (via the structureequations (4.11)) that the essential torsion of the system ( e I , e Ω ) is zero on the open set U . Itfollows from Proposition A.11 that the system ( e I , e Ω ) is involutive at p if and only if the tableau A p is involutive.To show that this is the case, we need to know the reduced Cartan characters of the tableau A p , and the dimension of the first prolongation, A (1) p , of A p . Proposition 4.5. The first prolongation of the tableau A p is an affine-linear space of dimension .Proof. See Appendix B.2. (cid:3) Proposition 4.6. The system ( e I , e Ω) has reduced Cartan characters s ′ = 6 , s ′ = 6 , s ′ = 5 , s ′ = 2 . Proof. Let p ∈ S with e y ( p ) , . . . , e y ( p ) all non-zero. Equation (4.14) may then be looked on asdefining one of the 1-forms, say e π , in terms of the other seven. Note that, since the thirty1-forms { e ω i , e θ ab , e ρ a , e µ a , e ν a } must span the cotangent space at each point of the twenty-nine-dimensional manifold S , it follows that the linear relation (4.14) is the only relation obeyed bythese 1-forms on S . As such, once we have substituted for e π , say, the remaining differential forms { e π , . . . , e π , e ρ a , e µ a , e ν a } that appear in the matrix π are linearly-independent on S .We then consider the tableau matrix, π := π mod e θ , e ω , and we wish to calculate the reducedCartan characters. This should be computed with respect to a generic basis of 1-forms { ω i } , sowe note that the tableau relative to a different basis, e ω a := P b (cid:0) σ − (cid:1) a b e ω b where σ ∈ GL(4 , R ),is given by π σ := πσ . Substituting for e π into the tableau matrix and noting that this is theonly relationship that our differential forms obey, we see that π then has six linearly-independent1-forms in its first column: e µ , e ρ − e π , e ρ − e π , e π , e π , e ν . Therefore s ′ = 6. The 1-forms in column three: e µ , − e ρ + e π , e π , − e ρ + e π , e π , e ν are then linearly-independent, and independent of those in column one. (In the preceding equation,we substitute for e π using equation (4.15).) Therefore s ′ = 6. If we then consider the linearcombination of α times column two and β times column four of (4.12), we gain the 1-forms α e µ + β e µ , α e π − β ( e ρ + e π ) , α ( e ρ + e π ) + β e π , α ( e ρ + e π ) − β ( e ρ + e π ) , α e ν + β e ν . If we then take α, β both non-zero, this gives five more linearly-independent 1-forms. Therefore s ′ = 5. Finally, s ′ + s ′ + s ′ + s ′ = 19, the number of linearly-independent 1-forms in π , whichfixes s ′ = 2.Note that the above is equivalent to taking σ = ∗ α ∗ ∗ β ∗ , where the last column is only constrained by the requirement that σ be non-singular. (cid:3) Proposition 4.7. The Pfaffian differential system ( e I , e Ω ) is involutive at p .Proof. s ′ + 2 s ′ + 3 s ′ + 4 s ′ = 6 + 12 + 15 + 8 = 41 = dim A (1) p . (cid:3) Theorem 4.8. Let X be an analytic manifold, and g an analytic Riemannian metric on X . Foreach x ∈ X , there exists a neighbourhood of x on which there exists an analytic coordinate systemin terms of which the metric g takes block-diagonal form.Proof. Given any point x ∈ M , choose a generic point p ∈ π − ( x ) ∈ S . By the previous Proposi-tion, the system ( e I , e Ω ) is involutive. Applying the Cartan–K¨ahler theorem (cf. Remark A.12), wededuce that there exists an integral manifold of the exterior differential system with independencecondition ( e I , e Ω) through p . This integral manifold corresponds to a section f : X → S and henceto an orthonormal coframe { ǫ i } on a neighbourhood of x that obeys equation (3.4). (cid:3) Remark . The solution to (3.4) is not unique but one has the freedom to independently makerotations in the ( ǫ , ǫ ) and ( ǫ , ǫ ) planes (equivalently, in the ( t, x ) and ( y, z ) planes of theproof of Proposition 3.1). This corresponds to the freedom to make rotations in the ( ω , ω ) and( ω , ω ) planes without changing ( I , Ω ). As a result the characteristic manifold is parameterisedby two functions of four variables, consistent with the result that s ′ = 2. LOCK DIAGONALISATION OF FOUR-DIMENSIONAL METRICS 13 Remark . The coordinate functions λ bca pull back to define functions on X that give thecomponents, { Γ ab } , of the Levi-Civita connection of the coframe { ǫ a } . The curvature of Γ , R Γ ,then automatically obeys the condition that R Γ + Γ (cid:0) Γ − Γ (cid:1) + Γ (cid:0) Γ − Γ (cid:1) + Γ (cid:0) Γ − Γ (cid:1) + Γ (cid:0) Γ − Γ (cid:1) = 0 . (4.16)In the present context, this condition is derived from pulling back the condition T ( p ) = 0 that wasrequired for our Pfaffian system on S to have integral elements. However, it can also be shownthat this condition arises directly from the symmetry requirements on the Levi-Civita connection(analogous to (4.3)) that follow from imposing (3.4).It turns out that (4.16) has a simple geometrical interpretation. Let R ⊥ denote the curvatureof the connection of the bundle normal to the ǫ ∧ ǫ plane. This is related to the full curvatureand the associated fundamental form A U by the Ricci equation g (cid:0) R ⊥ ( X , Y ) V , U (cid:1) = g ( R ( X , Y ) V , U ) − g ([ A U , A V ] X , Y ) . In the same way one can use the Ricci equation to obtain an expression for the curvature ˜ R ⊥ ofthe connection of the bundle normal to the ǫ ∧ ǫ plane. Then by adding the expressions for thetwo normal curvatures together one may write the curvature condition (4.16) in the alternativeform R ⊥ + ˜ R ⊥ = R . (4.17)So that the full curvature is just the sum of the two normal curvatures.4.2. The Lorentzian case. Although we have carried out all of our calculations for the case ofa Riemannian four-manifold, the calculations carry through, essentially unchanged, if the metrichas Lorentzian signature. We can easily obtain the geometric condition corresponding to (4.17) byusing the Newman–Penrose null formalism (see e.g. [PR87]). We start by introducing a (complex)basis of null 1-forms ( ℓ , n , m , m ). Then in terms of this basis the condition (3.4) that the metriccan be block diagonalised is given by ℓ ∧ n ∧ d ℓ = 0 , ℓ ∧ n ∧ d n = 0 , m ∧ m ∧ d m = 0 , m ∧ m ∧ d m = 0 . From equation (4.13.44) in [PR87], the above conditions result in reality constraints on the spincoefficients given by ρ = ρ, ρ ′ = ρ ′ , τ ′ = τ, τ ′ = τ . (4.18)We now make use of the Newman–Penrose equations (4.11.12) in [PR87] to obtain the equation D ′ ρ − δ ′ τ + Dρ ′ − δτ ′ = 2 ρρ ′ − (cid:0) τ τ + τ ′ τ ′ (cid:1) + ρ ( γ + γ ) + ρ ′ ( γ ′ + γ ′ ) − (cid:0) τ ( α + α ′ ) + τ ( α ′ + α ) (cid:1) − − + κκ ′ − σσ ′ ) . Because of the reality conditions on the spin coefficients (4.18), we see that the imaginary part ofthe left hand side of this equation must vanish. Similarly all the terms but the final one on theright hand side are real and have vanishing imaginary part. It must therefore be the case that thefinal term also has vanishing imaginary part so thatIm (Ψ + κκ ′ − σσ ′ ) = 0 . (4.19)Therefore, our block-diagonalisation condition necessarily implies that this constraint must besatisfied. Note that both (4.18) and (4.19) are invariant under spin and boost transformationswhich reflects the fact that the 2-forms ℓ ∧ n and m ∧ m are invariant under such transformations.To relate this condition to equation (4.17) above we introduce the complex curvature of thesurface spanned by m ∧ m which is given by the formula K = σσ ′ − Ψ − ρρ ′ + Φ + Λ . Twice the real part of this gives the Gaussian curvature while twice the imaginary part gives thecurvature of the connection of the normal bundle, which in view of the reality conditions on thespin coefficients is given by Im K = Im ( σσ ′ − Ψ )The corresponding curvature of the connection of the normal bundle to ℓ ∧ n is obtained byapplying the Sachs ∗ -operation (which has the effect of swapping m ∧ m with ℓ ∧ n ). Under thisoperation we have σ ∗ = − κ, σ ′∗ = κ ′ , Ψ ∗ = Ψ , so that the normal curvature is this time given byIm K ∗ = Im ( − κκ ′ − Ψ )Finally we note that the full curvature for the orthonormal frame corresponding to the Newman–Penrose null tetrad is given by R T XY Z = − . Hence condition (4.17) becomesIm K + Im K ∗ = Im Ψ . Substituting for Im K and Im K ∗ we again obtain equation (4.19). Therefore, the constraintobtained from the Newman–Penrose equations agrees with that obtained from the prolongationprocess.Finally, with reference to Remark 3.2, it should be noted that the constraints (4.16) and (4.19)that have arisen via the prolongation procedure are both preserved under conformal transforma-tions of the metric, g . This is, again, a manifestation of the fact that our problem is actually aproblem in conformal, rather than Riemannian/Lorentzian, geometry.5. Doubly biorthogonal coordinates The problem of diagonalising a metric in 3-dimensions is equivalent to that of finding threefamilies of 2-surfaces f i ( x , x , x ) = c i , i = 1 , , x i ′ = f i ( x , x , x )brings the metric to diagonal form. Darboux [Dar98] (see also Eisenhart [Eis60])) was able tofind all triply orthogonal systems for the flat metric by first giving a condition on two families of2-surfaces that guaranteed the existence of a third family orthogonal to both.Let f ( x, y, z ) = a = constant ,g ( x, y, z ) = b = constantbe two 1-parameter families of 2-surfaces S a and S b . The normal 1-form to S a is df and thenormal 1-form to S b is dg . We require these to be orthogonal so that g ( df, dg ) = 0 . (5.1)We now construct a 1-form ω orthogonal to both S a and S b ω = ⋆ ( df ∧ dg ) . (5.2)In order for there to be a 2-surface mutually orthogonal to both S a and S b we require ω to besurface forming and hence d ω ∧ ω = 0 . (5.3)Substituting for (5.2) into (5.3) gives the condition d ( ⋆ ( df ∧ dg )) ∧ ( df ∧ dg ) = 0 . (5.4)When written out in components (5.4) takes the form ǫ cab ǫ cde { ( ∇ b ∇ d f )( ∇ e g ) + ( ∇ d f )( ∇ b ∇ e g ) } ǫ akl ( ∇ k f )( ∇ l g ) = 0 , LOCK DIAGONALISATION OF FOUR-DIMENSIONAL METRICS 15 which can be simplified to read ǫ abc ∇ b f ∇ c g (cid:2) ( ∇ d g )( ∇ d ∇ a f ) − ( ∇ d f )( ∇ d ∇ a g ) (cid:3) = 0 . (5.5)On the other hand differentiating (5.1) gives( ∇ b ∇ a f )( ∇ b g ) + ( ∇ a f )( ∇ b ∇ a g ) = 0 . (5.6)We can now use (5.6) to replace the second derivatives of g in (5.5) by second derivatives of f toobtain: ǫ abc ( ∇ b f )( ∇ c g )( ∇ d g )( ∇ d ∇ a f ) = 0 . Now since ∇ d g is normal to S b , it is tangent to S a . Hence if we are given some function f thatdefines a family of surfaces S a , any surface S b that intersects it orthogonally with the mutuallyorthogonal direction surface forming, must intersect S a in a line with tangent direction X a thatsatisfies ǫ abc ( ∇ b f ) X c X d ( ∇ d ∇ a f ) = 0 . (5.7)This is just the classical result that the surfaces intersect in lines of curvature [Dar98, Eis60].The significant point about this is that given f we can solve (5.7) to give X a algebraically interms of first and second derivatives of f . Since X a is tangent to both S a and S b it is normal tothe third surface and must satisfy the surface orthogonal condition ǫ abc ( ∇ a X b ) X c = 0 . Substituting for X a we obtain a third-order partial differential equation for f ; the Darboux equa-tion [Dar98], see also Eisenhart [Eis60] for details.We see from the above that the coordinate surface of a triply orthogonal system must satisfyDarboux’s equation. Conversely, given a solution f ( x, y, z ) of the Darboux equation one cancalculate the lines of curvature of the surfaces S a given by f ( x, y, z ) = a , and then find anorthogonal family of surfaces S b which intersects S a orthogonally along these lines. One thenknows that the direction orthogonal to both normals is surface orthogonal and hence one has atriply orthogonal system of surfaces. (Note in practice it is often simpler to perform the last twosteps in the opposite order.) Hence all triply orthogonal surface are determined by solutions tothe third-order Darboux partial differential equation.In the case of ‘doubly biorthogonal’ coordinate systems we proceed in a similar manner. Wefirst ask when there exists a family of two surfaces orthogonal to a given two-parameter family of2-surfaces.Let the given two-parameter family of two surfaces S a,b be given by f ( x, y, z, w ) = a, g ( x, y, z, w ) = b. Since df and dg are both co-normals to S we require ω = ⋆ ( df ∧ dg ) to be surface-orthogonal. Bythe Frobenius theorem this is the condition( ⋆d ω ) ∧ ⋆ ω = 0 , which, in components, takes the form ǫ ijkl ( ∇ j f )( ∇ k g ) { ( ∇ m f )( ∇ m ∇ l g ) − ( ∇ m g )( ∇ m ∇ l f ) } = 0 . (5.8)If one contracts (5.8) with ∇ i f or ∇ i g then the expression vanishes whatever the value of thefinal term. On the other hand if one contracts it with an element µ i that is not in the linearspan of ∇ i f and ∇ i g then Y i = ǫ ijkl µ i ∇ j f ∇ k g is a non-zero vector orthogonal to ∇ i f and ∇ i g .Furthermore any vector Y i orthogonal to ∇ i f and ∇ i g can be obtained in this way by choosing µ i suitably. Hence we require Y i (cid:8) ( ∇ j f )( ∇ j ∇ i g ) − ( ∇ j g )( ∇ j ∇ i f ) (cid:9) = 0 for all Y i such that Y i ∇ i f = Y i ∇ i g = 0 . (5.9)This gives a pair of coupled second-order equations for f and g . Note that, unlike the case oftriply orthogonal systems, g ij ∇ i f ∇ j g = 0 in general since we cannot be expected to diagonaliseone of the 2 × g infavour of derivatives of f as was done in three dimensions. Indeed (5.9) implies (5.8) and hence that ω = ⋆ ( df ∧ dg ) is surface orthogonal. Thus (5.9) is a necessary and sufficient condition forthe existence of a doubly biorthogonal coordinate system. Proposition 5.1. All doubly biorthogonal systems are determined by solutions to the pair ofcoupled second-order partial differential equations Y i (cid:8) ( ∇ j f )( ∇ j ∇ i g ) − ( ∇ j g )( ∇ j ∇ i f ) (cid:9) = 0 for all Y i such that Y i ∇ i f = Y i ∇ i g = 0 . Appendix A. Results from the theory of exterior differential systems We now recall some standard definitions and results from the theory of exterior differentialsystems. For more information, see [BCG + M be an arbitrary smooth manifold of dimension n . Let Ω p ( M )denote the space of C ∞ sections of V p T ∗ M and Ω ∗ ( M ) := L np =0 Ω p ( M ).An exterior differential system , I , on M consists of a two-sided, homogeneous differential ideal, I ⊂ Ω ∗ ( M ). In particular, we have • Given α ∈ I , then α ∧ β ∈ I and β ∧ α ∈ I for all β ∈ Ω ∗ ( M ). • I = L I q where I q := I ∩ Ω q ( M ) and, for any α ∈ I , the part of α ∈ I lying in I q alsolies in I , for q = 0 , . . . , n . • For all α ∈ I we have d α ∈ I .Given a point x ∈ M , a k -dimensional linear subspace E k ⊆ T x M (where k ∈ { , . . . , n } ) is an integral element of ( I , Ω ) (of dimension k ) based at x if ϕ | E k = 0 for all ϕ ∈ I , where α | E k denotes the restriction of a form α to E k . The set of integral elements of I of dimension k isdenoted V k ( I ).An exterior differential system with independence condition , ( I , Ω ), on M consists of an exteriordifferential system I ⊂ Ω ∗ ( M ), and a non-vanishing differential form Ω ∈ Ω p ( M ). Given a point x ∈ M , an p -dimensional linear subspace E p ⊆ T x M is an integral element of ( I , Ω ) based at x if ϕ | E p = 0 for all ϕ ∈ I and Ω | E p = 0. The set of integral elements of ( I , Ω ) is denoted V p ( I , Ω ). Definition A.1. An integral manifold of ( I , Ω ) is an immersed sub-manifold i : N → M with theproperty that i ∗ ϕ = 0, for all ϕ ∈ I , and i ∗ Ω = 0. Equivalently, i ∗ ( T x N ) ⊂ T i ( x ) M should be anintegral element of ( I , Ω ), for each x ∈ N . Definition A.2. An integral flag of ( I , Ω ) based at x is a nested sequence of subspaces (0) x ⊂ E ⊂ E ⊂ · · · ⊂ E p ⊆ T x M , with the properties that • E k is of dimension k , for k = 0 , . . . , p − • E p is an integral element of ( I , Ω ). Definition A.3. Let e , . . . , e k be a basis for E k ⊆ T x M . The polar space of E is the vectorspace H ( E ) = n v ∈ T x M : ϕ ( v , e , . . . , e k ) = 0 for all ϕ ∈ I k +1 (cid:12)(cid:12) x o . Definition A.4. Let (0) x ⊂ E ⊂ E ⊂ · · · ⊂ E p ⊆ T x M be an integral flag of ( I , Ω ) based at x ∈ M . We define the integers { c k : k = − , , . . . , p } as follows: c k = k = − , codim H ( E k ) k = 0 , . . . , p − M − p k = p. We now quote the first half of Theorem 1.11 from Chapter III of [BCG + LOCK DIAGONALISATION OF FOUR-DIMENSIONAL METRICS 17 Proposition A.5. Let ( I , Ω ) be an exterior differential system with independence condition onmanifold M , where I contains no non-zero forms of degree . Let (0) x ⊂ E ⊂ E ⊂ · · · ⊂ E p ⊂ T x M be an integral flag of ( I , Ω ) . Then V p ( I , Ω ) ⊆ Gr p ( T M ) is of codimension at least c + c + · · · + c p − at E p . If there exists a neighbourhood, U of E p in Gr p ( T M ) such that V p ( I , Ω ) ∩ U is a smoothsub-manifold of codimension c + c + · · · + c p − in U at E p , then we say that the integral flag E p passes Cartan’s test .The key result is the following: Theorem A.6 (Cartan–K¨ahler Theorem: [BCG + . Let ( I , Ω ) bean analytic differential ideal on a manifold M . Let E p ⊂ T x M be an integral element of ( I , Ω ) that passes Cartan’s test. Then there exists an integral manifold of ( I , Ω ) through x , the tangentspace to which, at x , is E p . A.1. Linear Pfaffian systems. A Pfaffian system is an exterior differential system with inde-pendence condition, ( I , Ω ), on a manifold M such that I is generated, as a differential ideal, bysections of a sub-bundle I ⊂ T ∗ M . (It is assumed that I is of constant rank, s .) The indepen-dence condition, Ω , may be characterised by a sub-bundle J ⊂ T ∗ M , with I ⊂ J ⊂ T ∗ M andrank J/I = n , in which case Ω corresponds to a non-vanishing section of ∧ n ( J/I ). Such a Pfaffiansystem is linear if dI ≡ J. Locally, we may choose a coframe { θ , . . . , θ s , ω , . . . , ω n , π , . . . , π t } on M such that I x =span ( θ , . . . , θ s ), J x = span ( θ , . . . , θ s , ω , . . . , ω n ). In this case, the condition that the Pfaffiansystem be linear is that there exist functions A aεi , c aij on M such that d θ a ≡ X ε,i A aεi π ε ∧ ω i + 12 X i,j c aij ω i ∧ ω j mod θ . (A.1)Under a change of coframe of the form( θ σ , ω i , π ε ) ( θ σ , ω i , π ε + X i p εi ω i ) , (A.2)the coefficients c aij transform according to the rule c aij c aij + X ε ( A aεi p εj − A aεj p εi ) . We define two collections of coefficients c aij , e c aij to be equivalent if there exists parameters p εi such that e c aij = c aij + P ε ( A aεi p εj − A aεj p εi ), and denote the corresponding equivalence class ofcoefficients by [ c ]. [ c ] is the essential torsion of the linear Pfaffian system ( I , Ω ). If it is possibleto choose the p εi such that e c aij = 0 (i.e. there is no essential torsion) then we say that the torsioncan be absorbed . Given a point x ∈ M , there exists an integral element of ( I , Ω ) based at x if andonly if [ c ] ( x ) = 0.In the terminology of Olver [Olv95, pp. 351], the degree of indeterminacy , r (1) , of the abovecoframe is the number of the number of solutions of the homogeneous problem X ε ( A aεi p εj − A aεj p εi ) = 0 . Equivalently, it is the number of transformations of the form (A.2) that leave the structure equa-tions (A.1) unchanged.If the torsion vanishes on an open neighbourhood, U , of x , then we write (A.1) in the form d θ a ≡ X i π ai ∧ ω i mod θ , (A.3)where π ai ≡ P ε,i A aεi π ε mod { θ , ω } . To determine the involutivity of a torsion-free linear Pfaffian system at x ∈ M , we need toconsider its tableau A x , which is a linear subspace of I ∗ x ⊗ ( J x /I x ). For our purposes, however, itis simpler (but equivalent) to consider the corresponding tableau matrix: Definition A.7. Given a linear Pfaffian system with structure equations as in (A.3) and a point x ∈ M , the tableau matrix at x is the s × n matrix of elements of T ∗ x M/J x given by π x = ( π ai ( x )) mod { θ ( x ) , ω ( x ) } . The reduced Cartan characters, s ′ , . . . , s ′ , of the tableau A x are defined by s ′ + · · · + s ′ k = the number of linearly-independent 1-forms in the first k columns of π x , for a generic choice of the 1-forms { ω i } .In order to check for involutivity of the system ( I , Ω ) at x ∈ M , we need to know the dimensionof the first prolongation, A (1) , of the tableau A x . We do not give a formal definition of A (1) , butcontent ourselves with the following characterisation, which gives us sufficient information tocalculate its dimension: Proposition A.8 ([IL03], Proposition 5.7.1) . Let x ∈ M and π ai ∈ T ∗ x M satisfy d θ a ≡ π ai ∧ ω i mod θ . Then the first prolongation, A (1) , of the tableau A x may be identified with the space of -forms e π ai ≡ π ai mod θ such that d θ a ≡ e π ai ∧ ω i mod θ .Remark A.9 . Proposition A.8 implies that dim A (1) is equal to the degree of indeterminacy, r (1) of the coframe. Therefore, in this notation, a Pfaffian system is involutive if it satisfies s ′ + 2 s ′ + · · · + ns ′ n = r (1) . Proposition A.10 ([BCG + . The first prolongation of the tableau, A x , and thereduced Cartan characters obey the inequality dim A (1) ≤ s ′ + 2 s ′ + · · · + ns ′ n . The tableau, A x , is involutive if equality holds in this equation. Proposition A.11 ([BCG + . The linear Pfaffian system ( I , Ω ) isinvolutive at x ∈ M if and only if (i) [ c ]( x ) = 0 ; (ii) the tableau A is involutive.Remark A.12 . If the system ( I , Ω ) is involutive at x ∈ M , then the Cartan–K¨ahler theoremimplies the existence of an integral manifold of the system ( I , Ω ) through the point x . Appendix B. Absorption formulae B.1. Explicit absorption procedures. The structure equations for the Pfaffian system ( I , Ω )on the manifold M (1) are given in equation (4.6). We can absorb most of the torsion in the original LOCK DIAGONALISATION OF FOUR-DIMENSIONAL METRICS 19 problem by setting π = dλ + T ω + T ω + T ω , π = dλ + T ω + T ω , π = dλ + T ω , π = dλ , π = dλ + T ω + T ω + T ω , π = π = dλ + T ω , π = dλ − T ω + T ω , π = π = dλ , π = dλ + T ω + T ω + T ω , π = π = dλ + T ω , π = dλ − T ω − T ω , π = dλ − T ω + T ω + T ω , π = dλ − T ω + T ω , π = π = dλ + (cid:0) T + T (cid:1) ω , π = dλ − T ω + T ω + T ω , π = dλ − T ω − T ω , π = dλ + T ω + T ω + T ω , π = dλ + T ω + T ω , π = dλ + T ω , π = dλ . The structure equations then take the form given in equation (4.9). Note that the quantityon the left-hand-side of equation (4.7) is invariant under transformations of the form π bca → π bca + δ π bca with δ π bca = P d Π abcd ω d that preserve the required symmetries of the π bca (i.e. π = π ). As such, it follows that, at points of M (1) at which T ( x, g, λ ) = 0, there remainsessential torsion in the system that cannot be absorbed into a redefinition of the 1-forms π bca .B.2. Calculation of degree of indeterminacy. We let X := (cid:0) y , . . . , y (cid:1) ∈ R , with thesplit-signature metric q ( X , X ) := 2 (cid:0) y y − y y + y y − y y (cid:1) . Then our constraint equation (4.10) takes the T ( x, g, X ) := q ( X , X ) + R ( x, g ) = 0 . (B.1)We then need to consider the pull-back to S of the exterior derivative of T , and find that i ∗ ( dT ) = e y e π + e y e π − e y e π − e y e π + e y e π + e y e π − e y e π − e y e π + X a Ψ a e ω a ≡ e θ . (B.2)Note that the 1-forms { e π α , e ρ a , e µ a , e ν a } are not uniquely determined by the structure equations (4.11)and (4.13). In particular, we are free to consider variations of the form e π α e π α + δ e π α , e ρ i e ρ i + δ e ρ i , (B.3a) e µ a e µ a + δ e µ a , e ν a e ν a + δ e ν a (B.3b)with δ e π α , δ e ρ a , δ e µ a , δ e ν a ≡ e ω a , (B.4) as long as they preserve (4.11) and (4.13). We first wish to show that, in the generic case where e y , . . . , e y are all non-zero, we may use such transformations to absorb the P a Ψ a e ω a term in (B.2)into a redefinition of the 1-forms e π α , e ρ i .Firstly, it is straightforward to show that the most general variation that preserves the structureequations (4.11) and (4.12) is of the form (from now on, we drop tildes on all quantities) δ π = α ω + β ω + γ ω + δ ω ,δ π = ǫ ω + ζ ω + δ ω + η ω ,δ π = θ ω + δ ω + ι ω + κ ω ,δ π = δ ω + λ ω + µ ω + ν ω , along with δ π = ξ ω + o ω + 12 ( λ − θ ) ω + 12 ( π − ρ ) ω ,δ π = σ ω + τ ω + 12 ( υ − φ ) ω + 12 ( λ − θ ) ω ,δ π = 12 ( γ − η ) ω + 12 ( υ − π ) ω + χ ω + ψ ω ,δ π = 12 ( φ − ρ ) ω + 12 ( γ − η ) ω + ω ω + Ω ω , and δ ρ = ( ζ − ξ ) ω + ( o + ǫ ) ω + 12 ( λ + θ ) ω + 12 ( π + ρ ) ω ,δ ρ = ( β − σ ) ω + ( τ + α ) + 12 ( υ + φ ) ω + 12 ( λ + θ ) ω ,δ ρ = − 12 ( γ + η ) ω − 12 ( υ + π ) ω − ( χ + ν ) ω − ( φ + µ ) ω ,δ ρ = − 12 ( φ + ρ ) ω − 12 ( γ + η ) ω − ( ω + κ ) ω − ( δ − Ω) ω , where α, . . . , ω and Ω are 25 free parameters. We now wish to find a transformation of theform (B.3a) with the property that y δ π + y δ π − y δ π − y δ π + y δ π + y δ π − y δ π − y δ π = − X a Ψ a ω a . Using the form of δ π α given above, this implies that we need to find vectors Y , . . . , Y of theform Y = (cid:18) α, ξ, ǫ, σ, θ, 12 ( γ − η ) , δ, 12 ( φ − ρ ) (cid:19) , Y = (cid:18) β, o, ζ, τ, δ, 12 ( φ − ρ ) , λ, 12 ( γ − η ) (cid:19) , Y = (cid:18) γ, 12 ( λ − θ ) , δ, 12 ( υ − φ ) , ι, χ, µ, ω (cid:19) , Y = (cid:18) δ, 12 ( π − ρ ) , η, 12 ( λ − θ ) , κ, ψ, ν, Ω (cid:19) , with the property that q ( X , Y i ) = − Ψ i , i = 1 , . . . , . (B.5)In the generic case where y , . . . , y are all non-zero, these equations may be solved for four of thefree parameters in the Y i , and hence will yield the required transformation (B.3a) in terms of theremaining 21 free parameters. Substituting these expressions into δ π α , we therefore generate a21-parameter family of 1-forms π ′ α := π α + δ π α , ρ ′ i := ρ i + δ ρ i in terms of which the constraintequation (B.2) takes the required form y π ′ + y π ′ − y π ′ − y π ′ + y π ′ + y π ′ − y π ′ − y π ′ ≡ θ . (B.6) LOCK DIAGONALISATION OF FOUR-DIMENSIONAL METRICS 21 Finally, based on the preceding calculations, we deduce Proposition 4.5: Proof of Proposition 4.5. Since we are dealing with a linear Pfaffian system, the first prolongationof A p is necessarily an affine-linear space (cf. [BCG + r (1) , the degree of indeterminacy of our coframe. By definition, r (1) is equal to the number of parameters in a change of the 1-forms as in equations (B.3a), (B.3b)and (B.4) that preserve the form of the structure equations (4.11) and (4.12). Setting Ψ a = 0 inthe calculations above, we see that there exists a 21-parameter family of 1-forms, δ e π α , δ e ρ a on S that satisfy these conditions. 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Grant: Fakult¨at f¨ur Mathematik, Universit¨at Wien, Nordbergstrasse 15, 1090 Wien,Austria E-mail address : [email protected] J.A. Vickers: School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK E-mail address ::