Flat deformation of a spacetime admitting two Killing fields

It is shown that given an analytic Lorentzian metric on a 4-manifold, $g$, which admits two Killing vector fields, then it exists a local deformation law $\eta = a g + b H$, where $H$ is a 2-dimensional projector, such that $\eta$ is flat and admits the same Killing vectors. We also characterize the particular case when the projector $H$ coincides with the quotient metric. We apply some of our results to general stationary axisymmetric spacetimes


Introduction
It has been recently proved [1], [2] that given a semi-Riemannian analytic metric g ab on a manifold M, locally there exist two scalars a and b and a 2-dimensional projector H ab -i.e. H ab g bc H cd = H ad and (H ab g ab ) = 2-such that the deformed metric η ab := ag ab + bH ab (1) is flat. We call this formula the deformation law associated with (a, b, H ab ).
The 2-dimensional projector H ab defines an almost-product structure [3], [4] g ab = H ab + K ab and the deformation law (1) differently scales the plane H ab , by a factor ϕ = a + b, and the plane K ab , by a factor a.
We also proved in ref. [2] that in case that g ab admits a Killing field X a , then a deformation law can be found such that η ab also admits X a as a Killing field.
Assume now that g ab admits a wider Lie algebra G of Killing fields, i.e. dimG > 1. Is it possible to find a deformation law such that η ab admits any X ∈ G as a Killing field? Notice that, as η ab is flat, the Lie algebra of its isometries is maximal, i. e. the Poincaré algebra P. Therefore, in order that the answer to the question above is affirmative, it is necessary that G ⊂ P. We thus advance the following conjecture: If g ab admits a Killing algebra G and G ⊂ P, then there exist deformation laws such as (1) such that η ab is flat and any X ∈ G is a Killing field of η ab The problem we shall tackle in the present paper is a little bit simpler: we shall consider an analytic 1 metric g ab admitting two commuting Killing fields and we shall see that a deformation law (1) exists such that η ab is flat and admits the same Killing fields. We shall confine ourselves to the case in which the metric induced on the plane spanned by the Killing vectors is non-degenerate and hyperbolic, that is: timelike orbits (whereas the elliptic case can be dealt in a similar way, the degenerate case is quite different). The paper is structured as follows: in section 2 we present the formalism and prove some intermediate results 2 which we sall apply in section 3 to prove the stated result.
The formalism allows a reformulation of the proof in the quotient 2-manifold, so that a dimensional reduccion occurs. In section 4 we study the specially simple case when the almost-product structure implicit in the deformation law coincides with the almost-product structure induced by the Killing fields and we apply the above results to the case of stationary axisymmetric spacetimes. 1 As the Cauchy-Kovalewski theorem is invoked at some point in the proof, the validity of the results presented here is restricted to the analytic cathegory 2 With a different notation, this formalism was developed in [6] and we present it here in a way suited to our purposes 2 Spacetimes admiting two commuting Killing vectors Let M be a spacetime with a metric η ab admitting two commuting Killing vectors X a A , L X A η ab = 0 , A = 1, 2.
Note that, at this point, η ab does not designate necessarily a flat metric. Through any point x ∈ M there is an integral submanifold V x , i.e. T x V x = span{X a A , A = 1, 2}, which we call the orbit trough x. Let {e a , a = 1 . . . 4} be a base in T x M and {ω b , b = 1 . . . 4} its dual base. We denote by λ AB the metric products: and define that is, ξ A a , A = 1, 2 is the dual base for X a A , A = 1, 2, on T x V x . The metric induced by η ab on the orbit V x is λ AB ξ A a ξ B b = ξ A a ξ A b which, as already mentioned, will be assumed hyperbolic, that is, and that, in a terminology borrowed from principal bundles theory [8], we shall call the vertical metric. Note that, from its definition and the fact that the Killing vectors commute, it is immediate to see that λ AB is preserved by X A , i. e. L X A λ AB = 0.
Let us assume that the set of all orbits of X a A , A = 1, 2, is a 2-manifold, i. e. the quotient manifold S. The tensor projects then vectors in T M onto vectors that are orthogonal to the orbits. Again, in analogy with the principal bundles terminology, vectors that are orthogonal to the orbits will be called horizontal.
There is a one-to-one correspondence [5] between tensor fields T ′ a... b... on S and horizontal tensor fields on M, i. e. those T a... b... fulfilling that is, tensor fields that are horizontal and Lie-constant along X a A . Following Geroch [5] «While it is useful conceptually to have the two-dimensional manifold S, it plays no further logical role in the formalism. We shall hereafter drop the primes: we shall continue to speak of tensor fields being on S, merely as a shorthand way of saying that the field (formally, on M) satisfies (5)» From L X A η ab = 0 and the fact that the Killing vectors commute, it follows that the horizontal metric is preserved by X A , A = 1, 2. By the above mentioned correspondence, it induces a metric on S and, as the vertical metric is hyperbolic, h ab is elliptic. We shall denote the inverse horizontal metric as h ab , one then has h ab := η ab − ξ A a X b A and h ab h bc = h a c .

The Killing equation
As the Killing vectors commute, it follows easily that: where dτ = τ a ω a and a stroke | denotes differentiation. Hence, λ BC and τ are functions on S and λ BC|a , τ b are 1-forms on S.
A further consequence of (8) and the commutativity of the Killing vectors is that L X B ξ A a = 0, whence it follows that As S has only 2 dimensions, dξ A = θ A ǫ, where ǫ is the volume tensor (see Appendix A) and we call the scalar θ A , the twist of X A . Then, including (7), we have that with θ A := λ AB θ B .

2.2
The Levi-Civita connection on the quotient manifold S and the Riemann tensor Given a horizontal tensor T a... b... we define From (105), it follows that for a horizontal vector w b : It can be easily proved that D is a symmetric linear connection on the quotient manifold. Moreover, since D a h bc = 0, it is the Levi-Civita connection for h bc in S.
The Riemann tensor R cdab for the connection D can be derived from the Ricci identities,

and one thus obtains
where Due to the symmetries of the Riemann tensor and the low dimensionality, we also have that and a similar expression for R ⊥ dcab . Hence, (12) implies that To derive the remaining components of R cdab , namely those having some vertical indices, we use that, since X b D is a Killing vector [7], and, after some algebra we obtain: The Riemann tensor for η ab can be reconstructed from these components according to: where Ω ab and ǫ dc are defined in (101) and (102) (see Appendix A) and It is trivial to see that in expression (18) the first Bianchi identity is componentwise satisfied.
Equations, (19-24) are relations between the quotient metric and the covariant kinematical invariants of the Killing fields, on the one hand, and the ambient metric on the other. They must also be taken as equations to solve in the so called 'reconstruction problem' (see next section).

The reconstruction problem
It consists in reconstructing an ambient metric η ab from a given quotient metric h ab provided that η ab admits two Killing vectors X b A , A = 1, 2. It is particularly interesting the case in which the final ambient metric is required to have some prescribed geometric properties, e. g. being flat, which is the case she shall ultimately be interested in.
It is easy to prove that giving a metric η ab on M is equivalent to providing: (a.i) two covectors ξ Aa ∈ Λ 1 M such that L X B ξ Aa = 0 and that λ AB := ξ Aa X a B is a non-degenerate matrix, and (a.ii) the quotient metric on S. (The signatures of both h ab and λ AB must be chosen so that the signature of η ab is (+3, −1).) On their turn these conditions are equivalent to giving: ii) a 2-squared symmetric non-degenerate matrix λ AB ∈ Λ 0 S and (b.iii) the quotient metric on S.

Reconstructing a flat metric with two prescribed Killing vectors
Assume now that we want the ambient metric to be flat. Are there any further restrictions on h ab , ξ A a and λ AB that are derived from the flatness of η ab ? As η ab X a B = ξ Ba and X a A , A = 1, 2, are Killing vectors, the results derived in section 2.1 apply. Therefore, L X B ξ A a = 0, L X B λ AB = 0 and equations similar to (10) do hold. Thus, although λ AB ∈ Λ 0 (S), ξ Aa and ξ A a are not covectors on S because they are not orthogonal to X b B . Let us assume however that we are given two covectors ξ A a ∈ Λ 1 (M) such that ξ Ab X b B = δ A B , A, B = 1, 2 and that L X B ξ A a = 0. Then ξ A a can be written as We shall call κ A a the shift covectors, differentiating and taking (10) into account, we arrive at: Bearing this result in mind, the expressions (19-24) imply a second order partial differential system on the variables λ AB , κ A a i h ab , namely which has to be solved on S and the solutions are to be used as the data (b.i) to (b.iii) necessary to reconstruct η ab .
In Appendix B we prove that equations (27) imply that h ab , ξ A a and λ AB are constrained by the following conditions • The horizontal metric h ab must be either (i) flat or, if not, (ii) its Ricci scalar must satisfy • In case (i), take θ A = 0 and whereλ AB andq AB are constant matrices fulfillingq ABλ AB =q ABλ BCq CD = 0 , and withm 2n1 −m 1n2 = 1 and Φ ± fulfilling (133).
• In case (ii) choose two constants α = 0 and C = 0 and take In both cases we still have to determine ξ A ∈ Λ 1 M. To this aim, we first choose two 1-forms which follows from (10) and is always integrable due to the fact that dim S = 2.

Flat deformation
The central result of the present paper is the following theorem.
Theorem 1 Let g ab be a Lorentzian metric admitting two commuting Killing vector fields, X a A , A = 1, 2. Then, there exist two functions a, b and an elliptic 2-dimensional projector H a b such that the deformed metric is flat and admits X a A , A = 1, 2, as Killing vector fields with vanishing twists.
The proof spreads all over the present section but we first need to review some useful results.
Lemma 1 Let X a be a Killing vector for g ab and let η ab be defined by (32) with b = 0, then: for a proof.

2.
With respect to the Killing vectors, the given metric g ab splits into its horizontal and vertical parts as follows: where h ab is the quotient metric and λ AB ξ A a ξ B b is the metric on the orbits, with On its turn, the sought-after deformed metric η ab may also be split into its horizontal and vertical parts with ξ A a and λ AB defined as in (2). Since X a A are commuting Killing vectors for both g ab and η ab , we have that L X A ξ B a = L X A ξ B a = 0 and, as a consequence, we can introduce the shift covectors As commented above -conditions (b.i) to (b.iii) in subsection 2.3-to determine η ab is equivalent to finding an elliptic horizontal metric h ab , the hyperbolic matrix λ AB , A, B = 1, 2, and two covectors

The unknowns
To relate these objects with the unknowns a, b and H ab in the deformation law (32), we shall take into account that the elliptic 2-dimensional projector and can be written as: with m a m b g ab = n a n b g ab = 1 and m a n b g ab = 0 The covectors m a and n a can then be written in terms of their respective vertical and horizontal components (relatively to the vectors X a B and the metric g ab ): In Appendix C -equations (136), (137) and proposition 5-we prove that the covectors m a and n a may be chosen so that: Also in Appendix C it is shown -equation (143)-that the shift covectors are Following the guidelines advanced in Appendix B to ensure that the components (27) of the Riemann tensor for η ab do vanish we choose 3 We confine ourselves to the generic case xy = 0. The fully degenerate case x = y = 0 is studied in detail in section 4 (a) θ A = 0, A = 1, 2 and, including (37), the exterior derivatives of the shift covectors are θ A and ǫ ab respectively being the twist of ξ A b and the volume tensor for h ab .
Furthermore, using (41) and (140-142) in Appendix C, we obtain gives H AB in terms of Φ ± , a and b.
(c) A flat horizontal metric h ab which, according to equation (150) in Appendix C, is where z 1 and z 2 are given by (146), {μ a ,ν a } is an h-orthonormal base and The Ricci scalar is Finally, from equation (144) in Appendix C, we have that

The equations
Equation (44) Let us first examine the equations (42) which, on account of (41), can be takien as differential equations on µ a and ν a . Since {μ a ,ν a } is an h-orthonormal base of T S, we can write where D is the Levi-Civita connection for h ab ; therefore Let {α 3 a ,α 4 a } be a given h-orthonormal base and let {ê a 3 ,ê a 4 } be the dual base. We then have that it exists ψ ∈ Λ 0 S, such that From this and (50) we readily obtain that (ψ a + ω a )ν b +μ c γ c ab = 0 and write: where γ c a b are the connection coefficients for D. Then, combining (41), (42),(47) and (51), we arrive at and, writing ω = Ω 1μ + Ω 2ν , we finally obtain where withμ b := h bcμ c and so on.
Equations (54) can be solved providing expressions for Ω 1 and Ω 2 in terms of a, b, t and f : where (139) has been included. In their turn, the components of ω a in the given orthonormal base are: Once ω a is known, we may substitute it in equations (53) which then become a partial differential system on ψ a . As too many derivatives of the unknowns are specified, an integrability condition is Now, taking (56) and (57) into account, the above is a second order partial differential equation on the unknowns a, b, Φ ± and ψ.
We are thus led to solving the partial differential system constituted by equations (30), (48), (53), (58) and the flatness of h ab : The number of equations exceeding by far the number of unknowns, we shall deal much in the same way as it is usually done with Einstein equations: considering a certain subset of distinguished equations as the reduced PDS, and treating the remaining equations as constraints; the existence of solutions will then be studied in terms of a Cauchy problem.
We choose a hypersurface Σ in S which will act as a Cauchy hypersurface for the partial differential system (Σ is actually a curve because S has two dimensions), and take Gaussian h-normal coordinates (y 3 , y 4 ) in a neighbourhood of Σ, so that y 4 = 0 on Σ and Thus, the above mentioned h-orthonormal may be taken to bê We can now consider (59) as a system of differential equations in the five unknowns: Φ ± , a, b and ψ, and separate: the reduced system, namely the constraints: Proof: By differentiating (30) and (53), we easily obtain (recall that we have chosen q = 1) and, particularly, since we are dealing with a solution of the reduced system (61), we obtain for the constraints: which is a linear, homogeneous, partial differential system to be fulfilled by the constraints, whence it follows that the vanishing of the constraints on Σ propagates to an open neighbourhood of Σ.

The reduced system. Characteristic determinant
To decide whether Σ is a non-characteristic hypersurface for the reduced system (61) we must study its characteristic determinant [13]. To this end we must consider only the principal part of its equations, i. e. the part containing the highest order derivatives of the unknowns. Particularly, and ∂ 4 ψ, and we easily obtain that: where ∼ = means "equal apart from non-principal terms".
The principal parts of the remaining two equations are not so simple; they look like It easily follows that the characteristic determinant of the reduced system is and we do not need to calculate explicitly all the coefficients in the principal part of W and R. A detailed, heavy-going calculation yields where f (a) := 1 − a(z 1 + z 2 ) + a 2 z 1 z 2 and, according with (146), Then, in order that Σ is a non-characteristic hypersurface, Cauchy data must be chosen so that

The constraints
The Cauchy data, namely Φ ± ,Φ ± := ∂ 4 Φ ± , a, b,ȧ,ḃ and ψ on Σ, must be chosen so that χ = 0 and the constraints (62) are fulfilled. Σ is a curve and the coordinate u := y 3 acts as a curve parameter; the constraints can thus be written as (c = 3, 4), and we must replace D 3 Φ ± by dΦ ± du , D 4 Φ ± byΦ ± and so on. Γ c ed and Γ c ed are the Christoffel symbols for the connections D and D, respectively, and they depend on Φ ± , a, b, their first order derivatives and ψ.
We can therefore prescribe arbitrary values for a,ȧ, b andḃ on Σ, because there is no constraint on them, and then substitute them into (67) which can be taken as an ordinary differential system on the remaining Cauchy data: Φ ± ,Φ ± and ψ on the curve Σ. This system admits a solution for any given initial data Φ ± (x 0 ), D 3 Φ ± (x 0 ) and ψ(x 0 ), for a given point x 0 ∈ Σ.
As for the remaining constraint,

Summary of the proof
We now show how to construct the deformed metric η ab from a solution to the above Cauchy problem.

A particularly simple case
We shall now consider the fully degenerate case x = y = 0, which implies that H ab is a horizontal tensor and, as it is a 2-dimensional projector in a 2-dimensional space, H ab = h ab . Therefore the original and deformed metrics respectively are g ab = h ab + k ab and η ab = ϕh ab + ak ab with k ab := λ AB ξ A a ξ B b and ϕ := a + b. In this case, which we shall refer to as a degenerate deformation law, the 2-planes of the almost-product structure g ab = H ab + K ab which, by a biconformal deformation yields the flat metric η ab , coincide with the almost-product structure associated to the orbits of the Killing fields. This is indeed a non-generic case: a metric g ab with two commuting Killing vectors does not, in general, admit a degenerate deformation law yielding a flat η ab . We shall here characterize the metrics g ab admitting a degenerate deformation.
From (68) we have that whence it follows that where the fact that ǫ ab = ϕǫ ab has been included.
Since η ab is flat and has two commuting Killing vectors (see Appendix B), only two possibilities are left: (a) θ 1 = θ 2 = 0 which, by (70), implies that θ 1 = θ 2 = 0 and Notice thatλ AB := τ −1 λ AB √ 2 = τ −1 λ AB √ 2 =λ AB and, as thisλ AB corresponds to the metric η ab which is flat, the results derived in Appendix B apply. Particularly from (115) we have that where q AB (f ) is derived from dλ AB as indicated in proposition 3.

Case (a): From proposition 3 in Appendix B and equations (21) and (71), we have that
which is a necessary condition to be fulfilled by g ab in order to admit a degenerate deformation law.
Thus we must first check whether R 3 = 0 and then take q = 0 if det(q AB ) = 0 or q = sign (det(q AB )) otherwise.
(a.1) If q = 0 we have that [equation (29)]λ AB =λ AB =λ AB +q AB F , withλ AB andq AB constant, λ ABq AB = 0 and det(λ AB = −1. Hence, from dλ AB =q AB dF it is immediate to determinê q AB and dF (appart form a constant factor). Now, by (29) we also have that τ = τ 0 constant and D b F c = 0, which leads to where ψ := log ϕ and the relation between both connections, D and D has been included.
On the one hand, the second equation implies that which is a constraint on F and, on the other, it allows to obtain that is, ψ = log F 2 + constant.
Combining now this equation with (19) and including that h ab = e ψ h ab , we arrive at [11] which is a further necessary condition connecting F and R.
(a.2) If q = −1 [see Appendix B, right after (123)], then λ AB must be constant and this implies that a ∈ Λ 0 S must exist such that aλ AB = λ AB = constant. In this case equation (19) becomes a condition on the conformal factor ϕ = e ψ , namely [11] R + h bc D bc ψ = 0 wherem A ,n A are constant andm 2n1 −m 1n2 = 1. This is a necessary condition to be fulfilled by λ AB which will ensure that (24) is satisfied and will allow to derive f ,m A andn A .
The functions f and t = log τ must fulfill (126) and R 2 = 0, which respectively amount to: Furthermore, the condition (19) implies that (see [11] Since f is known, equations (79) and (83) allow to determine On its turn, equation (82)  (D a f a ) and If the twists θ A do not vanish we are compelled to try with case (b) and (70) imposes a first retriction, namely, a couple of constants (k 1 , k 2 ) = (0, 0) must exist such that Furthermore, from proposition 2 in Appendix B we have that λ ∈ Λ 0 S andλ AB constant must exist such that λ AB = k A k B λ +λ AB . Taking in consideration (69), this is equivalent to [as λ AB is non-degenerated, (p 1 , p 2 ) = (0, 0)], which in turn is equivalent to If this happens, the factor a ∈ Λ 0 S is Now, from (108) and (70) we have that This factor must furthermore fulfill the additional conditions implied by (112) and (113) and where ψ := log ϕ, τ = aτ and τ 2 := τ b τ c h bc .
Summarizing, if the twists θ A do not vanish we must first check whether (88)

Example: stationary axisymmetric spacetimes
We now consider the case of a stationary axisymmetric spacetime [12] whose line element is where N , U and K are arbitrary functions of ρ and z. The Killing vectors are X 1 = ∂ φ and X 2 = ∂ t and the associated 1-forms are Therefore we have that h bc = e −2U +2K δ bc and the determinant is τ = √ 2 ρ and the inverse matrix is It can be easily checked that where L(N ) is an arbitrary function of the variable N .
If Q = 0, by conveniently choosing the sign of φ we get L ′ = 1 or L = N + C with C = constant.
Then, the case q = 0 in Appendix B applies and, from equation (125), we have that and F = −L −1 . The results for the case (a.1) in section 4 also apply and we have that Besides, we must take in consideration that h bc := e ψ h bc = e ψ−2U +2K δ bc is flat, which is equivalent to [11] δ bc ∂ bc (4U + log H) = 0 (100) Summarizing, a degenerated deformation law exists that transforms the stationary axisymmetric metric (94) into a flat metric iff: (i) R 3 = 0, (ii) a constant C exists such that ρe −2U = N + C, (iii) λ AB /ρ has the form (98) and (iv) U simultaneously fulfills (99) and (100). In such a case, the biconformal factors are a = τ 0 /τ and ϕ = e ψ with ψ given by (99).
and ǫ ab ǫ cb = h a c . It is obvious that ǫ cb is horizontal and Lie-constant, hence ǫ cb ∈ Λ 2 S.
The dual bivectors respectively are: Furthermore, if w a is a vector on S, then If w b is a vector field on S, from [X A , w] = 0 it follows that and also, where ∇ A stands for X a A ∇ a . Now, using the identity: d(log |detλ AB |) = dλ AB λ AB , from (3) we have

Appendix B
Our goal here is to see how equations (27), namely constrain the possible values of λ AB , κ A a and h ab , In this case one also has θτ = C and − τ 2 2 = αλ +δ (108) with C, α andδ constant.
Proof: Indeed, by (22) P Ab = 0 implies 2dθ A + θ C λ CT dλ T A = 0. Multiplying it by θ A ′ , A ′ = A, and using that Q T b = 0 amounts to θ 1 dλ T 2 = θ 2 dλ T 1 , one readily obtains that θ 1 dθ 2 = θ 2 dθ 1 , which implies (a) either θ 1 = θ 2 = 0 or Furthermore, substituting this into θ 1 dλ A2 = θ 2 dλ A1 and taking the symmetry of λ AB into account, we obtain that dλ AB = k A k B dλ, with λ ∈ Λ 0 S, and therefore The inverse matrix λ AB is and we also have that Substituting these into P Ab = 0, we obtain 2dθ + θk C λ CT k T dλ = 0, which implies that d θ |det(λ AB )| 1/2 = 0 or θτ = C , constant, where (3) has been included. 2 Let us now study the implications of the remaining curvature equations, R 1 = R 2 = R 3 = 0 and P DcAb = 0. Consider first the case (b): θ A = k A θ and dλ AB = k A k B dλ. Equations R 2 = 0 and R 3 = 0 are identically satisfied and do not imply any further condition on θ, τ or k A . Then, taking (110) and (108) into account, equation R 1 = 0 implies that and equation P DcAb = 0 amounts to This is a partial differential system that is integrable provided that the Ricci scalar is (112). Combining now equations (112) and (113) we arrive at which is a condition to be fulfilled by R, the Ricci scalar of the given metric h ab on S, in order that the ambient flat metric η ab exists. Proof: 1E|cλ2T |b , whereλ ABλ BC = δ A C , and therefore, A short calculation then proves that this is equivalent to the existence of F ∈ Λ 1 S such that dλ AB ∝ F . Now, since dimS = 2, F is integrable, i.e. proportional to du for some u ∈ Λ 0 S, whence it follows that dλ AB = Q AB du for some Q AB ∈ Λ 0 S and the integrability conditions imply that Then (115) follows taking f = u and q AB = Q AB , if Q(u) := det(Q AB ) = 0, or taking df = |Q| du Notice that neither f nor q AB vanish except in the trivial caseλ AB =constant.
q AB is a symmetric, traceless, 2-square matrix of functions on S. Since the number of dimensions is 2, using the characteristic polynomial we have that Consider now the following quadratic form on the space of symmetric 2-square matrices: It can be easily seen that it is non-degenerate and has signature(+ + −). We can then complete a base {λ AB , q AB , w AB } in this space of symmetric matrices such that and, besides, det(q AB ) = − det(w AB ) = q and det(λ AB ) = −1.
{λ AB , q AB , w AB } is thus a rigid base for the quadratic form (117): an orthogonal base in the case q = 0 and a base containing two conjugate null vectors in the case q = 0. In all instances, w AB is thoroughly determined byλ AB and q AB . We thus have the following differential equations [the first one comes from (115)]: where q = ±1 or 0 and u(f ) and v(f ).
As the quadratic form (117) can be associated with a non-degenerate metric product in the 3-space of symmetric 2-square matrices, these equations can be seen as a sort of "Frênet-Serret equations".
Using this, equations (122) read: whereλ AB andq AB are constant matrices satifying (118). Now, equation (126) is a partial differential system where all the derivatives of the unknowns t and f are specified. The subsequent integrability conditions do not imply any new condition.
Moreover, equation (123) implies a further restriction which is compatible with (126); indeed, D a R 2 + t a R 2 = 0, and provided that R 2 vanishes at x 0 ∈ S it vanishes in some open neighbourhood of x 0 .
We now introduce the new variables Φ ± := e (t±f )/2 and equations (126) and (129) become From (128) it then follows immediately that With a little of algebra it can be seen that, as a consequence of (118), there existm A andn A such thatm 2n1 −m 1n2 = 1 and that Therefore, (131) finally yields with Φ ± fulfilling (130)

Summary: How to proceed? Guidelines
We start from a given Riemannian metric h ab on S.   (111)] and take In both cases the covectors ξ A ∈ Λ 1 M can be determined as indicated in subsection 2.4, i. e. by solving equations (25) and (26).

Appendix C
Proposition 4 The covectors m a and n a in the expression (38) for the 2-dimensional elliptic projector H ab can be chosen so that L X A m a = L X B n a = 0 Indeed, let m ′ a and n ′ b be a couple of covectors fulfilling (38). As they are determined except for a rotation ζ ∈ Λ 0 M, the couple of covectors m a = m ′ a cos ζ + n ′ a sin ζ and n a = −m ′ a sin ζ + n ′ a cos ζ (134) also fulfill (38). Since L X A H ab = 0, there exist γ A ∈ Λ 0 M, A = 1, 2, such that L X A m ′ a = γ A n ′ a and L X B n ′ a = −γ B m ′ a . We then choose for ζ any solution of From the definitions of m A and n A , and the fact that both Killing vectors commute, it follows immediately that L X A m B = L X A n B = 0, A, B = 1, 2.
We shall thus take and it is then easy to prove that L X B µ a = L X B ν a = 0.
There is still left the residual freedom to rotate an angleζ such that X Aζ = 0, i. e.ζ ∈ Λ 0 S, which can be used to show that Proposition 5 The covectors m a and n a in (38) can be chosen so that λ AB m A n B = 0, where m A := m a X a A and n B := n a X a B .
We have thus proved that two g-orthonormal covectors m a and n a can be found such that H ab = m a m b + n a n b , with L X A m a = L X A n a = 0 and λ AB m A n B = 0 As a consequence, the covectors µ a and ν a in (135) fulfill where x := λ AB m A m B and y := λ AB n A n B . As λ AB is hyperbolic and h ab is elliptic, m a and n a can be chosen so that x < 0 < y < 1. Furthermore we have that 5 Then it easily follows that (m 1 n 2 − m 2 n 1 ) 2 = xy det(λ AB ) and, appropriately choosing the signs of m A and n B , we have that m 1 n 2 − m 2 n 1 = −τ −2xy and m 1 n 2 − m 2 n 1 = τ τ 2 −2xy (139) Substituting now (135) into (38), we obtain the splitting of H ab in its vertical, horizontal and cross components: where H AB := H ab X a A X b B = m A m B + n A n B . From equations (32) and (140) it readily follows and, taking (138) and (140) into account, we have that the vertical metric is the inverse of which is λ AB = − 2 τ 2 σ AC σ BD λ CD where σ AB = −σ BA and σ 12 = −1. Moreover, since λ AB is hyperbolic and a, a + b and y are positive, it must be a/x + b < 0, which implies that −a/b < x.
It follows immediately from (141) that the shift covector (37) is Comparing the expression (142) for λ AB with equation (43) we have that it exists ζ ∈ Λ 0 S such that As a consequence, z 1 = (a + bx) −1 and z 2 = (a + by) −1 are the solutions to the equation z 2 − λ AB λ AB z + τ 2 /τ 2 = 0. Therefore, assuming b > 0, we have that z 1 > z 2 and where λ AB λ AB has to be understood as [see equations (131) and (142)] Hence, equations (146) permit to derive as functions of a, b and Φ ± .
An expression of ζ in terms of these variables is also obtained from (145): Substituting then (140), (142), (143) and (32) into (36), we arrive at where it has been used that λ AB m A m B = x a + bx , λ AB n A n B = y a + by and λ AB m A n B = 0 Now, according to (137), there exists an h-orthonormal base {μ a ,ν a } of T S such that µ a = µ a √ 1 − x and ν a =ν a √ 1 − y which, substituted into equation (149), yields with z 1 and z 2 given by (146).