Weyl metrics and Wiener-Hopf factorization

We consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics in four space-time dimensions. We present, for the first time, a rigorous proof of the remarkable fact that the canonical Wiener-Hopf factorization of a matrix obtained from a general (possibly unbounded) monodromy matrix, with respect to an appropriately chosen contour, yields a solution to the non-linear gravitational field equations. This holds regardless of whether the dimensionally reduced metric in two dimensions has Minkowski or Euclidean signature. We show moreover that, by taking advantage of a certain degree of freedom in the choice of the contour, the same monodromy matrix generally yields various distinct solutions to the field equations. Our proof, which fills various gaps in the existing literature, is based on the solution of a second Riemann-Hilbert problem and highlights the deep role of the spectral curve, the normalization condition in the factorization and the choice of the contour. This approach allows for the explicit construction of solutions, including new ones, to the non-linear gravitational field equations, using simple complex analytic results. As an illustration, we show that by factorizing a simple rational diagonal matrix, we explicitly obtain a class of Weyl metrics which includes, in particular, the solution describing the interior region of the Schwarzschild black hole, a cosmological Kasner solution and the Rindler metric.


Introduction
The field equations of gravitational theories in D space-time dimensions are a system of non-linear PDE's for the space-time metric which are, in general, very difficult to solve. Exact solutions can however be found under simplifying assumptions, for instance spherical symmetry. One such exact solution, of significant physical and mathematical interest, is the well-known Schwarzschild solution which, in D = 4 dimensions in coordinates (t, r, θ, φ), takes the form By restricting to the subspace of solutions that only depend on two of the D space-time coordinates, various approaches to solving the field equations become available. One such approach is the Riemann-Hilbert approach, which can be applied to gravitational theories that satisfy certain requirements, see for instance [1][2][3][4][5] and references therein.
When reducing to two dimensions, there are two distinct cases to consider, which we will distinguish through a parameter σ: either all the directions over which one reduces are space-like (σ = −1), or one of them is time-like and the others are space-like (σ = 1).
In this paper, for the first time, a unified Riemann-Hilbert approach to both cases is presented. Namely, we give a rigorous proof that the canonical Wiener-Hopf factorization (defined in Section 2) of a general monodromy matrix M, with respect to an appropriately chosen contour Γ in the complex plane, always yields exact solutions to the gravitational field equations, for σ = ±1. Note that a Wiener-Hopf factorization must be defined with respect to a given contour [6][7][8].
This proof, which is based on the formulation and solution of another associated Riemann-Hilbert problem, highlights the power of the Riemann-Hilbert approach and clarifies the roles played by the so-called spectral curve and the properties of the Wiener-Hopf factorization. In this way we revisit and generalize the results obtained in [1][2][3][4], also filling a few gaps in the presentation given there.
As a result one finds, in particular, that from rational n × n monodromy matrices M, whose canonical Wiener-Hopf factorization can be constructed explicitly and in a computationally simple manner, one obtains explicit exact solutions that would be very difficult to obtain through other approaches. This is the case of the novel solutions presented in [5], whose construction, based on the Riemann-Hilbert approach of [4], is hereby rigorously justified.
We also show in this paper that by taking advantage of the possible choices of factorization contours Γ and appropriate changes of coordinates, each monodromy matrix gives rise not to one solution, but to a whole class of exact solutions. We illustrate this surprising result by showing that from the canonical Wiener-Hopf factorization of a rational diagonal matrix of a very simple kind that is easily factorizable, one obtains a wide class of metrics that includes all type A space-time metrics, a cosmological Kasner solution and the Rindler metric, as well as solutions whose metric tensor is continuous but not smooth due to the presence of null hypersurfaces [9][10][11][12]. This wide class includes the solutions that describe the exterior and the interior regions of the Schwarzschild black hole. In doing so, we moreover put in evidence the essential difference between the case σ = 1 and σ = −1. Previous recent work that addresses this difference includes [13,14]. This paper is organized as follows. We chose to keep the introduction as brief as possible, supplementing it by a summary of the main results in Section 2. In Section 3 various preliminary results are presented that will be subsequently used. In Section 4 we revisit the Breitenlohner-Maison linear system and the integrability of the field equations. In Section 5 we introduce and discuss monodromy matrices. In Section 6 we prove the main theorem of the paper, which states that the canonical Wiener-Hopf factorization of a monodromy matrix always yields exact solutions, in both cases σ = ±1. In Section 7 we discuss the possibility to obtain such solutions from other types of matrix factorizations, allowing for poles in the factors. In Section 8 we present a thorough study of the exact solutions that can be obtained from the canonical factorization, for σ = ±1, of a very simple rational diagonal monodromy matrix, given by (2.19) with = 1. We call this monodromy matrix the Schwarzschild monodromy matrix. We show that by using the different allowed choices of the factorization contour, one can construct a wide class of metrics, including for instance the solution that describes the interior region of the Schwarzschild black hole. In Section 9 we discuss the canonical factorization of the monodromy matrix (2.19) with = 0. The resulting solutions include the Rindler metric and a cosmological Kasner solution. In Appendix A we discuss jumps of the transverse extrinsic curvature that arise in the presence of null hypersurfaces in space-time across which the metric tensor is only continuous. In Appendix B we summarize the class of A-metrics, which here is obtained from the canonical factorization of the monodromy matrix (2.19).

Summary of the main results
To be able to use a Wiener-Hopf factorization to obtain solutions of the field equations of gravitational theories in D space-time dimensions, we consider gravitational theories that satisfy certain requirements, as follows. Firstly, we restrict to gravitational theories in the absence of a cosmological constant, that is, gravitational theories that have Minkowski space-time as a vacuum solution. Secondly, we focus on the subspace of solutions possessing sufficiently many commuting isometries so that the theory can be dimensionally reduced, first to three dimensions, and subsequently to two dimensions. Thirdly, we take the dimensionally reduced theory in three dimensions to be described by a scalar sigmamodel coupled to three-dimensional gravity, such that the target space of the sigma-model is a symmetric space G/H. This target space is endowed with an involution , associated with an involutive Lie algebra automorphism of the Lie algebra of G. This involution, also called 'generalized transposition', leaves the coset representative M ∈ G/H invariant, i.e M = M . For instance, for H ⊂ O(n), acts by matrix transposition, while for H ⊂ U (n) it acts by Hermitian conjugation. The involution acts anti-homomorphically on g ∈ G, i.e. (g 1 g 2 ) = g 2 g 1 (see [4] for a review).
The problem of solving the gravitational field equations in D dimensions is then reduced to solving a system of non-linear second order PDE's depending on two coordinates, which we denote by (ρ, v) ∈ R 2 , with ρ > 0 and v ∈ R. These coordinates are called Weyl coordinates, and accordingly we will denote the (ρ, v) upper-half plane by Weyl upper-half plane. The system of non-linear second order PDE's to be solved is given by [15,16] where A denotes the matrix one-form Here, denotes the Hodge star operator in two dimensions, which maps a one-form to a one-form, and satisfies We recall that σ = ±1 distinguishes between the case where both ρ and v are space-like coordinates (σ = 1), and the case where one of them is a time-like and the other one is space-like (σ = −1). Given a solution of (2.1), one then obtains a solution to the gravitational field equations. Note that, if M is a diagonal matrix, M −1 also provides a solution to the field equations. As an illustration, let us consider solutions to the field equations of General Relativity in four dimensions that possess two commuting isometries. Accordingly, we take the space-time metric to have the Weyl-Lewis-Papapetrou form with ∆ > 0, where ds 2 2 denotes a flat, two-dimensional line element, which we take to be either ∆, B and ψ are functions of the Weyl coordinates (ρ, v) only. By dimensionally reducing over (y, φ) to two dimensions, the functions ∆ and B become encoded in the matrix M , which in this case is a two-by-two matrix. Thus, given a solution M of (2.1), one immediately reads off the expressions for ∆ and B. The function ψ in (2.4) is determined by integration [15,16], Therefore, the central question is: how do we determine a solution M (ρ, v) of (2.1)?
To do so, we proceed in three steps.
Step 1: We use the fact that the non-linear PDE's (2.1) form an integrable system, i.e. they are the solvability conditions [17] for a certain Lax pair, the so-called Breitenlohner-Maison (BM) linear system [1]. This is an auxiliary linear system in two dimensions given by specified in terms of the matrix one-form A = M −1 dM and a function ϕ belonging to where (ρ 0 , v 0 ) is a point in the neighbourhood of which the linear system is to be solved. If the BM linear system has a non-trivial solution satisfying certain invertibility and differentiability conditions, then A = M −1 dM satisfies (2.1). Note that ± √ −σ are the fixed points of the involution ι σ in C\{0}, which will play a fundamental role.
Step 2: If we take ϕ, ϕ ∈ T , and if X and X denote solutions to the corresponding BM linear system, then we can prove that the matrix Thus, the matrix M is independent of the Weyl coordinates, even though all the individual factors depend on (ρ, v). The expression (2.11) shows that M can be constructed by a product involving solutions to the BM linear system and a matrix M that solves the field equations (2.1).
In what follows, we take M to be -invariant, i.e. M = M.
Step 3: Now we turn to the reverse question: given a -invariant matrix M that is independent of the Weyl coordinates, can we construct a factorization of the form (2.11), such that the middle factor M (ρ, v) solves the field equations, and the factor X provides a solution to the BM linear system? If this can be done, we say that M is a monodromy matrix for M (ρ, v). This is where Wiener-Hopf factorization comes into play. We will have to make certain (though very general) assumptions, as follows.
Assumption 1: Let M(u) be an -invariant, invertible matrix function of the complex variable u, and consider the matrix that is obtained by composition with i.e. (2.14) For each (ρ, v), the relation (2.13) is an algebraic curve in the complex variables u and τ , called the spectral curve. It plays a fundamental role in this study. Note also that if we replace τ by any ϕ ∈ T , the relation (2.13) is satisfied for u = ω. In (2.14), we consider (ρ, v) as an arbitrary pair of parameters in a neighbourhood of (ρ 0 , v 0 ), and take τ as the the (complex) independent variable. To emphasize this aspect, we will denote the matrix M in (2.14) by M (ρ,v) (τ ).

Assumption 2:
There exists an open set S in the Weyl upper-half plane such that, for every (ρ 0 , v 0 ) ∈ S, one can find a simple closed curve Γ in the τ -plane, which is ι σ -invariant and encircles the origin, such that: open set O in the τ -plane, containing Γ (see Figure 1). b) M (ρ,v) (τ ) admits a canonical Wiener-Hopf factorization with respect to Γ, ) and its inverse are analytic and bounded in the open set D + ∪ O (respectively D − ∪ O). Here, D + denotes the simply connected interior region of Γ, and Consider an n × n matrix function M(τ ), τ ∈ Γ, such that both M and M −1 are continuous on Γ. A representation of M(τ ), τ ∈ Γ, as a product where d is a diagonal matrix of the form d(τ ) = diag(τ k j ) j=1,2,...,n with k j ∈ Z, and where M ±1 + (respectively M ±1 − ) admit bounded analytic extensions to D + (respectively D − ), is called a (bounded) Wiener-Hopf factorization. When d = I, this factorization is called canonical. The latter, if it exists, is unique, up to a constant matrix factor which can be fixed by imposing a normalization condition, such as M + (0) = I [7,18,19]. As shown in [4], (2.15) can be written as Under these assumptions, we have the following main theorem: The proof of this theorem gives, moreover, an affirmative answer to the question raised in Step 3.
By using this result, we study the solutions to the field equations of General Relativity in four dimensions that arise from the canonical Wiener-Hopf factorization of a particular type of matrices. These monodromy matrices are chosen to be of the simplest possible, non-constant kind, namely, they are diagonal, rational, with det M = 1, possessing only one zero and one pole in the extended u-plane, as follows, Note that the matrix M (ρ,v) (τ ), obtained from (2.19) by composition with (2.13), always admits a canonical Wiener-Hopf factorization [5]. From that factorization we obtain, by choosing different possible factorization contours, a wide class of solutions that includes all type A space-time metrics, a cosmological Kasner solution and the Rindler metric, as well as solutions whose metric tensor is continuous but not smooth due to the presence of null hypersurfaces [9][10][11][12].
Although the matrices M (ρ,v) (τ ) in the cases σ = 1 and σ = −1 are very similar, the behaviour of the corresponding factors M (ρ, v) is remarkably different, due to the presence of square roots that may vanish along certain lines in the (ρ, v) upper-half plane when σ = −1. Such lines are absent when σ = 1. Obtaining the A-metrics for σ = −1 also involves extending the real-valued solutions by affine transformations in the (ρ, v) upperhalf plane, followed by appropriate changes of coordinates.
So far, our discussion focussed on canonical factorizations of monodromy matrices. What about other types of factorizations? Do other types of matrix factorizations, where different analyticity and normalization conditions are imposed on the factors (allowing them to be meromorphic, for instance), also yield solutions to the field equations, at least in certain cases? To address this question, we consider the example of a cosmological Kasner solution which, when expressed in terms of Weyl coordinates, belongs to the class σ = −1. By solving the BM linear system for this solution, we construct an infinite set of monodromy matrices. We then pick one of them, and we show that this particular monodromy matrix possesses a meromorphic factorization [20] that gives back the Kasner solution, whereas its canonical factorization gives rise to a different solution to the field equations. Both solutions to the field equations are, however, related by a certain transformation, which we give in Section 7. To our knowledge, this is the first time that it is shown that a meromorphic factorization of a monodromy matrix can give rise to a solution of the gravitational field equations.

Preliminary results
We begin by introducing the involution ι σ in C\{0}, It has two fixed points, which we denote by ±p F : ±i if σ = 1, and ±1 if σ = −1. We will denote the set of fixed points by F P σ . Next, we introduce the set W ρ,v = v ± √ −σ ρ for any (ρ, v) in the Weyl upper-half plane. We can write this set as W ρ,v = {v ± p F ρ}.

Remark:
Let 3.1 Properties of ϕ ∈ T Proposition 3.1. Let (ρ 0 , v 0 ) be a point in the Weyl upper half-plane. For all ω ∈ C\W ρ 0 ,v 0 , there exists a branch of the square root such that ϕ and ϕ = −σ/ϕ, given by are of class C ∞ in a neighbourhood of (ρ 0 , v 0 ).
Proof. Let (ρ 0 , v 0 ) be any point in the Weyl upper half-plane. We start by noting that Let us first consider the case σ = 1. We choose the principal branch of √ z (i.e. arg z ∈ ] − π, π]), with branch cut R − . We have (ω − v 0 ) 2 + σρ 2 0 ∈ R iff ω I = 0 or ω R = v 0 . If ω I = 0, the real part of (3.4) is positive. If ω R = v 0 , the real part of (3.4) is also positive unless |ω I | > ρ 0 (we exclude v 0 ±iρ 0 ∈ W ρ 0 ,v 0 from our considerations). Therefore, with the chosen branch of and |ω I | > ρ 0 . Since in the latter case the real part of (3.4) is negative, we choose the branch of √ z with arg z ∈ [0, 2π[ for this case.
When σ = −1, a similar reasoning leads to the choice of the latter branch for √ z, unless ω I = 0 and |ω R − v 0 | > ρ 0 , in which case we choose the principal branch.

Contour properties
The following result will be needed in subsequent sections.
Proposition 3.4. Let Γ be a simple closed curve in the complex τ -plane, such that it encircles the origin of the τ -plane, and such that it is invariant under the involution ι σ . Let D + denote the simply connected interior region of Γ (i.e. 0 ∈ D + ), and let D − = Proof. We take Γ to be positively oriented. Consider a point w 0 ∈ C\ ({0} ∪ Γ). Then, the winding number of Γ around w 0 is either zero or one. Now, for any such w 0 , where we used the invariance of Γ under ι σ . Hence, if the winding number around w 0 is zero, the winding number around ι σ (w 0 ) is one, and vice-versa.
Corollary 3.4.1. Let Γ be a simple closed curve in the complex τ -plane, such that it encircles the origin of the τ -plane, and such that it is invariant under the involution ι σ . Then, Γ passes through the fixed points of ι σ .
Proof. This is a simple consequence of Proposition 3.4. Let w 0 denote one of the fixed points. If w 0 / ∈ Γ, then either w 0 ∈ D + or w 0 ∈ D − , in which case w 0 and ι σ (w 0 ) have different winding numbers by Proposition 3.4, which contradicts the assumption that w 0 is a fixed point of ι σ (i.e. w 0 = ι σ (w 0 )).

The Breitenlohner-Maison linear system
The non-linear equations (2.1) are an integrable system, i.e., they are the solvability conditions [17] for a certain Lax pair, the so-called Breitenlohner-Maison (BM) linear system [1]. This is an auxiliary linear system in the Weyl upper-half plane given by It is defined in terms of a function ϕ and a matrix one-form A. The latter takes the form given in (2.2). We assume that the components of A are one-forms, whose coefficient functions are continuously differentiable functions in an open set in the Weyl upper-half plane. We will refer to this requirement by saying that A is of class C 1 . The function ϕ is taken from the set T defined in (3.5).
Given A and ϕ, we seek solutions X with the following properties: the matrix X is invertible; X is twice continuously differentiable (i.e. of class C 2 ) and X −1 is continuously differentiable (i.e. of class C 1 ) with respect to (ρ, v).
Next, let us discuss the solvability of the BM linear system (4.1), following [4,16]. We will make use of the relations and where we recall that ρ > 0.
Proof. ⇒: We multiply (4.1) by X −1 on the right, to obtain Using (2.3), we obtain for its Hodge dual, Multiplying by ϕ gives Their sum simplifies to and multiplying by X on the right gives, Applying the Hodge star operator to it, and multiplying it by ϕ, respectively, we obtain the following equations, Subtracting the second equation from the first, we get which, when divided by (ϕ 2 + σ), gives the BM linear system Now we show that the solvability of the linear system (4.1) implies the field equations (2.1).
Theorem 4.2. Let ϕ ∈ T . If the equation for the BM linear system, is satisfied, then A is a solution to the field equations Proof. Using Proposition 4.1, and taking the differential of (4.4), we get By (4.2) and (4.3), the previous equation becomes (4.17) Now we use the following relations for one-forms B, C, valid in two dimensions, as well as the relation dA + A ∧ A = 0 satisfied by A = M −1 dM . These relations, together with (3.8), lead to a simplification of (4.18), This can be written as Then, using (2.3), this results in

Monodromy matrix
The BM linear system (4.1) uses, as an input, a matrix one-form A = M −1 dM and a function ϕ ∈ T . These quantities are defined on the Weyl upper half-plane. Given a solution X(ρ, v) to the BM linear system, Breitenlohner and Maison constructed [1] a matrix M that is independent of the Weyl coordinates (ρ, v). Here we revisit and generalize their construction. In doing so, we refrain from explicitly indicating the dependence of ϕ ∈ T on the Weyl coordinates (ρ, v), for ease of notation. As mentioned earlier, the involution acts on matrices N by 'generalized transposition'. We will assume that this 'generalized transposition' commutes with differentiation (see Chapter 9 of [21]) (5.1) We follow the notation used in [4]. The coset representative M ∈ G/H takes the form M = V V . Since we take A to be of class C 1 in an open set, V is of class C 2 in this set. Since M is locally invertible, so is V . We decompose dV V −1 into one-forms that under the involution are either invariant or anti-invariant, Given a solution X to the BM linear system with input (A, ϕ), we define [1] Now, instead of ϕ ∈ T , consider picking ϕ ∈ T with ϕ = −σ/ϕ. Given any solution X to BM linear system with input (A, ϕ), let P denote We then use P andP to define the matrix M by Note that X is not necessarily a priori related to X, except for the relation between ϕ and ϕ. Hence, (5.7) generalizes the definition given in [1].
Theorem 5.1. Let ϕ, ϕ ∈ T , satisfying the relation (3.6). Let X and X denote solutions to the corresponding BM linear system based on Proof. We consider the BM linear system with input (A, ϕ) and solution X. Then, by Proposition 4.1, X, A, ϕ satisfy, Using (5.5), we compute where we used the relation (5.9) as well as (5.3) and (5.4).
Similarly, for the BM linear system with input (A, ϕ) and solution X, we obtain Next, using (5.7) and the property (5.1), we compute which, using that acts as an anti-homomorphism on matrices, can be written as Substituting the expression (5.10) and (5.11) into this gives Then, using (5.3), and taking into account that ϕ = −σ/ϕ, we obtain Thus, we have shown that the matrix M is independent of the Weyl coordinates (ρ, v) for any X and X that solve (4.1) with input ϕ and ϕ, respectively, with ϕ and ϕ satisfying the relation (3.6).
Thus, the matrix M defined in (5.7) is independent of the Weyl coordinates, even though the individual factors depend on (ρ, v). This result, which generalizes previous results from (Section 3 of) [1] and from (Section 5 of) [4], suggests that all information on M (ρ, v) may be lost after multiplication by X and X on the right and on the left, respectively; indeed we will later give an example in which, for a given M (ρ, v), it is possible to obtain in this way an arbitrary constant matrix M, in particular the identity matrix. In case we have a matrix M, independent of (ρ, v), from which one can obtain, via an appropriate factorization, a solution M (ρ, v) of the field equations, we say that M is a monodromy matrix for M (ρ, v). This will be addressed in the next section.

Canonical factorization gives a solution to the BM linear system
In the previous section, we saw that, by (5.7), one can construct matrices M that are independent of the Weyl coordinates, from given solutions X and X of the BM linear system with input (A, ϕ) and (A, ϕ), respectively, where ϕ = −σ/ϕ, ϕ ∈ T . Therefore, we obtain a factorization of M in terms of X, X and M .
In this section, we study the reverse question. Namely, we obtain A(ρ, v) (or equivalently, M (ρ, v)) from a factorization of a matrix function M of the form (6.2) given below. Naturally, to do so, we must make certain (though very general) assumptions, as follows.
Assumption 1: Let M(u) be an -invariant, invertible matrix function of the complex variable u, and consider the matrix that is obtained by composition with For later convenience, we note the relation of (6.1), called the spectral curve, with property (3.7).
Now we recall the definition (3.1) of the involution ι σ . Assumption 2a: There exists an open set S such that, for every (ρ 0 , v 0 ) ∈ S, one can find a simple closed curve Γ in the τ -plane, which is ι σ -invariant and encircles the origin, such that: For all (ρ, v) in a neighbourhood of (ρ 0 , v 0 ), the matrix (6.2), as well as its inverse, is analytic in a region (i.e. in an open, connected set) in the τ -plane containing Γ, which we denote by O and which we require to be invariant under ι σ (see Figure 1). We denote by D + the simply connected interior region of Γ (hence 0 ∈ D + ) and by D − = C\(D + ∪ Γ) the exterior region. Recall that with these assumptions, we have, for p ∈ C\{0}, that p ∈ D + ⇔ ι σ (p) ∈ D − , see Proposition 3.4, and Γ necessarily passes through the fixed points of the involution ι σ , which are ±i when σ = 1 and ±1 when σ = −1, see Corollary 3.4.1.
Now consider (ρ, v) as an arbitrary pair of parameters in a neighbourhood of (ρ 0 , v 0 ), and take τ as the independent (complex) variable. To emphasize this aspect, we will denote the matrix in (6.2) by M (ρ,v) (τ ).
Assumptions 2b, 2c): With the same notation as in Assumption 2a, for any (ρ, v) in a neighbourhood of (ρ 0 , v 0 ), M (ρ,v) (τ ) admits a canonical factorization of Wiener-Hopf (or Birkhoff ) type with respect to Γ, where M + (ρ,v) (τ ) (respectively M − (ρ,v) (τ )) and its inverse are analytic and bounded in D + ∪ O (respectively D − ∪ O), and we assume the normalization condition M + (ρ,v) (0) = I. We set Such a factorization, if it exists, is unique; necessary and sufficient conditions for its existence were given in [4] (see also the references therein and [22]). A result given in [5] states that, for any scalar function f (u) that is continuous, as well as its inverse, on Γ u (here Γ u denotes the image of Γ under (6.1)), the function f (ρ,v) (τ ) always admits a canonical factorization with respect to Γ. It follows from this result that a canonical Wiener-Hopf factorization always exists when the monodromy matrix is triangular [7,18]. It is also clear that only boundedness of M (ρ,v) (τ ) in the variable τ , on Γ, is assumed. M(u) may be unbounded, see for example Section 9. Note that it follows from Assumption 2a that the representation (6.3) holds for all τ ∈ O.
Moreover, it was shown in [4] that under Assumption 1, (6.3) can be written as Our last assumption is as follows. Assumption 3: M (ρ, v) is of class C 2 , and for each τ ∈ D + ∪ O the matrix function X is of class C 2 as a function of (ρ, v), and ∂X/∂ρ and ∂X/∂v are analytic as functions of τ in the domain D + ∪ O.
We now state the main result of this section as a theorem. Proof. Take any point in S, which we will denote by (ρ 0 , v 0 ). In the following, (ρ, v) will denote an arbitrary point in a neighbourhood of (ρ 0 , v 0 ). We take the contour Γ to have the properties described under Assumption 2a. In the factorization (6.4), τ varies along Γ ⊂ O, while (ρ, v) is kept fixed. Substituting τ by ϕ of the form (3.5) with ω ∈ C ρ 0 ,v 0 , we obtain M(ω), which is independent of (ρ, v), and hence differentiating with respect to (ρ, v) gives zero, i.e. d(M(ω)) = 0, where d denotes the differential with respect to (ρ, v). Then, multiplying by (X ) −1 on the left and by X −1 on the right, we obtain, Next, we multiply this system of differential equations by ϕ + σ/ϕ, where we recall that ϕ = 0 and ϕ 2 + σ = 0. Using the property (5.1), we obtain We evaluate (6.14) Inserting these expressions into (6.10) results in Now evaluate (6.15) at (ρ 0 , v 0 ). This holds for any function ϕ satisfying the assumptions.
Recall that any such function is labelled by ω ∈ C ρ 0 ,v 0 . Keeping (ρ 0 , v 0 ) fixed, and varying over all ω ∈ C ρ 0 ,v 0 results in scanning over all values τ ∈ O\F P σ . In this way, we can reinterpret (6.15) as an equality that holds for all τ ∈ Γ\F P σ at fixed (ρ 0 , v 0 ). For ease of notation, we now denote (ρ 0 , v 0 ) simply by (ρ, v), taking into account that (ρ 0 , v 0 ) was arbitrarily fixed. Hence By continuity, this equality can also be extended to the two fixed points of Γ. Hence, (6.16) holds ∀τ ∈ Γ.
We now rewrite (6.16) as where we have separated terms in such a manner that the left hand side of the equality is analytic in D − (by Proposition 3.4, if X(τ, ·, ·) is analytic in D + then X(− σ τ , ·, ·) is analytic in D − ) except for a pole of order one at τ = ∞, while the right hand side is analytic in D + except for a pole of order one at τ = 0. Since (6.17) holds for τ ∈ Γ, it constitutes a matricial Riemann-Hilbert problem that is associated to the canonical Wiener-Hopf factorization (6.4).
This associated Riemann-Hilbert problem is solved by means of a generalization of Liouville's theorem. Namely, consider a function f that: equals f + in D + and is analytic in D + except for a simple pole at τ = 0; that equals f − in D − and is analytic in D − except for a simple pole at τ = ∞; and that satisfies f + = f − for all τ on Γ. Then, f can only be of the form where A, B, C are constants. In our case, A, B, C are matrix one-forms that are independent of τ . We then infer two equations from (6.17). The first one reads where the left hand side is analytic in D − except for a simple pole at τ = ∞. The second equation reads where the right hand side is analytic in D + except for a simple pole at τ = 0. First we focus on (6.20) and rewrite it as The left hand side of this equation is analytic in D + , and hence the right hand side has to be analytic at τ = 0. Thus, the expression in the bracket on the right hand side must vanish for τ = 0. Now recall (6.6) and (6.7). Therefore we conclude Applying a similar reasoning to (6.19) and demanding analyticity at τ = ∞, we infer Hence, we conclude that both sides (6.17) are equal to a τ -independent matrix one-form B. Since this is valid for any (ρ, v) ∈ S, we denote B ≡ G(ρ, v). Now recall that (6.21) holds ∀τ ∈ Γ, and in particular at the fixed points τ 2 = −σ of the involution (3.1). Let us denote the two fixed points by {p F ,p F = −p F }. We introduce two projections which satisfy P + + P − = id, P + − P − = σ p F , P 2 ± = P ± , P + P − = 0. Then, using (2.3), we infer dρ + p F dv = 2P − dρ as well as P ± = ∓p F P ± .

Meromorphic factorizations: a case study
So far, we discussed canonical Wiener-Hopf factorizations of monodromy matrices. We focussed on canonical factorizations, since for these we have Theorem 6.1, which guarantees that this type of factorization yields a solution to the field equations. We do not have an analogous theorem for other types of factorizations, such as meromorphic factorizations [20]. This does not mean that those other types of factorization may not also yield solutions to the field equations. In this section, we discuss an example for which this happens. This is the example of a Kasner solution, which is a cosmological solution to Einstein's field equations in four dimensions. It belongs to the class σ = −1. We proceed as follows. We explicitly solve the BM linear system for a particular Kasner solution, and obtain the explicit expression for X(ρ, v), c.f. (7.9). We then use X(ρ, v) to construct matrices M associated to this Kasner solution which are independent of the Weyl coordinates (ρ, v). We then pick a particular matrix M, which has a factorization of the form (7.14), with M (ρ) corresponding to the Kasner solution. However, this factorization is not a canonical Wiener-Hopf factorization, since X and X are not analytic in the interior and exterior region, respectively, of any contour Γ satisfying Assumption 2a, although X is analytic in a neighbourhood of τ = 0. However, the matrix M we picked also admits a canonical factorization. The resulting factor M (ρ, v) yields a different solution of the field equations. This new solution and the Kasner solution turn out to be related by the transformation given in Proposition 3.3. Therefore, in certain circumstances, by performing a canonical factorization of M and applying an appropriate transformation to the associated solution, one obtains a new solution to the field equations that results from a meromorphic factorization of M. We consider the field equations of General Relativity in vacuum. They admit a cosmological Kasner solution given by where the exponents p i obey the following conditions: In the following, we take all the p i to be non-vanishing. Without loss of generality, we will assume that 0 < p 1 < 1. We now bring (7.1) into Weyl-Lewis-Papapetrou form (2.4) with σ = −1, Note that for t > 0, ρ > 0, the function ρ(t) = t 1−p 1 is one-to-one. The associated matrix M ∈ G/H = SL(2, R)/SO(1, 1) is diagonal and takes the form , (7.5) and the resulting matrix one-form A = M −1 dM is given by We now specialize to the case in which case e ψ = 9ρ 8 and We pick any element ϕ ∈ T , c.f. (3.5), and we explicitly solve the associated BM linear systen (4.1) for X. We obtain as can be verified by substituting this expression into (4.1). Here, c 1 , c 2 , c 3 , c 4 ∈ C are arbitrary integration constants.
For the theory at hand, the involution acts as transposition on matrices. We therefore obtain for the solution X of the linear system (4.1) based on ϕ = 1/ϕ, where thec i ∈ C denote an independent set of integration constants, unrelated to the c i . Next we compute the combination which is independent of the Weyl coordinates (ρ, v). Thus, we may obtain an arbitrary constant matrix M, in particular the identity matrix, and the information about M (ρ, v) is lost. However, as mentioned before, by choosing an appropriate matrix M that is independent of (ρ, v), we can recover a solution M (ρ, v) either by Wiener-Hopf or by meromorphic factorization. To illustrate this, we now describe how this arises in a specific example.
Let us now choose these constants as follows: c 2 =c 2 = c 3 =c 3 = 0 and c 1 =c 1 = (2ω) 2 , c 4 =c 4 = (2ω) −2 , with ω ∈ C. Then, (7.11) becomes where, for all (ρ, v) taken in a neighbourhood of (ρ 0 , v 0 ), ϕ ω ∈ T with ω = v 0 ± ρ 0 , c.f. (3.5). The expression for X(τ, ρ, v), is obtained by making use of the algebraic relation (3.7) in (7.9). Also using it on the left hand side of (7.12) yields (7.14) Now, when viewed as a factorization of M v + ρ 2 (1+τ 2 ) τ with respect to a contour Γ, (7.14) describes a meromorphic factorization: although X is normalized at τ = 0, i.e. X(τ = 0, ρ, v) = 1, it has a double pole in the interior of the curve Γ, located at one of the zeroes of 2v τ ρ + 1 + τ 2 . The two zeroes of this quadratic polynomial in τ are at τ = (−v ± v 2 − ρ 2 )/ρ. Let us denote them by τ 1 = (−v + v 2 − ρ 2 )/ρ and τ 1 = (−v − v 2 − ρ 2 )/ρ. They satisfy τ 1 τ 1 = 1. Since the curve Γ is ι σ -invariant, then, by Proposition 3.4, if τ 1 is located in the interior of the curve Γ, τ 1 is located in the exterior region, and vice-versa (recall that τ 1 and τ 1 cannot lie on Γ). Hence, we have obtained the Kasner solution (7.5) from a meromorphic factorization of M v + ρ 2 (1+τ 2 ) τ . Now let us discuss the canonical factorization of the same matrix M v + ρ 2 (1+τ 2 ) τ with respect to Γ. If τ 1 is located in the exterior region of Γ, the resulting factors in (6.4) are (7.15) and analogously if τ 1 is in the exterior region of Γ. The subscript c refers to the canonical factorization. The matrix M c (ρ, v) in (7.15) describes a solution to the field equations (2.1) that is different from the Kasner solution M (ρ) described by (7.5). Both matrices are related by This is a particular instance of the following result: we can construct new solutions to the field equations by multiplying M c (ρ, v) by with ϕ ∈ T . The resulting matrix M as well as (7.17) satisfy the field equations (2.1) by virtue of Proposition 3.3. The case (7.16) corresponds to taking α = 0, β = 4, K = 1 and ϕ(ρ, v) = τ 1 (ρ, v). Note that neither the canonical factorization nor the solution given by (7.17) are valid on the lines v = ±ρ, for which the points τ 1 , τ 1 coincide with the fixed points ±1. However, upon multiplication by a matrix as in (7.16), we obtain a solution that is valid for all (ρ, v) on the Weyl upper half-plane. More generally, given matrix M 1 and M 2 that solve the field equations (2.1), the product M 1 M 2 will also solve the field equations (2.1) provided that This is the case when both M 1 and M 2 are diagonal matrices, or for instance, when

The Schwarzschild monodromy matrix
In this section, we discuss the family of solutions to the field equations of General Relativity in vacuum that results from the canonical factorization of the monodromy matrix (2.19) with = 1, which we call the Schwarzschild monodromy matrix, for both cases σ = ±1. The involution acts as transposition on matrices. As we shall see, the case σ = −1 is more intricate than the case σ = 1.
Substituting u by the expression on the right hand side of the spectral curve relation (6.1), the resulting matrix is Now we perform the canonical factorization of M (ρ,v) (τ ) with respect to a closed contour Γ in the τ -plane, satisfying the following requirements: 1. Γ passes through the fixed points of the transformation τ → − σ τ ; 2. Γ encircles τ = 0;

is analytic in an open set containing Γ.
For any (ρ, v) in the Weyl upper half-plane such that τ 1 (ρ, v), τ 2 (ρ, v) / ∈ F P σ (where we recall that F P σ denotes the set of fixed points of the involution (3.1)), the contour Γ can be chosen in such a way as to bypass τ 1 and τ 2 . There are four possible classes of contours from which Γ can be chosen: (i) τ 1 and τ 2 are inside the contour Γ; (ii) τ 1 is outside and τ 2 is inside of Γ; (iii) both τ 1 and τ 2 are outside of Γ; (iv) τ 1 is inside and τ 2 is outside of Γ.
Factorizing with respect to Γ we obtain, for each of these cases, and where i) for a contour in class (i) ii) for a contour in class (ii) iii) for a contour in class (iii) iv) for a contour in class (iv) The resulting four-dimensional space-time metrics are, in Weyl-Lewis-Papapetrou form, given by with ∆ > 0, with ds 2 2 given by either (2.5) or (2.6), and with ψ obtained by solving (2.7). We will now first discuss the case σ = 1, and subsequently the more intricate case σ = −1.

σ = 1
When σ = 1, we have which are real for any (ρ, v) in the Weyl upper half-plane. Hence, τ 1 and τ 2 can never coincide with the two fixed points ±i of the involution τ → −1/τ . As a consequence, any of the solutions to the field equations obtained by canonical factorization of M (ρ,v) (τ ) is valid in all of the Weyl upper half-plane. The relative positions of τ 1 and τ 2 on the real axis of the τ -plane depends on (ρ, v) in the Weyl upper half-plane, as follows: Possible choices of contours for the four classes of contours (i) − (iv) are depicted in Figure  2.
Case (i): for Γ belonging to class (i), we obtain where ψ = ψ i is obtained by solving (2.7), up to a constant which we set to zero, and where K = K i is the the Kretschmann scalar K = R µνδη R µνδη . Note that the expressions (8.14) are invariant under v → −v. We note that ∆ i is bounded for all (ρ, v) in the Weyl upper half-plane, and bounded away from zero except in a neighbourhood of the segment ρ = 0, −m < v < m, where ∆ i → 0 as ρ → 0 with v fixed.
Using the bijection from the domain r > 2m, 0 < θ < π in spherical coordinates onto the Weyl upper half-plane, given by ∆ i becomes ∆ i = 1 − 2m/r, which results in the line element Case (ii): for Γ belonging to class (ii), we obtain where ψ = ψ ii is obtained by solving (2.7), up to a constant which we set to zero, and where K = K ii is the the Kretschmann scalar K = R µνδη R µνδη . Let us consider these expressions in the limit ρ → 0 with v fixed. For fixed v ∈]−m, m[, ∆ ii and K ii are bounded away from 0, while ψ ii → 0. For v > m, we have ∆ ii → 0, K ii → 3/(4m 4 ), while for v < −m we have ∆ ii → ∞, K ii → ∞. Thus, for ρ = 0, v > m there is a Killing horizon, and for ρ = 0, v < −m a curvature singularity.
Using the change of coordinates with ∈]0, 2m[, ϑ ∈]0, ∞[, the space-time metric in four dimensions takes the form which describes the 'interior' region of the AII-metric (B.3). Although this metric resembles the Schwarzschild metric (8.19), it has a different isometry group.
Case (iii): for Γ belonging to class (iii), we obtain (8.20) Using the bijection from the domain r > 0, 0 < θ < π in spherical coordinates onto the Weyl upper half-plane, given by ρ = r 2 + 2mr sin θ , v = (r + m) cos θ , (8.21) results in the line element This is the 'negative mass' Schwarzschild solution [12], which has a naked curvature singularity at r = 0. This solution is an example of an AI-metric, c.f. Appendix B.
Case (iv): we note that the transformation τ Hence, for Γ belonging to class (iv), we obtain Since (8.17) and (8.23) are related by the coordinate transformation v → −v, the solution described by (8.23) is the same as (8.19).

σ = −1
When σ = −1 we have, from (8.3), (8.24) Note that, differently from the case σ = 1, τ 1 and τ 2 can now take complex values and may, moreover, coincide with the fixed points ±1 ∈ F P −1 through which the curve Γ has to pass. This occurs for (ρ, v) on four half-lines, namely Imposing that τ 1 and τ 2 are real, so as to ensure that the solutions obtained from the canonical factorization of M (ρ,v) (τ ) are real, we find that we have to restrict (ρ, v) to lie in the open set consisting of the three regions in the Weyl upper-half plane denoted by I, A, B in Figure 3. In Figure 3, we denote by L ± the lines described by ρ = ±(v + m) and Figure 3: Regions in the Weyl upper-half plane for which both τ 1 and τ 2 are real.
by R ± the lines described by ρ = ±(v − m). Considering now the cases (i) − (iv) corresponding to the various possible choices of the contour Γ, as described at the beginning of Section 8, we start by noting that in each of the cases (i) − (iv) we have ∆ > 0 if (ρ, v) is taken to lie in region I, while ∆ < 0 for region A and region B.
We will first consider the case when (ρ, v) takes values in region I, in which case τ 1 > 1 and −1 < τ 2 < 0. Possible choices of contours for the four classes of contours (i) − (iv) are depicted in Figure 4.

Region I
Here we take ρ to be a time-like coordinate. Accordingly, we take the four-dimensional space-time metric to be given by (8.11) and (2.5) (with σ = −1), i.e.
We obtain the following expressions for ∆ and ψ and for the four-dimensional Kretschmann scalar K: Case (i): for Γ belonging to class (i), we obtain where ψ = ψ i is obtained by solving (2.7), up to a constant which we set to zero. Note that the expressions (8.27) are invariant under v → −v.
and for ρ = 0 and −m < v < m, This solution has thus a Killing horizon on the segment ρ = 0, −m < v < m.
Case (ii): for Γ belonging to class (ii), we obtain where ψ = ψ ii is obtained by solving (2.7), up to a constant which we set to zero. For (ρ, v) taking values in region I, we obtain the following boundary values: on R − , they coincide with those for the case (i) given in (8.28) and for ρ = 0 and −m < v < m, This solution has no Killing horizon and no curvature singularity.
Case (iii): for Γ belonging to class (iii), we obtain For (ρ, v) taking values in region I, we obtain the following boundary values: on R − , and for ρ = 0 and −m < v < m, Thus, this solution has a curvature singularity on the segment ρ = 0, −m < v < m.
Hence, for Γ belonging to class (iv), we obtain For (ρ, v) taking values in region I, we obtain the following boundary values: on R − , and for ρ = 0 and −m < v < m, This solution has no Killing horizon and no curvature singularity.

Extending solutions: the interior region of the Schwarzschild solution
In Subsection 8.1 we used the change to spherical coordinates (8.21) to show that one of the solutions to the field equations that we obtained in Weyl coordinates describes the exterior region of the Schwarzschild black hole. If we write (8.21) as i.e. it coincides on I S with the metric describing the interior region of the Schwarzschild solution.
It is thus natural to ask whether it is possible to extend ∆ i in region I into neighbouring regions of the Weyl upper-half plane in a continuous manner, in such a way as to recover the whole interior through appropriate changes of coordinates. This extension can indeed be implemented by performing affine coordinate changes (3.18) of Weyl coordinates. The whole interior region of the Schwarzschild solution is obtained by extending the solution in region I to the quadrangle depicted in Figure 6, keeping ρ as the time-like coordinate. We now proced to explain this extension.
We can define bijections from the triangle I onto each of the four triangles I, II, III, IV represented in Figure 6. This is implemented using the affine coordinate transformations given in Table 1. Note that the affine transformations are such that they leave the lines ρ = m + v and ρ = m − v invariant. Taking into account the results of Subsection 8.2.1, we may therefore extend ∆ i (ρ, v) continuously to the rectangle −m < v < m, 0 < ρ < 2m by where ∆ ii (B −1 (ρ, v)) and ∆ iv (D −1 (ρ, v)) can in turn be further extended naturally to the triangles II and IV , respectively, in Figure 6, since are well-behaved and positive also in these regions.
In order to ensure that ρ is a time-like coordinate in the quadrangle depicted in Figure  6, the affine transformations of Table 1 from region I into regions II and IV need to be accompanied by the transformation ds 2 2 → −ds 2 2 , which is consistent with the fact that ds 2 2 can take the form (2.5) or (2.6). Then, expressing (8.46) in spherical coordinates (r, θ), with 0 < θ < π, 0 < r < 2m, according to Table 2, we obtain (8.45), which describes the whole interior region of the Schwarzschild black hole. This solution is an example of an AI-metric, c.f. Appendix B. If we now consider ∆ ii or ∆ iii instead of ∆ i in triangle I of the Weyl upper-half plane, we obtain, by an appropriate change of coordinates, the Schwarzschild metric (8.45) in one of the triangles of Figure 5, which we can then extend in an analogous manner. As an example, consider ∆ ii in region I and subject it to the following sequence of coordinate transformations, This yields the Schwarzschild solution (8.45) in region II S , see Figure 5, provided the transformation (8.48) is accompanied by the transformation ds 2 2 → −ds 2 2 , to ensure that ρ is a time-like coordinate.
Clearly, we could have defined other continuous extensions of ∆ i in region I, keeping ρ as the time-like coordinate, by using the same affine transformations. There are three such possible extensions when taking into account the behaviour of ∆ i on the boundary lines ρ = ±v + m. They are given by However, unlike the extension (8.46), these three extensions have a jump in the transverse extrinsic curvature [10,23] across the lines ρ = ±v + m. We discuss this in Appendix A.

Regions A and B
Now we take (ρ, v) to lie in regions A and B in Figure 3. Then, as mentioned at the beginning of Subsection 8.2, from the factorization of M (ρ,v) (τ ) with σ = −1 we obtain four different solutions ∆(ρ, v), corresponding to the cases (i) − (iv), with ∆ < 0. Although this violates our initial assumption that ∆ > 0 (see (2.4)), this is easily overcome by changing the overall sign of the monodromy matrix (8.1). This implies that we can take over the expressions for ∆, ψ and K for the cases (i) − (iv) of Subsection 8.2, merely changing the sign of the expression for ∆.
We begin by considering solutions in region A.
First case: let us first consider the solution ∆ A iv = 1 τ 1 τ 2 , where τ 1 and τ 2 are given by (8.24), This solution possesses a Killing horizon for we obtain a solution in region A (represented in Figure 7) which continuously extends ∆ A iv to A∪A , and is actually also defined and valid in the region A which comprises the points between the lines v = −m and ρ = v + m. we see thatÃ is bijectively mapped onto A by andÃ is bijectively mapped onto A by With this change of coordinates, the previous solution given by As before, to ensure that ρ is a time-like coordinate in region A , the affine transformation (ρ, v) → (−m − v, −m − ρ) needs to be accompanied by the transformation ds 2 2 → −ds 2 2 . The resulting four-dimensional metric reads Second case: in an analogous manner, we can extend the solution ∆ A ii = 1/(∆ A iv ) in region A to region A ∪ A ∪ A by The associated space-time metric has a curvature singularity at ρ = 0, v < −m. By using the change of coordinates ρ = 2 + 2m sinh ϑ , v = −(m + ) cosh ϑ , (8.59) defined for > 0, ϑ > 0, the regions {( , ϑ) : > 0, ϑ > 0, < −m+m cosh ϑ} and {( , ϑ) : > 0, ϑ > 0, > −m + m cosh ϑ} are bijectively mapped into A and A , respectively, and in these new coordinates ∆(ρ, v) is given by ∆(ρ, v) = 1 + 2m/ , which results in the four-dimensional metric ds 2 4 = 1 + 2m dt 2 − 1 + 2m −1 d 2 + 2 dϑ 2 + sinh 2 ϑ dφ 2 . (8.60) This is the 'negative mass' version of the AII-metric (8.57) [12], c.f. Appendix B.
Note that the other two vacuum solutions on A ∪ A given by are identical to (8.55) and (8.58), respectively, via (ρ, v) → (−m − v, −m − ρ). Now let us consider solutions defined in region B. We note that solutions on B ∪ B obtained in an analogous manner as above are mapped into the ones on A ∪ A discussed above, so that we do not have to discuss them separately.
Finally, we note that, as in (8.49), we can define other continuous extensions of ∆ A i , . . . , ∆ A iv . These other extensions will suffer from a jump in the transverse extrinsic curvature across the line ρ = −m − v.
9 The monodromy matrix with = 0 In this section, we discuss the canonical factorization of the monodromy matrix (2.19) with = 0, for both cases σ = ±1. Note that this monodromy matrix is unbounded in the complex u-plane. The involution acts again as transposition on matrices. Using the spectral curve (2.13), we obtain where τ 0 = ϕ α=0 (ρ, v), with ϕ α (ρ, v) given in (8.4), and where τ 0 = −σ/τ 0 . Next, let us factorize (9.2) with respect to a contour Γ that satisfies Assumption 2a, c.f. Section 6. There are two possible classes of contours, from which Γ can be chosen: (i) τ 0 is inside the contour Γ (and hence τ 0 is outside of Γ); (ii) τ 0 is inside the contour Γ (and hence τ 0 is outside of Γ).
Factorizing with respect to Γ we obtain, for each of these cases, where i) for a contour in class (i) ii) for a contour in class (ii) (9.7) We take the four-dimensional space-time metric to be given by ds 2 4 = −σ∆ dt 2 + ∆ −1 e ψ ds 2 2 + ρ 2 dφ 2 , (9.8) with ds 2 2 given by either (2.5) or (2.6). ψ is obtained by solving (2.7), and we demand ∆ > 0. Inspection of (9.6) and (9.7) however shows that there are regions in the Weyl upper-half plane where ∆ < 0. This can be dealt with by changing the overall sign of the monodromy matrix (9.1).
We will now first discuss the case σ = 1, and subsequently the more intricate case σ = −1.
Case ii): for Γ belonging to class (ii), we obtain which requires changing the overall sign of the monodromy matrix (9.1), as mentioned above. The expression for ψ ii (ρ, v) is, up to an integration constant, given by Since the coordinate transformation v → −v converts the expressions for ∆ ii and ψ ii into those for ∆ i and ψ i given above, the resulting space-time metric agrees with the one of Case i).
A similar reasoning applies to the solution obtained from M −1 (ρ, v).

σ = −1
When σ = −1, we have Imposing that τ 0 and τ 0 are real, so as to ensure that ∆ is real, requires restricting (ρ, v) to lie in one of the following regions of the Weyl upper-half plane: either v > ρ or v < −ρ.
Finally, let us consider the case when (ρ, v) takes values in the region v < −ρ, or equivalently,ṽ = −v > ρ. Noting that we see that the discussion of the case v < −ρ gets mapped to the previous discussion of the case v > ρ.