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Predictability of subluminal and superluminal wave equations

It is sometimes claimed that Lorentz invariant wave equations which allow superluminal propagation exhibit worse predictability than subluminal equations. To investigate this, we study the Born-Infeld scalar in two spacetime dimensions. This equation can be formulated in either a subluminal or a superluminal form. Surprisingly, we find that the subluminal theory is less predictive than the superluminal theory in the following sense. For the subluminal theory, there can exist multiple maximal globally hyperbolic developments arising from the same initial data. This problem does not arise in the superluminal theory, for which there is a unique maximal globally hyperbolic development. For a general quasilinear wave equation, we prove theorems establishing why this lack of uniqueness occurs, and identify conditions on the equation that ensure uniqueness. In particular, we prove that superluminal equations always admit a unique maximal globally hyperbolic development. In this sense, superluminal equations exhibit better predictability than generic subluminal equations.


Introduction
Many Lorentz invariant classical field theories permit superluminal propagation of signals around non-trivial background solutions. It is sometimes claimed that such theories are unviable because the superluminality can be exploited to construct causality violating solutions, i.e., "time machines". The argument for this is to consider two lumps of non-trivial field with a large relative boost: it is claimed that there exist solutions of this type for which small perturbations will experience closed causal curves [1]. However, this argument is heuristic: the causality-violating solution is not constructed, it is simply asserted to exist. This means that the argument is open to criticism on various grounds [2,3,4].
The reason that causality violation would be problematic is that it implies a breakdown of predictability. In this paper, rather than focusing on causality violation, we will investigate predictability. Our aim is to determine whether there is any qualitative difference in predictability between Lorentz invariant classical theories which permit superluminal propagation and those that do not.
We will consider quasilinear scalar wave equations for which causality is determined by an effective metric g(u, du) which depends on the scalar field u and its first derivative du. In the initial value problem we specify initial data (S, u, du) where S is the initial hypersurface and u, du are chosen on S such that S is spacelike w.r.t. g(u, du). We can now ask: what is the largest region M of spacetime in which the solution is uniquely determined by the data on S? Uniqueness requires that (M, g) should be globally hyperbolic with Cauchy surface S, i.e., the solution should be a globally hyperbolic development (GHD) of the data on S. This suggests that the "largest region in which the solution is unique" will be a GHD that is inextendible as a GHD, i.e., it is a maximal globally hyperbolic development (MGHD).
Our aim, then, is determine whether there is any qualitative difference between MGHDs for subluminal and superluminal equations.
In section 2, we will introduce the class of scalar wave equations that we will study, and define what we mean by "subluminal" and "superluminal" equations. Note that the standard linear wave equation is both subluminal and superluminal according to our definition.
In section 3 we will study an example of a Lorentz invariant equation in 1 + 1 dimensions, namely the Born-Infeld scalar field. The general solution of this equation is known [5,6]. This equation can be formulated in either a subluminal or superluminal form. One can consider the interaction of a pair of wavepackets in these theories. If the amplitude of the wavepackets is not too large then the wavepackets merge, interact, and then separate again [6]. In the subluminal theory they emerge with a time delay, in the superluminal theory there is a time advance. The MGHD is the entire 2d Minkowski spacetime in both cases.
For larger amplitude, it is known that the solution can form singularities in the subluminal theory [6]. Singularities can also form in the superluminal theory. In both cases, the formation of a singularity leads to a loss of predictability because MGHDs are extendible across a Cauchy horizon, and the solution is not determined uniquely beyond a Cauchy horizon. However, there is a qualitative difference between the subluminal and superluminal theories. In the superluminal theory there is a unique MGHD. However, in the subluminal theory, MGHDs are not unique: there can exist multiple distinct MGHDs arising from the same initial data. This is worrying behaviour. Given a solution defined in some region U , we can ask: in which subset of U is the solution determined uniquely by the initial data? In the superluminal case, this region is simply the intersection of U with the unique MGHD, or, equivalently, the domain of dependence of the initial surface within U . This can be determined from the solution itself. However, in the subluminal case there is, in general, no such method of determining the appropriate subset of U . To determine the region in which the solution is unique, one has to construct all other solutions arising from the same initial data! In section 4 we will discuss the existence and uniqueness of MGHDs for a large class of quasilinear wave equations (in any number of dimensions). We start by proving a theorem asserting that two GHDs defined in regions U 1 and U 2 will agree in U 1 ∩ U 2 provided U 1 ∩ U 2 is connected. Thus if one can show that U 1 ∩U 2 is always connected then one always has uniqueness. We will prove that this is the case for any equation with the property that there exists a vector field which is timelike w.r.t. g(u, du) for all (u, du). For such an equation, and for a suitable initial surface, we prove that there exists a unique MGHD. Note that any superluminal equation admits such a vector field so for any superluminal equation there exists a unique MGHD.
Our Born-Infeld example demonstrates that one cannot expect a unique MGHD for a general subluminal equation. One can define the maximal region in which solutions are unique, which we call the maximal unique globally hyperbolic development (MUGHD). Unfortunately, as mentioned above, there is no simple characterization of the MUGHD: given a solution defined in a region U , there is no simple general method for determining which part of U belongs to the MUGHD. As we will show, one can establish some partial results e.g. for a solution defined in U , the solution is unique in the subset of U corresponding to the domain of dependence of the initial surface detemined w.r.t. the Minkowski metric. However, this is rather a weak result especially for equations with a speed of propagation considerably less than the speed of light.
An important application of the notion of a MGHD is Christodoulou's work on shock formation in relativistic perfect fluids [7]. Given that this work concerns subluminal equations, one might wonder whether the MGHD constructed in Ref. [7] suffers from the lack of uniqueness dicussed above. We will prove that if a MGHD "lies on one side of its boundary" then it is unique. This provides a method for demonstrating uniqueness of a MGHD once it has been constructed. In particular, this implies that there is a unique MGHD for the initial data considered in Ref. [7]. However, we emphasize that the equations of Ref. [7] are likely to exhibit non-uniqueness of MGHDs for more complicated choices of initial data.
Of course we have not answered the question which motivated the present work, namely whether it is possible to "build a time machine" in any Lorentz invariant theory which admits superluminal propagation. However, our work does show that the object that one would have to study in order to address this question, namely the MGHD, is well-defined in a superluminal theory. Smooth formation of a time machine would require that there exist generic initial data belonging to some suitable class (e.g. smooth, compactly supported, data specified on a complete surface extending to spatial infinity in Minkowski spacetime) for which the MGHD is extendible, with a compactly generated [8] Cauchy horizon. 1 In section 3.2 we explain why this is not possible in 1+1 dimensions. Whether this is possible in a higher dimensional superluminal theory (let alone all such theories) is an open question.

.1 Subluminal and superluminal equations
Consider a scalar field u : R d+1 → R in (d + 1)-dimensional Minkowski spacetime. Assume that the field satisfies a quasilinear equation of motion 2 g µν (u, du)∂ µ ∂ ν u = F (u, du) (2.1) where F is a smooth function and (2.1) is written with respect to the canonical coordinates x µ on R d+1 . We will say that (M, u) is a hyperbolic solution if M is a connected open subset of R d+1 and u : M → R is a smooth solution of the above equation for which g µν (u, du) has Lorentzian signature. For such a solution we can define g µν (u, du) as the inverse of g µν and then (M, g) is a spacetime. Causality for the scalar field is determined by the metric g so we will be studying the causal properties of the spacetime (M, g). Now assume that we have a Minkowski metric m µν on R d+1 , with inverse m µν . We call the above equation subluminal if, whenever g µν is Lorentzian, the null cone of g µν lies on, or inside, the null cone of m µν . We call the equation superluminal if, whenever g µν is Lorentzian, the null cone of g µν lies on, or outside, the null cone of m µν . Most equations are neither subluminal nor superluminal e.g. because the null cones of g and m may not be nested or because the relation between the null cones of g and m may be different for different field configurations. Note also that the standard wave equation (g µν = m µν ) is both subluminal and superluminal according to our definitions.
Since M is a subset of R d+1 it follows that M is orientable because an orientation (d+1)-form of R d+1 can be restricted to M . In the superluminal case, any vector field T µ that is timelike w.r.t. m µν must also be timelike w.r.t. g µν . It follows that (M, g) is time orientable in the superluminal case. In the subluminal case, note that the null cone of g µν lies on or outside the null cone of m µν hence the 1-form dx 0 (for inertial frame coordinates x µ ) is timelike w.r.t. g µν . Therefore T µ = −g µν (dx 0 ) ν = −g 0µ defines a time orientation so (M, g) is time orientable. Furthermore, this shows that x 0 is a global time function which implies that (M, g) is stably causal in the subluminal case [2].

The initial value problem
Let's now discuss the initial value problem for an equation of the form (2.1). Consider prescribing smooth initial data (S, u, du) where S is a hypersurface in R d+1 and (u, du) are specified on S. Local well-posedness of the initial value problem requires that initial data is chosen so that g(u, du) is Lorentzian and that S must be spacelike w.r.t. g(u, du). Given such data, one expects a unique hyperbolic solution of (2.1) to exist locally near S. 3 We'll say that a hyperbolic solution (M, u) is a development of the data on S if S ⊂ M and the solution (M, u) is consistent with the data on S. To discuss predictability, we would like to know whether (M, u) is uniquely determined by the initial data (S, u, du). A necessary condition for such uniqueness is that (M, g) should be globally hyperbolic with Cauchy surface S. If (M, g) is not globally hyperbolic then the solution in the region of M beyond the Cauchy horizons H ± (S) is not determined uniquely by the data on S. We will say that a hyperbolic solution (M, u) is a globally hyperbolic development (GHD) of the initial data iff (M, g) is globally hyperbolic with Cauchy surface S.
A GHD (M, u) is extendible if there exists another GHD (M , u ) with M M and u = u on M . We say that (M, u) is a maximal globally hyperbolic development (MGHD) of the initial data if (M, u) is not extendible as a GHD of the specified data on S. Note that a MGHD might be extendible but the extended solution will not be a GHD of the data on S: it will exhibit a Cauchy horizon for S.
MGHDs play an important role in General Relativity. In General Relativity, given initial data for the Einstein equation, there exists a unique (up to diffeomorphisms) MGHD of the data [12]. This MGHD is therefore the central object of interest in GR because it is the largest region of spacetime that can be uniquely predicted from the given initial data. Any well-defined question in the theory can be formulated as a question about the MGHD. 4 Surprisingly, the subject of maximal globally hyperbolic developments for equations of the form (2.1) has not received much attention. By analogy with the Einstein equation one might expect a unique MGHD for such an equation. We will see that this is indeed the case for super-luminal equations but it is not always true for subluminal equations. The reason that this does not occur for the Einstein equation is that solving the Einstein equation involves constructing the differential structure on spacetime whereas in solving (2.1) the spacetime manifold has a fixed differential structure. It is this rigidity which leads to non-uniqueness of MGHDs for subluminal equations.
3 Born-Infeld scalar in two dimensions

Two dimensions
Let's now consider Lorentz invariant equations. By this we mean that we pick a Minkowski (i.e. flat, Lorentzian) metric m µν on R d+1 , with constant components in the canonical coordinates x µ , and we demand that isometries of m µν map solutions of the equation to solutions of the equation. We will assume that our equation has the form (2.1) where now g = g(m, u, du) and F = F (m, u, du) depend on the choice of m.
The two-dimensional case is special because if m is a Minkowski metric then so iŝ Using this fact we can relate subluminal and superluminal equations. Definê Now u satisfies (2.1) if, and only if, it satisfieŝ We view this equation as describing a scalar field in 2d Minkowski spacetime with metricm. It is easy to see that if (2.1) is a subluminal equation then (3.5) is superluminal, and vice-versa. Since the above transformation reverses the overal sign of m and g, it maps timelike vectors to spacelike vectors and vice-versa, i.e., the causal "cones" of the two theories are the complements of each other. This means that any solution of a superluminal equation arises from a solution of the corresponding subluminal equation simply by interchanging the definitions of timelike and spacelike. For example, if one draws a spacetime diagram for a solution of the subluminal equation, with time running from bottom to top, then the same diagram describes a solution of the superluminal equation, with time running from left to right (or right to left: one still has the freedom to choose the time orientation).

Superluminal equations in two dimensions
In this section we will consider causal properties of superluminal equations in 1 + 1 dimensions. The low dimensionality imposes strong restrictions on the causal structure of solutions. We will review some results on causality in 1 + 1 dimensions and explain why it is not possible to violate causality in a smooth way in a finite region of spacetime. This section is not essential for following the rest of the paper.
Assume we have a hyperbolic solution u defined on some open subset M of R 2 . If M is simply connected then M is homeomorphic to R 2 , which implies that (M, g) is stably causal [10,Theorem 3.43]. Hence a violation of stable causality requires that M is not simply connected i.e. M must have holes or punctures. We assume that (M, u) is inextendible, i.e., it is not possible to extend u as a hyperbolic solution onto a connected open set strictly larger than M . Hence the non-trivial topology of M must be associated with u developing some pathological feature when we attempt to extend to points of ∂M , for example, u or its derivative might blow up, or g might fail to be Lorentzian at such points.
This looks bad for the possibility of smoothly violating causality (i.e. "forming a time machine"). But maybe the above pathological features are consequences of the time machine, i.e., they lie to the future of the causality violating region. This is not the case: we will explain why some such pathology must occur before causality is violated. One cannot form a time machine smoothly in a two-dimensional superluminal theory.
Pick inertial coordinates (t, x) for Minkowski spacetime so that Now dx is spacelike w.r.t. m µν , which implies that it is also spacelike w.r.t. g µν (because, for a superluminal equation, the null cone of g µν lies on, or inside, that of m µν ). Hence x is a global space function for g, i.e., a function with everywhere spacelike (non-zero) gradient. The transformation of the previous subsection relates it to a global time function of the corresponding subluminal equation. Consider a null geodesic of g. Then x must be monotonic along the geodesic. To see this, let V be tangent to the geodesic. Then V x = V µ (dx) µ and this cannot vanish because V is null w.r.t. g and dx is spacelike w.r.t. g. 5 Non-vanishing of V x implies that x is monotonic along the geodesic. It follows that a null geodesic of g cannot be closed and cannot intersect itself. (Again this is easy to understand using the transformation of the previous section.) It is also easy to see that there cannot be a smooth closed future-directed causal curve (w.r.t. g) which is simple, i.e., does not intersect itself. This is because there will be a point on any such curve at which the tangent vector is timelike and past directed w.r.t. the Minkowski metric m µν and hence also timelike and past directed w.r.t. g µν , contradicting the fact that the curve is future-directed. Hence a closed future-directed causal curve must be non-smooth or non-simple. Now let S be a partial Cauchy surface, i.e., a surface (actually a line) which, viewed as a subset of (M, g), is closed, achronal and edgeless [11]. The future domain of dependence of S is D + (S) and the future Cauchy horizon is H + (S) = D + (S) − I − (D + (S)). If causality is violated to the future of S then this must occur outside D + (S), so H + (S) is non-empty. A standard result states that H + (S) is achronal and closed, and that every p ∈ H + (S) lies on a null geodesic contained in H + (S) which is past inextendible without a past endpoint in M [11].
Consider following a generator of H + (S) to the past. Since x is monotonic it must either diverge or approach a finite limit along this generator. If x diverges then the generator originates from infinity in R 2 . Consider the case that x approaches a finite limit in the past. From the fact that the generator is null w.r.t. g and hence non-timelike w.r.t. m we have |dt/dx| ≤ 1, which implies (via integration) that t also approaches a finite limit. Hence the generator has an endpoint p in R 2 . But it cannot have an endpoint in M so p / ∈ M . Since we are assuming that (M, u) is inextendible, p must correspond either to a singularity of the spacetime (M, g), or to a "point at infinity" in (M, g). In the latter case, g would have to blow up at p, which is singular behaviour from the point of view of the Minkowski spacetime.
This proves that generators of H + (S) must emanate either from infinity in Minkowski spacetime or from a point of R 2 that is singular w.r.t. (M, g) or "at infinity" w.r.t. (M, g). None of these possibilities corresponds to what is usually regarded as the condition for creation of a time machine in a bounded region of space, namely a "compactly generated" Cauchy horizon [8] (one whose generators remain in a compact region of (M, g) when extended to the past). If the generator does not emanate from infinity in R 2 then it remains in a compact region of R 2 but not a compact region of M : in M it "emerges from a singularity" or "from infinity".
To violate causality in a smooth way, the generators of H + (S) would have to emanate from infinity. This can happen even for the linear wave equation if S extends to left and/or right past null infinity in 2d Minkowski spacetime. In this case, H + (S) exists because information can enter the spacetime from past null infinity without crossing S. This is rather uninteresting (unrelated to any violation of causality) so consider instead the case of S extending to (left and right) spatial infinity in 2d Minkowski spacetime. For such S there is no Cauchy horizon for the linear wave equation so now consider such S for a nonlinear equation of the form (2.1). Assume that the initial data (u, du) is compactly supported on S. Under time evolution, the u field can propagate out to future null infinity. In 2d, even for the linear wave equation solutions do not decay at null infinity, so u does not necessarily decay near future null infinity. This implies that g may not approach m near future null infinity. So perhaps causality violation could originate at infinity with a Cauchy horizon forming at left and/or right future null infinity and propagate into the interior of the spacetime along null geodesics of g which are spacelike w.r.t. m. It would be interesting to find an example for which this behaviour occurs.

Born-Infeld scalar
In two dimensional Minkowski spacetime, consider a scalar field with equation of motion obtained from the Born-Infeld action where c is a constant. By rescaling the coordinates we can set c = ±1. The case c = 1 is the standard Born-Infeld theory. This theory is referred to as "exceptional" because, unlike in most nonlinear theories, a wavepacket in this theory propagates without distortion and never forms a shock [14]. The equation of motion is with indices raised using m µν . The inverse of g µν is g µν = m µν + c∂ µ Φ∂ ν Φ (3.10) We will refer to this as the "effective metric". A calculation gives Hence g is a Lorentzian metric (i.e. the equation of motion is hyperbolic) if, and only if, In the language of section 2.1, a hyperbolic solution must satisfy this inequality. Consider a vector V µ . Note that If c = 1 then the final term is non-positive. Hence if V is causal w.r.t. g µν then V is causal w.r.t. m µν , i.e., the null cone of g lies on or inside that of m. However, for c = −1, the null cone of m lies on or inside that of g. Hence the c = +1 theory is subluminal and the c = −1 is superluminal according to the definitions of section 2.1.
The two theories are related by the transformation (c, m, g) → (−c, −m, −g) with Φ fixed. This is the map described in section 3.1.

Relation to Nambu-Goto string
It is well-known that the c = 1 theory is a gauge-fixed version of an infinite Nambu-Goto string whose target space is 2 + 1 dimensional Minkowski spacetime. The same is true for c = −1 except that the target space now has + − − signature, i.e., two time dimensions. The action of such a string is where with G AB = diag(−1, 1, c) (c = ±1), x µ are worldsheet coordinates, and X A (x) are the embedding coordinates of the string. It is assumed that the worldsheet of the string is timelike, i.e., that g µν has Lorentzian signature. Fixing the gauge as and defining Φ(x) = X 2 (x), the action reduces to that of the Born-Infeld scalar described above, and the worldsheet metric g µν is the same as the effective metric given by equation (3.10). Note that the c = ±1 theories are mapped to each other under the transformation (G, g) → (−G, −g). From the worldsheet point of view, this corresponds to interchanging the definitions of timelike and spacelike, as discussed above.
Although the Born-Infeld scalar can be obtained from the Nambu-Goto string, we will not regard them as equivalent theories. We will view the BI scalar as a theory defined in a global 2-dimensional Minkowski spacetime. No such spacetime is present for the Nambu-Goto string. Of course any solution of the BI scalar theory can be "uplifted" to give some solution for the Nambu-Goto string. However, the converse is not true because not all solutions of the Nambu-Goto string can be written in the gauge (3.16). In particular, string profiles which "fold back" on themselves as in Fig. 1 are excluded by this gauge choice. From the BI perspective, such configurations will look singular. Of course such singularities can be eliminated by returning to the Nambu-Goto picture. However, we will not do this: the point is that the BI scalar is our guide to possible behaviour of nonlinear scalar field theories in 2d Minkowski spacetime, and most such theories do not have any analogue of the Nambu-Goto string interpretation.

Non-uniqueness
We can use the Nambu-Goto string to explain heuristically why there is a problem with the subluminal Born-Infeld scalar theory. Consider a left moving and a right moving wavepacket propagating along the string. As we will review below, if the wavepackets are sufficiently strong, when they intersect then the string can fold back on itself as described above. This is shown in Fig. 1. When this happens, the field Φ "wants to become multi-valued". But this is not possible in the BI theory because Φ is a scalar field in 2d Minkowski spacetime so Φ must be single-valued.
Clearly we have to "choose a branch" of the solution Φ at each point of 2d Minkowsi spacetime. We want to do this so that the solution is as smooth as possible. There are two obvious ways of doing this. We could start from the left of the string and extend until we reach the point A of infinite gradient as shown in Fig 1. But beyond this point we have to jump to the other branch, so the solution is discontinuous as shown in Fig. 2. If the discontinuity is approached from the left then the gradient of Φ diverges as we approach A. However, if approached from the right the gradient remains bounded up to the discontinuity at A. Following out this procedure for the full spacetime produces a globally defined solution of the Born-Infeld theory. After some time, the wavepackets on the Nambu-Goto string separate and the resulting Born-Infeld solution becomes continuous again. Now note that instead of starting on the left and extending to point A we could have started on the right and extended to point B. Now the discontinuity would occur at B instead of A. So now the solution appears as shown in Fig. 3. Approaching the discontinuity from the right, the gradient of Φ diverges at B. However approaching from the left, the gradient remains bounded up to the discontinuity at B. As above, this procedure gives a globally defined solution of the Born-Infeld theory. This is clearly a different solution from the solution discussed in the previous paragraph.  Starting from initial data prescribed on some line S in the far past, the above constructions produce two different solutions which agree with the data on S. Now non-uniqueness is to be expected because the solution Φ is singular (at A or B), so the corresponding spacetimes (M, g) will not be globally hyperbolic. Therefore lack of uniqueness is to be expected beyond the Cauchy horizon. However, we will show, in the subluminal case, that the lack of uniqueness occurs before a Cauchy horizon forms. In other words, the two solutions disagree in a region which belongs to D + (S) for both solutions. This implies that the two solutions cannot arise from the same MGHD of the data on S. Therefore MGHDs are not unique.
Clearly there are other ways we could construct Born-Infeld solutions from the Nambu-Goto solution: we do not have to take the discontinuity to occur at either point A or at point B, we could take it to occur at any point between A and B. This leads to an infinite set of possible solutions, and an infinite set of distinct MGHDs.
The above discussion was for the subluminal (c = 1) theory. We will show below that this problem does not occur for the superluminal theory. This is because, in the superluminal theory, A and B are timelike separated with B (say) occuring to the future of A. This implies that B lies to the future of the infinite gradient singularity at A hence B cannot belong to D + (S) if S is a surface to the past of A. Therefore there is a unique choice of branch in the superluminal theory. In this theory there is a unique MGHD.

General solution
The c = 1 (subluminal) BI scalar theory was solved by Barbashov and Chernikov [5,6]. We will follow the notation of Whitham [9], who gives a nice summary of their work. Because the superluminal and subluminal theories are related as discussed above, it is easy to write down the general solution for both cases. Write the Minkowski metric as and define null coordinates The solution is written in terms of a mapping Ψ : R 2 → R 2 given by and with Φ 1 (ρ) and Φ 2 (σ) arbitrary smooth functions. 6 Assuming that Ψ is invertible we can write ρ = ρ(ξ, η) and σ = σ(ξ, η) and the solution is given by Let's first review the Nambu-Goto string interpretation of this solution. We simply view (ρ, σ) as worldsheet coordinates and replace the LHS of (3.20) and (3.21) by X 1 − X 0 and X 1 + X 0 respectively. Together with X 2 = Φ = Φ 1 (ρ) + Φ 2 (σ) this specifies a globally well defined embedding of the string worldsheet. The worldsheet has R 2 topology with global coordinates (ρ, σ) and the worldsheet metric is so curves of constant ρ, σ are null geodesics w.r.t. g, i.e., characteristic curves (indeed the above solution was constructed using the method of characteristics). The solution describes a superposition of left moving and right moving wavepackets described by Φ 1 (ρ) and Φ 2 (σ), each travelling at the speed of light with respect to g. The worldsheet metric degenerates at points where Φ 1 (ρ)Φ 2 (σ) = −1. These correspond to "cusp" singularities at which the string worldsheet becomes null. Let's now forget about the Nambu-Goto string and discuss the interpretation of the above equations for the Born-Infeld scalar defined in 2d Minkowski spacetime. To specify the solution we need to specify Φ as a function of (ξ, η) and hence we need to know whether or not Ψ is actually invertible. The Jacobian of the map Ψ is In fact it can be shown that Ψ is globally invertible when this condition is satisfied [6]. The argument goes as follows. First use (3.21) to write and then substitute into (3.20) to obtain We want to use this equation to determine ρ as a function of ξ, η. A calculation gives So (3.25) implies that F is a strictly increasing function of ρ and hence there exists at most one solution ρ of (3.27). Given a solution for ρ, (3.26) determines σ uniquely. This proves that the map Ψ is injective. We define the subset M of Minkowski spacetime to be the image of Ψ. Then Ψ −1 is well-defined on M so (ρ, σ) is a global coordinate chart on M . Hence M is diffeomorphic to the (ρ, σ) plane, i.e., to R 2 . In such coordinates, the metric g is given by (3.23) so curves of constant ρ, σ are null geodesics w.r.t. g, i.e., characteristic curves. This metric is obviously conformal to the flat metric cdρdσ on R 2 and hence has the same (trivial) causal structure. Hence if (3.25) holds then g does not exhibit any causal pathologies for either sign of c. If, as well as (3.25), we make the assumption that Φ 1 and Φ 2 are compactly supported, or decay sufficiently rapidly at large ρ, σ, then for large |ρ|, G(ρ, η) differs from η by a constant and the RHS of (3.28) approaches 1 at large |ρ|. Hence F (ρ, η) → ±∞ for ρ → ±∞. In this case, there exists exactly one solution for ρ, σ so the map Ψ is a bijection hence M is all of Minkowski spacetime. So in this case, the above construction defines a global solution on Minkowski spacetime.

Choosing a branch
We will now discuss what happens when the condition (3.25) is violated. We will assume that Φ 1 and Φ 2 decay for large values of their argument (as will be the case if Φ 1 and Φ 2 are compactly supported), so if (3.25) is violated then somewhere in the (ρ, σ) plane we have Φ 1 (ρ)Φ 2 (σ) = with = ±1. Our assumption that Φ 1 and Φ 2 decay implies that this equation defines a closed contour in the (ρ, σ) plane. Such a contour might be disconnected, with multiple components, so let's focus on one such component, call it C.
Consider the subluminal c = 1 case. If = −1 then C corresponds to a cusp singularity on the Nambu-Goto string. Since C is closed, it describes the appearance of a cusp singularity at some point of the string, which splits into two such singularities, and after some time these recombine into a single singularity, which immediately disappears. The case = +1 corresponds to the string remaining smooth but the gradient becoming infinite on C. Again, since C is closed, what happens is that the gradient becomes infinite at some point of the string, this then splits into two points with infinite gradient (as shown in Fig 1), and after some time these recombine into a single point of infinite gradient, which immediately disappears.
We can now examine the consequences of this behaviour for the c = 1 Born-Infeld scalar. It is easy to see from Fig 1 that the above behaviour implies that Ψ is not injective: different values of (ρ, σ) (three in the figure) map to the same point in spacetime, i.e., the same (ξ, η), and hence Φ is multivalued there [6]. But in the Born-Infeld theory, Φ is supposed to be a scalar field in Minkowski spacetime, so it isn't allowed to be multivalued! We are only allowed to pick one of the possible values for Φ, i.e., we have to "choose a branch" as in Figs. 2 and 3. Let's now discuss how to do this in general, for either value of c.
To choose a branch, we simply restrict Ψ to an open region O of the (ρ, σ) plane on which Ψ is injective and Φ 1 (ρ) 2 Φ 2 (σ) 2 = 1 so the Jacobian (3.24) is non-zero. Defining M = Ψ(O) we can then use (ρ, σ) ∈ O as global coordinates on M . In these coordinates, the metric g is given by (3.23). Let (M, Φ) be a hyperbolic solution constructed this way. Now O must be connected: if it were not then, using the injectivity and continuity of Ψ, it would follow that M is not connected. However, M is connected by definition.
By definition, O cannot contain any point where Φ 1 (ρ) 2 Φ 2 (σ) 2 = 1. However such points may arise on the boundary of O and correspond to points where Φ is singular, as discussed above. To see this, assume that (ρ 0 , σ 0 ) ∈ ∂O and Φ 1 (ρ 0 )Φ 2 (σ 0 ) = ∈ {−1, 1} and let p = Ψ(ρ 0 , σ 0 ). Let's calculate the gradient of Φ w.r.t. the Minkowski coordinates (ξ, η) as we approach p. The result is Hence if = +1 then the derivative of Φ diverges at p. This corresponds to the Nambu-Goto string developing an infinite gradient at p. For = −1, the gradient of Φ is bounded as we approach p but the second derivative diverges at p. For example which generically diverges at p. This corresponds to a cusp on the Nambu-Goto string.
(The latter possibility is excluded if O contains points near infinity since we're assuming that Φ 1 , Φ 2 vanish at infinity. ) We can now investigate causality of solutions constructed using the method just described. The vector fields ∂/∂ρ and ∂/∂σ are null w.r.t. g. Let's determine whether they are future or past directed. Recall (section 2.1) that a time-orientation of g is induced by the one of Minkowski space. For the subluminal case c = 1 we know (section 2.1) that x 0 is a global time function for (M, g) (so (M, g) is stably causal). From this it can be shown that ∂/∂ρ is past-directed and ∂/∂σ is future-directed w.r.t. g.
In the superluminal case c = −1, ∂/∂x 1 is timelike w.r.t. m so, from the argument of section 2.1, we can choose ∂/∂x 1 as a time-orientation on (M, g). Making this choice, by calculating inner products with ∂/∂x 1 w.r.t. g we find that if Φ 1 (ρ) 2 Φ 2 (σ) 2 < 1 then ∂/∂ρ and ∂/∂σ are both future directed w.r.t. g whereas if Φ 1 (ρ) 2 Φ 2 (σ) 2 > 1 then they are both past-directed. In the former case, let V = ∂/∂ρ + ∂/∂σ, which is future-directed and timelike w.r.t. g. We This proves that solutions constructed using the method described above cannot exhibit any violation of causality. Note that the general arguments of section 3.2 already showed that causality violation cannot occur smoothly in a finite region of spacetime, i.e., any violation of causality must originate either from a singularity or from infinity. The arguments of the present section show that for superluminal Born-Infeld theory, if we construct solutions by "choosing a branch" as described above then even these possibilities are excluded: causality is never violated.
However, we note that the above method of starting from a global Nambu-Goto solution and "choosing a branch" may not be the most general way of constructing solutions of Born-Infeld theory. The general solution of Born-Infeld theory described in section 3.6 is a local solution, i.e., any solution of the theory can be written in this form locally. But there is no guarantee that this must hold globally as we have assumed above. It might be necessary to cover spacetime with multiple charts O α , each with corresponding coordinates (ρ α , σ α ). In any given chart the solution will take the form described above. With multiple charts, it is not obvious how to construct a time function (for the superluminal theory) so the above proof of stable causality no longer works.
For the rest of this paper, we will be interested in the initial value problem and globally hyperbolic developments (GHDs). We will explain in the next section, why the above "choosing a branch" method is always valid when constructing a GHD.

The initial value problem
Given a hyperbolic solution (M, Φ) we now want to ask: in what subset of M is Φ determined uniquely by initial data (Φ, dΦ) prescribed on a partial Cauchy surface S ⊂ M ? We will assume that S extends to left and right spatial infinity in 2d Minkowski spacetime.
We will start by using a standard argument to show that the null coordinates (ρ, σ) can be defined globally throughout the domain of dependence D(S) = D + (S)∪D − (S) in the spacetime (M, g). The argument goes as follows. Global hyperbolicity implies that D(S) has topology R × S. Since S is non-compact, this is the same as R 2 . This implies that the future-directed null geodesics in D(S) can be divided into two families F 1 and F 2 such that each point of D(S) lies on exactly one geodesic in each family [10, proposition 3.41]. We can call these families "right" and "left" moving. 7 Global hyperbolicity of D(S) implies that each member of F 1 or F 2 intersects S exactly once. Let z be a coordinate on S. We now define a function u by setting u(p) equal to the value of z at which the member of F 1 through p intersects S. Similarly we define a function v using F 2 .
We now want to use (u, v) as coordinates on D(S) but to do this we must check that (u, v) is injective, i.e., that we can't have two points p, q ∈ D(S) with the same (u, v). This happens if, and only if, p, q lie on the same pair of left and right moving null geodesics γ, γ . We assume, without loss of generality, that q ∈ J + (p) and also that q is the first intersection of γ and γ to the future of p. ConsiderJ + (p). Near p, this is generated by γ and γ . IfJ + (p) does not extend as far as q then some generator ofJ + (p) must have an endpoint on, say, γ. But this is not possible because this would be a boundary ofJ + (p) andJ + (p) cannot have a boundary. So the generators ofJ + (p) must coincide with γ, γ at least as far as q. Let T denote the union of the images of γ and γ between p and q inclusive. Note that T has the topology of a circle.
Time orientability implies that there exists a future directed timelike vector field V through every point. Within D(S) each integral curve of V must intersect S exactly once so by following the integral curves of V we can define a map α from T to S. T is achronal because T ⊂J + (p). This implies that α is injective. Hence α is a homeomorphism from T to α(T ) ⊂ S. T is compact hence α(T ) is compact. But S is non-compact, which implies that α(T ) must have a non-trivial boundary, which is impossible because α is a homeomorphism and T does not have a boundary.
We have shown that we can't have two points p, q ∈ D(S) with the same (u, v). Hence (u, v) can be used as global coordinates on D(S). These coordinates are characteristic, i.e., constant along null geodesics of g. Since the solution of section 3.6 was constructed using the method of characteristics, it follows that (ρ, σ) are related to (u, v) simply by reparametrization ρ = ρ(u), σ = σ(v). Hence in D(S) the solution can be written in terms of a single coordinate chart (ρ, σ) and the solution is constructed as in section 3.6. Since our interest is in globally hyperbolic developments of the initial data on S, there is no loss of generality in restricting to solutions constructed as described in section 3.6.
Consider first the case in which (3.25) is satisfied and Φ 1 , Φ 2 are compactly supported. As explained above, Ψ : R 2 → R 2 is a bijection so the solution is defined on the entire (ρ, σ) plane and M is all of Minkowski spacetime. Using (ρ, σ) as coordinates on M , the metric is given by (3.23) which is a conformal to the trivial metric cdρdσ on R 2 . Hence the spacetime (M, g) is globally hyperbolic, and surfaces of constant σ − cρ are Cauchy surfaces. The initial data on such a surface determines Φ 1 and Φ 2 , which then uniquely determines the solution everywhere else. So there is a unique MGHD when (3.25) is satisfied.
We will now discuss solutions for which (3.25) is not satisfied. We will do this using an example.
Consider a situation for which there exists a simple smooth closed curve C in the (ρ, σ) plane the open region enclosed by C. An example of functions Φ 1 and Φ 2 with this property is given by These functions are plotted in Fig. 4. Note that Φ 1 and Φ 2 both decay exponentially fast towards infinity and The curve C is plotted in Figure 6 using coordinates Consider the function F (ρ, η) defined by (3.27). For given (ξ, η), ρ is determined by solving Figure 5: A typical example of F (ρ, η) for fixed η. For given (ξ, η), the value of ρ is determined by solving ξ = F (ρ, η). The section highlighted yellow is in D, the section highlighted green is in U . ξ = F (ρ, η) and σ is then given by (3.26). For fixed η, equation (3.28) shows that the graph of F against ρ has positive gradient for (ρ, σ) outside C and negative gradient for (ρ, σ) ∈ D.
There is a range of values for η such that F (ρ, η) has the form shown in Fig. 5. From this plot it is clear that, for a range of ξ, several different values of ρ (and hence σ) correspond to the same (ξ, η) hence Ψ is not injective. For example the distinct points p, q and r of the (ρ, σ) plane map to the same point of Minkowski spacetime. Note that the point r belongs to D but p and q lie outside C.     Figure). Hence the region of Minkowski spacetime on which Ψ is not invertible is simply X ≡ Ψ(D). For the above example, this region is shown in Fig. 7. This region "starts" and "ends" at a pair of points belonging to Ψ(C). At these points the gradient of Φ diverges, i.e., they are singular points. Outside X, Ψ is invertible and hence the solution Φ is uniquely defined on R 2 \X. We will define U to be the region of the (ρ, σ) plane which maps to this region: U is the maximal region of the (ρ, σ) plane on which Φ is uniquely defined without having to "choose a branch". In Fig. 1, it consists of all points to the right of A or to the left of B. For the example (3.31), the region U is the region outside the dotted curves in Fig. 6 and the image Ψ(U ) of U in Minkowski spacetime is the region outside the dotted curves in Fig. 7.
Since Ψ is invertible on U , we have a hyperbolic solution (Ψ(U ), Φ) and a corresponding spacetime (Ψ(U ), g). We can now ask: how much of this solution is uniquely determined by initial data on a partial Cauchy surface S? We choose S to be a line of constant Minkowski time x 0 to the past of C, as shown in Fig. 7. We define Σ = Ψ −1 (S). This gives Σ as shown in Fig. 6. Note that S is spacelike with respect to both g and the Minkowski metric. We want to determine the domain of dependence of S in the spacetime (Ψ(U ), g).
The past domain of dependence D − (S) is trivial: it is simply the entire region to the past of S. This is easy to see by using (ρ, σ) coordinates because the metric in these coordinates is manifestly conformally flat from (3.23) and so the causal structure of g is manifest, with null geodesics at 45 • to the horizontal when plotted in coordinates (y 0 , y 1 ) as in Fig. 6.
The future domain of dependence D + (S) is non-trivial: it is bounded to to the future by a pair of Cauchy horizons emanating from the boundary of U , as shown in Fig. 6, and also by sections of the red and blue dotted curves. These sections are spacelike w.r.t. g. In Minkowski spacetime, the Cauchy horizons appear as shown in Fig. 7.
The full domain of dependence D(S) = D + (S) ∪ D − (S) of S in the spacetime (Ψ(U ), g) can be written as D(S) = Ψ(U ) where U is the region of the (ρ, σ) plane shown in Fig. 6. The hyperbolic solution (Ψ(U ), Φ) is a globally hyperbolic development (GHD) of the initial data on S. However, it is not a maximal globally hyperbolic development (MGHD) because, as we will show, it can be extended into the region X whilst maintaining global hyperbolicity.
To extend the solution into X we have to "choose a branch". Consider first extending to include the region V of the (ρ, σ) plane as shown in Figure 8. To do this let P = U ∪ V \C. Then Ψ is injective on P . The image Ψ(P ) is shown in Fig. 9. It covers all of Minkowski spacetime except for the line Ψ(V ∩ C) shown in red. The gradient of Φ diverges as this line is approached from the region X, i.e., from the left. Hence the solution Φ cannot be extended to this line so the hyperbolic solution (Ψ(P ), Φ) is inextendible. Approaching the red line from the right, the gradient remains bounded but the solution Φ is discontinuous on the red line. This corresponds to the behaviour sketched in Fig. 2.   We now ask what region of the hyperbolic solution (Ψ(P ), Φ) is predictable from the initial surface S. This is easiest to understand in the (ρ, σ) plane shown in Fig. 8, because null geodesics of g are at 45 • . Clearly there are past directed causal curves in P that end on C or ∂W , and do not cross Σ. The image (under Ψ) of points on such curves do not belong to the domain of dependence of S in the spacetime (Ψ(P ), g). This domain of dependence is Ψ(P ) where P is the union of U and the hatched region ofV shown in Fig. 8. P is bounded to the future by a pair of Cauchy horizons shown in light blue and orange, as well as by (spacelike w.r.t. g) parts of the solid and dotted red curves. The region Ψ(P ) of Minkowski spacetime is shown in Fig. 9. Note that the Cauchy horizons both emanate from the singular curve shown in red, the behaviour near this curve is shown in more detail in Fig. 10.
The hyperbolic solution (Ψ(P ), Φ) is a GHD of the initial data on S. Note that Ψ(U ) is a subset of Ψ(P ) and the two solutions agree on this subset. Therefore the GHD (Ψ(P ), Φ) is an extension of the GHD (Ψ(U ), Φ). 8 In constructing this extension we made the choice to extend the solution from U into V . However we could instead choose to extend the solution from U into W as in Figure 11. Let Q = U ∪ W \C. As before, Ψ is injective on Q so this defines a hyperbolic solution (Ψ(Q), Φ). This gives a different extension of the solution (Ψ(U ), Φ) into the region X. The image Ψ(Q) is shown in Fig. 12. It covers the whole of Minkowski spacetime except for the line Ψ(W ∩C) shown in blue. The gradient of Φ diverges as this line is approached from the region X, i.e., from the right. Hence the solution Φ cannot be extended to this line so the hyperbolic solution (Ψ(Q), Φ) is inextendible. Approaching the blue line from the left, the gradient remains bounded but the solution Φ is discontinuous on the red line. This corresponds to the behaviour sketched in Fig.  3.  As before, we can now define the subset Q ⊂ Q such that Ψ(Q ) is the domain of dependence of the initial surface S in the spacetime (Ψ(Q), g) and hence (Ψ(Q ), Φ) is a GHD of the initial data on S. Q is bounded to the future by the Cauchy horizons shown in Fig. 11 and by spacelike (w.r.t. g) portions of the solid and dotted blue curves. The behaviour of these Cauchy horizons in Minkowski spacetime is shown in Fig. 12.
We have now constructed several different GHDs of the initial data on S, namely Ψ(U ), Ψ(P ) and Ψ(Q ). Ψ(P ) and Ψ(Q ) are both extensions of Ψ(U ). Note that the solutions in Ψ(P ) and Ψ(Q ) differ in the region X. Hence we have constructed two GHDs of the same initial data that differ in some region. This implies that there cannot be some larger GHD that contains both Ψ(P ) and Ψ(Q ) and hence there cannot exist a unique MGHD of the data on S.
Consider the intersection Ψ(P ) ∩ Ψ(Q ). Note that this is disconnected, consisting of two connected components. One component contains S but no points of X and the other component is a subset of X. The two solutions agree on the former component but they disagree on the latter component. In Section 4 we will prove that this disconnectedness is a necessary condition for two GHDs to differ in some region.
Another point to emphasize is that the boundary of Ψ(P ) consists of a section (along the right boundary of X, between the lower black dot and the lower cross of Fig. 10), which can be approached from both sides (either the left or the right) within Ψ(P ). In other words Ψ(P ) lies on both sides of its boundary. (The same is true for Ψ(Q ).) In section 4 we will show that this property is a necessary condition for non-uniqueness of MGHDs.
We have constructed two GHDs which differ in X. This demonstrates that there must exist at least two distinct MGHDs. In fact one can argue that there are infinitely many distinct MGHDs. Consider again the GHD Ψ(U ). Previously we extended this by either extending from U into V or from U into W . But we could instead extend from U into part of V and part of W . This can be done preserving injectivity of Ψ. In Minkowski spacetime, this corresponds to simultaneously extending Ψ(U ) across the dotted red and blue curves, in a neighbourhood of the lower black dot of Fig. 7. This corresponds to extending simultaneously from both the left and the right in Fig. 1. If one extends "as far as possible" then the resulting solution will have a discontinuity at some location other than A or B (on Fig. 1), and there are clearly infinitely many points where one could choose to locate the discontinuity. A MGHD constructed this way will be discontinuous along a curve in region X starting at the lower black dot of Fig. 7. The gradient of the solution will remain bounded as this curve is approached from either side (although the gradient diverges at the black dot). There are infinitely many different locations for this curve, corresponding to the infinitely many different MGHDs.
All of the different MGHDs agree in the region Ψ(U ) but they differ in X. Therefore it seems appropriate to define the maximal unique globally hyperbolic development (MUGHD) (R, Φ) of the initial data on S as follows. R is the largest open subset of Minkowski spacetime on which the solution is uniquely determined by the data on S. Such a development is necessarily globally hyperbolic with Cauchy surface S. For the above example, we have R = Ψ(U ). As we have seen, the solution (R, Φ) can be extended, whilst maintaining global hyperbolicity, but not in a unique way. From Figs 6 and 7 we see that the future boundary of R consists of a singular point (the lower black dot in Fig. 7) from which emanate a pair of spacelike (w.r.t. g) curves which connect to a pair of future Cauchy horizons (which are null w.r.t. g). The solution can be smoothly, but non-uniquely extended across the spacelike curves and the Cauchy horizons.
The extendibility across the spacelike curves is a new kind of breakdown of predictability. Fig. 7 suggests that we should view these spacelike curves (the early time sections of the red and blue dotted curves) as a "consequence" of the formation of a singularity (the black dot). This interpretation is suggested if one uses x 0 as a time function (e.g. in a numerical simulation). However, since these curves are spacelike, they are not in causal contact with the singularity. Furthermore, it is just as legitimate to use y 0 as a time function. From this point of view, Fig.  6 shows that the spacelike curves form before (i.e. at earlier y 0 ) the singular point. So it is incorrect to ascribe the breakdown of predictability to the formation of the singularity.
This behaviour is worrying. Given a development of the data on S, there is no general way of determining, from the solution itself, which region of it belongs to the MUGHD. To determine this region one has to construct all GHDs with the same initial data! This is much worse than the failure of predictability associated with the formation of a Cauchy horizon because the location of a Cauchy horizon within a development can be determined from the solution itself.
How would the non-uniqueness of MGHDs manifest itself in, say, a numerical simulation? The answer is that the solution will depend not just on the initial data but also on the choice of time function. To see this, consider the globally hyperbolic development (Ψ(P ), Φ). Since S is a Cauchy surface we can choose a global time function for (Ψ(P ), Φ) such that S is a surface of constant time. We can do the same for (Ψ(Q ), Φ). Of course these two time functions are different but either could be used for a numerical evolution starting from the data on S. For points in the MUGHD Ψ(U ), the results of these two numerical evolutions will agree. However, for points in X, the results will disagree. In practice one would not know a priori which points belong to the MUGHD, i.e., one would not know in what region the results of the numerical evolution are independent of the choice of time function. 9 Note that, for any solution, the domain of dependence of S defined using the Minkowski metric m is a subset of the domain of dependence of S defined using g. Hence a solution which is globally hyperbolic w.r.t. g is also globally hyperbolic w.r.t. m. We could therefore ask about uniqueness of MGHDs defined w.r.t. m instead of w.r.t. g. We'll refer to these as m-MGHDs. For the above example, there is indeed a unique m-MGHD: it is bounded to the future by two future-directed null (w.r.t. m) lines emanating from the lower black dot in Fig. 7. We'll prove in Section 4 that any subluminal equation always admits a unique m-MGHD, which is a subset of the MUGHD. However, if the speed of propagation w.r.t. g is much less than the speed of propagation w.r.t. m then the m-MGHD may not contain a large part of the MUGHD.
We have used the Born-Infeld scalar as an example exhibiting non-uniqueness of MGHDs. This example is rather artificial because there is a "more fundamental" underlying theory, namely the Nambu-Goto string, for which there is no problem with predictability. However, our point is that if this pathological feature can occur for a particular scalar field theory then it is to be expected to occur also for other scalar field theories for which there is no analogue of the Nambu-Goto string interpretation.

Superluminal case, c = −1
Non-uniqueness of MGHDs is not a problem in the superluminal case. To see this we first consider the way in which two points in the (ρ, σ) plane that map to the same point in Minkowski space are related. Consider two points p, q in the (ρ, σ) plane such that Ψ(p) = Ψ(q). Nonuniqueness arose in the subluminal theory because we could choose to include either p or q when constructing a GHD. We will argue that this is not possible in the superluminal theory because either p or q must lie to the future of the singular set C and hence one of these points cannot belong to a GHD of data prescribed on a surface S to the past of C.

Consider a GHD (Ψ(O), Φ) of the data on S = Ψ(Σ) where
O is an open region of the (ρ, σ) plane on which Ψ is injective, as discussed in section 3.7. Of course we need Σ ⊂ O. By global hyperbolicity, any past-directed causal (w.r.t. g) curve in Ψ(O) must cross S hence any past-directed causal curve in O must cross Σ.
Consider the straight line connecting r to p in the (ρ, σ) plane. This line crosses C and it is timelike outside C. Furthermore the portion of this line from C to p is future directed because the time function y 1 = (ρ + σ)/2 increases along the line (since both ρ and σ increase). Hence in the (ρ, σ) plane we have a future-directed timelike line from C to p. Equivalently we have a past-directed timelike line from p to C. Now if p belonged to O then any past-directed causal (w.r.t. g) curve from p would have to intersect Σ. But the straight line just discussed does not intersect Σ because it hits the singular set C when extended to the past! Therefore p cannot belong to O. Only q can belong to O. More generally, if there are multiple points of the (ρ, σ) plane lying outsideD and mapping to the same point of Minkowski spacetime then only the earliest such point, i.e. the one with the smallest value of y 1 , can belong to O.
This can be illustrated using the same example (3.31) as we studied the subluminal case. In view of the discussion in section 3.1 we can use the same spacetime diagrams but with time running horizontally rather than vertically. However, it is more convenient to rotate the diagrams so that time runs up the page. So in Fig. 13 we plot the (ρ, σ) plane with the y 1 axis vertical and y 0 horizontal. Similarly in Minkowski spacetime we plot in Fig. 14 the x 1 axis vertically and x 0 horizontally (c.f. 3.17). We will take our partial Cauchy surface S = Ψ(Σ) where Σ is a surface of constant y 1 to the past of the singular set C as shown in Fig. 13.
The regions U, V, W, D, X are exactly the same as for the subluminal case. The difference now, as it clear from Fig. 13, is that the region W lies to the future of the singular set C. Hence, in contrast with the subluminal case, W cannot form part of any GHD of the initial data on S.
To construct a MGHD we extend into region V as shown in Fig. 13. The MGHD is bounded to the future by a spacelike part of C (where the gradient of Φ diverges) and a pair of Cauchy horizons. The MGHD in Minkowski spacetime is shown in Fig. 14.
In summary, we have explained why in the superluminal theory there exists a unique MGHD, in contrast with the subluminal theory. Hence the superluminal theory is more predictive than  the subluminal theory. Of course the formation of a Cauchy horizon indicates that there is still a loss of predictability in the superluminal theory in the sense that the solution can be extended beyond the Cauchy horizon in a non-unique way. This means that the analogue of the strong cosmic censorship conjecture is false in this theory. However, in contrast with the subluminal theory, it is clear why this breakdown of predictability occurs: the Cauchy horizons emerge from a singularity so the loss of predictability is related to not knowing what boundary conditions to impose at the singularity. This is in contrast with the subluminal theory, where we saw that uniqueness can be lost along a spacelike curve.

Higher dimensions
It is easy to see that this pathological behaviour in the subluminal case is not restricted to two spacetime dimensions. The Born-Infeld scalar field theory in (d + 1)-dimensional Minkowski spacetime is defined by generalizing the action (3.7) to d + 1 dimensions. The two dimensional theory can be obtained trivially from the d + 1 dimensional theory by assuming that Φ does not depend on d − 1 of the spatial coordinates. Hence our 2d solutions can be interpreted as solutions in d + 1 dimensions with translational invariance in d − 1 directions. Such solutions are unphysical because they do not decay at infinity. However, given initial data for such a solution, one could modify the data outside a ball of radius R so that it becomes compactly supported. In the subluminal case, the resulting solution would be unchanged in the region inside the ingoing Minkowski lightcone emanating from the surface of this ball. Hence if R is chosen large enough then the evolution of the solution inside the ball will behave as above for long enough to see the non-uniqueness discussed above. Therefore the subluminal Born-Infeld scalar in d + 1 dimensions will not always admit a unique MGHD.
In the superluminal case, there does exist a unique MGHD: we will prove below that any superluminal equation always admits a unique MGHD.
4 Uniqueness properties of the initial value problem for quasilinear wave equations

Introduction
In this section we consider a quasilinear wave equation of the form where u : R d+1 ⊇ U → R, g is a smooth Lorentz metric valued function, 10 F is smooth with F (0, 0) = 0, and the coordinates used for defining (4.39) are the canonical coordinates x µ on R d+1 . Let S ⊆ R d+1 be a connected hypersurface of R d+1 . Initial data for (4.39) on S consists of a smooth real valued function f 0 : S → R and a smooth one form α 0 (with values in T * R d+1 ) along S such that X(f 0 ) = α 0 (X) holds for all vectors X tangent to S and such that the hypersurface S is spacelike with respect to the Lorentzian metric g(f 0 , α 0 ). A globally hyperbolic development (GHD) of initial data (f 0 , α 0 ) on a hypersurface S for (4.39) consists of a smooth solution u : U → R of (4.39) with S ⊆ U and u| S = f 0 , du| S = α 0 , and such that U is globally hyperbolic with respect to the Lorentzian metric g(u, du) with Cauchy hypersurface S.
As we will show/recall in the following, the initial value problem for the equation (4.39) with initial data given on a hypersurface S is locally well-posed. Here, we mean by this that the following two properties hold: 1. there exists a globally hyperbolic development u : U → R of the initial data 2. given two globally hyperbolic developments u 1 : U 1 → R and u 2 : U 2 → R of the same initial data, then there exists a common globally hyperbolic development (CGHD), that is, a globally hyperbolic development v : Note that the second property is only a weak version of what one might understand under 'local uniqueness', since it allows for the existence of a third globally hyperbolic development u 3 : U 3 → R of the same initial data such that there exists an x ∈ V ∩ U 3 with u 3 (x) = u 1 (x) = u 2 (x). 11 The aim of this section of the paper is to investigate the uniqueness properties for solutions of quasilinear wave equations. In Section 4.2 we first prove the second property of the local well-posedness statement from above and then establish the main theorem of this section: two globally hyperbolic developments of the same initial data agree on the intersection of their domains if this intersection is connected. Section 4.3 then specialises to quasilinear wave equations (4.39) with the property that there exists a vector field T on R d+1 such that T is timelike with respect to g µν (u, du) for all u, du. (4.40) In particular superluminal equations have this property. We show that for such equations the intersection of the domains of two globally hyperbolic developments of the same initial data is always connected -and we thus obtain that any two globally hyperbolic developments agree on the intersection of their domains. The case of subluminal equations is considered in Section 4.4.
Here, we show that if one of the two globally hyperbolic developments is also globally hyperbolic with respect to the Minkowski metric, then again, the intersection of the domains is connected -and we can thus apply our main theorem from Section 4.2. The next three sections deal with existence questions: Section 4.5 proves the first property of the above local well-posedness statement, Section 4.6 establishes the existence of a unique maximal globally hyperbolic development for quasilinear wave equations with the property (4.40), and Section 4.7 considers subluminal equations and shows the existence of a maximal region on which solutions are unique and which is globally hyperbolic (i.e. a MUGHD).
The final section, Section 4.8, present a uniqueness criterion for general quasilinear wave equations of a very different flavour. It states that if there exists a maximal globally hyperbolic development with the property that its domain of definition always lies to just one side of its boundary, then this maximal globally hyperbolic development is the unique one. In particular this implies uniqueness of the MGHD constructed in Ref. [7].

Uniqueness results for general quasilinear wave equations
Proposition 4.41 (Local uniqueness). Let u 1 : U 1 → R and u 2 : U 2 → R be two globally hyperbolic developments for (4.39) of the same initial data prescribed on a hypersurface S ⊆ R d+1 . Then there exists a common globally hyperbolic development v : V → R.
Proof. For p ∈ S let W p ⊆ R d+1 be an open neighbourhood of p on which there exists slice coordinates for S and in which the Lorentzian metric g(f 0 , α 0 ) given by the initial data is C 0 -close to the Minkowski metric. Moreover, we require W p ⊆ U 1 ∩ U 2 . Let S p be an open neighbourhood of p in S the closure of which is compactly contained in W p . The standard literature methods (see for example [17]) ensure that there is an open neighbourhood DS p ⊆ W p of S p with the property that any two solutions, which are defined on DS p and attain the given 11 We will discuss how this might happen at the end of Section 4.7.
initial data on S p , agree, and such that DS p is globally hyperbolic with Cauchy hypersurface S p . It thus follows that u 1 | DSp = u 2 | DSp . We now set V = p∈S DS p . It is immediate that u 1 and u 2 agree on this set and that V is globally hyperbolic with Cauchy hypersurface S.
One can now ask whether global uniqueness holds, which is the property that if u 1 : U 1 → R and u 2 : U 2 → R are two globally hyperbolic developments of the same initial data, then u 1 and u 2 agree on U 1 ∩ U 2 . Note that 'global' refers to the property that 'the two solutions agree in all of U 1 ∩ U 2 ' -in contrast to the local result provided by Proposition 4.41, which only guarantees uniqueness in some smaller subset of U 1 ∩ U 2 .
The last author sketched an idea for a proof of global uniqueness in Section 1.4.1 of [13]. However, this sketch has the flaw that it tacitly assumes that given two globally hyperbolic developments u 1 : U 1 → R and u 2 : U 2 → R of the same initial data, that U 1 ∩ U 2 is then connected -which is in general not true as illustrated by the example presented in Section 3.8 of this paper. The necessity of the assumption of connectedness enters in the sketch as follows: One starts by considering the maximal globally hyperbolic region W contained in U 1 ∩ U 2 on which u 1 and u 2 agree (i.e. the maximal common globally hyperbolic development (MCGHD)) and one would like to show that this region coincides with U 1 ∩ U 2 . Assuming W U 1 ∩ U 2 one can find a boundary point of W in U 1 ∩ U 2 provided U 1 ∩ U 2 is connected. The argument then proceeds by constructing a spacelike slice through a suitable boundary point and appealing to the local uniqueness result in order to conclude that u 1 and u 2 also agree on a neighbourhood of this slice and thus on an even bigger globally hyperbolic region than W -a contradiction to the maximality of W . This is roughly how one proves global uniqueness under the condition that U 1 ∩ U 2 is connected. Note that if U 1 ∩ U 2 is disconnected, the same argument shows that the domain W of the MCGHD equals the connected component of U 1 ∩ U 2 that contains S.
For the Einstein equations one does not need to condition the global uniqueness statement, since one has the freedom to construct the underlying manifold -there is no fixed background. We will explain this in the following: Given two globally hyperbolic developments u 1 and u 2 for the Einstein equations one constructs a bigger one in which both are contained (and thus proves global uniqueness) by glueing u 1 and u 2 together along the MCGHD of u 1 and u 2 . However, in the case that u 1 and u 2 are two globally hyperbolic developments of a quasilinear wave equation on a fixed background such that U 1 ∩ U 2 is disconnected, glueing them together along the MCGHD (which equals the connected component of U 1 ∩ U 2 which contains the initial data hypersurface), would yield a solution which is no longer defined on a subset of R d+1 , but instead on a manifold which projects down on U 1 ∪ U 2 ⊆ R d+1 and contains the other connected components of U 1 ∩ U 2 twice. Of course this is not allowed if we insist that solutions of (4.39) should be defined on a subset of R d+1 . So the key difference between the Einstein equations and a quasilinear wave equation (4.39) is that for the former the underlying manifold is constructed along with the solution whereas for the latter, it is fixed a priori. This is the reason why one does not need to condition the global uniqueness statement for the Einstein equations.
Theorem 4.42. Let u 1 : U 1 → R and u 2 : U 2 → R be two globally hyperbolic developments of (4.39) arising from the same initial data given on a connected hypersurface S ⊆ R d+1 . Assume that U 1 ∩ U 2 is connected. Then u 1 and u 2 agree on U 1 ∩ U 2 .

Proof.
Step 1: Given two globally hyperbolic developments u 1 : U 1 → R and u 2 : U 2 → R of (4.39) arising from the same initial data on S, we consider the set {v α : V α → R | α ∈ A} of all common globally hyperbolic developments. By Proposition 4.41 we know that this set is non-empty. We define v 0 : V 0 → R, where V 0 := α∈A V α and v 0 (x) = v α (x) for x ∈ V α . It is immediate that this is well-defined and that v 0 : V 0 → R is a common globally hyperbolic development with the property that any other common globally hyperbolic development is a subset of V 0 . We call v 0 : V 0 → R the maximal common globally hyperbolic development. We now set out to show that V 0 = U 1 ∩ U 2 , from which the theorem follows.
Step 2: Assume that V 0 U 1 ∩ U 2 . Since we assume that U 1 ∩ U 2 is connected, there exists then a point q ∈ ∂V 0 ∩ U 1 ∩ U 2 . 12 Without loss of generality we assume that q ∈ J + g(u 1 ,du 1 ) (S, U 1 ). We show in the following that there exists a point p ∈ ∂V 0 ∩ U 1 ∩ U 2 such that holds. Such a point p can be thought of as a point where the boundary is spacelike. In the following the causality relations are with respect to the metric g(u 1 , du 1 ). If (4.43) holds for q = p, we are done -hence, we assume that there is a second point r ∈ J − (q, U 1 ) ∩ ∂V 0 ∩ J + (S, U 1 ). The global hyperbolicity of V 0 with Cauchy hypersurface S together with the openness of the timelike relation implies that ∂V 0 ∩ J + (S, U 1 ) is achronal. Hence, the past directed causal curve γ : [0, 1] → U 1 with γ(0) = q and γ(1) = r is a null geodesic. Using the global hyperbolicity of V 0 it follows that γ([0, 1]) ⊆ ∂V 0 ∩ U 1 ∩ U 2 , since if there were a t ∈ (0, 1) with γ(t) ∈ V 0 , then we would also obtain γ(1) ∈ V 0 -and if γ(t) ∈ U 1 \V 0 , then by the openness of the timelike relation one could find a past directed timelike curve starting from a point in V 0 close to q that lies completely in J + (S, U 1 ) and ends at a point in U 1 \ V 0 close to γ(t). Moreover, γ([0, 1]) ⊆ U 2 , since γ(0) ∈ U 2 and by the smoothness of u 2 , γ is also a past directed null geodesic in the globally hyperbolic U 2 -hence, it cannot leave U 2 without first crossing S.
We now extend γ maximally in U 1 to the past. The global hyperbolicity of U 1 entails that γ has to intersect S, thus entering V 0 and leaving ∂V 0 . We now consider γ −1 ∂V 0 ∩J + (S, U 1 ) . The argument from the last paragraph shows that this is a connected interval, and the closedness of ∂V 0 ∩ J + (S, U 1 ) in J + (S, U 1 ) together with S ⊆ V 0 implies that γ −1 ∂V 0 ∩ J + (S, U 1 ) =: [0, a] for some a > 0. Note also that it follows from the last paragraph that γ(a) ∈ U 2 . We claim that p := γ(a) ∈ J + (S, U 1 ) satisfies (4.43).
Assume p = γ(a) does not satisfy (4.43). Then there is a point s ∈ J − (p, U 1 )∩∂V 0 ∩J + (S, U 1 ). As before, one can connect p and s by a past directed null geodesic that is contained in ∂V 0 ∩ U 1 . However, by the definition of a, this null geodesic cannot be the continuation of γ| [0,a] . We can thus connect q and s by a broken null geodesic, and thus by a timelike curve in U 1 -contradicting the achronality of ∂V 0 ∩ J + (S, U 1 ).
Step 3: Let p ∈ ∂V 0 ∩ U 1 ∩ U 2 be as in (4.43). We claim that for every open neighbourhood W ⊆ U 1 of p there exists a point q ∈ I + (p, W ) such that J − (q, To show this, let p be as above and assume the claim was not true. Then there exists a neighbourhood W ⊆ U 1 of p such that for all q ∈ I + (p, W ) there exists a pointq ∈ J − (q, In particular, let us choose a sequence q j ∈ I + (p, U 1 ) and q j ∈ J − (q, U 1 ) ∩ (U 1 \ V 0 ) ∩ J + (S, U 1 ) ∩ (U 1 \ W ) with q j ∈ I − (q 0 , U 1 ) for all j ∈ N and q j → p.
Step 5: Since u 1 and u 2 agree on V 0 , by continuity they (and their derivatives) also agree on Σ ⊆ V 0 ∩ U 1 ∩ U 2 . Consider now a point in Σ ∩ ∂V 0 ∩ U 1 ∩ U 2 and take a simply connected neighbourhood W ⊆ U 1 ∩ U 2 thereof such that Σ W := Σ ∩ W is a closed hypersurface in W . By [16,Chapter 14,46. Corollary], Σ W is acausal in W .
Let D 1 Σ W ⊆ W ⊆ U 1 denote the domain of dependence of Σ W in W ⊆ U 1 and D 2 Σ W the domain of dependence of Σ W in W ⊆ U 2 . u 1 | D 1 Σ W and u 2 | D 2 Σ W are both globally hyperbolic developments of the same initial data on Σ W , and thus by Proposition 4.41 they agree in some small globally hyperbolic neighbourhood O ⊆ D 1 Σ W ∩ D 2 Σ W of Σ W . Note that O contains at least one point of ∂V 0 ∩ U 1 ∩ U 2 . Moreover, it is easy to see that V 0 ∪ O is globally hyperbolic with Cauchy hypersurface S. This, however, contradicts the maximality of V 0 . Remark 4.45. Let us remark that the proof in particular shows that under the assumptions of Theorem 4.42 the intersection U 1 ∩ U 2 is the maximal common globally hyperbolic development of u 1 : U 1 → R and u 2 : U 2 → R. If we fix a choice of normal of the initial data hypersurface S and stipulate that it is future directed in U 1 as well as in U 2 , then it follows in particular that a point x ∈ U 1 ∩ U 2 , which lies to the future of S in U 1 , also lies to the future of S in U 2 . Similarly for the past.
The following is an immediate consequence of the previous theorem. It shows that global uniqueness can only be violated for quasilinear wave equations in a specific way.
Corollary 4.46. Let u 1 : U 1 → R and u 2 : U 2 → R be two globally hyperbolic developments of (4.39) arising from the same initial data on a connected hypersurface S ⊆ R d+1 . If there exists an x ∈ U 1 ∩ U 2 with u 1 (x) = u 2 (x), then U 1 ∩ U 2 is not connected.
In particular, we recover that globally defined solutions are unique: Corollary 4.47. Let u 1 : R d+1 → R be a globally defined globally hyperbolic development of (4.39) arising from some initial data on a connected hypersurface S ⊆ R d+1 . Let u 2 : U 2 → R be another globally hyperbolic development of (4.39) of the same initial data. Then u 1 | U 2 = u 2 .
In the next two sections we consider two globally hyperbolic developments u 1 : U 1 → R and u 2 : U 2 → R of the same initial data and discuss criteria that ensure that U 1 ∩ U 2 is connected. Here, the choice of the initial data hypersurface S plays an important role. This can already be seen from the special case of the linear wave equation in Minkowski space: consider a spacelike but not achronal hypersurface that winds up around the x 0 -axis in R×(B d 2 (0)\B d 1 (0)) ⊆ R d+1 . Prescribing generic initial data on this hypersurface, the extent of the future development restricts the extent of the past development. Given two globally hyperbolic developments, their intersection is in general not connected and global uniqueness does not hold. However, it is easy to show (see also Section 4.3) that for spacelike initial data hypersurfaces which are moreover achronal, this pathology for the linear wave equation in Minkowski space cannot occur. This example shows that any result demonstrating connectedness of U 1 ∩ U 2 for more general quasilinear equations will require some additional assumptions on the initial surface S analogous to the achronality assumption in Minkowski spacetime.

Uniqueness results for superluminal quasilinear wave equations
In the following we consider quasilinear wave equations (4.39) that enjoy property (4.40), i.e., that there exists a vector field T on R d+1 such that T is timelike with respect to g µν (u, du) for all u, du. In particular, superluminal equations enjoy this property, since one can take T = ∂/∂x 0 where x µ are inertial frame coordinates. We will show that for such equations the complication of U 1 ∩ U 2 being disconnected cannot arise, as long as the initial data is prescribed on a hypersurface S with the property that every maximal integral curve of T intersects S at most once. 13 Lemma 4.48. Assume that there exists a vector field T on R d+1 such that T is timelike with respect to g µν (u, du) for all u, du, where g is as in (4.39). Let u 1 : U 1 → R and u 2 : U 2 → R be two globally hyperbolic developments of (4.39) arising from the same initial data on a connected hypersurface S which has the property that every maximal integral curve of T intersects S at most once. Then U 1 ∩ U 2 is connected.
Proof. Let u 1 : U 1 → R, u 2 : U 2 → R be two globally hyperbolic developments arising from the same initial data on S and let x ∈ U 1 ∩ U 2 . Let γ be the maximal integral curve of T through x. By assumption, γ intersects S at most once. Since γ ∩ U 1 and γ ∩ U 2 are timelike curves in U 1 , U 2 , respectively, and U 1 , U 2 are globally hyperbolic with Cauchy hypersurface S, it follows that γ intersects S exactly once and that the portion of γ from x to γ ∩ S is contained in U 1 as well as in U 2 . This shows the connectedness of U 1 ∩ U 2 .
Let us remark, that one can replace in the above lemma the assumption that S is a connected hypersurface such that every maximal integral curve of T intersects S at most once, with the assumption that S is a hypersurface that separates R d+1 into two components. We leave the small modification of the proof to the interested reader.
Corollary 4.49. Assume that there exists a vector field T on R d+1 such that T is timelike with respect to g µν (u, du) for all u, du, where g is as in (4.39) and that initial data is posed on a connected hypersurface S which has the property that every maximal integral curve of T intersects S at most once.
Given two globally hyperbolic developments u 1 : U 1 → R and u 2 : U 2 → R, we then have Proof. This follows directly from Lemma 4.48 and Theorem 4.42.

Uniqueness results for subluminal quasilinear wave equations
We recall that a quasilinear wave equation of the form (4.39) is called subluminal iff the causal cone of g(u, du) is contained inside the causal cone of the Minkowski metric m = diag(−1, 1, . . . , 1). As shown in Section 3.8 of this paper, in general global uniqueness does not hold for subluminal quasilinear wave equations -even if the initial data is posed on the well-behaved hypersurface {x 0 = 0}. However, as we shall show below, developments are unique in regions that are globally hyperbolic with respect to the Minkowski metric. Recall the terminology introduced in Section 3.8: we say that a GHD of a subluminal quasilinear wave equation is a m-GHD iff it is also globally hyperbolic with respect to the Minkowski metric with Cauchy hypersurface S. As usual, S denotes here the initial data hypersurface.
Lemma 4.50. Let u 1 : U 1 → R and u 2 : U 2 → R be two GHDs of a subluminal quasilinear wave equation (4.39) arising from the same initial data given on a connected hypersurface S that is achronal with respect to the Minkowski metric m. Assume, moreover, that u 1 : U 1 → R is a m-GHD. Then U 1 ∩ U 2 is connected.
Proof. Let x ∈ U 1 ∩ U 2 and assume without loss of generality that x ∈ I + g(u 1 ,du 1 ) (S, U 1 ). We claim that this implies x ∈ I + g(u 2 ,du 2 ) (S, U 2 ). To see this, assume x ∈ I − g(u 2 ,du 2 ) (S, U 2 ). Hence, there exists a future directed timelike curve from S to x in U 1 and a future directed timelike curve from x to S in U 2 . They are both future directed timelike with respect to the Minkowski metric. Concatenating the two curves gives a contradiction to the achronality of S with respect to m. This shows x ∈ I + g(u 2 ,du 2 ) (S, U 2 ). Let γ be a curve in U 2 that starts at x and is timelike, past directed, and past inextendible w.r.t. g(u 2 , du 2 ). It thus intersects S. However, γ is also a past directed timelike curve with respect to m, and the global hyperbolicity of U 1 with respect to m implies that γ cannot leave U 1 without first intersecting S. Thus, the segment of γ from x to S is contained in U 1 ∩ U 2 . This shows the connectedness of U 1 ∩ U 2 .
Together with Theorem 4.42 the above lemma yields Corollary 4.51. Let u 1 : U 1 → R and u 2 : U 2 → R be two GHDs of a subluminal quasilinear wave equation (4.39) arising from the same initial data given on a connected hypersurface S that is achronal with respect to the Minkowski metric m. Assume, moreover, that u 1 : Remark 4.52. Let us remark that better bounds on the light cones of g µν (u, du) translate into an improvement of the uniqueness results. Above, we have only made use of the trivial Minkowski bound on the light cones for subluminal equations. If, for example, for a specific subluminal equation one can improve the a priori bound on the light cones of g µν (u, du) for certain initial data, then one can also improve the uniqueness result for these initial data.

Local existence for general quasilinear wave equations
This section provides the other half of the local well-posedness statement for quasilinear wave equations with data on general hypersurfaces: the local existence result. Moreover, this result is needed for the existence results of a unique maximal GHD for superluminal quasilinear wave equations and of a maximal unique GHD for subluminal quasilinear wave equations.
The proof of Theorem 4.53 proceeds by constructing solutions in local coordinate neighbourhoods around points of the initial data hypersurface and then patching them together. Note that this has to be carried out carefully to ensure that different local solutions agree on the intersection of their domains. Here we make use of Theorem 4.42 to guarantee uniqueness if the intersection of their domains is connected.
Proof. Given the initial data f 0 , α 0 on the hypersurface S (as discussed in Section 4.1) we choose a timelike normal N along S and extend it smoothly off S to yield a vector field which we also denote with N . There exists an open neighbourhood D of {0} × S in R × S and an open neighbourhood T ⊆ R d+1 of S such that the flow Φ of N is a diffeomorphism from D onto T . For p ∈ S let W p ⊆ T ⊆ R d+1 be an open neighbourhood of p on which there exists slice coordinates in which the Lorentzian metric g(f 0 , α 0 ) determined by the initial data is C 0 -close to the Minkowski metric. Let S p ⊆ S be a neighbourhood of p in S with closure that is compactly contained in W p . The standard energy methods in the literature (see for example [17]) yield that there exists a globally hyperbolic development u p : DS p → R for (4.39) of the initial data on S p , where DS p ⊆ W p . Moreover, by choosing DS p smaller if necessary, we can assume that N is timelike on DS p . We now claim that for all p, q ∈ S we have u p = u q on DS p ∩ DS q .
To show this, assume that DS p ∩DS q = ∅ and let A be a connected component of DS p ∩DS q . Consider an x ∈ A. The integral curve of N through x is a timelike curve in DS p as well as in DS q , and thus it has to intersect S p ∩ S q and, moreover, its segment from x to S p ∩ S q is contained in DS p ∩ DS q . This shows that A ∩ (S p ∩ S q ) is non-empty. It will follow a posteriori that A ∩ (S p ∩ S q ) is connected, but for the time being let S A be a connected component of A ∩ (S p ∩ S q ). We denote with D p S A , D q S A the domain of dependence of S A in A with respect to the Lorentzian metric arising from u p and u q , respectively. Since by the above argument involving the timelike integral curves of N , the intersection D p S A ∩D q S A is connected, Theorem 4.42 implies that we have Without loss of generality we assume that r lies to the future of S A . Let γ be any past directed and past inextendible causal curve in A with respect to the metric arising from u p that starts at r. The global hyperbolicity of D p S A ∩ D q S A (see Remark 4.45) implies and hence the part of γ to the causal future of S A is also a past directed causal curve in A with respect to u q . The global hyperbolicity of DS p and DS q shows that γ has to intersect S p ∩ S q , and by (4.54), γ in fact intersects S A . This, however, gives the contradiction r ∈ D p S A ∩ D q S A by definition of the domain of dependence. We thus conclude that D p S A ∩D q S A = A. Moreover, it now follows that u p = u q holds on DS p ∩ DS q . Hence, we can finish the proof by constructing a GHD u : U → R of the given initial data on S by setting U = p∈S DS p and u(x) = u p (x) for x ∈ DS p .

4.6
The existence of a unique maximal GHD for superluminal quasilinear wave equations Theorem 4.55. Assume that there exists a vector field T on R d+1 such that T is timelike with respect to g µν (u, du) for all u, du, where g is as in (4.39) and that initial data is posed on a connected hypersurface S which has the property that every maximal integral curve of T intersects S at most once. Given such initial data, there then exists a unique maximal globally hyperbolic development u max : U max → R, that is, a globally hyperbolic development u max : U max → R with the property that for any other globally hyperbolic development u : U → R of the same initial data we have U ⊆ U max and u max | U = u.
Proof. We consider the set {(u α , U α ) | α ∈ A} of all globally hyperbolic developments u α : U α → R arising from the given initial data on S as above. Note that this is a set and, moreover, it is non-empty by Theorem 4.53. We now define U max := α∈A U α and u max : U → R by u max (x) = u α (x) for α ∈ A with x ∈ U α . Note that the latter is well-defined by Corollary 4.49. In order to see that u max : U max → R is a globally hyperbolic development of (4.39) arising from the given initial data, consider an inextendible timelike curve γ : (a, b) → U max , where −∞ ≤ a < b ≤ ∞, and let a < t 0 < b. We have γ(t 0 ) ∈ U α 0 for some α 0 ∈ A. Let (a 0 , a 1 ) ⊆ I be the maximal interval containing t 0 such that γ| (a 0 ,a 1 ) maps into U α 0 . Since γ| (a 0 ,a 1 ) is an inextendible timelike curve in U α 0 , there exists a τ 0 ∈ (a 0 , a 1 ) with γ(τ 0 ) ∈ S. Thus, it remains to show that γ does not intersect S more than once. Without loss of generality we assume that γ (a 0 ,a 1 ) is future directed in U α 0 . We consider We already know that J is non-empty. Moreover, J is clearly open, since each U α is open. Let t n ∈ J be a sequence with t n → t ∞ ∈ (τ 0 , b) as n → ∞, and let α ∞ ∈ A be such that γ(t ∞ ) ∈ U α∞ . By the openness of U α∞ there is n 0 ∈ N with γ(t n 0 ) ∈ U α∞ . Since t n 0 ∈ J, there exists α n 0 ∈ A with γ [τ 0 , t n 0 ] ⊆ U αn 0 . It now follows from Remark 4.45 that γ(t n 0 ) must also lie to the future of S in U α∞ . Hence, S being a Cauchy hypersurface of U α∞ implies that t ∞ ∈ J. It thus follows that J = (τ 0 , b). We conclude that γ cannot intersect S again to the future of τ 0 . The analogous argument shows that it can neither intersect S again to the past of τ 0 . We thus conclude that U max is globally hyperbolic with Cauchy hypersurface S.
Finally, it is clear that any other globally hyperbolic development of the same initial data is contained in U max .
Remark 4.56. We note that the above construction of a unique maximal globally hyperbolic development is always possible provided the property of global uniqueness holds.

The existence of a maximal unique GHD for subluminal quasilinear wave equations
As mentioned before, for subluminal quasilinear wave equations there does not generally exist a unique maximal globally hyperbolic development. In this section we show existence of a globally hyperbolic development on the domain of which the solution is uniquely defined and which is maximal among all GHDs that have this property. But first we establish some terminology: We consider a subluminal quasilinear wave equation of the form (4.39) and consider initial data prescribed on a connected hypersurface S that is acausal with respect to the Minkowski metric m, i.e., there does not exist a pair of points on S that can be connected by a causal curve within the Minkowski spacetime. We call a GHD u 1 : U 1 → R a unique globally hyperbolic development (UGHD) iff for all other GHDs u 2 : U 2 → R we have u 1 = u 2 on U 1 ∩ U 2 . We note that any m-GHD is a UGHD by Corollary 4.51.
Theorem 4.57. Consider a subluminal quasilinear wave equation of the form (4.39). Given initial data on a connected hypersurface S that is acausal with respect to the Minkowski metric there exists a UGHD u : U → R with the property that the domain of any other UGHD is contained in U . The UGHD u : U → R is called the maximal unique globally hyperbolic development (MUGHD).
Proof. We consider the set {u α : U α → R | α ∈ A} of all UGHDs of the given initial data. Note that this set is non-empty: by Theorem 4.53 there exists a GHD u 1 : U 1 → R and we can now consider the domain of dependence of S in U 1 with respect to the Minkowski metric. By [16,Chapter 14,38. Theorem and 43. Lemma] this gives rise to a m-GHD. By Corollary 4.51 this is a UGHD.
We now set U := α∈A U α and u(x) := u α (x) for x ∈ U α . The latter is well-defined since each u α : U α → R is a UGHD. The same argument as in the proof of Theorem 4.55 shows that u : U → R is a globally hyperbolic development. To show that it is a UGHD, let u 2 : U 2 → R be a GHD and consider x ∈ U ∩ U 2 . There exists an α ∈ A with x ∈ U α , and since u α : U α → R is a UGHD it follows that u 2 (x) = u α (x) = u(x). Finally, it is clear by construction that the domain of any other UGHD is contained in U .
We summarise that given a GHD for a superluminal equation, one knows that it is contained in the unique maximal GHD. For subluminal equations, there are in general GHDs which are not contained in the maximal UGHD. However, given a m-GHD, it is contained in the maximal UGHD.
Furthermore, one can show that for subluminal equations there exists a unique maximal globally hyperbolic development, if one defines global hyperbolicity with respect to the Minkowski metric, i.e., a unique m-MGHD 14 . This object is then contained in the MUGHD, but in general the MUGHD is bigger.
Let us also remark that we expect that the analogue of Theorem 4.57 does not hold for more general quasilinear wave equations, i.e., ones which are neither subluminal nor superluminal. Indeed, even more strongly, we formulate the following This conjecture is based on the following scenario which we think might happen: there exists a quasilinear wave equation of the form (4.39) and initial data such that there exists an infinite family of GHDs the domains of which bend round back towards the initial data hypersurface S and approach it arbitrarily closely, as shown in Figure 15. This would imply that there is no neighbourhood of S on which the solution is uniquely defined. In particular, this would establish the sharpness of the local uniqueness statement of Proposition 4.41.

A uniqueness criterion for general quasilinear wave equations at the level of MGHDs
In this section we consider a general quasilinear wave equation of the form (4.39). Recall that a GHD u 1 : U 1 → R of given initial data posed on a hypersurface S is called a maximal globally hyperbolic development (MGHD) iff there does not exist a GHD u 2 : U 2 → R of the same initial data with U 1 U 2 . Note that by Theorem 4.42 any such GHD u 2 : U 2 → R would agree with u 1 on U 1 , and thus it would correspond to an extension of u 1 : U 1 → R. In other words, a MGHD is a GHD that cannot be extended as a GHD. The example from Section 3.8 shows that in general there can exist infinitely many MGHDs for given initial data. Consider now two such MGHDs u 1 : U 1 → R and u 2 : U 2 → R arising in the example of Section 3.8. Then U 1 ∩U 2 is disconnected. Let A denote the connected component containing S. Consider a point x ∈ U 1 ∩ U 2 which does not lie in A. The phenomenon of nonuniqueness, i.e., that u 1 (x) does not equal u 2 (x), arises, because the 'path of evolution' the second solution takes from A to reach x is blocked because the first solution is already defined in that very region. In the example of Section 3.8, this behaviour arises because U 1 (say) lies "on both sides of its boundary". The following theorem makes this precise and shows that this is the only mechanism at the level of MGHDs that leads to non-uniqueness for general quasilinear wave equations. It states that given an MGHD with the property that its domain of definition always lies to just one side of its boundary, i.e., the domain of definition cannot block evolution elsewhere, then it is the unique MGHD.

(4.60)
Then u 1 : U 1 → R is the unique MGHD, i.e., any other GHD u : U → R satisfies U ⊆ U 1 and thus also u 1 | U = u.
Note that in order to apply this theorem to a concrete example one has to first construct a/the whole MGHD and is only then able to infer a posteriori that the evolution was indeed unique. In particular, this theorem applies to the explicitly constructed MGHD in [7].
Proof. Let u 1 : U 1 → R be a MGHD of given initial data such that (4.60) is satisfied. Let u 2 : U 2 → R be a second GHD of the same initial data and, to obtain a contradiction, we assume that U 2 U 1 . Let us denote the connected component of U 1 ∩ U 2 that contains the initial data hypersurface S with A. A point in the boundary of ∂A cannot be contained in U 1 as well as in U 2 by definition of A. Since we have U 2 U 1 it follows that ∂A ∩ U 2 is non-empty and contained in the complement of U 1 . Thus, we obtain ∅ = ∂A ∩ U 2 ⊆ ∂U 1 . (4.61) Hence, we have exhibited a part of the boundary of the MGHD u 1 : U 1 → R to which the solution extends smoothly (from A). The idea is now to use property (4.60) to show that one can actually extend u 1 across this boundary to obtain a bigger GHD -thus violating the maximality of u 1 . 15 The construction is similar to the on in the proof of Theorem 4.42.
A slight variation of Remark 4.45 shows that the set A is the MCGHD of u 1 : U 1 → R and u 2 : U 2 → R. In particular, A is globally hyperbolic with Cauchy surface S. By (4.61), let q ∈ ∂A ∩ U 2 and assume without loss of generality that q ∈ J + g(u 2 ,du 2 ) (S, U 2 ). We are going to show that there exists a point p ∈ ∂A ∩ U 2 ∩ J + g(u 2 ,du 2 ) (S, U 2 ) with J − g(u 2 ,du 2 ) (p, U 2 ) ∩ ∂A ∩ J + g(u 2 ,du 2 ) (S, U 2 ) = {p} . (4.62) The proof of this is analogous to Step 2 in the proof of Theorem 4.42 and is only sketched in the following. Assume (4.62) does not hold for p = q. Then there exists another point r ∈ J − g(u 2 ,du 2 ) (p, U 2 ) ∩ ∂A ∩ J + g(u 2 ,du 2 ) (S, U 2 ). The global hyperbolicity of A implies the achronality of ∂A ∩ J + g(u 2 ,du 2 ) (S, U 2 ). Hence, the past directed causal curve connecting q with r is a null geodesic which lies in ∂A ∩ J + g(u 2 ,du 2 ) (S, U 2 ). We now extend this null geodesic maximally to the past and consider the point p where it leaves ∂A ∩ J + g(u 2 ,du 2 ) (S, U 2 ). This point p satisfies (4.62).
Step 3 of the proof of Theorem 4.42 applies literally unchanged if V 0 is replaced by A. Following Step 4 of the proof of Theorem 4.42 we now construct a spacelike (with respect to g(u 2 , du 2 )) hypersurface Σ ⊆ A ∩ U 2 that contains at least one point q ∈ ∂A ∩ U 2 ⊆ ∂U 1 .
By (4.60) we can now find a neighbourhood V of q together with a chart ψ : V → (−ε, ε) d+1 and a continuous function f : (−ε, ε) d → (−ε, ε) such that in this chart ∂U 1 ∩ V is given by the graph of f , U 1 ∩ V lies below the graph of f , and V \ U 1 lies above the graph of f . We can, after making V smaller if necessary, assume that V ⊆ U 2 and that the spacelike hypersurface Σ V := Σ ∩ V is a closed hypersurface in V . It follows from [16,Chapter 14,46. Corollary] that Σ V is acausal in V ⊆ U 2 . We consider now the domain of dependence DΣ V of Σ V in V ⊆ U 2 . Clearly, DΣ V contains points that lie above the graph of f in the chart ψ. We can now define u 3 : U 3 → R, U 3 := U 1 ∪ DΣ V , u 3 (x) := u 1 (x) for x ∈ U 1 and u 3 (x) := u 2 (x) for x ∈ DΣ V . This is well defined since the region below the graph of f in the chart ψ lies in A, where u 1 and u 2 agree. It is easy to see that u 3 : U 3 → R is a GHD the domain of which contains that of the MGHD u 1 : U 1 → R. This is a contradiction.