Communicated by P. Chrusciel

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.

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 [

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 a metric

Our aim, then, is determine whether there is any qualitative difference between MGHDs for subluminal and superluminal equations.

In Sect.

In Sect.

For larger amplitude, it is known that the solution can form singularities in the subluminal theory [

This is worrying behaviour. Given a solution defined in some region

In Sect.

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

An important application of the notion of a MGHD is Christodoulou’s work on shock formation in relativistic perfect fluids [

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 [

Consider a scalar field in -dimensional Minkowski spacetime. Assume that the field satisfies a quasilinear equation of motion

We will say that (

Now assume that we have a Minkowski metric on (i.e. a flat, Lorentzian metric), with inverse . We call the above equation

Most equations are neither subluminal nor superluminal e.g. because the null cones of and may not be nested or because the relation between the null cones of and may be different for different field configurations. Note also that the standard wave equation () is both subluminal and superluminal according to our definitions.

Clearly these definitions depends on the choice of . There are infinitely many Minkowski metrics on . An equation might be subluminal w.r.t. one choice of and superluminal w.r.t. some other choice. However, for many equations there exists no such that the equation is either subluminal or superluminal. In physics applications one usually has a preferred choice of , i.e., is “the” spacetime metric. In particular, this is the case for the class of Lorentz invariant equations (defined below).

Since

Let’s now discuss the initial value problem for an equation of the form (

We’ll say that a hyperbolic solution (

A GHD (

MGHDs play an important role in General Relativity. In General Relativity, given initial data for the Einstein equation, there exists a

Surprisingly, the subject of maximal globally hyperbolic developments for equations of the form (

Let’s now consider

The two-dimensional case is special because if

Since the above transformation reverses the overal sign of

In the Appendix we discuss some general properties of superluminal equations in two dimensions, in particular the question of whether solutions of such an equation can exhibit “causality violation”.

In two dimensional Minkowski spacetime, consider a scalar field with equation of motion obtained from the Born–Infeld action

The equation of motion is

Consider a vector . Note that

The two theories are related by the transformation with fixed. This is the map described in Sect.

It is well-known that the theory is a gauge-fixed version of an infinite Nambu–Goto string whose target space is dimensional Minkowski spacetime. The same is true for except that the target space now has signature, i.e., two time dimensions. The action of such a string is

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 (

We can use the Nambu–Goto string to explain heuristically why there is a problem with the

An example of the string folding back on itself. The gradient is infinite at points

Clearly we have to “choose a branch” of the solution 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

Now note that instead of starting on the left and extending to point

Solution with discontinuity at

Different solution with discontinuity at

Starting from initial data prescribed on some line

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

The above discussion was for the subluminal () theory. We will show below that this problem does not occur for the superluminal theory. This is because, in the superluminal theory, from the 2d Born–Infeld perspective,

The (subluminal) BI scalar theory was solved by Barbashov and Chernikov [

Assuming that is invertible we can write and and the solution is given by

Let and be smooth functions defined for all . Let

Clearly it will be important to determine whether or not is a diffeomorphism.

A necessary (although not sufficient) condition for to be a diffeomorphism is that either throughout

The Jacobian of the map is

A point on the boundary at which corresponds to a singularity:

Assume that is a diffeomorphism such that as for some . Let be a smooth curve with as . Then the gradient of the solution at the point diverges as .

A calculation gives

It can be shown similarly that points of where correspond to a divergence in the second derivative of although we will not need this result below.

We will be mainly interested in causal properties of the metric

Consider a Born–Infeld solution constructed as in Theorem

Direct calculation using (

Note that the vector fields and are null w.r.t.

Consider a Born–Infeld solution constructed as in Theorem

In the subluminal case () we know (Sect.

In the superluminal case (), is timelike w.r.t.

In the superluminal case, this proves that solutions constructed using Theorem

We are interested in globally hyperbolic developments of initial data. It is very easy to determine whether or not a solution constructed using Theorem

Consider a Born–Infeld solution constructed as in Theorem

This is an immediate consequence of (

Thus global hyperbolicity can be checked using the flat metric on

We will show that, given initial data on a surface

Let be distinct points such that . Then the straight line connecting

Let

Theorem

Let and be smooth functions on the real line such that the integrals in (

Following [

We now show that there exists exactly one solution of (

The solution of Theorem

This follows immediately from Lemma

As discussed above, we need to be a diffeomorphism for Eqs. (

We start by recording that the subluminal Born–Infeld scalar equation of motion (

To construct a solution of (

The region in which injectivity of fails can be determined numerically

Plot of the plane in coordinates defined by (

Minkowski spacetime with coordinates . The region

Lemma

We enlarge the GHD by pushing the left large dashed orange line of Fig.

Consider a curve approaching the right boundary of

The large dashed orange line on the left is a line of constant which is tangent to the boundary of

The region contains part of the left boundary of

We construct this new extension of as follows. Define to be the reflection of under . So is an extension of

We now have two different GHDs of the same intial data on

We will now discuss this result and highlight properties of our example that are relevant to the general results of Sect.

Consider the intersection shown in Fig.

The regions and are given by the right/left hatching respectively. The intersection of these regions is disconnected, with one component lying inside

Another point to emphasize is that the boundary of consists of a section (along the right boundary of

We have shown that there exist two distinct MGHDs arising from the same data on

The extendibility across the spacelike curves is a new kind of breakdown of predictability. Fig.

This behaviour is worrying. Given a development of the data on

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 . Since

Note that, for any solution, the domain of dependence of

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.

This ends the heuristic discussion of our example of non-uniqueness. We will now present a rigorous proof of the non-uniqueness of MGHDs.

For the Eq. (

We begin by remarking that we will prove the statement of the theorem with the hypersurface replaced by for . This represents no loss of generality since the Eq. (

We choose and as in (

We begin by noticing that the function

Next we investigate the qualitative behaviour of the intersection of the leaves of constant with the circle of radius . Let denote the restriction of to the circle of radius . Since the latter is compact, it follows that takes on its minimum and its maximum . Hence, the differential of must have at least two zeros.

We now parametrise the circle of radius by where , and we compute . It follows that

On the other hand we have

Similarly, by considering the lower arc of the circle and the plus sign in (

In conclusion, we have established the following qualitative picture: The curves of constant for are disjoint of the circle of radius and lie below it, the curve touches the circle in exactly one point in the quadrant , the curves of constant for intersect the circle in exactly two points, the curve touches the circle in exactly one point in the quadrant , and finally the curves of constant for are disjoint of the circle of radius and are lying above it. This behaviour is summarised in Fig.

The level sets of in the plane. Note that Lemmas

If is a diffeomorphism, then Theorem

For this, let us assume that for we have

The qualitative behaviour of for . The black dots show that for there are three solutions of

To understand the subsets of on which

Let now denote the point of contact of with the circle . We have and . We now define the region

To define the second GHD , we set

The domain

The domain

For this consider a curve . By continuity we can choose small enough such that the curve intersects the circle of radius at and such that and . Since , the qualitative analysis of below (

The functions and

We remark the following:

We emphasise that the two GHDs constructed in the proof of Theorem

We note that the two GHDs constructed in the proof of Theorem

The initial data constructed in the proof of Theorem

The following remark might be skipped and come back to when referred to later in Sect.

Recall that Eq. (

Non-uniqueness of MGHDs is

In the superluminal case, recall that the Minkowski metric (

The unique MGHD is obtained by taking

The reason that there is a unique MGHD in the superluminal case but not in the subluminal case was identified in Lemma

In the superluminal case we orient the plot so that the time function is the vertical axis. The large dashed green lines are lines of constant or that are tangent to the circle at their point of contact. The MGHD of the data on

Plot of in Minkowski spacetime, oriented so that is the vertical axis. The MGHD is the region bounded to the future by the spacelike (w.r.t.

It is easy to see that the pathological behaviour in the subluminal case is not restricted to two spacetime dimensions. The Born–Infeld scalar field theory in -dimensional Minkowski spacetime is defined by generalizing the action (

In the higher-dimensional superluminal case, there is a unique MGHD: we will prove below that

In this section we consider a quasilinear wave equation of the form

Let be a connected hypersurface of .

As we will show/recall in the following, the initial value problem for the Eq. (

there exists a globally hyperbolic development of the initial data

given two globally hyperbolic developments and of the same initial data, then there exists a

The aim of this section of the paper is to investigate the uniqueness properties for solutions of quasilinear wave equations. In Sect.

The next three sections deal with existence questions: Sect.

The final section, Sect.

(Local uniqueness). Let and be two globally hyperbolic developments for (

For let be an open neighbourhood of

One can now ask whether

The last author sketched an idea for a proof of global uniqueness in Section 1.4.1 of [

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 and for the Einstein equations one constructs a bigger one in which both are contained (and thus proves global uniqueness) by glueing and together along the MCGHD of and . However, in the case that and are two globally hyperbolic developments of a quasilinear wave equation on a fixed background such that is disconnected, glueing them together along the MCGHD (which equals the connected component of which contains the initial data hypersurface), would yield a solution which is no longer defined on a subset of , but instead on a manifold which projects down on and contains the other connected components of twice. Of course this is not allowed if we insist that solutions of (

Let and be two globally hyperbolic developments of (

The proof is based on ideas found in [

Given two globally hyperbolic developments and of (

We now set out to show that , from which the theorem follows. Assume that . Since we assume that is connected, there exists then a point .

In the following the causality relations are with respect to the metric . Let

We now extend maximally in to the past. The global hyperbolicity of entails that has to intersect

Assume does not satisfy (

To show this, let

Let be as in (

Let denote the domain of dependence of in and the domain of dependence of in . and are both globally hyperbolic developments of the same initial data on , and thus by Proposition

Let us remark that the proof in particular shows that under the assumptions of Theorem

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.

Let and be two globally hyperbolic developments of (

In particular, we recover that globally defined solutions are unique:

Let be a globally defined globally hyperbolic development of (

In the next two sections we consider two globally hyperbolic developments and of the same initial data and discuss criteria that ensure that is connected. Here, the choice of the initial data hypersurface

In the following we consider quasilinear wave equations (

Assume that there exists a vector field

Let , be two globally hyperbolic developments arising from the same initial data on

Let us remark, that one can replace in the above lemma the assumption that

Assume that there exists a vector field

Given two globally hyperbolic developments and , we then have for all .

This follows directly from Lemma

We recall that a quasilinear wave equation of the form (

Let and be two GHDs of a subluminal quasilinear wave equation (

Let and assume without loss of generality that . We claim that this implies . To see this, assume . Hence, there exists a future directed timelike curve from

Let be a curve in that starts at

Together with Theorem

Let and be two GHDs of a subluminal quasilinear wave equation (

Let us remark that better bounds on the light cones of 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 for certain initial data, then one can also improve the uniqueness result for these initial data.

This section provides the other half of the local well-posedness statement for quasilinear wave equations with data on

(Local existence). Given initial data for a quasilinear wave equation (

Moreover, this result is needed for the

Note that Theorem

Given the initial data on the hypersurface

To show this, assume that and let

Assume now that . Since

Assume that there exists a vector field

Given such initial data, there then exists a

We consider the set of all globally hyperbolic developments arising from the given initial data on

Finally, it is clear that any other globally hyperbolic development of the same initial data is contained in .

We note that the above construction of a unique maximal globally hyperbolic development is always possible provided the property of global uniqueness holds.

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 (

Consider a subluminal quasilinear wave equation of the form (

We consider the set of all UGHDs of the given initial data. Note that this set is non-empty: by Theorem

We now set and for . The latter is well-defined since each is a UGHD. The same argument as in the proof of Theorem

Consider a subluminal quasilinear wave equation of the form (

One considers the set of all

We summarise that given a GHD for a

Let us also remark that we expect that the analogue of Theorem

There are quasilinear wave equations of the form (

This conjecture is based on the following scenario which we think might happen: there exists a quasilinear wave equation of the form (

A possible mechanism for a resolution of Conjecture

In this section we consider a general quasilinear wave equation of the form (

The example from Sect.

Let be a MGHD of given initial data for a quasilinear wave equation of the form (

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

Let be a MGHD of given initial data such that (

A slight variation of Remark

Step 3 of the proof of Theorem

By (

We conclude with presenting a simple criterion that ensures that condition (

Let be a GHD of given initial data posed on an open and connected subset

As an application of the lemma and of Theorem

Before we give the proof, let us also emphasise that condition (

Let us introduce the notation with and let . It thus follows that . Without loss of generality let us assume that and that is future directed. We first show that .

Assume it was not the case and there existed a with . We can then find a point sufficiently close to

In particular, it follows that . Let now

We are grateful to M. Dafermos for useful discussions. FCE is supported by an STFC studentship. HSR is partially supported by STFC grant No. ST/P000681/1.

In this Appendix we will consider causal properties of superluminal equations in dimensions. The low dimensionality imposes strong restrictions on the causal structure of solutions. We will review some results on causality in dimensions and explain why it is not possible to violate causality in a smooth way in a finite region of spacetime.

Assume we have a hyperbolic solution

This looks bad for the possibility of smoothly violating causality (i.e. “forming a time machine”). But maybe the above pathological features are

Pick inertial coordinates (

Consider a null geodesic of

It is also easy to see that there cannot be a smooth closed future-directed causal curve (w.r.t.

Now let

Consider following a generator of to the past. Since

This proves that generators of must emanate either from infinity in Minkowski spacetime or from a point of that is singular w.r.t. (

To violate causality in a smooth way, the generators of would have to emanate from infinity. This can happen even for the linear wave equation if

The word “generic” is included to reflect the condition that the time machine should be stable under small perturbations of the initial data.

Everything we say in the next few sections applies also to a quasilinear

Here, and throughout this paper, ‘smooth’ means .

In fact, for a general equation this is expecting too much. We will discuss this in Sect.

For example, the strong cosmic censorship conjecture asserts that, for suitable initial data, the MGHD is generically inextendible. The weak cosmic censorship conjecture asserts that, for asymptotically flat initial data, the MGHD generically has a complete future null infinity.

The only exceptions we are aware of are the sketches in [

This solution was obtained using the method of characteristics which only works when the equation is hyperbolic so only hyperbolic solutions are obtained using this method.

The latter assumption could be relaxed by replacing the infinite limits of the integrals by finite constants.

These plots were determined using the

In the Nambu–Goto string interpretation,

In the Nambu–Goto string interpretation, the string worldsheet on a surface of constant intersecting

In the Nambu–Goto string interpretation, this corresponds to Fig.

Since we are dealing with a subluminal theory, one could just declare that is a preferred time function and ignore the above problems. However this is unsatisfactory: if one uses as the time function (with

Note that the regions , etc in the proof of this theorem are defined slightly differently from the regions defined in the discusion above.

I.e., the component which contains points of arbitrarily large and negative -coordinate.

The reason for including in (

Equivalently we could define

Note that the extendibility across the null sections of the boundary implies that the analogue of the strong cosmic censorship conjecture is false for the superluminal equation. But the behaviour is much better than in the subluminal case for which the object one needs to define to formulate this conjecture (the MGHD) is not even unique!

All the results presented in this section generalise literally unchanged to the setting of Sect.

We will discuss how this might happen at the end of Sect.

denotes the boundary of in .

The notation denotes the causal future of

For a superluminal equation with , any

In general, i.e., if property (

It is clear that

In 2d let

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