The future is not always open
Abstract
We demonstrate the breakdown of several fundamentals of Lorentzian causality theory in low regularity. Most notably, chronological futures (defined naturally using locally Lipschitz curves) may be nonopen and may differ from the corresponding sets defined via piecewise \(C^1\)curves \(\mathcal {C}^1\). By refining the notion of a causal bubble from Chruściel and Grant (Class Quantum Gravity 29(14):145001, 2012), we characterize spacetimes for which such phenomena can occur, and also relate these to the possibility of deforming causal curves of positive length into timelike curves (pushup). The phenomena described here are, in particular, relevant for recent synthetic approaches to lowregularity Lorentzian geometry where, in the absence of a differentiable structure, causality has to be based on locally Lipschitz curves.
Keywords
Causality theory Low regularity Chronological future Causal bubblesMathematics Subject Classification
53C50 83C751 Introduction
Until a decade ago, Lorentzian causality theory had mostly been studied under the assumption of a smooth spacetime metric. However, with the analytic viewpoint on general relativity becoming more and more prevalent, issues of regularity have increasingly come to the fore. In particular, while the geometric core of the theory is traditionally formulated in the smooth category, the analysis of the initial value problem, for example, requires one to consider metrics of low regularity.
Beginning in 2011 with the independent works [8, 10], a systematic study of Lorentzian causality theory under lowregularity assumptions began and has brought to light several, sometimes surprising, facts. In a nutshell, while some features of causality theory are rather robust and topological in nature, other results are usually proved by local arguments involving geodesically convex neighbourhoods, which do exist for \(\mathcal {C}^{1,1}\)metrics [22, 28]. Consequently for this regularity class, the bulk of causality theory [8, 23, 28] including the singularity theorems [14, 24, 25] remains valid. Moreover, arguments from causality theory which neither explicitly involve the exponential map nor geodesics have been found to extend to locally Lipschitz metrics (see [8, Thm. 1.25], [29, Rem. 2.5]). However, below \(\mathcal {C}^{1,1}\), explicit counterexamples show that convex neighbourhoods may no longer exist for metrics of Hölder regularity \(\mathcal {C}^{1,\alpha }\) for any \(\alpha <1\) [19, 37].
Moreover, Lorentzian causality theory has been generalized to a theory of cone structures, i.e. setvalued maps that assign a cone in the tangent space to each point on a manifold. This setting allows one to develop those aspects of the theory that merely depend on topological arguments under weak regularity assumptions. At the same time, this has led to the introduction of new methods to the field: Using weak KAM theory, existence results for smooth time functions have been derived for \(\mathcal {C}^0\)cone structures in [9, 10], while in [5] dynamical systems theory has been employed to derive similar results for merely upper semicontinuous cone structures. In a landmark paper [29], Minguzzi has extended these studies using methods from setvalued analysis, convexity and order topology to develop large parts of causality theory for locally Lipschitz cone structures.
 (1)
The pushup principle, i.e. \(\left( I^+\circ J^+ \right) \cup \left( J^+\circ I^+ \right) \subseteq I^+\), may cease to hold^{1};
 (2)
Light cones may cease to be hypersurfaces as they “bubble up” to have a nonempty interior.
Besides the above, a strong reason for basing causality theory entirely on locally Lipschitz curves comes from the desire to develop synthetic methods in Lorentzian geometry. In fact, following the pioneering works [1, 17], recently methods from synthetic geometry and, in particular, length spaces have been implemented in the Lorentzian setting and in causality theory [16, 21]. In this framework, which generalizes lowregularity Lorentzian geometry to the level of metric spaces, \(\mathcal {C}^1\)curves are not available. Thus, one needs a broad enough framework within which to address the problems highlighted by the examples given in Sect. 3 of the present paper.“In previous treatments of causality theory [4, 13, 18, 32, 33, 38] one defines futuredirected timelike paths as those paths \(\gamma \)which are piecewise differentiable, with\({\dot{\gamma }}\) timelike and future directed wherever defined; at break points one further assumes that both the leftsided and rightsided derivatives are timelike. This definition turns out to be quite inconvenient for several purposes. For instance, when studying the global causal structure of spacetimes one needs to take limits of timelike curves, obtaining thus — by definition — causal futuredirected paths. Such limits will not be piecewise differentiable most of the time, which leads one to the necessity of considering paths with poorer differentiability properties. One then faces the unhandy situation in which timelike and causal paths have completely different properties. In several theorems, separate proofs have then to be given. The approach we present avoids this, leading — we believe — to a considerable simplification of the conceptual structure of the theory.”
This article is structured as follows. In Sect. 2, we define the basic notions of causality theory in several variants and establish some fundamental relations between them. We then introduce tools tailored to low regularity, which enable us to characterize openness of chronological futures and pasts, and relate nonbubbling with pushup and clarify several aspects of pathological behaviour of continuous spacetimes. In Sect. 3, we provide explicit continuous spacetimes that display various pathologies. In particular, we give for the first time examples where the chronological future is not open and where the chronological future defined via smooth curves and the one defined via Lipschitz curves differ. This answers several open questions raised in [8]. The final section provides an overview of the interdependence of the various causality notions studied in this work.
To conclude this introduction, we introduce some basic notation. Throughout this paper, M will denote a smooth connected second countable Hausdorff manifold. Unless otherwise stated, g will be a continuous Lorentzian metric on M, and we will assume that (M, g) is timeoriented (i.e. there exists a continuous timelike vector field \(\xi \), that is, \(g(\xi ,\xi )<0\) everywhere). We then call (M, g) a continuous spacetime. We shall also fix a smooth complete Riemannian metric h on M and denote the induced (length) metric by \(d_h\) and the induced norm by \(\Vert .\Vert _h\). Given Lorentzian metrics \(g_1\), \(g_2\), we say that \(g_2\) has strictly wider light cones than \(g_1\), denoted by \(g_1\prec g_2\), if for any tangent vector \(X \ne 0\), \(g_1(X,X) \leqslant 0\) implies that \(g_2(X,X)<0\) (cf. [30, Sec. 3.8.2], [8, Sec. 1.2]). Thus, any \(g_1\)causal vector is timelike for \(g_2\).
2 Curves, bubbles, and open questions
For Lorentzian metrics of regularity at least \(\mathcal {C}^2\), chronological futures and pasts are usually defined via piecewise \(\mathcal {C}^1\) (or \(\mathcal {C}^\infty \))curves (e.g. [4, 18, 30, 32]). Especially when working in lower regularity, several authors have employed more general classes of curves, the widest one being that of absolutely continuous curves, basically due to the fact that this is the largest class of functions for which the Fundamental Theorem of Calculus holds. This in turn is required to obtain a reasonable notion of length of curves and to control the curves via their causality. To see this, let \(c:[0,1]\rightarrow [0,1]\) be the Cantor function (which has bounded variation) and consider the curve \(\gamma :\)\([0,1]\ni t\mapsto (c(t),0)\) in twodimensional Minkowski spacetime \({\mathbb {R}}^2_1\). Then, \(\gamma \) parameterizes the timelike curve \([0,1]\ni t\mapsto (t,0)\), but its tangent is (0, 0) almost everywhere, i.e. spacelike.
Furthermore, absolutely continuous causal curves always possess a locally Lipschitz reparameterization (cf. Lemma 2.8). On the other hand, irrespective of the regularity of the metric, the limit curve theorems ([27]), which are of central importance in causality theory, even when applied to families of smooth causal curves typically yield curves that are merely locally Lipschitz continuous. It is therefore natural to define causal futures and pasts using such curves, and indeed it was demonstrated in [7, 8, 23, 28] for spacetimes, as well as in the significantly more general setting of cone structures in [5, 10, 29] that a fully satisfactory causality theory can be based on locally Lipschitz causal curves. As we shall see below, the situation is more involved in the case of timelike curves.
To discuss the dependence of causality theory on the underlying notions of timelike, respectively, causal curves, we introduce the following notations:
Definition 2.1
 (1)
\(\mathcal {AC}\) the set of all absolutely continuous curves from an interval into M.
 (2)
\(\mathcal {L}\) the set of all locally Lipschitz curves from an interval into M.
 (3)
\({\mathcal {C}}^{1}_{\mathrm {pw}}\) the set of all piecewise continuously differentiable curves from an interval into M.
 (4)
\({\mathcal {C}}^\infty \) the set of all smooth curves from an interval into M.
Here, a curve is called absolutely continuous if its components in any chart are absolutely continuous, or, equivalently, if it is absolutely continuous as a map into the metric space \((M,d_h)\), where we recall that \(d_h\) is the metric on M induced by the Riemannian metric h on M (cf. the discussion preceding Thm. 6 in [28]).
The definition typically employed in the smooth case (cf. the above references), but sometimes also used for continuous metrics (e.g. in [35]), is the following.
Definition 2.2

timelike if \(g({\dot{\gamma }},{\dot{\gamma }})<0\) everywhere.

causal if \(g({\dot{\gamma }},{\dot{\gamma }})\le 0\) and \({\dot{\gamma }}\ne 0\) everywhere.
The usual definition of causal curves of regularity below piecewise \(\mathcal {C}^1\) is as follows (cf. [8, Def. 1.3]).
Definition 2.3
 (1)
timelike if \(g({\dot{\gamma }},{\dot{\gamma }})<0\) almost everywhere,
 (2)
causal if \(g({\dot{\gamma }},{\dot{\gamma }})\le 0\) and \({\dot{\gamma }}\ne 0\) almost everywhere.
Remark 2.4
In [5], there is one further notion of timelike curve that we should mention: The authors call a Lipschitz curve timelike if its Clarke differential lies in the open chronological cone in the tangent space at each parameter value. They then show ( [5, Lem. 2.11]) that, using this definition, the chronological futures and pasts of a point \(p\in M\) are precisely the sets \(I^\pm _{{\mathcal {C}}^\infty }(p)\) (see Definition 2.5). Therefore, there is no need for a separate treatment of this approach.
Henceforth, we will adhere to the following convention: Whenever a curve is in \(\mathcal {AC}\), we will use Definition 2.3; however, if a curve is explicitly noted to be in \({\mathcal {C}}^{1}_{\mathrm {pw}}\), then we use Definition 2.2.
We now can define the chronological and causal future and past of a point depending on the class of curves chosen.
Definition 2.5
Definition 2.6
 (i)
\((\varphi _*g)(0) = \eta \), the Minkowski metric.
 (ii)
There exists some \(C>1\) such that \(\eta _{C^{1}}\prec \varphi _* g \prec \eta _C\) on \(L\times V\).
Note that point (ii) above implies that \(\frac{\partial }{\partial t}\) is timelike and \(\frac{\partial }{\partial x^i}\) (\(1\leqslant i\leqslant n\)) is spacelike on U. By [8, Prop. 1.10], every point p lies in the domain of a cylindrical chart. Such a domain is called a cylindrical neighbourhood.
It was shown in the proof of [8, Prop. 1.10] that in terms of a cylindrical chart \((\varphi ,U)\) (where we usually will suppress \(\varphi \) notationally), \(\partial J_{\mathcal {L}}^+(p,U)\) is given as the graph of a Lipschitz function \(f_ :V \rightarrow L\), that \(I_{\mathcal {L}}^+(p,U)\) is contained in the epigraph \(\mathrm {epi}(f_):=\{(t,x):t\geqslant f_(x) \} = J_{\mathcal {L}}^+(p,U)\) of \(f_\) and that the interior of \(J_{\mathcal {L}}^+(p,U)\), denoted by \(J_{\mathcal {L}}^+(p,U)^\circ \), equals the strict epigraph \(\mathrm {epi}_S(f_):=\{(t,x):t > f_(x) \} = J^+(p,U)^\circ \) of \(f_\). Here, \(f_\) is defined as the pointwise limit of the (Lipschitz) graphing functions of \(J^+_{{\hat{g}}_k}(p,U)\), where the \({\hat{g}}_k\succ g\) are smooth Lorentzian metrics converging locally uniformly to g as \(k\rightarrow \infty \). Moreover, the graphing function of \(\partial {\check{I}}^+(p,U)\) is denoted by \(f_+\).
Lemma 2.7
Proof
The following Lemma is an analogue of [20, Lem. 3.2.1] (providing a distinguished local parameterization for continuous causal curves in smooth spacetimes) for continuous metrics and cylindrical neighbourhoods:
Lemma 2.8
Proof
The previous result allows us to conclude that absolutely continuous causal curves always possess a reparameterization that is Lipschitz continuous. This fact was already noticed in [29, Rem. 2.3]. We include an alternative proof based on cylindrical neighbourhoods for convenience.
Lemma 2.9
Let (M, g) be a continuous spacetime. Then, any causal curve in \(\mathcal {AC}\) is locally Lipschitz continuous and hence lies in \(\mathcal {L}\). Thus, for any \(p\in M\), \(I^\pm _{\mathcal {AC}}(p) = I^\pm _{\mathcal {L}}(p)\) and \(J^\pm _{\mathcal {AC}}(p) = J^\pm _{\mathcal {L}}(p)\).
Proof
Based on this result, for (M, g) a continuous spacetime we shall henceforth define the chronological, respectively, causal future/past of a point by \(I^\pm (p):=I^\pm _{\mathcal {AC}}(p)=I^\pm _{\mathcal {L}}(p)\) and \(J^\pm (p) :=J^\pm _{\mathcal {AC}}(p)=J^\pm _{\mathcal {L}}(p)\). This is in accordance with the conventions used in e.g. [7, 8, 10, 12, 14, 15, 23, 34], and, for the case of \(J^\pm (p)\) also with [5, 28, 29].
For chronological (i.e. possessing no closed timelike curves) spacetimes, we have the following characterization of openness of chronological futures and pasts:
Theorem 2.10
 (i)
For all \(p\in M\), \(I^\pm (p)\) is open.
 (ii)
For all \(p\in M\), \(\partial I^\pm (p)\) is achronal.
 (iii)
For all \(p\in M\), \(\partial I^\pm (p)\) is an achronal Lipschitzhypersurface.
 (iv)
If F is a future/past set with \(\partial F\not =\varnothing \), then \(\partial F\) is an achronal Lipschitzhypersurface.
Proof

(iii)\(\Rightarrow \)(ii) is clear.

(ii)\(\Rightarrow \)(i): Suppose that \(I^+(p)\) were not open for some \(p\in M\). Then there exists some \(q\in I^+(p)\cap \partial I^+(p)\). As M is chronological, \(p\not \in I^+(p)\). Since clearly \(p\in \overline{I^+(p)}\), it follows that \(p\in \partial I^+(p)\). But then the elements p and q of \(\partial I^+(p)\) can be connected by a timelike curve, contradicting achronality.

(i)\(\Rightarrow \)(iv): Using cylindrical neighbourhoods, this follows exactly as in the smooth case (cf., e.g. [32, Cor. 14.27]), cf. also [11, Proof of Thm. 2.6]. For a detailed proof, see [26, Thm. 2.3.5].

(iv)\(\Rightarrow \)(iii): One only has to note that each \(\partial I^+(p)\) is nonempty, which follows from M being chronological, cf. the argument in (ii)\(\Rightarrow \)(i) above.
Using Lemmas 2.7 and 2.9, it follows that \(\mathcal {B}^+_{\mathrm {int}}(p,U)\) consists of those points that can be reached from p by a futuredirected Lipschitz timelike curve, but not by a futuredirected piecewise \(\mathcal {C}^1\) timelike curve. It remained an open problem in [8] whether spacetimes with nontrivial interior bubble sets actually exist. We will give examples of this phenomenon in the next section.
The exterior bubbling set is closely connected to pushup properties. To relate these to the concepts introduced above, let us first give a formal definition:
Definition 2.11
A continuous spacetime (M, g) is said to possess the pushup property if the following holds: Whenever \(\gamma :[a,b]\rightarrow M\) is a (absolutely continuous) future/pastdirected causal curve from \(p=\gamma (a)\) to \(q=\gamma (b)\) and if \(\{t\in [a,b]: {\dot{\gamma }}(t) \text { exists and is future/past directed timelike} \}\) has nonzero Lebesgue measure, then in any neighbourhood of \(\gamma ([a,b])\) there exists a future/pastdirected timelike Lipschitz curve connecting p and q. In particular, \(q\in I^+(p)\), respectively, \(q\in I^(p)\).
This property implies all the commonly used versions of pushup results. In particular, for an absolutely continuous causal curve \(\gamma :[a,b]\rightarrow M\) the set \(\{t \in [a,b]: {\dot{\gamma }}(t)\) exists and is future/past directed timelike\(\}\) has nonzero Lebesgue measure if and only if \(\gamma \) has positive length. A spacetime that possesses the pushup property in the sense of Definition 2.11 also satisfies what in [7, Lemma 2.4.14] and [8, Lemma 1.22] are called pushup Lemma I (i.e. \(I^+(J^+(\Omega )) = I^+(\Omega )\) for any \(\Omega \subseteq M\)) and pushup Lemma II, cf. [7, Lemma 2.9.10] and [8, Lemma 1.24]. The following result shows that pushup is in fact equivalent to the nonexistence of exterior bubbling:
Theorem 2.12
 (i)
For each \(p\in M\) and each cylindrical chart U centred at p, \(\mathcal {B}^\pm _{\mathrm {ext}}(p,U)=\varnothing \).
 (ii)
For each \(p\in M\) and each cylindrical chart U centred at p, \(\partial I^\pm (p,U) = \partial J^\pm (p,U)\).
 (iii)
(M, g) possesses the pushup property.
Proof

(i)\(\Rightarrow \)(iii): By covering \(\gamma ([a,b])\) with cylindrical charts U contained in the given neighbourhood of \(\gamma ([a,b])\), it suffices to show pushup for a curve \(\gamma :[a,b]\rightarrow U\) emanating from p that is futuredirected causal and such that the set \(A:=\{t\in [a,b] : {\dot{\gamma }}(t) \text { exists and is timelike} \}\) has positive Lebesgue measure \(\lambda (A)\). Moreover, by Lemma 2.9, we only need to work with Lipschitz curves. So we need to establish that \(q:=\gamma (b)\in I^+(p,U)\). To begin with, suppose that \(q\in J^+(p,U)^\circ = \mathrm {epi}_S(f_)\). Since \(I^+(p,U)\) is dense in \(J^+(p,U)\) by assumption and since \(\eta _{C^{1}}\prec g\) on U, we can find an element \(q'\in I^+(p,U)\) such that q lies in the \(\eta _{C^{1}}\)future of \(q'\), and thereby in \(I^+(p,U)\) itself. Note that this argument in fact shows that \(J^+(p,U)^\circ \subseteq I^+(p,U)\). We are therefore left with the case \(q\in \partial J^+(p,U)\). Suppose, then, that such a q were not contained in \(I^+(p,U)\).
We claim that there exists a nontrivial interval \([c,d]\subseteq (a,b)\) such that \(\gamma ([c,d])\subseteq I^+(p,U)\) and that \({\bar{s}}\in A\) for some \({\bar{s}}\in (c,d)\). To see this, we adapt an argument from the proof of [8, Prop. 1.21]. As was shown there, if \(t\in A\) and \(\gamma (t)\in \partial J^+(p,U)\) then there exists some \(\varepsilon >0\) such that \(\gamma ((t,t+\varepsilon ))\) lies in \(J^+(p,U)^\circ \) and thereby in \(I^+(p,U)\) by the above. Arguing analogously to the past, it follows that \(\varepsilon \) can be chosen so small that also \(\gamma ((t\varepsilon ,t))\) is disjoint from \(\partial J^+(p,U)\) (in fact, lies in \(\mathrm {hyp}_S(f_)\)). To prove our claim, it suffices to show that there exists some \(s\in A\) with \(\gamma (s)\not \in \partial J^+(p,U)\) (hence \(\gamma (s)\in \mathrm {epi}_S(f_)\)), as then also a nontrivial interval around s is mapped by \(\gamma \) to \(\mathrm {epi}_S(f_)\). So it only remains to exclude the possibility that \(\gamma (s)\in \partial J^+(p,U)\) for each \(s\in A\). To do this, we note that since \(\lambda (A)>0\), A must contain accumulation points of A (otherwise A would consist of isolated points and thereby be countable). However, if \(s\in A\) is such an accumulation point and \(s_k\) is a sequence in A converging to s, then choosing \(\varepsilon \) as above for s we obtain that \(s_k\in (s\varepsilon ,s+\varepsilon ){\setminus }\{s\}\) for k large, contradicting the assumption that \(\gamma (A)\subseteq \partial J^+(p,U)\).
Now let \(f_q^\) be the graphing function of \(\partial J^(q,U)\).^{3} Then no point r in \(\gamma ([c,d])\) can lie strictly below the graph of \(f_q^\) since otherwise \(r \in I^(q,U) \cap I^+(p,U)\), and a fortiori \(q\in I^+(p,U)\), contrary to our assumption. Hence, \(\gamma _{[c,d]}\) must lie on the graph of \(f_q^\). But then the same argument as above, using \(f_q^\) and working in the past direction leads to a contradiction since at \(\gamma ({\bar{s}})\) the curve must enter \(I^(q,U)\). Summing up, we conclude that \(q\in I^+(p,U)\), as claimed.

(iii)\(\Rightarrow \)(ii): Since \(\partial _t\) is future directed timelike, (iii) implies that \(J^+(p,U)^\circ = \mathrm {epi}_S(f_) \subseteq I^+(p,U)\), so \(\partial I^+(p,U) = \overline{I^+(p,U)}{\setminus }I^+(p,U)^\circ \subseteq \partial J^+(p,U)\). Conversely, considering a vertical line through any point q in \(\partial J^+(p,U)\) and using pushup again, it follows that in any neighbourhood of q there lie points from \(I^+(p,U)^\circ \) as well as points from the complement of \(\overline{I^+(p,U)}\), so \(q\in \partial I^+(p,U)\).

(ii)\(\Rightarrow \)(i): Since \(J^+(p,U)=\mathrm {epi}(f_) = \overline{\mathrm {epi}_S(f_)}=\overline{J^+(p,U)^\circ }\), it suffices to show that \(J^+(p,U)^\circ \subseteq I^+(p,U)\). Suppose, to the contrary, that there exists some \(q\in J^+(p,U)^\circ {\setminus }I^+(p,U)\). We can then connect q via a continuous path \(\alpha \) that runs entirely in \(J^+(p,U)^\circ \) with a point in \(I^+(p,U)\) (e.g. with any point on the positive taxis). But then \(\alpha \) has to intersect \(\partial I^+(p,U)\), producing a point in \(\partial I^+(p,U){\setminus } \partial J^+(p,U)\), a contradiction. \(\square \)
Remark 2.13
We note that the equivalence between the pushup property and the absence of causal bubbles was first observed, even in the more general setting of closed cone structures (with chronological futures/pasts defined via piecewise \({\mathcal {C}}^1\)curves) by E. Minguzzi in [29, Thm. 2.8].
Concerning the relationship between the various bubble sets, we have:
Lemma 2.14
Proof
By [8, Prop. 1.10] and [8, Prop. 1.21], the bubble set \(\mathcal {B}^+(p,U)\) is the intersection of \(\mathrm {epi}_S(f_)\) and the strict hypograph \(\mathrm {hyp}_S(f_+)\) of the graphing function \(f_+\) of \(\partial {\check{I}}^+(p,U)\). From this, the second inclusion follows immediately. To see the first one, note that by the properties of cylindrical neighbourhoods detailed before Lemma 2.7, any \(p\in \mathcal {B}^+(p,U)\) is contained in \(J^+(p,U){\setminus } \overline{{\check{I}}^+(p,U)}\). Now suppose, in addition, that \(p\not \in \overline{\mathcal {B}^+_{\mathrm {int}}(p,U)} \supseteq \overline{I^+(p,U)}{\setminus } \overline{{\check{I}}^+(p,U)}\). Then \(p\in J^+(p,U){\setminus } \overline{I^+(p,U)} = \mathcal {B}^+_{\mathrm {ext}}(p,U)\). \(\square \)
From Lemma 2.7 and Lemma 2.9, we obtain:
Theorem 2.15
 (i)For any \(p\in M\),$$\begin{aligned} I^\pm _{\mathcal {C}^\infty }(p) = I^\pm _{{\mathcal {C}}^{1}_{\mathrm {pw}}}(p) = I^\pm _{\mathcal {L}}(p) = I^\pm _{\mathcal {AC}}(p). \end{aligned}$$
 (ii)
There is no internal bubbling, i.e. \(\mathcal {B}^{\pm }_{\mathrm {int}}(p,U)=\varnothing \) for every \(p\in M\) and every cylindrical neighbourhood U of p.
 (iii)
\(I^\pm (p)=\check{I}^\pm (p)\) for every \(p\in M\).
Corollary 2.16
 (i)
(M, g) is causally plain.
 (ii)
There is neither internal nor external bubbling, i.e. \(\mathcal {B}^{\pm }_{\mathrm {int}}(p,U) = \mathcal {B}^{\pm }_{\mathrm {ext}}(p,U)=\varnothing \), for all \(p \in M\).
Proof
Let (M, g) be causally plain. By [8, Prop. 1.21], we have \(\check{I}^{\pm }(p) = I^{\pm }(p)\) for all \(p \in M\), so there is no interior bubbling. By [8, Lemma 1.22], (M, g) has the pushup property, so, by Theorem 2.12, we have \(\mathcal {B}^{\pm }_{\mathrm {ext}}(p,U) = \varnothing \).
The other direction follows directly from Lemma 2.14. \(\square \)
Any spacetime with a Lipschitz continuous metric is causally plain by [8, Cor. 1.17], and so, by Corollary 2.16, such a spacetime has neither internal nor external bubbling. Moreover, by Theorem 2.15, for spacetimes without internal bubbling, it does not matter which type of curves is used in the definition of chronological futures and pasts.
 (Q1)
Is \(I^\pm (p)\) open for each p?
 (Q2)
Is \(I^\pm (p)={\check{I}}^\pm (p)\) for each p?
 (Q2\({}^{\prime }\))

Is it true that, for all \(p\in M\), \(I^\pm _{\mathcal {C}^\infty }(p) = I^\pm _{{\mathcal {C}}^{1}_{\mathrm {pw}}}(p) = I^\pm _{\mathcal {L}}(p) = I^\pm _{\mathcal {AC}}(p)\)?
 (Q3)
Does an affirmative answer to (Q1) for a given spacetime imply the same for (Q2)?
 (Q4)
Is \(\check{I}^+(p)\) dense in \(I^+(p)\) i.e. is \(I^\pm (p) \subseteq \overline{{\check{I}}^\pm (p)}\), for all \(p \in M\)?
 (Q5)
Given a noncausally plain spacetime, are all bubble sets of the same type (interior or exterior)?
3 Counterexamples
Example 3.1
The generator constructed above reaches the taxis at a finite value \(t_1>t_0\) and can then be continued vertically up the taxis as a null generator of \(\partial J^+(p)\). Indeed, we note that any absolutely continuous, futuredirected causal curve from a point on the taxis cannot enter the region \(x>0\). In particular, let \(\alpha :[0, \infty ) \rightarrow {\mathbb {R}}^2\) be an absolutely continuous, futuredirected causal curve with \(\left( x \circ \alpha \right) (0) = 0\) and assume that there exists \(s_0 > 0\) such that \(\left( x \circ \alpha \right) (s_0) > 0\). Then, there exists a subset \(B \subseteq [0, \infty )\) with positive Lebesgue measure such that for all \(s \in B\), \(\frac{\mathrm {d}}{\mathrm {d}s} \left( x \circ \alpha \right) (s)\) exists and is strictly positive. (If this were not the case, then \(\left( x \circ \alpha \right) (s_0) = 0 + \int _0^{s_0} \frac{\mathrm {d}}{\mathrm {d}s} \left( x \circ \alpha \right) (s) \, \mathrm{d}s \leqslant 0\).) However, any vector of the form \(\left( \frac{\mathrm{d}t}{\mathrm{d}s}, \frac{\mathrm{d}x}{\mathrm{d}s} \right) \) with \(\frac{\mathrm{d}t}{\mathrm{d}s} \geqslant 0\) and \(\frac{\mathrm{d}x}{\mathrm{d}s} > 0\) is not futuredirected causal in the region \(x \geqslant 0\). This contradicts the assumption that the curve \(\alpha \) is futuredirected causal, i.e. that \(\dot{\alpha }( s)\) is futuredirected causal almost everywhere.
The preceding observation implies that the set \(J^+(p)\) consists of the light blue region in Fig. 2, along with the vertical null generator from p and the rightmoving null generator from p. (In particular, the subset of the taxis with \(t \geqslant t_1\) is part of the boundary of \(J^+(p)\).)

The chronological future of any point with \(x < 0\) is not an open subset of \({\mathbb {R}}^2\). In particular, \(I^+(p)\) consists of the light blue region in Fig. 2, including the subset of thetaxis with\(t > t_1\). This answers (Q1) in the negative.

\(\check{I}^+(p)\) consists of the light blue region in Fig. 2, excluding the subset of the taxis with \(t > t_1\). As such, \(\check{I}^+(p) \ne I^+(p)\). This answers (Q2) in the negative.

The exceptional curve c constructed in the example may be taken to be smooth, with only one point at which it is null. It may be extended in a timelike fashion to give a curve that is smooth except at one point of nondifferentiability, and timelike everywhere where its derivative is defined (at which point, the tangent vector from the past direction is null and the tangent vector to the future is timelike). As such, the curve is not timelike in the sense of Definition 2.2, but it is timelike in the sense of Definition 2.3.

Since \(\check{I}^+(p)\) is not equal to \(I^+(p)\), it follows from Theorem 2.15 and Corollary 2.16 that the spacetime cannot be causally plain, so there must exist a point \(r \in M\) with \(\mathcal {B}^{\pm }(r, U) \ne \varnothing \), for some cylindrical neighbourhood U of r. Indeed, while the bubble set of p is empty, every point on the taxis has a nonempty past bubble set. This behaviour will be even more pronounced in the following example.
Example 3.2

The set \(I^+(q)\) is open for all points \(q \in {\mathbb {R}}^2\). However, for the particular point p in Fig. 3, we have that \(I^\pm (p) \ne {\check{I}}^\pm (p)\). This shows that the answer to (Q3) is negative.

For the point p in Fig. 3, the set \(\check{I}^+(p)\) is contained in the region \(x < 0\), while \(I^+(p)\) contains a subset of the region \(x>0\) with nonempty interior. It follows that \(\check{I}^+(p)\) is not a dense subset of \(I^+(p)\) in this spacetime. Therefore, this example answers (Q4) in the negative.
Example 3.3
 For \((t, x) \not \in B\), we define$$\begin{aligned} \theta (t, x) := {\left\{ \begin{array}{ll} 0, &{}x < 1, \\ \arccos x^{\alpha }, &{}1 \leqslant x \leqslant 1, \\ 0, &{} x > 1; \end{array}\right. } \end{aligned}$$
 For \((t, x) \in A\), let$$\begin{aligned} \theta (t, x) := \arctan t^{\lambda }; \end{aligned}$$
 We choose the function \(\theta \) on the set \(B {\setminus } A\) in such a way that
 (i)
\(\theta \) is continuous on \(M \equiv {\mathbb {R}}^2\);
 (ii)
For each fixed \(t \in [1, 1]\), the angle \(\theta (t, x)\) is a nondecreasing function of \(x \in [1, 0]\), i.e. for each fixed \(t \in \left[  \rho , \rho \right] \), the function \(x \mapsto \theta (t, x)\) is nondecreasing;
 (iii)The light cones match up to those on the boundary of the sets A and B. In particular, we requireand$$\begin{aligned} \theta \left( t, 1 \right) = 0, \qquad \theta \left( t, \frac{1}{2} \right) = \arccos \left( \frac{1}{2} \right) ^{\alpha }, \qquad \forall t \in [1, 1], \end{aligned}$$$$\begin{aligned} \theta \left( t, \frac{5}{6} \right) = \theta \left( t, \frac{2}{3} \right) = \arctan t^{\lambda }, \qquad \forall t \in [\rho , \rho ]. \end{aligned}$$
 (i)
Remark 3.4
The function \(\theta (t, x)\) may be chosen to be smooth away from the taxis and the points of the set A that lie on the xaxis. Globally, we then have that \(\theta \) has Hölder regularity \(\mathcal {C}^{0, \beta }\), with \(\beta := \min (\alpha , \lambda )\).
Let \(q = (0, x_0)\) be a point in the interior of the set A. We denote by \(\gamma _0\) the rightmoving null generator from q that immediately leaves the xaxis at q, i.e. \(t(\gamma _0(s)) > 0\) for all \(s > 0\). However, rightmoving null generators from q can travel along the xaxis for a finite time and then leave the xaxis. We denote by \(\gamma _1\) the rightmoving null generator from q that leaves the xaxis at the largest value of x, i.e. \(\gamma _1\) leaves the xaxis at the point \((0, x_1)\) where \(x_0< \frac{2}{3} \leqslant x_1 < \frac{1}{2}\). The null generators \(\gamma _0\), \(\gamma _1\) later intersect the taxis at the points \((t_0, 0)\), \((t_1, 0)\), respectively, where \(t_1 < t_0\). The set \(J^+(q) {\setminus } \overline{I^+(q)}\) contains the area between \(\gamma _0\) and \(\gamma _1\) up to the taxis and hence has nonempty interior. As in Example 3.2, for any \(\overline{t} > t_0\), the point \((\overline{t}, 0) \in I^+(q)\) and \(I^+(\overline{t}, 0)\) contains a subset of the set \(x > 0\) with nonempty interior. Since \(\check{I}^+(q)\) is contained in the set \(x < 0\), it follows that \(I^+(q) {\setminus } \check{I}^+(q)\) has nonempty interior as well.
This example therefore answers (Q5) in the negative and indeed shows that there exist points q in the spacetime such that \(\mathcal {B}^+_{\mathrm {ext}}(q, U)\) is nonempty for some cylindrical neighbourhood U of q, and \(\check{I}^+(q) \ne I^+(q)\).
Remark 3.5
In all of the above examples, the exceptional curves can, in fact, be chosen to be \(\mathcal {C}^1\) with nontimelike tangent vector at only one point. Heuristically, one might expect that the seemingly innocuous change of allowing \(\mathcal {C}^1\) curves that have timelike tangent vector except at a finite number of points at which the tangent vector may be null should not affect chronological pasts and futures. However, our examples show that in fact all of the pathologies exhibited above would persist if this definition were adopted.
4 Conclusions
Here, \(\textcircled {1}\) was shown in [8, Cor. 1.17]. Irreversibility of this arrow can be seen by considering a metric that is conformally equivalent to the Minkowski metric via a nonLipschitz factor. Of more interest is the fact that warped products with onedimensional base (i.e. metrics of the form \(\mathrm{d}t^2+f^2 h\) with h any Riemannian metric and \(f:(a,b) \rightarrow (0,\infty )\) continuous) are causally plain for continuous, not necessarily Lipschitz, warping functions f, cf. [2]. Equivalence \(\textcircled {2}\) is Corollary 2.16. Irreversibility in \(\textcircled {3}\), \(\textcircled {4}\) and \(\textcircled {8}\), as well as the claims in \(\textcircled {9}\) follow from [8, Ex. 1.12] and Example 3.2, respectively. Equivalence \(\textcircled {5}\) holds by Theorem 2.15, while \(\textcircled {6}\) is a consequence of Theorem 2.12. The claims in \(\textcircled {7}\) follow from [8, Ex. 1.12] and Example 3.1. Finally, Example 3.2 entails irreversibility of \(\textcircled {8}\).
The watershed in this chain of implications is the case of causally plain spacetimes. Indeed, for such metrics all versions of chronological futures and pasts coincide. Moreover, there are no bubble sets, chronological futures and pasts are open and the standard pushup properties hold. Indeed, it was demonstrated in [21, Sec. 5.1] that strongly causal and causally plain spacetimes form strongly localizable Lorentzian length spaces, implying that their causality theory is optimal in the sense of synthetic lowregularity Lorentzian geometry.

In order to derive the maximal benefit from limit curve theorems, set \(J^\pm (p):=J^\pm _{\mathcal {AC}}(p) = J^\pm _{\mathcal {L}}(p)\) (cf. Lemma 2.9).

To guarantee openness of chronological futures and pasts and to avoid interior bubbling, set \(I^\pm (p):=I^\pm _{{\mathcal {C}}^{1}_{\mathrm {pw}}}(p)\). Then in fact \(I^\pm (p)={\check{I}}^\pm (p) = I^\pm _{{\mathcal {C}}^\infty }(p)\) by Lemma 2.7.
While this strategy returns us to the “unhandy situation in which timelike and causal paths have completely different properties” [7, p. 14], it appears to us to be the most fitting approach. Moreover, note that, in fact, several fundamental works on lowregularity Lorentzian geometry have adopted these conventions, in particular [5, 28, 29] and, in the timelike case, [11, 35]. On the other hand, in synthetic approaches to lowregularity Lorentzian geometry (such as [16, 21]), where this strategy is not an option (due to the absence of a differentiable structure), phenomena such as the ones laid out in Sect. 3 have to be taken into consideration.
Footnotes
 1.
Here \(I^+\) and \(J^+\) denote the chronological and the causal relation, respectively, and e.g. \(I^+\circ J^+ =\{(p,r)\in M \times M: \exists q\in M\) with \((p,q)\in J^+\) and \((q,r)\in I^+ \}\).
 2.
We note that the phenomenon of bubbling, albeit not under this name, was first observed in [6].
 3.
It follows from the proof of [8, Prop. 1.10] that the cylindrical neighbourhood U of p can be chosen ‘locally uniformly’ in the sense that each \(q\in U\) possesses a cylindrical chart with domain U.
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). We are indebted to two anonymous referees for several remarks that have led to considerable improvements of the manuscript.
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