Inextendibility of spacetimes and Lorentzian length spaces
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Abstract
We study the lowregularity (in)extendibility of spacetimes within the syntheticgeometric framework of Lorentzian length spaces developed in Kunzinger and Sämann (Ann Glob Anal Geom 54(3):399–447, 2018). To this end, we introduce appropriate notions of geodesics and timelike geodesic completeness and prove a general inextendibility result. Our results shed new light on recent analytic work in this direction and, for the first time, relate lowregularity inextendibility to (synthetic) curvature blowup.
Keywords
Length spaces Lorentzian length spaces Causality theory Synthetic curvature bounds Triangle comparison Metric geometry InextendibilityMathematics Subject Classification
53C23 53C50 53B30 53C80 83C751 Introduction
One can distinguish between two main lines of research in lowregularity geometry. One approach is analytical, where one lowers the differentiability assumptions on, for example, (pseudo)Riemannian metrics below the level where curvature can be classically defined. For example, one can study geometrical properties of (pseudo)Riemannian metrics that have regularity \(C^0\), \(C^{0, \alpha }\) or \(C^{1, 1}\), etc., or socalled Geroch–Traschen metrics, for which the Christoffel symbols are \(L^2_{\mathrm {loc}}\), and the curvature is welldefined as a distribution [16, 28, 39]. The other approach to studying lowregularity geometries is by “synthetic” or metric space methods. Here, curvature bounds for Alexandrov spaces and CAT(k) spaces are defined in terms of comparison properties of geodesic triangles.
In the context of lowregularity Riemannian geometry, examples of a result of an analytical nature would be DeTurck and Kazdan’s study concerning harmonic coordinates [9], Taylor’s results on regularity of isometries [42] and Lytchak and Yaman’s result [30] that minimising curves for \(C^{0, \alpha }\) Riemannian manifolds are \(C^{1, \beta }\) curves, where \(\beta = \frac{\alpha }{2\alpha }\). Examples in this direction in the Lorentzian setting are the positive mass theorem for distributional curvature [19, 27], work on cone structures [6, 11, 32] and the recent work of extending the classical singularity theorems to \(\mathcal {C}^{1,1}\)regularity [17, 25, 26], which in turn builds on previous results in lowregularity Lorentzian geometry and causality [7, 23, 24, 31, 37].
In the synthetic direction, the theory of Alexandrov spaces with curvature bounded above and/or below is welldeveloped as an appropriate generalisation of Riemannian geometry with sectional curvature bounds (see, for instance, [2, 4, 36]), and the work of Lott–Villani–Sturm gives a generalisation of the notion of a Riemannian metric with lower bound on the Ricci curvature to metric measure spaces [29, 40, 41].
In this paper, we will concentrate on a generalisation of Lorentzian geometry suitable for the lowregularity setting. More precisely, we shall be interested in the problem of finding lowregularity extensions of spacetimes. Concerning this question, approached from the analytical side, several fundamental contributions have appeared recently. Of particular relevance to us, Sbierski has shown the \(C^0\)inextendibility of the Schwarzschild solution [38]. Building upon Sbierski’s work, Galloway, Ling and Sbierski established that global hyperbolicity combined with timelike geodesic completeness implies \(C^0\)inextendibility. Further developments in this direction are due to Galloway–Ling and Graf–Ling (see below). In a related direction, Dafermos and Luk have recently shown \(C^0\)extendibility of the interior of the Kerr solution [10].
In this paper, we will concentrate on the syntheticgeometric approach to extendibility. In [22], the theory of Lorentzian length spaces has been developed, which will form the framework of the present work. In this more axiomatic approach, there is a notion of a geodesic (as a locally lengthmaximising curve), which is not available in the more analytical direction of research. Therefore, it is possible to mimic the classical proof that geodesic completeness implies inextendibility (see, for example, [3, Prop. 6.16]). Moreover, within this picture, it becomes clear precisely what minimal geometric properties are underlying certain analytical extension results. In particular, for the first time, our approach allows us to directly relate lowregularity inextendibility with (synthetic) curvature blowup. Such a result does not appear to be feasible in a purely analytical approach, due to the lack of a notion of a curvature for the extended spacetime.
An additional advantage of our synthetic approach is that there is no requirement for the introduction of coordinate systems, and regularity conditions (such as existence of smooth structures, or a certain level of differentiability) never arise. In this regard, it should perhaps be noted that in the analytical work on lowregularity extensions, one has to carry out standard geometrical constructions on the original manifold. As such, even though one works in a coordinate chart of the extended manifold in which the metric is merely continuous, the metric on the intersection of the original manifold with the coordinate chart must be \(C^2\)regular.^{1} One could compare this situation with, for example, the fact that the Nash–Kuiper theorem [21, 34] implies that the flat metric on \(T^2\) can be induced from a \(C^1\) map \(T^2 \rightarrow \mathbb {R}^3\).^{2} In the coordinate system in which the map is \(C^1\), the induced metric will be merely \(C^0\), even though we know that there exists a coordinate system in which the metric is smooth. As such, one could consider a more general notion of \(C^0\) extensions of spacetimes, where one allows the regularity of the metric on the original manifold to drop. On the contrary, in our approach, such issues never arise. In fact, the extensions that we consider need not even be manifolds.
Our main references for Lorentzian geometry and causality theory are [3, 8, 33, 35], as well as [7] for the case of continuous Lorentzian metrics.
The plan of the paper is as follows: In Sect. 2 we briefly recall some main concepts and results on Lorentzian length spaces. Section 3 introduces extensions of Lorentzian (pre)length spaces, relates them to extensions of spacetimes and shows that the future or past boundary of an extension is nonempty. In Sect. 4 we define geodesics in the synthetic setting and show that this notion reduces precisely to that of pregeodesics for spacetimes. We also demonstrate that, as in the smooth case, extendibility as a geodesic is equivalent to continuous extendibility. In Sect. 5 we define an analogue of timelike completeness: a Lorentzian prelength space is said to have property (TC) if all inextendible timelike geodesics have infinite length. This is the key property on which our main inextendibility result (Theorem 5.3) rests. We then establish connections between inextendibility and the occurrence of synthetic causal curvature singularities. Finally, in Sect. 6 we relate the results of the present work to the recent advances in the study of the lowregularity inextendibility of spacetimes.
2 A short introduction to Lorentzian length spaces
Here we briefly recall some basic notions and results from the theory of Lorentzian length spaces, following [22], to which we refer for further details and proofs.
A set X endowed with a preorder \(\le \) and a transitive relation \(\ll \) contained in \(\le \) is called a causal space. We write \(x<y\) if \(x\le y\) and \(x\ne y\). If \(x\ll y\), respectively, \(x\le y\) we call x and y timelike, respectively, causally related. Chronological and causal futures and pasts \(I^\pm (x)\), \(J^\pm (x)\) of a point x are then defined in the usual manner based on these relations.
If X is, in addition, equipped with a metric d and a lower semicontinuous map \(\tau :X\times X \rightarrow [0, \infty ]\) that satisfies the reverse triangle inequality \(\tau (x,z)\ge \tau (x,y) + \tau (y,z)\) (for all \(x\le y\le z\)), as well as \(\tau (x,y)=0\) if \(x\nleq y\) and \(\tau (x,y)>0 \Leftrightarrow x\ll y\), then \((X,d,\ll ,\le ,\tau )\) is called a Lorentzian prelength space and \(\tau \) is called the time separation function of X. Note that lower semicontinuity of \(\tau \) implies that \(I^{\pm }(x)\) is open, for any \(x \in X\).
Standard causality conditions (chronology, (strong) causality, global hyperbolicity, etc.) can also be imposed on Lorentzian prelength spaces, and substantial parts of the causal ladder [33] continue to hold in this general setting. A Lorentzian prelength space X is called causally pathconnected if for all \(x,y\in X\) with \(x\ll y\) (respectively, \(x<y\)) there is a futuredirected timelike (respectively, causal) curve from x to y. A neighbourhood U of x is called causally closed if the relation \(\le \) is closed in \(\bar{U}\times \bar{U}\), and X itself is called locally causally closed if every point has a causally closed neighbourhood.
 (i)
The dlength of all causal curves contained in \(\Omega _x\) is uniformly bounded.
 (ii)
There is a continuous map \(\omega _x:\Omega _x \times \Omega _x\rightarrow [0,\infty )\) such that \((\Omega _x, d_{\Omega _x\times \Omega _x},\) \(\ll _{\Omega _x\times \Omega _x},\le _{\Omega _x\times \Omega _x}, \omega _x)\) is a Lorentzian prelength space, and for every \(y\in \Omega _x\) we have \(I^\pm (y)\cap \Omega _x\ne \emptyset \).
 (iii)
For all \(p,q\in \Omega _x\) with \(p<q\) there is a futuredirected causal curve \(\gamma _{p,q}\) from p to q that is maximal in \(\Omega _x\) and satisfies \(L_\tau (\gamma _{p,q}) = \omega _x(p,q) \le \tau (p,q)\).
If, in addition, the neighbourhoods \(\Omega _x\) can be chosen such that
 (iv)
Whenever \(p,q\in \Omega _x\) satisfy \(p\ll q\) then \(\gamma _{p,q}\) is timelike and strictly longer than any futuredirected causal curve in \(\Omega _x\) from p to q that contains a null segment,
Any smooth strongly causal spacetime is an example of a regular Lorentzian length space (with metric \(d = d^h\) induced by any Riemannian metric h on the spacetime). More generally, any spacetime with a continuous, strongly causal and causally plain metric (see the remark preceding Corollary 5.5) is a (strongly) localisable Lorentzian length space. Further examples are provided by certain Lorentz–Finsler spaces in the sense of [32] or, for the nonmanifold setting, causal Fermion systems [12, 13].
 (i)
\(\tau _{U\times U}\) is finite and continuous.
 (ii)
Whenever \(x,y \in U\) with \(x < y\), there exists a causal curve \(\alpha \) in U with \(L_\tau (\alpha ) = \tau (x,y)\).
 (iii)
If (x, y, z) is an admissible causal geodesic triangle in U, realised by maximal causal curves (or a constant curve, respectively) \(\alpha , \beta , \gamma \) whose side lengths satisfy timelike size bounds for K, and if \((\bar{x},\bar{y},\bar{z})\) is a comparison triangle of (x, y, z) in \(M_K\) realised by causal geodesics (or a constant curve) \(\bar{\alpha }\), \(\bar{\beta }\), \(\bar{\gamma }\), then whenever p, q are points on the timelike sides of (x, y, z) and \(\bar{p}\), \(\bar{q}\) are corresponding points of the timelike sides of \((\bar{x},\bar{y},\bar{z})\), we have \(\tau (p,q)\le \bar{\tau }(\bar{p}, \bar{q})\) (respectively, \(\tau (p,q)\ge \bar{\tau }(\bar{p}, \bar{q}))\).
3 Extensions
We start the main part of our work by defining the notion of an extension of a Lorentzian prelength space, requiring only conditions that are natural within our setting. This concept is fully compatible with the usual notion of extension for spacetimes, see Proposition 3.5.
Definition 3.1
 (i)
the metric space \((\tilde{X}, \tilde{d})\) is connected,
 (ii)
there exists an isometry \(\iota :(X,d)\rightarrow (\tilde{X}, \tilde{d})\) of metric spaces,
 (iii)
the image \(\iota (X)\) is a proper, open subset of \(\tilde{X}\),
 (iv)
\(\iota \) preserves \(\ll \) and \(\le \), i.e. \(\forall x,y\in X\): if \(x\le y\) then \(\iota (x)\ \tilde{\le }\ \iota (y)\) and if \(x\ll y\) then \(\iota (x)\ \tilde{\ll }\ \iota (y)\), and
 (v)a curve \(\gamma :I \rightarrow X\) is timelike (respectively, causal) if and only if \(\iota \circ \gamma \) is timelike (respectively, causal) in \((\tilde{X},\tilde{d},\tilde{\ll },\tilde{\le },\tilde{\tau })\). Furthermore, \(\iota \) preserves \(\tau \)lengths, i.e. for any \(\le \)causal curve \(\gamma :I\rightarrow X\) we have$$\begin{aligned} L_\tau (\gamma ) = L_{\tilde{\tau }}(\iota \circ \gamma )\,. \end{aligned}$$(2)
Remark 3.2
Of course, this definition also applies to Lorentzian length spaces, i.e. a Lorentzian length space is extendible if there is a Lorentzian length space \((\tilde{X},\tilde{d},\tilde{\ll },\tilde{\le },\tilde{\tau })\) and \(\iota :(X,d)\rightarrow (\tilde{X}, \tilde{d})\) with the above properties 3.1(v). In this case conditions (iv) and (v) slightly simplify.
Lemma 3.3
Let \((\tilde{X},\tilde{d},\tilde{\ll },\tilde{\le },\tilde{\tau })\) be an extension of \((X,d,\ll ,\le ,\tau )\), where both are Lorentzian length spaces. Then \(\tilde{\tau } \circ (\iota \times \iota ) \ge \tau \).
Proof
Let \(p,q\in X\) with \(\tau (p,q)>0\) (if \(\tau (p,q)=0\) there is nothing to do). Let \(\gamma \) be a futuredirected \(\le \)causal curve from p to q (which exists due to \(p\le q\) and the causal pathconnectedness of X). Then \(\iota \circ \gamma \) is \(\tilde{\le }\)causal and \(L_\tau (\gamma )=L_{\tilde{\tau }}(\iota \circ \gamma )\le \tilde{\mathcal {T}}(\iota (p),\iota (q)) = \tilde{\tau }(\iota (p),\iota (q))\). Taking the supremum over all futuredirected \(\le \)causal curves from p to q, we get \(\mathcal {T}(p,q)\le \tilde{\tau }(\iota (p),\iota (q))\) and since \(\mathcal {T}=\tau \) the claim follows. \(\square \)
The following lemma shows that condition (v) of Definition 3.1 required of an extension is in fact not too strong. Moreover, it demonstrates that for smooth strongly causal spacetimes the time separation function determines the metric completely.
Lemma 3.4
Let (M, g) and \((\tilde{M},\tilde{g})\) be smooth spacetimes (of the same dimension) with time separation functions \(\tau \) and \(\tilde{\tau }\), respectively. Let (M, g) be strongly causal and let \(\iota :M\rightarrow \tilde{M}\) be onto. Then \(\iota \) is an isometry if and only if \(\iota \) preserves causal curves and their lengths, i.e. a curve \(\gamma \) is causal in M if and only if \(\iota \circ \gamma \) is causal in \(\tilde{M}\) and for such curves, \(L_g(\gamma ) = L_{\tilde{g}}(\iota \circ \gamma )\).
Proof
It is a classical result that goes back to Hawking, King and McCarthy [20] (cf. [33, Prop. 3.34] or [3, Thm. 4.17]) that \(\iota \) is an isometry if and only if it preserves \(\tau \). By definition of the time separation functions in spacetimes, this latter condition is, in turn, implied by \(\iota \) preserving the glengths of causal curves. \(\square \)
Furthermore, in the case of spacetimes the above result implies that there is no difference between an extension in our sense, and in the usual sense of an isometric embedding (cf. [38, Def. 2.15]. To be precise, we have the following result:
Proposition 3.5
Let (M, g) and \((\tilde{M}, \tilde{g})\) be smooth, strongly causal spacetimes (of the same dimension) and let \(\iota :M\rightarrow \tilde{M}\) be a map such that \(\iota (M)\subset \tilde{M}\). Then the induced Lorentzian length space of \((\tilde{M}, \tilde{g})\) extends the one coming from (M, g) via \(\iota \) if and only if \(\iota \) is a (smooth) isometric embedding.
Proof
We start with the following observation: Let \(\tilde{h}\) be any Riemannian metric on \(\tilde{M}\) with induced metric \(d^{\tilde{h}}\). This fixes the induced Lorentzian length space in the following sense: Any other Riemannian metric on \(\tilde{M}\) also induces the manifold topology and the notion of locally Lipschitz continuous curves is preserved (cf. [8, Prop. 2.3.1]), thus fixing the spacetime \((\tilde{M}, \tilde{g})\) and any Riemannian background metric determines the resulting Lorentzian length space.
To illustrate that one can have extensions that are not manifolds, we consider the following example, which is a Lorentzian version of [2, Ex. 4.2.5].
Example 3.6
Let \(\mathbb {R}^2_1\) be twodimensional Minkowski space and embed it into \(\mathbb {R}^3\) as a plane through the origin orthogonal to the zdirection, i.e. \(N:=\{(t,x,0): (t,x)\in \mathbb {R}^2\}\). We now add a halfray to the origin and give the resulting space the structure of a Lorentzian length space. Let \(\Gamma :=\{(0,0,z):z\ge 0\}\) and set \(\tilde{M}:= N \cup \Gamma \) (Fig. 1). On N we use the relations from Minkowski space and on \(\Gamma \) we define \(Z_1:=(0,0,z_1)\ll Z_2:=(0,0, z_2)\) if \(z_1 < z_2\), and \(Z_1\le Z_2\) if \(Z_1\ll Z_2\) or \(Z_1=Z_2\). For \(p=(t,x,0)\in N\) and \(Z\in \Gamma \) we define \(p\ll Z\) if \((t,x)\ll 0\) in \(\mathbb {R}^2_1\) and analogously for the causal relation. We define the time separation function \(\tau \) as the time separation function coming from Minkowski space on N, for points on \(\Gamma \) we set \(\tau ((0,0,z_1),(0,0,z_2)):=z_2z_1\) if \(z_1\le z_2\) (zero otherwise) and for \(p=(t,x,0)\in N\) and \(Z=(0,0,z)\) we set \(\tau (p,Z):=\sqrt{t^2x^2} + z\) if \(p\le Z\) (and zero otherwise). As \(\tau \) is continuous this gives a Lorentzian prelength space. In fact, this construction gives a Lorentzian length space as it is clearly pathconnected and locally causally closed. Moreover, it is regularly localisable since maximal causal curves always exist (they are the, possibly broken, straight lines) and the induced length agrees with the \(\tau \)length by construction. Furthermore, it is not hard to see that \(\tilde{M}\) is strongly causal. In this space maximal curves branch: every maximal curve from \(J^(0)\) to \(J^+(0)\) has 0 as a branching point, as the curve is allowed to continue into N or \(\Gamma \). This implies via [22, Cor. 4.13] that \(\tilde{M}\) has timelike curvature unbounded below, i.e. a curvature singularity in the sense of [22, Def. 4.20]. Finally, \(\tilde{M}\) extends \(M\backslash \{(0,0)\}\), thereby providing an example of a nonmanifold extension. Note that \(\tilde{M}\) does not extend M since M is not embedded into \(\tilde{M}\) as an open subset.
At this point we can introduce the past and future boundary of Lorentzian prelength spaces with respect to an extension in complete analogy to the case of spacetime extensions, see [14, Def. 2.1].
Definition 3.7
Let \((\tilde{X},\tilde{d},\tilde{\ll },\tilde{\le },\tilde{\tau })\) be a Lorentzian prelength space extending the Lorentzian prelength space \((X,d,\ll ,\le ,\tau )\) via the embedding \(\iota \). The future/past boundary \(\partial ^+(X)\) / \(\partial ^(X)\) of X is defined as the set of all points \(\tilde{p}\in \partial \iota (X)\) that can be reached by a future/pastdirected \(\tilde{\ll }\)timelike curve \(\gamma :[0,1]\rightarrow \tilde{X}\) such that \(\gamma ([0,1))\subseteq \iota (X)\) and \(\gamma (1)=\tilde{p}\).
The following result establishes that for any extension of a Lorentzian length space the future or past boundary is nonempty. It is a direct analogue of [38, Lemma 2.17].
Lemma 3.8
Let \((\tilde{X},\tilde{d},\tilde{\ll },\tilde{\le },\tilde{\tau })\) be an extension of \((X,d,\ll ,\le ,\tau )\), where both are Lorentzian length spaces, and denote the corresponding isometry by \(\iota \). Then there is a \(\tilde{\ll }\)timelike curve \(\tilde{\gamma }:[0,1]\rightarrow \tilde{X}\) such that \(\tilde{\gamma }([0,1))\subseteq \iota (X)\) and \(\tilde{\gamma }(1)\in \tilde{X}\backslash \iota (X)\), i.e. \(\partial ^+(X)\cup \partial ^(X)\ne \emptyset \).
Proof
Since \(\iota (X)\) is a proper and open subset of \(\tilde{X}\) and \(\tilde{X}\) is connected, we get that \(\partial \iota (X)\ne \emptyset \). Let \(\tilde{p}\in \partial \iota (X)\) and let \(\tilde{\Omega }\) be a localising neighbourhood of \(\tilde{p}\) in \(\tilde{X}\). Then, \(\tilde{I}^\pm (\tilde{p})\cap \tilde{\Omega }\ne \emptyset \) and let \(\tilde{q}\in \tilde{I}^(\tilde{p})\cap \tilde{\Omega }\). We now consider two cases. First, if \(\tilde{q}\in \iota (X)\), then since \(\tilde{q}\tilde{\ll }\tilde{p}\) there is a \(\tilde{\ll }\)timelike curve \(\tilde{\gamma }:[0,1]\rightarrow \tilde{X}\) such that \(\tilde{\gamma }(0)=\tilde{q}\), \(\tilde{\gamma }(1)=\tilde{p}\). Set \(s_0:=\sup \{s\in [0,1]: \tilde{\gamma }([0,s])\subseteq \iota (X)\}\), then since \(\iota (X)\) is open and \(\tilde{p}\in \partial \iota (X)\) we have \(\tilde{\gamma }(s_0)\in \tilde{X}\backslash \iota (X)\). Reparametrising \(\tilde{\gamma }_{[0,s_0]}\) to [0, 1] yields the result. The second case is when \(\tilde{q}\in \tilde{X}\backslash \iota (X)\). Now \(\tilde{I}^+(\tilde{q})\cap \tilde{\Omega }\) is a neighbourhood of \(\tilde{p}\in \partial \iota (X)\), thus \(\iota (X)\cap (\tilde{I}^+(\tilde{q})\cap \tilde{\Omega })\ne \emptyset \). Let \(\tilde{r}\in \iota (X)\cap (\tilde{I}^+(\tilde{q})\cap \tilde{\Omega })\), then \(\tilde{q}\tilde{\ll }\tilde{r}\) and the result follows as in the first case by arguing into the past. \(\square \)
4 Geodesics
In this synthetic approach we have the tools at hand to define causal geodesics as locally lengthmaximising curves. Furthermore, we establish that for spacetimes the synthetic notion is compatible with the analytical one.
Definition 4.1
Let \((X,d,\ll ,\le ,\tau )\) be a localising Lorentzian prelength space and let \(\gamma :I\rightarrow X\) be a futuredirected causal curve. Then \(\gamma \) is a geodesic if for every \(t_0\in I\) there exists a localising neighbourhood \(\Omega \) of \(\gamma (t_0)\) and a neighbourhood \(J=[c,d]\) of \(t_0\) in I such that \(\gamma _{J}\) is maximal in \(\Omega \) from \(\gamma (c)\) to \(\gamma (d)\).
Remark 4.2
To show that for a smooth and strongly causal spacetime this notion is equivalent to the notion of causal pregeodesics, we need the following lemma stating a general property of strongly causal Lorentzian length spaces.
Lemma 4.3
Let \((X,d,\ll ,\le ,\tau )\) be a strongly causal Lorentzian length space. Then for all \(x\in X\) and every localising neighbourhood \(\Omega \) of x with local time separation function \(\omega \) there is a neighbourhood U of x, \(U\subseteq \Omega \) such that \(\omega _{U\times U}\) is completely determined by \(\tau \): \(\forall p,q\in U:\ \omega (p,q)=\tau (p,q)\). In particular, \(\tau \) is continuous on a neighbourhood of the diagonal in \(X\times X\).
Proof
Let \(x\in X\) and let \(\Omega \) be a localising neighbourhood of x with local time separation function \(\omega \). By strong causality and [22, Lemma 2.38(iii)] there is a neighbourhood U of x with \(U\subseteq \Omega \) such that all causal curves with endpoints in U are contained in \(\Omega \). Let \(p,q\in U\) with \(p<q\), then by the properties of \(\Omega \) (see Sect. 2) there is a causal curve \(\gamma _{pq}\) that is maximal in \(\Omega \) from p to q with \(L_\tau (\gamma _{pq}) = \omega (p,q)\). As \(p,q\in U\), any causal curve connecting these points is contained in \(\Omega \). Thus, \(\gamma _{pq}\) is maximal even in X, and consequently, we have \(\tau (p,q) = \mathcal {T}(p,q) = L_\tau (\gamma _{pq}) = \omega (p,q)\). The neighbourhood of the diagonal can be chosen to be the union of all such \(U\times U\) as above. \(\square \)
With the above lemma we can now establish the promised compatibility.
Theorem 4.4
Let (M, g) be a smooth, strongly causal spacetime and let \((M,d^h,\ll ,\le ,\tau )\) be the induced Lorentzian length space [22, Ex. 3.24(i)]. Then a causal pregeodesic of (M, g) is a geodesic in the sense of Definition 4.1 and vice versa.
Proof
Note that the above proof also shows that the property of being timelike agrees for causal pregeodesics of (M, g) and geodesics in the sense of Definition 4.1 (contrary to the case for arbitrary curves, cf. [22], Ex. 2.22]).
Definition 4.5
Let \((X,d,\ll ,\le ,\tau )\) be a localising Lorentzian prelength space and let \(\gamma :[a,b)\rightarrow X\) be a futuredirected geodesic. Then \(\gamma \) is extendible as a geodesic if there exists a (futuredirected) geodesic \(\bar{\gamma }:[a,b]\rightarrow X\) with \(\bar{\gamma }_{[a,b)}=\gamma \). Otherwise, \(\gamma \) is called inextendible as a geodesic.
A wellknown property of geodesics in smooth semiRiemannian manifolds is the fact that extendibility as a geodesic is equivalent to continuous extendibility. Its standard proof relies on the existence of convex neighbourhoods. The following result is an analogue in the setting of Lorentzian prelength spaces, with localising neighbourhoods working as a substitute.
Proposition 4.6
Let \((X,d,\ll ,\le ,\tau )\) be a strongly causal and localising Lorentzian prelength space and let \(\gamma :[a,b)\rightarrow X\) be a futuredirected geodesic. Then \(\gamma \) is extendible as a geodesic if and only if it is extendible as a continuous curve to [a, b].
Proof
5 Timelike completeness and inextendibility
As discussed in the introduction, our approach allows us to mimic the proof from the smooth case that geodesic completeness implies inextendibility, i.e. [3, Prop. 6.16]. We first introduce an appropriate notion of timelike geodesic completeness for Lorentzian prelength spaces.
Definition 5.1
Let \((X,d,\ll ,\le ,\tau )\) be a localising Lorentzian prelength space, then X is said to have property (TC) if all inextendible timelike geodesics have infinite \(\tau \)length.
This notion is equivalent to timelike geodesic completeness in the case of smooth and strongly causal spacetimes:
Lemma 5.2
Let \((M,d^h,\ll ,\le ,\tau )\) be the Lorentzian length space induced by a smooth and strongly causal spacetime (M, g). Then (M, g) is timelike geodesically complete if and only if \((M,d^h,\ll ,\le ,\tau )\) has property (TC).
Proof
First, let (M, g) be not timelike geodesically complete, so that there exists an inextendible timelike geodesic (without loss of generality inextendible to the future) \(\gamma :[a,b)\rightarrow M\), with \(b<\infty \), thus \(L_g(\gamma )<\infty \). Since \(L_g=L_\tau \) by [22, Prop. 2.32], Theorem 4.4 implies that property (TC) cannot hold. Conversely, let (M, g) be timelike geodesically complete and let \(\gamma :[0,b)\rightarrow M\) be an inextendible timelike geodesic (in the sense of Definition 4.1). Then by Theorem 4.4 \(\gamma \) is a timelike pregeodesic of (M, g), hence by completeness \(L_g(\gamma )=\infty \) (cf. [35, p. 154]). Since \(L_g = L_\tau \), property (TC) follows. \(\square \)
Property (TC) does guarantee inextendibility, as the following result shows.
Theorem 5.3
Let \((X,d,\ll ,\le ,\tau )\) be a strongly causal Lorentzian length space that has property (TC). Then \((X,d,\ll ,\le ,\tau )\) is inextendible as a regular Lorentzian length space.
Proof
Assume, to the contrary, that there exists a regular Lorentzian length space \((\tilde{X},\tilde{d},\tilde{\ll },\tilde{\le },\tilde{\tau })\) that extends \((X,d,\ll ,\le ,\tau )\). By Lemma 3.8 there is a (without loss of generality) futuredirected \(\tilde{\ll }\)timelike curve
We can now relate the lowregularity inextendibility to a blowup of curvature. More precisely, we have the following result.
Theorem 5.4
Let \((X,d,\ll ,\le ,\tau )\) be a strongly causal Lorentzian length space that has property (TC). If X is extendible, the extension has a causal curvature singularity [22, Def. 4.20]. Specifically, the extension cannot have bounded upper causal curvature.
Proof
Let \((X,d,\ll ,\le ,\tau )\) be a Lorentzian length space that is strongly causal and has property (TC). Assume that there exists a Lorentzian length space \((\tilde{X},\tilde{d},\tilde{\ll },\tilde{\le },\tilde{\tau })\) extending \((X,d,\ll ,\le ,\tau )\) and having causal curvature bounded above. Then [22, Rem. 4.16, Thm. 4.17 and Thm. 4.18] yield that \((\tilde{X},\tilde{d},\tilde{\ll },\tilde{\le },\tilde{\tau })\) is regular. This contradicts the inextendibility result Theorem 5.3 and yields that X has a curvature singularity in the sense of [22, Def. 4.20]. \(\square \)
We now specialise to the case where the object to be extended is a smooth spacetime. Firstly, recall that causally plain spacetimes are precisely those that do not exhibit the bubbling phenomenon. Roughly speaking, a metric is bubbling if it contains a point where the boundary of the future null cone has nonempty interior. (For a precise definition, see [7, Definition 1.16]; cf. also the discussion preceding Lemma 5.6 in [22].) Spacetimes (M, g) with g a Lipschitz metric are causally plain [7, Corollary 1.17].
The following result is now a direct corollary of Theorem 5.3.
Corollary 5.5
Let (M, g) be a smooth, strongly causal and timelike geodesically complete spacetime and let \((M,d^h,\ll ,\le ,\tau )\) be its induced Lorentzian length space. Then \((M,d^h,\ll ,\le ,\tau )\) is inextendible as a regular Lorentzian length space, and hence also inextendible in the class of continuous, strongly causal and causally plain spacetimes that are regular.
Proof
By Lemma 5.2 \((M,d^h,\ll ,\le ,\tau )\) has property (TC) and strong causality is the same notion for spacetimes and the corresponding Lorentzian length spaces by [22, Lemma 2.21(i),(ii) and Lemma 2.38(iii)]. Thus, Theorem 5.3 applies, showing that \((M,d^h,\ll ,\le ,\tau )\) is inextendible as a regular Lorentzian length space. Furthermore, by [22, Thm. 5.12] every continuous strongly causal and causally plain spacetime \((\tilde{M}, \tilde{g})\) gives rise to a Lorentzian length space. \(\square \)
Also in this case of spacetimes we obtain the result that timelike geodesic completeness forces the extension to have a curvature singularity, even though curvature cannot be defined in the usual sense via the Riemann tensor.
Corollary 5.6
Let (M, g) be a smooth, strongly causal and timelike geodesically complete spacetime and let \((M,d^h,\ll ,\le ,\tau )\) be its induced Lorentzian length space. If \((M,d^h,\ll ,\le ,\tau )\) is extendible as a Lorentzian length space then, the extension has a causal curvature singularity. (It cannot have causal curvature bounded above.)
Remark 5.7
In [1], Alexander and Bishop introduced sectional curvature bounds for general semiRiemannian manifolds. Moreover, they characterised these curvature bounds via triangle comparison with small triangles in model spaces (i.e. the spaces \(M_K\) from (1) in the Lorentzian setting), see [1, Thm. 1.1]. As was shown in [22, Ex. 4.9], our definitions in Sect. 2 are compatible with these curvature bounds in this sense and in particular a curvature singularity in our sense implies that there cannot be a corresponding sectional curvature bound in the sense of [1]. Corollary 5.6 therefore implies that if the extension is assumed to be a smooth and strongly causal spacetime itself, then its sectional curvature as defined in [1] must be unbounded above.
To conclude this section we note that it is an interesting open question whether one can characterise completeness of timelike geodesics in Lorentzian length spaces via condition (TC), analogous to the smooth case, cf. [35, p. 154].
6 Relation to other results on lowregularity inextendibility
In this final section we relate our work to further current results on the lowregularity inextendibility of spacetimes.

It allows the original spacetime to be of low regularity (continuous and causally plain) as well.

There might be continuous strongly causal, causally plain spacetimes inducing a regular Lorentzian length space where the metric is not locally Lipschitz continuous.

It applies even to nonmanifold extensions, and

It relates inextendibility with curvature blowup (Theorem 5.4).
It should also be noted that in our framework one can define future/past oneconnectedness [38, Def. 2.13] and future/past divergence [14, Def. 2.4(2)] as for spacetimes. Since being extendible forces the future or past boundary to be nonempty by Lemma 3.8, a further line of study could be to see if, as for spacetimes, future (past) oneconnectedness together with future (past) divergence yields empty future (past) boundary (cf. [14, Thm. 2.5]).
To summarise, we have developed a framework where we can show inextendibility of spaces that resemble timelike geodesically complete spacetimes, in a similar spirit as the classical result [3, Prop. 6.16]. Our approach provides a new and unified perspective on the recent results [15, 18], see the discussion above. Moreover, for the first time we can relate lowregularity inextendibility with a (synthetic) curvature blowup—a fact that fits well with physical expectations. Finally, it shows that timelike geodesic completeness is a very robust property, which carries over even to spaces that are not spacetimes or even manifolds.
Footnotes
Notes
Acknowledgements
This work was supported by research Grants P26859 and P28770 of the Austrian Science Fund FWF. The work of J.G. was partially supported by STFC Consolidated Grant ST/L000490/1.
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