1 Introduction↩︎

In 1931 H.Hopf and his student W.Rinow published their famous completeness result for Riemannian manifolds. Throughout we consider a manifold to be smooth, paracompact, Hausdorff, connected, finite-dimensional, and without boundary.

Originally, Hopf and Rinow called (a\(_0\)) the abatability postulate (Abtragbarkeitspostulat), (a) the infinity postulates (Unendlichkeitspostulat), (b) the completeness postulate (Vollständigkeitspostulat), and (c) the compactness postulate (Kompaktheitspostulat). Most differential geometers focus on and apply the equivalence (a)\(\Longleftrightarrow\)(b). The metric equivalence (b)\(\Longleftrightarrow\)(c) already became more central in a generalization of the Hopf–Rinow Theorem to locally compact length metric spaces by S.Cohn-Vossen. We refer to Theorem 1 (b)\(\Longleftrightarrow\)(c) also as the metric Hopf–Rinow Theorem or the Riemannian completeness-compactness Theorem. For semi-Riemannian manifolds geodesic completeness is rare and difficult to obtain. Even compact Lorentzian manifolds need not be geodesically complete (the Clifton–Pohl torus is a counterexample, see ON?*p. or GHL:Rie?*Ex.), and strong additional conditions such as homogeneity (obtained by Marsden Ma72? in 1972) or flatness (obtained by Carrière Ca89? in 1989) are required to imply it in general settings. The compact case shows that (c)\(\Longrightarrow\)(a) is false in semi-Riemannian geometry. Since no canonical distance function exists on semi-Riemannian manifolds, a generalization of the Hopf–Rinow Theorem involving (b) was also considered to be impossible in this setting.

The aim of the present manuscript is to show that (and in which sense precisely) the equivalence (b)\(\Longleftrightarrow\)(c) is robust when leaving the positive definite Riemannian world in negative directions.

We proceed in three steps. First, we prove that our recent Lorentzian result BGH2?*Theorem 1.4] is a generalization of Theorem 1 (b)\(\Longleftrightarrow\)(c) in Section 3 (the necessary Lorentzian background is provided in Section 2). Second, we extend the relevant definitions and this result to proper cone structures in Section 4. Third, we introduce the notion of semi-Riemannian spacetimes, discuss their existence and other properties, and show that they carry proper cone structures in Section 5. All steps combined imply, in particular, that the metric Hopf–Rinow Theorem 1 (b)\(\Longleftrightarrow\)(c) generalizes to a large class of (noncompact) semi-Riemannian manifolds. In the remaining part of this introduction we present and discuss our results in more detail.

The Lorentzian setting. In recent work L.García-Heveling and the author proved the following Lorentzian completeness-compactness theorem with one negative tangent direction (see Section 2 for all notions).

Can we view Theorem 2 as the Lorentzian analogue or generalization of the metric Hopf–Rinow Theorem 1?

In the present manuscript we show that the answer is yes. At first, the metric Hopf–Rinow Theorem immediately appears to be more special and fragile than Theorem 2. This is because the notions (time functions, null distance, global hyperbolicity) relevant for (b’) and (c’) are conformally invariant while the Riemannian distance function is only preserved by isometries. In Section 3 we demonstrate that Theorem 2 is indeed the correct extension of the metric Hopf–Rinow Theorem 1 (b)\(\Longleftrightarrow\)(c) by proving that both properties are (independently) equivalent to their Lorentzian counterparts when restricted to Lorentzian products.

What about geodesic completeness? In the product case it is known that (a) geodesic completeness of \((\Sigma,\sigma)\) is equivalent to (a’) geodesic completeness of \((\Sigma',\sigma')\), and also equivalent to global hyperbolicity of \((\Sigma',\sigma')\) (see, for instance, BEE?*Theorems 3.66 and 3.67). Thus in the Lorentzian product case (and only then!) all properties are equivalent independent of each other:

\[\begin{tikzcd}[row sep=huge, column sep = huge] (a_0) \arrow[Leftrightarrow]{r}{\text{Thm.~\ref{HRthm}}} &(a) \arrow[Leftrightarrow]{r}{\text{Thm.~\ref{HRthm}}} \arrow[Leftrightarrow,dotted]{d}[swap]{\text{\cite{BEE}*{Thm.~3.67}}} \arrow[Leftrightarrow,dashed]{rd}[sloped,below]{\text{\cite{BEE}*{Thm.~3.66}}} & (c) \arrow[Leftrightarrow]{d}{\text{Thm.~\ref{prop:equivLR} (i)}} \arrow[Leftrightarrow]{r}{\text{Thm.~\ref{HRthm}}} & (b) \arrow[Leftrightarrow]{d}{\text{Thm.~\ref{prop:equivLR} (ii)}} \\& (a') & (c') \arrow[Leftrightarrow]{r}[swap]{\text{Thm.~\ref{BGHthm}}} & (b') \end{tikzcd}\]

While the equivalence of (a)\(\Longleftrightarrow\)(c’) carries over to warped products \(M \times_f \Sigma\) for \((M,g)\) a spacetime, even for warped products with compact Riemannian slices (a)\(\not\Longrightarrow\)(a’) (see BEE?*Theorem 3.68 and preceding discussion). More generally, the singularity theorems of Penrose and Hawking (see, for instance, HE?*Chapter 9) from the 1960s reveal that (c’)\(\not\Longrightarrow\)(a’) and due to Anti-de Sitter space (a’)\(\not\Longrightarrow\)(c’). It was very recently shown by Sánchez San:gh?*Section 6.4 that for general spacetimes also (c’)\(\not\Longrightarrow\)(a) for its slices. These observations show that on semi-Riemannian manifolds we are forced to let go of geodesic completeness (a) and (a’) also in the noncompact setting, and that new tools are needed.

The semi-Riemannian setting. That lifting and projecting the properties relevant for the metric Hopf–Rinow Theorem 1 is doable in the Lorentzian setting raises the hope that it is possible to iterate this procedure and add arbitrarily many negative tangent directions. The first obstacle we face is the lack of an analogue of causality theory for semi-Riemannian manifolds, meaning that none of the notions used in (b’) and (c’) yet exist. To this end we introduce and analyze in Section 5 a new subclass of semi-Riemannian manifolds by generalizing the concept of a Lorentzian \((n-1,1)\)-spacetime to semi-Riemannian manifolds of any index \(0 \leq \nu \leq n\) as follows.

Whether a given manifold \(M\) admits such a spacetime structure is a purely topological question. In 1935 E.Stiefel asked in his seminal thesis Sti:parallel?, supervised by H.Hopf, precisely this question in the closed manifold case (translated from German):

Is there a system of \(\nu\) continuous vector fields on \(M\), which are linearly independent in each point of \(M\)?

Such a system of of vector fields is called a tangent \(\nu\)-frame in the literature, and its existence equivalent to the existence of a semi-Riemannian \((n-\nu,\nu)\)-spacetime structure on \(M\). (This is also the reason why we call it a time frame in Definition 4, even though the order of the \(X_i\) is irrelevant for us.)

For \(\nu=1\), the Poincaré–Hopf Theorem Ho:vf? from 1927 fully answers the above question in the closed manifold case, characterizing existence by \(\chi(M)=0\). In 1955 Markus Mar? showed that no conditions are needed in the open case.

For \(\nu>1\) Stiefel’s question is in its generality still open, but led to many exciting developments in algebraic topology. The first necessary conditions were formulated in terms of Stiefel–Whitney classes Sti:parallel?, Whi:top?, and later characteristic classes. In the 1960s and 1970s several breakthrough results were obtained for closed manifolds in special cases (in particular, for \(\nu=2,3\), such as the work of Atiyah and Dupot AtDu?, Du:K?, and \(\nu=n\)), which we discuss in more detail in Section 5.1. More recently, Bökstedt–Dupont–Svane BDS:cobord? also give a brief overview and present new results for \(\nu=4,5,6,7\) (under additional assumptions on \(M\) and \(n\)).

Note that the above investigations all concern the closed case and often allow finite singularities. From our semi-Riemannian perspective, we also immediately understand that the situation for \(\nu>1\) is more subtle than for \(\nu=1\) because not every manifold admitting a semi-Riemannian metric also admits a (possibly different) semi-Riemannian spacetime metric (\(\mathbb{S}^2 \times \mathbb{S}^2\) is a counterexample, see Example 41). Nonetheless, it is this open case that we are interested in and ultimately even forced to consider in the present paper, simply because closed manifolds do not admit time functions. Thus the tools developed in this paper may in the future also help to shed light on the existence problem of a tangent \(\nu\)-frame in the noncompact case. By establishing the following result we show, in particular, that \((n-\nu,\nu)\)-spacetimes can be studied with techniques from the theory of cone structures developed in recent years by Fathi and Siconolfi FaSi?, Fa:time?, Bernard and Suhr BeSu:time?, BeSu:gh?, and Minguzzi Min:cone?.

In the framework of cone structures the notions of time functions and global hyperbolicity appearing in Theorem 2 have already been introduced and are well studied. In Section 4 we show that also the null distance can be extended to proper cone structures. The main result of this manuscript is an extension of Theorem 2, and thus of the metric Hopf–Rinow Theorem, to cone structures.

Together with Theorem 5 this characterization yields a conformal completeness-compactness theorem in semi-Riemannian geometry.

Above we have already seen that the Riemannian equivalence (b)\(\Longleftrightarrow\)(c) is a special case of the Lorentzian equivalence (b’)\(\Longleftrightarrow\)(c’). Can we in the same way relate \(\nu\) and \(\nu+1\) if \(\nu \geq 1\)?

In Section 5.4 we show that the answer is no. For one, if we extend a \((n-\nu,\nu)\)-spacetime \((M,g)\) orthogonally to a \((n-\nu,\nu+1)\)-spacetime \((M',g')=(\mathbb{R}\times M,-dt^2\oplus g)\), then the corresponding null distances cannot be directly related because they are both conformally invariant. In fact, even the existence of a null distance on \((M',g')\) requires additional properties and is not a given as in the Lorentzian product case. For the same reason global hyperbolicity does not carry over. Nonetheless, we can still recover parts of Theorem 3 in semi-Riemannian geometry as summarized in the following result obtained in Section 5.

In Example 65 we demonstrate that the first implication in Theorem 8 cannot be reversed if \(\nu < n\). Naively speaking, one could even say that this is already evident in the \(\nu = 0\) case since all manifolds admit a Riemannian metric and all Riemannian manifolds are globally hyperbolic (strictly speaking, Riemannian manifolds have empty/degenerate cones and are usually not considered), but already in Theorem 3 (i) globally hyperbolic Lorentzian products require completeness of the Riemannian slice.

In spite of such problems we have shown through our approach that many tools from Lorentzian geometry (cone structures, conformal methods etc.) can successfully be introduced in the semi-Riemannian setting and be utilized to prove interesting new results like Corollary 7 generalizing parts of the Hopf–Rinow Theorem. Thus, semi-Riemannian geometry is not only close to Lorentzian and Riemannian geometry in a differential geometric sense (as already emphasized by O’Neill ON?) but also in a metric context. In fact, it prompts the question what the true Lorentzian (vs.general semi-Riemannian) features of recent nonsmooth Lorentzian geometric theories and approaches to quantum gravity are. Answering this question will be crucial for ultimately linking these approaches to the smooth setting of general relativity. Along these lines it could also be insightful to investigate the links between semi-Riemannian and Lorentzian geometry in PDE theory. After all, the notion of global hyperbolicity was introduced in this context by J.Leray in 1952 and is crucial for proving global uniqueness of solutions to wave equations.

Outline. In Section 2 we recall the basic notions of Lorentzian geometry and causality theory including global hyperbolicity, time functions, and the null distance. We also derive new results for Lorentzian products needed in subsequent sections. In Section 3 we show that the Lorentzian Theorem 2 generalizes the metric Hopf–Rinow Theorem 1 (b)\(\Longleftrightarrow\)(c) by establishing Theorem 3 for Lorentzian products. In Section 4 we recall basic results of the theory of cone structures and show when the null distance is a well-defined concept in this framework. We also prove the cone completeness-compactness Theorem 6. In Section 5 we introduce the notion of \((n-\nu,\nu)\)-spacetime structures for semi-Riemannian manifolds, discuss their existence and links to differential/algebraic topology. We then proceed to show that \((n-\nu,\nu)\)-spacetimes are continuous proper cone structures (Theorem 5) from which Corollary 7 follows. Finally, we relate the notions of (stable) causality and of global hyperbolicity for different \(\nu\) and \(n-\nu\) for products appearing in Theorem 8.

Notation. Throughout we denote Riemannian manifolds by \((\Sigma,\sigma)\), Lorentzian and semi-Riemannian manifolds by \((M,g)\). We denote semi-Riemannian products by \((M',g') = (\mathbb{R}\times M,-dt^2\oplus g)\) and \((M'',g'') = (\mathbb{R}\times M,dt^2\oplus g)\). If a background Riemannian metric is used it is denoted by \(h\). We use the signature convention \((+,\ldots,+)\) for Riemannian manifolds and \((+,\ldots,+,-)\) for Lorentzian manifolds etc.

2 Causal and metric properties of Lorentzian manifolds↩︎

In this section we recall basics notions and results of Lorentzian geometry (for more details see ON?, BEE?, Min:LCT?) and the more recent null distance of Sormani and Vega SV?. Readers familiar with Lorentzian geometry or the null distance can skip Sections 2.1 or 2.2, respectively. In Section 2.3 we obtain new results about Lorentzian products that we apply in Section 3.

2.1 Causality and time↩︎

A semi-Riemannian manifold of signature \((n-\nu,\nu)\) is an \(n\)-dimensional manifold \(M\) equipped with metric tensor \(g\), i.e., a covariant \(2\)-tensor field that is symmetric and nondegenerate, with constant index \[\nu = \max \{ \dim S \, ; \, S \text{ subspace of } T_pM \text{ for which } g_p|_S \text{ is negative definite} \}.\] A Lorentzian manifold has signature \((n-1,1)\). In this section we assume that \((M,g)\) is a Lorentzian manifold, in Section 5 we consider the general semi-Riemannian setting.

A tangent vector \(v \in T_pM \setminus \{0\}\) is timelike if \(g(v,v)<0\), null if \(g(v,v)=0\), and spacelike if \(g(v,v)>0\). A tangent vector is causal if it is timelike or null. By convention we consider \(v=0\) to be spacelike. A Lorentzian manifold \((M,g)\) with a given continuous timelike vector field \(X\in\mathfrak{X}(M)\) is said to be time oriented by \(X\) and called a spacetime BEE?*p.. It is well known that a manifold admits a spacetime structure if and only if it is either noncompact or compact with vanishing Euler characteristic (see also Introduction).

A causal vector \(v \in T_pM \setminus \{0\}\) in a spacetime \((M,g)\) is said to be future directed if \(g_p(v,X(p)) < 0\). The class of locally Lipschitz curves (with respect to any background Riemannian metric) induces two relations on a spacetime, the chronological relation \[\begin{align} p \ll q \Longleftrightarrow \exists \text{ future directed timelike curve from } p \text{ to } q, \end{align}\] and the causal relation \[\begin{align} p \leq q \Longleftrightarrow \exists \text{ future directed causal curve from } p \text{ to } q, \text{ or } p=q, \end{align}\] with the notation that \(p < q\) if \(p \leq q\) and \(p\neq q\). We write \[\begin{align} I^+(p) &= \{ q\in M \, ; \, p \ll q \}, \\ J^+(p) &= \{ q\in M \, ; \, p \leq q \} \end{align}\] for the chronological future and causal future of \(p\), respectively. Analogous definitions apply for the past (with \(-\)). The chronological relation is open, transitive, and contained in the causal relation, which itself is transitive and reflexive.

Every closed (compact without boundary) Lorentzian manifold admits closed timelike curves, and thus allows time travel. For physical and geometric reasons we would like to exclude this setting and implicitly then focus on the noncompact case.

We require some mild additional property from our spacetimes, namely that they admit time functions.

The existence of a time function conveniently implies that a spacetime is causal, i.e., does not admit closed causal curves (and hence the causal relation is antisymmetric, thus an order relation). To be precise, the existence of a time function is equivalent to causality being a stable property, as shown by Hawking Haw:time? (see also Proposition 27) who also argues that any physically reasonable theory of gravity must be stably causal. The following more refined property of a spacetime is inevitable for a well-posed initial value problem for the Einstein equations in general relativity, the singularity theorems of Penrose and Hawking, Lorentzian splitting theorems, and indeed for most results in Lorentzian geometry.

There are many important characterizations and implications of global hyperbolicity which have been obtained since the 1950s, see BGH2?*Section 1 for an outline and Theorem 29 (for proper cone structures) for some characterizations relevant for this work.

2.2 Null distance↩︎

Sormani and Vega SV? showed that any stably causal spacetime \((M,g)\) can be equipped with a conformally invariant length metric space structure that respects and encodes the causal structure and topology of a spacetime. We recall some basic constructions below. See SV?, AB?, Ve?, SaSo?, BGH2?, G:null?, Al1? for more details and insights, and Section 4.3 for some proofs in the more general setting of cone structures.

The chronological relation being open implies that any two points \(p,q \in M\) can be joined by a locally Lipschitz path \(\beta \colon I \to M\) which is piecewise causal, i.e., it is either future or past directed on each subinterval of a partition \(\inf I = a_0 < a_1 < \ldots < a_k = \sup I\) of the interval \(I\) SV?*Lemma 3.5. The class of piecewise causal curves on \(M\) is denoted by \(\hat{\mathcal{A}}\). Since \((M,g)\) is a stably causal spacetime is can be equipped with a time function \(\tau\). The null length of \(\beta \in \hat{\mathcal{A}}\) is defined as \[\hat{L}_\tau(\beta) = \sum_{i=1}^k |(\tau \circ \beta)(a_i) - (\tau \circ \beta)(a_{i-1})|.\] It is easy to see that the null length is additive, continuously depends on the curve parameter and is invariant under reparametrizations. Based on this length structure one can obtain a length metric.

By definition, \(\hat{d}_\tau\) is a conformally invariant pseudometric on \((M,g)\). That the null distance is distinguishing asks slightly more of the time function used. Such time functions always exist already in the stably causal setting, for example, one may use temporal functions BeSa:orthsplit?.

The author and B.Allen have shown the following important connection to completeness that was used in the proof of Theorem 2 (c’)\(\Longrightarrow\)(b’).

Note that the completeness needed in the assumption of the statement is only the metric completeness property (b) in the Hopf–Rinow Theorem 1, in particular, the proof does not require the use of the infinity postulate (a), as can be seen from the fact that it holds for any complete metric that induces the manifold topology AB?*Theorem 1.6. The author and L.García-Heveling have in BGH2?*Section 1 established more connections of the null distance to Riemannian geometry. It should be noted, in particular, that weak temporal functions are precisely those time functions \(\tau\) that induce locally equivalent metrics \(\hat{d}_\tau\) BGH2?*Corollary 1.8.

2.3 Lorentzian products↩︎

Every Riemannian manifold \((\Sigma,\sigma)\) can be studied as a Lorentzian manifold by considering the canonical product \[(\Sigma',\sigma') = (\mathbb{R}\times \Sigma, -dt^2 \oplus \sigma).\] In what follows we prove some results about such Lorentzian product spacetimes needed in Section 3. Although our focus is on the above case it is easy to see that most results also hold for products \(N \oplus \Sigma\) for \(N\) a suitable Lorentzian manifold and that similar results are true also for warped product spacetimes (see also AB?, BEE?).

The canonical projection \(t = \pi_0 \colon \Sigma' \to \mathbb{R}\), \(t(p_0,p_\Sigma) := p_0\), satisfies \[dt_p(v) = - g(v,\partial_t) = v_0 > 0\] for future directed causal vectors \(v \in T_p\Sigma' \cong \mathbb{R}\times T_{p_\Sigma}\Sigma\). Thus \(t\) is a smooth temporal function of \((M,g)\). The null distance \(\hat{d}_t\) on \(\Sigma'\) is thus well-defined by Theorem 12 and by AB?*Lemma 4.4 of the form \[\begin{align} \label{dtau} \hat{d}_t(p,q) = \begin{cases} |t(q) - t(p)|, & q \in J^\pm(p), \\ d_\sigma(p_\Sigma,q_\Sigma), & \text{otherwise}. \end{cases} \end{align}\tag{1}\] We show slightly more in the following result.

We use Lemma 14 to characterize the boundary of the causal future.

The above findings are interesting independent of completeness or the Hopf-Rinow Theorem. For us they are crucial for proving Theorem 3 in Section 3 because we do not assume geodesic completeness of \((\Sigma,\sigma)\) nor any knowledge of the Hopf–Rinow Theorem 1 in parts or as a whole (otherwise we would obtain a circular argument). It is useful to note, however, that with the additional assumption of geodesic completeness a lot more can be shown.

Geodesic completeness of \((\Sigma,\sigma)\) can furthermore be employed constructively for proofs about the null distance on \(\mathbb{R}\times \Sigma\).

On the other hand, it is easy to construct examples of Lorentzian products with incomplete fibers that do not satisfy 2 . Consider, for example, \(\mathbb{R}\times (\mathbb{R}^n \setminus \{0\})\) equipped with the metric induced from Minkowski metric and the canonical time function. Nevertheless, any null distance on any spacetime encodes causality locally as has been shown independently by Sakovich and Sormani SaSo?*Theorem 1.1 and García-Heveling and the author BGH2?*Theorem 3.4.

3 Theorem 2 implies the metric Hopf–Rinow Theorem↩︎

In this section we are solely concerned with Lorentzian products, i.e., spacetimes of the form \[(\Sigma',\sigma')=(\mathbb{R}\times\Sigma,-dt^2\oplus \sigma)\] with \((\Sigma,\sigma)\) a Riemannian metric. We prove Theorem 3 (i) and (ii) in Sections 3.1 and 3.2 respectively. In other words we show that the metric Hopf–Rinow Theorem 1 (b)\(\Longleftrightarrow\)(c) is a special case of Theorem 2, and thus can be also viewed as a special case of Theorem 6.

3.1 The Heine–Borel property vs.global hyperbolicity↩︎

The product \((\Sigma',\sigma')\) is always a stably causal spacetime independent of properties of \((\Sigma,\sigma)\), thus it remains to relate the Riemannian compactness postulate to the corresponding conformal compactness postulate for Lorentzian products.

Strictly speaking we prove nothing new, but our proof is new: It is well-known that geodesic completeness of the Riemannian fiber \((\Sigma,\sigma)\) is equivalent to be the Lorentzian product \(I \times \Sigma\) being globally hyperbolic (see Remark 16), and together with the Hopf–Rinow Theorem 1 (a)\(\Longleftrightarrow\)(b) we would therefore be done. But we need to abolish the sophisticated analytic notion of geodesic completeness and just relate the two topological conditions directly. By doing away with (geodesic) completeness we can avoid a circular argument and thus ensure that Theorem 2 really implies Theorem 1 (b)\(\Longleftrightarrow\)(c).

3.2 Metric completeness↩︎

In the same spirit as in Section 3.1 we directly relate the metric completeness property of \(\Sigma\) to that of \(\Sigma' = \mathbb{R}\times \Sigma\).

4 Cone structures↩︎

The aim of this section is to introduce all notions of and to prove Theorem 6. In Sections 4.1 and 4.2 we recall some definitions and results of the theory of cone structures. Readers familiar with the recent theory of cone structures as in Min:cone?, FaSi?, Fa:time?, BeSu:time?, BeSu:gh? may skip this part. In Section 4.3 we introduce the null distance and show that it is a well-defined metric on proper cone structures. In Section 4.4 we prove that the Lorentzian Theorem 2 also has a cone version, namely Theorem 6. These extensions are straightforward but nonetheless have to be checked carefully. The main asset of setting up everything in the cone framework is, apart from its own benefit, that we can efficiently apply it to our new notion of semi-Riemannian spacetimes in Section 5.

4.1 Cone structures↩︎

We use mostly the conventions of Minguzzi Min:cone? which builds on the pioneering work of Fathi and Siconolfi FaSi?, Fa:time? and results of Bernard and Suhr BeSu:time?, BeSu:gh?.

Note that for a closed or proper cone structure it is not sufficient to interpret these notions fiberwise, i.e, that each cone \(C_p\) is closed or proper! Instead, these notions require the use of the topology of the cone bundle. Higher regularity generally helps. For many properties, approximation by wider and more regular cones is sufficient. We write \(C \prec C'\) if \(C \subseteq \operatorname{int}C'\) and \(C \preccurlyeq C'\) if \(C \subseteq C'\).

4.2 Causality and time functions↩︎

By using the differential inclusions \[\dot{\gamma}(t) \in C_{\gamma(t)}, \qquad \dot{\gamma}(t) \in (\operatorname{int}C)_{\gamma(t)}\] on a closed or proper cone structure \((M,C)\) one can define a chronological relation \(I^+\) and a causal relation \(J^+\) on \((M,C)\) that resembles the corresponding Lorentzian relations (cf.Section 2.1). In addition to recalling the notation of Min:cone?*Section 2.1 we add some elements. For various reasons it is convenient and sufficient to work within the class of locally absolutely continuous (or locally Lipschitz) paths Bur?, Min:cone?.

Based on Definition 24 one proceeds as in Section 2.1 to define the causal relation \(J^+\) and chronological relation \(I^+\). The usual properties follow. We review these results briefly to remind the reader that (and why) several require proper cone structures as opposed to only closed ones.

In order to extend the notion of null distance to semi-Riemannian manifolds, we recall that time and temporal functions for closed cone structures are defined in the same way as in Definition 9. One can also define locally anti-Lipschitz time functions in the same way as in Theorem 12. See also Min:cone?*Section 2.2 for more notions.

As in Lorentzian geometry the existence of time functions trivially requires that \((M,C)\) is causal. Fathi and Siconolfi FaSi?*Theorem 1.1 were the first to show that stable causality is sufficient. Minguzzi’s proof is based on the \(K^+\)-relation of Sorkin and Woolgar SoWo? and Levin’s Theorem Lev?.

Several important characterizations of global hyperbolicity are already known for closed and proper cone structures as well as the stability of this notion. For continuous cone structures these results were obtained in the pioneering work of Fathi and Siconolfi FaSi?. We only state the results that we will use and refer to the literature FaSi?, Fa:time?, BeSu:time?, BeSu:gh?, Min:cone? for proofs.

The proof of Theorem 29 for closed and proper cone structures is based on the earlier works of Fathi and Siconolfi FaSi?*Theorem 1.3 for (i)\(\Longleftrightarrow\)(ii) and of Bernard and Suhr BeSu:time?*Theorem 3, Corollary 1.8 for (i)\(\Longleftrightarrow\)(iii) and the splitting. Note that in (iii) we use the recent notation of Bernard and Suhr BeSu:gh?*Definition 1.2 rather than calling it \(h\)-steep as in Min:cone?*Section 2.2.

Being a closed cone structure does, of course, not imply that the causal relation is closed. But global hyperbolicity and properness suffices.

This result thus shows that global hyperbolicity is stronger than \(K\)-causality (because \(K^+ = J^+\), which is antisymmetric if causal) in the framework of proper cone structures independent of the use of time functions. It is also clear that other notions on the causal ladder can be defined equally well for cone structures, however, none of those notions are relevant for us.

4.3 The null distance for cone structures↩︎

Equipped with a causal structure and time function we are finally in a position to introduce a cone version of the null distance of Sormani and Vega SV?. The definitions of piecewise causal paths, null lengths, and null distance of Section 2.2 carry over verbatim to closed cone structures. For the corresponding results to hold, however, the properness of cone structures is crucial.

Lemma 31 immediately implies that \(\hat{d}_\tau\) is a pseudometric on \(M\) but even in Minkowski space more is needed. The following result is the cone version of Theorem 12.

Theorem 32 furthermore implies, as already established in AB?*Theorem 1.1, Proposition 3.8 in the Lorentzian setting, that the length structure respects the manifold topology and that the null distance is an intrinsic metric.

4.4 A completeness-compactness theorem for cone structures↩︎

As in Theorem 13 it follows that the null distances of “complete” time functions are complete. Time functions that satisfy this condition exist for globally hyperbolic cone structures by Theorem 29. Because both results are valid for proper cone structures we obtain a natural extension to a refined Lorentzian completeness-compactness result of García-Heveling and the author BGH2?*Theorem 4.2. Theorem 6 is a simplified version of the following result.

5 Semi-Riemannian spacetimes↩︎

Thanks to Theorem 3 we have seen that Theorem 2 is a conformal generalization of the metric Hopf–Rinow Theorem 1 (b)\(\Longleftrightarrow\)(c) to Lorentzian signature. A natural question that arises is whether one can iterate this procedure and also obtain a semi-Riemannian version of Theorem 2. In this section we show that the answer is yes. To this end we introduce semi-Riemannian spacetimes and initiate their study in Section 5.1. We provide several important examples and see that their existence is intimately tied to a deep and largely open problem in differential/algebraic topology. In Section 5.2 we answer the above question by showing that semi-Riemannian spacetimes admit continuous proper cone structures that are conformally invariant. Thus Theorem 6 yields a true conformal and semi-Riemannian version of the metric Hopf–Rinow Theorem, namely Corollary 7. In Section 5.4 we further investigate if and how closely (stable) causality as well as global hyperbolicity are related for products (with regards to dimension and signature) and find, amongst others, that some but not all parts of Theorem 3 (i) can be recovered.

5.1 Definition, existence, and examples↩︎

Recall from the introduction that we consider the following class of semi-Riemannian manifolds \((M,g)\) with signature \((n-\nu,\nu)\), which interpolates between all manifolds equipped with a metric of positive definite signature \((n,0)\) and parallelizable manifolds equipped with a metric of negative definite signature \((0,n)\).

In this section we discuss a vast list of examples and nonexamples of semi-Riemannian spacetimes and obtain some statements about the existence of such semi-Riemannian structures. We will see that there is a novel and interesting link of semi-Riemannian geometry to a challenging problem studied in differential and algebraic topology.

By definition, any Riemannian manifold is time orientable and a \((n,0)\)-spacetime with \(X = \emptyset\). Every Lorentzian manifold that can be time oriented by a timelike vector field is a \((n-1,1)\)-spacetime and a spacetime in the usual Lorentzian sense (cf.Section 2). The use of the word "spacetime" in Definition 34 is fully justified in Section 5.2 where we see that the cone structure imposed by \(X\) coincides with the usual one. It is well-known that not every Lorentzian manifold is time orientable, and that not every manifold even admits a Lorentzian metric. Using homotopy theory Steenrod first characterized the existence of a semi-Riemannian metric on a given (closed) manifold by the existence of a corresponding continuous tangent subbundle. We recall his argument in modern terminology.

In general, the frame bundle associated to the tangent subbundle (ii) does not admit a global section, and thus the semi-Riemannian metric obtained in Theorem 35 may not be time frame orientable. Interestingly, if \(\nu = 1\) the existence of a Lorentzian metric implies the existence of a time oriented Lorentzian metric (the converse being trivially true). After having discussed topological constraints characterizing such an existence we show in Example 41 that the existence of a semi-Riemannian metric of index \(\nu \geq 2\) on \(M\) does not imply the existence of a \((n-\nu,\nu)\)-spacetime structure on \(M\). We can, however, independently characterize the existence of a spacetime metric by the existence of a suitable global partial frame. Note that the condition of admitting such structures is invariant under \(C^1\) diffeomorphisms.

Parallelizable manifolds are useful for obtaining general semi-Riemannian spacetimes via the following (warped) product construction.

Let us return to the question of existence of a \((n-\nu,\nu)\)-spacetime structure on a given manifold \(M\). The corresponding topological problem, i.e., that of characterizing the existence of a tangent \(\nu\)-frame in condition (ii) of Theorem 36, is well-known and still largely open since almost 100 years. We shall provide some necessary conditions and discuss some progress that has been made since, and recall some of the important notions that have been developed in this context.

The most basic necessary condition for the existence of a tangent \(\nu\)-frame is an immediate consequence of the Poincaré–Hopf Theorem Ho:vf? (an extension of the Hairy Ball Theorem of Poincaré Po:S2? for \(\mathbb{S}^2\) from 1885 and of Brouwer Br:?, Sn? for \(\mathbb{S}^{2n}\) from 1912) since we certainly demand continuity and nowhere vanishing of at least one vector field. The full characterization in the \(\nu=1\) case was shown in Markus Mar?*Theorem 3.

If \(\nu=1\) one can furthermore show that the existence of Lorentzian metric on a given manifold implies the existence of a time oriented Lorentzian metric by moving to a double covering and applying Theorem 40 (see, for instance, ON?*p.). This is not the case for \(\nu \geq 2\) as the following simple example demonstrates (unless, for instance, \(M=\mathbb{S}^n\) and \(2\nu \leq n\) Ste?*Theorem 27.16).

The refined notion of parallelizability of degree \(\nu\) used in Theorem 36 (ii) already appears in the seminal thesis of E.Stiefel Sti:parallel? from 1935, supervised by H.Hopf, which he opens precisely by posing the following question in the closed case:

When does an \(n\)-dimensional manifold \(M\) admit a tangent \(\nu\)-frame for \(1 \leq \nu \leq n\)?

Neither Stiefel nor followers did so far succeed in fully characterizing the existence of a tangent \(\nu\)-frame for \(\nu >1\). But already Stiefel obtained landmark results on the existence of tangent \(\nu\)-frames with certain types of singularities (points where the vector fields become linearly dependent or discontinuous) and introduced a sequence of obstruction classes in cohomology (independently and shortly afterwards Whitney Whi:top? studied the analogous classes for sphere bundles, the modern axiomatic definition of the Stiefel–Whitney classes for vector bundles is due to Hirzebruch). Although we are not interested in understanding the structure of singularities in this work, Stiefel’s and subsequent results could provide a very fruitful starting point to pursue in the future. The reason for this is that we already know that understanding degeneracies in Lorentzian manifolds is related to finding an admissible notion for topology change of acausal slices, a topic that is highly relevant for quantum gravity and nonsmooth Lorentzian geometry (see, for instance, BDGS?, BGH1?, GH:top?, Hor?).

One of the most basic results for the existence of a tangent \(\nu\)-frame without singularities can in modern terminology and thanks to Theorem 36 be formulated as follows.

The necessary condition of Theorem 42 is very weak and far from being sufficient. For instance, any unit sphere \(\mathbb{S}^n\) satisfies \(w_1(T\mathbb{S}^n)=\ldots=w_n(T\mathbb{S}^n)=0\) because \(w_0(T\mathbb{S}^n)=1\) and the total Stiefel–Whitney class is \(w(T\mathbb{S}^n)=1\) Hus?*Chapter 16, Proposition 4.4, but we have already mentioned in Example 38 that only spheres of dimension \(n=1,3,7\) are parallelizable. In 1962 Adams managed to settle the question on the maximal degree of parallelizability for spheres, based on earlier necessary criteria due to the Hurwitz–Radon–Eckman Theorem in linear algebra and James.

Much less is known for general manifolds. Let us briefly recall the progress that has been made beyond Theorem 42 since the 1930s. For certain classes of manifolds with particular dimensional restrictions \(n\) (often \(\operatorname{mod} 4\), spin, etc.) and for small \(\nu\) knowledge about characteristic classes provides important necessary conditions. Thomas Tho:vfs? reviews many classical results of Atiyah, Bott, Frank, Hirzebruch, Hopf, Kervaire, Mayer, Milnor, Steenrod, Whitehead, Wu and many others from the 1950s and 1960s and discusses topological invariants that govern whether one can turn a tangent \(\nu\)-field with finite singularities into a regular one without singularities and collects restrictions and obstructions for the existence of a tangent \(2\)-field with singularities. In the 1970s Atiyah and Dupont AtDu?, Du:K? from the 1970s, using the Atiyah–Singer Index Theorem (see also At?), obtain particularly strong results for the existence of tangent \(2\)- and \(3\)-fields for closed oriented manifolds allowing finite singularities. See also Crabb and Steer CrSt? using the same technique and Koschorke Ko:vflecture? treating these cases for nonorientable closed manifolds using a different approach. The more recent paper by Bökstedt–Dupont–Svane BDS:cobord? gives a nice overview of known results and for \(\nu=4,5,6,7\) (under additional assumptions on \(M\) and \(n\), such as spin) computes the index, whose vanishing characterizes the existence of a \(\nu\)-frame, and thereby shows that it is a global invariant.

Despite the huge amount of fascinating and deep mathematical work that has been carried out on the existence of tangent \(\nu\)-frames and singularities in the closed manifold setting the equivalent problem for and also the topological structure of open (noncompact without boundary) manifolds is much less understood. For us the noncompact situation is significantly more important and in the hope to spark some interest in this problem we collect some known constructions and basic ideas that could be useful for analyzing the noncompact problem.

5.2 Semi-Riemannian spacetimes admit conformal proper cone structures↩︎

Given a spacetime structure on a manifold there are several ways to distinguish between future and past, and study causality. We pick a natural way to do so which for \(\nu =1\) agrees with the usual Lorentzian one and show that it leads to proper cone structures.

Nonetheless, in analogy to the Lorentzian setting, we show that the definition of future/past directed causal vectors defines a proper cone structure on \(M\). The regularity of the metric tensor and the vector fields controls the regularity of the cone structure and ensures that it is topologically well behaved. We restate Theorem 5 of the introduction in a more refined way as follows.

Before we prove Theorem 49 a few remarks are in order. Let us first recall the definition of continuity of set-valued maps that is used in the statement (see, for instance, Aubin and Cellina AuCe:setvalued?*Chapter 1).

For cone structures \(p \mapsto C_p\) on a manifold \(M\) these properties can be checked on coordinate patches (see Min:cone?*p.): At every \(p \in M\) we have a local coordinate system \(\{x^\alpha\}\) on a neighborhood \(U\) of \(p\). Via the local trivialization of the tangent bundle \(TU\) we obtain a splitting \(U \times \mathbb{R}^n\) and can compare sets over different tangents spaces this way. The notion of upper and lower semicontinuity are then applied to the set-valued map \[F(p) = [ C_p \cup \{0\}] \cap \mathbb{B}^n,\] where \(\mathbb{B}^n\) denotes the closed unit ball of \(\mathbb{R}^n\) (instead one could also work with the sphere subbundle and compare the set values \(C_p \cap \mathbb{S}^n\) using the Hausdorff distance on \(\mathbb{S}^n\); one can also say when \(F\) is locally Lipschitz then).

Furthermore, recall that properness of a cone structure requires a nonempty interior bundle. We thus construct a vector in the interior of the cone explicitly by making use of the negative definite version of the following result.

Thanks to Theorem 49 we can apply the tools of the theory of cone structures from Section 4 to semi-Riemannian spacetimes using \(p\mapsto C_p\). For instance, the causal and chronological relation behave as desired and we can use the most important tools of Lorentzian causality theory in the semi-Riemannian context¹. The steps on the causal ladder and, in particular, global hyperbolicity can be defined directly via the theory of cone structures and Theorems 29 and 6 apply. We obtain Corollary 7 directly from Theorems 33 and 36.

That we actually gain information in our semi-Riemannian setup in contrast to general cone structures is evident from its conformally invariant setup.

Finally, note that the stability of global hyperbolicity (as shown for continuous proper cone structures by Fathi and Siconolfi FaSi?*Theorem 1.2 and later extended by others, see also Min:cone?*Theorem 2.39) can in the semi-Riemannian context be interpreted not only in terms of a nearby metric tensor \(\tilde{g}\) but also in terms of a nearby time frame orientation. We show a result via the second approach.

5.3 Semi-Riemannian vector spaces↩︎

In this section we are concerned with the flat spacetime \(\mathbb{R}^{n-\nu,\nu}\). We first fix the canonical \((n-\nu,\nu)\)-spacetime structure on \(\mathbb{R}^n\) (and \(\mathbb{C}^n\)) by using the standard orthonormal global coordinate frame \((E_i)_i\).

We discuss and derive several properties of \(\mathbb{R}^{n-\nu,\nu}\) by hand. All cases \(0 < \nu \leq n\) are covered. If \(\nu = n\) the empty sum convention \(\sum_{j=n+1}^n = 0\) is used.

\(I^+\) and \(J^+\) relations↩︎

Two points \(p,q \in \mathbb{R}^{n-\nu,\nu}\) are future/past directed timelike/causally/lightlike related if and only if the straight line connecting them has the same causal character, i.e., if the corresponding vector \(q-p \in \mathbb{R}^{n-\nu,\nu} \cong T_0 \mathbb{R}^{n-\nu,\nu}\) is future/past directed timelike/causal/lightlike. The convex hull operation can be carried out explicitly to characterize the causal relation in this setting (convex combinations can be split up with the Minkowski inequality, which singles out the \(1\)-norm on \(\operatorname{span}(E_1,\ldots,E_\nu)\) as necessary and sufficient upper bound for convexity). For \(p = \sum_{i=1}^n p^i E_i\) and \(q = \sum_{i=1}^n q^i E_i\) \[\begin{align} p \leq q \Longleftrightarrow\, & q^i \geq p^i \text{ for all }i=1,\ldots,\nu, \text{ and } \\ &\sum_{i=1}^\nu q^i-p^i \geq \sqrt{\sum_{j=\nu+1}^{n} (q^j - p^j)^2}. \end{align}\] If \(p\neq q\) then automatically at least one \(q^i > p^i\) (see also Lemma 46). Analogous characterizations hold for timelike and lightlike with all \(>\) and at least one \(=\) compared to the above, respectively (recall that lightlike means \(q-p \in \partial C\)).

Canonical time function↩︎

The function \[T(p) := \sum_{i=1}^\nu p^i\] is a smooth time function because by the above \[\begin{align} p < q \Longrightarrow \, & q^i \geq p^i \text{ for all } i=1,\ldots,\nu, \text{ and }\\ & q^k > p^k \text{ for at least one } k=1,\ldots, \nu, \\ \Longrightarrow \, & T(q) > T(p). \end{align}\] It is easy to see that \(T\) is, in fact, completely uniform Cauchy temporal: Since \(dT = \sum_{i=1}^\nu E^\flat_i\), for any \(v \in T_p \mathbb{R}^{n-\nu,\nu}\) future directed causal \(v^i \geq 0\), \(i=1,\ldots,\nu\), and \[\begin{align} dT(v) = \sum_{i=1}^\nu v^i \geq \sqrt{\sum_{i=\nu+1}^n (v^i)^2}, \end{align}\] which implies that it is bounded below by the Euclidean norm of \(v\) since \[\begin{align} 2 dT(v) \geq \sum_{i=1}^\nu v^i + \sqrt{\sum_{i=\nu+1}^n (v^i)^2}\geq \sqrt{\sum_{i=1}^n (v^i)^2} = \| v\|_2. \end{align}\]

If \(\nu=1\) then \(T\) is the canonical time function used for Minkowski space.

Global hyperbolicity↩︎

Since \(T\) is a Cauchy time function it follows from Theorem 29 (ii)\(\Longrightarrow\)(i) that \(\mathbb{R}^{n-\nu,\nu}\) is globally hyperbolic. In connection with Theorem 3 that we aim to (partly) generalize in Section 5.4 it is insightful to prove global hyperbolicity also with “bare hands” using the Heine–Borel property of Euclidean space \(\mathbb{R}^n\).

For any \(p \in \mathbb{R}^{n-\nu,\nu}\) \[J^+(p) = \left(\bigcap_{j=1}^{\nu} \{ x^j \geq p^j \}\right) \cap \left\{ \sum_{i=1}^ \nu x^i-p^i \geq \sqrt{\sum_{i=\nu+1}^{n} (x^i-p^i)^2} \right\},\] and thus the causal future and past sets \(J^\pm(p)\) are closed as an intersection of closed sets. Hence for any \(p\) and \(q\) the causal diamond \(J^+(p) \cap J^-(q)\) is closed. In what follows we show that \(J^+(p) \cap J^-(q)\) is a bounded subset of the Euclidean space \(\mathbb{R}^{n}\). Since \[J^+(p) \cap J^-(q) \subseteq \bigcap_{j=1}^{\nu} \{ q^j \geq x^j \geq p^j \}\] the set is bounded in the first \(\nu\) coordinate directions by future and past directedness. If \(\nu=n\) we have shown boundedness of \(J^+(p) \cap J^-(q)\). If \(\nu < n\) boundedness in the remaining \(n-\nu\) directions follows from causality. For \(x \in J^+(p) \cap J^-(q)\) and \(i=1,\ldots,\nu\) \[|q^i - x^i| + |x^i - p^i| = q^i - p^i \leq \max_{1\leq j\leq \nu} q^j - p^j =: \frac{R}{\nu}\] and thus \[\sqrt{\sum_{i=\nu+1}^{n} (x^i-q^i)^2} + \sqrt{\sum_{i=\nu+1}^{n} (x^i-p^i)^2} \leq 2R,\] meaning that \(x\), and thus \(J^+(p) \cap J^-(q)\), is also bounded in the remaining \(n-\nu\) coordinate directions.

Finally, we apply the Heine–Borel property of the Euclidean space and conclude that \(J^+(p) \cap J^-(q)\) is compact.

Null distance↩︎

By Theorem 52 the null distance \(\hat{d}_T\) is a conformally invariant metric that induces the topology of \(\mathbb{R}^{n-\nu,\nu}\). Sormani and Vega SV?*Proposition 3.4 have shown that Minkowski space can easily be equipped with a non-locally anti-Lipschitz time function so that the corresponding null distance is only a pseudometric. We show that \(\tau = T^{2k+1}\), \(k \in \mathbb{N}\), are analogous examples for \(\mathbb{R}^{n-\nu,\nu}\) as long as \(\nu < n\). Let \(p = (0,\ldots,0)\) and \(q = (0,\ldots,0,1)\). By the above \(p \nleq q\). The paths \(\gamma^\pm \colon [0,1]\to M\), \[\gamma^\pm(t) \to (\pm t,0,\ldots,t)\] are causal, with \(\gamma^+\) being future directed and \(\gamma^-\) being past directed. By rescaling and concatenating such future and past directed causal paths we obtain piecewise causal paths \(\beta_j \colon [0,1] \to [0,\frac{1}{2j}] \times \{ 0 \} \times \ldots \times [0,1]\) from \(p\) to \(q\) with \(2j\) pieces of \(\tau\)-height \((\frac{1}{2j})^{2k+1}\), i.e., \[\hat{L}_\tau (\beta_j) = \frac{2j}{(2j)^{2k+1}} = \frac{1}{4^k j^{2k}}.\] Thus \[\hat{d}_\tau (p,q) \leq \lim_{j\to\infty} \hat{L}_\tau(\beta_j) = 0,\] which means that \(\hat{d}_\tau\) is not distinguishing with respect to any \(\tau = T^{2k+1}\).

This example furthermore shows that the case \(\nu= n\) is special, which foreshadows some results of the following section.

After having discussed \(\mathbb{R}^{n-\nu,\nu}\) with the canonical time frame orientation induced by \(E_1,\ldots,E_\nu\) we conclude with a simple modification thereof, demonstrating that the choice (and orientation) of vector fields is crucial for the global causal properties of a semi-Riemannian spacetime.

5.4 Global hyperbolicity iterated↩︎

In Theorem 3 (i) we have related the Heine–Borel property of a Riemannian manifold \((\Sigma,\sigma)\) directly to the compactness of causal diamonds of the corresponding Lorentzian product spacetime \((\mathbb{R}\times \Sigma, -dt^2 \oplus \sigma)\). Can we prove a similar connection for the related notions of global hyperbolicity for different \(\nu\)?

Since we have already treated the \(\nu=0\) case we can restrict to \(0 < \nu \leq n\). We first attempt a naive approach. By that we mean that to go from a spacetime structure on \((M,g)\) given by vector fields \(X_1,\ldots,X_\nu\) we define one on the product \[\begin{align} \label{Mprime} (M',g') = (\mathbb{R}\times M, -dt^2 \oplus g) \end{align}\tag{11}\] by using the additional vector field \(X_{\nu+1} =\partial_t\). We call this \((n-\nu,\nu+1)\)-spacetime structure on \((M',g')\) the orthogonally extended \((n-\nu,\nu+1)\)-spacetime. We also consider the product \[\begin{align} \label{Mdprime} (M'',g'') = (\mathbb{R}\times M, dt^2 \oplus g) \end{align}\tag{12}\] equipped with the same spacetime structure but viewed as a \((n+1-\nu,\nu)\)-spacetime.

We show that the answer to the above question is no, i.e., that the notions of global hyperbolicity for different \(\nu\) are in general unrelated. For instance, we demonstrate that global hyperbolicity of \((M,g)\) does not imply global hyperbolicity of \((M',g')\). On the other hand, we succeed in showing that the top-down implication for \((M',g')\) to \((M,g)\) still holds. In this section we prove Theorem 8 and provide a counterexample for the missing bottom-up implication. The reason why going from \((M,g)\) to \((M',g')\) does not work is, in short, because the boundary \(\partial C'\) of the corresponding cone structure includes all of \(C\) and is “too wide”. In other words, \(M'\) leaves more room than in \(M\) because \(g'(v,v) \leq 0\) does not imply \(g(v,v) \leq 0\) (this was not an issue for positive definite \(g\)). The metric perspective and the fact that we crucially exploited the direct (and not just conformal) relation between \(d_\sigma\) and \(\hat{d}_t\) in Section 3 also offers an explanation for why our proof from Theorem 3 cannot be extended.

Before jumping right into the analysis of global hyperbolicity in this framework it is insightful and necessary to consider (stable) causality first. Indeed, we see that problems already occur at this level.

The implications of ?? allow us to efficiently relate the causal structures of \(M\), \(M'\), and \(M''\), which ultimately cumulates to the prove of Theorem 8.

Note that Proposition 60 is already weaker than in the Riemannian vs.Lorentzian case (\(\nu=0\)) where \(M'\) is (even stably!) causal completely independent of \(M\). From the proof it is also clear which implication is the most fragile one. Indeed, we lose it when stepping up on the causal ladder.

Recall that the reason why the second implication (\(M \Longrightarrow M''\)) in Proposition 61 holds is that compared to \(M\) the projections of the cones in \(M''\) to \(M\) are narrower (due to more positive directions) or remain the same (if the new variable remains fixed). On the other hand, the projections of the cones in \(M'\) become strictly wider than the cones on \(M\) (due to more negative tangent directions) even if the new variable is fixed (unless \(\nu =n\), in which case they remain the same). In other words, if \(\gamma\) is a future directed \(g'\)-causal curve then \(\pi_M \circ \gamma\) need not be future directed \(g\)-causal. This also explains why the implication (\(M \Longrightarrow M'\)) only works for \(\nu=n\).

Moreover, unlike from the Riemannian to the Lorentzian case the coordinate \(t\) is not a time function for \(M'\) if \(X_{\nu+1} = \partial_t\). We show that if one mildly perturbs the vector field \(\partial_t\) one can nonetheless ensure that \(t\) is a time function and thus safe stable causality for \(M'\) while leaving the first \(\nu\) time frame orienting vector fields intact. As in Section 2.3 (stable) causality of \(M\) is then also not required.

Having established the equivalence of causality in Proposition 60 and having shown how time functions carry over from one spacetime to another in Propositions 61 and 62, we finally turn to Cauchy time functions and global hyperbolicity and prove the remaining implications of Theorem 8.

We start with the easier situation \(M''\). Adding positive tangent directions, meaning \(+dt^2\), is not a problem as no additional time frame orientation defining vector fields are needed and projections and pullbacks behave as desired.

We conclude with an example that demonstrates that, in general, the implication of Proposition 64 cannot be reversed if \(\nu<n\). This should come as no surprise because in Lemma 58 we have observed that the cones in \(M'\) include the base \(M\) if we choose \(X_{\nu+1}=\partial_t\). However, we argue that even when turning them into true cones, for instance, by using \(X_{\nu+1} = \partial_t - \varepsilon (X_1 + \ldots + X_\nu)\) for \(\varepsilon >0\) as in Proposition 62 for enforcing stable causality on \(M'\), global hyperbolicity of \(M\) does not, in general, imply global hyperbolicity of \(M'\). This is in stark contrast to the Riemannian and Lorentzian equivalence obtained in Theorem 3.

All in all, we have shown Theorem 8. The implications are established in Propositions 63 and 64, and Example 65 shows that the missing implication is indeed false.

Acknowledgments↩︎

The author would like to thank Jim Isenberg for encouraging her to explore the semi-Riemannian setting and Steffen Sagave for discussing some references in algebraic topology. This project was supported by the Dutch Research Council (NWO), Project number VI.Veni.192.208.

Statements and declarations↩︎

This work was supported by the Dutch Research Council (NWO), Project number VI.Veni.192.208. The author states that there is no conflict of interest. No data was generated or used in this manuscript.

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1. With Lipschitz regularity the (for \(1 < \nu < n\)) nonconvex precausal cone map \(p\mapsto \mathcal{C}_p\) could also be used to obtain solutions to the corresponding differential inclusion but limit curves, which we implicitly use via Min:cone?, are generically only in the relaxed convexified differential inclusion AuCe:setvalued?*Sections 2.3 and 2.4.↩︎