2. Preliminaries

Here we follow Wald (Reference Wald1984) and Malament (Reference Malament2012). An $n$ -dimensional, general relativistic spacetime (for $n \ge 2$ ) is a pair of mathematical objects $\left( {M,{g_{ab}}} \right)$ where $M$ is a smooth, connected, $n$ -dimensional, Hausdorff manifold and ${g_{ab}}$ is a smooth metric of Lorentz signature $\left( { + , - , \ldots , - } \right)$ defined on $M$ . In what follows, let ${\cal U}$ be the collection of all spacetimes. We say two spacetimes $\left( {M,{g_{ab}}} \right)$ and $\left( {M{\rm{'}},g{{\rm{'}}_{ab}}} \right)$ are isometric if there is a diffeomorphism $\varphi :M \to M{\rm{'}}$ such that ${\varphi ^{\rm{*}}}\left( {g{{\rm{'}}\!_{ab}}} \right) = {g_{ab}}$ .

Fix a model $\left( {M,{g_{ab}}} \right)$ . For each point $p \in M$ , the metric assigns a cone structure to the tangent space ${M_p}$ . Any tangent vector ${\xi ^a}$ in ${M_p}$ will be timelike if ${g_{ab}}{\xi ^a}{\xi ^b} \gt 0$ , null if ${g_{ab}}{\xi ^a}{\xi ^b} = 0$ , or spacelike if ${g_{ab}}{\xi ^a}{\xi ^b} \lt 0$ . Null vectors create the cone structure; timelike vectors fall inside the cone while spacelike vectors fall outside. A time orientable model is one that has a continuous timelike vector field on $M$ . In what follows, we assume that models are time orientable and that an orientation has been chosen.

For some connected interval $I \subset \mathbb{R}$ , a smooth curve $\gamma :I \to M$ is timelike if its tangent vector ${\xi ^a}$ at each point in $\gamma \left[ I \right]$ is timelike. Similarly, a curve is null if its tangent vector at each point is null. A curve is causal if its tangent vector at each point is either null or timelike. A causal curve is future-directed if its tangent vector at each point falls in or on the future lobe of the light cone. A causal curve $\gamma :I \to M$ is closed if the tangent vector is nowhere vanishing and there are distinct $s,s{\rm{'}} \in I$ such that $\gamma \left( s \right) = \gamma \left( {s{\rm{'}}} \right)$ . $\left( {M,{g_{ab}}} \right)$ satisfies chronology if it does not contain a closed timelike curve; it satisfies causality if it does not contain a closed causal curve.

We write $p \ll q$ (respectively, $p \lt q$ ) if there exists a future-directed timelike (respectively, causal) curve from $p$ to $q$ . For any point $p \in M$ , we define the timelike future of $p$ as the set ${I^ + }\left( p \right) = \left\{ {q:p \ll q} \right\}$ . Similarly, the causal future of $p$ is the set ${J^ + }\left( p \right) = \{ q:p \lt q\} $ . The timelike and causal pasts of $p$ , denoted ${I^ - }\left( p \right)$ and ${J^ - }\left( p \right)$ , are defined analogously. The spacetime $\left( {M,{g_{ab}}} \right)$ satisfies distinguishability if there do not exist distinct points $p,q \in M$ such that ${I^ - }\left( p \right) = {I^ - }\left( q \right)$ or ${I^ + }\left( p \right) = {I^ + }\left( q \right)$ . We say $\left( {M,{g_{ab}}} \right)$ admits a global time function if there is a smooth function $t:M \to \mathbb{R}$ such that, for any distinct points $p,q \in M$ , if $p \in {J^ + }\left( q \right)$ , then $t\left( p \right) \gt t\left( q \right)$ . $\left( {M,{g_{ab}}} \right)$ satisfies global hyperbolicity if it is causal and, for any points $p,q \in M$ , the set ${J^ + }\left( p \right) \cap {J^ - }\left( q \right)$ is compact.

A curve $\gamma :I \to M$ is maximal if there is no curve $\gamma {\rm{'}}:I{\rm{'}} \to M$ such that $I$ is a proper subset of $I{\rm{'}}$ and $\gamma \left( s \right) = \gamma {\rm{'}}\left( s \right)$ for all $s \in I$ . The curve $\gamma :I \to M$ is a geodesic if ${\xi ^a}{\nabla _a}{\xi ^b} = 0$ , where ${\xi ^a}$ is its tangent vector and ${\nabla _a}$ is the unique derivative operator compatible with ${g_{ab}}$ . A maximal geodesic $\gamma :I \to M$ is incomplete if $I \ne {\mathbb R}$ . A spacetime is geodesically incomplete if it harbors an incomplete geodesic and geodesically complete otherwise; one can define causal geodesic (in)completeness in an analogous way.

Let the energy–momentum tensor ${T_{ab}}$ for the spacetime $\left( {M,{g_{ab}}} \right)$ be defined by Einstein’s equation: ${R_{ab}} - {1 \over 2}R{g_{ab}} = 8\pi {T_{ab}}$ , where ${R_{ab}}$ is the Ricci tensor and $R$ the scalar curvature associated with ${g_{ab}}$ . We say that $\left( {M,{g_{ab}}} \right)$ is a vacuum solution if ${T_{ab}} = 0$ . The null energy condition is satisfied if, for any null vector ${\chi ^a}$ , we have ${T_{ab}}{\chi ^a}{\chi ^b} \ge 0$ . The weak energy condition is satisfied if, for each timelike vector ${\xi ^a}$ , we have ${T_{ab}}{\xi ^a}{\xi ^b} \ge 0$ . The strong energy condition is satisfied if, for any unit timelike vector ${\xi ^a}$ , we have $\left( {{T_{ab}} - {1 \over 2}T{g_{ab}}} \right){\xi ^a}{\xi ^b} \ge 0$ . Finally, the dominant energy condition is satisfied if, for any future-directed unit timelike ${\xi ^a}$ , the vector $T_{{\rm{\;\;}}b}^a{\xi ^b}$ is causal and future-directed.

3. Inextendibility

A spacetime $\left( {M{\rm{'}},g{{\rm{'}}_{ab}}} \right)$ is an extension of the spacetime $\left( {M,{g_{ab}}} \right)$ if there is a proper subset $N \subset M{\rm{'}}$ such that the spacetimes $( {N,g{{\rm{'}}\!_{ab}}} )$ and $(Mg_{ab})$ are isometric. A spacetime is extendible if it has an extension and inextendible otherwise. One can show that any extendible spacetime has an inextendible extension (Geroch Reference Geroch, Carmeli, Fickler and Witten1970b). This fact helps to underpin the nearly universally held position that “any [physically] reasonable space-time should be inextendible” (Clarke Reference Clarke1993, 8). John Earman summarizes and responds to the usual line of argument (cf. Penrose Reference Penrose1969; Geroch Reference Geroch, Carmeli, Fickler and Witten1970b):

Setting aside the metaphysical issues outlined here, we see that the inextendibility condition also faces an important conceptual difficulty: the standard formulation is defined relative to the background “possibility space” ${\cal U}$ (the collection of all spacetimes) despite the fact that within ${\cal U}$ lurk “physically unreasonable” members (Manchak Reference Manchak2011, Reference Manchak2020). Shouldn’t a spacetime which is extendible according to the standard definition count as “inextendible” if none of its extensions are “physically reasonable”? Even if we cannot pin down, once and for all, a single collection of “physically reasonable” spacetimes, one can still explore variant formulations of the inextendibility condition defined relative to subcollections of ${\cal U}$ (Geroch Reference Geroch, Carmeli, Fickler and Witten1970b; Manchak Reference Manchak2016). For any collection ${\cal P} \subseteq {\cal U}$ , consider the following: a ${\cal P}$ -spacetime is a spacetime in ${\cal P}$ ; a ${\cal P}$ -spacetime $\left( {M{\rm{'}},g{{\rm{'}}\!_{ab}}} \right)$ is a ${\cal P}$ - extension of a ${\cal P}$ -spacetime $\left( {M,{g_{ab}}} \right)$ if $\left( {M{\rm{'}},g{{\rm{'}}\!_{ab}}} \right)$ is an extension of $\left( {M,{g_{ab}}} \right)$ ; a ${\cal P}$ -spacetime is ${\cal P}$ -extendible if it has a ${\cal P}$ -extension and is ${\cal P}$ -inextendible otherwise. It is trivial that for any collection ${\cal P} \subset {\cal U}$ of inextendible spacetimes (e.g., the collection of geodesically complete spacetimes) a ${\cal P}$ -inextendible spacetime must be inextendible. The general situation is quite different, however. For each ${\cal P} \subset {\cal U}$ , consider the following statement:

Let $\left( V \right),\left( {DEC} \right),\left( {SEC} \right),\left( {WEC} \right),\left( {NEC} \right) \subset {\cal U}$ be, respectively, the collections of vacuum solutions and spacetimes satisfying the dominant, strong, weak, and null energy conditions; note that $\left( V \right) \subset \left( {DEC} \right) \subset \left( {WEC} \right) \subset \left( {NEC} \right)$ and $\left( V \right) \subset \left( {SEC} \right) \subset \left( {NEC} \right)$ . Let $\left( {GH} \right),\left( {TF} \right),\left( {Dist} \right),\left( {Caus} \right),\left( {Chron} \right) \subset {\cal U}$ be, respectively, the collections of spacetimes which are globally hyperbolic, admit a global time function, are distinguishing, causal, and chronological. Of course, $\left( {GH} \right) \subset \left( {TF} \right) \subset \left( {Dist} \right) \subset \left( {Caus} \right) \subset \left( {Chron} \right)$ . Let $\left( {GI} \right) \subset {\cal U}$ be the collection of geodesically incomplete spacetimes. We have the following proposition (Manchak Reference Manchak2017, Reference Manchak, Madarász and Székely2021).

The status of (*) is still unknown for ${\cal P} = \left( V \right)$ and ${\cal P} = \left( {Chron} \right)$ (cf. Krasnikov Reference Krasnikov2018). Indeed, the ${\cal P} = \left( {Chron} \right)$ case has been one focus of the “time travel” literature for some time but remains difficult to settle (cf. Krasnikov Reference Krasnikov2018). But in general, the proposition suggests that we should carefully attend to the differences between the standard definition of inextendibility and other variants. In formulating a version of the “cosmic censorship” conjecture, Wald (Reference Wald1984, 304–305) does just this when he appreciates that while some “maximal Cauchy developments … are known to be extendible” it may be that all such extensions fail to be ${\cal P}$ -extensions for some carefully chosen collection ${\cal P} \subset {\cal U}$ of “physically reasonable” spacetimes.

4. Stability

In their influential book The Large Scale Structure of Space-Time, Hawking and Ellis wrote:

It is of some interest that despite the claim that a suitable topology can be put on the entire collection ${\cal U}$ , no one has yet done this even after almost fifty years (Fletcher Reference Fletcher2016). Instead, various topologies have been defined on each collection ${\cal L}\left( M \right) \subset {\cal U}$ of all spacetimes with underlying manifold $M$ . The most commonly used are the “ ${C^k}$ fine” topologies (also called the “ ${C^k}$ open” topologies) for $k \ge 0$ , which we shall consider here (cf. Geroch Reference Geroch and Rainer1971; Hawking and Ellis Reference Hawking and Ellis1973).

Let $\left( {M,{g_{ab}}} \right)$ and $\left( {M,g{{\rm{'}}\!_{ab}}} \right)$ be spacetimes, let ${h^{ab}}$ be any positive definite metric on $M$ , and let ${\nabla _a}$ be the unique derivative operator compatible with ${h^{ab}}$ . At each point in $M$ , the distance function $d\left( {{g_{ab}},g{{\rm{'}}\!_{ab}},{h^{ab}},k} \right)$ between the $k$ th partial derivatives (for $k \ge 0$ ) of the Lorentzian metrics ${g_{ab}}$ and $g{{\rm{'}}\!_{ab}}$ on $M$ relative to ${h^{ab}}$ is given by

$${[{h^{ac}}{h^{bd}}\left( {{g_{ab}} - g{{\rm{'}}\!_{ab}}} \right)\left( {{g_{cd}} - g{{\rm{'}}\!_{cd}}} \right)]^{1/2}}{\rm{\;for\;\;}}k = 0,$$

$${[{h^{ac}}{h^{bd}}{h^{{r_1}{s_1}}} \cdots {h^{{r_k}{s_k}}}\left( {{\nabla _{{r_1}}} \cdots {\nabla _{{r_k}}}\left( {{g_{ab}} - g{{\rm{'}}\!_{ab}}} \right)} \right)\left( {{\nabla _{{s_1}}} \cdots {\nabla _{{s_k}}}\left( {{g_{cd}} - g{{\rm{'}}\!_{cd}}} \right)} \right)]^{1/2}}{\rm{\;for\;\;}}k \gt 0\;.$$

A ${C^k}$ fine neighborhood of a spacetime $\left( {M,{g_{ab}}} \right)$ is any collection ${\cal N} \subset {\cal L}\left( M \right)$ which includes all spacetimes $\left( {M,g{{\rm{'}}\!_{ab}}} \right)$ such that ${\rm{Su}}{{\rm{p}}_M}\left[ {d\left( {{g_{ab}},g{{\rm{'}}\!_{ab}},{h^{ab}},j} \right)} \right] \lt \varepsilon $ for $j = 0, \ldots ,k$ , where ${h^{ab}}$ is a positive definite metric on $M$ and $\varepsilon $ is a positive number. For all ${\cal Q} \subseteq {\cal P} \subseteq {\cal U}$ , we say the property ${\cal Q}$ is ${C^k}$ stable relative to the collection ${\cal P}$ if, for each ${\cal Q}$ -spacetime $\left( {M,{g_{ab}}} \right)$ , there is a ${C^k}$ fine neighborhood of $\left( {M,{g_{ab}}} \right)$ such that every ${\cal P}$ -spacetime in the neighborhood is a ${\cal Q}$ -spacetime. Immediately we see that, for all ${\cal Q} \subseteq {\cal P} \subseteq {\cal U}$ , if property ${\cal Q}$ is ${C^k}$ stable relative to the collection ${\cal P}$ , then ${\cal Q}$ is ${C^l}$ stable relative to ${\cal P}$ for all $l \ge k$ .

We know that even the coarsest of all of the ${C^k}$ fine topologies is still quite fine: If $\left( {M,{g_{ab}}} \right)$ is a spacetime and $M$ is non-compact, then the collection $\left\{ {\left( {M,\lambda {g_{ab}}} \right):\lambda \in \left( {0,\infty } \right)} \right\}$ does not represent a ${C^0}$ fine continuous curve; in addition, the induced topology on the collection is discrete (Geroch Reference Geroch and Rainer1971). It seems the ${C^k}$ fine topologies have too many open sets to capture, once and for all, what it means for one spacetime to be “nearby” another. On the other hand, this means that instability results are all the more significant. Early results concerned two important causal properties (Hawking Reference Hawking1969; Geroch Reference Geroch1970a).

Consider a few remarks concerning Proposition 2. First, the $\left( {TF} \right)$ case tells us that any spacetime $\left( {M,{g_{ab}}} \right)$ with a global time function is “stably causal” in the sense that one can find a ${C^0}$ neighborhood of $\left( {M,{g_{ab}}} \right)$ such that each spacetime in the neighborhood admits a global time function and is therefore causal. Second, there were significant gaps in the proof concerning the $\left( {GH} \right)$ case which were filled in only recently (Navarro and Minguzzi Reference Navarro and Minguzzi2011). Finally, a simple but physically significant corollary to the proposition ensures that the ${C^k}$ stability of $\left( {TF} \right)$ and $\left( {GH} \right)$ with respect to ${\cal U}$ will “transfer down” to the ${C^k}$ stability of $\left( {TF} \right) \cap {\cal P}$ and $\left( {GH} \right) \cap {\cal P}$ with respect to any “physically reasonable” collection ${\cal P} \subseteq {\cal U}$ . In general, we have the following.

What about the stability of other important spacetime properties? Consider the collection $\left( {GC} \right) \subset {\cal U}$ of geodesic complete spacetimes. The following claim was made in the first edition of Beem and Ehrlich (Reference Beem and Ehrlich1981).

Soon after there was to be a dramatic turn of events. Ehrlich later recounted the following:

From Williams (Reference Williams1984) we have the following result, which is all the more remarkable given how fine even the ${C^0}$ topologies have been shown to be.

It is of some interest that this result fails within the Riemannian context where both geodesic completeness and geodesic incompleteness are ${C^k}$ stable for all $k \ge 0$ (Beem and Ehrlich Reference Beem and Ehrlich1987). By the time the second edition of their book was published, Beem and Ehrlich had worked to salvage the stability of geodesic (in)completeness by restricting attention to special cases. Consider the following proposition, which is representative of this effort (Beem et al. Reference Beem, Ehrlich and Easley1996).

Here, the physical significance is limited given that attention is restricted to globally hyperbolic spacetimes and the ${C^0}$ case is not considered. Are more general results available? Nothing so far—even today, we do not have a good understanding of the (in)stability properties of geodesic (in)completeness and closely related properties (cf. Manchak Reference Manchak2018; Doboszewski Reference Doboszewski2020).

5. Stability and Inextendibility

What is known concerning the (in)stability of the inextendibility properties? Very little. What we do have is due to Beem and Ehrlich (Reference Beem and Ehrlich1987). Using their work concerning the stability of geodesic completeness, and drawing on on the fact that geodesic completeness implies inextendibility, they show the following.

It is somewhat remarkable that, even after restricting attention to Minkowski spacetime, the ${C^0}$ case is unsettled. Are general results available? One would love to know the status of the following conjecture, for example.

Given the dramatic twists and turns so far concerning the (in)stability of geodesic (in)completeness—and the many surprises throughout the history of global Lorentzian geometry more generally—the status of the conjecture is anyone’s guess. But even if it were true, there is a sense in which its physical significance would seem to be quite limited since there is no assurance here that the stability of inextendibility relative to ${\cal U}$ would “transfer down” to the stability of ${\cal P}$ -inextendibility relative to some “physically reasonable” collection ${\cal P} \subset {\cal U}$ . Indeed, consider the following.

Proposition 7 There are collections ${\cal Q} \subset {\cal P} \subset {\cal U}$ such that ${\cal P}$ -inextendibility is ${C^k}$ stable relative to ${\cal P}$ for all $k \ge 0$ but ${\cal Q}$ -inextendibility fails to be ${C^k}$ stable relative to ${\cal Q}$ for all $k \ge0$ . Moreover, ${\cal P}$ can be chosen so that each member is a globally hyperbolic vacuum solution.

Proof. We work in two dimensions to simplify the presentation, but one can generalize in the natural way. In first stage of the proof, we define the collections ${\cal Q},{\cal P} \subset {\cal U}$ . Consider the smooth bump function $u:\left[ { - 2,2} \right] \to \mathbb{R}$ given by

For each $n \in {\mathbb{Z}^ + }$ , we let ${f_n},{F_n}:\left[ { - 2,2} \right] \to \mathbb{R}$ be the functions ${f_n}\left( t \right) = {[1 - u\left( t \right)/n]^{1/2}}$ and ${F_n}\left( t \right) = \mathop \int \nolimits_0^t {f_n}\left( x \right){\rm{\;}}dx$ . The graphs of the functions ${f_1}\left( t \right)$ and ${F_1}\left( t \right)$ are given in Figure 1. Note that, for all $n$ , ${F_n}\left( t \right)$ is strictly increasing and thus invertible. When no confusion arises, we will abuse notation and consider the functions ${f_n}\left( t \right)$ and ${F_n}\left( t \right)$ where the domain is restricted to $\left( { - 2,2} \right)$ .

Figure 1.

The functions ${f_1}\left( t \right)$ and ${F_1}\left( t \right)$ .

Let $M = \{ \left( {t,\varphi } \right) \in \mathbb{R} \times {S^1}: - 2 \lt t \lt 2\} $ . For each $n \in {\mathbb{Z}^ + }$ , let $\left( {M,{g_{ab}}\left( n \right)} \right)$ be the spacetime defined by setting ${g_{ab}}\left( n \right) = f_n^2\left( t \right){\nabla _a}t{\nabla _b}t - {\nabla _a}\varphi {\nabla _b}\varphi $ . Let $\left( {M,{g_{ab}}\left( \dagger \right)} \right)$ be the spacetime defined by setting ${g_{ab}}\left( \dagger \right) = {\nabla _a}t{\nabla _b}t - {\nabla _a}\varphi {\nabla _b}\varphi $ . Finally, let $\left( {M,{g_{ab}}\left( \ddagger \right)} \right)$ be the spacetime defined by setting ${g_{ab}}\left( \ddagger \right) = f_\ddagger ^2\left( t \right){\nabla _a}t{\nabla _b}t - {\nabla _a}\varphi {\nabla _b}\varphi $ where ${f_\ddagger }:\left( { - 2,2} \right) \to \mathbb{R}$ is given by ${f_\ddagger }\left( t \right) = \pi {\rm{se}}{{\rm{c}}^2}\left( {\pi t/4} \right)/2$ . Let ${\cal P} \subset {\cal U}$ be the collection $\left\{ {\left( {M,{g_{ab}}\left( \dagger \right)} \right),\left( {M,{g_{ab}}\left( \ddagger \right)} \right)} \right\} \cup \left\{ {\left( {M,{g_{ab}}\left( n \right)} \right):n \in {\mathbb{Z}^ + }} \right\}$ ; let ${\cal Q} \subset {\cal P}$ be the collection ${\cal P} - \left\{ {\left( {M,{g_{ab}}\left( \ddagger \right)} \right)} \right\}$ .

In the second stage of the proof, we establish the following facts: (i) for all $n$ , $\left( {M,{g_{ab}}\left( n \right)} \right)$ is ${\cal Q}$ -extendible (and hence ${\cal P}$ -extendible); (ii) $\left( {M,{g_{ab}}\left( \dagger \right)} \right)$ is ${\cal Q}$ -inextendible but ${\cal P}$ -extendible; (iii) $\left( {M,{g_{ab}}\left( \ddagger \right)} \right)$ is ${\cal P}$ -inextendible; and (iv) each member of ${\cal P}$ is a globally hyperbolic vacuum solution. All of these facts will follow easily once we define, for each spacetime in ${\cal P}$ , an isometric variant.

First, consider $\left( {M{\rm{'}}\left( \ddagger \right),g{{\rm{'}}\!_{ab}}} \right)$ , where $M{\rm{'}}\left( \ddagger \right) = \mathbb{R} \times {S^1}$ and $g{{\rm{'}}\!_{ab}} = {\nabla _a}t{\rm{'}}{\nabla _b}t{\rm{'}} - {\nabla _a}\varphi {\rm{'}}{\nabla _b}\varphi {\rm{'}}$ . We find that $\left( {M,{g_{ab}}\left( \ddagger \right)} \right)$ is isometric to $\left( {M{\rm{'}}\left( \ddagger \right),g{{\rm{'}}\!_{ab}}} \right)$ ; to see this, use the diffeomorphism ${{\rm{\Psi }}_\ddagger }:M \to M{\rm{'}}\left( \ddagger \right)$ defined by ${{\rm{\Psi }}_\ddagger }\left( {\left( {t,\varphi } \right)} \right) = \left( {2{\rm{tan}}\left( {\pi t/4} \right),\varphi } \right)$ and note that ${\nabla _a}\left( {2{\rm{tan}}\left( {\pi t/4} \right)} \right) = {f_\ddagger }\left( t \right){\nabla _a}t$ . Next, consider $\left( {M{\rm{'}}\left( \dagger \right),g{{\rm{'}}\!_{ab}}} \right)$ , where $M{\rm{'}}\left( \dagger \right) = \{ \left( {t{\rm{'}},\varphi {\rm{'}}} \right) \in M{\rm{'}}\left( \ddagger \right): - 2 \lt t{\rm{'}} \lt 2\} $ and the domain of $g{{\rm{'}}\!_{ab}}$ is restricted in the natural way. We find that $\left( {M,{g_{ab}}\left( \dagger \right)} \right)$ is isometric to $\left( {M{\rm{'}}\left( \dagger \right),g{{\rm{'}}\!_{ab}}} \right)$ ; to see this, just use the identity map ${{\rm{\Psi }}_\dagger }:M \to M{\rm{'}}\left( \dagger \right)$ . Finally, consider the spacetime $\left( {M{\rm{'}}\left( n \right),g{{\rm{'}}\!_{ab}}} \right)$ for all $n \in {\mathbb{Z}^ + }$ , where $M{\rm{'}}\left( n \right) = \{ \left( {t{\rm{'}},\varphi {\rm{'}}} \right) \in M{\rm{'}}\left( \ddagger \right): - {F_n}\left( 2 \right) \lt t{\rm{'}} \lt {F_n}\left( 2 \right)\} $ and once again the domain of $g{{\rm{'}}_{ab}}$ is restricted in the natural way. One can verify that ${F_1}\left( 2 \right) \approx 1.88$ (see Figure 1) and, for all $n$ , we have ${F_1}\left( 2 \right) \lt {F_n}\left( 2 \right) \lt 2$ . Now, for all $n$ , we find that $\left( {M,{g_{ab}}\left( n \right)} \right)$ is isometric to $\left( {M{\rm{'}}\left( n \right),g{{\rm{'}}\!_{ab}}} \right)$ ; to see this, use the diffeomorphism ${{\rm{\Psi }}_n}:M \to M{\rm{'}}\left( n \right)$ defined by ${{\rm{\Psi }}_n}\left( {\left( {t,\varphi } \right)} \right) = \left( {{F_n}\left( t \right),\varphi } \right)$ and note that ${\nabla _a}{F_n}\left( t \right) = {f_n}\left( t \right){\nabla _a}t$ .

It is immediate that, for all $n$ , $M{\rm{'}}\left( n \right)$ . is a proper subset of $M\left( \dagger \right)$ , which is, in turn, a proper subset of $M\left( \ddagger \right)$ . So $\left( {M{\rm{'}}\left( \ddagger \right),g{{\rm{'}}\!_{ab}}} \right)$ is an extension of $\left( {M{\rm{'}}\left( \dagger \right),g{{\rm{'}}\!_{ab}}} \right)$ , which is an extension of $\left( {M{\rm{'}}\left( n \right),g{{\rm{'}}\!_{ab}}} \right)$ for all $n$ . Moreover, these spacetimes are globally hyperbolic vacuum solutions since they are just portions of two-dimensional Minkowski spacetime which has been “rolled up” in the spacelike direction (see Figure 2). Using the isometries established above, we find that $\left( {M,{g_{ab}}\left( \ddagger \right)} \right)$ is an extension of $\left( {M,{g_{ab}}\left( \dagger \right)} \right)$ , which is an extension of $\left( {M,{g_{ab}}\left( n \right)} \right)$ for all $n$ . Moreover, each of these ${\cal P}$ -spacetimes must be a globally hyperbolic vacuum solution. So it follows that (i)–(iv) are true.

Figure 2.

The collections ${\cal P}$ and ${\cal Q}$ .

In the third stage of the proof, we show that the property of ${\cal P}$ -inextendibility is ${C^0}$ stable (and hence ${C^k}$ stable for all $k \ge 0$ ) relative to ${\cal P}$ . Since $\left( {M,{g_{ab}}\left( \ddagger \right)} \right)$ is the only ${\cal P}$ -inextendible spacetime in ${\cal P}$ , we are done if we can find a ${C^0}$ fine neighborhood ${\cal N} \subset {\cal U}$ of $\left( {M,{g_{ab}}\left( \ddagger \right)} \right)$ such that none of the ${\cal P}$ -extendible spacetimes can be found in ${\cal N}$ . Let ${h^{ab}}$ be the positive definite metric on $M$ given by ${h^{ab}} = {(\partial /\partial t)^a}{(\partial /\partial t)^b} + {(\partial /\partial \varphi )^a}{(\partial /\partial \varphi )^b}$ . Let ${\cal N} \subset {\cal L}\left( M \right)$ be the ${C^0}$ fine neighborhood of $\left( {M,{g_{ab}}\left( \ddagger \right)} \right)$ defined as the collection of all spacetimes $\left( {M,{g_{ab}}} \right)$ such that ${\rm{Su}}{{\rm{p}}_M}\left[ {d\left( {{g_{ab}}\left( \ddagger \right),{g_{ab}},{h^{ab}},0} \right)} \right] \lt 1$ . We now show that each ${\cal P}$ -extendible spacetime fails to make it into ${\cal N}$ .

Consider the ${\cal P}$ -extendible spacetime $\left( {M,{g_{ab}}\left( n \right)} \right)$ for any $n$ . We find that ${g_{ab}}\left( \ddagger \right) - {g_{ab}}\left( n \right)$ is just $\left( {f_\ddagger ^2\left( t \right) - f_n^2\left( t \right)} \right){\nabla _a}t{\nabla _b}t$ . So ${h^{ab}}\left( {{g_{ab}}\left( \ddagger \right) - {g_{ab}}\left( n \right)} \right) = f_\ddagger ^2\left( t \right) - f_n^2\left( t \right)$ . But one can verify that, for all $t \in \left( { - 2,2} \right)$ , we have $f_\ddagger ^2\left( t \right) \ge f_\ddagger ^2\left( 0 \right) = {\pi ^2}/4 \gt 2$ and $f_n^2\left( t \right) \le 1$ . It follows that ${\rm{Su}}{{\rm{p}}_M}\left[ {d\left( {{g_{ab}}\left( \ddagger \right),{g_{ab}}\left( n \right),{h^{ab}},0} \right)} \right] \gt 1$ and therefore $\left( {M,{g_{ab}}\left( n \right)} \right)$ fails to be in ${\cal N}$ for all $n$ . The argument for the remaining ${\cal P}$ -extendible spacetime $\left( {M,{g_{ab}}\left( \dagger \right)} \right)$ is analogous: We find that ${g_{ab}}\left( \ddagger \right) - {g_{ab}}\left( \dagger \right)$ is just $\left( {f_\ddagger ^2\left( t \right) - 1} \right){\nabla _a}t{\nabla _b}t$ . So ${h^{ab}}\left( {{g_{ab}}\left( \ddagger \right) - {g_{ab}}\left( \dagger \right)} \right) = f_\ddagger ^2\left( t \right) - 1$ . Since $f_\ddagger ^2\left( t \right) \gt 2$ for all $t \in \left( { - 2,2} \right)$ , it follows that ${\rm{Su}}{{\rm{p}}_M}\left[ {d\left( {{g_{ab}}\left( \ddagger \right),{g_{ab}}\left( \dagger \right),{h^{ab}},0} \right)} \right] \gt 1$ and therefore $\left( {M,{g_{ab}}\left( \dagger \right)} \right)$ fails to be in ${\cal N}$ . So, we have established that each ${\cal P}$ -extendible spacetime fails to make it into ${\cal N}$ and therefore ${\cal P}$ -inextendibility is ${C^0}$ stable (and hence ${C^k}$ stable for all $k \ge 0$ ) relative to ${\cal P}$ .

In the final stage of the proof, we show that the property of ${\cal Q}$ -inextendibility is not ${C^k}$ stable relative to ${\cal Q}$ for all $k \ge 0$ . We restrict attention to the $k = 1$ case to simplify the presentation but one can generalize in the natural way. Since $\left( {M,{g_{ab}}\left( \dagger \right)} \right)$ is ${\cal Q}$ -inextendible, we are done if we can show that, for any ${C^1}$ fine neighborhood of $\left( {M,{g_{ab}}\left( \dagger \right)} \right)$ , there is some $n$ such that the ${\cal Q}$ -extendible spacetime $\left( {M,{g_{ab}}\left( n \right)} \right)$ is in the neighborhood.

Let ${h^{ab}}$ be any positive definite metric on $M$ and $\varepsilon $ any positive number. The smooth scalar fields ${\alpha _0},{\alpha _1}:M \to \mathbb{R}$ are defined by

$${\alpha _0}\left( {t,\varphi } \right) = u\left( t \right){h^{ab}}{\nabla _a}t{\nabla _b}t,$$

$${\alpha _1}\left( {t,\varphi } \right) = {[{h^{ac}}{h^{bd}}{h^{rs}}\left( {{\nabla _r}\left( {u\left( t \right){\nabla _a}t{\nabla _b}t} \right)} \right)\left( {{\nabla _s}\left( {u\left( t \right){\nabla _c}t{\nabla _d}t} \right)} \right)]^{1/2}}.$$

The quantity ${g_{ab}}\left( \dagger \right) - {g_{ab}}\left( n \right)$ is just $(1 - f_n^2(t)){\nabla _a}t{\nabla _b}t = \left( {u\left( t \right)/n} \right){\nabla _a}t{\nabla _b}t$ for all $n$ . So $d\left( {{g_{ab}}\left( \dagger \right),{g_{ab}}\left( n \right),{h^{ab}},0} \right) = {\alpha _0}/n$ and $d\left( {{g_{ab}}\left( \dagger \right),{g_{ab}}\left( n \right),{h^{ab}},1} \right) = {\alpha _1}/n$ . Let $N$ be the compact region $\left\{ {\left( {t,\varphi } \right) \in M: - 1 \le t \le 1} \right\}$ . By construction, ${\alpha _0}$ and ${\alpha _1}$ vanish on $M - N$ . So ${\rm{Su}}{{\rm{p}}_{M - N}}\left[ {d\left( {{g_{ab}}\left( \dagger \right),{g_{ab}}\left( n \right),{h^{ab}},k} \right)} \right] = 0$ for $k = 0,1$ . Now consider $N$ . Because this region is compact, we know that there is an $m \in \mathbb{R}$ such that ${\alpha _0}\left( p \right) \lt m$ and ${\alpha _1}\left( p \right) \lt m$ for all $p \in N$ . So, for each $n$ , we know ${\rm{Su}}{{\rm{p}}_N}\left[ {d\left( {{g_{ab}}\left( \dagger \right),{g_{ab}}\left( n \right),{h^{ab}},k} \right)} \right] \lt m/n$ for $k = 0,1$ . But $m/n \lt \varepsilon $ for large enough $n$ . It follows that for $k = 0,1$ we have ${\rm{Su}}{{\rm{p}}_M}\left[ {d\left( {{g_{ab}}\left( \dagger \right),{g_{ab}}\left( n \right),{h^{ab}},k} \right)} \right] \lt \varepsilon $ for large enough $n$ . So, for any ${C^1}$ fine neighborhood of $\left( {M,{g_{ab}}\left( \dagger \right)} \right)$ , there is some $n$ such that the ${\cal Q}$ -extendible spacetime $\left( {M,{g_{ab}}\left( n \right)} \right)$ is in the neighborhood. So the property of ${\cal Q}$ -inextendibility fails to be ${C^1}$ stable relative to ${\cal Q}$ . QED

To highlight the physical significance of the proposition, suppose, for example, that it is true that “all physically reasonable spacetimes are globally hyperbolic” (Wald Reference Wald1984, 304). And suppose that one were able to show that the property of $\left( {GH} \right)$ -inextendibility is ${C^k}$ stable relative to the collection $\left( {GH} \right)$ for some $k \ge0$ . Because there remain “physically unreasonable” spacetimes lurking within $\left( {GH} \right)$ , one would also want assurance that the ${C^k}$ stability of $\left( {GH} \right)$ -inextendibility “transfers down” to the ${C^k}$ stability of ${\cal P}$ -inextendibility for any collection ${\cal P} \subset \left( {GH} \right)$ . The proposition tells us that we do not have this assurance. Moreover, the predicament persists even if we further restrict attention to spacetimes which are well-behaved locally. Indeed, it is difficult to see how one might rule out as “physically unreasonable” a collection of globally hyperbolic vacuum solutions without invoking an inextendibility property of some kind.