1. Introduction

A space-time is counted as inextendible if, intuitively, it is “as large as it can be.” It has been argued that inextendibility is a “reasonable physical condition to be imposed on models of the universe” (Geroch Reference Geroch, Carmeli, Fickler and Witten1970, 262) and that a space-time must be inextendible if it is “to be a serious candidate for describing actuality” (Earman Reference Earman1995, 32). Here, in a variety of ways, we register some skepticism with respect to such positions.

2. Preliminaries

We begin with a few preliminaries concerning the relevant background formalism of general relativity.Footnote ¹ An n-dimensional, relativistic space-time (for $n\ge 2$) is a pair of mathematical objects (M, g_ab), where M is a connected n-dimensional Hausdorff manifold (without boundary) that is smooth and g_ab is a smooth, nondegenerate, pseudo-Riemannian metric of Lorentz signature (−, +, …, +) defined on M. We say two space-times (M, g_ab) and (M′, $g_{\mathrm{ab}}^{\prime }$) are isometric if there is a diffeomorphism $\phi :M\to M^{\prime }$ such that ${\phi }_{*}{g}_{\mathrm{ab}}=g_{\mathrm{ab}}^{\prime }$. Two space-times (M, g_ab) and (M′, $g_{\mathrm{ab}}^{\prime }$) are locally isometric if, for each point $p\in M$, there is an open neighborhood O of p and an open subset O′ of M′ such that O and O′ are isometric, and, correspondingly, with the roles of (M, g_ab) and (M′, $g_{\mathrm{ab}}^{\prime }$) interchanged.

A space-time (M, g_ab) is extendible if there exists a space-time (M, g_ab) and a proper isometric embedding $\phi :M\to M^{\prime }$. Here, the space-time (M′, $g_{\mathrm{ab}}^{\prime }$) is an extension of (M, g_ab). A space-time is inextendible if it has no extension. A $P$-space-time is a space-time with property $P$. A $P$-space-time (M′, $g_{\mathrm{ab}}^{\prime }$) is a $P$-extension of a $P$-space-time (M, g_ab) if (M′, $g_{\mathrm{ab}}^{\prime }$) is an extension of (M, g_ab). A $P$-space-time is $P$-extendible if it has a $P$-extension and is $P$-inextendible otherwise. We say (M, g_ab, F) is an n-dimensional framed space-time if (M, g_ab) is an n-dimensional space-time and F is an orthonormal n-ad of vectors {ξ₁, … , ξ_n} at some point $p\in M$. We say an n-dimensional framed space-time (M′, $g_{\mathrm{ab}}^{\prime }$, F′) is a framed extension of the n-dimensional framed space-time (M, g_ab, F) if there is a proper isometric embedding $\phi :M\to M^{\prime }$, which takes F into F′.

For each point $p\in M$, the metric assigns a cone structure to the tangent space M_p. Any tangent vector ξ^a in M_p will be time-like if $g_{\mathrm{ab}}{\xi }^a{\xi }^b<0$, null if $g_{\mathrm{ab}}{\xi }^a{\xi }^b=0$, or space-like if $g_{\mathrm{ab}}{\xi }^a{\xi }^b>0$. Null vectors create the cone structure; time-like vectors are inside the cone, while space-like vectors are outside. A time orientable space-time is one that has a continuous time-like vector field on M. In what follows, it is assumed that space-times are time orientable.

For some connected interval $I\subseteq ℝ$, a smooth curve $\gamma :I\to M$ is time-like if the tangent vector ξ^a at each point in γ[I] is time-like. Similarly, a curve is null (respectively, space-like) if its tangent vector at each point is null (respectively, space-like). A curve is causal if its tangent vector at each point is either null or time-like. A causal curve is future directed if its tangent vector at each point falls in or on the future lobe of the light cone. We say a time-like curve $\gamma :\left[s,s^{\prime }\right]\to M$ is closed if $\gamma \left(s\right)=\gamma \left(s^{\prime }\right)$. A space-time (M, g_ab) satisfies chronology if it does not contain a closed time-like curve. For any two points p, $q\in M$, we write $p\ll q$ if there exists a future-directed time-like curve from p to q. This relation allows us to define the time-like past of a point p: $I^{-}\left(p\right)=\left\{q:q\ll p\right\}$. We say a space-time (M, g_ab) satisfies past distinguishability if there do not exist distinct points p, $q\in M$ such that $I^{-}\left(p\right)=I^{-}\left(q\right)$. We say a set $S\subset M$ is achronal if there do not exist p, $q\in S$ such that $p\in I^{-}\left(q\right)$.

An extension of a curve $\gamma :I\to M$ is a curve ${\gamma }^{\prime }:I^{\prime }\to M$ such that I is a proper subset of I′ and $\gamma \left(s\right)={\gamma }^{\prime }\left(s\right)$ for all $s\in I$. A curve is maximal if it has no extension. A curve $\gamma :I\to M$ in a space-time (M, g_ab) is a geodesic if ${\xi }^a{\nabla }_a{\xi }^b=0$, where ξ^a is the tangent vector and ∇_a is the unique derivative operator compatible with g_ab. A point $p\in M$ is a future endpoint of a future-directed causal curve $\gamma :I\to M$ if, for every neighborhood O of p, there exists a point $t_0\in I$ such that $\gamma \left(t\right)\in O$ for all $t>t_0$. A past endpoint is defined similarly. A causal curve is inextendible if it has no future or past endpoint. A causal geodesic $\gamma :I\to M$ in a space-time (M, g) is past incomplete if it is maximal and there is an $r\in ℝ$ such that $r<s$ for all $s\in I$.

For any set $S\subseteq M$, we define the past domain of dependence of S, written D ⁻(S), to be the set of points $p\in M$ such that every causal curve with past endpoint p and no future endpoint intersects S. The future domain of dependence of S, written D ⁺(S), is defined analogously. The entire domain of dependence of S, written D(S), is just the set $D^{-}\left(S\right)\cup D^{+}\left(S\right)$. The edge of an achronal set $S\subset M$ is the collection of points $p\in S$ such that every open neighborhood O of p contains a point $q\in I^{+}\left(p\right)$, a point $r\in I^{-}\left(p\right)$, and a time-like curve from r to q that does not intersect S. A set $S\subset M$ is a slice if it is closed, achronal, and without edge. A space-time (M, g_ab) that contains a slice S such that $D\left(S\right)=M$ is said to be globally hyperbolic.

Given a space-time (M, g_ab), let T_ab be defined by $(1/8\pi )\left(R_{\mathrm{ab}}-\frac{1}{2}{\mathrm{Rg}}_{\mathrm{ab}}\right)$, where R_ab is the Ricci tensor and R the scalar curvature associated with g_ab. We say that $(M,g_{\mathrm{ab}})$ satisfies the weak energy condition if, for each time-like vector ξ^a, we have $T_{\mathrm{ab}}{\xi }^a{\xi }^b\ge 0$. We say that (M, g_ab) is a vacuum solution if $T_{\mathrm{ab}}=0$.

Let S be a set. A relation ≤ on S is a partial order if, for all a, b, $c\in S$, (i) $a\le a$; (ii) if $a\le b$ and $b\le c$, then $a\le c$; and (iii) if $a\le b$ and $b\le a$, then $a=b$. If ≤ is a partial ordering on a set S, we say a subset $T\subseteq S$ is totally ordered if, for all a, $b\in T$, either $a\le b$ or $b\le a$. Let ≤ be a partial ordering on S, and let $T\subseteq S$ be totally ordered. An upper bound for T is an element $u\in S$ such that for all $a\in T$, $a\le u$. A maximal element of S is an element $m\in S$ such that for all $c\in S$, if $m\le c$, then $c=m$. Zorn’s lemma (which is equivalent to the axiom of choice) is as follows: Let ≤ be a partial order on S. If each totally ordered subset $T\subseteq S$ has an upper bound, there is a maximal element of S.

3. Definition

Recall the standard definition of space-time inextendibility.

The definition requires that an inextendible space-time be “as large as it can be” in the sense that one compares it to a background class of all “possible” space-times. Standardly, one takes this class to be the set of all (smooth, Hausdorff) Lorentzian manifolds as defined in the previous section. But what should this class be? The answer is unclear.

Consider Misner space-time (Hawking and Ellis Reference Hawking and Ellis1973). Let Misner* be the globally hyperbolic “bottom half” of Misner space-time. By the standard definition of inextendibility, Misner* is extendible and cannot be extended and remain globally hyperbolic (see below). But suppose that a version of the cosmic censorship conjecture is correct and all physically reasonable space-times are globally hyperbolic (Penrose Reference Penrose, Hawking and Israel1979). Then should Misner* not be considered “as large as it can be”? It follows that whether Misner* space-time should count as inextendible depends crucially on the outcome of this version of the cosmic censorship conjecture—a conjecture that is far from settled (Earman Reference Earman1995; Penrose Reference Israel1999) and perhaps may never be settled (Manchak Reference Manchak2011).

Because of examples like these, one is tempted to revise the definition of inextendibility.Footnote ² But Geroch (Reference Geroch, Carmeli, Fickler and Witten1970, 278) has argued that a revision is less urgent if one can show, for a variety of physically reasonable properties $P$, that the following statement is true.

The significance of (*) is this: if a property $P$ satisfies (*), then any $P$-space-time is inextendible if and only if it is $P$-inextendible. Effectively, it makes no difference in such cases whether one defines inextendibility relative to the standard class of all space-times or a revised class of all $P$-space-times. Accordingly, one would like to investigate (*) with respect to a variety of properties $P$. Already from the Misner* example above, we have the following proposition (a proof is provided in the appendix).

Are there physically reasonable properties $P$ that render (*) true? Geroch (Reference Geroch, Carmeli, Fickler and Witten1970, 289) has suggested a number of good candidates, including being a vacuum solution, satisfying chronology, and satisfying an energy condition. The first two cases are still open. Here, we settle the case in which $P$ is the weak energy condition (a proof is provided in the appendix).

We see that the prospect of avoiding the need to revise to the definition of inextendibility does not look good. In the meantime, we may conclude that it is not yet clear that the standard definition captures the intuitive idea that an inextendible space-time is “as large as it can be.”

4. Metaphysics

A number of experts in general relativity (Penrose Reference Penrose1969; Geroch Reference Geroch, Carmeli, Fickler and Witten1970; Clarke Reference Clarke1976) seem to be committed to the following statement. “Metaphysical considerations suggest that to be a serious candidate for describing actuality, a spacetime should be [inextendible]. For example, for the Creative Force to actualize a proper subpart of a larger spacetime would seem to be a violation of Leibniz’s principles of sufficient reason and plenitude. If one adopts the image of spacetime as being generated or built up as time passes then the dynamical version of the principle of sufficient reason would ask why the Creative Force would stop building if it is possible to continue” (Earman Reference Earman1995, 32).

These metaphysical views are underpinned by an important result due to Geroch (Reference Geroch, Carmeli, Fickler and Witten1970).

The result (which makes use of Zorn’s lemma) seems to show that the Creative Force can always build space-time until it is no longer possible to build. But of course, this interpretation presupposes that we have been working with the proper definition of inextendibility. And as we have noted, it is not yet clear that we are. Accordingly, one would like to know, for a variety of physically reasonable properties $P$, whether the following version of the Geroch (Reference Geroch, Carmeli, Fickler and Witten1970) result is true.

With a bit of work (and Zorn’s lemma), one can show the following proposition (a proof is provided in the appendix).

We see that if we revise the definition of inextendibility to be relative to the class of all chronological space-times (rather than the standard class of all space-times), we have an analogue of the Geroch (Reference Geroch, Carmeli, Fickler and Witten1970) result. This is certainly good news for those committed to the metaphysical views expressed above. But are there physically reasonable properties $P$ that render (**) false? There are.

Of course, space-time properties may be considered physically reasonable in various senses. Let us conservatively restrict attention to a property usually taken to be satisfied by models of our own universe: the property of having every inextendible time-like geodesic be past incomplete. Let us call this the big bang property, given that it is satisfied by all of the standard “big bang” cosmological models. We are now in a position to state the following proposition (Manchak Reference Manchak2016b).

We see that if we revise the definition of inextendibility to be relative to the class of all space-times with the big bang property (rather than the standard class of all space-times), we do not have an analogue of the Geroch (Reference Geroch, Carmeli, Fickler and Witten1970) result. It is not yet clear that the Creative Force always has the option of building space-time to be “as large as it can be.”

5. Epistemology

What observational evidence is there (or could there be) in support of the position that space-time is “as large as it can be”? Following Malament (Reference Malament, Earman, Glymour and Stachel1977), let us say that a space-time (M, g_ab) is observationally indistinguishable from another space-time (M′, $g_{\mathrm{ab}}^{\prime }$) if, for every point $p\in M$, there is a point $p^{\prime }\in M^{\prime }$ such that I ⁻(p) and I ⁻(p′) are isometric. One can show the following proposition (Manchak Reference Manchak2011).

Under the standard definition of inextendibility, it seems that any observer in a chronological space-time is not in a position to know that her space-time is “as large as it can be.” But this interpretation presupposes that we have been working with the proper definition of inextendibility. And as we have noted, it is not yet clear that we are. Accordingly, one would like to know, for a variety of physically reasonable properties $P$, whether the following version of the Manchak (Reference Manchak2011) result is true.

It turns out that a large class of physically reasonable properties render (***) true. Following Manchak (Reference Manchak2011), let us say that a property $P$ on a space-time is local if, given any two locally isometric space-times (M, g_ab) and (M′, $g_{\mathrm{ab}}^{\prime }$), (M, g_ab) has $P$ if and only if (M′, $g_{\mathrm{ab}}^{\prime }$) has $P$. Local properties include being a vacuum solution and satisfying the weak energy condition. We are now in a position to state the following proposition (a proof is provided in the appendix).

We see that, whenever $P$ is a local property, if we revise the definition of inextendibility to be relative to the class of all $P$-space-times (rather than the standard class of all space-times), we have an analogue of the Manchak (Reference Manchak2011) result. It is not yet clear that we can ever have observational evidence that space-time is “as large as it can be.”

6. Conclusion

We have registered some skepticism with respect to the position that space-time must be inextendible—that it must be “as large as it can be” in some sense. We have done this in a variety of ways. First we have shown that it is not yet clear that the standard definition of inextendibility captures the intuitive idea that an inextendible space-time is “as large as it can be.” Second, we have shown, by exploring some plausible revisions to the definition of inextendibility, that it is not yet clear that a space-time can always be extended to be “as large as it can be.” Finally we have shown, by exploring a class of plausible revisions to the definition of inextendibility, that it is not yet clear that we can ever know that space-time is “as large as it can be.”