1 Introduction

Global spacetime structure concerns the more foundational aspects of general relativity (e.g. the topological and causal structure of spacetime). Upon investigation, it is often the case that seemingly plausible statements concerning global spacetime structure turn out to be false. Indeed, even after the shift to a relativistic worldview it seems “we are still somewhat over-conditioned to Minkowski spacetime” (Reference Geroch, Horowitz, Hawking and IsraelGeroch & Horowitz, 1979, p. 215). This Element can be viewed as a kind of manual to help us unlearn what we think we know concerning the global structure of spacetime. A large number of example spacetimes (with diagrams) are central to the presentation and serve to demonstrate just how much is permitted under general relativity. Along the way, open questions are highlighted and periodic exercises can be used to test one’s understanding (sample solutions are given in the Appendix).

Section 2 concerns the basic structure of spacetime. A number of preliminary definitions are presented to get things started. The cut-and-paste method is also introduced, which is used throughout to construct a vast array of example spacetimes. Although such spacetimes may seem artificial in some sense, we find that “the mere existence of a space-time having certain global features suggests that there are many models - some perhaps quite reasonable physically - with very similar properties” (Reference Geroch and SachsGeroch, 1971a, p. 78). Section 3 covers the causal structure of spacetime. It follows a fairly conventional presentation of the hierarchy of causality conditions (Reference Hawking and EllisHawking & Ellis, 1973; Reference WaldWald, 1984). But some nonstandard topics of interest are also explored including the so-called Malament-Hogarth spacetimes allowing for “supertasks” of a certain kind (Reference Earman and NortonEarman & Norton, 1993).

Section 4 concerns the singular structure of spacetime. An example sin-gularity theorem is presented showing a sense in which some “physically reasonable” spacetimes have singularities (cf. Reference Hawking and PenroseHawking & Penrose, 1970). This raises a difficulty in how to sort singular spacetimes into physically reasonable and physically unreasonable varieties. Two families of conditions are investigated that are meant to do the sorting. One family primarily concerns the causal structure of spacetime and forbids “naked” singularities of various types; the other family primarily concerns the modal structure of spacetime and forbids spacetime “holes” of various types. After considering a rich collection of examples, the upshot seems to be that what counts as a physically reasonable spacetime is far from clear (Reference EarmanEarman, 1995, p. 86).

As we leave old intuitions behind, a rather basic question arises: What can we know concerning the global structure of spacetime? Building on a trio of papers from Reference Geroch, Earman, Glymour and StachelGeroch (1977), Reference Glymour, Earman, Glymour and StachelGlymour (1977), and Reference Malament, Earman, Glymour and StachelMalament (1977a), Section 5 explores the epistemic structure of spacetime. It seems that even after we have (i) taken into consideration all possible observational data we could ever (even in principle) gather and (ii) inductively fixed the local features of any unobservable regions of spacetime, a type of “cosmic underdetermination” keeps us from pinning down the global structure of the universe. And if we take seriously the idea that we cannot come to know the global structure of spacetime through observation, queer possibilities present themselves. Does our universe allow for “time travel” of a certain kind? Do spacetime “holes” exist in our universe? This suggests that perhaps we have been too quick to discount as physically unreasonable some of the more peculiar global spacetime properties since, for all we know, such properties obtain in our own universe.

In Section 6, the modal structure of spacetime is explored through the lens of the inextendibility condition. This is the requirement that the universe be as large as possible relative to a standard background collection of spacetimes. But the inextendibility condition would seem to be physically significant only insofar as the background collection coincides with physically reasonable possibilities (Reference Geroch, Carmeli, Fickler and WittenGeroch, 1970a). And because what counts as a physically reasonable spacetime is not clear - especially given the underdetermination results just mentioned - it seems natural to consider various nonstandard definitions of inextendibility in a pluralistic way. Upon investigation, we find that foundational claims concerning inextendibility can fail to hold up under some modified definitions. For example, it can happen that a spacetime is “extendible” and yet has no “inextendible extension” - a strange state of affairs with the potential to clash with various Leibniz-inspired metaphysical principles in favor of the “maximality” of spacetime (Reference EarmanEarman, 1995, p. 32). In addition, the demand for modified forms of inextendibility can lead to situations in which a spacetime is forced into having global properties of interest. A so-called time machine represents one example along these lines, but other “machine” spacetimes can also be studied (cf. Reference Earman, Wuthrich, Manchak and ZaltaEarman et al., 2016). Stepping back, we find that the prospect of a clear distinction between physically reasonable and physically unreasonable spacetimes is more elusive than ever.

2 Preliminaries

A (general relativistic) spacetime is a pair $(M,g_{ab})$ where $M$ is a smooth, connected, Hausdorff, paracompact, $n$ -dimensional ($n\ge 2$ ) manifold and $g_{ab}$ is a smooth metric on $M$ of Lorentz signature $(+,-,\dots ,-)$ . Under the assumption of Einstein’s equation (see p. 6), a spacetime is a model of general relativity and represents a possible universe compatible with the theory. Details concerning the relevant background mathematics (including the “abstract index” notation used throughout) can be found in Reference Hawking and EllisHawking and Ellis (1973), Reference WaldWald (1984), or Reference MalamentMalament (2012). Here, we follow Reference Geroch, Horowitz, Hawking and IsraelGeroch & Horowitz (1979) in avoiding technical machinery wheneverpossible.

We begin with the notion of a manifold, which, unless otherwise stated, is taken to be smooth, connected, Hausdorff, and paracompact (see the Appendix for basic topological definitions). All of the topological structure of a spacetime $(M,g_{ab})$ is given by the manifold $M$ ; it fully captures the shape of the model. Locally, a manifold looks like plain old ${ℝ}^n$ although globally it may have a very different structure. A number of manifolds are easy to visualize. For example, consider the sphere $S^2$ . Despite its round shape, if one zooms in on the vicinity of any point, one finds it has the same topological structure as the plane ${ℝ}^2$ (see Figure 1). Other two-dimensional manifolds include the cylinder $S^1\times ℝ$ and the torus $S^1\times S^1$ . In addition, the result of taking any manifold and removing from it a closed proper subset also counts as a manifold. For example, a new manifold ${ℝ}^2-\left\{\right(0,0\left)\right\}$ can be constructed by excising the origin fromthe plane.

Figure 1 : The sphere $S^2$ has the same topological structure as the plane ${ℝ}^2$ in the vicinity of each point.

We say the $n$ -dimensional manifolds $M$ and $N$ are diffeomorphic if there is a bijection $\phi :M\to N$ such that both it and its inverse are smooth. Diffeomorphic manifolds have identical topological and smoothness properties. It turns out that every non-compact manifold of two dimensions or more admits some Lorentzian metric. One can also show that the compact manifold $S^n$ for $n\ge 2$ admits a Lorentzian metric if and only if $n$ is odd (Reference Geroch, Horowitz, Hawking and IsraelGeroch & Horowitz, 1979). We also have the useful result that any manifold $M\times N$ admits a Lorentzian metric if either $M$ or $N$ does. And of course, if $M$ admits a Lorentzian metric, then so does $M-C$ where $C$ is any closed proper subset of $M$ .

Exercise 1 Find a manifold M and a point p ∈ M such that M and M-{p}are diffeomorphic.

Each point on a manifold represents an idealized possible event in spacetime (e.g. one’s birth). The Lorentzian metric tells us how such events in spacetime are related to one another. Consider a spacetime $(M,g_{ab})$ . At each point $p\in M$ , the metric $g_{ab}$ assigns to each vector ${\xi }^a$ in the tangent space of $p$ a length given by ${\xi }^a{\xi }^b{g}_{ab}={\xi }^a{\xi }_a\in ℝ$ . This creates a type of double cone structure in the tangent space of each point. Positive-length vectors are timelike and fall inside the cone, negative-length vectors are spacelike and fall outside the cone, and zero-length vectors are null and make up the boundary of the cone (see Figure 2).

Figure 2 A three-dimensional double cone structure at the point $p$ . A timelike vector ${\tau }^a$ , a null vector ${\nu }^a$ , and a spacelike vector ${\sigma }^a$ are depicted.

One can think of the cone structure at each point as representing the speed of light in all directions there; timelike and spacelike vectors represent, respectively, velocities that are slower and faster than light. For this reason, we often refer to these structures as light cones in what follows. Now consider a smooth curve $\gamma :I\to M$ where $I$ is some connected interval of $ℝ$ . (In what follows, curves are understood to be smooth unless otherwise stated.) If each of its tangent vectors ${\xi }^a$ is timelike according to $g_{ab}$ , then we say the curve $\gamma$ is timelike. Timelike curves represent the possible trajectories of massive objects. Analogous definitions can be given for spacelike and null curves; a causal curve has no spacelike tangent vectors (see Figure 3).

Figure 3 A pair of causal curves in a three-dimensional spacetime. One is timelike (solid line) and one is null (dotted line).

Associated with $g_{ab}$ is a unique derivative operator ${\nabla }_a$ on $M$ that is compatible with the metric in the sense that ${\nabla }_a{g}_{bc}=0$ . We say that a given curve $\gamma :I\to M$ is a geodesic if, for each each point along the curve, the tangent vector ${\xi }^a$ is such that ${\xi }^a{\nabla }_a{\xi }^b=0$ . One can think of a geodesic as a curve that is as straight as possible according to a given metric. Timelike geodesics represent the possible trajectories of non-accelerating (freely falling) massive objects; null geodesics represent the possible trajectories of light. In any spacetime $(M,g_{ab})$ , one can always find some open neighborhood $O\subseteq M$ around any point $p\in M$ such that any two points $q,r\in O$ can be connected by a unique geodesic whose image is contained in $O$ .

Exercise 2 Find a spacetime (M, g_ab)and a pair of points p, q ∈ M that can be connected by spacelike and null geodesics but not by a timelike geodesic.

A curve $\gamma :I\to M$ in a spacetime $(M,g_{ab})$ is maximal if there is no curve ${\gamma }^{\prime }:I^{\prime }\to M$ such that $I$ is properly contained in I' and $\gamma \left(s\right)={\gamma }^{\prime }\left(s\right)$ for all $s\in I$ . If a maximal geodesic $\gamma :I\to M$ is such that $I\ne ℝ$ , then we say it is incomplete. A spacetime that harbors an incomplete geodesic is geodesically incomplete; otherwise it is geodesically complete. An incomplete timelike geodesic can be considered a type of singularity since it represents a possible trajectory of a freely falling massive object whose existence is cut short in either the past or future direction (cf. Reference GerochGeroch, 1968a; Reference CurielCuriel, 1999). By excising points from the manifold, one can easily create examples of geodesically incomplete spacetimes (see Figure 4).

Figure 4 The timelike geodesic $\gamma$ is maximal but incomplete since it cannot be extended through the missing point.

Given a spacetime $(M,g_{ab})$ , one can use its associated derivative operator ${\nabla }_a$ to define the Riemann tensor $R_{bcd}^a$ where $R_{bcd}^a{\xi }^b=-2{\nabla }_{[c}{\nabla }_{d]}{\xi }^a$ for all smooth vector fields ${\xi }^a$ . Here, the square brackets indicate the antisymmetrization operation. In this case, we find that $-2{\nabla }_{[c}{\nabla }_{d]}{\xi }^a=-({\nabla }_c{\nabla }_d-{\nabla }_d{\nabla }_c){\xi }^a$ (see Reference MalamentMalament, 2012, p. 33). The Riemann tensor encodes all of the curvature of spacetime at each point in $M$ . A spacetime is flat if its Riemann tensor vanishes everywhere. The contraction of the Riemann tensor leads to the Ricci tensor $R_{ab}=R_{abc}^c$ and the Ricci scalar $R=R_a^a$ (see Reference MalamentMalament, 2012, 84). The distribution of matter in spacetime can be represented by the energy-momentum tensor $T_{ab}$ defined via Einstein’s equation: $R_{ab}-(1/2)R{g}_{ab}=8\pi T_{ab}$ . Here, we have ignored the possibility of a nonzero “cosmological constant” term in Einstein’s equation (see Reference EarmanEarman, 2001). Indeed, within the field of global structure there is a general lack of concern with the details of Einstein’s equation; we find that “things which can happen in the absence of this equation can usually also happen in its presence” (Reference Geroch, Horowitz, Hawking and IsraelGeroch & Horowitz, 1979, p. 215). If a spacetime is such that its corresponding energy-momentum tensor vanishes everywhere, then it is vacuum. It turns out that any two-dimensional spacetime is vacuum (see Reference Fletcher, Manchak, Schneider and WeatherallFletcher et al., 2018). In dimension three or greater, a spacetime is vacuum if and only if its associated Ricci tensor vanishes everywhere. Of course, any flat spacetime is necessarily vacuum.

We are now in a position to define Minkowski spacetime – it is any flat, geodesically complete spacetime with manifold ${ℝ}^n$ . In standard $(t,x)$ coordinates, two-dimensional Minkowski spacetime comes out as $({ℝ}^2,g_{ab})$ where $g_{ab}={\nabla }_{a}t{\nabla }_{b}t-{\nabla }_{a}x{\nabla }_{b}x$ . This is the spacetime of special relativity and the vanilla model of general relativity. In what follows, we use Minkowski spacetime as our basic tool to construct various examples; we cut it, glue it, bend it, and warp it in order to get what we need. In a representation of Minkowski spacetime in standard coordinates, the light cones are uniformly oriented throughout and all geodesics appear as straight lines (see Figure 5).

Figure 5 A timelike geodesic (solid line) and a null geodesic (dotted line) in two-dimensional Minkowski spacetime.

Exercise 3 Find a flat spacetime such that every maximal timelike geodesic is incomplete but some maximal null and spacelike geodesics are complete.

Some spacetimes $(M,g_{ab})$ admit a continuous timelike vector field ${\xi }^a$ on $M$ and some do not. Those that do (e.g. Minkowski spacetime) allow for a consistent global distinction between the “past” and “future” temporal directions since the continuous timelike vector field picks out one of two “lobes” of the light cone at each point. Such spacetimes are said to be time-orientable. One can show that any spacetime $(M,g_{ab})$ is time-orientable if $M$ is simply connected. A classic example of a spacetime that fails to be time-orientable can be constructed by starting with a Möbius strip manifold and orienting the light cones in such a way that any would-be continuous timelike vector field is flipped when transported around the strip (see Figure 6). In the following, we assume that spacetimes are time-orientable and that a temporal direction has been chosen. A causal curve $\gamma :I\to M$ in a spacetime $(M,g_{ab})$ is future-directed if its tangent vector at each point falls in or on the future lobe of the light cone or vanishes; an analogous definition can be given for past-directed causal curves. Unless otherwise stated, causal curves are understood to befuture-directed.

Figure 6 A spacetime that fails to be time-orientable since the flip in the Möbius strip precludes any continuous timelike vector field.

Exercise 4 Find a spacetime (M, g_ab)for some M ⊂ ℝ² that fails to be timeorientable.

Consider a spacetime $(M,g_{ab})$ and a pair of points $p$ and $q$ in $M$ that, respectively, represent the past event of one’s birth and the future event of one’s reading of this sentence. One’s trajectory through spacetime from the first event to the second can be represented by a future-directed timelike curve $\gamma :I\to M$ connecting $p$ to $q$ . The metric $g_{ab}$ assigns a length $\Vert \gamma \Vert =\int {\left(g_{ab}{\xi }^a{\xi }^b\right)}^{1/2}ds$ to this curve by adding up the lengths of all the tangent vectors ${\xi }^a$ along the curve. This length represents the elapsed time between $p$ and $q$ along $\gamma$ . It follows that the elapsed time between any two events will depend on how one moves through spacetime from one to the other. Some trajectories with velocity vectors “close to the speed of light” will have a short elapsed time relative to others. Indeed, continuity considerations require that if two points can be connected by a timelike curve, then for any $ϵ>0$ , there is a timelike curve connecting the points with length less than $ϵ$ . It turns out that some spacetimes (e.g. Minkowski spacetime) are such that if two points can be connected by a timelike curve, then there is a longest curve connecting the points that must be a geodesic (see Figure 7).

Figure 7 The points $p$ and $q$ can be connected by a short timelike curve (dotted line), but the longest such curve will be a geodesic (solid line).

A point $p\in M$ in a spacetime $(M,g_{ab})$ is a future endpoint of a future-directed causal curve $\gamma :I\to M$ if, for every open neighborhood $O$ of $p$ , there exists a point $s^{\prime }\in I$ such that $\gamma \left(s\right)\in O$ for all $s>s^{\prime }$ . A past endpoint is defined analogously. We say that a causal curve is future-inextendible if it has no future endpoint and analogously for past-inextendible. A causal curve is inextendible if it is both future-inextendible and past-inextendible. A causal curve that is inextendible must be maximal, but the converse is false. In Minkowski spacetime, a timelike curve can “wiggle” faster and faster as a future endpoint (which is not part of the curve) is approached (cf. Reference PenrosePenrose, 1972, p. 3). The curve counts as maximal since any extension through the endpoint must fail to be smooth (see Figure 8).

Figure 8 A maximal timelike curve with future endpoint $p$ .

Given an $n$ -dimensional spacetime $(M,g_{ab})$ , a set $S\subset M$ is a spacelike surface if $S$ is an $(n-1)$ -dimensional sub-manifold of $M$ such that every curve whose image is contained in $S$ is spacelike. A set $S\subset M$ in a spacetime $(M,g_{ab})$ is achronal if no two points in $S$ can be connected by a timelike curve. The edge of a closed, achronal set $S\subset M$ is the collection of points $p\in S$ for which every open neighborhood $O$ of $p$ contains points $q$ and $r$ such that future-directed timelike curves exist from $q$ to $p$ , from $p$ to $r$ , and from $q$ to $r$ where the last curve fails to intersect $S$ (see Figure 9). A slice is a closed, achronal set with an empty edge. In Minkowski spacetime in standard $(t,x)$ coordinates, each $t=$ constant surface counts as a slice. But not all spacetimes admit slices. For example, consider the spacetime $(S^1\times ℝ,g_{ab})$ where $g_{ab}={\nabla }_{a}t{\nabla }_{b}t-{\nabla }_{a}x{\nabla }_{b}x$ and $0\le t\le 2\pi$ ; this is just Minkowski spacetime that has been “rolled up” along the time direction. Let this spacetime be called time-rolled Minkowski spacetime. In an analogous way, one can also construct other two-dimensional models: space-rolled, null-rolled, and (time and space)–rolled Minkowski spacetimes.

Figure 9 The point $p$ is in the edge of the closed, achronal set $S$ since every open neighborhood $O$ of $p$ contains points $q$ and $r$ such that future-directed timelike curves exist from $q$ to $p$ , from $p$ to $r$ , and from $q$ to $r$ where the last curve fails to intersect $S$ .

Exercise 5 Find a spacelike surface in Minkowski spacetime that fails to be achronal.

A diffeomorphism $\phi :M\to M^{\prime }$ between the spacetimes $(M,g_{ab})$ and $(M^{\prime },{g^{\prime }}_{ab})$ is an isometry if ${\phi }^{*}\left({g^{\prime }}_{ab}\right)=g_{ab}$ where ${\phi }^{*}$ is the map associated with $\phi$ , which, for any point $p\in M$ , pulls back the tensor ${g^{\prime }}_{ab}$ at $\phi \left(p\right)\in M^{\prime }$ to the tensor ${\phi }^{*}\left({g^{\prime }}_{ab}\right)$ at $p\in M$ (Reference MalamentMalament, 2012, p. 36). Spacetimes $(M,g_{ab})$ and $(M^{\prime },{g^{\prime }}_{ab})$ are isometric if there is an isometry between them. Isometric spacetimes have fully equivalent structure and share all of the same physical properties; indeed, when no confusion arises, we often take isometric spacetimes to be the same spacetime in what follows. Consider the spacetimes $(M,g_{ab})$ and $(M^{\prime },{g^{\prime }}_{ab})$ . If there is a proper subset $O$ of M' such that $(M,g_{ab})$ and $(O,{g^{\prime }}_{ab})$ are isometric, then we say that $(M,g_{ab})$ is extendible and $(M^{\prime },{g^{\prime }}_{ab})$ is an extension of $(M,g_{ab})$ . A spacetime that is not extendible is inextendible.

Exercise 6 Find a pair of non-isometric spacetimes such that each counts as an extension of the other.

It turns out that every geodesically complete spacetime (e.g. Minkowski spacetime) is inextendible. But the other direction does not hold. To see this, consider Minkowski spacetime in standard $(t,x)$ coordinates and remove two slits $S_n=\left\{\right(0,n):n\le x\le n+1/2\}$ for $n=1,2$ . Excluding the four slit boundary points, identify the top edge of each slit with the bottom edge of the other (Reference Hawking and EllisHawking & Ellis, 1973, p. 58; Reference Geroch, Earman, Glymour and StachelGeroch, 1977, p. 89). The resulting spacetime is such that an observer entering one slit from below must emerge from the other slit from above. Because the four slit boundary points are “missing” from the spacetime, there are incomplete geodesics (see Figure 10). But one can show that this spacetime cannot be extended.

Figure 10 The top edge of the slit $S_1$ is identified with the bottom edge of the slit $S_2$ and vice versa. A maximal geodesic (dotted line) that approaches one of the four missing points must be incomplete.

Exercise 7 Find a flat, inextendible spacetime (ℝ², g_ab)that is not isometric to Minkowski spacetime.

Any spacetime with compact manifold must be inextendible (Reference O’NeillO’Neill, 1983, p. 155). Moreover, one can show (using Zorn’s lemma) that every spacetime is either inextendible or has an inextendible extension (Reference Geroch, Carmeli, Fickler and WittenGeroch, 1970a, p. 277). In general, an extension to a given extendible spacetime is not unique. But given any inextendible spacetime $(M,g_{ab})$ and any point $p\in M$ , every extension of the extendible spacetime $(M-\{p\},g_{ab})$ is isometric to $(M,g_{ab})$ (see Reference Manchak, Madarasz and SzekelyManchak, forthcoming). So we do have unique extensions (up to isometry) in some cases.

Spacetimes $(M,g_{ab})$ and $(M^{\prime },{g^{\prime }}_{ab})$ are locally isometric if for each point $p\in M$ there is an open set $O\subset M$ containing $p$ and an open set $O^{\prime }\subset M^{\prime }$ such that $(O,g_{ab})$ and $(O^{\prime },{g^{\prime }}_{ab})$ are isometric, and, correspondingly, with the roles of $(M,g_{ab})$ and $(M^{\prime },{g^{\prime }}_{ab})$ interchanged. We say that a spacetime property is local if, given any pair of locally isometric spacetimes, one spacetime has the property if and only if the other does as well; a spacetime property is global if it is not local (Reference ManchakManchak, 2009). One can verify that the property of being vacuum comes out as local. On the other hand, consider two copies of Minkowski spacetime and remove a point from one of them. These spacetimes are locally isometric. But although Minkowski spacetime is geodesically complete by definition (and therefore inextendible), Minkowski spacetime with one point removed is clearly extendible (and therefore geodesically incomplete). So geodesic completeness and inextendibility count as global spacetime properties.

Exercise 8 Is being time-orientable a global property? Is being two-dimensional?

3 Causality

We begin an exploration of the causal structure of spacetime by defining a pair of two-place relations on $M$ for every spacetime $(M,g_{ab})$ . For each $p,q\in M$ , we write $p\ll q$ if there is a future-directed timelike curve from $p$ to $q$ ; we write $p<q$ if a future-directed causal curve exists from $p$ to $q$ . Immediately, we see that if $p\ll q$ , then $p<q$ . The other direction does not hold in general since, for example, a future-directed null geodesic from the point $p$ to the point $q$ in Minkowski spacetime will be such that $p<q$ but $p\text{ nll}q$ (see Figure 11). One can show that for any spacetime $(M,g_{ab})$ and any points $p,q\in M$ , if $p<q$ and $p\text{ nll}q$ , then any causal curve connecting $p$ and $q$ must be a null geodesic.

Figure 11 The points $p$ and $q$ can be connected by a future-directed causal curve (dotted line) but cannot be connected by a timelike curve.

The relation < is always reflexive: for any spacetime $(M,g_{ab})$ and any point $p\in M$ , we have $p<p$ . To see this, consider that one can always define a trivial curve $\gamma :I\to M$ to be such that $\gamma \left(s\right)=p$ for all $s\in I$ ; the curve has a vanishing tangent vector everywhere and therefore counts as a null curve that is both past and future directed. The relation $\ll$ can sometimes be reflexive as in time-rolled Minkowski spacetime (see Figure 12) and can sometimes fail to be reflexive as in Minkowski spacetime where $p\ll p$ for no point $p$ . The relations < and $\ll$ are always transitive (O’Neill, 1983, p. 402): for any spacetime $(M,g_{ab})$ and for any points $p,q,r\in M$ , if both $p<q$ and $q<r$ , it follows that $p<r$ (and analogously for the $\ll$ relation). Some spacetimes such as time-rolled Minkowski spacetime have symmetric relations < and $\ll$ : for any points $p,q\in M$ , if $p<q$ , then $q<p$ (and analogously for the $\ll$ relation). But in other spacetimes – Minkowski spacetime is one example – one can find a pair of points $p,q\in M$ , such that $p<q$ but $q\nless p$ (and analogously for the $\ll$ relation).

Figure 12 Time-rolled Minkowski spacetime is such that a future-directed timelike curve $\gamma$ exists from any point $p$ back to itself.

Exercise 9 Find a spacetime (M, g_ab) and points p, q ∈ M such that p ≪ p, q ≪ q, and p ≪ q but q ≪ q.

We say that a pair of spacetimes $(M,g_{ab})$ and $(M,{g^{\prime }}_{ab})$ are conformally equivalent if there is some smooth, everywhere positive, scalar field $\Omega :M\to ℝ$ such that ${g^{\prime }}_{ab}={\Omega }^2{g}_{ab}$ . Here, the scalar field is called the conformal factor. Conformally equivalent spacetimes $(M,g_{ab})$ and $(M,{g^{\prime }}_{ab})$ have identical causal structure in the sense that for all $p,q\in M$ , $p<q$ in $(M,g_{ab})$ if and only if $p<q$ in $(M,{g^{\prime }}_{ab})$ and analogously for the $\ll$ relation. One can show that if a pair of conformally equivalent spacetimes $(M,g_{ab})$ and $(M,{g^{\prime }}_{ab})$ assign the same length $\Vert \gamma \Vert$ to every timelike curve $\gamma :I\to M$ , then the two spacetimes are, in fact, isometric (Reference MalamentMalament, 2012, p. 137).

When constructing spacetimes with particular properties, it is often useful to consider conformally equivalent versions of a simple model. Here is one famous example (Reference GerochGeroch, 1968a, p. 531). Suppose one wanted to find a spacetime with some timelike incomplete geodesics but no spacelike or null incomplete geodesics. Start with Minkowski spacetime $(M,g_{ab})$ in standard $(t,x)$ coordinates and consider the conformally equivalent spacetime $(M,{\Omega }^2{g}_{ab})$ where $\Omega :M\to ℝ$ is such that (i) $\Omega (t,x)=\Omega (t,-x)$ , (ii) $\Omega =1$ for $\left|x\right|>1$ , and (iii) $\Omega (t,0)\to 0$ as $t\to \infty$ . The symmetry of (i) ensures that the maximal timelike curve at $x=0$ is a geodesic. From (iii), we know that this geodesic will be incomplete if $\Omega$ is chosen to approach zero sufficiently fast. But (ii) requires that any null or spacelike maximal geodesic must escape the region $\left|x\right|\le 1$ in both directions and thus end up being complete (see Figure 13).

Figure 13 A maximal timelike geodesic (solid line) is incomplete due to the chosen conformal factor $\Omega$ , but any maximal null or spacelike geodesic is complete since it must escape the region between the dotted lines.

Consider a spacetime $(M,g_{ab})$ and a point $p\in M$ . The timelike future of $p$ is the set $I^{+}\left(p\right)=\{q:p\ll q\}$ . Similarly, the causal future of $p$ is the set $J^{+}\left(p\right)=\{q:p<q\}$ ; the timelike past $I^{-}\left(p\right)$ and causal past $J^{-}\left(p\right)$ are defined analogously. For any set $S\subseteq M$ , we define $I^{+}\left(S\right)$ to be $\cup \left\{I^{+}\right(p):p\in S\}$ and analogously for $I^{-}\left(S\right)$ , $J^{+}\left(S\right)$ , and $J^{-}\left(S\right)$ . The causal (respectively, timelike) future of a point represents the region of spacetime that can be possibly influenced by particles (respectively, massive particles) at the point. For any $p\in M$ , one can show that the regions $I^{+}\left(p\right)$ and $I^{-}\left(p\right)$ are open. But although the regions $J^{+}\left(p\right)$ and $J^{-}\left(p\right)$ can sometimes be closed (e.g. in Minkowski spacetime), they are not closed in general. To see this, consider Minkowski spacetime in standard $(t,x)$ coordinates and remove the point $(0,0)$ . The point $p=(1,1)$ in the resulting spacetime is such that $J^{-}\left(p\right)$ is not closed (see Figure 14). A useful result states that for any $p,q,r\in M$ , if either (i) $q\in J^{+}\left(p\right)$ and $r\in I^{+}\left(q\right)$ or (ii) $q\in I^{+}\left(p\right)$ and $r\in J^{+}\left(q\right)$ , then $r\in I^{+}\left(p\right)$ . And from this one can show that for any $p\in M$ , the regions $I^{+}\left(p\right)$ and $J^{+}\left(p\right)$ share identical boundaries and closures. Analogous results hold for the past direction.

Figure 14 The point $p$ is such that $J^{-}\left(p\right)$ is not closed. No future-directed causal curve exists from any point on the dotted line to $p$ due to the missing point.

Exercise 10 Find a geodesically complete spacetime (M, g_ab) and a point p ∈ M such that J⁻(p)is not closed.

We say a causal curve $\gamma :I\to M$ is closed if there are distinct points $s,s^{\prime }\in I$ such that $\gamma \left(s\right)=\gamma \left(s^{\prime }\right)$ and $\gamma$ has no vanishing tangent vectors. It is immediate that a spacetime $(M,g_{ab})$ has a closed timelike curve through a point $p\in M$ if and only if $p\in I^{-}\left(p\right)$ . A closed timelike curve allows for “time travel” of a certain kind; a massive object may both begin and end a journey through spacetime at the very same event. A spacetime free of closed timelike curves satisfies chronology. The chronology-violating region of a spacetime $(M,g_{ab})$ is the (necessarily open) set $\{p\in M:p\in I^{-}(p\left)\right\}$ . It has been conjectured that all physically reasonable spacetimes must have an empty chronology-violating region (cf. Reference HawkingHawking, 1992).

Minkowski spacetime satisfies chronology, but time-rolled Minkowski spacetime does not. Let $(M,g_{ab})$ be time-rolled Minkowski spacetime in $(t,x)$ coordinates. The curve $\gamma \text{}:[0,2\pi ]\to M$ defined by $\gamma \left(s\right)\text{}=(s,0)$ is a closed timelike geodesic. But not all spacetimes with closed timelike curves have closed timelike geodesics (Gödel, 1949). It turns out that if a spacetime $(M,g_{ab})$ is such that $M$ is a compact manifold, then it must have a non-empty chronology-violating region (Reference GerochGeroch, 1967). The converse does not hold, however; indeed, one can find a chronology-violating spacetime with the manifold ${ℝ}^n$ for all $n\ge 3$ . But given any non-compact manifold of two dimensions or more, one can find a chronological spacetime with that underlying manifold (Reference Penrose, DeWitt and WheelerPenrose, 1968). A spacetime $(M,g_{ab})$ is totally vicious if its chronology-violating region is all of $M$ . It is easy to verify that time-rolled Minkowski spacetime is totally vicious. One can show that if $(M,g_{ab})$ is totally vicious, then for all $p\in M$ we have $I^{-}\left(p\right)=I^{+}\left(p\right)=M$ (Reference Hounnonkpe and MinguzziMinguzzi, 2019, p. 113).

A spacetime satisfies causality if it is free of closed causal curves. It is immediate that any causal spacetime is chronological. But one can easily construct spacetimes that satisfy chronology but not causality; for example, consider null-rolled Minkowski spacetime (see Figure 15). One can show that a spacetime $(M,g_{ab})$ satisfies causality if and only if $J^{+}\left(p\right)\text{}\cap \text{}{J}^{-}\left(p\right)=\left\{p\right\}$ for all $p\in M$ . A spacetime $(M,g_{ab})$ satisfies distinguishability if, for all distinct $p,q\in M$ , both $I^{+}\left(p\right)\ne I^{+}\left(q\right)$ and $I^{-}\left(p\right)\ne I^{-}\left(q\right)$ hold. A spacetime satisfying distinguishability must satisfy causality and moreover cannot have “almost” closed causal curves of a certain kind. In particular, we find that a spacetime $(M,g_{ab})$ satisfies distinguishability if and only if, for all $p\in M$ and all sufficiently small open sets $O$ containing $p$ , there is neither a future-directed nor a past-directed timelike curve that begins at $p$ , leaves $O$ , and returns to $O$ (Malament 2012, Malament 2012, p. 133).

Figure 15 A closed null curve in a chronological spacetime.

Consider spacetimes $(M,g_{ab})$ and $(M^{\prime },{g^{\prime }}_{ab})$ that satisfy distinguishability. If there is a bijection $\theta :M\to M^{\prime }$ such that for all $p,q\in M$ we have $p\ll q$ if and only if $\theta \left(p\right)\ll \theta \left(q\right)$ , then the spacetimes are conformally equivalent (Reference MalamentMalament, 1977b). This means that, since the manifolds $M$ and M' must be diffeomorphic, if the causal structure of spacetime is sufficiently nice, then that structure alone determines the shape of the universe completely.

Exercise 11 Find a causal spacetime (M, g_ab) and a discontinuous bijection θ : M → M such that for all p, q ∈ M, p ≪ q if and only if θ(p) θ (q).

We say that a spacetime $(M,g_{ab})$ satisfies strong causality if, for all points $p\in M$ and all sufficiently small open sets $O\subseteq M$ containing $p$ , there is no future-directed timelike curve that begins in $O$ , leaves $O$ , and returns to $O$ (Reference MalamentMalament, 2012, p. 134). Any spacetime that satisfies strong causality also satisfies distinguishability. To see that the converse does not hold, consider time-rolled Minkowski spacetime in $(t,x)$ coordinates. One can delete the slits $S_1=\left\{\right(0,x):x\le 1\}$ and $S_1=\left\{\right(1,x):0\le x\}$ from the manifold so that distinguishability is saved but strong causality is not (see Figure 16). If a spacetime $(M,g_{ab})$ satisfies strong causality, then for every compact set $K\subset M$ , a causal curve $\gamma :I\to K$ must have both past and future endpoints in $K$ . So a strongly causal spacetime does not permit a (future or past) inextendible causal curve to be trapped within a compact region.

Figure 16 The removed slits $S_1$ and $S_2$ are chosen so that the spacetime is distinguishing but not strongly causal at the point $p$ .

A spacetime $(M,g_{ab})$ satisfies stable causality if there is a continuous timelike vector field ${\xi }^a$ on $M$ such that the spacetime $(M,g_{ab}+{\xi }_a{\xi }_b)$ satisfies chronology. This means that a stably causal spacetime remains free of closed timelike curves even if all of the light cones are opened by a small amount at each point. One can show that any simply connected, two-dimensional spacetime is stably causal (Reference Minguzzi, Sanchez, Alekseevsky and BaumMinguzzi & Sánchez, 2008). Any spacetime that satisfies stable causality also satisfies strong causality, but not the other way around. Indeed, one can define an infinite number of nonequivalent causal levels between strong causality and stable causality (Reference CarterCarter, 1971). We say that a spacetime $(M,g_{ab})$ admits a global time function if there is a smooth scalar field $t:M\to ℝ$ such that, for any distinct points $p,q\in M$ , if $p<q$ , then $t\left(p\right)<t\left(q\right)$ . One can think of the function $t$ as assigning a time to every point in $M$ such that it increases along every nontrivial future-directed causal curve. Remarkably, one can show that a spacetime satisfies stable causality if and only if it admits a global time function (Hawking 1969).

Exercise 12 Find a spacetime that satisfies strong causality but violates stable causality.

We say a spacetime $(M,g_{ab})$ is reflecting if for all $p,q\in M$ , $p$ is in the closure of $J^{-}\left(q\right)$ if and only if $q$ is in the closure of $J^{+}\left(p\right)$ (cf. Reference Kronheimer and PenroseKronheimer & Penrose, 1967). One can show that a spacetime $(M,g_{ab})$ is reflecting if and only if the following holds: for all $p,q\in M$ , $I^{+}\left(p\right)\subseteq I^{+}\left(q\right)$ if and only if $I^{-}\left(q\right)\subseteq I^{-}\left(p\right)$ (Reference Hawking and SachsHawking & Sachs, 1974). A useful result shows that a reflecting spacetime that is not totally vicious must be chronological (Reference Clarke and JoshiClarke & Joshi, 1988). A spacetime satisfies causal continuity if it is both distinguishing and reflecting. In a causally continuous spacetime, points that are close must have similarly close timelike pasts and futures. Every causally continuous spacetime must be stably causal, but the other direction does not hold.

A spacetime $(M,g_{ab})$ is causally closed if for all $p\in M$ , the sets $J^{+}\left(p\right)$ and $J^{-}\left(p\right)$ are closed. Every causally closed spacetime will be reflecting (Reference Hounnonkpe and MinguzziMinguzzi, 2019, p. 110). On the other hand, Minkowski spacetime with a point removed is reflecting but not causally closed. We say a spacetime satisfies causal simplicity if it satisfies causality and is causally closed. One can show that any causally simple spacetime must be causally continuous, but not the other way around (Reference Hawking and SachsHawking & Sachs, 1974). In dimension three or more, if a spacetime is not totally vicious, then it is causally simple if and only if it is causally closed (Reference Hounnonkpe and MinguzziHounnonkpe & Minguzzi, 2019).

Exercise 13 Find a spacetime that satisfies stable causality but violates causal continuity.

A spacetime $(M,g_{ab})$ is causally compact if for all $p,q\in M$ , the region $J^{-}\left(p\right)\text{}\cap \text{}{J}^{+}\left(q\right)$ is compact. Any causally compact spacetime must be causally closed and therefore reflecting. But the $x>0$ portion of Minkowski spacetime in standard $(t,x)$ coordinates shows that a causally closed spacetime need not be causally compact. The so-called transverse ladder of causal conditions can be summarized as follows: causal compactness $\Rightarrow$ causal closedness $\Rightarrow$ reflectivity (Reference Hounnonkpe and MinguzziMinguzzi, 2019, p. 142). All three of these conditions can be satisfied in causally misbehaved models such as the totally vicious (time and space)-rolled Minkowski spacetime. But under the assumption of various minimal causal conditions, the satisfaction of any of the transverse ladder conditions ensures an extremely well-behaved causal structure. As we have seen, any reflecting and distinguishing spacetime is causally continuous (and thus stably causal) and any causally closed and causal spacetime is causally simple (and thus causally continuous). Let us now consider the case of causalcompactness.

We say that a spacetime satisfies global hyperbolicity if it satisfies causality and is causally compact (Reference Bernal and SanchezBernal & Sánchez, 2007). In dimension three or more, we find that if a spacetime is either (i) non-compact or (ii) not totally vicious, then it is globally hyperbolic if and only if it is causally compact (Reference Hounnonkpe and MinguzziHounnonkpe & Minguzzi, 2019). Any globally hyperbolic spacetime is causally simple, but there are spacetimes showing that the converse does not hold. Stepping back, the hierarchy of causal conditions considered here can be summarized as follows: global hyperbolicity $\Rightarrow$ causal simplicity $\Rightarrow$ causal continuity $\Rightarrow$ stable causality $\Rightarrow$ strong causality $\Rightarrow$ distinguishability $\Rightarrow$ causality $\Rightarrow$ chronology $\Rightarrow$ non-totally vicious.

One example of a spacetime that is causally simple but not globally hyperbolic is is anti-de Sitter spacetime – a model with manifold ${ℝ}^n$ and light cones that open up rapidly as they approach spatial infinity (see Figure 17). In $(t,x)$ coordinates, two-dimensional anti-de Sitter spacetime comes out as $({ℝ}^2,g_{ab})$ where $g_{ab}={\cosh }^{2}x{\nabla }_{a}t{\nabla }_{b}t-{\nabla }_{a}x{\nabla }_{b}x$ .

Figure 17 Anti-de Sitter spacetime is causally simple but not globally hyperbolic since the points $p$ and $q$ are such that the region $J^{-}\left(p\right)\cap J^{+}\left(q\right)$ is not compact.

The timelike past of every point in anti-de Sitter spacetime contains the image of a past-extendible timelike curve with infinite length. This is a curious property that seems to permit a “supertask” of a certain kind (Reference Earman and NortonEarman & Norton, 1993; Reference Earman, Wuthrich, Manchak and ZaltaManchak & Roberts, 2016). Let us say that a spacetime $(M,g_{ab})$ is Malament-Hogarth if there is a point $p\in M$ and a past-extendible timelike curve $\gamma :I\to M$ such that $\Vert \gamma \Vert =\infty$ and the image of $\gamma$ is contained in $I^{-}\left(p\right)$ . In such a spacetime, an observer at the event $p$ can “see” an observer along $\gamma$ who has an infinite amount of future time in which to complete a “super task” such as checking all possible counterexamples to Goldbach’s conjecture (i.e. the claim that every even integer greater than two is the sum of two primes). It is immediate that any spacetime that violates chronology will be Malament-Hogarth. And although one can find causally simple spacetimes that are Malament-Hogarth (e.g. anti-de Sitter spacetime), no globally hyperbolic ones exist (Reference HogarthHogarth, 1992).

Exercise 14 Find a Malament-Hogarth spacetime that is flat and satisfies chronology.

To get a better grip on the physical significance of global hyperbolicity, we turn to an equivalent formulation of the condition that concerns causal determinism of a certain kind. Consider a spacetime $(M,g_{ab})$ and set $S\subseteq M$ . We define the future domain of dependence of $S$ , denoted $D^{+}\left(S\right)$ , to be the set of points $p\in M$ such that every past-inextendible causal curve through $p$ intersects $S$ . The past domain of dependence is defined analogously. The full domain of dependence of $S$ is the set $D\left(S\right)=D^{-}\left(S\right)\cup D^{+}\left(S\right)$ . Since any causal influence at a point in $D\left(S\right)$ must register on the set $S$ , one can think of the physical situation on $D\left(S\right)$ as fully determined by the physical situation on $S$ (Reference Choquet-Bruhat and GerochChoquet-Bruhat & Geroch, 1969; Reference EarmanEarman, 1986). In Minkowski spacetime, the domain of dependence of a closed, achronal surface will often be diamond shaped; in Minkowski spacetime with a point removed, a notch can appear (see Figure 18).

Figure 18 The domain of dependence $D\left(S\right)$ of a closed, achronal surface $S$ in Minkowski spacetime with a point removed. Points on the dotted line are in the boundary of $D\left(S\right)$ but not in $D\left(S\right)$ itself.

Exercise 15 In Minkowski spacetime (M, g_ab), find slices S, S' ⊂ M such that D(S) ∩ D(S') = ∅ but D(S) ∪ D(S') = M.

If a closed, achronal set $S\subset M$ in a spacetime $(M,g_{ab})$ is such that $D\left(S\right)=M$ , then we say $S$ is a Cauchy surface. The physical situation on a Cauchy surface would seem to determine the physical situation everywhere in spacetime. Remarkably, one can show that a spacetime satisfies global hyperbolicity if and only if it admits a Cauchy surface; moreover, a globally hyperbolic spacetime $(M,g_{ab})$ will be such that $M$ is homeomorphic to $ℝ\times N$ where $N\subset M$ is any Cauchy surface (Reference GerochGeroch, 1970b). This captures a sense in which global hyperbolicity forbids topology change of a certain kind (cf. Reference GerochGeroch, 1967). For Minkowski spacetime in standard $(t,x)$ coordinates, each $t=$ constant slice is a Cauchy surface. On the other hand, Minkowski spacetime with a point removed fails to have a Cauchy surface (see Figure 19).

Figure 19 The slice $S$ is not a Cauchy surface. The point $p$ is not contained in $D\left(S\right)$ since the inextendible timelike curve $\gamma$ passing through $p$ approaches the missing point and thereby fails to intersect $S$ .

Exercise 16 Find a manifold M that admits a Lorentzian metric but is such that every spacetime (M, g_ab) fails to have a Cauchy surface.

Given a spacetime $(M,g_{ab})$ and a closed, achronal set $S\subset M$ , the future Cauchy horizon of $S$ is the region $H^{+}\left(S\right)$ defined by taking the closure of $D^{+}\left(S\right)$ and removing the points in $I^{-}\left(D^{+}\right(S\left)\right)$ . The past Cauchy horizon is defined analogously. One can show that $H^{+}\left(S\right)$ and $H^{-}\left(S\right)$ are both closed and achronal. In addition, every $p\in H^{+}\left(S\right)$ is the future endpoint of some null geodesic contained in $H^{+}\left(S\right)$ that is either past-inextendible or has a past endpoint on the edge of $S$ (see Figure 20). Analogous results hold for $H^{-}\left(S\right)$ . The full Cauchy horizon of $S$ is the set $H\left(S\right)=H^{+}\left(S\right)\cup H^{-}\left(S\right)$ . For any $S$ that is closed and achronal, we find that $H\left(S\right)$ is the boundary of $D\left(S\right)$ and therefore closed. Moreover, for any non-empty $S$ that is closed and achronal, $H\left(S\right)$ is empty if and only if $S$ is a Cauchy surface.

Figure 20 The future Cauchy horizon $H^{+}\left(S\right)$ of the closed, achronal surface $S$ . The point $p$ is the future endpoint of a null geodesic contained in $H^{+}\left(S\right)$ that has past endpoint $q$ on the edge of $S$ ; the point $p$ is also the future endpoint of a past-inextendible null geodesic contained in $H^{+}\left(S\right)$ that emerges from the missing point.

4 Singularities

The various singularity theorems show senses in which a physically reasonable spacetime can have incomplete timelike geodesics (Reference PenrosePenrose, 1965; Reference Hawking and PenroseHawking & Penrose, 1970). To obtain such results, one must suppose some local constraint on the distribution of matter in the form of “energy conditions” (see Reference Curiel, Lehmkuhl, Schie-mann and ScholzCuriel, 2017). We say a spacetime $(M,g_{ab})$ satisfies the weak energy condition if, for all timelike vectors ${\xi }^a$ at each point in $M$ , we have $T_{ab}{\xi }^a{\xi }^b\ge 0$ . This condition asserts that energy density cannot be negative. A spacetime satisfies the strong energy condition if, for all timelike vectors ${\xi }^a$ at each point in $M$ we have $(T_{ab}-\frac{1}{2}T{g}_{ab}){\xi }^a{\xi }^b\ge 0$ for $T=T_a^a$ . This condition asserts that “gravity attracts.” Finally, a spacetime satisfies the dominant energy condition if it satisfies the weak energy condition and, in addition, for all timelike vectors ${\xi }^a$ at each point in $M$ , the vector $T^{ab}{\xi }_a$ is causal. This condition asserts that matter cannot travel faster than light. Indeed, one can show that if a spacetime $(M,g_{ab})$ satisfies the dominant energy condition and $T_{ab}$ vanishes on some achronal set $S\subset M$ , then $T_{ab}$ vanishes on all of $D\left(S\right)$ (see Figure 21).

Figure 21 If the dominant energy condition is satisfied and $T_{ab}=0$ on some achronal set $S$ , then $T_{ab}=0$ throughout $D\left(S\right)$ .

It is immediate that every vacuum spacetime (e.g. a flat or two-dimensional spacetime) satisfies all of the energy conditions. Let us now restrict attention to four-dimensional spacetimes $(M,g_{ab})$ satisfying the constant curvature condition: $R_{abcd}=(1/12)R(g_{ac}{g}_{bd}-g_{ad}{g}_{bc})$ (see Reference Hawking and EllisHawking & Ellis, 1973, p. 124). One can show that spacetimes satisfying the constant curvature condition must have a constant Ricci scalar $R$ . Examples of such spacetimes include Minkowski spacetime for which $R=0$ , anti-de Sitter spacetime for which $R<0$ , and de Sitter spacetime for which $R>0$ (see p. 35). A four-dimensional spacetime satisfying the constant curvature condition is such that $T_{ab}=-(1/32\pi )R{g}_{ab}$ , which provides an easy way to construct spacetimes violating one or more of the energy conditions. For example, a four-dimensional version of anti-de Sitter spacetime for which $R=-32\pi$ is such that $T_{ab}=g_{ab}$ and $T=T_a^a=n=4$ , showing that the strong energy condition must be violated.

Exercise 17 Find a spacetime that satisfies the strong energy condition but violates the weak energy condition.

We say a spacetime satisfies the generic condition if each causal geodesic with tangent ${\xi }^a$ encounters some effective curvature in the sense that there is a point at which ${\xi }_{[a}{R}_{b\left]mn\right[c}{\xi }_{d]}{\xi }^m{\xi }^n\ne 0$ . Only spacetimes with very special symmetries (e.g. any flat spacetime) will fail to be generic. If a spacetime satisfies the generic condition and is such that $R_{ab}{\xi }^a{\xi }^b\ge 0$ for all timelike vectors ${\xi }^a$ – a requirement that is equivalent to the strong energy condition in four dimensions – we find a sense in which nearby timelike geodesics will tend to “cross” if they are complete (Reference Geroch, Horowitz, Hawking and IsraelGeroch & Horowitz, 1979, p. 264). If a timelike geodesic is crossed in this way, one can always find a pair of points along it and a timelike curve connecting the points that has a longer length than the geodesic (see Figure 22).

Figure 22 A timelike curve (dotted line) from $p$ to $q$ that cuts the corner near the crossing timelike geodesics (solid lines) has a longer length than the geodesic from $p$ to $q$ .

We see that if a spacetime is four-dimensional and satisfies the generic and strong energy conditions, there will be a timelike geodesic that is either incomplete or does not maximize length in the sense just given. One can rule out the latter possibility by appealing to various global properties of interest. For example, here is a singularity theorem along these lines: any four-dimensional, stably causal spacetime with compact slice that also satisfies the generic and strong energy conditions must have an incomplete timelike geodesic (Reference Geroch, Horowitz, Hawking and IsraelGeroch & Horowitz, 1979, p. 265).

Exercise 18 Find a four-dimensional, stably causal spacetime with compact slice that satisfies the strong energy condition but is geodesically complete.

Because the singularity theorems show senses in which physically reasonable spacetimes can be geodesically incomplete, a difficulty arises in how to rule out pathological examples. Consider a spacetime $(M,g_{ab})$ and any point $p\in M$ ; how does one prohibit the seemingly artificial spacetime $(M-\{p\},g_{ab})$ ? Requiring geodesic completeness will do the job, but this route is too heavy-handed given the singularity theorems. Instead, several other interrelated global conditions (e.g. inextendibility) have been suggested – none entirely satisfactory – to sort singular spacetimes into physically reasonable and physically unreasonable varieties. Let’s take a look at a few examples.

We first consider a pair of definitions concerning so-called naked singularities that primarily concern the causal structure of spacetime. We say a future-inextendible causal geodesic $\gamma :I\to M$ is future-incomplete if there is a $r\in ℝ$ such that $r>s$ for all $s\in I$ ; a past-incomplete geodesic is defined analogously. A spacetime $(M,g_{ab})$ has a detectable naked singularity if there is a point $p\in M$ and a future-incomplete timelike geodesic $\gamma :I\to M$ such that the image of $\gamma$ is contained in $I^{-}\left(p\right)$ (cf. Reference Geroch, Horowitz, Hawking and IsraelGeroch & Horowitz, 1979, p. 274). In such a spacetime, the singularity is naked in the sense that an observer at $p$ can see it. It is not difficult to verify that a spacetime with a point removed from its manifold will always have a detectable naked singularity (see Figure 23). One can also show that any spacetime with a detectable naked singularity will not be globally hyperbolic.

Figure 23 The point $p$ is such that the future-incomplete timelike geodesic $\gamma$ approaching the missing point is contained in the region $I^{-}\left(p\right)$ .

Exercise 19 Find a causally simple spacetime with detectable naked singularity.

A second type of naked singularity concerns the evolution of some initial data. We say a spacetime $(M,g_{ab})$ has an evolved naked singularity if there is a slice $S\subset M$ and a point $p\in H^{+}\left(S\right)$ such that the region has compact closure. Here, the requirement that have compact closure ensures that the slice $S$ is not poorly chosen; for Minkowski spacetime in standard $(t,x)$ coordinates, a poorly chosen slice includes the hyperboloid $S=\left\{\right(t,x):t=-{(x^2+1)}^{1/2}\}$ , which has a non-empty future Cauchy horizon (cf. Reference EarmanEarman, 1995, p. 75). Minkowski spacetime with one point removed is an example of a spacetime with an evolved naked singularity (see Figure 24). A spacetime with an evolved naked singularity need not have a detectable naked singularity and vice versa. But as before, any globally hyperbolic spacetime is free of evolved naked singularities.

Figure 24 Due to the missing point, the slice $S$ is such that there is a point $p\in H^{+}\left(S\right)$ for which $I^{-}\left(p\right)\cap S$ has compact closure.

Exercise 20 Find a spacetime with detectable naked singularity but no evolved naked singularity; find a spacetime with an evolved naked singularity but no detectable naked singularity.

A number of cosmic censorship conjectures have been suggested that serve to rule out nakedly singular spacetimes of various types including the two considered here. One quite strong version of this conjecture that precludes both detectable and evolved naked singularities is simply the assertion that all physically reasonable spacetimes must be globally hyperbolic (Reference PenrosePenrose, 1969, Reference Penrose, Hawking and Israel1979). But this position seems far from secure (Reference EarmanEarman, 1995; Reference PenrosePenrose, 1999).

Other conditions used to rule out physically unreasonable singularities concern the modal structure of spacetime; they rule out spacetime “holes” in the sense that they require that spacetime be as large as possible in various ways (cf. Reference EarmanEarman, 1989, pp. 159–163). Inextendibility is one example of this type of condition. But inextendibility alone is not strong enough to rule out all examples of seemingly artificial singularities (recall Figure 10). To handle many of these other cases, another condition can be used. We say a spacetime $(M,g_{ab})$ is hole-free if, for every achronal surface $S\subset M$ and every isometric embedding $\phi :D\left(S\right)\to M^{\prime }$ into a spacetime $(M^{\prime },{g^{\prime }}_{ab})$ , we have $\phi \left[D\right(S\left)\right]=D\left(\phi \right[S\left]\right)$ (Reference Geroch, Earman, Glymour and StachelGeroch, 1977, p. 87). It is immediate that Minkowski spacetime with a point removed is not hole-free since the domain of dependence of some achronal surfaces could have been larger in Minkowski spacetime (see Figure 25).

Figure 25 Due to the missing point, the achronal surface $S$ is such that $D\left(S\right)$ can be isometrically embedded via $\phi$ in such a way that a point $p$ is contained in $D\left(\phi \right[S\left]\right)$ but not in $\phi \left[D\right(S\left)\right]$ .

Despite the intuitive appeal of the hole-freeness condition, a surprising result shows that Minkowski spacetime actually fails to satisfy it (Reference KrasnikovKrasnikov, 2009). Take an achronal surface in two-dimensional Minkowski spacetime that has an open domain of dependence and isometrically embed it into space-rolled Minkowski spacetime. If the embedding is well chosen, the domain of dependence of the image of the surface now has a portion of its boundary included (see Figure 26).

Figure 26 The achronal surface $S$ is such that $D\left(S\right)$ can be isometrically embedded via $\phi$ in such a way that a point $p$ is contained in $D\left(\phi \right[S\left]\right)$ but not in $\phi \left[D\right(S\left)\right]$ .

One can fix up the definition of hole-freeness in a variety of ways (cf. Reference MinguzziMinguzzi, 2012). Consider a globally hyperbolic spacetime $(M,g_{ab})$ and an isometric embedding $\phi :M\to M^{\prime }$ into a spacetime $(M^{\prime },{g^{\prime }}_{ab})$ . We say $(M^{\prime },{g^{\prime }}_{ab})$ is an effective extension of $(M,g_{ab})$ if, for some Cauchy surface $S$ in $(M,g_{ab})$ , $\phi \left[M\right]$ is a proper subset of the interior of $D\left(\phi \right[S\left]\right)$ and $\phi \left[S\right]$ is achronal. We say a spacetime $(M,g_{ab})$ is hole-free* if, for every $K\subseteq M$ such that $(K,g_{ab})$ is a globally hyperbolic spacetime with Cauchy surface $S$ , if $(K^{\prime },g_{ab})$ is not an effective extension of $(K,g_{ab})$ where K' is the interior of $D\left(S\right)$ , then $(K,g_{ab})$ has no effective extension. One intuitively satisfying consequence of this definition is this: for any spacetime $(M,g_{ab})$ and any point $p\in M$ , the spacetime $(M-\{p\},g_{ab})$ is not hole-free* (cf. Reference MinguzziMinguzzi, 2012). It is not difficult to see that a hole-free* spacetime need not be inextendible and vice versa. But one can show that a spacetime is hole-free* if it is either (i) inextendible and globally hyperbolic or (ii) geodesically complete (Reference ManchakManchak, 2014a). It is an open question whether global hyperbolicity in (i) can be weakened to causal simplicity (cf. Reference MinguzziMinguzzi, 2012). We do know that the geodesic completeness condition in (ii) can be weakened significantly to a type of local inextendibility condition (see p. 28)