1. Introduction

Recent results show one sense in which the global structure of space-time cannot be fully established (Manchak Reference Manchak2009a). It seems that, excluding certain pathological examples, every cosmological model is empirically underdetermined; no amount of observational data we could ever (even in principle) accumulate can force one and only one cosmological model on us. Additionally, one can show that even under the assumption of an inductive principle—that the physical laws we determine locally are applicable throughout the universe—these general epistemological difficulties remain.

However, it may be that we are able to make partial determinations concerning important global properties. For example, it might be possible to discover whether our universe is causally well behaved in some respect. The first task of this article will be to demonstrate that for many global properties of interest, even this partial determination is impossible. This certainly exacerbates the epistemic plight of the cosmologist. But, in addition, these underdetermination results seem to have implications for what counts as a “physically reasonable” cosmological model.

Cosmologists often use certain global properties (e.g., those relating to causal misbehavior) to exclude some models from serious consideration. But, on what grounds should these properties be regarded as physically unreasonable if we cannot rule out, even with a robust type of inductive reasoning, the possibility of the properties obtaining in our own universe? The second task of this article is to carefully investigate this question.

2. Background Structure

We start by reviewing the relevant background formalism of general relativity.Footnote ¹ A relativistic space-time is a pair of mathematical objects , where is a connected four-dimensional manifold (without boundary) that is smooth (infinitely differentiable). Here, is a smooth, nondegenerate, pseudo-Riemannian metric of Lorentz signature defined on . Each point in the manifold represents an “event” in space-time. For each point in the manifold, the metric assigns a light cone structure in the tangent space . Any tangent vector in will be timelike (if ), null (if ), or spacelike (if ). Null vectors create the “cone” structure. Timelike vectors are inside the cone, while spacelike vectors are outside. A time-orientable space-time is one that has a continuous timelike vector field on . A time-orientable space-time allows us to distinguish between the future and the past lobes of the light cone. In what follows, we assume that space-times are time orientable.

For some interval , a smooth curve is timelike if the tangent vector at each point in is timelike. Similarly, a curve is null (respectively, spacelike) if its tangent vector at each point is null (respectively, spacelike). A timelike curve is future directed if its tangent vector at each point lies in the future lobe of the light cone. For any two points , is to the timelike future of (written ) if there exists a timelike, future-directed curve from to . A future-directed curve from to that is either timelike or null indicates that is to the causal future of (written ). These relations allow us to define the following sets of points in : , , , and . The set will be used extensively throughout the article and is called the observational past of . This set represents those points that can possibly be observed from .Footnote ²

A point is a future endpoint of a future-directed causal curve if, for every neighborhood of , there exists a point such that for all . A past endpoint is defined similarly. For any set , we define the past domain of dependence of (written ) to be the set of points such that every causal curve with past endpoint and no future endpoint intersects . The future domain of dependence of (written ) is defined analogously. The entire domain of dependence of (written ) is just the set .

A set is achronal if no two points in can be connected by a timelike curve. The edge of a closed, achronal set is the collection of points such that every open neighborhood of contains a point , a point , and a timelike curve from to that does not intersect . A set is a slice if it is closed, achronal, and without edge. A set is a spacelike hypersurface if is an three-dimensional submanifold such that every curve in is spacelike.

Two space-times and are isometric if there is a diffeomorphism such that . For ease of presentation, we will sometimes say that two manifolds and are isometric when it is clear which metrics are associated with and . Two space-times and are locally isometric if, for each point , there is an open neighborhood of and an open subset of such that and are isometric and, correspondingly, with the roles of and interchanged.

3. Observational Indistinguishability

We are now prepared to consider the notion of observationally indistinguishable space-times. Intuitively, one space-time is observationally indistinguishable from another if no observer in the first space-time has grounds for deciding which of the two she inhabits.Footnote ³ Formally, we have the what follows.

It should be clear from the definition that, given that a space-time is observationally indistinguishable from another space-time , no empirical data could (even in principle) allow an observer in to distinguish between the two models.

It has been shown that, except for certain pathological models, not only is every space-time observationally indistinguishable from some other, but the result holds even if we require the two space-times to have identical local properties. In other words, cosmologists inherit a serious epistemic predicament even under the inductive assumption that “the normal physical laws we determine in our space-time vicinity are applicable at all other spacetime points” (Ellis Reference Ellis1975, 246). In order to present the result, consider the following definition.

Under this definition, it should be clear that, for example, the property of satisfying the standard energy conditions counts as local, while the property of being stably causal is classified as global.Footnote ⁴

Now, consider a special class of space-times with the property that for some point in , . Such space-times necessarily contain closed timelike curves and, in addition, have the property that all of space-time may be observed from one point. We will call these space-times causally bizarre.

Now we are ready to introduce the underdetermination result discussed above.Footnote ⁵

So, even if one fixes the local structure of space-time completely, there is a sense in which the global structure of every model is underdetermined. This poses quite a problem for the cosmologist. But, as we are about to see, the epistemic situation is even worse than the theorem above seems to suggest.

4. Underdetermination of Global Properties

In this section, we restrict our attention to four global properties of interest: inextendibility, isotropy, global hyperbolicity, and hole-freeness. Let us define each of them in turn.

We say a space-time is inextendible if it is not possible to properly embed it isometrically into another space-time. So, the property of inextendibility ensures that space-time is “as large as it can be.”

A space-time is (spatially) isotropic if there is a timelike vector field on such that, for all and any two unit vectors and at that are orthogonal to , there is an isometry that leaves and the field fixed but rotates into . Intuitively, isotropic space-times, sometimes called the “Friedmann models,” have no preferred spatial directions.Footnote ⁶ A space-time is globally hyperbolic if there is a slice in such that . Roughly, a globally hyperbolic space-time allows one to determine, from initial conditions on , the physical situation on all of . For this reason, global hyperbolicity is usually taken to be a necessary condition of Laplacian determinism.

A space-time is hole-free if, for any spacelike hypersurface in there is no isometric embedding into another space-time such that . So, in a hole-free space-time, the Cauchy development of every spacelike surface is “as large as it can be.”

With these definitions in place, we are ready to strengthen the underdetermination result from above. Let be the set of global properties consisting of inextendibility, isotropy, global hyperbolicity, and hole-freeness. We have what follows.

We can understand the theorem to be saying that, not only is it impossible to fully establish the global structure of space-time, but we cannot even make partial determinations concerning a handful of space-time properties of interest. It seems that, although our universe may be inextendible, isotropic, globally hyperbolic, and hole-free, we can never know that it is. We mention in passing, however, that if it turns out that our universe fails to have the properties in , it may be possible to determine that this is the case (see Malament Reference Manchak1977).

5. Physical Reasonableness

Some may insist that the result just presented has little or no physical significance.Footnote ⁷ One view is to hold that the space-time constructed in the proof cannot be physically reasonable because it was assembled via a seemingly artificial “cut-and-paste” process. But, such an objection must be formulated carefully because any space-time can be fabricated by such a construction (see Geroch Reference Geroch1971b, 78).

Some conditions for ruling out manufactured examples have been proposed. But, none seem to be entirely satisfactory. Inextendibility is not strong enough to forbid many cut-and-paste examples (see Earman Reference Earman1995, 98). However, the property of local inextendibility, introduced by Hawking and Ellis (Reference Joshi1973, 59), was later shown to be much too strong (see Beem Reference Beem1980). And although the condition of hole-freeness seems to be adequate for many purposes, as we will see below, it carries with it some significant problems as well.

It must be remembered, too, that the theorem can have physical relevance, even if the particular model constructed in the proof does not. Geroch (Reference Geroch1971b, 78) explains: “The space-times obtained by cutting and patching are not normally considered as serious models of our universe. However, the mere existence of a space-time having certain global features suggests that there are many models—some perhaps quite reasonable physically—with similar properties.” Some have held that any space-time that fails to have one or more of the properties in is ipso facto physically unreasonable. In other words, the conditions of inextendibility, isotropy, global hyperbolicity, and hole-freeness have all been taken, at one time or another, to be satisfied by all reasonable models of our universe. But how does one justify such a position? We know that, given the theorem above, this justification cannot be due to any observational data we have collected or likewise any considerations of the local structure of space-time. In the remainder of this section, we will briefly examine the rationale in support of each of the properties under investigation. We hope to show that, in each case, the justification is dubious in certain respects.

6. Conclusion

It seems that one can certainly find principled reasons for believing that all physically reasonable cosmological models are inextendible, isotropic, globally hyperbolic, and hole-free. Indeed, metaphysical considerations, the Copernican principle, and a desire to preserve causal determinism all lead one to certain conclusions concerning the global structure of space-time.

But as we have seen, these guiding principles, as well as the conclusions drawn from them, are far from uncontroversial. It is our position that the existence of this controversy, coupled with the underdetermination result above, requires a certain modesty with regard to the situation. One should simply be open to the possibility that our own universe is not best represented by an inextendible, isotropic, globally hyperbolic, hole-free model. Of course, this possibility demands that we apply great care when labeling as “physically unreasonable” space-times that fail to have some or all of these properties.

We close with some final thoughts concerning physical reasonableness and underdetermination. We have proceeded under the rather basic assumptions that space-time is temporally orientable, the manifold is Hausdorff, and so on. But, do we know that all physically reasonable space-times possess these properties? It seems that we do not—a construction similar to the one used above shows this.

So, it may be that our universe is not time orientable, or Hausdorff, and so on (see Earman Reference Earman2002, Reference Earman and Dieks2008). But, the point is that we need not relax these standard assumptions of physicality in order to arrive at our findings. Indeed, we have attempted to strengthen our underdetermination results as much as possible by greatly limiting the class of admissible cosmological models and then showing that the epistemological predicament persists.

A few matters are still unresolved. Which global properties can be assumed while maintaining the underdetermination of space-time structure? In other words, for which sets of global properties will the following statement be true?

One can straightforwardly show that if = {inextendibility, stable causality}, the proposition holds. What about other properties thought to be characteristic of physically reasonable space-time? In particular, if contains hole-freeness* or global hyperbolicity, does the underdetermination result fall apart? These questions remain open.