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On the Boundary of the Cosmos

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Abstract

Intuitively, the totality of physical reality—the Cosmos—has a beginning only if (i) all parts of the Cosmos agree on the direction of time (the Direction Condition) and (ii) there is a boundary to the past of all non-initial spacetime points such that there are no spacetime points to the past of the boundary (the Boundary Condition). Following a distinction previously introduced by J. Brian Pitts, the Boundary Condition can be conceived of in two distinct ways: either topologically, i.e., in terms of a closed boundary, or metrically, i.e., in terms of the Cosmos having a finite past. This article proposes that the Boundary Condition should be posed disjunctively, modifies and improves upon the metrical conception of the Cosmos’s beginning in light of a series of surprising yet simple thought experiments, and suggests that the Direction and Boundary Conditions should be thought of as more fundamental to the concept of the Cosmos’s beginning than classical Big Bang cosmology.

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Notes

  1. I will often speak as though spatio-temporal regions are point sets, but friends of the Aristotelian conception of continua can interpret at least some of my point set talk as involving a kind of pragmatic fiction. Moreover, the Aristotelian account of continua lacks the resources to distinguish between closed and open sets and, for that reason, do not have the resources to develop the Topological Conception, since the Topological Conception requires the notion of a point set with a closed boundary. However, the Boundary Condition is defined disjunctively and the second disjunct – the Metrical Conception – is the only natural way for Aristotelians to develop a conception of the Cosmos’s beginning. For that reason, the Boundary Condition does not assume the truth of either the Aristotelian or Cantorian accounts.

  2. Although I will describe spacetime in terms of the pair (M, g), I should not be interpreted as siding with what has sometimes been called the “angle bracket school” [14] or indeed any other approach to the foundations of spacetime theories; instead, I only intend that spacetime can be mathematically represented by the pair (M, g).

  3. For defenses of metrical conventionalism, see [13, 15,16,17].

  4. My comments depend upon two facts: first, that there is no topological feature that distinguishes the intervals (0, 1] and \((-\infty , 1]\). Second, that there is a topological feature that distinguishes (0, 1] and [0, 1]. What does it mean to say that there is no topological feature that distinguishes two intervals? X and Y are said to be topologically equivalent just in case there exists a continuous transformation from X to Y and which has a continuous inverse. In order to prove that (0, 1] and \((-\infty , 1]\) are topologically equivalent, it suffices to construct a suitable continuous transformation and to show that the inverse of the transformation is continuous. I will assume that the topology on \((-\infty , 1]\) and (0, 1] is the subspace topology, that is, the topology inherited from the standard topology on \({\mathbb {R}}\). Consider the function \(f(x) = -1/x + 2\). Trivially, f(x) is a monotonically increasing function that maps the interval (0, 1] to \((-\infty , 1]\); moreover, f(x) trivially has a continuous inverse, at least on the interval in question. Having established the first result, I move to considering the second, namely, that there is a topological feature that distinguishes the intervals (0, 1] and [0, 1]. Here, I will draw upon a more general result, namely, that there is a topological distinction between an open and a closed boundary. One can prove that compact (closed and bounded) sets can be mapped by continuous functions only to compact sets. Consequently, there is no continuous function mapping the compact set [0, 1] to the non-compact set (0, 1]; hence, (0, 1] and [0, 1] are not topologically equivalent [18, p. 121]. An anonymous referee claimed that the intervals (0, 1] and \((-\infty , 1]\) are topologically distinct even though the two intervals are homeomorphic. To be sure, if we consider the two intervals as sub-intervals of \({\mathbb {R}}\), then there are topological features that the two intervals do not share. Here are three examples: (i) the interval \((-\infty , 1]\) is closed in the real numbers \({\mathbb {R}}\), whereas (0, 1] is not; (ii) the complement of \((-\infty , 1]\) in \({\mathbb {R}}\) is connected, whereas the complement of (0, 1] is not; (iii) the closure of (0, 1] in \({\mathbb {R}}\) is compact, while the closure of \((-\infty , 1]\) is not. (Thanks to Brian Harbourne and Nung Kwan Yip for helpful discussion of this point.) While one may use any one of these features to claim that \((-\infty , 1]\) and (0, 1] are not “topologically equivalent”, all three are extrinsic features that depend upon how the intervals \((-\infty , 1]\) and (0, 1] are embedded within \({\mathbb {R}}\). Due to the fact that \((-\infty , 1]\) and (0, 1] are homeomorphic, there is no intrinsic way to distinguish the two. Note that I am using the intervals \((-\infty , 1]\) and (0, 1] as “stand-ins” for the Cosmos’s history; since the Cosmos is the totality of physical reality, the only features that matter for my purposes are the features that are intrinsic to an interval, and so, for my purposes, there is no relevant topological distinction between \((-\infty , 1]\) and (0, 1].

  5. Bimetric theories indistinguishable from standard General Relativity have been considered in [19,20,21,22,23, 24, pp. 335–336] A similar—though in principle observationally distinguishable—theory was considered in [21, 22, 25]; that theory approximates standard General Relativity arbitrarily well given a sufficiently small graviton mass. Bimetric theories have also been considered in [26,27,28].

  6. This claim is easy to motivate. Consider, for example, the case where we begin with Minkowski space and then construct a new spacetime \(S'\) using the procedure I’ve described. And now suppose, for reductio, that there exists a closed space-like boundary \({\mathcal {B}}\) shared by all of the time-like curves in \(S'\). There are two cases that we can consider: first, the case where \({\mathcal {B}}\) is “parallel” to \(\Sigma\), that is, the case where all of the time-like geodesics connecting \({\mathcal {B}}\) and \(\Sigma\) measure the same proper time at the point of intersection with \(\Sigma\), and, second, the case where \({\mathcal {B}}\) is not parallel to \(\Sigma\). In the first case, since all of the time-like geodesics measure the same proper time when they intersect \(\Sigma\), we are guaranteed that there are at least two time-like geodesics that measure the same age at the point at which they respectively intersect with \(\Sigma\). But, by construction, no two time-like geodesics have the same age at their respective points of intersection with \(\Sigma\). In the second case, \({\mathcal {B}}\) is tilted with respect to \(\Sigma\); in that case, there is some volume where \({\mathcal {B}}\) and \(\Sigma\) intersect. Past the volume where \({\mathcal {B}}\) and \(\Sigma\) intersect, \(\Sigma\) does not exist. For that reason, the spacetime will include at least one point that is not in the causal future or the causal past of some point in \(\Sigma\), so that \(\Sigma\) is not actually a spacetime-wide surface. Again, one of our assumptions has been violated. Thus, since the spacetime’s closed boundary cannot be a space-like surface in either case and the two cases are mutually exclusive and exhaustive, the spacetime’s closed boundary cannot be a space-like surface. While this result is much more difficult to establish when we allow for arbitrary spacetime curvature, a single example suffices to establish that one can construct spacetimes with the features I’ve described.

  7. That is, \(\varepsilon\) is a bijection from the positive real numbers to the set of time-like curves. We are guaranteed that this bijection exists because there is a bijection from n-dimensional space to the set of positive real numbers. Nonetheless, bijections between n-dimensional space and the positive real numbers are not generally smooth, thereby lending another reason why the boundary described is “jagged”.

  8. Skow cashes out his view in terms of absolute time, but indicates that he intends for his view to be generalizable to relativistic spacetimes.

  9. On some quantum gravity theories—such as causal set theory [45,46,47,48,49,50]—the spacetime metric appears only in the theory’s continuum limit, thereby allowing for the possibility that there are regions of the Cosmos where the spacetime metric is inapplicable. However, we should not necessarily think of those regions as amorphous in the sense discussed in this section. Consider, for example, Brightwell and Gregory’s [50] construction of the continuum limit for a spacetime interval spanned by a number of spacetime atoms “linked” together in a chain. When the chain is sufficiently long, the spacetime interval is proportional to the number of links in the chain. As causal set theorists like to say, in causal set theory, metrical facts are determined by counting. For that reason, supposing that there are only a small number of spacetime atoms in some region, so that the continuum limit does not apply in the region, we need only consider a larger region to recover relevant metrical facts. In any case, recall that the Boundary Condition for the Cosmos to have a beginning is disjunctive. If the initial portion of the Cosmos is correctly described by causal set theory, then, since causal sets always have closed boundaries, the Cosmos would satisfy the first disjunct—by having a topological beginning—and so would have a beginning.

  10. Nothing crucial in this example hangs on whether time is absolute. The example can be reconstructed for relativistic spacetimes. To construct a relativistic spacetime without metrical structure, first consider a spacetime S with metric \(g_{\mu \nu }\). And now consider the metric \({\tilde{g}}_{\mu \nu }\) produced from \(g_{\mu \nu }\) by the conformal transformation \({\tilde{g}}_{\mu \nu } = \Omega ^2 g_{\mu \nu }\) where \(\Omega\) is a positive and smooth but otherwise arbitrary scalar function. For relativistic spacetimes, multiplication by \(\Omega ^2\) leaves the spacetime’s light cone structure unaltered. Call the resulting spacetime \({\tilde{S}}\). Two spacetimes that are related by such a transformation, e.g., S and \({\tilde{S}}\), are said to be conformally equivalent A spacetime without metrical structure can then be constructed by identifying all of the members of a given class of conformally equivalent spacetimes. Let’s call the spacetime that results from identifying all of the members of a given class of conformally equivalent spacetimes \(S_C\). Since the conformal transformation left the light cone structure unaltered, \(S_C\) is equipped with light cone structure but not metrical structure and so \(S_C\) is an example of a relativistic metrically amorphous spacetime. To construct a relativistic spacetime analogous to the spacetime inhabited by Pam and Jim, one can “glue” a metrically amorphous spacetime region R between two regions that are not metrically amorphous.

  11. A similar point has been previously made in various places, but, in particular, see [51, pp. 125–126, 131] As Earman argues, in order for the Cosmos to have an objectively finite past, there must be an objective best choice of global time function according to which the past is finite. In contrast, Hermann Weyl [52] maintained that the choice of time scale is, is to a certain degree, conventional. In more technical terms, Weyl argued that there is gauge freedom in one’s choice of metric tensor so that the metric tensor is determined only up to a conformal factor, as in footnote 10. In any case, were Weyl’s theory correct, time scale would not correspond to any objective physical fact [53, p. 451]. Additional technical details for Weyl’s theory can be found in [54].

  12. An analogous construction can be produced using relativistic physics.

  13. Ideally, the account should also be consistent with a future quantum gravity theory, but, given that we do not yet possess a successful quantum gravity theory, the account that I offer here will need to be provisional.

  14. What do I mean by the ‘objective’ or ‘fundamental’ qualifier? Recall that one way to motivate a bimetric theory takes inspiration from Poincaré’s thought experiments in which there is an apparent metric due to the presence of universal forces on our measuring devices. By an ‘objective’ or ‘fundamental’ metric I mean a metric that is not merely apparent and that truly has metaphysical significance. I do not adopt a stance in this article on what precisely is requird for a metric to truly have metaphysical significance.

  15. The first spacetime atom in each temporal series of spacetime atoms need not be numerically identical to the first spacetime atom in any other temporal series of spacetime atoms. Note that this allows for an interesting possibility. There could be a discrete spacetime with a jagged edge, i.e., a discrete spacetime such that there are only a finite number of spacetime atoms to the past of every non-initial spacetime atom and yet there is no upper bound to the number of spacetime atoms to the past of non-initial spacetime atoms. For example, for every spacetime atom with n previous atoms, there may be another with \(n+1\) previous atoms.

  16. Perhaps the reader will object that one way that a series can have a boundary involves the series having a first member and having a first member has to do with the ordinal structure of the series. Nonetheless, the conjunction of the Direction Condition and the Boundary Condition successfully captures the Cosmos’s ordinal structure; for example, the Cosmos might have a closed boundary—and so satisfy the Topological Conception—and that closed boundary might be the Cosmos’s first moment in virtue of satisfying the Direction Condition. Moreover, the conjunction of the Direction Condition and the Boundary Condition capture a broader range of cases than if we defined the Boundary Condition in terms of the ordinal structure of spacetime.

  17. For some of the problems involved with utilizing b-incompleteness as the definitive feature of singular spacetimes, see chapter 2 in [61]; also see [69].

  18. Here, \({\mathcal {B}}\) need not be a space-like surface. Instead, I will understand \({\mathcal {B}}\) as the set of closed initial points for all of the time-like and light-like curves in the spacetime. In the case of a spacetime with a “jagged” boundary, there may be no simple relationship – and possibly no continuity – between the points in the set.

  19. This result will not necessarily follow for any curve that is not located in \({\mathcal {B}}\). For example, suppose that spacetime has a closed boundary but that the initial portion of the Cosmos has the “fractal” metrical properties discussed above. In that case, any time-like or light-like curve not located in \({\mathcal {B}}\) will have infinite backwards extension.

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Linford, D. On the Boundary of the Cosmos. Found Phys 53, 76 (2023). https://doi.org/10.1007/s10701-023-00718-6

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  • DOI: https://doi.org/10.1007/s10701-023-00718-6

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