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«John Byron Manchak*y A number of models of general relativity seem to contain “holes” that are thought to be “physically unreasonable.” One ...»

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Epistemic “Holes” in Space-Time

John Byron Manchak*y

A number of models of general relativity seem to contain “holes” that are thought to be

“physically unreasonable.” One seeks a condition to rule out these models. We examine

a number of possibilities already in use. We then introduce a new condition: epistemic

hole-freeness. Epistemic hole-freeness is not just a new condition—it is new in kind. In

particular, it does not presuppose a distinction between space-times that are “physically reasonable” and those that are not.

1. Introduction. A number of models of general relativity seem to contain “holes” that are thought to be “physically unreasonable.” One seeks a condition to rule out these models. We examine a number of possibilities already in use. We then introduce a new condition: epistemic hole-freeness.

Epistemic hole-freeness is not just a new condition—it is new in kind. In particular, it does not presuppose a distinction between space-times that are “physically reasonable” and those that are not.

2. Preliminaries. We begin with a few preliminaries concerning the relevant background formalism of general relativity.1 An n-dimensional, relativistic space-time ðfor n ≥ 2Þ is a pair of mathematical objects ðM ; gab Þ.

Object M is a connected n-dimensional manifold ðwithout boundaryÞ that is smooth ðinfinitely differentiableÞ. Here, gab is a smooth, nondegenerate, pseudo-Riemannian metric of Lorentz signature ð1, 2,..., 2Þ defined on M. Note that M is assumed to be Hausdorff; for any distinct p; q ∈ M, one Received January 2015; revised July 2015.

* To contact the author, please write to: Department of Logic and Philosophy of Science, University of California, Irvine; e-mail: jmanchak@uci.edu.

y Thanks to Jeff Barrett, Thomas Barrett, Erik Curiel, David Malament, Sarita Rosenstock, Chris Smeenk, Jim Weatherall, Chris Wüthrich, and a number of anonymous reviewers for helpful suggestions on previous drafts.

1. The reader is encouraged to consult Hawking and Ellis ð1973Þ, Wald ð1984Þ, and Malament ð2012Þ for details. An outstanding ðand less technicalÞ survey of the global structure of space-time is given by Geroch and Horowitz ð1979Þ.

Philosophy of Science, 83 (April 2016) pp. 265–276. 0031-8248/2016/8302-0006$10.00 Copyright 2016 by the Philosophy of Science Association. All rights reserved.

This content downloaded from 128.200.138.056 on March 29, 2016 10:55:06 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).

266 JOHN BYRON MANCHAK can find disjoint open sets Op and Oq containing p and q, respectively. We say two space-times ðM ; gab Þ and ðM′; g′ab Þ are isometric if there is a dif feomorphism φ : M →M′ such that φ* ðgab Þ 5 gab.

For each point p ∈ M, the metric assigns a cone structure to the tangent space Mp. Any tangent vector ξa in Mp will be time-like if gab ξa ξb 0, null if gab ξa ξb 5 0, or space-like if gab ξa ξb 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. A time orientable space-time allows one to distinguish between the future and past lobes of the light cone. In what follows, it is assumed that space-times are time orientable.

For some open ðconnectedÞ interval I ⊆ ℝ, a smooth curve γ : I → M is time-like if the tangent vector ξa at each point in γ½IŠ is time-like. Similarly, a curve is null ðrespectively, space-likeÞ if its tangent vector at each point is null ðrespectively, space-likeÞ. 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.

An extension of a curve γ : I → M is a curve γ′ : I′ → M such that I is a proper subset of I′ and γðsÞ 5 γ′ðsÞ for all s ∈ I. A curve is maximal if it has no extension. A curve γ : I → M in a space-time ðM ; gab Þ is a geodesic if ξa ∇a ξb 5 0, where ξa is the tangent vector and ∇a is the unique derivative operator compatible with gab. Let γ : I → M be a time-like curve with unit tangent vector ξb. The acceleration vector is αb 5 ξa ∇a ξb, and the magnitude of acceleration is a 5 ð−αb αb Þ1=2. The total acceleration of γ is ∫γ ads, where s is elapsed proper time along γ.

For any two points p; q ∈ M, we write p ≪ q if there exists a future directed time-like curve from p to q. We write p q if there exists a future directed causal curve from p to q. These relations allow us to define the time-like and causal pasts and futures of a point p: I − ð pÞ 5 fq : q ≪ pg, I 1ð pÞ 5 fq : p ≪ qg, J − ð pÞ 5 fq : q pg, and J 1 ðpÞ 5 fq : p qg. Naturally, for any set S ⊆ M, define J 1 ½SŠ to be the set ∪fJ 1 ðxÞ : x ∈ Sg, and so on. A set S ⊂ M is achronal if S ∩ I − ½SŠ 5 ∅. A space-time satisfies chronology if, for each p ∈ M, p ∉ I − ð pÞ.

A point p ∈ M is a future endpoint of a future directed causal curve γ : I →M if, for every neighborhood O of p, there exists a point t0 ∈ I such that γðtÞ ∈ O for all t t0. A past endpoint is defined similarly. A causal curve is future inextendible ðrespectively, past inextendibleÞ if it has no future ðrespectively, pastÞ endpoint.





For any set S ⊆ M, we define the past domain of dependence of S, written D − ðSÞ, to be the set of points p ∈ 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 1 ðSÞ, is defined analogously. The entire domain of dependence of S, written DðSÞ, is just the set D − ðSÞ∪D 1 ðSÞ. The edge

–  –  –

of an achronal set S ⊂ M is the collection of points p ∈ S such that every open neighborhood O of p contains a point q ∈ I 1 ðpÞ, a point r ∈ I − ðpÞ, and a time-like curve from r to q that does not intersect S. A set S ⊂ M is a slice if it is closed, achronal, and without edge. A space-time ðM ; gab Þ that contains a slice S such that DðSÞ 5 M is said to be globally hyperbolic.

3. A Condition to Disallow Holes? Consider the following example ðsee fig. 1Þ.

Example 1. Let ðM ; gab Þ be Minkowski space-time and let p be any point in M.

Consider the space-time ðM − f pg; gab Þ.

The space-time seems to have an artificial “hole.” One seeks to find a ðsimple, physically meaningfulÞ condition to disallow the example. ðThe condition need not be a sufficient condition for “physical reasonableness”; it need only be necessary.Þ But “although one perhaps has a good intuitive idea of what it is that one wants to avoid, it seems to be difficult to formulate a precise condition to rule out such examples” ðGeroch and Horowitz 1979, 275Þ.

Many of the conditions used to rule out the “hole” in example 1 require that certain regions of ðor curves inÞ space-time be “as large as they can be.” For example, geodesic completeness requires every geodesic to be as large as it can be in a certain sense. Hole-freeness essentially requires the domain of dependence of every space-like surface to be as large as it can be.

Inextendibility requires the entirety of space-time to be as large as it can be.

Let us examine each of these three conditions in more detail. First, consider geodesic completeness.

Figure 1. Minkowski space-time with a point removed from the manifold.

This content downloaded from 128.200.138.056 on March 29, 2016 10:55:06 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).

268 JOHN BYRON MANCHAK Definition. A space-time ðM ; gab Þ is geodesically complete if every maximal geodesic γ : I → M is such that I 5 ℝ. A space-time is geodesically incomplete if it is not geodesically complete.

If an incomplete geodesic is time-like or null, there is a useful distinction one can introduce ðwhich we will need later onÞ. We say that a future directed time-like or null geodesic γ : I → M without a future endpoint is future incomplete if there is an r ∈ ℝ such that s r for all s ∈ I. A past incomplete time-like or null geodesic is defined analogously. Next, consider inextendibility.

–  –  –

Finally, consider hole-freeness. Initially, one defined ðGeroch 1977Þ a space-time ðM ; gab Þ to be hole-free if, for every space-like surface S ⊂ M and every isometric embedding φ : DðSÞ → M ′ into some other space-time ðM ′; g′ b Þ, we have φðDðSÞÞ 5 DðφðSÞÞ. The definition seemed to be satisa factory. But surprisingly, it turns out the definition is too strong; Minkowski space-time fails to be hole-free under this formulation ðKrasnikov 2009Þ.

But one can make modifications to avoid this consequence ðManchak 2009Þ.

Let ðK; gab Þ be a globally hyperbolic space-time. Let φ : K → K′ be an isometric embedding into a space-time ðK′; g′ b Þ. We say ðK′; g′ab Þ is an a effective extension of ðK; gab Þ if, for some Cauchy surface S in ðK; gab Þ, φ½KŠ ⊊ int ðDðφ½SŠÞÞ and φ½SŠ is achronal. Hole-freeness can then be defined as follows.

Definition. A space-time ðM ; gab Þ is hole-free if, for every set K ⊆ M such that ðK; gabjK Þ is a globally hyperbolic space-time with Cauchy surface S, if ðK′; gabjK′ Þ is not an effective extension of ðK; gabjK Þ where K′ 5 int ðDðSÞÞ, then there is no effective extension of ðK; gabjK Þ.

What is the relationship between the three conditions? There are only two implication relations between them ðManchak 2014Þ.

PROPOSITION 1. Any space-time that is geodesically complete is hole-free and inextendible.

Now, any of the three conditions can be used to rule out the “hole” in example 1. But due to the singularity theorems ðHawking and Penrose 1970Þ, geo

–  –  –

desic completeness is now considered to be much too strong a condition; it seems to be violated by “physically reasonable” space-times. In what follows, let us focus on the remaining two conditions that are usually taken to be satisfied by all “physically reasonable” space-times. Indeed, these two conditions are still in use ðsee Earman 1995Þ. Might hole-freeness or inextendibility ðor their conjunctionÞ be the condition we are looking for? Consider the following example.

Example 2. Let ðM ; gab Þ be Minkowski space-time, and let p be any point in M.

Let Ω : M − f pg → ℝ be a smooth positive function that approaches zero as the point p is approached. Now consider the space-time ðM − f pg;

Ω2 gab Þ.

The space-time in example 2 is inextendible and hole-free. Nonetheless, it seems there is still an artificial “hole” in the space-time. One seeks a ðsimple, physically meaningfulÞ condition to rule out even these holes.

4. A New Condition. Consider the following definition form.

Definition. A space-time ðM ; gab Þ has an epistemic hole if there are two future inextendible time-like curves γ and γ′ with the same past endpoint and which ____ such that I − ½γŠ is a proper subset of I − ½γ′Š.

The physical significance of the definition form is this: Suppose two observers are both present at some event. Now suppose ðsubject to the restrictions in the blankÞ they go their separate ways. If it is the case that one observer can eventually know everything the other can eventually know and more, then there is a kind of epistemic “hole” preventing the latter observer from knowing the extra bit. One might require the region of space-time that an observer can eventually know to be “as large as it can be.” In other words, one might require space-time to be free of epistemic holes.

If no restrictions are given in the blank, examples 1 and 2 count as having epistemic holes as we would hope. But, unfortunately, this version of the condition is too strong; it rules out space-times that are usually thought to be “physically reasonable” in some sense. Take Minkowski space-time, for example. It counts as having epistemic holes. ðConsider any point in the Minkowski space-time. Now consider any observer at the point who, with infinite total acceleration, reaches “null infinity” and another observer at the point who does not. See fig. 2.Þ In order to not count Minkowski space-time as having epistemic holes, one seeks to fill the blank with reasonable restrictions. Let us consider two natural possibilities: “are geodesics” and “have finite total acceleration.” Let EHðgÞ and EHðf Þ respectively denote these two versions of the epistemic This content downloaded from 128.200.138.056 on March 29, 2016 10:55:06 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).

270 JOHN BYRON MANCHAK Figure 2. Observers γ1 and γ2 in Minkowski space-time. The set I − ½γ2 Š ðthe shaded areaÞ is a proper subset of I − ½γ1 Š ðthe entire manifoldÞ.

hole definition. In addition, if a space-time fails to have an EHðgÞ, let us say it is EHðgÞ-free ðand respectively for the EHðf Þ caseÞ.

Clearly, if a space-time is EHðf Þ-free, then it also EHðgÞ-free.2 And as we would hope, examples 1 and 2 each have an EHðgÞ and therefore an EHðf Þ ðsee fig. 3Þ. Indeed, acausal examples aside, it seems almost every artificially mutilated space-time will have an epistemic hole of some type. Now, Minkowski space-time is EHðf Þ-free and EHðgÞ-free by construction. What about other “physically reasonable” space-times? The Schwarzchild solution is a good test case; its future inextendible time-like curves have event horizons that might allow for epistemic holes.3 But this is not the case; it and its Kruskal extension count as EHðf Þ-free and EHðgÞ-free ðsee fig. 4Þ.



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