FREE ELECTRONIC LIBRARY - Dissertations, online materials

Pages:   || 2 |

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

-- [ Page 1 ] --

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 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 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 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.

Pages:   || 2 |

Similar works:

«Title Authenticity in learning for the twenty-first century: Bridging the formal and the informal Author(s) David Hung, Lee Shu-Shing and Kenneth Y T Lim Source Educational Technology Research and Development, 60(6), 1071-1091 Published by Springer This document may be used for private study or research purpose only. This document or any part of it may not be duplicated and/or distributed without permission of the copyright owner. The Singapore Copyright Act applies to the use of this document....»

«Vol. 11, Issue 4, August 2016 pp. 124–130 Democracy Is a Design Problem Dana Chisnell Co-Director Center for Civic Design The Butterfly Ballot Changed Everything 5443 Tates Bank Road Cambridge, MD 21613 It was a form—an election ballot—that changed everything United States about design in elections in the United States. It came to be dana@civicdesign.org called the butterfly ballot, and it was used in Palm Beach County, Florida in the presidential election in 2000. Up to this point, a...»

«CITY COUNCIL, CITY OF ROCKFORD JOURNAL OF PROCEEDINGS JANUARY 20, 2009 COUNCIL CONVENED AT 6:09 P.M.1. The Clerk called the meeting to order in the absence of the Mayor, and recognized Alderman Wasco who moved that Alderman Mark serve as Mayor Pro Tem for the meeting. Said motion was seconded by Alderman Holt. MOTION PREVAILED (Ald. Bell, Jacobson, Thompson-Kelly, McNeely absent). 2. The invocation was given by Father Ron Montayne, St. Sebastian Orthodox Catholic Church/Police Chaplain and the...»

«Archbishop Molloy High School’s 28 th Annual Jim Kinnier Stanner Golf Classic Monday, May 5, 2014. Plandome CC North Hills CC Congratulations to Archbishop Molloy High School, Chairman Bob Metzger '88, Honorary Chairman Jack Conley, and the 28th Annual Jim Kinnier Stanner Golf Classic 26th Annual 28th Annual Jim Kinnier Stanner Golf Classic Jim Kinnier Stanner Golf Classic The Proceeds from this year’s Stanner Golf Classic will benefit the at Proceeds from this year’s Stanner Golf...»

«The Financial Review 38 (2003) 515529 What Can “Nine-Eleven” Tell Us about Closed-end Fund Discounts and Investor Sentiment? Timothy R. Burch∗ University of Miami Douglas R. Emery University of Miami Michael E. Fuerst University of Miami Abstract We use the horrific events of September 11, 2001 (“nine-eleven”) as a natural test of the hypothesis that closed-end mutual fund discounts from fund net asset values reflect small investor sentiment. Because nine-eleven was a sudden,...»

«OXFORD DIOCESAN ADVISORY COMMITTEE FOR THE CARE OF CHURCHES MINUTES of the meeting of the Committee held in the McKenna Room at Christ Church, Oxford at 10am on Monday 12th January PRESENT : Mr C Baker (Chairman), the Archdeacon of Buckingham, the Venerable C Allsopp, Mrs G Argyle, Mrs J Christie, Mr D Clark, Professor B Kemp, Mr B Martin, Professor J Missenden, Mr J Munby, Canon T Stirling, the Reverend J West, the Reverend D Witchell, the Reverend A Doig, Ms C Townsend (acting secretary),...»

«GUNVOR GUTTORM Árbediehtu (Sami traditional knowledge) – as a concept and in practice I would like to start this article with a brief narrative from the Swedish part of Sámi Land (Sápmi), from an area we can call Jávregaska. Jávregaska consists of two large lakes. Between the lakes there is a muotki (strip of land), which is also the dividing line between the areas where two separate groups of Sami people have lived and shared the surrounding natural resources. Jávregaska is an area of...»

«Strong opto-electro-mechanical coupling in a silicon photonic crystal cavity Alessandro Pitanti,1,2 Johannes M. Fink,1 Amir H. Safavi-Naeini,1,3 Jeff T. Hill,1,3 Chan U. Lei,1 Alessandro Tredicucci,2,4 and Oskar ∗ Painter1, 1 Institutefor Quantum Information and Matter and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA 2 NEST and Istituto Nanoscienze CNR, Scuola Normale Superiore, Piazza San Silvestro 12, 56127 Pisa,...»

«ER IK NO O N AN A Post-Punk Prose Milieu Law of Desire, by Andrej Blatnik, translated by Tamara M. Soban. Champaign, Illinois: Dalkey Archive Press, 2014. Once upon a time, in a fantastical Cold War Central and Eastern Europe, which only existed within the imaginations of outsiders nourished on foreign books and music, nothing transmitted the spirit of freedom like rock and prose. The artists who’d created these works knew something about liberation—or so the narrative went, anyway— that...»

«MEETING OF COUNCIL Edited minutes of the meeting held on Wednesday 21 May 2014 Council Chamber, Churchill House Items which remain (at least for the time being) confidential to Council are not included in these minutes Members attending: Dr J-P v an Besouw, President Dr J Nolan Dr D M Nolan Dr J A Langton Dr L Brennan Dr J Colv in Professor J R Sneyd Dr V R Alladi Dr A Batchelor Dr E J Fazackerley Dr K Grady Dr S Fletcher Professor D Row botham Professor M Mythen Professor R Mahajan Dr P Kumar...»

«Introduction I should have my eyes awaked towards culture through travel. People deviating from their daily life could be friendly felt in each other.. And, Friends live in various countries, have thought in different and have shared human life. They are all culture creators. Beautiful scenery is still vivid in my eyes. The next scene took my breath away a few times. People drew round to press the shutter and stare at “dancing Australian aborigines only wearing a short pant.”. I was also...»

«6.1 Asteroid Eros This NASA, NEAR image of the surface of the asteroid Eros was taken on February 12, 2001 from an altitude of 120 meters (Credit: Dr. Joseph Veverka/ NEAR Imaging Team/Cornell University). The image is 6 meters wide. The scale of an image is found by measuring with a ruler the distance between two points on the image whose separation in physical units you know. In this case, we are told the image width is 6.0 meters. Step 1: Measure the width of the image with a metric ruler....»

<<  HOME   |    CONTACTS
2016 www.dissertation.xlibx.info - Dissertations, online materials

Materials of this site are available for review, all rights belong to their respective owners.
If you do not agree with the fact that your material is placed on this site, please, email us, we will within 1-2 business days delete him.