Determinism and General Relativity
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2008. Pruning Some Branches from 'Branching Spacetimes'
CHAPTER 10 Pruning Some Branches from “Branching Spacetimes” John Earman* Abstract Discussions of branching time and branching spacetime have become com- mon in the philosophical literature. If properly understood, these concep- tions can be harmless. But they are sometimes used in the service of debat- able and even downright pernicious doctrines. The purpose of this chapter is to identify the pernicious branching and prune it back. 1. INTRODUCTION Talk of “branching time” and “branching spacetime” is wide spread in the philo- sophical literature. Such expressions, if properly understood, can be innocuous. But they are sometimes used in the service of debatable and even downright per- nicious doctrines. The purpose of this paper is to identify the pernicious branching and prune it back. Section 2 distinguishes three types of spacetime branching: individual branch- ing, ensemble branching, and Belnap branching. Individual branching, as the name indicates, involves a branching structure in individual spacetime models. It is argued that such branching is neither necessary nor sufficient for indeterminism, which is explicated in terms of the branching in the ensemble of spacetime mod- els satisfying the laws of physics. Belnap branching refers to the sort of branching used by the Belnap school of branching spacetimes. An attempt is made to sit- uate this sort of branching with respect to ensemble branching and individual branching. Section 3 is a sustained critique of various ways of trying to imple- ment individual branching for relativistic spacetimes. Conclusions are given in Section 4. * Department of History and Philosophy of Science, University of Pittsburgh, Pittsburgh, USA The Ontology of Spacetime II © Elsevier BV ISSN 1871-1774, DOI: 10.1016/S1871-1774(08)00010-7 All rights reserved 187 188 Pruning Some Branches from “Branching Spacetimes” 2. -
The Reduction Problem for Einstein's Equations I
168 THE REDUCTION PROBLEM FOR EINSTEIN'S EQUATIONS Vincent Moncrief I. MOTIVATION h has long been known that the vacuum (or electro-vacuum) Einstein equations for spacetimes admitting a single (spacelike) Killing field can be locally reexpressed as a 2 + 1 dimensional system of Einstein-harmonic map equations [1, 2] . In this formulation the unknown fields are (in the vacuum case) a Lorentzian metric on the (locally defined) space of orbits of the Killing field and the norm and twist potential of this Killing field which define a harmonic map from the space of orbits to hyperbolic two-space. A natural problem is to study this formulation globally on suitably chosen manifolds and to apply standard methods of elliptic to reduce the system to its "minimal form". In physics terminology one would like to solve an elliptic set of equations for all the dependent or "non-propagating" variables and be left with an unconstrained hyperbolic system for only the independent "physical degrees of freedom" of the gravitational field. One mathematical motivation for this reduction program is that it aims to take maximal advantage of the elliptic aspects of Einstein's equations, for which the needed global analysis techniques are already at hand, and to reduce the complementary (and presumably more difficult) hyperbolic aspects of the system as much as possible. The ultimate aim of this project is to study the long time existence properties of Einstein's equations and the hope is that the reduction program should help to make such an analysis more tractable. 169 Another more "practical" motivation for this program is that it suggests a numerical method for approximately solving the Einstein equations which avoids the notorious problem of the "drifting of the constraints". -
The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 53, Number 4, October 2016, Pages 513–554 http://dx.doi.org/10.1090/bull/1544 Article electronically published on August 2, 2016 THE EMERGENCE OF GRAVITATIONAL WAVE SCIENCE: 100 YEARS OF DEVELOPMENT OF MATHEMATICAL THEORY, DETECTORS, NUMERICAL ALGORITHMS, AND DATA ANALYSIS TOOLS MICHAEL HOLST, OLIVIER SARBACH, MANUEL TIGLIO, AND MICHELE VALLISNERI In memory of Sergio Dain Abstract. On September 14, 2015, the newly upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW) signal, emitted a billion light-years away by a coalescing binary of two stellar-mass black holes. The detection was announced in February 2016, in time for the hundredth anniversary of Einstein’s prediction of GWs within the theory of general relativity (GR). The signal represents the first direct detec- tion of GWs, the first observation of a black-hole binary, and the first test of GR in its strong-field, high-velocity, nonlinear regime. In the remainder of its first observing run, LIGO observed two more signals from black-hole bina- ries, one moderately loud, another at the boundary of statistical significance. The detections mark the end of a decades-long quest and the beginning of GW astronomy: finally, we are able to probe the unseen, electromagnetically dark Universe by listening to it. In this article, we present a short historical overview of GW science: this young discipline combines GR, arguably the crowning achievement of classical physics, with record-setting, ultra-low-noise laser interferometry, and with some of the most powerful developments in the theory of differential geometry, partial differential equations, high-performance computation, numerical analysis, signal processing, statistical inference, and data science. -
Lost in the Tensors: Einstein's Struggles with Covariance Principles 1912-1916"
JOHN EARMAN and CLARK GL YMOUR LOST IN THE TENSORS: EINSTEIN'S STRUGGLES WITH COVARIANCE PRINCIPLES 1912-1916" Introduction IN 1912 Einstein began to devote a major portion of his time and energy to an attempt to construct a relativistic theory of gravitation. A strong intimation of the struggle that lay ahead is contained in a letter to Arnold Sommerfeld dated October 29, 1912: At the moment I am working solely on the problem of gravitation and believe 1 will be able to overcome all difficulties with the help of a local, friendly mathemat- ician. But one thing is certain, that I have never worked so hard in my life, and that I have been injected with a great awe of mathematics, which in my naivet~ until now I only viewed as a pure luxury in its subtler forms! Compared to this problem the original theory of relativity is mere child's play.' Einstein's letter contained only a perfunctory reply to a query from Sommerfeld about the Debye-Born theory of specific heats. Obviously disappointed, Som- merfeld wrote to Hilbert: 'My letter to Einstein was in vain . Einstein is evidently so deeply mired in gravitation that he is deaf to everything else? Sommerfeld's words were more prophetic than he could possibly have known; the next three years were to see Einstein deeply mired in gravitation, sometimes seemingly hopelessly so. In large measure, Einstein's struggle resulted from his use and his misuse, his understanding and his misunderstanding of the nature and implications of covariance principles. In brief, considerations of general covariance were bound up with Einstein's motive for seeking a 'generalized' theory of relativity; mis- understandings about the meaning and implementation of this motivation threatened to wreck the search; and in the end, the desire for general covariance helped to bring Einstein back onto the track which led to what we now recognize *Present address c/o Department of Philosophy, University of Minnesota, Minneapolis, Minn, U.S.A. -
THE GLOBAL EXISTENCE PROBLEM in GENERAL RELATIVITY 3 Played a Vital Role As Motivation and Guide in the Development of the Results Discussed Here
THE GLOBAL EXISTENCE PROBLEM IN GENERAL RELATIVITY LARS ANDERSSON1 Abstract. We survey some known facts and open questions concerning the global properties of 3+1 dimensional spacetimes containing a com- pact Cauchy surface. We consider spacetimes with an ℓ–dimensional Lie algebra of space–like Killing fields. For each ℓ ≤ 3, we give some basic results and conjectures on global existence and cosmic censorship. Contents 1. Introduction 1 2. The Einstein equations 3 3. Bianchi 15 4. G2 23 5. U(1) 30 6. 3+1 34 7. Concluding remarks 39 Appendix A. Basic Causality concepts 40 References 41 1. Introduction arXiv:gr-qc/9911032v4 28 Apr 2006 In this review, we will describe some results and conjectures about the global structure of solutions to the Einstein equations in 3+1 dimensions. We consider spacetimes (M,¯ g¯) with an ℓ–dimensional Lie algebra of space– like Killing fields. We may say that such spacetimes have a (local) isometry group G of dimension ℓ with the action of G generated by space–like Killing fields. For each value ℓ 3 of the dimension of the isometry group, we state the reduced field equations≤ as well as attempt to give an overview of the most important results and conjectures. We will concentrate on the vacuum case. In section 2, we present the Einstein equations and give the 3+1 decom- position into constraint and evolution equations, cf. subsection 2.2. Due to Date: June 27, 2004. 1Supported in part by the Swedish Natural Sciences Research Council, contract no. F-FU 4873-307, and the NSF under contract no. -
Quantization, Nonlinear Partial Differential Equations,And Operator
http://dx.doi.org/10.1090/pspum/059 Other Titles in This Series 59 William Arveson, Thomas Branson, and Irving Segal, editors, Quantization, nonlinear partial differential equations, and operator algebra (Massachusetts Institute of Technology, Cambridge, June 1994) 58 Bill Jacob and Alex Rosenberg, editors, K-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara, July 1992) 57 Michael C. Cranston and Mark A. Pinsky, editors, Stochastic analysis (Cornell University, Ithaca, July 1993) 56 William J. Haboush and Brian J. Parshall, editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991) 55 Uwe Jannsen, Steven L. Kleiman, and Jean-Pierre Serre, editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R. O. Wells, Jr., editor, The mathematical heritage of Hermann Weyl (Duke University, Durham, May 1987) 47 Paul Fong, editor, The Arcata conference on representations of finite groups (Humboldt State University, Arcata, California, July 1986) 46 Spencer J. -
Lawrence Sklar
LAWRENCE SKLAR Born: June 25, 1938 in Baltimore, MD Married to: Elizabeth S. Sklar; one child Education Oberlin College, B.A., 1958 Princeton University, M.A., 1960; Ph.D., 1964 Fellowships, Awards and National Offices Held Undergraduate Ford Foundation Early Admission Scholarship Honors List (all years) Phi Beta Kappa (elected in junior year) Sigma Xi (associate member) Graduate Woodrow Wilson Fellowship, 1959-60 Chancellor Green Fellowship, 1960-61 Charlotte Elizabeth Proctor Advanced Fellowship (awarded to top ten students in third year graduate class), 1961-62 National Science Foundation Cooperative Fellowship, 1962-63 Post-Graduate American Council of Learned Societies Study Fellowship (held at Oxford University), 1965-66 John Simon Guggenheim Memorial Foundation Fellowship, 1974-75 Franklin J. Matchette Prize. Awarded by the American Philosophical Association to Space, Time, and Spacetime as outstanding philosophical book of 1973 and 1974 National Science Foundation Research Grants, 1977-78, 1979-80, 1982, 1984-85, 1986-87, 1988-89, 1998-2001, 2002-03 Rackham Foundation Summer Research Fellowship, 1983, 1994 2 Nelson Fellow, Philosophy Department, University of Michigan, l991-l994, 1995- James B. and Grace J. Nelson Professorship, Philosophy Department, University of Michigan, 1994-95 National Endowment for the Humanities Fellowship, 1995-96 Faculty Recognition Award, University of Michigan, 1995-98 William K. Frankena Collegiate Professorship, University of Michigan, 1995-2002 Lakatos Award. Awarded to Physics and Chance as outstanding book in the philosophy of science for 1995. Physics and Chance selected by Choice Magazine as Outstanding Academic Book in philosophy of science for 1995 Fellow, American Academy of Arts and Sciences John Locke Lectureship in Philosophy, 1998, Oxford University Visiting Fellowship, All Souls College, Oxford University, 1998 Michigan Humanities Award, 1998-99. -
Arxiv:1901.02380V1 [Math-Ph]
A Euclidean Signature Semi-Classical Program Antonella Marini Department of Mathematics, Yeshiva University, 500 West 185th Street, New York, NY 10033, USA. and Department of Mathematics, University of L’Aquila, Via Vetoio, 67010 L’Aquila, AQ ITALY. E-mail address: [email protected] Rachel Maitra Department of Applied Mathematics, Wentworth Institute of Technology, 550 Huntington Avenue, Boston, MA 02115-5998, USA. E-mail address: [email protected] Vincent Moncrief Department of Physics and Department of Mathematics, Yale University, P.O. Box 208120, New Haven, CT 06520, USA. E-mail address: [email protected] arXiv:1901.02380v1 [math-ph] 8 Jan 2019 In this article we discuss our ongoing program to extend the scope of certain, well- developed microlocal methods for the asymptotic solution of Schr¨odinger’s equation (for suitable ‘nonlinear oscillatory’ quantum mechanical systems) to the treatment of several physically significant, interacting quantum field theories. Our main focus is on applying these ‘Euclidean-signature semi-classical’ methods to self-interacting (real) scalar fields of renormalizable type in 2, 3 and 4 spacetime dimensions and to Yang-Mills fields in 3 and 4 spacetime dimensions. A central argument in favor of our program is that the asymptotic methods for Schr¨odinger operators developed in the microlocal literature are far superior, for the quantum mechanical systems to 2 which they naturally apply, to the conventional WKB methods of the physics litera- ture and that these methods can be modified, by techniques drawn from the calculus of variations and the analysis of elliptic boundary value problems, to apply to cer- tain (bosonic) quantum field theories. -
On the Existence of Killing Fields in Smooth Spacetimes with a Compact Cauchy Horizon
On the existence of Killing fields in smooth spacetimes with a compact Cauchy horizon Mart´ın Reiris Ithurralde [email protected] Ignacio Bustamante Bianchi [email protected] Centro de Matem´atica, Universidad de la Rep´ublica, Montevideo, Uruguay. Abstract We prove that the surface gravity of a compact non-degenerate Cauchy horizon in a smooth vacuum spacetime, can be normalized to a non-zero constant. This result, combined with a recent result by Oliver Petersen and Istv´anR´acz,end up proving the Isenberg-Moncrief conjecture on the existence of Killing fields, in the smooth differentiability class. The well known corollary of this, in accordance with the strong cosmic censorship conjecture, is that the presence of compact Cauchy horizons is a non-generic phenomenon. Though we work in 3 + 1, the result is valid line by line in any n + 1-dimensions. 1 Introduction This article discusses the existence of Killing fields on smooth (i.e. C1) time-orientable smooth vacuum 3 + 1 - spacetimes (M; g) having a compact connected Cauchy horizon C. To make the setup clear from the start, the horizon C is assumed to divide M into two connected regions, one of which, H, is a globally hyperbolic spacetime having a smooth closed three-manifold as a Cauchy surface. Such C is known to be always a smooth [4], [6], [5], totally geodesic null hypersurface of M, ruled by null geodesics [16]. We will assume throughout that C is non-degenerate, namely, that there is on it at least one future or past incomplete null geodesic, (recall that an inextensible geodesic is incomplete if it has finite affine length). -
Indeterminism Is a Modal Notion: Branching Spacetimes and Earman’S Pruning
Indeterminism is a modal notion: branching spacetimes and Earman’s pruning Tomasz Placek and Nuel Belnap Contents 1 Three types of branching 4 1.1 Ensemble and individual branching defined . 5 1.2 BST branching . 6 2 BT/BST branching 8 2.1 BST: Our World and its point events . 8 2.2 BST: histories . 9 2.3 BST: axioms . 10 2.4 BST: space-like relatedness . 10 2.5 BST: modal thickness and thinness . 10 2.6 BST: applications . 11 2.7 Spatiotemporal locations . 12 3 Physically-motivated BST models 13 3.1 Minkowskian Branching Structures . 13 3.2 Defining MBS’s . 18 3.3 Takingstock............................ 22 3.4 Historical remarks . 23 4 Further replies to Earman 24 4.1 BST: Hausdorffproperty..................... 24 4.2 The thin red line . 31 4.3 Semantic rule (R)......................... 33 4.4 Past/future asymmetry . 35 5 Indeterminism 37 1 6 Final 40 2 Abstract The paper defends an Aristotelian notion of indeterminism, as rig- orously formulated in the framework of branching space-times (BST) of Belnap (1992), against criticism by Earman (2008) based on a model-theoretic characterization of indeterminism. It delineates BST branching against the background provided by Earman’s (2008) dis- tinction between individual vs. ensemble branching. Partly in order to motivate our responses to Earman, it describes a construction of physically-motivated BST models, in which histories are isomorphic to Minkowski spacetime. Finally it responds to Earman’s criticisms leveled against BST by addressing a topological issue, the question of an actual future, the past/future asymmetry, and some semantical questions. -
Aspects of Determinism in Modern Physics
ASPECTS OF DETERMINISM IN MODERN PHYSICS John Earman 1 INTRODUCTION The aims of this chapter are to review some aspects of determinism that are famil- iar to physicists but are little discussed in the philosophical literature and to show how these aspects connect determinism to issues about symmetries in physics, the structure and ontological status of spacetime, predictability, and computability.1 It will emerge that in some respects determinism is a robust doctrine and is quite hard to kill, while in other respects it is fragile and requires various enabling as- sumptions to give it a fighting chance. It will also be seen that determinism is far from a dead issue. Whether or not ordinary non-relativistic quantum mechanics (QM) admits a viable deterministic underpinning is still a matter of debate. Less well known is the fact that in some cases QM turns out to be more deterministic than its classical counterpart. Quantum field theory (QFT) assumes determinism, at least at the classical level, in order to construct the field algebra of quantum observables. Determinism is at the heart of the cosmic censorship hypothesis, the most important unsolved issue in classical general relativity theory (GTR). And issues about the nature and status of determinism lie at the heart of key foundation issues in the search for a theory of quantum gravity. 2 PRELIMINARIES 2.1 The metaphysics of determinism The proposal is to begin by getting a grip on the doctrine of determinism as it was understood pre-GTR and pre-QM, and then subsequently to try to understand how the doctrine has to be adjusted to accommodate these theories. -
Intermediate Philosophy of Physics Reading List
Intermediate Philosophy of Physics Reading List James Read [email protected] This is James Read’s reading list for the Finals paper, Intermediate Philosophy of Physics. If you have any questions, comments, or suggestions, please email me at the above address. 1 1 Special Relativity As preparation for the special relativity section of the paper, you might consider reading: (Warning: All of these books are stellar, but some of the later entries are very technical!) 1. N. David Mermin, It’s About Time: Understanding Einstein’s Relativity, Princeton: Prince- ton University Press, 2009. 2. Tim Maudlin, Philosophy of Physics Volume I: Space and Time, Princeton: Princeton Uni- versity Press, 2012. 3. Hans Reichenbach, The Philosophy of Space and Time, New York: Dover, 1957. 4. Harvey R. Brown, Physical Relativity: Spacetime Structure from a Dynamical Perspective, Oxford: Oxford University Press, 2005. 5. Roberto Torretti, Relativity and Geometry, New York: Dover, 1996. 6. Michael Friedman, Foundations of Space-Time Theories, Princeton: Princeton University Press, 1983. 2 1.1 Newton’s laws State Newton’s laws of motion and define all terms therein. How (if at all) do the laws depend upon one another? Do the laws together imply that Newtonian mechanics is Galilean invariant? Core reading 1. Herbert Pfister and Markus King, Inertia and Gravitation, Heidelberg: Springer, 2015. xx1.1-1.3. 2. Roberto Torretti, Relativity and Geometry, New York: Dover, 1996. Ch. 1. 3. Harvey R. Brown, Physical Relativity: Spacetime Structure from a Dynamical Perspective, Oxford: Oxford University Press, 2005. xx2.2, 3.1, 3.2. 4. Michael Friedman, Foundations of Space-Time Theories, Princeton, NJ: Princeton Univer- sity Press, 1983.