DOCSLIB.ORG

The Newtonian Limit of Geometrostatics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Newtonian Limit of Geometrostatics Dissertation The Newtonian Limit of Geometrostatics Carla Cederbaum betreut von Prof. Dr. Gerhard Huisken eingereicht am Fachbereich Mathematik und Informatik der Freien Universit¨at Berlin angefertigt am Max-Planck-Institut fur¨ Gravitationsphysik (Albert-Einstein-Institut) Vorgelegt am 10.Mai 2011 1. Gutachter: Prof. Dr. Gerhard Huisken 2. Gutachter: Prof. Dr. Jan Metzger Datum der Disputation: 13. Juli 2011 Betreuung: Prof. Dr. Gerhard Huisken, Max-Planck-Institut fur¨ Gravitationsphysik (Albert-Einstein-Institut) Freie Universit¨at Berlin Zusammenfassung Diese Arbeit besch¨aftigt sich mit dem Newtonschen Limes der Allgemeinen Re- lativit¨atstheorie (ART). Dabei geht es um die Frage, ob und in welcher Weise die klassische Newtonsche Gravitationstheorie als Grenzwert fur¨ im Verh¨altnis zur Lichtgeschwindigkeit kleine Geschwindigkeiten aus der ART hervorgeht. Diese Frage ist einerseits aus Konsistenzgrunden¨ relevant, andererseits hat sie auch praktische Grunde,¨ wird doch die Newtonsche Gravitationstheorie auch heute noch fur¨ astrophysikalische, astronomische und technische Berechnungen und Beobachtungen eingesetzt. Des Weiteren kann ein vertieftes Verst¨andnis des Newtonschen Limes relativistische Modellierung, numerische Simulation und physikalische Interpretation verbessern und vereinfachen. Die mathema- tische Untersuchung des Newtonschen Limes erfolgt dabei in der Sprache der Rahmentheorie, die in den 1980er Jahren von Jurgen¨ Ehlers [Ehl81] vorgeschla- gen wurde. Diese erm¨oglicht eine einheitliche Beschreibung der Newtonschen (koordinatenabh¨angigen) und der (koordinatenunabh¨angigen) ART. Die vorliegende Arbeit nutzt die Ehlerssche Rahmentheorie, um speziell den Newtonschen Limes physikalischer Gr¨oßen wie Masse und Schwerpunkt eines relativistischen Systems zu analysieren. Dabei beschr¨ankt sie sich auf die Un- tersuchung statischer isolierter relativistischer Systeme, deren Materie einen kompakten Tr¨ager besitzt. Fur¨ diese schl¨agt sie den Namen Geometrostatik vor, um die Bedeutung der Geometrie hervorzuheben und die Theorie von der allgemeineren Geometrodynamik abzugrenzen. Die Betrachtung des New- tonschen Limes der Geometrostatik fuhrt¨ dabei durch Analogiebildung mit der Newtonschen Theorie auch zu einem vertieften Verst¨andnis der Geometrostatik selbst. In engem Zusammenhang damit steht die pseudo-Newtonsche Gravita- tionstheorie, die durch eine konforme Transformation aus der Geometrostatik hervorgeht und sich besonders gut fur¨ die Untersuchung des Newtonschen Li- mes derselben eignet. Mit deren Hilfe lassen sich viele Newtonsche Konzepte auf die Geometrostatik ubertragen.¨ So werden in dieser Arbeit beispielsweise ein Zweites pseudo-Newtonsches Bewegungsgesetz formuliert, eine Charakte- risierung von Aquipotentialfl¨ ¨achen vorgenommen und Eindeutigkeitsfragen in Bezug auf geometrische und physikalische Gr¨oßen beantwortet. Im Rahmen von Geometrostatik und pseudo-Newtonscher Gravitationstheo- rie leitet diese Arbeit außerdem neue quasi-lokale Definitionen und Formeln fur¨ die Masse und den Schwerpunkt eines physikalischen Systems her. Diese werden mit dem asymptotischen Verhalten der geometrostatischen Variablen (vgl. bspw. [Bei80, KM95]) sowie mit den bereits in der ART vorhandenen asymptotischen Konzepten von Masse und Schwerpunkt (vgl. bspw. [HY96, Bar86]) in Beziehung gesetzt. Die neuen Begriffe unterscheiden sich von den bisherigen dadurch, dass sie nicht erst asymptotisch, sondern direkt in der unmittelbaren Umgebung der Materie bestimmt werden k¨onnen und stellen damit ein neues Werkzeug zur Analyse statischer Systeme dar. Gleichzeitig erm¨oglichen sie den Nachweis der Konvergenz von Masse und Schwerpunkt im Newtonschen Limes, der im letzten Kapitel dieser Arbeit erbracht wird. i The Newtonian Limit in a Nutshell1 1 c Einsteinbild: bpk / Orren Jack Turner; layout: Sascha Rieger, Milde Marketing iii Contents 0 Introduction 1 1 Mathematical Preliminaries 5 1.1 Notation and Conventions . 5 1.2 Submanifold Geometry . 7 1.3 Lie Derivatives, Connections, and Killing Vectors . 9 1.4 Weighted Sobolev Spaces . 12 1.4.1 The Faster Fall-Off Trick . 15 2 Isolated Relativistic Systems 18 2.1 Setting and Notation . 18 2.2 3+1 Decomposition and Cauchy Problem . 19 2.3 Asymptotically Flat Ends and Their Properties . 22 2.4 Mass and Energy . 24 2.5 The Center of Mass . 26 3 Geometrostatics 32 3.1 The Static Metric Equations . 32 3.2 Well-Known Properties of Geometrostatic Systems . 38 3.3 The Asymptotic Center of Mass . 42 3.4 Asymptotic Uniqueness in Geometrostatics . 49 4 Pseudo-Newtonian Gravity 56 4.1 Pseudo-Newtonian Gravity . 56 4.2 The Pseudo-Newtonian Mass . 63 4.3 The Pseudo-Newtonian Center of Mass . 67 5 Linking Physics to Geometry 72 5.1 The Static Positive Mass Theorem . 72 5.2 The Pseudo-Newtonian Second Law of Motion . 73 5.3 Surfaces of Equilibrium . 75 5.4 Photon Spheres . 83 6 The Newtonian Limit 88 6.1 Jurgen¨ Ehlers' Frame Theory . 88 6.2 Static Isolated Systems in Frame Theory . 93 6.3 The Newtonian Limit of Geometrostatics . 101 6.4 The Newtonian Limit of Mass and its Center . 106 v Contents Bibliography 112 Index 120 List of Symbols 124 vi 0 Introduction Gravity is one of the four fundamental physical forces in our universe. It gives rise to the orbits of the planets around the sun and to many other equally omnipresent phenomena. Starting with Isaac Newton in the 18th century, natural philosophers and physicists have theorized on and experimented with gravity and its astronomical consequences throughout history. Two major gravitational theories have evolved that are still relevant today. One of them is the so-called Newtonian theory of gravity or \Newtonian gravity (NG)" for short. It builds upon the Newtonian laws of motion and characterizes gravitation as a force field acting instantly and at a distance. In modern language, it is often formulated in terms of a (Newtonian) potential U that satisfies a Poisson equation 4U = 4πGρ relating it to the matter density ρ via the gravitational constant G on the Euclidean space R3. We have this formulation of Newtonian gravity at the back of our minds throughout this thesis. The other major theory of gravitation that remained important and is moreover still vividly researched on today is Albert Einstein's theory of general relativity or \general relativity (GR)" for short dating back to the early years of the twentieth century. It takes a very different approach eliminating the concept of force altogether and unifying space and time into a curved Lorentzian 4-manifold (L4; g) called spacetime. The \Einstein 1 8πG equations" Ric − 2 R g = c4 T then relate the curvatures Ric and R of the Lorentzian metric to an \energy-momentum" or \matter tensor field” T , also coupling them via the gravitational constant G. In addition, however, another constant called c enters the game. c denotes the speed of light; it gave rise to the birth of the theory of (special) relativity in the first place and holds responsible for the coupling of space and time in GR. In this thesis, we are interested in shedding light on the relationship of these two theories. This relationship has been discussed and studied by numerous scientists and philosophers and from many different perspectives. Before we turn to its physical and mathematical aspects, let us have a quick look at some of its philosophical facets following Jurgen¨ Ehlers in [Ehl81, Ehl86, Ehl89]. First of all, the relationship between the theories NG and GR is one of the most important examples of how a “finalized” theory is replaced (or not so?) by a more comprehensive one in science. It can thus add to the theory of science debate which tries to settle the issue of whether such change comes by revolution or rather occurs in small steps. Secondly, and maybe even deeper, the two gravitational theories provide a good example for studying the epistemological question whether progress in science can at all be made or whether what we as scientists consider to be an \instant of progress" is just a \change of taste or opinion". Coming back to the physical aspects of the relationship of NG and GR, it is important to realize that many if not most empirical observations and measurements which have been designed to test GR have been pursued in weakly coupled relatively isolated systems moving slowly with respect to each other. Moreover, predictions and measurement metho- dology are often devised and calculated within a Newtonian framework of thought and 1 0 Introduction in a perturbative approach starting with Newtonian concepts and equations. This kind of approach in fact implicitly presumes that GR \goes over to NG as c ! 1". This, however, is not a prerequisite but a claim which is intuitively convincing, suggestive, and desirable for the purpose of perturbative methods but needs to be proven rigorously. In the last thirty years, mathematicians and physicists have put a lot of effort into formalizing and proving that these perturbative approaches (which go by the names of \Newtonian limit" (c ! 1) and \post-Newtonian expansions"(power series in 1=c2 or 1=c)) are actually ju- stified and that there are reasonable examples for their usefulness (cf. e. g. Alan D. Rendall [Ren92a], Todd Oliynyk [Oli07], Martin Lottermoser [Lot88], and Jurgen¨ Ehlers [Ehl89] as well as references therein). Their arguments rely on a mathematical frame theory devised and named after Jurgen¨ Ehlers, cf. [Ehl89]. This frame theory allows to not only formally compare Newtonian and relativistic equations or specific components of relevant tensors but empowers
Load more

Recommended publications
  • The Rebirth of Cosmology: from the Static to the Expanding Universe
    Physics Before and After Einstein 129 M. Mamone Capria (Ed.) IOS Press, 2005 © 2005 The authors Chapter 6 The Rebirth of Cosmology: From the Static to the Expanding Universe Marco Mamone Capria Among the reasons for the entrance of Einstein’s relativity into the scientific folklore of his and our age one of the most important has been his fresh and bold approach to the cosmological problem, and the mysterious, if not paradoxical concept of the universe as a three-dimensional sphere. Einstein is often credited with having led cosmology from philosophy to science: according to this view, he made it possible to discuss in the pro- gressive way typical of science what had been up to his time not less “a field of endless struggles” than metaphysics in Kant’s phrase. It is interesting to remember in this connection that the German philosopher, in his Critique of Pure Reason, had famously argued that cosmology was beyond the scope of science (in the widest sense), being fraught with unsolvable contradictions, inherent in the very way our reason functions. Of course not everybody had been impressed by this argument, and several nineteenth-century scientists had tried to work out a viable image of the universe and its ultimate destiny, using Newtonian mechanics and the principles of thermodynamics. However, this approach did not give unambiguous answers either; for instance, the first principle of thermodynamics was invoked to deny that the universe could have been born at a certain moment in the past, and the second principle to deny that its past could be infinite.
    [Show full text]
  • 1 Clifford Algebraic Computational Fluid Dynamics
    Clifford Algebraic Computational Fluid Dynamics: A New Class of Experiments. Dr. William Michael Kallfelz 1 Lecturer, Department of Philosophy & Religion, Mississippi State University The Philosophy of Scientific Experimentation: A Challenge to Philosophy of Science-Center for Philosophy of Science, University of Pittsburgh October 15-16, 2010. October 24, 2010 Abstract Though some influentially critical objections have been raised during the ‘classical’ pre- computational simulation philosophy of science (PCSPS) tradition, suggesting a more nuanced methodological category for experiments 2, it safe to say such critical objections have greatly proliferated in philosophical studies dedicated to the role played by computational simulations in science. For instance, Eric Winsberg (1999-2003) suggests that computer simulations are methodologically unique in the development of a theory’s models 3 suggesting new epistemic notions of application. This is also echoed in Jeffrey Ramsey’s (1995) notions of “transformation reduction,”—i.e., a notion of reduction of a more highly constructive variety. 4 Computer simulations create a broadly continuous arena spanned by normative and descriptive aspects of theory-articulation, as entailed by the notion of transformation reductions occupying a continuous region demarcated by Ernest Nagel’s (1974) logical-explanatory “domain-combining reduction” on the one hand, and Thomas Nickels’ (1973) heuristic “domain- preserving reduction,” on the other. I extend Winsberg’s and Ramsey’s points here, by arguing that in the field of computational fluid dynamics (CFD) as well as in other branches of applied physics, the computer plays a constitutively experimental role—supplanting in many cases the more traditional experimental methods such as flow- visualization, etc. In this case, however CFD algorithms act as substitutes, not supplements (as the notions “simulation” suggests) when it comes to experimental practices.
    [Show full text]
  • MATTERS of GRAVITY, a Newsletter for the Gravity Community, Number 3
    MATTERS OF GRAVITY Number 3 Spring 1994 Table of Contents Editorial ................................................... ................... 2 Correspondents ................................................... ............ 2 Gravity news: Open Letter to gravitational physicists, Beverly Berger ........................ 3 A Missouri relativist in King Gustav’s Court, Clifford Will .................... 6 Gary Horowitz wins the Xanthopoulos award, Abhay Ashtekar ................ 9 Research briefs: Gamma-ray bursts and their possible cosmological implications, Peter Meszaros 12 Current activity and results in laboratory gravity, Riley Newman ............. 15 Update on representations of quantum gravity, Donald Marolf ................ 19 Ligo project report: December 1993, Rochus E. Vogt ......................... 23 Dark matter or new gravity?, Richard Hammond ............................. 25 Conference Reports: Gravitational waves from coalescing compact binaries, Curt Cutler ........... 28 Mach’s principle: from Newton’s bucket to quantum gravity, Dieter Brill ..... 31 Cornelius Lanczos international centenary conference, David Brown .......... 33 Third Midwest relativity conference, David Garfinkle ......................... 36 arXiv:gr-qc/9402002v1 1 Feb 1994 Editor: Jorge Pullin Center for Gravitational Physics and Geometry The Pennsylvania State University University Park, PA 16802-6300 Fax: (814)863-9608 Phone (814)863-9597 Internet: [email protected] 1 Editorial Well, this newsletter is growing into its third year and third number with a lot of strength. In fact, maybe too much strength. Twelve articles and 37 (!) pages. In this number, apart from the ”traditional” research briefs and conference reports we also bring some news for the community, therefore starting to fulfill the original promise of bringing the gravity/relativity community closer together. As usual I am open to suggestions, criticisms and proposals for articles for the next issue, due September 1st. Many thanks to the authors and the correspondents who made this issue possible.
    [Show full text]
  • Hamilton's Ricci Flow
    The University of Melbourne, Department of Mathematics and Statistics Hamilton's Ricci Flow Nick Sheridan Supervisor: Associate Professor Craig Hodgson Second Reader: Professor Hyam Rubinstein Honours Thesis, November 2006. Abstract The aim of this project is to introduce the basics of Hamilton's Ricci Flow. The Ricci flow is a pde for evolving the metric tensor in a Riemannian manifold to make it \rounder", in the hope that one may draw topological conclusions from the existence of such \round" metrics. Indeed, the Ricci flow has recently been used to prove two very deep theorems in topology, namely the Geometrization and Poincar´eConjectures. We begin with a brief survey of the differential geometry that is needed in the Ricci flow, then proceed to introduce its basic properties and the basic techniques used to understand it, for example, proving existence and uniqueness and bounds on derivatives of curvature under the Ricci flow using the maximum principle. We use these results to prove the \original" Ricci flow theorem { the 1982 theorem of Richard Hamilton that closed 3-manifolds which admit metrics of strictly positive Ricci curvature are diffeomorphic to quotients of the round 3-sphere by finite groups of isometries acting freely. We conclude with a qualitative discussion of the ideas behind the proof of the Geometrization Conjecture using the Ricci flow. Most of the project is based on the book by Chow and Knopf [6], the notes by Peter Topping [28] (which have recently been made into a book, see [29]), the papers of Richard Hamilton (in particular [9]) and the lecture course on Geometric Evolution Equations presented by Ben Andrews at the 2006 ICE-EM Graduate School held at the University of Queensland.
    [Show full text]
  • The Motion of a Body in Newtonian Theories1
    The Motion of a Body in Newtonian Theories1 James Owen Weatherall2 Logic and Philosophy of Science University of California, Irvine Abstract A theorem due to Bob Geroch and Pong Soo Jang [\Motion of a Body in General Relativity." Journal of Mathematical Physics 16(1), (1975)] provides the sense in which the geodesic principle has the status of a theorem in General Relativity (GR). Here we show that a similar theorem holds in the context of geometrized Newtonian gravitation (often called Newton-Cartan theory). It follows that in Newtonian gravitation, as in GR, inertial motion can be derived from other central principles of the theory. 1 Introduction The geodesic principle in General Relativity (GR) states that free massive test point particles traverse timelike geodesics. It has long been believed that, given the other central postulates of GR, the geodesic principle can be proved as a theorem. In our view, though previous attempts3 were highly suggestive, the sense in which the geodesic principle is a theorem of GR was finally clarified by Geroch and Jang (1975).4 They proved the following (the statement of which is indebted to Malament (2010, Prop. 2.5.2)): Theorem 1.1 (Geroch and Jang, 1975) Let (M; gab) be a relativistic spacetime, with M orientable. Let γ : I ! M be a smooth, imbedded curve. Suppose that given any open subset O of M containing γ[I], there exists a smooth symmetric field T ab with the following properties. 1. T ab satisfies the strict dominant energy condition, i.e. given any future-directed time- ab ab like covectors ξa, ηa at any point in M, either T = 0 or T ξaηb > 0; ab ab 2.
    [Show full text]
  • Inverse Mean Curvature Evolution of Entire Graphs
    INVERSE MEAN CURVATURE EVOLUTION OF ENTIRE GRAPHS PANAGIOTA DASKALOPOULOS AND GERHARD HUISKEN n Abstract. We study the evolution of strictly mean-convex entire graphs over R by Inverse Mean Curvature flow. First we establish the global existence of starshaped entire graphs with superlinear growth at infinity. The main result in this work concerns the critical case of asymptotically conical entire convex graphs. In this case we show that there exists a time T < +1, which depends on the growth at infinity of the initial data, such that the unique solution of the flow exists for all t < T . Moreover, as t ! T the solution converges to a flat plane. Our techniques exploit the ultra-fast diffusion character of the fully-nonlinear flow, a property that implies that the asymptotic behavior at spatial infinity of our solution plays a crucial influence on the maximal time of existence, as such behavior propagates infinitely fast towards the interior. Contents 1. Introduction 1 2. The geometric equations and preliminaries 5 3. Self-similar solutions 10 4. The super-linear case and short time existence 13 5. Lp bounds on 1=H 23 6. L1 estimates on 1=H 29 7. Long time existence Theorem 1.1 40 References 41 arXiv:1709.06665v1 [math.DG] 19 Sep 2017 1. Introduction n n+1 We consider a family of immersions Ft : M ! R of n-dimensional mean convex hypersurfaces n+1 n in R . We say that Mt := Ft(M ) moves by inverse mean curvature flow if @ (1.1) F (z; t) = H−1(z; t) ν(z; t); z 2 M n @t where H(z; t) > 0 and ν(z; t) denote the mean curvature and exterior unit normal of the surface Mt at the point F (z; t).
    [Show full text]
  • The Newtonian Limit of General Relativity
    The Newtonian Limit of General Relativity Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften an der Fakult¨atMathematik und Physik der Eberhard-Karls-Universit¨atT¨ubingen vorgelegt von Maren Reimold aus N¨ordlingen Unterst¨utztdurch das Evangelische Studienwerk Villigst e.V. T¨ubingen,September 3, 2010 Contents Deutsche Zusammenfassung 3 Introduction 5 0.1 Transitions from tangent and cotangent space and induced connections 8 0.2 Concepts of curvature . 9 0.3 Newton's theory of gravitation and Einstein's theory of relativity . 13 1 Frame theory 19 1.1 The structure of the frame theory . 19 1.2 Linear Algebra . 23 1.3 Transfer to the frame theory . 30 1.4 The case λ =0 ............................. 34 2 The Newtonian limit: definition and existence 63 2.1 Definition of the Newtonian limit . 63 2.2 Extension of spacetimes . 66 2.3 Examples . 66 2.4 Existence of a limit . 75 2.5 Static and spherically symmetric spacetimes . 82 3 Existence of genuine Newtonian limits 95 3.1 Transformation of coordinates . 96 3.2 Conditions for the curvature tensor . 103 3.3 Asymptotically flat spacetimes . 106 A Appendix 121 A.1 The Schwarzschild spacetime . 121 A.2 The Kerr spacetime . 128 Index 151 Bibliography 153 1 Deutsche Zusammenfassung So lange es die Allgemeine Relativit¨atstheoriegibt, so lange gibt es auch die zugeh¨origeFrage, inwieweit man die Newtonsche Gravitationstheorie als einen Spezialfall oder doch wenigstens als eine Grenzlage der Allgemeinen Relativit¨ats- theorie auffassen kann. Schon am 18. November 1915, eine Woche bevor
    [Show full text]
  • Predictiun in General Relativity Notes L
    David Malament ~~~~ROBERTGEROCH~~~~ is taken by "unrolling" M. M and M' are obseivationally indistinguishable since no obseivational past of any future-inextendible cuive in either extends beyond the excision barriers. But only M ' admits a global time function. Predictiun in General Relativity Notes l. See also Clark Clymour, ''Topology, Cosmology and Convention," Synthese 24 (1972), 195--218. 2. A compre hensive treatment of work on th e g1obal structure of (relativist.ic) space·limes is given in' S. W. Hawking and C. f . R. Ellis, The Large Scale Stn1cture of Space-Time 1. Introduction (Cambridge' Cambridge Univer>ity Press, 1973). See also Roger Penrose, Techniques of Differential Topology in Relativity (Philadelphi3' Society for Industrial and Applied Mathe­ There are at least two contexts within which one might place a discus­ matics. 1972). More accessible than either is Robert .Geroch, "Space-Time Structure from a Global Viewpoint," in B. K. Sachs, ed., General Relativity and Cosmology (New York sion of the possibilities for making predictions in physics. In the first, one Academic Press, 1971). is concerned only with the actual physical world: one imagines that he has 3. A future end point need not be a point on the curve. The definition is this: lf M is a somehow learned what some physical system is like now, and one wishes space-time, / a connected subset of R, and u: I-. Ma future-directed causal curve, a point x is the future end point of u if for every neighborhood 0 of x there is a t0 EI such that u(t) E 0 to determine what that system will be like in the future.
    [Show full text]
  • Geometric Analysis
    IAS/PARK CITY MATHEMATICS SERIES Volume 22 Geometric Analysis Hubert L. Bray Greg Galloway Rafe Mazzeo Natasa Sesum Editors American Mathematical Society Institute for Advanced Study Geometric Analysis https://doi.org/10.1090//pcms/022 IAS/PARK CITY MATHEMATICS SERIES Volume 22 Geometric Analysis Hubert L. Bray Greg Galloway Rafe Mazzeo Natasa Sesum Editors American Mathematical Society Institute for Advanced Study Hubert Lewis Bray, Gregory J. Galloway, Rafe Mazzeo, and Natasa Sesum, Volume Editors IAS/Park City Mathematics Institute runs mathematics education programs that bring together high school mathematics teachers, researchers in mathematics and mathematics education, undergraduate mathematics faculty, graduate students, and undergraduates to participate in distinct but overlapping programs of research and education. This volume contains the lecture notes from the Graduate Summer School program 2010 Mathematics Subject Classification. Primary 53-06, 35-06, 83-06. Library of Congress Cataloging-in-Publication Data Geometric analysis / Hubert L. Bray, editor [and three others]. pages cm. — (IAS/Park City mathematics series ; volume 22) Includes bibliographical references. ISBN 978-1-4704-2313-1 (alk. paper) 1. Geometric analysis. 2. Mathematical analysis. I. Bray, Hubert L., editor. QA360.G455 2015 515.1—dc23 2015031562 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society.
    [Show full text]
  • Global Properties of Physically Interesting Lorentzian Spacetimes
    Global properties of physically interesting Lorentzian spacetimes Deloshan Nawarajan and Matt Visser School of Mathematics and Statistics, Victoria University of Wellington; PO Box 600, Wellington 6140, New Zealand. E-mail: [email protected]; [email protected] Abstract: Under normal circumstances most members of the general relativity community focus almost exclusively on the local properties of spacetime, such as the locally Euclidean structure of the manifold and the Lorentzian signature of the metric tensor. When combined with the classical Einstein field equations this gives an extremely successful empirical model of classical gravity and classical matter | at least as long as one does not ask too many awkward questions about global issues, (such as global topology and global causal structure). We feel however that this is a tactical error | even without invoking full-fledged \quantum gravity" we know that the standard model of particle physics is also an extremely good representation of some parts of empirical reality; and we had better be able to carry over all the good features of the standard model of parti- cle physics | at least into the realm of semi-classical quantum gravity. Doing so gives us some interesting global features that spacetime should possess: On physical grounds spacetime should be space-orientable, time-orientable, and spacetime-orientable, and it should possess a globally defined tetrad (vierbein, or in general a globally defined vielbein/n-bein). So on physical grounds spacetime should be parallelizable. This strongly suggests that the metric is not the fundamental physical quantity; a very good case can be made for the tetrad being more fundamental than the metric.
    [Show full text]
  • Geometric Relativity
    GRADUATE STUDIES IN MATHEMATICS 201 Geometric Relativity Dan A. Lee 10.1090/gsm/201 Geometric Relativity GRADUATE STUDIES IN MATHEMATICS 201 Geometric Relativity Dan A. Lee EDITORIAL COMMITTEE Daniel S. Freed (Chair) Bjorn Poonen Gigliola Staffilani Jeff A. Viaclovsky 2010 Mathematics Subject Classification. Primary 53-01, 53C20, 53C21, 53C24, 53C27, 53C44, 53C50, 53C80, 83C05, 83C57. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-201 Library of Congress Cataloging-in-Publication Data Names: Lee, Dan A., 1978- author. Title: Geometric relativity / Dan A. Lee. Description: Providence, Rhode Island : American Mathematical Society, [2019] | Series: Gradu- ate studies in mathematics ; volume 201 | Includes bibliographical references and index. Identifiers: LCCN 2019019111 | ISBN 9781470450816 (alk. paper) Subjects: LCSH: General relativity (Physics)–Mathematics. | Geometry, Riemannian. | Differ- ential equations, Partial. | AMS: Differential geometry – Instructional exposition (textbooks, tutorial papers, etc.). msc | Differential geometry – Global differential geometry – Global Riemannian geometry, including pinching. msc | Differential geometry – Global differential geometry – Methods of Riemannian geometry, including PDE methods; curvature restrictions. msc | Differential geometry – Global differential geometry – Rigidity results. msc — Differential geometry – Global differential geometry – Spin and Spin. msc | Differential geometry – Global differential geometry – Geometric evolution equations (mean curvature flow,
    [Show full text]
  • The Universe of General Relativity, Springer 2005.Pdf
    Einstein Studies Editors: Don Howard John Stachel Published under the sponsorship of the Center for Einstein Studies, Boston University Volume 1: Einstein and the History of General Relativity Don Howard and John Stachel, editors Volume 2: Conceptual Problems of Quantum Gravity Abhay Ashtekar and John Stachel, editors Volume 3: Studies in the History of General Relativity Jean Eisenstaedt and A.J. Kox, editors Volume 4: Recent Advances in General Relativity Allen I. Janis and John R. Porter, editors Volume 5: The Attraction of Gravitation: New Studies in the History of General Relativity John Earman, Michel Janssen and John D. Norton, editors Volume 6: Mach’s Principle: From Newton’s Bucket to Quantum Gravity Julian B. Barbour and Herbert Pfister, editors Volume 7: The Expanding Worlds of General Relativity Hubert Goenner, Jürgen Renn, Jim Ritter, and Tilman Sauer, editors Volume 8: Einstein: The Formative Years, 1879–1909 Don Howard and John Stachel, editors Volume 9: Einstein from ‘B’ to ‘Z’ John Stachel Volume 10: Einstein Studies in Russia Yuri Balashov and Vladimir Vizgin, editors Volume 11: The Universe of General Relativity A.J. Kox and Jean Eisenstaedt, editors A.J. Kox Jean Eisenstaedt Editors The Universe of General Relativity Birkhauser¨ Boston • Basel • Berlin A.J. Kox Jean Eisenstaedt Universiteit van Amsterdam Observatoire de Paris Instituut voor Theoretische Fysica SYRTE/UMR8630–CNRS Valckenierstraat 65 F-75014 Paris Cedex 1018 XE Amsterdam France The Netherlands AMS Subject Classification (2000): 01A60, 83-03, 83-06 Library of Congress Cataloging-in-Publication Data The universe of general relativity / A.J. Kox, editors, Jean Eisenstaedt. p.
    [Show full text]