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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
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