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Martin Bojowald the Universe: a View from Classical and Quantum Gravity

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Martin Bojowald the Universe: a View from Classical and Quantum Gravity Martin Bojowald The Universe: A View from Classical and Quantum Gravity Related Titles Erdmenger, J. (ed.) Morsch, O. String Cosmology Quantum Bits and Quantum Modern String Theory Concepts from the Secrets Big Bang to Cosmic Structure How Quantum Physics is 2009 Revolutionizing Codes and Computers ISBN: 978-3-527-40862-7 2008 ISBN: 978-3-527-40710-1 Ng, T.-K. Introduction to Classical and Scully, R. J., Scully, M. O. Quantum Field Theory The Demon and the Quantum From the Pythagorean Mystics to 2009 Maxwell’s Demon and Quantum Mystery ISBN: 978-3-527-40726-2 2007 ISBN: 978-3-527-40688-3 Forshaw, J., Smith, G. Dynamics and Relativity Liddle, A. 2009 An Introduction to Modern ISBN: 978-0-470-01459-2 Cosmology 2003 Fayngold, M. ISBN: 978-0-470-84835-7 Special Relativity and How it Works Fayngold, M. 2008 Special Relativity and Motions ISBN: 978-3-527-40607-4 Faster than Light 2002 ISBN: 978-3-527-40344-8 Martin Bojowald The Universe: A View from Classical and Quantum Gravity WILEY-VCH Verlag GmbH & Co. KGaA The Author All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and Martin Bojowald publisher do not warrant the information Pennsylvania State University contained in these books, including this book, to Physics Department be free of errors. Readers are advised to keep in University Park mind that statements, data, illustrations, USA procedural details or other items may inadvertently be inaccurate. Author photography on backcover: Library of Congress Card No.: © Silke Weinsheim applied for ([email protected]) British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Regis- tered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Composition le-tex publishing services GmbH, Leipzig Printing and Binding Markono Print Media Pte Ltd, Singapore Cover Design Adam-Design, Weinheim Printed in Singapore Printed on acid-free paper ISBN 978-3-527-41018-7 V Contents 1TheUniverseI1 1.1 Newtonian Gravity 1 1.2 Planets and Stars 43 1.3 Cosmology 52 2 Relativity 61 2.1 Classical Mechanics and Electrodynamics 61 2.2 Special Relativity 74 2.3 General Relativity 134 3TheUniverseII173 3.1 Planets and Stars 174 3.2 Black Holes 190 3.3 Cosmology 201 4 Quantum Physics 217 4.1 Waves 218 4.2 States 235 4.3 Measurements 264 5 The Universe III 283 5.1 Stars 284 5.2 Elements 290 5.3 Particles 296 6 Quantum Gravity 305 6.1 Quantum Cosmology 308 6.2 Unification 312 6.3 Space-Time Atoms 324 7TheUniverseIV339 7.1 Big Bang 340 7.2 Black Holes 347 7.3 Tests 351 VI Contents Acknowledgement 357 Index 359 VII for A VIII You delight in laying down laws, Yetyoudelightmoreinbreakingthem. Like children playing by the ocean who build sand-towers with constancy and then destroy them with laughter. But while you build your sand-towers the ocean brings more sand to the shore, and when you destroy them the ocean laughs with you. Kahlil Gibran: The Prophet 1 1 The Universe I Observations of the stars and planets have always captured our imagination, in myths as well as methods. While the myths have lost much in importance and relevance, traces of inspiration from the stars can still be seen in modern thinking and technology. 1.1 Newtonian Gravity A concept in physics that, despite its age, enjoys enormous importance, yet has often been remodeled, is the one of the force. In colloquial speech, it is associat- ed with violence or, in personal terms, egotism: “It is a necessary and general law of nature to rule whatever one can.”1) Wespeakofforcesofnaturethatcausein- escapable calamity, or the personal force of despots small and large. Physicists, as human beings, know about these facets of force, but they have also developed it into a powerful and impartial method for predicting motion of all kinds. Force In physics, forces play a more important role than realized colloquially, a role perhaps most powerful and at the same time, most innocent. Forces in physics are embodied by laws of nature. If there is a force acting on some material object, this object has to move in a certain way dictated by the force. The action is most powerful because it cannot be evaded, and it is most innocent because it happens without a purpose, without a hidden (or any) agenda. This concept of a force, al- though it has been somewhat eroded by quantum mechanics2) is one of the central pillars of our description of nature. It serves to explain motion of all kinds, to pre- dict new phenomena, and to understand the workings of useful technology. Forces govern what we do on Earth, but the sky is no exception. A force causes acceleration, or a change of velocity, in a precise and qualitative way: For any given object, the acceleration due to a force is proportional to the magnitude of the force; 1) Thucydides: History of the Peloponnesian War. 2) Nothing in physics, it appears, is permanent or unquestionable, but the demise of one piece of thinking often begets a new, stronger term. The ocean laughs with us, and in its playful way guides us to higher learning. The Universe: A View from Classical and Quantum Gravity, First Edition. Martin Bojowald. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2013 by WILEY-VCH Verlag GmbH & Co. KGaA. 2 1TheUniverseI force equals mass times acceleration. This is a mathematical equality which tells us that its two sides, force and the product of mass times acceleration, are two sides of the same coin. Knowing the force in a given situation is as much as knowing the value which the product of mass and acceleration takes, for they can never be different. Newton’s second law In formula, F D ma,whereF stands for the value of the force acting on an object, m for the object’s mass, and a for the acceleration caused. This equality is often called Newton’s second law. (The first law, some kind of precursor of the second law, states that the action performed on an object equals the reaction caused. Action, in this sense, has only received vague defini- tionsinphysics,andsotheroleofthefirstlawwanedinthecourseofhistory.) One might think that the concept of a force seems to be redundant because we could always replace it with the product of mass times acceleration. However, the distinction of the two sides of an equation is important. We interpret the force, one side of the equation, as causing an action on the object, while the acceleration, on the other side of the equation, is the effect. Although both sides always take the same numerical value, their physical interpretations are not identical to each other. While we see and measure the effect of a force by the acceleration it causes on objects (or by deformations of their shapes), the value of the force itself is derived from the expectation we have for the strength of a certain action. This expectation may come from the physical strength of our arms which we know from experience, or it may be based on a detailed theory for a general physics notion such as the gravitational force. In this way, the distinction of the two sides of a mathematical equation becomes an important conceptual one: ingrained features of an actor in the play of nature, and effects that the action causes. Both sides of Newton’s second law can be as different as psychology and history. As in a literary play, one often understands the characters by watching their ac- tions and reactions. After some time, one begins to form a theory about their na- ture; their actions reflect back on them by the way we judge them and expect how theymaybehaveinfuturescenes.Inphysics,forcesarefirstunderstoodbytheef- fects they cause, such as making an apple fall to the ground; physicists then begin to judge the force, and try to understand what makes it pull or push. One forms a theory about the force, that is, more mathematical equations that allow one to compute the value of the force from other ingredients, independent of the acceler- ations caused on objects. This theory, in turn is tested by using it to compute forces in new situations and comparing with observed accelerations. New predictions can be made to see what the same force may do under hitherto unseen circumstances. The theory is being tested, and can be trusted if many tests are passed. Acceleration Forces can be determined by the accelerations caused (or deforma- tions, as a secondary effect of acceleration of parts of elastic materials). We feel this interplay physically when we attempt to exert a force: our muscles contract, and our 1.1 Newtonian Gravity 3 body begins to move. For acceleration or general causes of forces, we have a more direct sense than for the forces themselves, reflected in the way physics determines forces by measuring accelerations of objects. We notice acceleration whenever our velocity changes. Here, we have a second relationship involving acceleration: it is the ratio of the change of velocity in a given interval of time by the magnitude of that interval, assumed small.
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