Operational Spacetime Interactions and Particles
Operational Spacetime: Interactions and Particles provides readers with a basic understanding of the mutual conditioning of spacetime and interactions and matter. The spacetime manifold will be looked at to be a reservoir for the parametrizati
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Fundamental Theories of Physics
An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application
Series Editors: GIANCARLO GHIRARDI, University of Trieste, Italy VESSELIN PETKOV, Concordia University, Canada TONY SUDBERY, University of York, UK ALWYN VAN DER MERWE, University of Denver, CO, USA
Volume 163 For other titles published in this series, go to http://www.springer.com/series/6001
Heinrich Saller
Operational Spacetime Interactions and Particles
123
Heinrich Saller MPI f¨ur Physik Werner-Heisenberg-Institut F¨ohringer Ring 6 80805 M¨unchen Germany [email protected]
ISBN 978-1-4419-0897-1 e-ISBN 978-1-4419-0898-8 DOI 10.1007/978-1-4419-0898-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009939325 c Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Contents 0 Introduction and Orientation
1
1 Einstein’s Gravity 1.1 Geometrization of Gravity . . . . . . . . . . . . . . . . . . . . 1.2 Schwarzschild–Kruskal Spacetime . . . . . . . . . . . . . . . . 1.3 Friedmann and de Sitter Universes . . . . . . . . . . . . . . .
17 17 20 24
2 Riemannian Manifolds 2.1 Differentiable Manifolds . . . . . . . . . . . . . . . . 2.1.1 External Derivative . . . . . . . . . . . . . . 2.2 Riemannian Operation Groups . . . . . . . . . . . . 2.2.1 Metric-Induced Isomorphisms . . . . . . . . . 2.2.2 Tangent Euclidean and Poincaré Groups . . . 2.2.3 Global and Local Invariance Groups . . . . . 2.2.4 Riemannian Connection . . . . . . . . . . . . 2.3 Affine Connections . . . . . . . . . . . . . . . . . . . 2.3.1 Torsion, Curvature, and Ricci Tensor . . . . . 2.3.2 Cartan’s Stuctural Equations . . . . . . . . . 2.4 Lie Groups as Manifolds . . . . . . . . . . . . . . . . 2.4.1 Lie Group Operations . . . . . . . . . . . . . 2.4.2 Lie Algebra Operations . . . . . . . . . . . . 2.4.3 The Poincaré Group of a Lie Group . . . . . 2.4.4 Lie–Jacobi Isomorphisms for Lie Groups . . . 2.4.5 Examples . . . . . . . . . . . . . . . . . . . . 2.4.6 Adjoint and Killing Connection of Lie Groups 2.5 Riemannian Manifolds . . . . . . . . . . . . . . . . . 2.5.1 Lorentz Covariant Derivatives . . . . . . . . . 2.5.2 Laplace–Beltrami Operator . . . . . . . .
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