Foundations of Special Relativity: Kinematic Axioms for Minkowski Space-Time
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John W. Schutz Monash University, Clayton, Victoria/Australia
Foundations of Special Relativity: Kinematic Axioms for Minkowski Space-Time
Springer-Verlag Berlin· Heidelberg· New York 1973
AMS Subject Classifications (1970): Primary: 70A05, 83A05, 83F05 Secondary: 50-00, 50A05, 50 AIO, 50C05, 50D20, 53C70 ISBN 3-540-06591-1 Springer-Verlag Berlin, Heidelberg· New York ISBN 0-387-06591-1 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin . Heidelberg 1973 . Library of Congress Catalog Card Number 73-20806. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
To
Amina
PREFACE
The aim of this monograph is to give an axiomatic development of Einstein's theory of special relativity from axioms which describe intuitive concepts concerning the kinematic behaviour of inertial particles and light signals.
I am grateful to Professor G. Szekeres and Dr. E.D. Fackerell for their encouragement and constructive suggestions during the preparation of this monograph.
John W. Schutz Monash University
TABLE OF CONTENTS
CHAPTER 1.
INTRODUCTION
1
CHAPTER 2.
KINEMATIC AXIOMS FOR MINKOWSKI SPACE-TIME
7
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13
Primitive Notions Existence of Signal Functions The Temporal Order Relation The Triangle Inequality Signal Functions are Order-Preserving The Coincidence Relation. Events Optical Lines Axiom of the Intermediate Particle The Isotropy of SPRAYs The Axiom of Dimension The Axiom of Incidence The Axiom of Connectedness Compactness of Bounded sub-SPRAYs
CHAPTER 3. 3.1 3.2 3.3 3.4 3.5 3.6 3.7
CONDITIONALLY COMPLETE PARTICLES
7 8 9
12 13
14-
17 24 25 33 35 36 38 42
Conditional Completion of a Particle 42 Properties of Extended Signal Relations and Functions44 Generalised Triangle Inequalities 47 Particles Do Not Have First or Last Instants 48 Events at Which Distinct Particles Coincide 50 Generalised Temporal Order. Relations on the Set of Events. Observers. 52 Each Particle is Dense in Itself 57
VIII CHAPTER 4. 4.1 4.2 4.3 4.4
Collinearity. The Two Sides of an Event The Intermediate Instant Theorem Modified Signal Functions and Modified Record Functions Betweenness Relation for n Particles
CHAPTER 5. 5.1 5.2 5.3 5.4 5.5
6.4
8.4
THEORY OF PARALLELS
Divergent and Convergent Parallels The Parallel Relations are Equivalence Relations Coordinates on a Collinear Set Isomorphisms of a Collinear Set of Particles Linearity of Modified Signal Functions
CHAPTER 8. 8.1 8.2 8.3
COLLINEAR PARTICLES
Basic Theorems The Crossing Theorem Collinearity of Three Particles. Properties
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