Orthogonality and Spacetime Geometry

  • Robert Goldblatt

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Robert Goldblatt
    Pages 1-17
  3. Robert Goldblatt
    Pages 18-72
  4. Robert Goldblatt
    Pages 73-111
  5. Robert Goldblatt
    Pages 112-136
  6. Robert Goldblatt
    Pages 137-160
  7. Back Matter
    Pages 161-190

About this book

Introduction

This book examines the geometrical notion of orthogonality, and shows how to use it as the primitive concept on which to base a metric structure in affine geometry. The subject has a long history, and an extensive literature, but whatever novelty there may be in the study presented here comes from its focus on geometries hav­ ing lines that are self-orthogonal, or even singular (orthogonal to all lines). The most significant examples concern four-dimensional special-relativistic spacetime (Minkowskian geometry), and its var­ ious sub-geometries, and these will be prominent throughout. But the project is intended as an exercise in the foundations of geome­ try that does not presume a knowledge of physics, and so, in order to provide the appropriate intuitive background, an initial chapter has been included that gives a description of the different types of line (timelike, spacelike, lightlike) that occur in spacetime, and the physical meaning of the orthogonality relations that hold between them. The coordinatisation of affine spaces makes use of constructions from projective geometry, including standard results about the ma­ trix represent ability of certain projective transformations (involu­ tions, polarities). I have tried to make the work sufficiently self­ contained that it may be used as the basis for a course at the ad­ vanced undergraduate level, assuming only an elementary knowledge of linear and abstract algebra.

Keywords

Mathematica Matrix Vector space algebra constant correlation form geometry knowledge linear algebra projective transform threefold transformation

Authors and affiliations

  • Robert Goldblatt
    • 1
  1. 1.Department of MathematicsVictoria UniversityWellingtonNew Zealand

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4684-6345-3
  • Copyright Information Springer-Verlag New York 1987
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-96519-2
  • Online ISBN 978-1-4684-6345-3
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • About this book