Author | Goldblatt, Robert. author |
---|---|

Title | Orthogonality and Spacetime Geometry [electronic resource] / by Robert Goldblatt |

Imprint | New York, NY : Springer US, 1987 |

Connect to | http://dx.doi.org/10.1007/978-1-4684-6345-3 |

Descript | X, 194 p. online resource |

SUMMARY

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 havยญ 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 varยญ ious sub-geometries, and these will be prominent throughout. But the project is intended as an exercise in the foundations of geomeยญ 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 maยญ trix represent ability of certain projective transformations (involuยญ tions, polarities). I have tried to make the work sufficiently selfยญ contained that it may be used as the basis for a course at the adยญ vanced undergraduate level, assuming only an elementary knowledge of linear and abstract algebra

CONTENT

1 A Trip On Einsteinโ{128}{153}s Train -- 2 Planes -- 2.1 Affine Planes and Fields -- 2.2 Metric Vector Spaces -- 2.3 Metric Planes -- 2.4 The Singular Plane -- 2.5 The Artinian Plane -- 2.6 Constants of Orthogonality -- 2.7 The Three Real Metric Planes -- 3 Projective Transformations -- 3.1 Projective Planes -- 3.2 Projectivities and Involutions -- 3.3 Matrix-Induced Projectivities -- 3.4 Projective Collineations -- 3.5 Correlations and Polarities -- 4 Threefolds -- 4.1 Affine Spaces -- 4.2 Metric Affine Spaces -- 4.3 Singular Threefolds -- 4.4 Nonsingular Threefolds -- 5 Fourfolds -- 5.1 Artinian Four-Space -- 5.2 Affine Fourfolds and Projective Three-Space -- 5.3 Nonsingular Fourfolds -- 5.4 The Three Real Fourfolds -- Appendix A Metageometry

Mathematics
Geometry
Mathematics
Geometry