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Orthogonality and Spacetime Geometry: Universitext

Autor Robert Goldblatt
en Limba Engleză Paperback – 13 mai 1987
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.
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Specificații

ISBN-13: 9780387965192
ISBN-10: 038796519X
Pagini: 194
Ilustrații: X, 194 p.
Dimensiuni: 155 x 235 x 11 mm
Greutate: 0.29 kg
Ediția:Softcover reprint of the original 1st ed. 1987
Editura: Springer
Colecția Springer
Seria Universitext

Locul publicării:New York, NY, United States

Public țintă

Research

Descriere

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.

Cuprins

1 A Trip On Einstein’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.