Ebook: Orthogonality and Spacetime Geometry

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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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 focus of the book is on geometries having lines which are self-orthogonal, or even singular (orthogonal to all lines). The most significant examples concern the four-dimensional spacetime of special relativity, however no knowledge of physics is presumed. An initial chapter has been included which explains the physical interpretation of the different orthogonality relations. The mathematical background needed is basic abstract and linear algebra.

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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 focus of the book is on geometries having lines which are self-orthogonal, or even singular (orthogonal to all lines). The most significant examples concern the four-dimensional spacetime of special relativity, however no knowledge of physics is presumed. An initial chapter has been included which explains the physical interpretation of the different orthogonality relations. The mathematical background needed is basic abstract and linear algebra.
Content:
Front Matter....Pages i-ix
A Trip on Einstein’s Train....Pages 1-17
Planes....Pages 18-72
Projective Transformations....Pages 73-111
Threefolds....Pages 112-136
Fourfolds....Pages 137-160
Back Matter....Pages 161-190

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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 focus of the book is on geometries having lines which are self-orthogonal, or even singular (orthogonal to all lines). The most significant examples concern the four-dimensional spacetime of special relativity, however no knowledge of physics is presumed. An initial chapter has been included which explains the physical interpretation of the different orthogonality relations. The mathematical background needed is basic abstract and linear algebra.
Content:
Front Matter....Pages i-ix
A Trip on Einstein’s Train....Pages 1-17
Planes....Pages 18-72
Projective Transformations....Pages 73-111
Threefolds....Pages 112-136
Fourfolds....Pages 137-160
Back Matter....Pages 161-190
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