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This unique book presents a particularly beautiful way of looking
at special relativity. The author encourages students to see beyond
the formulas to the deeper structure. The unification of space and
time introduced by Einstein's special theory of relativity is one
of the cornerstones of the modern scientific description of the
universe. Yet the unification is counterintuitive because we
perceive time very differently from space. Even in relativity, time
is not just another dimension, it is one with different properties
The book treats the geometry of hyperbolas as the key to
understanding special relativity. The author simplifies the
formulas and emphasizes their geometric content. Many important
relations, including the famous relativistic addition formula for
velocities, then follow directly from the appropriate (hyperbolic)
trigonometric addition formulas. Prior mastery of (ordinary)
trigonometry is sufficient for most of the material presented,
although occasional use is made of elementary differential
calculus, and the chapter on electromagnetism assumes some more
advanced knowledge. Changes to the Second Edition The treatment of
Minkowski space and spacetime diagrams has been expanded. Several
new topics have been added, including a geometric derivation of
Lorentz transformations, a discussion of three-dimensional
spacetime diagrams, and a brief geometric description of "area" and
how it can be used to measure time and distance. Minor notational
changes were made to avoid conflict with existing usage in the
literature. Table of Contents Preface 1. Introduction. 2. The
Physics of Special Relativity. 3. Circle Geometry. 4. Hyperbola
Geometry. 5. The Geometry of Special Relativity. 6. Applications.
7. Problems III. 8. Paradoxes. 9. Relativistic Mechanics. 10.
Problems II. 11. Relativistic Electromagnetism. 12. Problems III.
13. Beyond Special Relativity. 14. Three-Dimensional Spacetime
Diagrams. 15. Minkowski Area via Light Boxes. 16. Hyperbolic
Geometry. 17. Calculus. Bibliography. Author Biography Tevian Dray
is a Professor of Mathematics at Oregon State University. His
research lies at the interface between mathematics and physics,
involving differential geometry and general relativity, as well as
nonassociative algebra and particle physics; he also studies
student understanding of "middle-division" mathematics and physics
content. Educated at MIT and Berkeley, he held postdoctoral
positions in both mathematics and physics in several countries
prior to coming to OSU in 1988. Professor Dray is a Fellow of the
American Physical Society for his work in relativity, and an
award-winning teacher.
There are precisely two further generalizations of the real and
complex numbers, namely, the quaternions and the octonions. The
quaternions naturally describe rotations in three dimensions. In
fact, all (continuous) symmetry groups are based on one of these
four number systems. This book provides an elementary introduction
to the properties of the octonions, with emphasis on their
geometric structure. Elementary applications covered include the
rotation groups and their spacetime generalization, the Lorentz
group, as well as the eigenvalue problem for Hermitian matrices. In
addition, more sophisticated applications include the exceptional
Lie groups, octonionic projective spaces, and applications to
particle physics including the remarkable fact that classical
supersymmetry only exists in particular spacetime dimensions.
This unique book presents a particularly beautiful way of looking
at special relativity. The author encourages students to see beyond
the formulas to the deeper structure. The unification of space and
time introduced by Einstein's special theory of relativity is one
of the cornerstones of the modern scientific description of the
universe. Yet the unification is counterintuitive because we
perceive time very differently from space. Even in relativity, time
is not just another dimension, it is one with different properties
The book treats the geometry of hyperbolas as the key to
understanding special relativity. The author simplifies the
formulas and emphasizes their geometric content. Many important
relations, including the famous relativistic addition formula for
velocities, then follow directly from the appropriate (hyperbolic)
trigonometric addition formulas. Prior mastery of (ordinary)
trigonometry is sufficient for most of the material presented,
although occasional use is made of elementary differential
calculus, and the chapter on electromagnetism assumes some more
advanced knowledge. Changes to the Second Edition The treatment of
Minkowski space and spacetime diagrams has been expanded. Several
new topics have been added, including a geometric derivation of
Lorentz transformations, a discussion of three-dimensional
spacetime diagrams, and a brief geometric description of "area" and
how it can be used to measure time and distance. Minor notational
changes were made to avoid conflict with existing usage in the
literature. Table of Contents Preface 1. Introduction. 2. The
Physics of Special Relativity. 3. Circle Geometry. 4. Hyperbola
Geometry. 5. The Geometry of Special Relativity. 6. Applications.
7. Problems III. 8. Paradoxes. 9. Relativistic Mechanics. 10.
Problems II. 11. Relativistic Electromagnetism. 12. Problems III.
13. Beyond Special Relativity. 14. Three-Dimensional Spacetime
Diagrams. 15. Minkowski Area via Light Boxes. 16. Hyperbolic
Geometry. 17. Calculus. Bibliography. Author Biography Tevian Dray
is a Professor of Mathematics at Oregon State University. His
research lies at the interface between mathematics and physics,
involving differential geometry and general relativity, as well as
nonassociative algebra and particle physics; he also studies
student understanding of "middle-division" mathematics and physics
content. Educated at MIT and Berkeley, he held postdoctoral
positions in both mathematics and physics in several countries
prior to coming to OSU in 1988. Professor Dray is a Fellow of the
American Physical Society for his work in relativity, and an
award-winning teacher.
Differential Forms and the Geometry of General Relativity provides
readers with a coherent path to understanding relativity. Requiring
little more than calculus and some linear algebra, it helps readers
learn just enough differential geometry to grasp the basics of
general relativity. The book contains two intertwined but distinct
halves. Designed for advanced undergraduate or beginning graduate
students in mathematics or physics, most of the text requires
little more than familiarity with calculus and linear algebra. The
first half presents an introduction to general relativity that
describes some of the surprising implications of relativity without
introducing more formalism than necessary. This nonstandard
approach uses differential forms rather than tensor calculus and
minimizes the use of "index gymnastics" as much as possible. The
second half of the book takes a more detailed look at the
mathematics of differential forms. It covers the theory behind the
mathematics used in the first half by emphasizing a conceptual
understanding instead of formal proofs. The book provides a
language to describe curvature, the key geometric idea in general
relativity.
There are precisely two further generalizations of the real and
complex numbers, namely, the quaternions and the octonions. The
quaternions naturally describe rotations in three dimensions. In
fact, all (continuous) symmetry groups are based on one of these
four number systems. This book provides an elementary introduction
to the properties of the octonions, with emphasis on their
geometric structure. Elementary applications covered include the
rotation groups and their spacetime generalization, the Lorentz
group, as well as the eigenvalue problem for Hermitian matrices. In
addition, more sophisticated applications include the exceptional
Lie groups, octonionic projective spaces, and applications to
particle physics including the remarkable fact that classical
supersymmetry only exists in particular spacetime dimensions.
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