2017;
403 pp;
Hardcover

MSC: Primary 83; 01;

Print ISBN: 978-1-4704-3483-0

Product Code: MBK/102

List Price: $75.00

AMS Member Price: $60.00

MAA Member Price: $67.50

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**Electronic ISBN: 978-1-4704-3669-8
Product Code: MBK/102.E**

List Price: $75.00

AMS Member Price: $60.00

MAA Member Price: $67.50

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#### Supplemental Materials

# It’s About Time: Elementary Mathematical Aspects of Relativity

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*Roger Cooke*

This book has three main goals. First, it
explores a selection of topics from the early period of the theory of
relativity, focusing on particular aspects that are interesting or
unusual. These include the twin paradox; relativistic mechanics and
its interaction with Maxwell's laws; the earliest triumphs of general
relativity relating to the orbit of Mercury and the deflection of
light passing near the sun; and the surprising bizarre metric of Kurt
Gödel, in which time travel is possible. Second, it provides an
exposition of the differential geometry needed to understand these
topics on a level that is intended to be accessible to those with just
two years of university-level mathematics as background. Third, it
reflects on the historical development of the subject and its
significance for our understanding of what reality is and how we can
know about the physical universe. The book also takes note of
historical prefigurations of relativity, such as Euler's 1744 result
that a particle moving on a surface and subject to no tangential
acceleration will move along a geodesic, and the work of Lorentz and
Poincaré on space-time coordinate transformations between two
observers in motion at constant relative velocity.

The book is aimed at advanced undergraduate mathematics, science,
and engineering majors (and, of course, at any interested person who
knows a little university-level mathematics). The reader is assumed to
know the rudiments of advanced calculus, a few techniques for solving
differential equations, some linear algebra, and basics of set theory
and groups.

#### Readership

Undergraduate and graduate students and general readers interested in mathematical aspects of relativity.

#### Reviews & Endorsements

This book is both pedagogical and humanistic in nature...in a historical setting, he gives a wealth of mathematical tools and many applications to astronomy, physics, and cosmology.

-- Alan S. McRae, Mathematical Reviews

Roger Cooke has successfully presented a wealth of fascinating ideas from the realm of physics, astronomy and cosmology while developing a range of powerful mathematical tools...This is an encyclopaedic discourse on relativity in a mathematical, philosophical and 'humanistic' setting...Being inexpert in this field myself, I was captivated by Roger Cooke's introduction to relativity. His book will appeal to a wide readership and it should provide the basis for a taught course at some suitable stage at the undergraduate level and beyond.

-- Peter Ruane, MAA Reviews

#### Table of Contents

# Table of Contents

## It's About Time: Elementary Mathematical Aspects of Relativity

- Cover Cover11
- Title page i2
- Contents iii4
- Preface vii8
- Part 1 . The Special Theory 122
- Chapter 1. Time, Space, and Space-Time 324
- 1. Simultaneity and Sequentiality 324
- 2. Synchronization in Newtonian Mechanics 627
- 3. An Asymmetry in Newtonian Mechanics: Electromagnetic Forces 1940
- 4. The Lorentz Transformation 2041
- 5. Contraction of Length and Time 2647
- 6. Composition of Parallel Velocities 3051
- 7. The Twin Paradox 3253
- 8. Relativistic Triangles 3556
- 9. Composition of Relativistic Velocities as a Binary Operation* 3960
- 10. Plane Trigonometry* 4566
- 11. The Lorentz Group* 4869
- 12. Closure of Lorentz Transformations under Composition* 5273
- 13. Rotational Motion and a Non-Euclidean Geometry* 5778
- 14. Problems 6485

- Chapter 2. Relativistic Mechanics 7192
- 1. The Kinematics of a Particle 7192
- 2. From Kinematics to Dynamics: Mass and Momentum 7596
- 3. Relativistic Force 79100
- 4. Work, Energy, and the Famous 𝐸=𝑚𝑐² 85106
- 5. Newtonian Potential Energy 87108
- 6. Hamilton’s Principle 92113
- 7. The Newtonian Lagrangian 93114
- 8. The Relativistic Lagrangian 96117
- 9. Angular Momentum and Torque 98119
- 10. Four-Vectors and Tensors* 101122
- 11. Problems 113134

- Chapter 3. Electromagnetic Theory* 115136

- Part 2 . The General Theory 125146
- Introduction to Part 2 127148
- Chapter 4. Precession and Deflection 129150
- 1. Gravitation as Curvature of Space 131152
- 2. First Analysis: Newtonian Orbits 132153
- 3. Second Analysis: Newton’s Law with Relativistic Force 137158
- 4. Third Analysis: Newtonian Orbits as Geodesics 140161
- 5. Fourth Analysis: General Relativity 155176
- 6. Einstein’s Law of Gravity 161182
- 7. Computation of the Relativistic Orbit 166187
- 8. The Speed of Light 177198
- 9. Deflection of Light Near the Sun 179200
- 10. Problems 183204

- Chapter 5. Concepts of Curvature, 1700–1850 189210
- Chapter 6. Concepts of Curvature, 1850–1950 225246
- Chapter 7. The Geometrization of Gravity 303324

- Part 3 . Historical and Philosophical Context 329350
- Chapter 8. Experiments, Chronology, Metaphysics 331352
- 1. Experimental Tests of General Relativity 332353
- 2. Chronology 336357
- 3. Space and Time 350371
- 4. The Reality of Physical Concepts 361382
- 5. The Harmony Between Mathematics and the Physical World 366387
- 6. Knowledge of Hypothetical Objects: An Example 376397
- 7. Knowledge of the Physical World 380401
- 8. A Few Words from the Discoverers 384405
- 9. Epilogue: The Reception of Relativity 386407

- Bibliography 389410
- Subject Index 393414
- Name Index 401422

- Back Cover Back Cover1426