A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity

Teach yourself the mathematics of relativity – THE AMAZON BESTSELLER
To really understand Einstein's theory of relativity – one of the cornerstones of modern physics – you have to get to grips with the underlying mathematics. A Most Incomprehensible Thing is aimed at the general reader who is motivated to tackle that not insignificant challenge. With a user-friendly style, clear step-by-step mathematical derivations, many fully solved problems and numerous diagrams, this self-study guide provides an accessible introduction to a fascinating but complex subject.

For those with minimal mathematical background, the first chapter gives a crash course in foundation mathematics. The reader is then taken gently by the hand and guided through a wide range of fundamental topics, including Newtonian mechanics; the Lorentz transformations; the all important metric tensor g_μν; tensor calculus; the Einstein field equations; the Schwarzschild solution (which gives a good approximation of the spacetime of our Solar System); black holes, relativistic cosmology and gravitational waves.

• Special relativity helps explain a huge range of non-gravitational physical phenomena and has some strangely counter-intuitive consequences. These include time dilation, length contraction, the relativity of simultaneity, mass-energy equivalence and an absolute speed limit.
• General relativity, the leading theory of gravity, is at the heart of our understanding of cosmology and black holes. Summed up in the words of eminent theoretical physicist John Archibald Wheeler: “Matter tells space how to curve. Space tells matter how to move.”

“I must observe that the theory of relativity resembles a building consisting of two separate stories, the special theory and the general theory. The special theory, on which the general theory rests, applies to all physical phenomena with the exception of gravitation; the general theory provides the law of gravitation and its relations to the other forces of nature.” – Albert Einstein, 1919
Understand even the basics of Einstein's amazing theory and the world will never seem the same again.

Contents
Preface
Introduction
1 Foundation mathematics
2 Newtonian mechanics
3 Special relativity
4 Introducing the manifold
5 Scalars, vectors, one-forms and tensors
6 More on curvature
7 General relativity
8 The Newtonian limit
9 The Schwarzschild metric
10 Schwarzschild black holes
11 Cosmology
12 Gravitational waves
Appendix: The Riemann curvature tensor
Bibliography
Acknowledgements

January 2019 – this revised printing includes new Chapter 11 (cosmology) sections on a scale factor and a spacetime diagram for our universe.