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It's About Time: Elementary Mathematical Aspects of Relativity

Roger Cooke
American Mathematical Society
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
P. N. Ruane
, on

Roger Cooke issues several disclaimers regarding his authorship of this inspirational book. He describes himself as a non-specialist in relativity, and says that his book is ‘neither a technical introduction to relativity, nor a systematic history of its development’. He says that his specific goal is ‘to simplify selected parts of relativity, to make them accessible to a person who has studied only the three basic semesters of calculus and two semesters of linear algebra’. From physics, knowledge of the basic principles of mechanics, electricity and magnetism is required of his readership (he cites middle undergraduates).

But 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. Readers with little or no knowledge of differential geometry may struggle to survive the compact introduction provided in this book. Within a hundred pages, this goes from the notion of curvature of a plane curve to the use of Lie brackets, parallel transport, Riemann curvature and k-forms on a manifold. It culminates with a definition of scalar curvature on a manifold as a contraction of a Ricci tensor, which is said to be one of the devices enabling the relativistic reformulation of mechanics.

This is an encyclopaedic discourse on relativity in a mathematical, philosophical and ‘humanistic’ setting. It consists of three volumes, the first of which is the hard copy now under review. Taking the form of an online resource, Volume 2 contains six appendices of mathematical background material. Volume 3 is available on the author’s website and is a compilation of the Mathematica notebooks that have been used to ease the labour of mathematical computation in the previous two volumes.

Although this book isn’t a history of relativity, its contents are introduced in an historical setting. Indeed, the name index lists around 160 contributors to the field of cosmological thinking — ranging from Archimedes, Aristotle and Ptolemy to later creative giants such as Kepler, Newton, Poincaré and Einstein himself. Among other mathematical luminaries there is discourse on the relevant contributions of Euler, Gauss, Riemann and Gödel. Chapter 5 identifies four phases in the emergence of curvature as an idea central to differential geometry. Euler and Gauss represent the period 1700–1850, while Riemann and Ricci are the main players in years 1850–1950.

For special relativity, discussion of simultaneity and sequentiality leads to the derivation of the Lorentz transformation. The underlying mathematics is simple enough, but the challenge is to visualize the relativistic concepts to which it is applied. Such concepts include the contraction of space and time, the resolution of the twin paradox, and relativistic triangles. Material built from these ideas includes ‘relativistic velocities as a binary operation’, plane trigonometry, the Lorentz group and non-Euclidean geometry. A novel feature of this book is the demonstration that the trigonometry of the hyperbolic plane follows from the Lorentz transformation (rather than the other way round).

The chapter on relativistic mechanics compares many classical ideas (Newtonian and Lagrangian) with their relativistic counterparts. For instance, the Newtonian concept of work (expressed as a line integral) is compared with the work done by a relativistic force expressed in terms of the speed of light (by the equation \(W=(m_{v_1}-m_{v_0})c^2\)). Chapter 3 provides a short account of relevant aspects of electromagnetic theory: it investigates the transformation of electric and magnetic fields, and invokes use of the Lorentz transformation to reveal the interaction of special relativity with Maxwell’s equations.

General relativity is the central theme of the book, and its evolution is encapsulated by the journey from Newton’s \(\mathbf{F}=m\mathbf{r}''\) to the Einstein field equation \(G_{\mu\nu}+\Lambda g_{\mu\nu}=(8\pi G/c^4)T_{\mu\nu}\). But readers aren’t thrown in at the theoretical deep end, because preliminary discussion of many important ideas, such as Einstein’s law of gravity, takes place in the specific context of discoveries that encouraged the development of the general theory. Chapter 4 fulfils this role via discussion of the enigma concerning the precession of Mercury and the relativistic deflection of light passing near to a star.

Having laid the physical and mathematical foundations in previous chapters, Roger Cooke brings things together in the penultimate chapter (Geometrization of Gravity). Two of the questions lying at the heart of this summary are:

  1. With respect to the principle that mechanical laws should be expressed as tensor equations involving at most 2nd order partial derivatives of the metric coefficients, Einstein was led to the Ricci tensor. But why not apply the Laplace-Beltrami operator to the those coefficients instead?
  2. How are the equations of motion to be determined in other aspects of relativistic mechanics? What replaces the famous \(\mathbf{F}=m\mathbf{a}\) when the gravitational field is produced by a continuous distribution of matter?

To consolidate the book’s ‘humanistic aims’, the very last chapter places relativity more firmly in a ‘Historical and Philosophical Context’. It includes a lengthy chronology of the major contributors to this discipline, and reflects upon our understanding of what reality is and how we can know about the physical universe.

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.


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Peter Ruane is retired from the field of mathematics education, which involved the training of primary and secondary school teachers. His postgraduate study included of algebraic topology and differential geometry, with applications to superconductivity.

See the table of contents in the publisher's webpage.