*The Biggest Ideas in the Universe: space, time, and motion*, by Sean Carroll, Dutton Books

Most popular science books are written to reach the largest audience possible, which comes with certain sacrifices. The assumption is that many readers might be deterred by technical topics and language, especially by equations that require higher mathematics. In physics one can therefore usually distinguish textbooks from popular physics books by flicking through the pages and checking for symbols.

The *Biggest Ideas in the Universe: space, time, and motion*, the first in a three-part series by Sean Carroll, goes against this trend. Written for “…people who have no mathematical experience than high-school algebra, but are willing to look at an equation and think about what it means”, there is no point in the book at which things are muddied because the maths becomes too advanced.

**Concepts and theories**

The first part of the book covers nine topics including conservation, space–time, geometry, gravity and black holes. Carroll spends the first few chapters introducing the reader to the thought process of a theoretical physicist: how to develop a sense for symmetries, the conservation of charges and expansions in small parameters. It also gives readers a fast introduction to calculus using geometric arguments to define derivatives and integrals. By the end of the third chapter, the concepts of differential equations, phase space and the principle of least action have been introduced.

The centre part of the book focusses on geometry. A discussion of the meaning of space and time in physics is followed by the introduction of Minkowski spacetime, with considerable effort given to the philosophical meaning of these concepts. The third part is the most technical. It covers differential geometry, a beautiful derivation of Einstein’s equation of general relativity and the final chapter uses the Schwarzschild solution to discuss black holes.

It is a welcome development that publishers and authors such as Carroll are confident that books like this will find a sizeable readership (another good, recent example of advanced popular physics texts is Leonard Susskind’s “A Theoretical Minimum” series). Many topics in physics can only be fully appreciated if the equations are explained and if chapters go beyond the limitations of typical popular science books. Carroll’s writing style and the structure of the book help to make this case: all concepts are carefully introduced and even though the book is very dense and covers a lot of material, everything is interconnected and readers won’t feel lost while reading. Regular reference to the historical steps in discovering theories and concepts loosen up the text. Two examples are the correspondence between Leibniz and Clarke about the nature of space and the interesting discussion of Einstein and Hilbert’s different approaches to general relativity. The whole series of books, of which two of the three parts will be published soon, is accompanied by recorded lectures that are freely available online and present the topic of every chapter, along with answers to questions on these topics.

It is difficult to find any weaknesses in this book. Figures are often labelled with symbols that readers not used to physics notation can find in the text, so more text in the figures would make them even more accessible. Strangely, the section introducing entropy is not supported by equations and, given the technical detail of all other parts of the book, Carroll could have taken advantage of the mathematical groundwork of the previous chapters here.

I want to emphasise that every topic discussed in *The Biggest Ideas in the Universe *is well established physics. No flashy but speculative theories or unbalanced focus on science-fiction ideas, which are often used to attract readers to theoretical physics, appear. It stands apart from similar titles by offering insights that can only be obtained if the underlying equations are explained and not just mentioned.

Anyone who is interested in fundamental physics is encouraged to read this book, especially young people interested in studying physics because they will get an excellent idea of the type of physical arguments they will encounter at university. Those who think their mathematical background isn’t sufficient will likely learn many new things, even though the later chapters are quite technical. And if you are at the other end of the spectrum, such as a working physicist, you will find the philosophical discussions of familiar concepts and the illuminating arguments included to elicit physical intuition most useful.