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The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions

Shing-Tung Yau and Steve Nadis
Basic Books
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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

This fascinating book may well have a similar impact to Stephen Hawking’s A Brief History of Time, but with a more restricted readership. This is because it more or less begins where Hawking left off, and because its emphasis is decidedly more mathematical. On the other hand, there is hardly a mathematical symbol on view, and many scientifically literate non-specialists will enjoy reading it.

The book is the tale of a work in progress, because string theory is some way off being a valid ‘theory of everything’, and the reasons for that come across clearly in its later chapters. And yet, despite the book’s central mathematical concept (Calabi-Yau manifolds) sharing the name of one of its authors, string theory is portrayed as being the joint product of many other researchers. Consequently, this book also forms an historical account of string theory, from the ideas of Kaluza and Klein in the 1920s to the present day.

The basic question is this: can there be a viable theory of quantum gravity that can unify quantum mechanics with general relativity in a self-consistent manner? Can such a theory be verified experimentally? Four main contenders for this role include the theories of loop quantum gravity, twistors, strings and M-theory (which unites five separate string theories into a single theory with 11 space-time dimensions). Yau and Nadis concern themselves with the last two of the four theories.

The Calabi-Yau version of string theory suggests that all particles and forces arise from vibrations of tiny strings that exist in a 10-dimensional universe. We can only perceive the 4 dimensions of space-time, because the other 6 dimensions are concealed in twisted Calabi-Yau manifolds. The manifolds are so small that they may always remain invisible, even though they have a proven mathematical existence. This 10-dimensional universe can be thought of as an infinite line representing spacetime, rather like a very thin garden hose. The Calabi-Yau manifolds themselves are hidden in cross-sections of this spacetime-hosepipe.

The existence of Calabi-Yau manifolds was established in 1976, and Yau was awarded the Fields medal for his proof of the underlying Calabi conjecture, which was formulated in the early 1950s, by Eugenio Calabi. It states that spaces satisfying certain topological conditions can be made into Riemannian manifolds whose Ricci curvature vanishes (they are “Ricci-flat”). They represent vacuum solutions to analogues of Einstein's equations for Riemannian manifolds with vanishing cosmological constant. Ricci-flat manifolds are therefore special cases of Einstein manifolds, where the cosmological constant may be non-zero.

Such is the arena of discussion in this book, whose story begins with an outline of geometrical thinking from the days of Plato to Einstein. It goes on to describe how physics became intertwined with geometry in a variety of historical contexts. So before readers are halfway through the book, they will be interpreting accounts of Calabi-Yau manifolds as belonging to SU(n) holonomy groups of Kähler manifolds of n complex dimensions. Readers are also required to imagine how compactification of these 10-dimensional manifolds can conceal 6 of the dimensions to leave only 4-dimensional space-time.

Despite joint authorship, the book is written in the first person singular, and in the words of Shing-Tung Yau. However, Steve Nadis has been instrumental in the process of rendering the material in a form that is amenable to non-specialists. In this respect, he has done a wonderful job, and I don’t see how the profound ramifications of this complex mathematical theory could have been further simplified.

This, of course, is consistent with Yau’s aim of making this book accessible to ‘educated lay people’. For this reason, the early chapters cover basic mathematical ideas, such as Pythagoras’ theorem, complex numbers, velocity-time graphs and visual 3d topology. But readers who need introductions to such basic stuff are likely to be overwhelmed by much of the subsequent mathematical narrative.

Generally, I found this introduction to string theory totally absorbing, and well worth re-reading. And yet, despite the high quality of the exposition, it is far from easy going. For one thing, concepts from particle physics, Calabi-Yau mathematics and cosmology are introduced thick and fast. Interwoven with this technical commentary is very frequent mention of many other mathematicians and physicists — not just with accounts of their contributions to string theory, but also with reference to their career progress. The final all-pervading strand consists of autobiographical observations on Yau’s personal and professional life. This is all very interesting, but sometimes at the expense of not being able to see the wood for the trees. Perhaps the addition of chapter summaries would ease one’s passage through this book.

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.

The table of contents is not available.