Mathematics holds the key to a unified theory of the Universe.
Hiding in the Mirror: The Mysterious Allure of Extra Dimensions, from Plato to String Theory and Beyond
- Lawrence M. Krauss
The search for the fundamental physical laws that govern the Universe has fascinated and driven humans for centuries. Its modern form was essentially launched by Isaac Newton, building on the pioneering work of his predecessors, notably Johannes Kepler and Galileo. From the publication of Newton's Principia onwards, it has been a remarkable story, which has delved into every realm of the physical world from the subatomic scale to that of the cosmos and the Big Bang. The theoretical framework that supported this great enterprise has, following the path firmly established by Newton, been based on mathematics. At every major step, physics has required, and frequently stimulated, the introduction of new mathematical tools and concepts. Our present understanding of the laws of physics, with their extreme precision and universality, is only possible in mathematical terms.
This mathematical take-over of physics has its dangers, as it could tempt us into realms of thought which embody mathematical perfection but might be far removed, or even alien to, physical reality. Even at these dizzying heights we must ponder the same deep philosophical questions that troubled both Plato and Immanuel Kant. What is reality? Does it lie in our mind, expressed in mathematical formulae, or is it ‘out there’. The recent developments in modern physics that go by the deceptively simple name of ‘string theory’ bring these age-old questions back to the fore, and are the focus of Hiding in the Mirror, a new book by Lawrence Krauss.
Most popular-science books are written by enthusiastic protagonists who seek to convey their excitement to the general public. This book is refreshingly different. On the big questions, Krauss remains a sceptic, a hard-nosed physicist questioning the mountain of mathematical theory that has yet to produce any experimental evidence. But this is no debunking exercise; Krauss makes a serious attempt to bring the reader to the very frontier of modern physics. He describes the intricate theoretical constructs that have been erected and he acknowledges that, at present, there is no alternative on the table: “it's the only game in town”.
One of the book's main themes is indicated by its subtitle, which refers to “the mysterious allure of extra dimensions”. This guides the reader through the ever-increasing complexity of physical theory. As Krauss points out, the mystery of extra dimensions exercised the imagination long before it entered serious physics. We are enchanted by the account of Edwin Abbott's Flatland, a nineteenth-century classic that combines a scientific and philosophical aim with social satire worthy of Jonathan Swift. The erudite but volatile British mathematician James Joseph Sylvester tried to refer to extra dimensions as ‘inconceivable’ (by analogy with ‘imaginary’ numbers). Fortunately for Einstein and subsequent physicists, the term never caught on.
Krauss gives pride of place to Einstein's revolutionary ideas that showed, first in his special theory of relativity and even more convincingly in his subsequent general theory, that space and time form an indissoluble four-dimensional continuum. This was undoubtedly a landmark in the history of human thought and fully justifies Einstein's iconic status. As Krauss explains, Einstein's general theory, which supplanted Newton's theory of gravitation, was reached by a process of pure thought — not by the pressure of unexplained experimental data. Einstein was motivated by aesthetic considerations, an impressive example of the power of beauty to act as a guiding light. On this point, Krauss quotes the mathematician Hermann Weyl: “My work always tried to unite the true with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.” Krauss comments that mathematicians, poets, writers and artists can all choose beauty over truth, but scientists do not have this luxury.
This is the final word in the book, but Krauss had already mentioned an episode involving Weyl in which beauty eventually triumphed. After Einstein had explained gravity in terms of the curvature of four-dimensional space-time, Weyl attempted to explain electromagnetism in similar terms. Einstein pointed out a fatal flaw in Weyl's explanation (not in the mathematics, but in its physics). Remarkably, Weyl did not withdraw the paper, and the paper was published together with Einstein's objection as an appendix. Clearly the beauty of the mathematics exercised its own appeal.
A few years later, after the appearance of quantum mechanics, a new physical interpretation of Weyl's mathematics was possible; this was the beginning of modern ‘gauge theory’, the basis of elementary particle physics. To a mathematician this seems a remarkable case of beauty winning through in the end, justifying Weyl's preference. After all, beauty is subjective, it is in the eye of the beholder, and we can be certain about ourselves. Truth is much more elusive and can change as new facts or ideas emerge.
Krauss's book is well written for a general audience and puts the scientific advances into a historical and philosophical context while keeping the technicalities under control. After a rapid overview of the past it focuses on the current aim of combining the two fundamental theories of twentieth-century physics: Einstein's general theory of relativity, and quantum mechanics, which between them deal with the very large and the very small. The need to unify these two theories is entirely aesthetic; there seems to be little need from the point of view of the experimentalist.
Over the past few decades, string theory has emerged as a serious contender to be such a unified theory. It involves extra dimensions galore: not just Einstein's four, or the five that also incorporates electromagnetism, but a total of 10 or 11. The extra dimensions are viewed as very small and curled up, so that, for most purposes, we are not aware of them. They only manifest themselves at very high energies, of the kind encountered in particle accelerators. Moreover, these extra dimensions are constrained by very precise symmetry requirements. The upshot of this is that string theorists can exhibit plausible models of a unified Universe, but unfortunately they cannot explain why we inhabit a particular one.
The mathematics involved in string theory is quite remarkable by any standards. In subtlety and sophistication it vastly exceeds previous uses of mathematics in physical theories. Almost every part of contemporary mathematics is involved somewhere in the story. Even more remarkable is that string theory has led to a whole host of amazing results in mathematics in areas that seem far removed from physics. To many this indicates that string theory must be on the right track. But Krauss is not a mathematician, so perhaps he is unaware of all this mathematical success, or maybe he discounts it as irrelevant. Time will tell.
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Atiyah, M. Pulling the strings. Nature 438, 1081–1082 (2005). https://doi.org/10.1038/4381081a