The Geometry of Time

Wiley VCH 2005

with 184 figures, 6 tables, and a glossary

by D.-E.Liebscher

Preface by Julian B.Barbour

As a boy of 12, Einstein encountered the wonder of Euclidean plane geometry in a little book that he called das heilige Geometrie-Büchlein (the holy geometry booklet). Something similar happened to Dierck Liebscher -- though admittedly not quite at the tender age of 12. As a physics student in Dresden, he heard lectures on projective geometry. The delight he got from these lectures has remained with him through his working life, and he has now passed on some of it in the present book.

It is a rather unusual book and all the better for it. One of the sad things about the hectic pace and competitiveness of modern scientific research is that truly beautiful discoveries and insights of earlier ages get completely forgotten. This is very true of projective geometry and the great synthesis achieved in the 19^th century by Cayley and Klein, who showed that the nine consistent geometries of the plane can all be derived from a common basis by projection. When Minkowski discovered that the most basic facts of Einstein's relativity can be expressed as the pseudo-Euclidean geometry of space and time, Klein hailed it as a triumph of his Erlangen program. For it showed that the trigonometry of pseudo-Euclidean space is the kinematics of relativity.

There is a very good reason why projective geometry is nevertheless not part of current physics courses. It can only be applied to spaces (or space-times) of constant curvature, and therefore fails in general relativity, in which the curvature in general varies from point to point. In such circumstances, one is forced (as in quantum mechanics) to use the analytical methods first introduced by Descartes. The beautiful synthetic methods of the ancient Greeks are not adequate. However, several of the most famous and important space-times that are solutions of Einstein's general relativity, notably Minkowski space and de Sitter (and anti de Sitter) space, do have constant curvature. One of the high points of Liebscher's book is the survey of all such spaces from the unified point of view of projective geometry. It yields insights lost to the analytic approach.

Perhaps the single most important justification for this book is the advent of computer graphics and the possibility of depicting on the page views of three-dimensional objects seen in perspective. Drawings and constructions may be distrusted as means to proofs, but they do give true insight that can be gained in no other way. The diagrams of this book constitute its real substance and yield totally new ways of approaching a great variety of topics in relativity and geometry. Especially interesting is the treatment of aberration, which is a vital part of relativity that gets far too little discussion in most textbooks.

This is not a textbook in any sense of the word. It is however a book that will instruct, deepen understanding, and open up new vistas. It will give delight to all readers prepared to make a modicum of effort. What more can one ask of a book?

South Newington, January 1999

Preface of the author

For the physicist, projective geometry is wonderland. I entered it once by the lectures of Rudolf Bereis in Dresden, and I was captured once and for all. When I found out that projective geometry yields a really exceptional path to the geometry of relativity, to all the curious behaviour of clocks and sticks that requires most of the time in any popularizing attempt, the excitement grew irrevocable. Projective geometry is the unifying point of view which renders many facts in relativity obvious because already familiar from Euclidean geometry. In the book ``Relativitätstheorie mit Zirkel und Lineal'' this has been shown extensively. Today, figures can be drawn and varied with ease by means of computers, and it is time to present comprehensively the the very wide possibilities of depicting the geometry of curved space, to include some relativistic cosmology, and to display something more of the connection between physics and geometry in general.

Famous philosophers, physicists and mathematicians wrote about the connection between physics and geometry; so did Kant, Helmholtz, Poincaré, Einstein and Hilbert. However, elementary illustrations of this fundamental question are rare. Here our book will enter. It considers the geometrical properties of space and time from the viewpoint of mechanics and cosmology. Contemplating just the boundary between geometry and physics, it will not aim at a fully detailed presentation of both disciplines. It will instead bring to focus that border region which is usually neglected in discourses on either fields. The reader is supposed to have simply the college acquaintance with geometry and mechanics, but also an eager mind for being led further into the world of both topics. By looking from either side the reader will recognize with surprise how much she or he can understand about the other side and how much each one depends on the other. Wherever possible, the text is held free of formulas. We believe the figures to allow the ``vide!'' of Euclides. We believe that the reader will not be unsensitive to the aesthetic side too. The formal aspects are offered in the appendices to the readers longing for a deeper understanding.

The book is not meant to give an axiomatic introduction either of mechanics nor of geometry. Instead, we shall try to mimic the path prom the elementary experiences to the deeper ones, and not only provide tha actual belief but also some of the intermediate steps. To speak with Einstein, we will first snuff with our nose on the ground before climbing the horse of generalization.

For the delight I found in writing the book, my gratitude shall cover a very wide span, which begins with the lessons in geometry I had the opportunity to listen at and ends with the equipment in the institute, including in between the innumerable occasions in which I enjoyed encouragement, discussion and immediate help. In particular, I want to thank E.Quaisser for important advice; H.-J.Treder for many intense discussions of the fundamentals, S.Liebscher for his skill in helping with every computer work and K.Liebscher for her invaluable support, infinite patience and extreme tolerance. It was joyful experience to see how the DAOS (Devil's Advocate Online Service) provided by S.Antoci added some italian spirit which the reader will meet at many places in the volume. The kind interest of J.B.Barbour saved me from troubles with the english language.

Some Figures Elastic billard collision a2 - b2 c2 Tiling of the pseudosphere Dr.Yi-nan Chin sent me the photo of a Zhi nan che

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Last updated: November 6, 1997