Einstein's Relativity and the Geometries of the Plane

Illustrations of the tie between geometry and physics

with 168 figures, 6 tables, and a glossary

by D.-E.Liebscher

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