Beyond Geometry: Classic Papers from Riemann to Einstein

by Peter Pesic (Editor)

Eight essays trace seminal ideas about the foundations of geometry that led to the development of Einstein's general theory of relativity. This is the only English-language collection of these important papers, some of which are extremely hard to find. Contributors include Helmholtz, Klein, Clifford, Poincaré, and Cartan.

Genres: Mathematics, History, Science, Nonfiction

224 pages, Paperback

First published December 15, 2006

Reviews

[Fraser Kinnear · Rating 3 out of 5]

Pretty difficult going, in spite of the intended wider audience. The opening essay, written by Reimann himself, sets the stage, while most of the subsequent essays are just commentary on Reimann, until you reach the five essays by Einstein at the end. The Helmholtz essay was probably the most approachable, and really fun as he delved into some cool "Flatland" type ideas of life in a finite closed universe.

Einstein's essays make the book, however, and the collection can be viewed in one way as laying the groundwork for the geometrical leap that Einstein relied upon for his theories of relativity. One core idea is that of the rigid body, and how it was first upended by Riemannian geometry and later proven not to exist in the material world by Einstein's Relativity theories. In fact, Einstein credits Reimann's discovery as empowering his own about SR & GR, as those tools were necessary for reconciling the invariant speed of light that comes out of Maxwell's equations with a geometry of spacetime.

Reimann himself makes some observations that come tantalizingly close to Einstein's great discovery. At the end of his essay, he speculates that forces (e.g., gravity) might have an impact on geometry in the real world:

... in a discrete manifold the principle of metric relations is already contained in the concept of the manifold, but in a continuous one it must come from something else. Therefore, either the reality underlying space must form a discrete manifold, or the basis for the metric relations must be sought outside it, in binding forces acting upon it.

Do we live in a discrete manifold or a continuous one? Reimann's earlier remarks opened the door to the possibility that our universe is discrete, and I wish this book was paired with some more contemporary essays, such as one about Planck length, which I take to be the funamental unit of measure for a discrete manifold-type universe.

I really wish I had read this book closer to when I had read Einstein's own book about GR & SR, as they pair nicely. However, there's still a lot I feel like I left on the table here. This book deserves another read one day.

[Silvio Curtis · Rating 5 out of 5]

Going into this book, I didn't have much idea of what exactly Riemann discovered in geometry. It turns out that he came up with the general formulation of geometry that includes both Euclidean and non-Euclidean geometry in any number of dimensions, and defined the curvature of space. (The curvature is zero in Euclidean geometry and positive or negative in non-Euclidean geometries). The essays in this book come from the late nineteenth century to the early twentieth. They trace how different people, including Hermann von Helmholtz, William Clifford, and Henri Poincaré, speculated on how the real world might be non-Euclidean at very small or large scales until Einstein founded the theory of relativity on non-Euclidean geometry. Riemann's lecture was pretty technical, and unfortunately I couldn't understand his original definition of curvature. Most of the other essays were more concerned with the physical and philosophical implications of the geometry than with proofs, and were clearer though still slow reading.