6 2 D O C . 1 0 R E V I E W O F W E Y L ’ S S P A C E – T I M E – M A T T E R

10.

Review of Hermann Weyl,

Space–Time–Matter: Lectures on General Relativity

[Einstein 1918h]

PUBLISHED 21 June 1918

IN: Die Naturwissenschaften 6 (1918): 373.

Weyl, Hermann, Raum–Zeit–Materie. Vorlesungen über allgemeine

Relativitätstheorie. Berlin, Julius Springer, 1918. VIII, 234 pp. Price, paper, 14.--

marks.

I am always tempted to read the individual parts of this book again, because every

page shows the amazingly steady hand of the master who has penetrated the subject

matter from the most diverse angles. I consider it a happy occasion that such a dis-

tinguished mathematician has taken care of this new field. He understood how to

combine mathematical rigor with graphic intuition. From this book, the physicist

can learn the foundations of geometry and the theory of invariants, and the mathe-

matician can learn those of electricity and the theory of gravitation.

The author begins with affine geometry built upon the concept of translation,

from which sprout the concepts of vector and tensor. By adding the basic concept

of metric (scalar product of two vectors) he obtains the Euclidean geometry. The

theory of tensors is auspiciously explained in mechanics and Maxwellian electro-

dynamics, where the latter finds a beautifully systematic presentation (first chap-

ter).

The second chapter is an introduction to the absolute differential calculus and

Riemannian geometry, resp. One especially sees there with amazement how the

most complicated becomes simple and self-evident under Weyl’s hand. The two

“non-Euclidean” geometries are represented first; then Gauss’s theory of surfaces

and Riemann’s generalization of it for multidimensional manifolds, a generaliza-

tion that constitutes the formal basis of the theory of general relativity. The advan-

ces of recent years, with the formal investigations about Riemann’s curvature

tensor due to Levi-Civita, Weyl, and Hessenberg are brilliantly brought into focus.

After these formal tools are completely mastered, the third chapter presents the

theory of special relativity and the fourth chapter the general theory; the special one

on 59 pages, the general one on 54 pages. It is here that Weyl not only demonstrates

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