Spacetime

Abstract

During the years 1917-1924, Weyl devoted a great deal of his energy to the development of the mathematical and philosophical foundations of Relativity Theory. Some of his most important contributions are:

1. The extension of Levi-Civita’s concept of parallel transport by means of an intrinsic characterization of this notion that does not require an embedding into a flat, higher dimensional metric space. This intrinsic formulation led to the construction of significant portions of differential geometry and of dynamics on the basis of parallel transport as the fundamental notion besides and even independently of the metric ((Weyl 1918c, 4 and 5 edn) and (Weyl, 1918b, 1918d, 1919b, 1923c)).

2. The ingenious construction of a unified field theory of gravitation and electromagnetism in terms of a gauge-invariant geometry ((Weyl 1918c, 4th and 5th edition) and (Weyl, 1918b, 1919b, 1920c, 1921b, 1921e)).

3. The construction of generalized affine, projective and conformal geometries (Weyl, 1921f, 1922a, 1923c, 1929e and Robertson and Weyl (1929)), which led to subsequent developments in differential geometry such as the concept of connections on principal fiber bundles.

4. The clarification of the role and significance of invariance, symmetry and relativity principles and the clarification of the role of coordinates and the distinction between active and passive transformations (Weyl, 1927[sic]/1966, 1938b, 1939a, 1939b, 1949a, 1949b).

5. The deep group-theoretical results concerning the uniqueness of the Pythagorean form of the metric, which Weyl referred to as the ‘Raumproblem’. This constitutes an important and interesting chapter in the history of the Riemann-Helmholtz-Lie-Weyl-Cartan problem of space (Weyl, 1922a, 1922b, 1923c, 1923f).

6. The discovery of the possibility of the geodesic orcausal-inertialmethod for determining the spacetime metric by first distinguishing between two primitive substructures of the pseudo-Riemannian manifold structure, namely, theconformalstructure (representing thecausal fieldgoverning light propagation) and theprojectivestructure (representing theinertialorguiding fieldgoverning all free (fall) motions) and then by showing that these structures uniquely determine the pseudo-Riemannian metric up to a constant positive factor (Weyl, 1921f, 1923c).

7. A realist field ontology of geometric structure and the analysis of the concept of motion and the role of Mach’s Principle (Weyl, 1918c, 1918d, 1920b, 1921b, 1921c, 1922c, 1922a, 1923c, 1927[sic]/1966, 1931b, 1949a, 1950a).

8. The prediction and computation of the cosmological¹³red shift, based on Weyl’s preferred de Sitter model, six years before the effect was empirically established by Hubble ((Weyl 1918c, 5 edn) and (Weyl, 1923b, 1930, 1934c)).

9. The invention of the idea of wormholes in connection with his analysis of mass in terms of electromagnetic field energy (Weyl, 1921c, 1924f).