Projective Relativity: Present Status and Outlook

TL;DR

This paper critically examines the development and misuse of projective relativity, especially in higher-dimensional Kaluza--Klein theories, proposing a new geometric interpretation and discussing implications for quantum field theory.

Contribution

It offers a new perspective on projective relativity by reinterpreting key conditions as geometric objects, challenging traditional field equation views and suggesting alternative directions.

Findings

Projective relativity was often misused in its own terms.

A reinterpretation of R5_{μν}=0 as a null-quadric instead of a vacuum Einstein equation.

Discussion of projective aspects in quantum (field) theory.

Abstract

We give a critical analysis of projective relativity theory. Examining Kaluza's own intention and the following development by Klein, Jordan, Pauli, Thiry, Ludwig and others, we conclude that projective relativity was abused in its own terms. Much more in the case of newer higher dimensional Kaluza--Klein theories with non-Abelian gauge groups. Reviewing the projective formulation of the Jordan isomorphy theorem yields some hints how one can proceed in a different direction. We can interpret the condition R5_{\mu\nu}=0 not as a field equation in a 5-dimensional Riemannian space, e.g. as vacuum Einstein-Hilbert equation, but can (or should) interpret it as a geometrical object, a null-quadric. Projective aspects of quantum (field) theory are discussed under this viewpoint.