# Editorial note to: Erwin Schr\"odinger, Dirac electron in the gravitational field I

## TL;DR

This paper provides a historical and mathematical analysis of Schr"odinger's 1932 work on the Dirac equation in curved spacetime, highlighting its significance and the development of related concepts like the spin connection.

## Contribution

It offers the first detailed historical and mathematical commentary on Schr"odinger's 1932 paper, clarifying its role in the development of the Schr"odinger-Lichnerowicz formula and spin connection.

## Key findings

- Schr"odinger's 1932 paper first derived the Schr"odinger-Lichnerowicz formula.
- Historical analysis of the debate on spin connection development.
- Clarification of the conflict between Schr"odinger and other physicists' approaches.

## Abstract

Editorial Note with a mathematical and historical introduction to a 1932 paper by Erwin Schr\"odinger on the generalization of the Dirac equation to a curved spacetime -- to appear in the 'Golden Oldie' section of the Journal of General Relativity and Gravitation alongside an English translation of that paper. The Schr\"odinger paper is of interest as the first place that the well-known formula $g^{\mu\nu}\nabla_\mu\nabla_\nu + m^2 + \frac{R}{4}$ was obtained for the 'square' of the Dirac operator in curved spacetime. This formula is known by a number of names and we explain why we favour the name 'Schr\"odinger-Lichnerowicz formula'. We also aim to explain how the modern notion of `spin connection' emerged from a debate in the physics journals in the period 1929-1933. We discuss the key contributions of Weyl, Fock and Cartan and explain how and why they were partly in conflict with the approaches of Schr\"odinger and several other authors. We reference and comment on some previous historical accounts of this topic.