Einstein Against Singularities: Analysis versus Geometry

TL;DR

This paper examines Einstein's perspective on singularities in spacetime, contrasting his analytic approach with the later geometrical methods used by relativists, highlighting the philosophical and mathematical differences.

Contribution

It offers a historical and conceptual analysis of Einstein's non-geometric methods and their role in the development of singularity theory in general relativity.

Findings

Einstein prioritized analytic methods over geometric structures.

Einstein aimed to eliminate singularities to reduce arbitrariness in physical theories.

He was willing to accept temporary singularities if they minimized arbitrariness.

Abstract

Einstein identified singularities in spacetimes, such as at the Schwarzschild radius, where later relativists only find a coordinate system assigning multiple values to a single spacetime event. These differing judgments derive from differences in mathematical methods. Later relativists employ geometrical structures to correct anomalies in the coordinate systems used in analytic expressions. Einstein took the analytic expressions to be primary and the geometrical structures as mere heuristics that could be overruled if physical assumptions required it. Einstein's non-geometric methods had a firm base in the history of mathematical methods. They continued the non-geometric orientation of Christoffel, Ricci and LeviCivita. Einstein's insistence that singularities must be eliminated marked a departure from earlier tolerance of singularities. It was founded upon his longterm project of eliminating arbitrariness from fundamental physical theories. However, Einstein was willing to theorize with singularities only temporarily if they were the least arbitrary approach then available.