The Hole Argument for Covariant Theories

TL;DR

This paper reformulates Einstein's hole argument using category theory and manifold invariance, providing a coordinate-free perspective and discussing implications for quantum gravity.

Contribution

It introduces a coordinate-free, categorical approach to the hole argument and explores its extensions to sets and relations relevant to quantum gravity.

Findings

Coordinate-free formulation of the hole argument

Avoidance of the hole argument using natural objects

Implications for quantum gravity theories

Abstract

The hole argument was developed by Einstein in 1913 while he was searching for a relativistic theory of gravitation. Einstein used the language of coordinate systems and coordinate invariance, rather than the language of manifolds and diffeomorphism invariance. He formulated the hole argument against covariant field equations and later found a way to avoid it using coordinate language. In this paper we shall use the invariant language of categories, manifolds and natural objects to give a coordinate-free description of the hole argument and a way of avoiding it. Finally we shall point out some important implications of further extensions of the hole argument to sets and relations for the problem of quantum gravity.