Translational Spacetime Symmetries in Gravitational Theories

TL;DR

This paper explores how to incorporate spacetime translations into gauge theories of gravity using affine bundles and Cartan connections, highlighting geometric structures and symmetry issues in Einstein-Cartan theory.

Contribution

It clarifies the geometric framework for including spacetime translations in gravitational gauge theories and discusses the implications of zero sections on gauge symmetry.

Findings

Affine connection splits into translational and homogeneous parts

Zero section reduces affine bundle to linear bundle

Alternative approaches to translational gauge symmetry are discussed

Abstract

How to include spacetime translations in fibre bundle gauge theories has been a subject of controversy, because spacetime symmetries are not internal symmetries of the bundle structure group. The standard method for including affine symmetry in differential geometry is to define a Cartan connection on an affine bundle over spacetime. This is equivalent to (1) defining an affine connection on the affine bundle, (2) defining a zero section on the associated affine vector bundle, and (3) using the affine connection and the zero section to define an "associated solder form," whose lift to a tensorial form on the frame bundle becomes the solder form. The zero section reduces the affine bundle to a linear bundle and splits the affine connection into translational and homogeneous parts; however it violates translational equivariance / gauge symmetry. This is the natural geometric framework for Einstein-Cartan theory as an affine theory of gravitation. The last section discusses some alternative approaches that claim to preserve translational gauge symmetry.