A proposal of foundation of spacetime geometry

TL;DR

This paper develops a unified framework for various spacetime geometries, including Minkowski and Poincaré, based on affine bundles, local frames, and simultaneity concepts, providing a foundation for gauge theories of gravity.

Contribution

It introduces a comprehensive approach to spacetime geometry starting from affine bundles, incorporating local frames and simultaneity, and derives Minkowski and Poincaré geometries as gauge theory backgrounds.

Findings

Constructs a unified spacetime framework from affine bundles.

Derives Minkowski metric and invariance conditions.

Links spacetime geometries to gauge theories of gravity.

Abstract

A common approach to metric-affine, local Poincar\'e, special-relativistic and Galilei spacetime geometry is developed. Starting from an affine composite bundle, we introduce local reference frames and their evolution along worldlines and we study both, absolute and relative simultaneity postulates, giving rise to alternative concepts of spacetime. In particular, the construction of the Minkowski metric, and its required invariance, allows either to reorganize the original affine bundle as a metric-affine geometry with explicit Lorentz symmetry, or to restrict it to a Poincar\'e geometry, both of them constituting the background of a wide class of gauge theories of gravity.