The generally covariant meaning of space distances

TL;DR

This paper introduces a covariant, geometric framework for defining space distances in General Relativity and extended theories, applicable even near horizons and singularities, aligning with astronomical measurement protocols.

Contribution

It extends the definition of space distances to non-test-bodies with horizons and singularities, compatible with standard and modified gravity theories.

Findings

Defines a covariant geometric approach to space distances.

Extends distance definitions through horizons and singularities.

Applicable to both General Relativity and Weyl geometries.

Abstract

We propose a covariant and geometric framework to introduce space distances as they are used by astronomers. In particular, we extend the definition of space distances from the one used between events to non-test-bodies with horizons and singularities so that the definition extends through the horizons and it matches the protocol used to measure them. The definition we propose can be used in standard General Relativity although it extends directly to Weyl geometries to encompass a number of modified theories, extended theories in particular.