General reference frames and their associated space manifolds

TL;DR

This paper formalizes the concept of reference frames in general spacetime, defining a unique associated space manifold, with implications for quantum mechanics and applications to G"odel's universe.

Contribution

It introduces a rigorous definition of reference frames as equivalence classes of charts with fixed time coordinates, linking them to unique space manifolds in general spacetime.

Findings

Defines a formal, rigorous notion of reference frames in general spacetime.

Associates each reference frame with a unique space manifold.

Discusses applications to G"odel's universe.

Abstract

We propose a formal definition of a general reference frame in a general spacetime, as an equivalence class of charts. This formal definition corresponds with the notion of a reference frame as being a (fictitious) deformable body, but we assume, moreover, that the time coordinate is fixed. This is necessary for quantum mechanics, because the Hamiltonian operator depends on the choice of the time coordinate. Our definition allows us to associate rigorously with each reference frame F, a unique "space" (a three-dimensional differentiable manifold), which is the set of the world lines bound to F. This also is very useful for quantum mechanics. We briefly discuss the application of these concepts to G\"odel's universe.