Local observers in stationary axisymmetric dust spacetimes

TL;DR

This paper develops a locally inertial reference system for observers in stationary axisymmetric dust spacetimes, enabling analysis of galactic kinematics within general relativity through photon frequency shifts and velocity reconstructions.

Contribution

It introduces a novel framework for defining local observers and reference systems in galactic models, facilitating relativistic kinematic studies.

Findings

Constructed a locally inertial reference system for dust spacetimes.

Analyzed photon frequency shifts between dust geodesics.

Outlined a method to reconstruct relative velocities relativistically.

Abstract

In this work, we construct a locally inertial reference system adapted to a geodesic observer in stationary, axisymmetric dust solutions of the Einstein equations employed as effective models of a portion of a galactic disc. To ensure a consistent spatial orientation among different local observers, we also introduce the radially locked reference system, in which one spatial axis is aligned with the radial direction defined by null geodesics passing through the galactic center. Within this framework, we analyze how the dust configuration is described by such observers by computing the frequency shift of photons exchanged between pairs of dust geodesics. Building on this construction, we outline a procedure to reconstruct spectroscopic and astrometric relative velocities with respect to locally inertial observers, providing a coherent foundation for the study of galactic kinematics in a fully general relativistic context.