On spacetime coordinates in special relativity

TL;DR

This paper derives key special relativity concepts such as time dilation, Doppler effect, length contraction, and Lorentz transformations using light clocks and spacetime diagrams, providing a clear coordinate-based interpretation.

Contribution

It offers a systematic derivation of relativistic spacetime relations from simple light clock setups, emphasizing coordinate interpretations and visualizations.

Findings

Derivation of time dilation and Doppler k-factor

Addition of velocities and length contraction formulas

Visualization of events via Minkowski diagrams

Abstract

Starting with two light clocks to derive time dilation expression, as many textbooks do, and then adding a third one, we work on relativistic spacetime coordinates relations for some simple events as emission, reflection and return of light pulses. Besides time dilation, we get, in the following order, Doppler k-factor, addition of velocities, length contraction, Lorentz Transformations and spacetime interval invariance. We also use Minkowski spacetime diagram to show how to interpret some few events in terms of spacetime coordinates in three different inertial frames.