Spacetime and Euclidean Geometry

TL;DR

This paper demonstrates that in a two-dimensional spacetime diagram, proper time and length squares relate directly to Euclidean areas, providing geometric proofs of the Minkowski metric using Euclidean geometry principles.

Contribution

It introduces a geometric approach to derive the Minkowski line element solely from Euclidean geometry and the principle of relativity, offering pedagogical insights.

Findings

Proper time and length squared are proportional to Euclidean areas in spacetime diagrams.

Two geometric proofs of the spacetime Pythagoras theorem are provided.

The Minkowski metric is derived using Euclidean geometric methods.

Abstract

Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding causal domain. We use this relation to derive the Minkowski line element by two geometric proofs of the "spacetime Pythagoras theorem".