On the differential geometry of curves in Minkowski space

TL;DR

This paper explores the differential geometry of curves in Minkowski space, establishing fundamental equations, proving key theorems, and applying these concepts to describe particle motion under electromagnetic fields.

Contribution

It introduces Minkowskian versions of classical curve theorems, derives the Serret-Frenet equations in Minkowski space, and connects curvature and torsion to electromagnetic field effects.

Findings

Established Serret-Frenet equations in Minkowski space

Proved Minkowskian versions of classical curve theorems

Linked curvature and torsion to electromagnetic field properties

Abstract

We discuss some aspects of the differential geometry of curves in Minkowski space. We establish the Serret-Frenet equations in Minkowski space and use them to give a very simple proof of the fundamental theorem of curves in Minkowski space. We also state and prove two other theorems which represent Minkowskian versions of a very known theorem of the differential geometry of curves in tridimensional Euclidean space. We discuss the general solution for torsionless paths in Minkowki space. We then apply the four-dimensional Serret-Frenet equations to describe the motion of a charged test particle in a constant and uniform electromagnetic field and show how the curvature and the torsions of the four-dimensional path of the particle contain information on the electromagnetic field acting on the particle.