The pedagogical value of the four-dimensional picture I: Relativistic mechanics of point particles

TL;DR

This paper explores the geometric nature of world lines in relativistic mechanics and demonstrates that Newton's second law can be seamlessly integrated into Minkowski spacetime without modification.

Contribution

It clarifies the geometric interpretation of particle trajectories and shows that relativistic dynamics do not require altering Newton's law, only embedding it in four-dimensional spacetime.

Findings

World lines are simple geometric objects in relativistic mechanics.

Newton's second law remains valid in relativistic context without modification.

Relativistic dynamics can be derived from classical laws within Minkowski spacetime.

Abstract

We outline two subjects of relativistic mechanics: (i) the set of allowable world lines, and (ii) the origin of the relativistic law of dynamics governing point particles. We show that: (i) allowable world lines in the classical theory of particles and fields are quite simple geometric objects as opposed to their associated three-dimensional trajectories, and (ii) Newton's second law requires neither modification nor generalization, it should be only smoothly embedded in the four-dimensional geometry of Minkowski spacetime to yield the dynamical law for relativistic particles.