# The hodograph method for relativistic Coulomb systems

## TL;DR

This paper introduces a velocity-space method for analyzing relativistic Coulomb systems, leveraging the linearity of velocity equations in hyperbolic geometry to simplify understanding of particle trajectories.

## Contribution

It presents a novel approach using velocity space geometry to analyze relativistic Coulomb systems, offering a simpler and more elegant alternative to traditional methods.

## Key findings

- Velocity space is a 3-D hyperboloid embedded in 3+1 space.
- Orbits in velocity space are classified and illustrated.
- The method simplifies the analysis of relativistic Coulomb trajectories.

## Abstract

Relativistic Coulomb systems are studied in velocity space, prompted by the fact that the study of Newtonian Kepler/Coulomb systems in velocity space provides a method much simpler (and more elegant) than the familiar analytic solutions in ordinary space. The key for the simplicity and elegance of the velocity-space method is the linearity of the velocity equation, which is a unique feature of $1/r$ interactions for Newtonian and relativistic systems alike, allowing relatively simple analytic discussion with coherent geometrical interpretations. Relativistic velocity space is a 3-D hyperboloid ($H^3$) embedded in a 3+1 pseudo-Euclidean space. The orbits in velocity space for the various types of possible trajectories are discussed, accompanied with illustrations.