The fate of Hamilton's Hodograph in Special and General Relativity

TL;DR

This paper explores the properties and behaviors of hodographs in various physical contexts, including non-relativistic, relativistic, and curved spacetime scenarios, revealing new geometric insights and explicit solutions.

Contribution

It introduces a unified geometric framework for hodographs across classical, relativistic, and curved spacetime physics, including explicit solutions and new properties.

Findings

Relativistic hodographs precess due to effective inverse cube forces.

Explicit hodograph solutions for photon orbits near black holes.

Non-relativistic hyperbolic and spherical space hodographs are circles.

Abstract

The hodograph of a non-relativistic particle motion in Euclidean space is the curve described by its momentum vector. For a general central orbit problem the hodograph is the inverse of the pedal curve of the orbit, (i.e. its polar reciprocal), rotated through a right angle. Hamilton showed that for the Kepler/Coulomb problem, the hodograph is a circle whose centre is in the direction of a conserved eccentricity vector. The addition of an inverse cube law force induces the eccentricity vector to precess and with it the hodograph. The same effect is produced by a cosmic string. If one takes the relativistic momentum to define the hodograph, then for the Sommerfeld (i.e. the special relativistic Kepler/Coulomb problem) there is an effective inverse cube force which causes the hodograph to precess. If one uses Schwarzschild coordinates one may also define a a hodograph for timelike or null geodesics moving around a black hole. Iheir pedal equations are given. In special cases the hodograph may be found explicitly. For example the orbit of a photon which starts from the past singularity, grazes the horizon and returns to future singularity is a cardioid, its pedal equation is Cayley's sextic the inverse of which is Tschirhausen's cubic. It is also shown that that provided one uses Beltrami coordinates, the hodograph for the non-relativistic Kepler problem on hyperbolic space is also a circle. An analogous result holds for the the round 3-sphere. In an appendix the hodograph of a particle freely moving on a group manifold equipped with a left-invariant metric is defined.