On the nonuniqueness of free motion of the fundamental relativistic rotator

TL;DR

This paper demonstrates that the fundamental relativistic rotator's free motion is non-unique and depends on an arbitrary function, revealing a fundamental inconsistency with classical determinism and classifying it as a pathological dynamical system.

Contribution

It provides a covariant solution to the rotator's equations and shows the non-uniqueness of its motion, highlighting its pathological nature as a dynamical system.

Findings

Motion depends on an arbitrary function of time.

The system violates the condition for unique Cauchy problem solutions.

Fundamental rotator's non-uniqueness contradicts classical determinism.

Abstract

Consider a class of relativistic rotators described by position and a single null direction. Such a rotator is called fundamental if both its Casimir invariants are intrinsic dimensional parameters independent of arbitrary constants of motion. As shown by Staruszkiewicz, only one rotator with this property exists (its partner with similar property can be excluded on physical grounds). We obtain a general solution to equations of free motion of the fundamental relativistic rotator in a covariant manner. Surprisingly, this motion is not entirely determined by initial conditions but depends on one arbitrary function of time, which specifies rotation of the null direction in the centre of momentum frame. This arbitrariness is in manifest contradiction with classical determinism. In this sense the isolated fundamental relativistic rotator is pathological as a dynamical system. To understand why this is so, we study the necessary condition for the uniqueness of the related Cauchy problem. It turns out that the fundamental relativistic rotator (together with its partner) can be uniquely characterized by violating this condition in the considered class of rotators.