Motion of a gyroscope on a closed timelike curve

TL;DR

This paper studies the behavior of gyroscopes moving along closed timelike curves in spacetime, revealing conditions for periodic spin-vector motion and implications for the consistency of such spacetimes.

Contribution

It demonstrates the existence of at least one periodic spin-vector on CTCs and analyzes the conditions under which gyroscopic motion is generically periodic or not.

Findings

Every CTC admits at least one T-periodic spin-vector.

Gyroscopes are either all T-periodic or none are, depending on the spacetime.

Periodicity of gyroscopic motion depends on the spacetime structure and CTC properties.

Abstract

We consider the motion of a gyroscope on a closed timelike curve (CTC). A gyroscope is identified with a unit-length spacelike vector - a spin-vector - orthogonal to the tangent to the CTC, and satisfying the equations of Fermi-Walker transport along the curve. We investigate the consequences of the periodicity of the coefficients of the transport equations, which arise from the periodicty of the CTC, which is assumed to be piecewise $C^2$. We show that every CTC with period $T$ admits at least one $T-$periodic spin-vector. Further, either every other spin-vector is $T-$periodic, or no others are. It follows that gyroscopes carried by CTCs are either all $T-$periodic, or are generically not $T-$periodic. We consider examples of spacetimes admitting CTCs, and address the question of whether $T-$periodicity of gyroscopic motion occurs generically or only on a negligible set for these CTCs. We discuss these results from the perspective of principles of consistency in spacetimes admitting CTCs.