Gyroscope precession in special and general relativity from basic principles

TL;DR

This paper explains gyroscope precession in special and general relativity using basic principles, simplifying traditional methods and extending analysis to curved spacetime and rotating frames.

Contribution

It introduces a simple law of rotation for an uncontracted grid, simplifying the analysis of gyroscope precession in relativity.

Findings

Derived a simple rotation law for uncontracted grids

Extended analysis to gyroscope precession in curved spacetime

Applied formalism to black hole orbit scenarios

Abstract

In special relativity a gyroscope that is suspended in a torque-free manner will precess as it is moved along a curved path relative to an inertial frame S. We explain this effect, which is known as Thomas precession, by considering a real grid that moves along with the gyroscope, and that by definition is not rotating as observed from its own momentary inertial rest frame. From the basic properties of the Lorentz transformation we deduce how the form and rotation of the grid (and hence the gyroscope) will evolve relative to S. As an intermediate step we consider how the grid would appear if it were not length contracted along the direction of motion. We show that the uncontracted grid obeys a simple law of rotation. This law simplifies the analysis of spin precession compared to more traditional approaches based on Fermi transport. We also consider gyroscope precession relative to an accelerated reference frame and show that there are extra precession effects that can be explained in a way analogous to the Thomas precession. Although fully relativistically correct, the entire analysis is carried out using three-vectors. By using the equivalence principle the formalism can also be applied to static spacetimes in general relativity. As an example, we calculate the precession of a gyroscope orbiting a static black hole. In an addendum the general reasoning is extended to include also rotating reference frames.