Gyroscope precession in special and general relativity from basic principles

Abstract

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.

A covariant formalism of spin precession with respect to a reference congruence

“…This is a perfect match with the result of the intuitive derivation performed in [2]. Analogously we may study (44) for the particular case of vanishing shear, thus considering a rigid congruence.…”

“…As an example we consider motion with fixed speed v along a circle in the xy-plane, with an angular frequency ω. Letting the groscope start at t = 0 at the positive x-axis, we get a set of coupled differential equations dS x dt = γ 2 v 2 ω sin(ωt) (S x cos(ωt) + S y sin(ωt)) (2) dS y dt = − γ 2 v 2 ω cos(ωt) (S x cos(ωt) + S y sin(ωt)) (3)…”

Visualization of Thomas–Wigner Rotations

“…"SymPy" script vtwr5bst.py calculates these matrices and their products in terms of T 12 and T 23 and inserts the result into Equation (17). The time component of Equation (17) We exclude the trivial solution γ = 1 and restrict ourselves to real values of T 12 and T 23 ; Equation (A13) then leads to Equation (22), a second order polynomial with respect to T 23 . The two solutions are given in Equation (23).…”

“…In the present paper, following Jonsson [22], an alternative route to visualize Thomas-Wigner rotations using active or "physical" boosts is attempted. G is accelerated starting from zero velocity in frame {1}, which is denoted by "laboratory frame" in the following.…”

Gyrations: The Missing Link Between Classical Mechanics with Its Underlying Euclidean Geometry and Relativistic Mechanics with Its Underlying Hyperbolic Geometry

“…We call it a gyrodistance function in order to emphasize the analogies it shares with its Euclidean counterpart, the distance function u − v in R n . Among these analogies is the gyrotriangle inequality according to which (15) u⊕v ≤ u ⊕ v for all u, v ∈ R n c . For this and other analogies that distance and gyrodistance functions share see [46,48].…”