## FANDOM

157 Pages

This entry is largely based on this paper.

Suppose we have an object going in a circle, such that its orbital angular momentum is $\vec{L}$. Suppose further that this object is rotating around itself with angular momentum $\vec{l}$. According to classical mechanics $\vec{l}$ should not change in time, but special relativistic corrections cause $\vec{l}$ to precess around $\vec{L}$. This is called Thomass precession.

The way to calculate the rate of precession is by considering the action of the centripetal force as a series of infinitesimal boosts in different directions. Classically, if a object was moving at velocity $\vec{\beta}_1$ and is then boosted by $\vec{\beta}_2$, its velocity (and hence direction) would be given by $\vec{\beta}_3 = \vec{\beta}_1 + \vec{\beta}_2$. In relativity, however, the application of two Lorentz boosts yields a slightly different velocity

$\vec{\beta}_4 = \frac{\gamma_1 \vec{\beta}_1 + \vec{\beta}_2 +\left(\gamma_1-1 \right ) \left(\vec{\beta}_1 \cdot \vec{\beta}_2 \right ) \vec{\beta}_1 / \beta_1^2}{\gamma_1 \left( 1 + \vec{\beta}_1 \cdot \vec{\beta}_2\right)}$

We will be interested in the limit where $\beta_2 \ll \beta_1 < 1$. The angle between the classical and relativistic directions is given by

$\theta \approx \tan \theta = \frac{|\vec{\beta}_3 \times \vec{\beta}_4|}{\vec{\beta}_3 \cdot \vec{\beta}_4} \approx \frac{\gamma_1 - 1}{\gamma_1} \frac{|\vec{\beta}_1\times\vec{\beta}_2|}{\beta_1^2} = \frac{\gamma_1}{\gamma_1 + 1} |\vec{\beta}_1\times\vec{\beta}_2|$

By dividing by the time interval in which the boost happens it is possible to get the precession rate, assuming $\theta = \Delta t \dot{\theta}$ and $\vec{\beta}_2 = \Delta t \vec{a}$ is the acceleration

$\dot{\theta} = \frac{\gamma_1}{\gamma_1+1} |\vec{\beta}_1 \times \vec{a}|$