Lense Thirring Precession

This entry is largely based on this paper.

We begin by considering the planetary model for the atom. We have a heavy, immobile nucleus, around which circles a light electron. In this case the electron moves around the nucleus in Keplerian orbits. Now, suppose that the nucleus has charge $Q$, radius $R$ and rotates around itself at velocity $\Omega$. The rotation creates magnetic fields, which causes the electron orbit to precess. This effect is known as Larmor precession. The angular velocity at which the orbit precesses is of the same order as the cyclotron frequency $\omega_c = \frac{q B}{m c}$, where $B$ is the magnetic field, $q$ is the charge of the electron, $m$ is the mass of the electron and $c$ is the speed of light. The magnetic field from the rotating nucleus (assuming its shape is a sphere) is given by

$B \approx \frac{Q}{c r^3} \omega R^2$

where $r$ is the distance between the nucleus and the electron. The precession frequency is therefore

$\omega_l \approx \frac{q}{m c} \frac{Q}{c r^3} \omega R^2$

The Lense Thirring precession is the gravitational equivalent to the Larmor precession. In order convert the expression above to gravity, we need to make a few changes. First, we have to include the gravitation constant $G$. Second, we replace charge by mass $q \rightarrow m$ and $Q \rightarrow M$. By applying these changes to $\omega_l$ we obtain the frequency for the Lense Thirring precession

$\omega_{lt} \approx \frac{G}{c} \frac{M \omega R^2}{c r^3} = \frac{G}{c^2 r^3} L$

where $L \approx M R^2 \omega$ is the angular momentum of the spin.