A bit of everything

# Centrifugal force in general relativity

One of the easiest comparison between the influence of coordinate change in general relativity and classical mechanics is the case of acceleration. The case of constant linear acceleration (Rindler coordinates) is fairly well-known, so let's investigate what happens in the rotating case.

We're dealing here with flat spacetime, so our original coordinates are Minkowski space, in cylindrical coordinates to facilitate things later on :

$$ds^2 = -dt^2 + dr^2 + dz^2 + r^2 d\theta^2$$

The Christoffel symbols are then

\begin{eqnarray} {\Gamma^r}_{\theta\theta} &=& -r\\ {\Gamma^\theta}_{r\theta} &=& \frac{1}{r} \end{eqnarray}

And our geodesic equation is

\begin{eqnarray} \ddot{t} &=& 0\\ \ddot{r} - r \dot{\theta}^2 &=& 0\\ \ddot{z} &=& 0\\ \ddot{\theta} + \frac{2}{r} \dot{r} \dot{\theta} &=& 0 \end{eqnarray}

This is, up to the non-relativistic limit, about what we'd expect from Newton's law in cylindrical coordinates. Now let's consider a rotating frame of reference. Our new coordinates are all identical, except

$$\bar{\theta} = \theta - \omega(t) t$$

with the inverse transformation $\theta = \bar{\theta} + \omega(t) t$. Our derivatives are therefore

$$\partial_t \bar{\theta} = - (\omega(t) + \dot{\omega}(t) t),\ \partial_t \theta = \omega(t) + \dot{\omega}(t) t$$

For now, let's consider $\dot{\omega} = 0$. These are the Born coordinates, and our new metric becomes

$$ds^2 = -(1 - \omega^2 r^2) dt^2 + 2 \omega r^2 d\theta dt + dr^2 + dz^2 + r^2 d\theta^2$$

The classical equivalent here will be some system rotating with constant velocity. We should then expect two things in the classical limit : the centrifugal force, and the Coriolis force. Our Christoffel symbols become :

\begin{eqnarray} {\Gamma^r}_{\theta\theta} &=& - r\\ {\Gamma^r}_{tt} &=& - r \omega^2\\ {\Gamma^r}_{\theta t} &=& - \omega r\\ {\Gamma^\theta}_{r\theta} &=& \frac{1}{r}\\ {\Gamma^\theta}_{rt} &=& \frac{\omega}{r} \end{eqnarray}

We gain three extra terms related to the rotation. This leads us to the geodesic equations

\begin{eqnarray} \ddot{t} &=& 0\\ r - r \dot{\theta}^2 - 2 \omega r \dot{\theta} \dot{t} - r \omega^2 \dot{t}^2 &=& 0\\ \ddot{z} &=& 0\\ \ddot{\theta} + \frac{2}{r} \dot{r} \dot{\theta} + \frac{2\omega}{r} \dot{r} \dot{t} &=& 0\\ \end{eqnarray}

Now as usual for the non-relativistic limit, we have $t = \beta \tau + t_0$

Posted on 2020-02-11 10:23:39
Tags : general-relativity , physics