 Geodesic equation in a uniformly rotating Minkowski chart

Set the inertial cylindrical angle to $\Phi =\varphi +\omega t$. The Minkowski metric becomes $d{s}^2=(c^2-{\omega }^2{r}^2)d{t}^2-2\omega r^{2}dtd\varphi -d{r}^2-r^{2}d{\varphi }^2-d{z}^2$. For a massive geodesic parametrized by proper time, its cyclic coordinates give conserved covariant four-momentum components $p_t=m[(c^2-{\omega }^2{r}^2)\dot{t}-\omega r^2\dot{\varphi }]$, $p_{\varphi }=-m{r}^2(\dot{\varphi }+\omega \dot{t})$ and $p_z=-m\dot{z}$. Their combination gives $\dot{t}=(p_t-\omega p_{\varphi })/(m{c}^2)$. The radial Euler-Lagrange equation gives the displayed coordinate acceleration, although the free worldline is straight in inertial coordinates. The conserved inertial energy is $p_t-\omega p_{\varphi }$, and the inertial axial angular momentum is $-p_{\varphi }$.