I understand that the coordinate transform for the Godel metric would look like this, ##x^{\alpha}=(t, x, y, z)=(t, r\cos{\phi}, r\sin{\phi}, z)## for cartesian and cylindrical coordinates. The cylindrical metric itself is given by ##g_{t,t}=c^2##,##g_{r,r}=1/(1+(r/2a)^2)##, ##g_{\phi,\phi}=...