I was wondering how can one change from Cartesian coordinate system to some other like polar coordinates or spherical coordinates, in the context of special relativity. For example, with the four-velocity, $$V^{\mu}=\frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau},\tag{1}$$ where $\mu=0,1,2,3$, $x^0=ct$, $x^1=x$, $x^2=y$ and $x^3=z$, if I want to change to cylindrical coordinates, can I simply use the transformations $x=r\cos (\theta)$, $y=r\sin(\theta)$, $z=z$, $t=t$ and then use the chain rule with partial derivatives to work out the four velocity in spherical coordinate, or I am missing something?
Another example, in the relativistic Lagrangian, $$\mathcal{L}=-mc^2\sqrt{1-\frac{|\mathbf{v}|^2}{c^2}}-U(\mathbf{r}),\tag{2}$$ it is correct to choose $r$ and $\theta$ as generalised coordinates, substitute $|\mathbf{v}|^2=\dot{r}^2+r^2\dot{\theta}^2$, to have $$\mathcal{L}=-mc^2\sqrt{1-\frac{\dot{r}^2+r^2\dot{\theta}^2}{c^2}}-U(r),\tag{3}$$ to create a "relativistic central force problem"?
Another question that I have is how this will affect the metric tensor? And the line element?