I'm calculating the effective metric for a vortex in polar coordinates. The velocity and the potential is:
\begin{equation} \mathbf{v}=\frac{A}{r} \hat{r} + \frac{B}{r}\hat{\theta} \end{equation}
So:
\begin{equation} \mathbf{v}=\boldsymbol{\nabla} \psi \longrightarrow \psi= A ~log r + B~\theta \end{equation}
And I have the line element in cartesian coordinates $(t,x^1,x^2,x^3)=(t,x,y,z)$:
\begin{equation} ds^2 = \dfrac{\rho_0}{c_s} \left[ - \left( c_s^2-v_0^2\right) dt^2 - v_0^i dt dx^i - v_0^j dt dx^j + \delta_{ij} dx^i dx^j \right] \end{equation}
I need to obtain the following line element, effective metric acoustic $(t,r,\theta)$:
\begin{equation} ds^2 = - \left( c_s^2-\frac{A^2+B^2}{r^2}\right) dt^2 +dr^2 - 2\frac{A}{r}dtdr + r^2d\theta-2Bdtd\theta \end{equation}
Without $z$ because vortex is axially symmetric. I don't know how can I do it. I would appreciate some help to get started, what do I do with the terms with $i$.