The Poincare-AdS$_3$ geometry is given in the Wikipedia article on Anti-de Sitter space as
$$ds^{2} = \frac{dr^{2}}{r^{2}} + r^{2}g_{\alpha\beta}dx^{\alpha}dx^{\beta}$$
$$=\frac{dr^{2}}{r^{2}} + r^{2}(-dt^{2} + dx^{2}),$$
where $(t,x)$ are the coordinates on the boundary of the cylinder. Therefore, using $x = r \theta$, we find that
$$ds^{2} = \frac{dr^{2}}{r^{2}} - r^{2}dt^{2} + r^{4}d\theta^{2}.$$
But I am not able to make a consistent dimensional analysis. For example, $dr^{2}/r^{2}$ is dimensionless, but $r^{2}dt^{2}$ has a mass dimension of $-4$ and $r^{4}d\theta^{2}$ also has a mass dimension of $-4$.
What am I doing wrong here?
