I have been reading Thomas Hartman's lecture notes on Quantum Gravity and Black Holes.
In page 97, he derives (9.4), which is the metric of AdS$_{3}$ in global coordinates:
$$ds^{2} = \ell^{2}(-\cosh^{2}\rho\ dt^{2} + d\rho^{2} + \sinh^{2}\rho\ d\phi^{2}).$$
In page 100, he states that, expanding the metric at large $r$ under the coordinate change \begin{align} t^{\pm} = t \pm \phi, \qquad \rho = \log(2r), \end{align} we can show that, to leading order, the induced metric on the hyperboloid AdS$_{3}$ becomes \begin{align} ds^{2} = \ell^{2} \left(\frac{dr^{2}}{r^{2}}-r^{2}dt^{+}dt^{-}\right). \end{align}
I find, under the coordinate change, that \begin{align} ds^{2} = \ell^{2} \left(\frac{dr^{2}}{r^{2}} - \frac{1}{4}dt^{+2} - \left(r^{2} + \frac{1}{16r^{2}} \right) dt^{+}dt^{-} - \frac{1}{4}dt^{-2} \right). \end{align}
Of course, the term in $1/r^{2}$ drops off at large $r$, but I am not able to get rid of the components in $dt^{+2}$ and $dt^{-2}$. Am I missing something here?
He then goes on to mention that these are Poincare coordinates, but does not make contact with the usual way in which the metric of AdS$_{3}$ is written in Poincare coordinates:
$$ds^{2} = \frac{\ell^{2}}{z^{2}}(dz^{2}-dt^{2}+d\vec{x}^{2}).$$
What am I missing here?