Ingoing BTZ coordinates
= Ingoing BTZ coordinates
{c}
For
$$
ds^2=-fdt^2+\frac{dr^2}{f}+r^2(d\phi-\Omega dt)^2,
$$
the transformations $dv=dt+dr/f$ and $d\chi=d\phi+\Omega\,dr/f$ give
$$
ds^2=-f\,dv^2+2\,dv\,dr+r^2(d\chi-\Omega\,dv)^2,
$$
which is regular at every horizon where $f$ vanishes while $\Omega$ remains finite.