 Solution

Substituting the two differential coordinate transformations cancels every coefficient singular as ${\Delta }^{-1}$, so the ingoing Kerr-like coordinates are regular at $r=r_{+}$. The inverse metric applied to the normal covector $dr$ hasIts norm is $g^{rr}=\Delta /\Sigma$, which vanishes at $r=r_{+}$. Thus that level set is a null hypersurface. On it the raised normal is proportional toSince the transformed stationary and axial Killing fields are $K={\partial }_v$ and $m={\partial }_{\chi }$, the horizon is a Killing horizon generated by