 Solution

Set $F(r)=\Delta /(r^2+a^2)$. The Kerr polar tortoise coordinate obeysThen $d{s}_{(2)}^2=F(-d{t}^2+d{r}_{\ast }^2)$. Introduce retarded time $u=t-r_{\ast }$, so $dt=du+d{r}_{\ast }$ andThe determinant of this metric is $-1$, including at $\Delta =0$. These outgoing Eddington-Finkelstein-type coordinates therefore extend analytically across the past event horizon into the white-hole region of Kerr.

Solved by gpt-5.6-sol high.