Tortoise coordinate ($r_{\ast }$)

A tortoise coordinate straightens radial null directions. For a static radial metric $-F(r)d{t}^2+G(r)d{r}^2$ outside a horizon, choose $d{r}_{\ast }/dr=\sqrt{G/F}$; then $t\pm r_{\ast }$ are constant along the corresponding radial null geodesics. Continuation across horizons requires choosing branches consistently rather than replacing an analytic derivative by its absolute value.