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For distinct horizon radii $a>b>0$, integrating $d{r}_{\star }/dr=f^{-1}$ gives $r_{\star }=r+\frac{a^3}{2(a^2-b^2)}\log |(r-a)/(r+a)|-\frac{b^3}{2(a^2-b^2)}\log |(r-b)/(r+b)|+C$. The tortoise coordinate defines constant retarded or advanced coordinates on radial null geodesics.

The ingoing Vaidya metric describes a spherically symmetric spacetime sourced by radial null matter, using advanced time $v$ and areal radius $r$. In geometrized units it is $d{s}^2=-(1-2M(v)/r)d{v}^2+2dvdr+r^{2}d{\Omega }^2$. A constant mass function reduces it to the Schwarzschild metric in ingoing null coordinates; a varying mass function changes the curvature and the radial null geodesics.