= Ingoing Kerr-Schild coordinates for Schwarzschild spacetime
{c}
{title2=$t'=t+(2\mu/c)\log|r/(2\mu)-1|$}
With $f=1-2\mu/r$, the time coordinate $t'=t+(2\mu/c)\log|r/(2\mu)-1|$ gives $ds^2=fc^2dt'^2-(4\mu c/r)dt'\,dr-(1+2\mu/r)dr^2-r^2d\Omega^2$. Its radial <metric tensor> block has <determinant> $-c^2$, so it extends regularly across the future <Schwarzschild event horizon>. The advanced null time of <ingoing Eddington-Finkelstein coordinates> is $v=t'+r/c$, not $t'$ itself. Radial null directions satisfy $dr/(c\,dt')=-1$ or $(1-2\mu/r)/(1+2\mu/r)$.
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