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Rotate any geodesic into an equatorial plane of the magnetically charged dilaton black hole. Conserved energy and angular momentum are $E=V\dot{t}$ and $L=R^2\dot{\varphi }$. Normalizing its tangent to $g(\dot{x},\dot{x})=-ϵ$, with $ϵ=1,0,-1$ for timelike, null and spacelike geodesics, gives ${\dot{r}}^2=E^2-V(ϵ+L^2/R^2)$. Thus its radial equation is ${\dot{r}}^2/2+\tilde{V}=0$. Radial null rays have $\dot{r}=\pm E$ and use the same Schwarzschild tortoise coordinate with $2M$ replaced by $r_{+}$.

Use geometrized units and metric signature $(-+++)$, with $M>0$. In the exterior of Schwarzschild spacetime, put $f=1-2M/r$. The Schwarzschild tortoise coordinate satisfiesThe retarded and advanced null coordinates $u=t-r_{\ast }$ and $v=t+r_{\ast }$ then give $d{s}^2=-fdudv+r^{2}d{\Omega }^2$. The logarithmic divergence of $r_{\ast }$ suggests exponentiating these null coordinates. In the right exterior define the Kruskal–Szekeres coordinatesTheir product eliminates $t$:Since $du=-4MdU/U$ and $dv=4MdV/V$, the Schwarzschild metric becomesHere $r$ is an implicitly defined function of $UV$. The derivative of the right side with respect to $r$ is $-r{e}^{r/(2M)}/(4M^2)$, which is nonzero at $r=2M$. The inverse function theorem therefore makes $r(UV)$ smooth across that surface, and the coefficient of $dUdV$ tends to $-16M^2/e$. Thus the Schwarzschild event horizon is a coordinate singularity of the original chart, while this Lorentzian metric remains regular there.

Extend the Kruskal–Szekeres coordinates to all real $U,V$ with $UV<1$. The signs give two exterior regions, $U<0<V$ and $V<0<U$, a future black hole region $U,V>0$, and a past white hole region $U,V<0$. The event horizons are $U=0$ or $V=0$, intersecting at the bifurcation surface. The boundary $UV=1$ has $r=0$ and is a genuine Schwarzschild singularity, as the Kretschmann scalar $48M^2/r^6$ diverges there.

Finally, $T=(V+U)/2$ and $X=(V-U)/2$ give $UV=T^2-X^2$ and a radial metric proportional to $-d{T}^2+d{X}^2$. Hence radial null geodesics have slopes $\pm 1$, the event horizons are $T=\pm X$, and the singular boundaries are $T^2-X^2=1$. This constructs the maximal Kruskal extension; a black hole produced by collapse need not contain the second exterior or the white hole of that eternal extension.

Use geometrized units $G=c=1$ and metric signature $(-,+,+,+)$. Put $f=1-2M/r$ in the Schwarzschild metric. The Schwarzschild tortoise coordinate satisfies $d{r}_{\ast }/dr=f^{-1}$, soSubstituting $dt=dv-f^{-1}dr$ gives the Ingoing Eddington-Finkelstein coordinates:The radial metric tensor has determinant $-1$ and inverse components $g^{vv}=0$, $g^{vr}=1$, $g^{rr}=f$. Thus it is nondegenerate and analytic at $r=2M$. The same expression defines a Lorentzian metric for every $r>0$, extending the exterior across the future Schwarzschild event horizon into the black hole. It does not include the other exterior or the white hole of the full Kruskal spacetime. At $r=0$, the Kretschmann scalar $R_{abcd}{R}^{abcd}=48M^2/r^6$ diverges, so this is a curvature singularity, not a removable coordinate singularity.

A radial null geodesic has $L=0$ and $ϵ=0$, so $\dot{r}=\pm E$ and $\dot{t}=E/V$. In the exterior, $E>0$ for a future-directed ray. Define the analogue of the Schwarzschild tortoise coordinate byThen an outgoing ray satisfies $\dot{u}=\dot{t}-\dot{r}/V=0$, whereas an ingoing ray satisfies $\dot{v}=\dot{t}+\dot{r}/V=0$. Thus $u=t-r_{\ast }$ is constant on outgoing rays and $v=t+r_{\ast }$ on ingoing rays. The radial two-dimensional metric becomes $-Vdudv$.

The ingoing coordinate is regular at the future horizon: substituting $t=v-r_{\ast }$ gives $d{s}^2=-Vd{v}^2+2dvdr+R^{2}d{\Omega }_2^2$. Inside the horizon, the words outgoing and ingoing refer to the continued null families; both future-directed families in the future interior decrease $r$.

WriteFor a radial null ray, the Schwarzschild spacetime line element givesOutside the horizon an ingoing ray has $dr/dt<0$, so it takes the minus sign. Since the Schwarzschild tortoise coordinate obeys $d{r}_{\ast }/dr=f^{-1}$,It follows thatis constant along every ingoing radial light ray.