= Solution
Take $M>0$ and write $f=1-2M/r$. Integrating $dr_*/dr=f^{-1}$ gives the <Schwarzschild tortoise coordinate>
$$
r_*=r+2M\log\left|\frac r{2M}-1\right|.
$$
The advanced and retarded <Eddington-Finkelstein coordinates> are respectively $v=t+r_*$ and $u=t-r_*$. Substituting $dt=dv-dr/f$ or $dt=du+dr/f$ gives
$$
\boxed{ds^2=-f\,dv^2+2\,dv\,dr+r^2d\Omega_2^2,}
\qquad
\boxed{ds^2=-f\,du^2-2\,du\,dr+r^2d\Omega_2^2.}
$$
Both radial-time blocks have determinant $-1$ and smooth coefficients at $r=2M$. Consequently the <Schwarzschild event horizon> is a coordinate singularity of the original chart, not a degeneracy of the geometry. The <Kretschmann scalar> $48M^2/r^6$ is finite there, while it diverges at $r=0$. Angular coordinate degeneracies at the polar axes are removed separately by ordinary sphere charts.
The <Ingoing Eddington-Finkelstein coordinates> extend the right exterior through its future <black hole> horizon. Radial <null geodesics> obey either $dv=0$ or $dr/dv=f/2$. Inside $r=2M$, the latter also moves towards smaller $r$ as $v$ increases, and the future-directed ingoing family has decreasing $r$. Thus neither family can escape to the exterior. The future horizon is the boundary separating points that can send signals to future <null infinity> from those that cannot. The retarded chart extends through the past horizon instead. Reversing the time orientation gives the <white hole>: future-directed signals can leave it, but exterior signals cannot enter it. These are distinct branches in the eternal extension; the white-hole branch need not occur in a spacetime formed by collapse.
For a simultaneous extension through both branches, start in the right exterior and define the <Kruskal–Szekeres coordinates>
$$
U=-e^{-u/(4M)},\qquad V=e^{v/(4M)}.
$$
Their product and the extended metric are
$$
\boxed{UV=\left(1-\frac r{2M}\right)e^{r/(2M)},\qquad
ds^2=-\frac{32M^3}{r}e^{-r/(2M)}\,dU\,dV+r^2d\Omega_2^2.}
$$
Indeed $du=-4M\,dU/U$ and $dv=4M\,dV/V$ transform $-f\,du\,dv$ into the displayed expression. The function of $r$ on the right of the product relation has derivative $-r e^{r/(2M)}/(4M^2)$, nonzero at $r=2M$, so $r$ is a smooth function of $UV$ across zero. The coefficient of $dU\,dV$ there is $-16M^2/e$, finite and nonzero. Allowing either sign of $U,V$ therefore gives the regular maximal extension with $UV<1$ and $r>0$.
With $T_K=(V+U)/2$ and $X_K=(V-U)/2$, the radial metric is a positive multiple of $-dT_K^2+dX_K^2$. The two exterior regions have $(U,V)=(-,+)$ and $(+,-)$. The future interior, or <black hole> region, has $(+,+)$; the past interior, or <white hole> region, has $(-,-)$. The <Killing horizons> are $U=0$ and $V=0$, meeting on the <bifurcation surface> at $U=V=0$.
For the <Penrose diagram>, compactify the null variables before forming time and space:
$$
\bar U=\arctan U,\qquad \bar V=\arctan V,\qquad
T=\frac{\bar V+\bar U}{2},\qquad X=\frac{\bar V-\bar U}{2}.
$$
This <Schwarzschild conformal compactification> preserves radial null directions because $dU\,dV=\sec^2\bar U\sec^2\bar V\,(dT^2-dX^2)$. The ranges $|\bar U|,|\bar V|<\pi/2$ bound the diagram, and $UV<1$ cuts off its future and past tips. On $UV=1$ with $U,V>0$, $\bar U+\bar V=\pi/2$, so the future <curvature singularity> is $T=\pi/4$. The past <curvature singularity> is similarly $T=-\pi/4$. Both have $|X|<\pi/4$ and are spacelike. The horizon lines remain $T=\pm X$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-56-schwarzschild-penrose.png]
{title=Penrose diagram of maximally extended Schwarzschild spacetime, with two exteriors, black and white hole interiors, horizons and conformal infinities}
Each point of this radial <Penrose diagram> represents a two-sphere, except at limiting boundaries. Each exterior has its own future and past <null infinity>, $\mathscr I^+$ and $\mathscr I^-$. For example in the right exterior, $V\to\infty$ at fixed $U<0$ gives $\mathscr I^+$, whereas $U\to-\infty$ at fixed $V>0$ gives $\mathscr I^-$. Its <spacelike infinity> is $(T,X)=(0,\pi/2)$; its future and past <timelike infinity> are the vertices $(\pi/4,\pi/4)$ and $(-\pi/4,\pi/4)$. The left exterior has the reflected counterparts. These conformal boundary points are ideal endpoints, not finite-distance spacetime events; the physical affine or proper parameters of escaping geodesics diverge.
The horizons are regular null surfaces, while the horizontal $r=0$ boundaries are genuine <curvature singularities>. The central point represents the regular <bifurcation surface>, not a singularity. Future-directed radial causal curves have nondecreasing $U,V$, so a curve cannot go from one exterior to the other: changing the requisite signs would require one null variable to decrease. Crossing a future horizon instead leads towards the future singularity. \b[The maximal extension has two causally disconnected exteriors, a future black-hole region and a past white-hole region.]
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