= Solution
Use <geometrized units> $G=c=1$ and positive <mass> $M$. The <Schwarzschild metric> initially describes an exterior region, not the whole <black hole>:
$$
ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,\qquad f(r)=1-\frac{2M}{r},\qquad r>2M.
$$
Here $r$ is the <areal radius> and $t$ is <Schwarzschild time>, normalized at infinity. We must distinguish a removable <coordinate singularity> from a genuine <curvature singularity>. The <Kretschmann scalar> is
$$
R_{abcd}R^{abcd}=\frac{48M^2}{r^6}.
$$
It is finite at $r=2M$ and diverges at $r=0$. Thus $r=2M$ can be crossed in a better chart, whereas $r=0$ cannot be made a regular point by changing the <coordinate chart>.
First integrate the <Schwarzschild tortoise coordinate>:
$$
\frac{dr_*}{dr}=\frac1f,\qquad r_*=r+2M\log\left|\frac r{2M}-1\right|.
$$
The radial metric becomes $f(-dt^2+dr_*^2)$. Set $v=t+r_*$; the <ingoing Eddington-Finkelstein coordinates> give
$$
ds^2=-f\,dv^2+2\,dv\,dr+r^2d\Omega^2.
$$
The radial block has determinant minus one and smooth coefficients at $r=2M$, so it extends through the future <Schwarzschild event horizon>. Its ingoing radial <null geodesics> have constant $v$, while the outgoing radial <null geodesics> obey $dr/dv=f/2$. In the future interior both future-directed radial null families decrease $r$; $r=0$ is therefore an unavoidable future boundary for causal motion there. The outgoing chart $u=t-r_*$ similarly gives $ds^2=-f\,du^2-2\,du\,dr+r^2d\Omega^2$, extending through the past horizon. A single one of these null charts does not display both horizons or the entire extension.
To regularize both horizons together, introduce <Kruskal–Szekeres coordinates> in the original exterior:
$$
u=t-r_*,\quad v=t+r_*,\qquad U=-e^{-u/(4M)},\quad V=e^{v/(4M)}.
$$
Then
$$
UV=\left(1-\frac r{2M}\right)e^{r/(2M)},\qquad
\boxed{ds^2=-\frac{32M^3}{r}e^{-r/(2M)}dU\,dV+r^2d\Omega^2.}
$$
For $r>0$ the right side defining $UV$ decreases monotonically from one to minus infinity. In fact its derivative is $-re^{r/(2M)}/(4M^2)$. Thus it determines a unique smooth $r=r(UV)$ wherever $UV<1$, including $UV=0$. At $r=2M$ the radial metric coefficient is finite and nonzero, proving actual regularity of both horizons and their intersection, rather than merely relabelling the old coordinate divergence. Allowing all real $U,V$ with $UV<1$ produces the simply connected <maximal analytic extension>, the <Kruskal spacetime>.
There are four regions of the <Kruskal spacetime>. The original exterior is $U<0,V>0$; a second asymptotically flat exterior is $U>0,V<0$. The future <black hole> has $U>0,V>0$ and $0<r<2M$; the past <white hole> has $U<0,V<0$ and the same range of <areal radius>. The horizon branches are $U=0$ and $V=0$. Their intersection $U=V=0$ is a regular <bifurcation surface>, a two-sphere of radius $2M$, not the centre of the spacetime. Each point of the radial diagram represents a symmetry two-sphere.
Put $T_K=(V+U)/2$ and $X_K=(V-U)/2$. Radial <null geodesics> have slopes $dT_K=\pm dX_K$, and the <curvature singularities> have
$$
T_K^2-X_K^2=1.
$$
The upper branch is the future spacelike <Schwarzschild singularity>; the lower branch is the <white hole> past singularity. The <Killing vector field> extends as $\partial_t=(V\partial_V-U\partial_U)/(4M)$. It is future directed in the original exterior and past directed in the other exterior for the chosen global <time orientation>; inside, this stationary vector is spacelike. In the future interior, $r$ is a time coordinate decreasing toward the singularity. For example, a radial infaller of unit specific <Killing energy> obeys $dr/d\tau=-\sqrt{2M/r}$ and reaches $r=0$ from the horizon in finite <proper time> $4M/3$. The curvature divergence and this geodesic endpoint show why the extension stops there.
For the global causal picture, compactify the null coordinates using $p=\arctan U$, $q=\arctan V$ and let $\mathcal T=2(p+q)/\pi$, $\mathcal X=2(q-p)/\pi$. The <conformal compactification> rescales the metric to remove the coordinate factors from the radial metric while preserving its <null directions>. Since $UV=1$ gives $p+q=\pm\pi/2$ on the two singular branches, they become the horizontal lines $\mathcal T=\pm1$. The resulting <Penrose diagram> displays the two exteriors, each with its own <past null infinity>, <future null infinity> and spatial infinity, plus the future and past interiors. The singularities are spacelike boundaries; the horizons are interior null surfaces. Future <event horizons> separate events that can escape to an exterior's future null infinity from events in the black-hole interior.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-62-extension.png]
{title=Kruskal and compactified diagrams of the maximal Schwarzschild extension, showing both exteriors, horizons and spacelike singularities}
The time-symmetric spatial slice has a bridge between the exteriors, but it is not a traversable passage. Future-directed causal curves have nondecreasing $U$ and $V$. Moving from the right exterior to the left would require $V$ to decrease from positive to negative; the reverse passage would require $U$ to decrease. Both are impossible. This proves the <non-traversability of the Schwarzschild bridge>. \b[The complete eternal vacuum extension has two exteriors, a <black hole> and a <white hole>; it has no causal route between its exteriors.] A <black hole> made by stellar collapse is a different global spacetime: its matter-filled past replaces the <white hole> region, and it normally has only one exterior. The vacuum extension constructed here should not be mistaken for the collapse history of a single star.
With <electric charge>, spherical electrovacuum instead gives the <Reissner-Nordstrom metric>, in an electromagnetic normalization absorbing the charge conversion constants:
$$
f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2},\qquad r_\pm=M\pm\sqrt{M^2-Q^2}.
$$
For $0<|Q|<M$, there are two simple horizons. Outside $r_+$ the metric is static; between $r_-$ and $r_+$, $f<0$ and $r$ is timelike. Inside $r_-$, $f>0$ again, so $r$ is spacelike and the $r=0$ singularity is timelike. The outer horizon is the <event horizon>; the inner horizon is a <Cauchy horizon>. The same logarithmic tortoise-coordinate and exponential-null-coordinate method crosses each simple horizon locally, but one exponential scale does not regularize both simultaneously: their <surface gravities> have different magnitudes,
$$
|\kappa_\pm|=\frac{r_+-r_-}{2r_\pm^2}.
$$
Continuing the exact metric through successive inner horizons produces repeated blocks with further exteriors, and evolution beyond a <Cauchy horizon> is not uniquely fixed by data on the original <Cauchy hypersurface>. The timelike singularity can be avoided by some trajectories in this ideal extension, unlike the unavoidable spacelike singularity of the future <Schwarzschild spacetime> interior. However, the enormous inner-horizon blueshift makes this exact continuation unstable: perturbations can cause <mass inflation>. This is an important qualification on the <global structure of a charged spherical black hole>.
At $|Q|=M$ the two horizons coalesce, $f=(1-M/r)^2$ and <surface gravity> is zero. The horizon is a <degenerate Killing horizon>, the tortoise-coordinate divergence becomes a pole and the ordinary nondegenerate <Kruskal spacetime> construction must be replaced. The exterior develops an infinitely long spatial throat and the formal <Hawking temperature> vanishes. For $|Q|>M$, no real zero of $f$ remains: the electrovacuum solution has a <naked singularity>, not a <black hole>. Consequently arbitrarily allowing charge changes the causal structure, not just the horizon radius.
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