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The advanced coordinate $v=t+r_{\ast }$ puts the radial Schwarzschild metric into the formwhich is regular at the future horizon.

These are the advanced, ingoing member of the Eddington-Finkelstein coordinates.

For the three cases, distinguish a curvature singularity from a coordinate singularity. The Reissner-Nordstrom Kretschmann scalar isIt diverges at $r=0$ when either $M$ or $Q$ is nonzero. Any positive root of $F$ instead admits Eddington-Finkelstein coordinates: set $v=t+\int dr/F$ to obtainThe determinant of its $(v,r)$ block is $-1$, including at $F=0$. Hence positive roots of $F$ are removable coordinate singularities of the static chart. The apparent degeneracy at the poles of the angular chart is also a coordinate singularity.

If $M^2=Q^2$ with $M>0$, thenThe curvature singularity remains at $r=0$. At $r=M$ the static chart has a coordinate singularity, removed by the Eddington-Finkelstein coordinates above. The Kretschmann scalar there equals $8/M^4$, which is finite. The coincident horizons form a degenerate Killing horizon, and its future outer branch is the event horizon of the extremal black hole. A double zero of $F$ does not remove the horizon.

In the extremal case, the squared norm of the Killing vector field isFor $M=|Q|>0$, it is negative for every regular $r\ne M$ and vanishes precisely at $r=M$ in the extended spacetime. ThusThe static chart cannot itself test the geometry at its horizon. In the regular Eddington-Finkelstein coordinates, the normal to $h=r-M=0$ is $n=dr$, andAt $r=M$ this gives $n^{♯}={\partial }_v=K$, a nonzero null vector tangent to the surface. Therefore $r=M$ is a null hypersurface, even though $F$ has the same sign on its two sides.

Take $M>0$ and write $f=1-2M/r$. Integrating $d{r}_{\ast }/dr=f^{-1}$ gives the Schwarzschild tortoise coordinateThe advanced and retarded Eddington-Finkelstein coordinates are respectively $v=t+r_{\ast }$ and $u=t-r_{\ast }$. Substituting $dt=dv-dr/f$ or $dt=du+dr/f$ givesBoth 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 coordinatesTheir product and the extended metric areIndeed $du=-4MdU/U$ and $dv=4MdV/V$ transform $-fdudv$ 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 $dUdV$ 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 $-d{T}_K^2+d{X}_K^2$. 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:This Schwarzschild conformal compactification preserves radial null directions because $dUdV={\sec }^2\bar{U}{\sec }^2\bar{V}(d{T}^2-d{X}^2)$. 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 $T=\pm X$.

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, ${\mathcal{I}}^{+}$ and ${\mathcal{I}}^{-}$. For example in the right exterior, $V\to \infty$ at fixed $U<0$ gives ${\mathcal{I}}^{+}$, whereas $U\to -\infty$ at fixed $V>0$ gives ${\mathcal{I}}^{-}$. 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. The maximal extension has two causally disconnected exteriors, a future black-hole region and a past white-hole region.

Write $r_s=2GM/c^2$ and $f=1-r_s/r$. The Schwarzschild tortoise coordinate satisfies $d{r}_{\ast }/dr=1/f$ and can be chosen as $r_{\ast }=r+r_s\ln |r/r_s-1|$. Define the advanced time $v=t+r_{\ast }/c$. Substituting $dt=dv-dr/(cf)$ into the Schwarzschild metric gives the ingoing Eddington-Finkelstein coordinates formThe coefficients are regular at $r=r_s$ and the radial-block determinant is $-c^2$, so the exterior Schwarzschild-time divergence is a coordinate singularity at the future event horizon. The physical curvature singularity at $r=0$ remains.

The two families of radial null trajectories in ingoing Eddington-Finkelstein coordinates obeyOutside the event horizon, the outgoing family increases $r$; at the horizon it generates the horizon itself; inside, even the outgoing future-directed family decreases $r$. An infaller crosses the future horizon at finite $v$ and finite proper time. The chart makes this causal geometry visible without mistaking Schwarzschild coordinate time for the infaller's clock.

For the Reissner-Nordstrom metric with $F=1-2M/r+Q^2/r^2$, the Kretschmann scalar isThe first expression follows by squaring the independent Riemann curvature tensor components in an orthonormal frame. This Kretschmann scalar diverges at $r=0$ for any nonzero $M$ or $Q$, but is finite at every positive root of $F$. Regular Eddington-Finkelstein coordinates show that those positive roots are coordinate singularities of the static chart. In the extremal case $Q^2=M^2>0$, its horizon value is $8/M^4$.

The Schwarzschild spacetime describes the spherically symmetric vacuum spacetime with zero cosmological constant. Its exterior chart, in units $G=c=1$ and metric signature $(-+++)$, isThe areal radius $r$ makes each symmetry sphere have area $4\pi r^2$; the mass parameter $M$ has length units in this convention. Birkhoff's theorem makes this the local vacuum geometry outside a spherical body, including when its interior changes spherically. A material star need not extend down to the Schwarzschild radius $2M$.

For the black-hole extension, $r=2M$ is an event horizon. The singular metric coefficients of this chart can be removed with Eddington-Finkelstein coordinates or Painlevé–Gullstrand coordinates. The invariant $R_{abcd}{R}^{abcd}=48M^2/r^6$ is finite there and diverges at $r=0$ in the extension. This Ricci flat metric has nonzero tidal forces. The $M=0$ limit recovers Minkowski spacetime in spherical coordinates.^[ref]