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A Finkelstein diagram plots areal radius $r$ against $T=v-r$ in Ingoing Eddington-Finkelstein coordinates. Its radial null directions show explicitly that both future light-ray families decrease $r$ inside a Schwarzschild black hole. It is not a compactified Penrose diagram.

Since $v=t+r_{\star }$, $dt=dv-f^{-1}dr$. Substitution into the radial metric cancels the $d{r}^2/f$ terms. The Ingoing Eddington-Finkelstein coordinates giveThe radial block has determinant $-1$, even when $f=0$. Its coefficients are analytic for every $r>0$, so this form extends through $r=r_{+}$ and also through $r=r_{-}$ to $0<r<r_{+}$. Ordinary angular coordinate singularities are removed by angular charts on the three-sphere. The genuine endpoint $r=0$ is not included in this extension.

The field $n=-{\partial }_r$ is an ingoing null tangent. It is future-directed in the original exterior, since its inner product with the future stationary Killing vector field is $-1$, and it remains future-directed by continuation. This fixes the time orientation used in the next part.

For an asymptotically flat spacetime with a chosen exterior component of future null infinity, the black-hole region isnamely the events from which no future-directed causal curve can reach that infinity. Here ${\mathcal{I}}^{+}$ is the infinity of the original $r>r_{+}$ exterior.

Let a future causal tangent in the Ingoing Eddington-Finkelstein coordinates have components $(\dot{v},\dot{r},\dot{\Omega })$. Its inner product with the future null field $n=-{\partial }_r$ is $-\dot{v}$, so $\dot{v}\ge 0$. The causal inequality isIf $\dot{v}>0$, this impliesIn the band $r_{-}<r<r_{+}$, $f<0$, so $\dot{r}<0$. If $\dot{v}=0$, the causal inequality forces $\dot{\Omega }=0$, and future direction means the remaining radial tangent is a nonnegative multiple of $n$, again giving $\dot{r}\le 0$. This is causal trapping between two spherical horizons: $r$ is nonincreasing along every future causal curve in the between-horizon band. No event there can escape outward. An event with $0<r\le r_{-}$ would also have to cross this band outward to reach the original exterior, which is impossible. Thus all $0<r<r_{+}$ events in this extension lie in the black-hole region.

Conversely, from any $r>r_{+}$ point, the outgoing radial null ray $u=\mathrm{constant}$ satisfies $dr/dv=f/2>0$. It reaches arbitrarily large $r$ and then the original future null infinity. Therefore the exterior does not intersect the black-hole region, and $r=r_{+}$ is its event horizon. This conclusion is relative to the selected asymptotic end; the maximal charged-black-hole extension can have other asymptotic ends.

In the Ingoing Eddington-Finkelstein coordinates, the stationary Killing vector field is $k={\partial }_v$ with norm $k^2=-f$. The normal vector to a level set of $r$ isAt $r=r_{+}$ it equals $k^a$, which is null there. Thus this null hypersurface is a Killing horizon of $k$. Its surface gravity is fixed by ${\nabla }_a(k^2)=-2\kappa k_a$ on the horizon. Since $k_a=(dr{)}_a$ there and ${\nabla }_a(k^2)=-f^{\prime }(r_{+})(dr{)}_a$,The positive sign follows from the future outer-horizon generator, normalized by the static time at infinity. It vanishes as the two radii approach the extremal limit.

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.

For radial free fall, let $E=f{\dot{t}}_A>0$ be Alice's conserved Killing energy. The inward branch of the timelike geodesic has $\dot{r}=-\sqrt{E^2-f}$, henceWith $\delta =r-2M$, the near-horizon geodesic equations giveThe constants in the logarithms are understood to make their arguments dimensionless. Therefore $u_A=t_A-r_{\ast }=2t_A+O(1)$. For an outgoing signal reaching the fixed-radius Bob, $t_B=u_A+r_{\ast }(r_B)$, so $t_B=2t_A+O(1)$. The redshift consequently behaves asThis is the answer when $t$ is Bob's Schwarzschild time, as appropriate to the stated observation. It is the Schwarzschild surface gravity $a=\kappa =1/(4M)$. If $t$ instead means the emission Schwarzschild time $t_A$, the same redshift is proportional to $e^{t_A/(2M)}$ and gives $M=1/(2a)$. The two answers use different clocks, not different dynamics. If the measured exponential uses Bob's proper time ${\tau }_B$, its rate is $1/(4M\sqrt{f_B})$. In SI units the reception-time result is $M=c^3/(4Ga)$.

The reception-time exponent also holds for smooth radial infall crossing the future Schwarzschild event horizon with finite nonzero $dr/d\tau$: regular Ingoing Eddington-Finkelstein coordinates give finite $v$ there, $u=v-2r_{\ast }\sim -4M\log ({\tau }_H-\tau )$. Thus no special value of $E$ is needed.

A future trapped surface is a smooth compact spacelike two-surface without boundary whose two future-directed orthogonal null expansions are strictly negative. In four-dimensional general relativity, the Penrose singularity theorem states: a time-oriented globally hyperbolic spacetime with a noncompact Cauchy hypersurface, a trapped surface, and the null convergence condition $R_{ab}{k}^a{k}^b\ge 0$ for every null vector is future null-geodesically incomplete. With the Einstein field equations, the null energy condition implies this null convergence condition; a cosmological constant drops out of the null contraction.

The Kruskal spacetime is an example. In its black hole interior, use future null normals in Ingoing Eddington-Finkelstein coordinates:For a round sphere of areal radius $r$, its area is $4\pi r^2$ and its null expansions areBoth are negative for $0<r<2M$. The vacuum Einstein field equations give $R_{ab}=0$, and a two-ended Kruskal Cauchy hypersurface is noncompact. The future radial null geodesics reaching $r=0$ in finite affine parameter provide precisely the incompleteness predicted by the Penrose singularity theorem. A trapped surface at $r<2M$ is strictly trapped; the horizon sphere has one zero null expansion instead.

FactorIntroduce $v=t+r_{\ast }$ with $d{r}_{\ast }/dr=f^{-1}$. Since $dt=dv-dr/f$,These are the charged analogue of Ingoing Eddington-Finkelstein coordinates. The $(v,r)$ block has determinant $-1$, so the metric and its inverse are nonsingular at $r=r_{+}$. For $M\ge Q$, $r_{+}$ is real and the metric extends analytically through it. The original potential becomes $A=-Qdv/r^2+Qdr/(r^{2}f)$; the radial term is locally exact, and a gauge transformation gives the horizon-regular potential $A=-Qdv/r^2$.