Eddington–Finkelstein Coordinates

Karl Schwarzschild's 1916 vacuum solution is the first and most important exact metric of general relativity, yet for four decades it carried a scar. In the natural curvature coordinates $(t,r,\theta,\phi)$ the coefficients misbehave at the Schwarzschild radius $r=2GM$ : the lapse $g_{tt}=-(1-2GM/r)$ falls to zero and $g_{rr}=(1-2GM/r)^{-1}$ blows up. To Schwarzschild, to Hilbert, and to most of the community into the 1950s, this surface looked like the physical edge of the world — a "Schwarzschild singularity" where space itself ended. That reading was wrong, and the tool that showed it wrong is a change of coordinates. The idea is to stop timing events with a distant observer's clock and start timing them with light. Radial photons are the one family of worldlines that couples cleanly to the geometry, and the coordinate built from them — Eddington sketched it in 1924, Finkelstein rediscovered and interpreted it in 1958 — glides straight through $r=2GM$ . Replacing $t$ by the advanced time $v=t+r_*$ turns the divergent $g_{rr}$ into a finite off-diagonal term, gives a metric whose determinant never vanishes for $r>0$ , and reveals $r=2GM$ for what it is: a smooth, one-way null surface, the event horizon — not a barrier but a point of no return. The same construction exposes deeper structure. The advanced chart regularizes the future horizon and covers the black-hole interior; the retarded chart, from $u=t-r_*$ , regularizes a past horizon and covers a white hole — together pointing straight at the maximal Kruskal–Szekeres extension. This is where the horizon stops being a paradox and becomes geometry.

Key takeaways

In Schwarzschild coordinates the metric appears to break down at $r=2GM$ ( $g_{tt}\to0$ , $g_{rr}\to\infty$ ), but the Kretschmann scalar stays finite there — the trouble is in the chart, not the spacetime.
The tortoise coordinate $r_*=r+2GM\ln|r/2GM-1|$ flattens radial light cones; the advanced null coordinate $v=t+r_*$ then yields the ingoing Eddington–Finkelstein metric, regular across the horizon.
Radial light cones tip over smoothly: at $r=2GM$ the outgoing cone edge is tangent to the surface, so nothing can climb outward — the horizon is a one-way null membrane.
Infalling particles cross $r=2GM$ in finite proper time; only the distant observer's coordinate time $t$ diverges, which is why the infaller's image "freezes."
The advanced chart ( $v$ ) describes a black hole; the retarded chart ( $u=t-r_*$ ) a time-reversed white hole — complementary patches of the maximal extension, not gauge copies.
Key idea: adopt coordinates whose constant-time surfaces are tied to radial null rays — replacing $t$ by $v=t+r_*$ removes the $1/(1-2GM/r)$ blowup, turning the apparent singularity at $r=2GM$ into a smooth one-way null surface.

The core insight

Adopt coordinates whose constant-time surfaces are tied to radial null rays: replacing $t$ by $v=t+r_*$ removes the $1/(1-2GM/r)$ blowup, turning the apparent singularity at $r=2GM$ into a smooth, one-way null surface — the event horizon.

Where this lives today

Eddington–Finkelstein coordinates and their cousins are the working chart of modern strong-gravity physics. In numerical relativity, evolving a black-hole spacetime demands slicings that do not develop coordinate singularities at the horizon, and the horizon-penetrating philosophy of the advanced chart — generalized to Kerr–Schild and Painlevé–Gullstrand form — is what makes the interior tractable. The "moving puncture" gauge behind the 2005 breakthrough simulations of binary black-hole mergers, and hence the waveform templates LIGO and Virgo matched against GW150914 in 2015, evolves data on slices that plunge smoothly through the horizons rather than freezing on them. The same coordinates underpin gravitational collapse: in Oppenheimer–Snyder dust collapse the advanced chart follows the stellar surface smoothly across $r=2GM$ , making horizon formation explicit where Schwarzschild time fails. They are the natural home of the null expansions $\theta_\pm$ that define trapped surfaces and apparent horizons — and thus the Penrose singularity theorem. And when the Event Horizon Telescope released images of M87 (2019) and Sgr A (2022), the ray-tracing that turns a simulated accretion flow into a predicted ring ran on exactly such horizon-penetrating grids. Wherever the horizon must be treated as geometry rather than a wall, Finkelstein's coordinate is in the toolkit.

The physical picture

Picture the radial light cones of the Schwarzschild geometry in a $(t,r)$ diagram. Far from the mass they are the familiar $45^\circ$ wedges of flat space. Sliding inward toward $r=2GM$ , the Schwarzschild cones pinch shut: outgoing rays climb ever more slowly, asymptoting to the horizon but never reaching it in coordinate time $t$ . This "freezing" is a property of the clock, not of the spacetime — a probe's own wristwatch shows a finite, uneventful crossing. Eddington–Finkelstein coordinates re-draw the same geometry so the cones keep a finite opening and simply tip inward through the horizon. The trick is to time events with ingoing light. Take $v=t+r_*$ as the new "time": surfaces of constant $v$ are ingoing spherical light fronts converging on the hole, and following one inward, $r$ decreases monotonically straight through $2GM$ with no stall. Anchored to infalling light, the horizon becomes just another value of $r$ on a smooth slice. The first figure is the diagram Finkelstein drew: ingoing rays run at a clean $45^\circ$ while the future light cones tilt over — vertical at the horizon, fully tipped toward $r=0$ inside.

Figure

Ingoing Eddington–Finkelstein diagram: replacing t by the advanced time v = t + r* straightens ingoing light rays to 45° (blue) while the future light cones tip inward as r falls — the outgoing edge (amber) is vertical and tangent to the horizon at r = 2GM and fully tilted toward r = 0 inside, so every timelike worldline (green) crosses the horizon and reaches the singularity

The defining feature of the horizon now emerges as pure geometry. On $r=2GM$ the outgoing edge of the future light cone is exactly tangent to the surface $r=\text{const}$ : a photon aimed radially "outward" there neither gains nor loses radius — it hovers. Aim any less steeply and it falls in; no timelike worldline can do better. That one-wayness is what separates a black hole from a merely strong field. The second figure attaches real numbers to the other half of the story — reconciling the infaller's finite proper time with the distant observer's endless wait — for Sagittarius A*, at the centre of the Milky Way.

Figure

For Sgr A* (M = 4.15×10⁶ M⊙) the event horizon sits at rₛ = 2GM/c² = 1.23×10¹⁰ m ≈ 0.082 AU: a body falling from rest crosses in finite proper time and reaches the singularity 27 s later (green), while in Schwarzschild coordinate time t the same infall only asymptotes to the horizon and never crosses (blue) — why a distant observer sees the image freeze and redshift away

Finally, the ingoing/outgoing asymmetry in the first diagram is the seed of the Kruskal picture. Choosing $v$ regularizes the future horizon and opens the black-hole interior below it; choosing $u=t-r_*$ regularizes a past horizon and opens a white-hole interior. Each chart sees only "half" of the maximally extended spacetime; only using both null coordinates at once, as in Kruskal–Szekeres coordinates, does a single chart cover all four regions — two exteriors, a black hole, and a white hole.

The core equations

Everything starts from the Schwarzschild line element, the unique static spherically symmetric vacuum solution. We work in units with $c=1$ , so the Schwarzschild radius is $r_s=2GM$ .

\mathrm{d}s^2=-\Big(1-\frac{2GM}{r}\Big)\mathrm{d}t^2+\Big(1-\frac{2GM}{r}\Big)^{-1}\mathrm{d}r^2+r^2\,\mathrm{d}\Omega^2 \qquad (1)

where: $G$ — Newton's constant (with $c=1$ ); $M$ — mass, so $2GM$ is the horizon radius; $\mathrm{d}\Omega^2=\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2$ the unit-sphere metric; $t$ — Schwarzschild (asymptotic-observer) time; $r$ — areal radius, so a sphere of coordinate $r$ has area $4\pi r^2$ . Reading it: the pathology sits entirely in the $tt$ and $rr$ coefficients at $r=2GM$ ; the angular part is smooth there. To flatten the radial light cones we integrate the radial null condition. The combination that appears is the tortoise coordinate $r_*$ , so named because it pushes $r=2GM$ off to $r_*=-\infty$ .

r_*=r+2GM\,\ln\!\left|\frac{r}{2GM}-1\right|,\qquad \frac{\mathrm{d}r_*}{\mathrm{d}r}=\Big(1-\frac{2GM}{r}\Big)^{-1} \qquad (2)

where: $r_*$ — tortoise coordinate; $r_*\to-\infty$ as $r\to2GM^+$ , $r_*\to+\infty$ as $r\to\infty$ . Reading it: the derivative is exactly the divergent factor $g_{rr}$ , which is why radial null rays become straight lines in $r_*$ . Radial null curves ( $\mathrm{d}\Omega=0$ , $\mathrm{d}s^2=0$ ) then satisfy $\mathrm{d}t=\pm\,\mathrm{d}r_*$ , so $t\mp r_*=\text{const}$ . These two constants are the null coordinates, the variables photons actually carry.

v=t+r_*\quad(\text{advanced}),\qquad u=t-r_*\quad(\text{retarded}) \qquad (3)

where: $v$ — advanced time, constant along each ingoing radial null ray; $u$ — retarded time, constant along each outgoing ray. Reading it: $v$ stays finite on the future horizon whereas $t$ diverges there — this single substitution is the whole construction. Substituting $\mathrm{d}t=\mathrm{d}v-\mathrm{d}r_*$ into (1) and cancelling the two $\mathrm{d}r^2$ terms yields the advanced (ingoing) Eddington–Finkelstein metric.

\mathrm{d}s^2=-\Big(1-\frac{2GM}{r}\Big)\mathrm{d}v^2+2\,\mathrm{d}v\,\mathrm{d}r+r^2\,\mathrm{d}\Omega^2 \qquad (4)

where: $g_{vv}=-(1-2GM/r)$ vanishes on the horizon but does not diverge; $g_{vr}=1$ is the cross term that keeps the metric nondegenerate. Reading it: there is no $(1-2GM/r)^{-1}$ anywhere — every coefficient is finite for all $r>0$ , so the chart reaches from infinity to the genuine singularity at $r=0$ . Regularity is proved by the determinant. A vanishing coefficient is harmless; only a vanishing or infinite determinant would signal breakdown, and here it does neither.

g_{\mu\nu}=\begin{pmatrix}-(1-2GM/r)&1&0&0\\1&0&0&0\\0&0&r^2&0\\0&0&0&r^2\sin^2\theta\end{pmatrix},\qquad \det g=-r^4\sin^2\theta \qquad (5)

where: $\det g=-r^4\sin^2\theta$ is finite and nonzero for every $r>0$ . Reading it: the $(v,r)$ block has determinant $-1$ independent of $r$ , so the horizon is no more special than any other radius for invertibility. Inverting the block gives $g^{vv}=0$ , $g^{vr}=1$ , and — crucial for the causal analysis — $g^{rr}=1-2GM/r$ . Setting $\mathrm{d}s^2=0$ in the advanced metric gives the two edges of the radial light cone — where the "tipping" becomes an equation.

\frac{\mathrm{d}v}{\mathrm{d}r}=0\ \ (\text{ingoing}),\qquad \frac{\mathrm{d}v}{\mathrm{d}r}=\frac{2}{1-2GM/r}\ \ (\text{outgoing}) \qquad (6)

where: the ingoing branch $v=\text{const}$ is regular for all $r$ ; the outgoing slope is positive outside, diverges at $r=2GM$ , and turns negative inside. Reading it: outside, "outgoing" rays truly move to larger $r$ ; at $r=2GM$ the outgoing photon is stuck; inside, even outgoing rays are dragged to smaller $r$ — nothing can raise its radius. The Kretschmann scalar, a coordinate-independent measure of curvature, certifies that $r=2GM$ is regular and $r=0$ is not.

K=R_{abcd}R^{abcd}=\frac{48\,G^2M^2}{r^6},\qquad K\big|_{r=2GM}=\frac{3}{4\,(GM)^4} \qquad (7)

where: $K$ is finite at the horizon and diverges as $r^{-6}$ toward $r=0$ . Reading it: a genuine singularity is where a curvature scalar blows up — only at $r=0$ . The horizon is smooth by this absolute test, confirming the coordinate diagnosis. For horizon-penetrating diagrams one needs $v$ explicitly in terms of $(t,r)$ — just (3) with $r_*$ substituted.

v=t+r+2GM\,\ln\!\left|\frac{r}{2GM}-1\right| \qquad (8)

where: $v$ is the ingoing null coordinate used as the vertical axis (often $v-r$ ) of the Finkelstein diagram. Reading it: at fixed $v$ , sending $r\to2GM$ forces $t\to+\infty$ — the finite- $v$ crossing is the infinite- $t$ freeze the second figure shows.

Derivation ABuilding the advanced chart and killing the coordinate singularity

The goal is to derive the advanced Eddington–Finkelstein metric (4) from the Schwarzschild metric (1) by the single substitution $t\to v=t+r_*$ , watching the divergent radial coefficient $g_{rr}=(1-2GM/r)^{-1}$ cancel exactly against the cross term the transformation produces, leaving every coefficient finite at $r=2GM$ . The "Schwarzschild singularity" is dismantled in three lines of calculus, and the off-diagonal $g_{vr}$ that replaces it is exactly what keeps the metric invertible on the horizon.

Step 1. From $v=t+r_*$ with $\mathrm{d}r_*/\mathrm{d}r=(1-2GM/r)^{-1}$ , differentiate: $\mathrm{d}t=\mathrm{d}v-(1-2GM/r)^{-1}\mathrm{d}r$ . Squaring, $\mathrm{d}t^2=\mathrm{d}v^2-2(1-2GM/r)^{-1}\mathrm{d}v\,\mathrm{d}r+(1-2GM/r)^{-2}\mathrm{d}r^2$ . Why: we re-express the misbehaving coordinate $t$ in terms of a coordinate $v$ finite on the horizon; every cancellation flows from this line.

Step 2. Multiply by $g_{tt}=-(1-2GM/r)$ : the $\mathrm{d}t^2$ term becomes $-(1-2GM/r)\mathrm{d}v^2+2\,\mathrm{d}v\,\mathrm{d}r-(1-2GM/r)^{-1}\mathrm{d}r^2$ . Why: the cross term has lost its denominator — the $(1-2GM/r)$ from $g_{tt}$ cancels the $(1-2GM/r)^{-1}$ from the transformation, giving the clean, horizon-regular $2\,\mathrm{d}v\,\mathrm{d}r$ .

Step 3. Add the untouched Schwarzschild radial piece $+(1-2GM/r)^{-1}\mathrm{d}r^2$ ; it is equal and opposite to the leftover $\mathrm{d}r^2$ term from Step 2, so the two divergent pieces annihilate, leaving $\mathrm{d}s^2=-(1-2GM/r)\mathrm{d}v^2+2\,\mathrm{d}v\,\mathrm{d}r+r^2\mathrm{d}\Omega^2$ . Why: this cancellation is the whole insight — the divergence of $g_{rr}$ was never physical but a Jacobian standing on its head, and choosing $v$ turns it right way up.

Step 4. Inspect the result at $r=2GM$ : $g_{vv}=0$ , $g_{vr}=1$ , and the $(v,r)$ block has determinant $-1\neq0$ . The metric is smooth and invertible; the surface is an ordinary point of the chart. Why: invertibility, not the vanishing of a single component, is the true test of a good coordinate system, and $g_{vr}=1$ supplies exactly the nondegeneracy that diagonal Schwarzschild coordinates lacked.

Go deeper · Derivation B · 1 minFinite proper time to cross the horizon

Derivation BFinite proper time to cross the horizon

The goal is to show quantitatively that a radially infalling particle reaches and passes $r=2GM$ — indeed reaches $r=0$ — in finite proper time, so the horizon is not a physical edge, while understanding why the same crossing takes infinite Schwarzschild coordinate time. We use the conserved energy, reduce the radial geodesic to first order, and integrate, getting a closed form smooth at $r=2GM$ and finite to $r=0$ .

Step 1. For radial timelike geodesics the normalization $u^au_a=-1$ with the conserved energy $E=-\xi_au^a$ ( $\xi=\partial_t$ ) gives $(\mathrm{d}r/\mathrm{d}\tau)^2=E^2-(1-2GM/r)$ . Why: this is the GR energy equation for radial free fall; the $(1-2GM/r)$ term is finite at the horizon — a first hint that nothing dramatic happens to $\tau$ there.

Step 2. Set $E=1$ (dropped from rest at infinity). Then $(\mathrm{d}r/\mathrm{d}\tau)^2=2GM/r$ , so $\mathrm{d}r/\mathrm{d}\tau=-\sqrt{2GM/r}$ . Why: the $1$ and the $-1$ cancel, leaving a Newtonian-looking free-fall law completely regular at $r=2GM$ — the horizon is invisible to proper-time dynamics.

Step 3. Integrate: $\Delta\tau=\int_{r_2}^{r_1}\sqrt{r/2GM}\,\mathrm{d}r=\tfrac{2}{3}(r_1^{3/2}-r_2^{3/2})/\sqrt{2GM}$ . The integrand is smooth through $r=2GM$ ; setting $r_2=0$ still gives a finite answer, and the horizon-to-singularity time is $\Delta\tau=\tfrac{2}{3}(2GM)^{3/2}/\sqrt{2GM}=\tfrac{4}{3}GM=\tfrac{2}{3}r_s$ . Why: an elementary finite integral settles the four-decade confusion — the crossing is quick, and the interior lifetime is set by the mass alone.

Step 4. Contrast the coordinate time: $\mathrm{d}t/\mathrm{d}r=-(1-2GM/r)^{-1}\sqrt{r/2GM}$ , so $\Delta t$ diverges logarithmically as $r_2\to2GM$ . But $v(r)$ stays finite, since $r_*\to-\infty$ absorbs $t\to+\infty$ . Why: the divergence lived only in $t$ ; both $\tau$ and $v$ sail through, which is why the advanced chart makes the infalling worldline smooth.

Go deeper · Derivation C · 1 minThe horizon is a null hypersurface and a Killing horizon

Derivation CThe horizon is a null hypersurface and a Killing horizon

The goal is to establish, coordinate-independently, that $r=2GM$ is a null hypersurface — its normal is a null vector, the precise statement that it is a one-way causal membrane — and to identify it as a Killing horizon on which the static Killing field goes null, the structure underwriting surface gravity and the Hawking temperature. The computation is short but load-bearing: it converts "cones tip over" into an invariant classification.

Step 1. Write the surface as the level set $\Phi(r)=r-2GM=0$ ; its normal covector is $n_\mu=\partial_\mu\Phi=\delta_\mu^{\,r}$ , and the character of the surface is fixed by $g^{\mu\nu}n_\mu n_\nu=g^{rr}$ . Why: a hypersurface is spacelike, timelike, or null according as its normal is timelike, spacelike, or null, and for a constant- $r$ surface that norm is just $g^{rr}$ .

Step 2. Inverting the advanced metric gives $g^{rr}=1-2GM/r$ , so $\|n\|^2=1-2GM/r$ : positive (timelike surface) outside, negative (spacelike surface) inside, exactly zero on $r=2GM$ . Why: $g^{rr}$ equals the factor that vanished in $g_{tt}$ — coordinate breakdown reappears as the honest statement that the normal goes null.

Step 3. A null normal means the surface is tangent to the local light cone and ruled by null generators; signals cross it one way only — the definition of an event horizon, the boundary of the region from which no future-directed causal curve reaches infinity. Why: "one-way membrane" is the consequence of a null normal — the outgoing cone edge lies in the surface, so future-directed motion carries you across, never back.

Step 4. The static Killing field $\xi=\partial_v$ has norm $\xi^a\xi_a=g_{vv}=-(1-2GM/r)$ , also vanishing on $r=2GM$ : a Killing field null on a null hypersurface makes it a Killing horizon, with surface gravity $\kappa=1/(4GM)$ . Why: this is the entry point to black-hole thermodynamics — $\kappa$ becomes the Hawking temperature $T_H=\kappa/2\pi$ .

Limiting cases

Behaviour of the advanced Eddington–Finkelstein metric (4) in physically distinct regimes:

Case	Result	Meaning
Far field $r\gg2GM$	$\mathrm{d}s^2\to-\mathrm{d}v^2+2\,\mathrm{d}v\,\mathrm{d}r+r^2\mathrm{d}\Omega^2$	With $v\to t+r$ this is flat Minkowski space in advanced null coordinates; the spacetime is asymptotically flat.
On the horizon $r=2GM$	$\mathrm{d}s^2\big\vert_{2GM}=2\,\mathrm{d}v\,\mathrm{d}r+(2GM)^2\mathrm{d}\Omega^2$	Finite and nondegenerate ( $\det g\neq0$ ); the $\mathrm{d}v^2$ term drops out, so $v=\text{const}$ is null — a null hypersurface.
Toward $r\to0$	$K=48G^2M^2/r^6\to\infty$	The Kretschmann scalar diverges; the chart stays valid down to but not at $r=0$ , the genuine curvature singularity.
Killing field on horizon	$\xi=\partial_v,\ \ \xi^a\xi_a=-(1-2GM/r)\to0$	The static Killing vector goes null on $r=2GM$ , a Killing horizon with surface gravity $\kappa=1/(4GM)$ — foundation of the Hawking temperature.
Retarded chart $u=t-r_*$	$\mathrm{d}s^2=-(1-2GM/r)\mathrm{d}u^2-2\,\mathrm{d}u\,\mathrm{d}r+r^2\mathrm{d}\Omega^2$	Same algebra, opposite cross-term sign ( $g_{ur}=-1$ ); regularizes the past horizon — a white hole, cones tip outward, matter can only emerge.

Common trapThe horizon is not a singularity, and the two null charts are not the same spacetime

Two traps trip up newcomers. First, $r=2GM$ is a coordinate singularity of the Schwarzschild chart, not a singularity of spacetime: invariants like $K=48G^2M^2/r^6$ are finite there and the advanced metric is smooth across it. The genuine curvature singularity is at $r=0$ , and no coordinate change removes it, because $K\to\infty$ is invariant. Second, the ingoing ( $v$ ) and outgoing ( $u$ ) charts are not two names for one geometry: the advanced chart regularizes the future horizon and covers a black-hole interior, the retarded chart a past horizon and a white-hole interior. They are complementary patches of the maximal (Kruskal–Szekeres) spacetime, and using the retarded chart to describe astrophysical collapse inverts the causal arrow and describes a physically different object.

Worked examples

1 · Sagittarius A*: horizon radius and the interior free-fall time (fully worked)

The Milky Way's central black hole, Sgr A, has mass $M=4.15\times10^6\,M_\odot$ (from the star S2's orbit). Locate its horizon, and compute the proper time for a probe falling from rest at infinity to travel from horizon to singularity. Contrast with what a distant observer sees.*

Step 1. The horizon is at $r_s=2GM/c^2$ . Using $GM_\odot/c^2=1.477$ km per solar mass, $r_s=2\times1.477\ \text{km}\times4.15\times10^6=\mathbf{1.23\times10^{10}\ m}$ , about $0.082$ AU or $17.6$ solar radii. This is the areal radius of the surface, and it fixes the scale of everything that follows.

Step 2. From Derivation 2 the proper time from horizon to singularity for an $E=1$ infall is $\Delta\tau=\tfrac{2}{3}r_s/c=\tfrac{2}{3}\times(1.23\times10^{10})/(3.00\times10^8)=\mathbf{27\ s}$ . The interior of a supermassive black hole is, for the infaller, brief and finite — under half a minute for Sgr A, and by the same formula about $11.9$ hours for the far larger M87 ( $6.5\times10^9\,M_\odot$ ).

Step 3. The coordinate time tells the opposite story. From (8), holding $v$ finite as $r\to2GM$ forces $t\to+\infty$ ; the crossing time diverges logarithmically, and every photon the probe emits on approach is redshifted, the last ones without bound. So the distant astronomer never records the crossing: the image slows, dims, reddens, and freezes on the horizon — the "frozen star," the old name for a black hole.

Step 4. There is no contradiction: $\tau$ and $t$ are different clocks. The wristwatch reads $27$ s across the interior, while $t$ labels light's arrival at infinity and runs to $\infty$ ; the advanced coordinate $v$ , finite at crossing, follows the probe rather than its escaping light. Result: $r_s=1.23\times10^{10}$ m, interior free-fall $\Delta\tau=27$ s, distant crossing time $t\to\infty$ .

2 · Reading the light cones: the outgoing slope and the stuck photon

Use (6) to evaluate the outgoing radial null slope $\mathrm{d}v/\mathrm{d}r=2/(1-2GM/r)$ at $r=4GM$ , $3GM$ , $2GM$ , $1.5GM$ , and interpret the sign changes. Then confirm an ingoing photon reaches $r=0$ at finite $v$ .

Step 1. Outside: at $r=4GM$ , $1-2GM/r=\tfrac12$ , so $\mathrm{d}v/\mathrm{d}r=\mathbf{4}$ ; at $r=3GM$ , $1-2GM/r=\tfrac13$ , so $\mathrm{d}v/\mathrm{d}r=\mathbf{6}$ . Both positive and finite: the outgoing photon climbs to larger $r$ , and the slope steepens toward the horizon — the cone pinching shut.

Step 2. On the horizon: at $r=2GM$ , $1-2GM/r=0$ , so $\mathrm{d}v/\mathrm{d}r\to\infty$ . The outgoing edge is vertical — the photon aimed outward stays exactly at $r=2GM$ , neither escaping nor falling. This tangency is what makes the horizon one-way.

Step 3. Inside: at $r=1.5GM$ , $1-2GM/r=-\tfrac13$ , so $\mathrm{d}v/\mathrm{d}r=\mathbf{-6}$ . The slope has flipped: the "outgoing" photon now moves to smaller $r$ . Both cone edges point inward — no future-directed ray can raise its radius.

Step 4. The ingoing branch is trivial: $\mathrm{d}v/\mathrm{d}r=0$ , valid for all $r>0$ with no sign change. A photon launched at $v=v_0$ stays there straight down to $r=0$ , arriving at finite advanced time. Result: outgoing slope $4\to6\to\infty\to-6$ across $r=4GM,3GM,2GM,1.5GM$ ; ingoing photons cross to $r=0$ at fixed $v$ .

Physics of the portrait

The regularization of the horizon is a story of a result found, forgotten, and found again — a coordinate Eddington wrote down for an unrelated reason in 1924, that nobody recognized, and that Finkelstein reintroduced in 1958 with the causal interpretation that finally made the black hole a respectable object of study.

Origins · 1 minDavid Finkelstein and how this was discovered

David Finkelstein

1929–2016 · the null-coordinate extension that made the Schwarzschild horizon a one-way membrane

David Finkelstein was an American physicist whose 1958 paper "Past-Future Asymmetry of the Gravitational Field of a Point Particle" (Physical Review 110, 965) did for the Schwarzschild solution what no amount of staring at the diagonal metric could: it showed that $r=2GM$ is "a perfect unidirectional membrane: causal influences can cross it in only one direction." He introduced the advanced-time coordinate $v=t+r_*$ , obtained the horizon-penetrating metric, and — the decisive step — read off its causal meaning, distinguishing the black-hole (future) from the white-hole (past) horizon. The coordinate was not new: Eddington had written essentially the same transformation in 1924, comparing Einstein's and Whitehead's theories, but with no inkling of horizons, and it was forgotten; Lemaître (1932) had likewise shown the surface passable. Finkelstein's null-coordinate form was the one that caught on, arriving as Wheeler's group revived the subject; within two years Kruskal and Szekeres had the maximal extension, and the modern theory of black holes was underway. This three-page paper, almost an aside, turned a forty-year-old "singularity" into geometry.

History

1916

— Schwarzschild publishes the exterior vacuum solution; the coefficient blow-up at r = 2GM is read as a physical edge

1924

— Eddington writes the advanced-time transformation, without recognizing the horizon

1932

— Lemaître shows the r = 2GM singularity is only apparent, removable by coordinates

1958

— Finkelstein introduces v = t + r*, the one-way membrane, and the past-future asymmetry

1960

— Kruskal and Szekeres give the maximal analytic extension covering all four regions

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §31.4 · Wald, General Relativity (1984), §6.4 and §12.5 (Killing horizons) · Carroll, Spacetime and Geometry (2004), §5.6–5.7 · Hartle, Gravity (2003), Ch. 12 · Schutz, A First Course in General Relativity (2nd ed., 2009), §11.4. Historical: Finkelstein, Phys. Rev. 110, 965 (1958); Eddington, Nature 113, 192 (1924); Lemaître (1933). Key results to keep in mind: the $t\to v$ substitution causes the two $\mathrm{d}r^2$ terms to cancel exactly, yielding (4); the metric determinant is $\det g=-r^4\sin^2\theta$ and the inverse component is $g^{rr}=1-2GM/r$ ; the Kretschmann scalar is $K=48G^2M^2/r^6$ , giving $K(2GM)=3/(4(GM)^4)$ , which is finite; the free-fall proper time is $\Delta\tau=\tfrac{2}{3}(r_1^{3/2}-r_2^{3/2})/\sqrt{2GM}$ for $E=1$ , with interior lifetime $\tfrac{2}{3}r_s$ ; and the surface gravity is $\kappa=1/(4GM)$ . For Sgr A*, using $M=4.15\times10^6\,M_\odot$ and $GM_\odot/c^2=1.477$ km, these give $r_s=1.23\times10^{10}$ m and $\Delta\tau=27$ s.

Builds on: The Schwarzschild Metric · Schwarzschild Radius and the Geometric Mass · Geodesic Equations in Schwarzschild Spacetime · Killing Vectors and Conserved Quantities in Schwarzschild. Leads to: Photon Orbits and the Photon Sphere · Birkhoff's Theorem · Gravitational Redshift in Schwarzschild Spacetime.

Ask the tutor: Ask me to walk through the $t\to v$ substitution line by line, plot the tipping light cones for your favourite black hole, or contrast the advanced chart with Kruskal–Szekeres and Painlevé–Gullstrand coordinates.

If r = 2GM is not a real singularity, why did it fool physicists for forty years?

Because the failure is subtle and the diagonal metric hides it: $g_{rr}=(1-2GM/r)^{-1}$ literally diverges at $r=2GM$ , so at a glance the geometry looks pathological. Telling a coordinate singularity from a physical one requires a coordinate-independent quantity — the Kretschmann scalar $K=48G^2M^2/r^6$ — and noticing it stays finite at the horizon while diverging only at $r=0$ . That diagnostic was not routine in 1916, the idea of a manifold covered by overlapping charts was still developing, and a star collapsing inside its own horizon seemed too extreme to take seriously. It took Finkelstein's 1958 causal reading to make the point stick.

What exactly is the tortoise coordinate, and why that name?

The tortoise coordinate is $r_*=r+2GM\ln|r/2GM-1|$ , defined so that $\mathrm{d}r_*=(1-2GM/r)^{-1}\mathrm{d}r$ . Its purpose is to make radial light rays into straight $45^\circ$ lines: in $(t,r_*)$ the radial null condition is simply $t=\pm r_*+\text{const}$ . The name nods to Zeno's Achilles and the tortoise — as $r\to2GM$ the logarithm sends $r_*\to-\infty$ , so the horizon is pushed infinitely far away and a light ray takes "forever" in $r_*$ to reach it. That infinite stretching encodes the freezing of Schwarzschild time; the advanced coordinate $v=t+r_*$ then repackages the $-\infty$ to cancel the $+\infty$ in $t$ , giving a finite crossing.

How can g_vv vanish at the horizon without anything breaking?

Because "the metric is fine" means the metric tensor is smooth and invertible, not that every component is nonzero. What matters is the determinant. In the advanced chart $g_{vv}=-(1-2GM/r)$ does vanish at $r=2GM$ , but the off-diagonal $g_{vr}=1$ keeps the $(v,r)$ block's determinant equal to $-1$ everywhere, so $\det g=-r^4\sin^2\theta$ is finite and nonzero for all $r>0$ . The vanishing of $g_{vv}$ is a statement, not a pathology: the line $v=\text{const}$ has become null on the horizon — the geometric fact that the surface is a null hypersurface. Contrast Schwarzschild coordinates, where the divergence of $g_{rr}$ genuinely destroys invertibility, which is why that chart fails and the advanced one does not.

Do Eddington–Finkelstein coordinates remove the r = 0 singularity too?

No, and nothing can. The horizon at $r=2GM$ is a coordinate singularity: an artifact of the Schwarzschild chart that disappears in the advanced chart, where every coefficient is finite. The point $r=0$ is a curvature singularity: the Kretschmann scalar $K=48G^2M^2/r^6$ diverges there, and because $K$ is a coordinate-independent invariant, no change of coordinates can make it finite. The advanced metric stays valid arbitrarily close to $r=0$ and describes the smooth infall right up to it, but at $r=0$ tidal forces are infinite and the classical theory breaks down. Removing the horizon's apparent pathology while leaving $r=0$ singular is exactly how the coordinates reveal which is which.

Why does the advanced chart describe a black hole and the retarded chart a white hole?

They regularize different horizons. The advanced coordinate $v=t+r_*$ is constant along ingoing light and stays finite on the future horizon; extending across it opens the black-hole interior, in the future of the exterior, from which nothing escapes and toward which cones tip inward. The retarded coordinate $u=t-r_*$ is constant along outgoing light and stays finite on a past horizon; extending across it opens a white-hole interior, in the past of the exterior, from which things can only emerge and toward which cones tip outward. Time-reverse a black hole and you get a white hole, and that reversal is exactly the swap $v\leftrightarrow u$ . Both live inside the maximally extended (Kruskal–Szekeres) spacetime; each chart is a one-sided view, and real collapse selects the advanced one.

Are these coordinates only a teaching device, or do physicists actually use them?

They are used constantly. Every numerical-relativity code evolving a black hole needs a slicing that passes smoothly through the horizon, or the grid stiffens and the run crashes; horizon-penetrating coordinates in the Eddington–Finkelstein family (and the related Kerr–Schild and Painlevé–Gullstrand forms) are the standard answer, and the "moving puncture" gauge behind modern binary-merger waveforms is built on that idea. The null expansions used to locate apparent horizons are computed most naturally in null coordinates, and the ray-tracing behind Event Horizon Telescope images follows photons across horizon-penetrating grids. Finkelstein's 1958 coordinate is embedded in the daily practice of strong-gravity astrophysics.