Horizons and Coordinate Singularities

The Schwarzschild metric describes the vacuum outside any static, spherical mass. For an ordinary star the solution applies only down to the stellar surface, well outside the radius $r_{s} = 2 GM / c^{2}$ . When the mass is compact enough that its surface lies inside $r_{s}$ , the vacuum solution extends across $r_{s}$ , and that surface becomes a one-way membrane: the event horizon. The metric written in the static coordinates $(t, r, θ, ϕ)$ misbehaves there, and separating the genuine physics from the coordinate artifact is the first task.

Two radii where the metric degenerates

In Schwarzschild coordinates with signature $(-, +, +, +)$ ,

d s^{2} = - (1 - \frac{r _{s}}{r}) c^{2} d t^{2} + (1 - \frac{r _{s}}{r})^{- 1} d r^{2} + r^{2} d Ω^{2}, d Ω^{2} = d θ^{2} + sin^{2} θ d ϕ^{2} .

Two radii break the line element. At $r = 0$ the coefficient of $c^{2} d t^{2}$ diverges; at $r = r_{s}$ the coefficient of $d r^{2}$ diverges while the coefficient of $c^{2} d t^{2}$ vanishes. A diverging metric component does not by itself signal anything physical, because a metric component is a coordinate-dependent number. A bad choice of coordinates degrades a component even in flat space: polar coordinates make $g_{ϕϕ} = r^{2} sin^{2} θ$ vanish on the axis, yet nothing is wrong there. The invariant question is whether a scalar built from the curvature diverges.

The relevant invariant is the Kretschmann scalar, the full contraction of the Riemann tensor with itself. For Schwarzschild,

K = R_{μν ρ σ} R^{μν ρ σ} = \frac{48 G ^{2} M ^{2}}{c ^{4} r ^{6}} = \frac{12 r _{s}^{2}}{r ^{6}} .

At $r = r_{s}$ this equals $12/ r_{s}^{4}$ , a finite number that grows as the hole shrinks but never blows up for a hole of nonzero mass. At $r = 0$ it diverges. The horizon is therefore a coordinate singularity — an artifact of the static chart — while $r = 0$ is a curvature singularity, a place where the geometry itself is unbounded and no coordinate change can repair it.

The tidal field at the horizon is set by $K^{1/2} \sim 1/ r_{s}^{2} \propto 1/ M^{2}$ . For a stellar-mass hole ( $r_{s}$ of a few kilometers) it is lethal; for the $4 \times 1 0^{6} M_{⊙}$ hole at the galactic center ( $r_{s} \approx 1.2 \times 1 0^{10} m$ ) it is gentler than Earth's tide at the horizon. Crossing the horizon is a local non-event; the coordinates, not the geometry, are what fail there.

Radial light rays and the tortoise coordinate

The pathology of the static chart shows up in the motion of light. For a radial null ray, $d s^{2} = 0$ and $d Ω = 0$ give

c \frac{d t}{d r} = \pm (1 - \frac{r _{s}}{r})^{- 1} .

As $r \to r_{s}^{+}$ the coordinate slope $d t / d r \to \infty$ : in the $(c t, r)$ diagram the light cones close up, and an infalling ray takes an infinite coordinate time $t$ to reach the horizon even though it is an ordinary null path. Integrating the slope defines the tortoise coordinate

r_{*} = r + r_{s} ln \frac{r}{r _{s}} - 1, c d t = \pm d r_{*},

which pushes the horizon to $r_{*} \to - \infty$ . Radial null rays are the straight lines $c t \mp r_{*} = const$ , but no finite coordinate patch covers the crossing.

In static Schwarzschild coordinates the radial light cones narrow to slivers as the horizon is approached, so an infalling ray needs unbounded coordinate time t to reach r_s even though its proper time is finite.

Eddington–Finkelstein coordinates

The tortoise coordinate suggests a null coordinate adapted to infalling light. Define the advanced time

v = c t + r_{*},

constant along each ingoing radial null ray. Replacing $c t$ by $v$ turns the line element into

d s^{2} = - (1 - \frac{r _{s}}{r}) d v^{2} + 2 d v d r + r^{2} d Ω^{2} .

Every coefficient is finite and smooth at $r = r_{s}$ ; the determinant of the $(v, r)$ block is $- 1$ , so the metric is non-degenerate there. The horizon was never a place where geometry broke; the static chart simply could not be continued across it. In these ingoing Eddington–Finkelstein coordinates the two families of radial null rays are

v = const (ingoing), \frac{d v}{d r} = \frac{2}{1 - r _{s} / r} (outgoing) .

The ingoing rays cross the horizon at finite $v$ and continue to $r = 0$ . The outgoing family has $d v / d r \to \infty$ at $r_{s}$ and, for $r < r_{s}$ , $d v / d r$ changes sign so that outgoing rays actually move to smaller $r$ . Inside the horizon both null directions point inward.

Ingoing Eddington–Finkelstein diagram with v as a slanted time axis. Ingoing rays are straight 45-degree lines that cross the horizon smoothly; the outgoing family (open cone edge) tips inward until, past r_s, both edges point to r=0.

An outgoing Eddington–Finkelstein coordinate $u = c t - r_{*}$ produces the time-reverse: a metric regular across a surface that only lets matter out. That patch describes a white hole, the time-reverse of a black hole, and it is a separate region of the fully extended geometry rather than a feature of any hole formed by collapse.

Maximal extension: Kruskal–Szekeres coordinates

The ingoing chart covers the black-hole interior; the outgoing chart covers the white-hole interior; neither covers both plus both asymptotic regions. The maximal analytic extension that shows the whole geometry at once uses the Kruskal–Szekeres coordinates $(T, X)$ . In the exterior $r > r_{s}$ ,

X = (\frac{r}{r _{s}} - 1)^{1/2} e^{r /2 r_{s}} cosh \frac{c t}{2 r _{s}}, T = (\frac{r}{r _{s}} - 1)^{1/2} e^{r /2 r_{s}} sinh \frac{c t}{2 r _{s}},

and the line element becomes

d s^{2} = \frac{4 r _{s}^{3}}{r} e^{- r / r_{s}} (- d T^{2} + d X^{2}) + r^{2} d Ω^{2},

with $r$ defined implicitly by

X^{2} - T^{2} = (\frac{r}{r _{s}} - 1) e^{r / r_{s}} .

The prefactor $4 r_{s}^{3} e^{- r / r_{s}} / r$ is finite and positive at $r = r_{s}$ , so the metric is regular across the horizon. In these coordinates radial null rays are exactly the $4 5^{\circ}$ lines $T = \pm X + const$ , so light cones are drawn the same way everywhere, as in flat spacetime.

The map has four branches, giving the diagram four regions separated by the two diagonals $T = \pm X$ (these lines are the horizon, $r = r_{s}$ ):

Region I ( $X > ∣ T ∣$ ): the exterior we live in, $r > r_{s}$ .
Region II ( $T > ∣ X ∣$ ): the black-hole interior. The curvature singularity $r = 0$ is the upper hyperbola $T^{2} - X^{2} = 1$ , a spacelike surface lying in the future of everything in Region II.
Region III ( $X < - ∣ T ∣$ ): a second, causally disconnected exterior.
Region IV ( $T < - ∣ X ∣$ ): the white-hole interior, the past time-reverse of Region II.

Kruskal–Szekeres diagram. Light cones are 45 degrees everywhere; the two diagonals are the horizon; regions I and III are the two exteriors, II the black-hole interior capped by the spacelike singularity, IV the white hole.

Regions III and IV do not exist for a hole formed by the collapse of a star: the collapsing matter replaces them, and the physical spacetime keeps only Region I, part of Region II, and the interior filled by the star. The full four-region diagram is the maximal extension of the eternal vacuum solution, useful because it displays the causal structure — including the fact that the singularity is a spacelike surface in the future, not a place ahead of an infalling observer in space.

What each observer sees

The horizon has opposite characters for the two natural observers.

The same infall in two clocks. Proper time along the worldline (left axis) reaches the horizon and the center at finite values; coordinate time measured at infinity (right curve) diverges at the horizon, and the received light reddens without limit.

The disagreement is not a paradox but a statement about the horizon's null character. The surface $r = r_{s}$ is generated by the outgoing light rays that never escape and never fall in; they hover at fixed $r$ forever. Coordinate time $t$ is the natural parameter along those rays, so it runs to infinity as any material worldline approaches them, while the proper time of a worldline that actually crosses stays finite. The next lesson keeps the horizon but adds two attributes a real astrophysical hole carries — rotation and charge — which split the single Schwarzschild surface into a richer structure.