---
title: Horizons and Coordinate Singularities
module: Black Holes
moduleNumber: 8
lessonNumber: 1
order: 801
summary: >
  The Schwarzschild radius is a coordinate singularity, not a curvature
  singularity: the metric blows up there only because the static coordinates
  fail, while the geometry stays finite. Eddington–Finkelstein and Kruskal–
  Szekeres coordinates cross the horizon smoothly and show the light cones
  tipping toward the center. A freely falling observer reaches the true
  singularity at r=0 in finite proper time, while a distant observer sees the
  infall freeze and redden at the horizon.
topics: [Black Holes]
draft: false
sources:
  - book: Hartle
    ref: "Gravity, Ch. 12 — Gravitational Collapse to Black Holes"
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §7 — The Schwarzschild Solution and Black Holes"
---

The [Schwarzschild metric](/relativity/the-schwarzschild-solution/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
$r_s = 2GM/c^2$. 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 $(t, r, \theta, \phi)$ 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 $(-,+,+,+)$,

$$
\d s^2 = -\left(1 - \frac{r_s}{r}\right)c^2\,\d t^2
       + \left(1 - \frac{r_s}{r}\right)^{-1}\d r^2
       + r^2\,\d\Omega^2,
\qquad \d\Omega^2 = \d\theta^2 + \sin^2\theta\,\d\phi^2.
$$

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 $g_{\phi\phi} = r^2\sin^2\theta$ 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,

$$
K = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}
  = \frac{48\,G^2 M^2}{c^4\,r^6}
  = \frac{12\,r_s^2}{r^6}.
$$

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.

> **Definition (Event horizon).** For the Schwarzschild geometry the event
> horizon is the null surface $r = r_s = 2GM/c^2$. It is the boundary of the
> region from which no causal curve reaches infinity: every future-directed
> timelike or null path that crosses it inward can never return to $r > r_s$.
> Its area is $A = 4\pi r_s^2 = 16\pi G^2 M^2/c^4$.

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
$4\times10^6\,M_\odot$ hole at the galactic center ($r_s \approx 1.2\times10^{10}\,\text{m}$)
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 $\d\Omega = 0$ give

$$
c\,\frac{\d t}{\d r} = \pm\left(1 - \frac{r_s}{r}\right)^{-1}.
$$

As $r \to r_s^+$ the coordinate slope $\d t/\d r \to \infty$: in the $(ct, r)$
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**

$$
r_\ast = r + r_s \ln\left|\frac{r}{r_s} - 1\right|,
\qquad c\,\d t = \pm\,\d r_\ast,
$$

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

$$
% caption: In static Schwarzschild coordinates the radial light cones narrow to
% slivers as the horizon is approached, so an infalling ray needs unbounded
% coordinate time t to reach r_s even though its proper time is finite.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,-0.3) -- (0,4.6) node[above] {ct};
  \draw[->, black] (-0.3,0) -- (5.6,0) node[right] {r};
  \draw[acc, very thick] (1.0,-0.15) -- (1.0,4.4);
  \node[acc, anchor=south west, rotate=90] at (1.02,1.3) {horizon r equals rs};
  % upward-opening light cones: opening angle widens with r (pinched at horizon)
  \foreach \x/\h in {1.5/0.12, 2.7/0.28, 3.9/0.46, 5.1/0.62} {
    \draw[black, thick] (\x,1.6) -- (\x+\h,2.35);
    \draw[black, thick] (\x,1.6) -- (\x-\h,2.35);
  }
  \node[black, anchor=west] at (2.5,3.6) {cones open far away};
  \node[black, anchor=west] at (1.3,0.7) {cones pinch at horizon};
\end{tikzpicture}
$$

## Eddington–Finkelstein coordinates

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

$$
v = ct + r_\ast,
$$

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

$$
\d s^2 = -\left(1 - \frac{r_s}{r}\right)\d v^2 + 2\,\d v\,\d r + r^2\,\d\Omega^2.
$$

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

$$
v = \text{const} \quad\text{(ingoing)},
\qquad
\frac{\d v}{\d r} = \frac{2}{\,1 - r_s/r\,} \quad\text{(outgoing)}.
$$

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.

$$
% caption: Ingoing Eddington–Finkelstein diagram with v as a slanted time axis.
% Ingoing rays are straight 45-degree lines that cross the horizon smoothly; the
% outgoing family (open cone edge) tips inward until, past r_s, both edges point
% to r=0.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.3,0) -- (6.0,0) node[right] {r};
  \draw[->, black] (0,-0.3) -- (0,4.7) node[above] {advanced time};
  \draw[very thick] (2.2,-0.15) -- (2.2,4.5);
  \node[anchor=south, rotate=90] at (2.05,2.2) {horizon};
  \draw[black] (0.2,-0.1) -- (0.2,4.5);
  \node[black, anchor=south, rotate=90] at (0.05,2.2) {singularity r equals 0};
  % ingoing null rays (straight, cross the horizon smoothly)
  \foreach \s in {2.9,3.7} {
    \draw[black, ->] (5.6,\s) -- (0.4,\s-2.4);
  }
  \node[black, anchor=west] at (4.2,3.5) {ingoing rays};
  % light cones: ingoing edge fixed at 45 deg, outgoing edge tips inward past horizon
  \foreach \x/\e in {4.6/0.7, 2.9/0.15, 1.2/-0.45} {
    \draw[acc, thick] (\x,1.6) -- (\x-0.7,2.3);
    \draw[acc, thick] (\x,1.6) -- (\x+\e,2.3);
  }
  \node[acc, anchor=west] at (3.1,0.85) {cones tip inward};
\end{tikzpicture}
$$

An **outgoing** Eddington–Finkelstein coordinate $u = ct - r_\ast$ 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 $r > r_s$,

$$
X = \left(\frac{r}{r_s} - 1\right)^{1/2} e^{r/2r_s}\cosh\frac{ct}{2r_s},
\qquad
T = \left(\frac{r}{r_s} - 1\right)^{1/2} e^{r/2r_s}\sinh\frac{ct}{2r_s},
$$

and the line element becomes

$$
\d s^2 = \frac{4\,r_s^3}{r}\,e^{-r/r_s}\left(-\d T^2 + \d X^2\right) + r^2\,\d\Omega^2,
$$

with $r$ defined implicitly by

$$
X^2 - T^2 = \left(\frac{r}{r_s} - 1\right)e^{r/r_s}.
$$

The prefactor $4r_s^3\,e^{-r/r_s}/r$ 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 $45^\circ$ lines $T = \pm X + \text{const}$, 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** ($X > \abs{T}$): the exterior we live in, $r > r_s$.
- **Region II** ($T > \abs{X}$): 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** ($X < -\abs{T}$): a second, causally disconnected exterior.
- **Region IV** ($T < -\abs{X}$): the white-hole interior, the past
  time-reverse of Region II.

$$
% caption: Kruskal–Szekeres diagram. Light cones are 45 degrees everywhere; the
% two diagonals are the horizon; regions I and III are the two exteriors, II the
% black-hole interior capped by the spacelike singularity, IV the white hole.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-3.2,0) -- (3.4,0) node[right] {X};
  \draw[->, black] (0,-3.0) -- (0,3.2) node[above] {T};
  % horizon diagonals
  \draw[acc, very thick] (-2.9,-2.9) -- (2.9,2.9);
  \draw[acc, very thick] (-2.9,2.9) -- (2.9,-2.9);
  \node[acc, anchor=south west] at (2.1,2.1) {horizon};
  % singularity hyperbolae (upper and lower)
  \draw[black!70, very thick] plot[domain=-1.55:1.55, samples=40] ({\x},{sqrt(\x*\x+1)});
  \draw[black!70, very thick, dashed] plot[domain=-1.55:1.55, samples=40] ({\x},{-sqrt(\x*\x+1)});
  \node[black!70, anchor=south] at (0,1.15) {singularity};
  % region labels
  \node[black] at (2.0,0) {I exterior};
  \node[black] at (-2.0,0) {III exterior};
  \node[black] at (0,2.0) {II hole};
  \node[black] at (0,-2.0) {IV white hole};
\end{tikzpicture}
$$

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.

> **Infalling observer.** A body released from rest at $r_0$ falls to the horizon
> and on to $r = 0$ in finite proper time. Integrating the radial timelike
> geodesic from the horizon to the center gives, for a hole of mass $M$,
> $$
> \Delta\tau = \int_0^{r_s}\frac{\d r}{c\sqrt{r_s/r}}
>            = \frac{2}{3}\,\frac{r_s}{c}
>            = \frac{4GM}{3c^3}.
> $$
> Nothing singular happens at the crossing; the traveler notices only the growing
> tidal stretch, which diverges only at $r = 0$, reached a fixed proper time
> later.

> **Distant observer.** Light emitted at radius $r$ and received at infinity is
> redshifted by
> $$
> 1 + z = \left(1 - \frac{r_s}{r}\right)^{-1/2},
> $$
> which diverges as $r \to r_s$. Signals from the infalling body arrive ever
> later and ever redder; the last photon emitted just outside the horizon takes
> infinite coordinate time to escape. The distant observer never sees the crossing:
> the image freezes at the horizon, dims exponentially on a timescale $\sim r_s/c$,
> and reddens to invisibility. The hole looks "frozen," the historical name
> _frozen star_.

$$
% caption: The same infall in two clocks. Proper time along the worldline
% (left axis) reaches the horizon and the center at finite values; coordinate
% time measured at infinity (right curve) diverges at the horizon, and the
% received light reddens without limit.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,-0.3) -- (0,4.6) node[above] {time};
  \draw[->, black] (-0.3,0) -- (5.8,0) node[right] {r};
  \draw[very thick] (1.4,-0.15) -- (1.4,4.4);
  \node[anchor=south, rotate=90] at (1.25,2.2) {horizon};
  \draw[black] (0.2,-0.1) -- (0.2,4.4);
  \node[black, anchor=south, rotate=90] at (0.05,2.2) {center};
  % proper time worldline: smooth finite curve from far r down to center
  \draw[acc, very thick] (5.4,0.4) .. controls (3.6,1.2) and (2.2,1.9) .. (1.4,2.4)
     .. controls (0.9,2.7) and (0.5,2.9) .. (0.2,3.0);
  \node[acc, anchor=south west] at (3.6,1.0) {proper time bounded};
  % coordinate time: diverges at horizon (asymptote)
  \draw[black, very thick, dashed] (5.4,0.5) .. controls (3.4,1.0) and (2.0,1.6) .. (1.7,2.6)
     .. controls (1.55,3.4) and (1.5,4.0) .. (1.47,4.4);
  \node[black, anchor=west] at (1.7,4.0) {coordinate time diverges};
\end{tikzpicture}
$$

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](/relativity/black-holes/rotating-and-charged-black-holes)
— which split the single Schwarzschild surface into a richer structure.

[^hartle-collapse]: **Hartle**, _Gravity_, Ch. 12 — gravitational collapse, the Schwarzschild horizon as a coordinate singularity, Eddington–Finkelstein coordinates, the light-cone structure near $r_s$, and the finite proper time to reach the singularity.

[^carroll-bh]: **Carroll**, _Lecture Notes on General Relativity_, §7 — the Kretschmann scalar distinguishing coordinate from curvature singularities, the tortoise coordinate, Kruskal–Szekeres coordinates and the maximally extended Schwarzschild geometry with its four regions.