---
title: Black Holes, Schwarzschild and Kerr
draft: false
module: Stellar Death and Compact Remnants
moduleNumber: 8
lessonNumber: 5
order: 805
summary: >
  Above the neutron-star mass limit gravity wins completely and the remnant is a
  black hole. The Schwarzschild solution gives the event horizon, gravitational
  redshift, and time dilation; the innermost stable circular orbit sets the
  efficiency of accretion. Rotating Kerr black holes drag spacetime and carry an
  ergosphere. Stellar-mass black holes are found in X-ray binaries, and the Event
  Horizon Telescope has imaged the shadow of a supermassive one.
topics: [Stellar Death and Compact Remnants]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 17 — General Relativity and Black Holes; §17.3 The Schwarzschild Solution; §17.4 Kerr Black Holes"
  - book: Maoz
    ref: "Ch. 4 — Stellar Death and Remnants; Ch. 6 — Interstellar Medium and Extragalactic Astronomy"
---

When a collapsing core exceeds the
[neutron-star mass
limit](/astrophysics-cosmology/stellar-death-and-compact-remnants/neutron-stars-and-pulsars),
no known pressure halts the collapse and the object becomes a black hole: a region of
spacetime from which nothing, not even light, escapes. Black holes are described
entirely by general relativity, and by only three numbers — mass, angular
momentum, and charge — the no-hair result. Astrophysical black holes carry no net
charge, so they are fixed by mass and spin alone. This lesson develops the
Schwarzschild geometry of a non-rotating black hole (horizon, redshift, time dilation),
the innermost stable circular orbit that governs how efficiently accretion extracts
energy, the Kerr geometry of a spinning black hole with its frame dragging and
ergosphere, and the observational signatures: stellar-mass black holes in X-ray
binaries and the imaged shadow of a supermassive one.

## The Schwarzschild radius and event horizon

The Schwarzschild solution is the unique spherically symmetric vacuum solution of
Einstein's equations. Outside a mass $M$ the metric is

$$
\d s^2 = -\left(1 - \frac{2GM}{rc^2}\right)c^2\,\d t^2
       + \left(1 - \frac{2GM}{rc^2}\right)^{-1}\d r^2
       + r^2\,\d\Omega^2,
$$

where $\d\Omega^2 = \d\theta^2 + \sin^2\theta\,\d\phi^2$. The metric coefficients
degenerate at the **Schwarzschild radius**

$$
R_S = \frac{2GM}{c^2}
    = 2.95\ \mathrm{km}\left(\frac{M}{M_\odot}\right).
$$

A Newtonian shortcut gives the same value: set the escape speed
$\sqrt{2GM/r}$ equal to $c$. For a $10\,M_\odot$ black hole $R_S \approx 30\
\mathrm{km}$; for the $4\times10^6\,M_\odot$ black hole at the Galactic center,
$R_S \approx 1.2\times10^7\ \mathrm{km}$, about a fifth of Mercury's orbit.

The surface $r = R_S$ is the **event horizon**. It is not a physical membrane — an
infalling observer crosses it without local incident — but a causal boundary: inside
it every future-directed path leads to smaller $r$, so no signal can climb back out.
The singularity in the metric at $r = R_S$ is a coordinate artifact, removable by a
change of coordinates; the genuine curvature singularity is at $r = 0$.

> **Definition (Event horizon).** The null surface $r = R_S = 2GM/c^2$ separating the
> region from which light can eventually reach a distant observer from the region from
> which it cannot. Its area $A = 4\pi R_S^2$ can only increase in classical processes,
> the area theorem.

$$
% caption: An embedding diagram of the Schwarzschild geometry: the spatial curvature
% deepens into a funnel toward the horizon, where the throat marks the Schwarzschild
% radius and beyond which no outgoing path exists.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% funnel: two profile curves and rings
\draw[very thick] (0.4,3.6) .. controls (2.6,3.4) and (3.4,2.0) .. (4.0,0.4);
\draw[very thick] (7.6,3.6) .. controls (5.4,3.4) and (4.6,2.0) .. (4.0,0.4);
% top rim ellipse
\draw[black] (4.0,3.55) ellipse (3.6 and 0.5);
% mid ring
\draw[black] (4.0,2.3) ellipse (1.7 and 0.28);
% throat ring
\draw[acc, very thick] (4.0,0.85) ellipse (0.55 and 0.16);
\node[acc, anchor=north, font=\scriptsize] at (4.0,0.7) {horizon};
\node[black, anchor=south, font=\scriptsize] at (7.0,3.6) {far away};
\end{tikzpicture}
$$

## Gravitational redshift and time dilation

Because the metric's time coefficient varies with radius, clocks run at
position-dependent rates. A photon emitted at radius $r$ with frequency
$\nu_{\mathrm{em}}$ and received far away arrives redshifted to

$$
\frac{\nu_{\infty}}{\nu_{\mathrm{em}}}
= \sqrt{1 - \frac{2GM}{rc^2}}
= \sqrt{1 - \frac{R_S}{r}}.
$$

As the emission radius approaches the horizon, $r \to R_S$, the received frequency
goes to zero: light from the horizon is infinitely redshifted. Equivalently, a clock
at rest at radius $r$ ticks slower than a distant clock by the same factor. An observer
watching an object fall in sees its light redshift and its clock slow without bound; the
object appears to freeze at the horizon and fade, never quite crossing it in the
distant observer's time. In the **infalling** frame, by contrast, the horizon is
crossed in finite proper time and the central singularity is reached shortly after.

> **Worked example.** Light climbing from just above the horizon of a $10\,M_\odot$
> black hole, at $r = 1.01\,R_S$, is redshifted by
> $\sqrt{1 - 1/1.01} = \sqrt{0.0099} \approx 0.10$: a factor of ten drop in frequency
> from a one-percent margin above the horizon. The redshift diverges as the margin
> closes, which is why the horizon glows ever fainter and redder to an outside watcher.

$$
% caption: The gravitational redshift factor rises from unity far out toward zero at
% the horizon, where the square-root of one minus the Schwarzschild radius over radius
% vanishes and emitted light is infinitely reddened.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {radius over RS};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {frequency ratio};
% horizon marker at r = R_S (x=1 -> plot x from 1)
\draw[black, densely dashed] (0.8,0) -- (0.8,4.0);
\node[black, anchor=south, rotate=90, font=\scriptsize] at (0.6,2.0) {horizon};
% sqrt(1 - 1/x) curve, starts 0 at x=1 rising toward 1
\draw[acc, very thick] (0.8,0.0) .. controls (1.4,2.0) and (2.6,3.1) .. (4.2,3.6)
  .. controls (5.8,3.85) and (7.2,3.95) .. (8.2,4.0);
\draw[black, densely dotted] (0,4.0) -- (8.2,4.0);
\node[black, anchor=south east, font=\scriptsize] at (8.1,4.0) {unity far away};
\end{tikzpicture}
$$

## The innermost stable circular orbit and accretion efficiency

Around a black hole, unlike in Newtonian gravity, circular orbits cease to be stable
inside a critical radius. The effective potential for orbital motion in the
Schwarzschild metric has a stable minimum only for orbits outside the **innermost
stable circular orbit (ISCO)**, at

$$
r_{\mathrm{ISCO}} = \frac{6GM}{c^2} = 3 R_S
$$

for a non-rotating black hole. Inside the ISCO, matter spirals in without further
orbiting. Accretion disks therefore terminate at the ISCO, and the binding energy of
that innermost orbit sets how much energy accretion can release.

The specific energy of a particle on the ISCO of a Schwarzschild black hole is
$E/mc^2 = \sqrt{8/9} \approx 0.943$, so the fraction of rest-mass energy radiated as
matter accretes from infinity to the ISCO is

$$
\eta = 1 - \sqrt{\frac{8}{9}} \approx 0.057,
$$

about $6\%$. Accretion onto a black hole is thus roughly ten times more efficient than
hydrogen fusion, which liberates $0.7\%$. For a rapidly spinning Kerr black hole the
ISCO shrinks toward the horizon and the efficiency rises to $\sim 42\%$, making
accretion the most efficient steady energy source known and powering the luminosity of
[active galactic nuclei](/astrophysics-cosmology/galaxies/active-galactic-nuclei-and-supermassive-black-holes).

$$
% caption: The innermost stable circular orbit shrinks and the accretion efficiency
% rises as the black-hole spin increases from a Schwarzschild value toward maximal
% Kerr, where the last stable orbit approaches the horizon.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {spin};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {energy yield};
% efficiency rising from ~0.057 at spin 0 to ~0.42 at maximal spin
\draw[acc, very thick] (0.5,0.6) .. controls (3.5,0.9) and (5.8,1.6) .. (7.4,3.9);
\node[acc, anchor=south, font=\scriptsize] at (2.6,0.75) {Schwarzschild about 6 percent};
\node[black, anchor=south east, font=\scriptsize] at (7.6,3.9) {maximal Kerr about 42 percent};
\draw[black, densely dotted] (7.4,0) -- (7.4,3.9);
\end{tikzpicture}
$$

## Kerr black holes and the ergosphere

A realistic black hole rotates, and rotation is described by the **Kerr solution**,
characterized by the mass $M$ and the angular momentum $J$, often written through the
spin parameter $a = J/Mc$ with $0 \le a \le GM/c^2$. Two surfaces appear.

- **The event horizon** shrinks with spin to
  $r_+ = (GM/c^2) + \sqrt{(GM/c^2)^2 - a^2}$, reaching $GM/c^2$ (half the
  Schwarzschild value) at the maximal spin $a = GM/c^2$.
- **The static limit** lies outside the horizon in the equatorial plane. Between the
  static limit and the horizon lies the **ergosphere**, a region where spacetime is
  dragged so strongly that no observer can remain at rest relative to distant stars;
  everything must co-rotate with the hole.

> **Definition (Frame dragging and the ergosphere).** A rotating mass drags inertial
> frames around with it (the Lense-Thirring effect). Close enough to a Kerr black hole
> the dragging becomes total: inside the **ergosphere** every timelike path must
> rotate in the direction of the spin. Energy can be extracted from the rotation by a
> particle that splits inside the ergosphere, the Penrose process, at the cost of the
> hole's angular momentum.

Spin leaves observable imprints. The reduced ISCO of a spinning hole raises the
accretion efficiency and hardens the disk spectrum, and the broadened, skewed profile
of the iron K-line reflected from the inner disk is used to measure black-hole spin.

$$
% caption: The Kerr geometry seen from the pole: the event horizon inside the static
% limit, with the ergosphere between them where frame dragging forbids any static
% observer.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% static limit (outer, oblate)
\draw[very thick] (4.0,2.0) ellipse (2.6 and 1.9);
% ergosphere fill between the two
\fill[acc!8] (4.0,2.0) ellipse (2.6 and 1.9);
\fill[white] (4.0,2.0) circle (1.2);
\draw[very thick] (4.0,2.0) circle (1.2);
\node[font=\scriptsize] at (4.0,2.0) {horizon};
\node[black, font=\scriptsize] at (4.0,3.4) {ergosphere};
\node[black, anchor=west, font=\scriptsize] at (6.7,2.0) {static limit};
% rotation arrow
\draw[->, black] (5.6,3.4) arc (40:-10:1.9);
\end{tikzpicture}
$$

## Stellar-mass black holes and the imaged shadow

Stellar-mass black holes announce themselves in **X-ray binaries**. A black hole in a
close orbit with a companion accretes gas through
[Roche-lobe overflow](/astrophysics-cosmology/binaries-and-gravitational-waves/binary-systems-and-mass-transfer);
the gas forms a disk that heats to X-ray temperatures near the ISCO. The classic case
is Cygnus X-1, where the radial-velocity curve of the supergiant companion yields a
mass function requiring an unseen object well above the neutron-star limit, near
$21\,M_\odot$, too massive to be anything but a black hole. Dozens of such
dynamically weighed black holes are now known, clustering around
$5$–$20\,M_\odot$; the gravitational-wave mergers detected by LIGO reach higher,
tens of solar masses, extending the mass function.

At the opposite scale, the **supermassive black holes** in galactic nuclei, $10^6$ to
$10^{10}\,M_\odot$, are weighed by the orbits of stars (Sgr A* at the Galactic center)
or of gas, and power [active galactic
nuclei](/astrophysics-cosmology/galaxies/active-galactic-nuclei-and-supermassive-black-holes).
The **Event Horizon Telescope**, a global array synthesizing an Earth-sized radio
dish, resolved the immediate surroundings of two supermassive black holes — M87* and
Sgr A* — imaging a bright ring of emission around a central dark region. The dark
region is the **black-hole shadow**: gravitational lensing wraps the image of the
horizon to an angular size of about $2.6\,R_S$ across, larger than the horizon itself,
ringed by a **photon ring** of light that orbited the hole before escaping. The
measured shadow diameters match the general-relativistic prediction for the
independently determined masses, a direct image test of the theory at the horizon
scale.[^co-bh]

$$
% caption: The black-hole shadow: photons on the last unstable circular orbit form a
% bright photon ring, lensing wraps the horizon into a central dark region of angular
% size about two and a half Schwarzschild radii.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% bright photon ring
\draw[acc, very thick] (4.0,2.0) circle (1.8);
\fill[acc!14] (4.0,2.0) circle (1.8);
% central shadow
\fill[black] (4.0,2.0) circle (1.15);
\draw[black] (4.0,2.0) circle (1.15);
\node[black, font=\scriptsize] at (4.0,2.0) {shadow};
\node[acc, anchor=west, font=\scriptsize] at (5.9,3.2) {photon ring};
\draw[->, black, densely dotted] (5.85,3.15) -- (5.15,2.9);
% horizon (smaller, dotted)
\draw[black, densely dashed] (4.0,2.0) circle (0.62);
\node[black, font=\scriptsize, anchor=north] at (4.0,1.35) {horizon};
\end{tikzpicture}
$$

## Summary

Above the neutron-star mass limit collapse produces a black hole, described by mass
and spin alone. The Schwarzschild solution places the event horizon at
$R_S = 2GM/c^2$, where light is infinitely redshifted and, to a distant observer,
infalling clocks freeze; the infalling observer, however, crosses in finite proper
time. Stable circular orbits exist only outside the ISCO at $3R_S$, whose binding
energy gives an accretion efficiency of $\sim 6\%$ for a non-rotating hole, rising to
$\sim 42\%$ for maximal Kerr spin. A rotating Kerr black hole drags spacetime and
carries an ergosphere where no observer can stay at rest and rotational energy can be
tapped. Stellar-mass black holes are weighed in X-ray binaries like Cygnus X-1, and
the Event Horizon Telescope has imaged the lensed shadow and photon ring of
supermassive black holes, confirming the horizon-scale predictions of general
relativity.

[^co-bh]: Carroll & Ostlie, §17.3–17.4 — the Schwarzschild and Kerr solutions, the event horizon, gravitational redshift, the innermost stable circular orbit, and the ergosphere.