Black Holes, Schwarzschild and Kerr

When a collapsing core exceeds the neutron-star mass limit, 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} = - (1 - \frac{2 GM}{r c ^{2}}) c^{2} d t^{2} + (1 - \frac{2 GM}{r c ^{2}})^{- 1} d r^{2} + r^{2} d Ω^{2},

where $d Ω^{2} = d θ^{2} + sin^{2} θ d ϕ^{2}$ . The metric coefficients degenerate at the Schwarzschild radius

R_{S} = \frac{2 GM}{c ^{2}} = 2.95 km (\frac{M}{M _{⊙}}) .

A Newtonian shortcut gives the same value: set the escape speed $2 GM / r$ equal to $c$ . For a $10 M_{⊙}$ black hole $R_{S} \approx 30 km$ ; for the $4 \times 1 0^{6} M_{⊙}$ black hole at the Galactic center, $R_{S} \approx 1.2 \times 1 0^{7} km$ , 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 $r = 0$ .

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.

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 $ν_{em}$ and received far away arrives redshifted to

\frac{ν _{\infty}}{ν _{em}} = 1 - \frac{2 GM}{r c ^{2}} = 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.

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.

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_{ISCO} = \frac{6 GM}{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 $E / m c^{2} = 8/9 \approx 0.943$ , so the fraction of rest-mass energy radiated as matter accretes from infinity to the ISCO is

η = 1 - \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.

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.

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 $a = J / M c$ with $0 \leq a \leq GM / c^{2}$ . Two surfaces appear.

The event horizon shrinks with spin to $r_{+} = (GM / c^{2}) + (GM / c^{2})^{2} - a^{2}$ , 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.

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.

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.

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; 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 $21 M_{⊙}$ , too massive to be anything but a black hole. Dozens of such dynamically weighed black holes are now known, clustering around $5$ – $20 M_{⊙}$ ; 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, $1 0^{6}$ to $1 0^{10} M_{⊙}$ , are weighed by the orbits of stars (Sgr A* at the Galactic center) or of gas, and power active galactic nuclei. 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.¹

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.

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 $R_{S} = 2 GM / c^{2}$ , 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 $3 R_{S}$ , 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.

Carroll & Ostlie, §17.3–17.4 — the Schwarzschild and Kerr solutions, the event horizon, gravitational redshift, the innermost stable circular orbit, and the ergosphere. ↩