A black hole forms when mass is compressed within its Schwarzschild radius $r_s = 2GM/c^2$, creating a region of spacetime from which nothing — not even light — can escape. Once a theoretical curiosity, black holes are now observed directly: their mergers detected by LIGO (2015), their shadows imaged by the Event Horizon Telescope (2019).

The event horizon

The event horizon at $r = r_s$ is not a physical surface — there’s no wall, no edge. A freely falling observer notices nothing special as they cross it. But once inside, all future-directed paths lead inward. The radial coordinate $r$ becomes timelike — moving to smaller $r$ is as inevitable as moving forward in time.

To see this, switch to Eddington-Finkelstein coordinates or Kruskal-Szekeres coordinates, which remove the coordinate singularity at $r = r_s$. The Schwarzschild coordinates break down at the horizon (like longitude at the poles), but the spacetime itself is smooth there.

Light bending and the photon sphere

At $r = 1.5\,r_s$ (the photon sphere), light can orbit the black hole in unstable circular orbits. Slightly inside, it spirals in; slightly outside, it escapes. This creates the bright ring seen in EHT images.

At $r = 3\,r_s$, the innermost stable circular orbit (ISCO) is the last stable orbit for massive particles. Matter in an accretion disc spirals inward until it reaches the ISCO, then plunges rapidly. The ISCO marks the inner edge of the accretion disc.

The singularity

At $r = 0$, the curvature invariant $R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma}$ diverges — this is a genuine singularity, not a coordinate artefact. Tidal forces become infinite. General relativity predicts its own breakdown here; a quantum theory of gravity is needed to describe what actually happens at the centre.

Penrose’s singularity theorem (1965, Nobel Prize 2020): once an event horizon forms, a singularity is inevitable — no equation of state for matter can prevent it. This was the first proof that singularities are generic features of GR, not artefacts of perfect symmetry.

Rotating black holes: the Kerr solution

Real astrophysical black holes spin. The Kerr metric (1963) describes a rotating black hole with mass $M$ and angular momentum $J$:

• Two horizons: an outer event horizon and an inner Cauchy horizon
• An ergosphere outside the event horizon where spacetime is dragged so strongly that nothing can remain stationary
• Frame dragging: space itself rotates around the black hole

The Kerr solution is characterised by just two parameters: $M$ and $J$ (plus charge $Q$ for the full Kerr-Newman family). This is the no-hair theorem: a black hole has no other distinguishing features. All the complexity of the collapsed star is radiated away; only mass, spin, and charge remain.

Observational evidence

• Gravitational waves (LIGO, 2015): Binary black hole merger GW150914. Masses ~36 + 29 $M_\odot$, producing ~62 $M_\odot$ (3 $M_\odot$ radiated as gravitational waves). Waveform matches GR prediction.
• Event Horizon Telescope (2019): Shadow of M87* — a 6.5 billion $M_\odot$ black hole. The dark region matches the predicted size from the Kerr metric.
• Stellar orbits: Stars orbiting Sgr A* (the Milky Way’s central black hole, ~4 million $M_\odot$) follow Keplerian orbits consistent with a point mass (Genzel & Ghez, Nobel Prize 2020).