Horizons and Singularities

The Schwarzschild and Kerr solutions each contain an event horizon — a one-way surface — and a singularity where curvature diverges. This page develops the global structure behind those features: what a horizon really is, how to see across the coordinate breakdown, how to draw causal structure compactly with Penrose diagrams, and what the singularity theorems and cosmic censorship say about where singularities must and must not appear.

We use $c=1$, keep $G$ explicit, mostly-plus signature.

Crossing the horizon: better coordinates

In Schwarzschild coordinates the metric degenerates at $r=2GM$, but the Kretschmann scalar is finite there — a coordinate, not physical, singularity. The defect is that Schwarzschild time $t$ is the proper time of a static observer, and no static observer exists at the horizon; $t\to \infty$ as an infalling body approaches, freezing it artificially.

Adapted coordinates remove the defect by following light rather than static observers:

• Eddington–Finkelstein. Replace $t$ by the ingoing null coordinate $v=t+r_{\ast }$, where $r_{\ast }=r+2GM\ln |r/2GM-1|$ is the "tortoise" coordinate. The metric becomes regular at $r=2GM$; ingoing light crosses smoothly, and the one-way character is manifest — inside the horizon all future light cones tip toward $r=0$.
• Kruskal–Szekeres. A further transformation to coordinates $(U,V)$ covers the maximal analytic extension: the exterior, the black-hole interior, a white hole (time-reverse of the black hole), and a second asymptotically flat region. Light cones are everywhere at $45^{\circ }$, so causal structure is read off directly. The eternal Schwarzschild spacetime thus has more structure than the exterior alone; realistic collapse, however, replaces the white hole and second region with the interior of the collapsing star.

The lesson generalizes: an event horizon is a perfectly regular place locally — a freely falling observer notices nothing special on crossing (the EEP) — but a globally defined boundary of no escape.

What a horizon is

Several inequivalent notions of "horizon" coincide for stationary black holes but diverge in dynamical settings:

• Event horizon. The boundary of the causal past of future null infinity — the set of events from which no signal ever reaches infinity. This is the "true" black-hole boundary, but it is teleological: locating it requires knowing the entire future, since whether a signal eventually escapes depends on the whole spacetime.
• Apparent horizon. The outermost surface on which outgoing light rays are momentarily not expanding (a marginally trapped surface). This is locally defined and is what numerical relativity tracks. It lies inside or on the event horizon and coincides with it in stationary spacetimes.
• Killing horizon. A null surface on which a Killing vector becomes null. For Schwarzschild the static Killing vector ${\partial }_t$ goes null at $r=2GM$; for Kerr the relevant Killing vector is ${\partial }_t+{\Omega }_H{\partial }_{\phi }$, rotating with the horizon's angular velocity ${\Omega }_H$. The surface gravity $\kappa$ — the acceleration, red-shifted to infinity, needed to hold a test body at the horizon — is defined on the Killing horizon and reappears as temperature in thermodynamics.

A trapped surface — a closed surface whose both ingoing and outgoing null congruences converge — is the sharp local signal that gravitational collapse has passed the point of no return, and is the key hypothesis of Penrose's theorem below.

Penrose (conformal) diagrams

To reason about global causal structure, Penrose diagrams conformally compress an entire spacetime into a finite figure while preserving light cones at $45^{\circ }$. Conformal rescaling $g\to {\Omega }^{2}g$ leaves the causal (null) structure intact, so "what can signal what" is read directly, with infinity brought to a finite boundary:

• ${\mathcal{I}}^{+}$ / ${\mathcal{I}}^{-}$ (scri, future/past null infinity) — where light rays end/begin;
• $i^{+}$, $i^{-}$ (future/past timelike infinity) — where massive worldlines end/begin;
• $i^0$ (spacelike infinity).

graph TD
  S["singularity r=0 (spacelike, jagged top)"]
  H["event horizon (45° null line)"]
  Ip["𝓘⁺ future null infinity"]
  Im["𝓘⁻ past null infinity"]
  S --- H
  H --- Ip
  Im --- H

On such a diagram a Schwarzschild black hole shows the singularity as a spacelike line across the top (a moment of time, not a place — reinforcing that inside the horizon, falling to $r=0$ is a matter of the future, not of direction), with the horizon a $45^{\circ }$ null line beneath it. Penrose diagrams for Kerr and Reissner–Nordström reveal their richer structure — timelike singularities, inner (Cauchy) horizons, and infinitely many asymptotic regions.

The singularity theorems

Before 1965 it was hoped that the Schwarzschild singularity was an artifact of exact spherical symmetry — that realistic, lumpy collapse would miss the center and avoid infinite density. The singularity theorems of Penrose (1965) and Hawking–Penrose (1970) demolished that hope:

Penrose's theorem. If a spacetime satisfies the null energy condition, contains a trapped surface, and has a non-compact Cauchy surface (an isolated system), then it is geodesically incomplete — some geodesic ends after finite affine parameter.

The proof is global and topological, using the focusing of null geodesics (Raychaudhuri) rather than any symmetry. Incompleteness is the rigorous definition of a singularity: a worldline simply stops, with nowhere in the manifold to continue. Consequences:

• Collapse generically produces singularities. Once a trapped surface forms (which the horizon guarantees), a singularity is inevitable regardless of symmetry or matter details.
• The Big Bang is a genuine singularity. The cosmological version (Hawking) applied to an expanding FLRW universe shows the initial singularity is unavoidable given the SEC.

This work earned Penrose the 2020 Nobel Prize. It also sharply delimits classical GR: the theory predicts its own breakdown, pointing to the need for quantum gravity at the singularities.

Cosmic censorship

Singularities are inevitable, but are they always safely hidden behind horizons? Penrose's cosmic censorship conjectures assert (roughly) yes:

• Weak cosmic censorship. Singularities formed in gravitational collapse from generic, physically reasonable initial data are always concealed behind event horizons — no naked singularity is visible to a distant observer. Physics outside remains predictable; the pathology is quarantined.
• Strong cosmic censorship. More stringently, generic spacetimes are globally hyperbolic: no observer can ever see a singularity, and the inner (Cauchy) horizons of Kerr/Reissner–Nordström — beyond which predictability would fail — are unstable and become singular, sealing off the exotic interior.

Both remain unproven and are among the deepest open problems in mathematical relativity. They matter physically: the entire predictive framework outside black holes, and the thermodynamic picture of horizons, presupposes that singularities stay censored. The over-extremal $a>GM$ Kerr naked singularity is exactly the configuration weak censorship forbids collapse from producing.

Summary

• The horizon at $r=2GM$ is a coordinate singularity; Eddington–Finkelstein and Kruskal–Szekeres coordinates cross it smoothly and reveal the maximal extension (interior, white hole, second region).
• "Horizon" has several notions: event (global, teleological), apparent (local, from trapped surfaces), Killing (with surface gravity $\kappa$); they coincide when stationary.
• Penrose diagrams conformally compactify spacetime, keeping light cones at $45^{\circ }$, to display causal structure and infinity (${\mathcal{I}}^{\pm }$, $i^{\pm }$, $i^0$); the Schwarzschild singularity is spacelike.
• The singularity theorems (NEC + trapped surface ⇒ geodesic incompleteness) make singularities generic and unavoidable — in collapse and at the Big Bang — signalling the breakdown of classical GR.
• Cosmic censorship (weak and strong, both open) conjectures that singularities stay hidden behind horizons, preserving predictability.

Next

Horizons behave startlingly like thermodynamic systems: the surface gravity $\kappa$ is a temperature and the area $A$ an entropy. Black-Hole Thermodynamics develops the four laws, Hawking radiation, and the information puzzle — the sharpest existing clue to quantum gravity.