Math 2.0

Relativity and gravitational theory Black holes Theorem

No-hair theorem

Builds on Black hole uniqueness theoremsAsymptotically flat spacetimeEvent horizonElectrovacuum solution

Statement

Let (M,g,F)(M, g, F) be a four-dimensional, stationary, asymptotically flat electrovacuum solution of the Einstein–Maxwell equations, with FF the electromagnetic field, which contains a black hole. Assume that gg is real analytic, that the domain of outer communications is globally hyperbolic, and that the event horizon is connected and non-degenerate. Then the domain of outer communications is isometric to that of a Kerr–Newman metric. In particular it is determined by three numbers: the mass mm, the angular momentum J=amJ = am and the electric charge QQ, with a2+Q2<m2a^2 + Q^2 < m^2.

Remarks

The proof combines Hawking's rigidity theorem with the black hole uniqueness theorems. Rigidity says that a rotating stationary black hole is also axisymmetric, and a non-rotating one is static; this is where analyticity is used. The uniqueness theorems then identify the spacetime in the static and in the axisymmetric case. The hypotheses on the field content matter. For the Einstein–Yang–Mills equations there are static, spherically symmetric black holes that are not determined by their mass and charge.

Going deeper

For a student

The no-hair theorem is the reason why gravitational wave observations of the final ringing of a merged black hole can test general relativity. The quasinormal mode frequencies of a Kerr black hole depend only on its mass and spin, so measuring two or more of them over-determines the black hole. The theorem concerns exactly stationary black holes; whether dynamical black holes do settle down to Kerr–Newman is the black hole stability problem.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Asymptotically flat spacetime, concept, no evidence levelAsymptotically flat s…conceptElectrovacuum solution, concept, no evidence levelElectrovacuum solutionconceptEvent horizon, concept, no evidence levelEvent horizonconceptKilling vector field, concept, no evidence levelKilling vector fieldconceptVacuum solution, concept, no evidence levelVacuum solutionconceptBlack hole uniqueness theorems, statement, no evidence levelBlack hole uniqueness…statementNo-hair theorem, statement, no evidence levelNo-hair theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Asymptotically flat spacetime concept
  2. Electrovacuum solution concept
  3. Event horizon concept
  4. Killing vector field concept
  5. Vacuum solution concept
  6. Black hole uniqueness theorems statement
  7. No-hair theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/No-hair_theorem
About this page
TagRAEHHEPX7F
URIm2://RAEHHEPX7F
Kindstatement, theorem
WritingW0 stub
MSC83C57, 83C50
StatusThis copy of the site has no build data. Run m2 status RAEHHEPX7F.
AboutBlack hole, Kerr–Newman metric
UsesBlack hole uniqueness theorems, Asymptotically flat spacetime, Event horizon, Electrovacuum solution
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Changed3 November 2026

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