Math 2.0

Relativity and gravitational theory Black holes Problem

Black hole stability problem

Builds on Kerr metricInitial value formulation of general relativityChoquet-Bruhat–Geroch theorem on maximal developments

Problem

Let m>0m > 0 and ∣a∣<m\abs{a} < m, and consider the initial data on a spacelike hypersurface induced by the Kerr metric of mass mm and spin aa. Take vacuum initial data sufficiently close to these data, and their maximal globally hyperbolic development under the vacuum Einstein field equations. This development exists by the Choquet-Bruhat–Geroch theorem. Show that it has a complete future null infinity. Show moreover that its domain of outer communications converges at late times to that of a Kerr metric. The limit should have parameters m′m' and a′a' close to mm and aa.

Remarks

Regge and Wheeler began in 1957 the study of linear perturbations of the Schwarzschild metric. The linear stability of Schwarzschild for the full linearised Einstein equations was proved by Dafermos, Holzegel and Rodnianski in 2016. Klainerman and Szeftel proved nonlinear stability of Schwarzschild for polarised axisymmetric perturbations. Dafermos, Holzegel, Rodnianski and Taylor proved in 2021 that a codimension-three family of data near Schwarzschild data converges to a nearby Schwarzschild metric. The codimension accounts for the three components of angular momentum, since general nearby data settle to a rotating black hole. Nonlinear stability of slowly rotating Kerr black holes, ∣a∣≪m\abs{a} \ll m, was announced by Giorgi, Klainerman and Szeftel in 2022. The full subextremal range ∣a∣<m\abs{a} < m is still open. The extremal case ∣a∣=m\abs{a} = m is different: there are instabilities along the horizon.

Reading path

To understand this page from the ground up, read these 30 pages in order. The foundations come first.

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Boyer–Lindquist coordinates, concept, no evidence levelBoyer–Lindquist coord…conceptCauchy surface, concept, no evidence levelCauchy surfaceconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptGlobally hyperbolic spacetime, concept, no evidence levelGlobally hyperbolic s…conceptHarmonic coordinate condition, concept, no evidence levelHarmonic coordinate c…conceptInitial value formulation of general relativity, concept, no evidence levelInitial value formula…conceptKilling vector field, concept, no evidence levelKilling vector fieldconceptRicci curvature, concept, no evidence levelRicci curvatureconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptSchwarzschild metric, example, no evidence levelSchwarzschild metricexamplePartial derivative, concept, no evidence levelPartial derivativeconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptKerr metric, example, no evidence levelKerr metricexamplePartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionChain rule in several variables, statement, no evidence levelChain rule in several…statementIterated derivative, definition, no evidence levelIterated derivativedefinitionJacobian determinant, definition, no evidence levelJacobian determinantdefinitionDerivative of the inverse function, statement, no evidence levelDerivative of the inv…statementC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementInverse function theorem, statement, no evidence levelInverse function theo…statementDiffeomorphism, definition, no evidence levelDiffeomorphismdefinitionChoquet-Bruhat–Geroch theorem on maximal developments, statement, no evidence levelChoquet-Bruhat–Geroch…statementBlack hole stability problem, problem, no evidence levelBlack hole stability …problem
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. Boyer–Lindquist coordinates concept
  2. Cauchy surface concept
  3. Euclidean space object
  4. Function concept
  5. Function composition concept
  6. Globally hyperbolic spacetime concept
  7. Harmonic coordinate condition concept
  8. Initial value formulation of general relativity concept
  9. Killing vector field concept
  10. Ricci curvature concept
  11. Fréchet derivative definition
  12. Function of several variables concept
  13. Schwarzschild metric example
  14. Partial derivative concept
  15. Limit of a function of several variables concept
  16. Kerr metric example
  17. Partial derivative definition
  18. Differentiable function of several variables definition
  19. Higher partial derivatives definition
  20. Jacobian matrix definition
  21. Differentiable on an open set definition
  22. Chain rule in several variables statement
  23. Iterated derivative definition
  24. Jacobian determinant definition
  25. Derivative of the inverse function statement
  26. C^k function characterised by continuous derivatives statement
  27. Inverse function theorem statement
  28. Diffeomorphism definition
  29. Choquet-Bruhat–Geroch theorem on maximal developments statement
  30. Black hole stability problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Black_hole
About this page
Tag6R0BJ6WS5A
URIm2://6R0BJ6WS5A
Kindproblem
WritingW0 stub
MSC83C57, 35Q75, 35B35
StatusThis copy of the site has no build data. Run m2 status 6R0BJ6WS5A.
AboutKerr metric, Black hole
UsesKerr metric, Initial value formulation of general relativity, Choquet-Bruhat–Geroch theorem on maximal developments
Changed3 November 2026

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