Math 2.0

Relativity and gravitational theory Initial value problem and Hamiltonian formulation Theorem

Choquet-Bruhat–Geroch theorem on maximal developments

Builds on Initial value formulation of general relativityGlobally hyperbolic spacetimeCauchy surfaceHarmonic coordinate conditionDiffeomorphism

Statement

Let (Σ,h,K)(\Sigma, h, K) be a smooth vacuum initial data set, as in the initial value formulation of general relativity. Then there is a development (M,g,ι)(M, g, \iota) of (Σ,h,K)(\Sigma, h, K) with the following property. For every development (M′,g′,ι′)(M', g', \iota') of the same data, there is an isometric embedding ψ ⁣:M′→M\psi \colon M' \to M with ψ∘ι′=ι\psi \circ \iota' = \iota. This maximal globally hyperbolic development is unique up to an isometry that commutes with the embeddings of Σ\Sigma.

Remarks

Every development is a globally hyperbolic spacetime, because ι(Σ)\iota(\Sigma) is a Cauchy surface. Local existence was proved by Choquet-Bruhat in 1952 with the harmonic coordinate condition, which turns the vacuum equations into a system of quasilinear wave equations. Local uniqueness says that two developments are isometric near Σ\Sigma, by a diffeomorphism that fixes the data. Choquet-Bruhat and Geroch then obtained the maximal development in 1969, by gluing all developments together with Zorn's lemma. Sbierski later gave a proof that avoids Zorn's lemma. Example: for the data (ℝ3,δ,0)(\R^3, \delta, 0) the maximal development is Minkowski space. The maximal development may still be extendible as a smooth spacetime beyond a Cauchy horizon, for example for Taub–NUT data; the extension is then no longer determined by the data.

Going deeper

For a student

The theorem makes general relativity a deterministic theory in the same sense as other hyperbolic equations. It gives the object on which global questions are asked: the nonlinear stability of Minkowski space concerns the maximal development of data near (ℝ3,δ,0)(\R^3, \delta, 0). The strong cosmic censorship conjecture asks whether, for generic data, the maximal development is inextendible.

Reading path

To understand this page from the ground up, read these 24 pages in order. The foundations come first.

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Cauchy 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…conceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionChain rule in several variables, statement, no evidence levelChain rule in several…statementHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated 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…statement
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. Cauchy surface concept
  2. Euclidean space object
  3. Function concept
  4. Function composition concept
  5. Globally hyperbolic spacetime concept
  6. Harmonic coordinate condition concept
  7. Initial value formulation of general relativity concept
  8. Fréchet derivative definition
  9. Function of several variables concept
  10. Limit of a function of several variables concept
  11. Partial derivative concept
  12. Differentiable function of several variables definition
  13. Partial derivative definition
  14. Differentiable on an open set definition
  15. Jacobian matrix definition
  16. Chain rule in several variables statement
  17. Higher partial derivatives definition
  18. Iterated derivative definition
  19. Jacobian determinant definition
  20. Derivative of the inverse function statement
  21. C^k function characterised by continuous derivatives statement
  22. Inverse function theorem statement
  23. Diffeomorphism definition
  24. Choquet-Bruhat–Geroch theorem on maximal developments statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Initial_value_formulation_(general_relativity)
About this page
Tag626MENG0QH
URIm2://626MENG0QH
Kindstatement, theorem
WritingW0 stub
MSC83C05, 35Q75, 35L70
StatusThis copy of the site has no build data. Run m2 status 626MENG0QH.
AboutInitial value formulation of general relativity
UsesInitial value formulation of general relativity, Globally hyperbolic spacetime, Cauchy surface, Harmonic coordinate condition, Diffeomorphism
Changed3 November 2026

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