theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported

open in the book · parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1893 · p. 1281

Rests on

depends_on definition 10.24 Well-posed problem ¶
- depends_on definition 10.1 Partial differential equation; order ¶
  - depends_on definition 7.98 Functions of class $C^{1}$ ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - depends_on definition 6.2 Open set ¶
    - depends_on definition 6.1 Topological space ¶
      - depends_on definition 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- depends_on definition 6.24 Metric ¶
  - depends_on definition 3.43 Map ¶
    - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.23 Predicate ¶
    - depends_on definition 3.28 Set ¶
  - depends_on definition 3.38 Ordered pair and Cartesian product ¶
depends_on proposition 44.44 Harmonic-gauge reduction ¶
- depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
  - depends_on definition 13.145 Affine connection ¶
    - depends_on definition 13.90 Mixed tensor ¶
      - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.88 General coordinate transformation ¶
    - depends_on definition 13.82 Vector field ¶
      - depends_on definition 13.81 Vector on a manifold ¶
  - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
    - depends_on definition 13.48 Differentiable manifold ¶
      - depends_on definition 13.47 Differentiable structure ¶
    - depends_on definition 13.84 Tensor ¶
      - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
  - depends_on definition 13.149 Metric compatibility ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
    - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
    - depends_on definition 13.146 Parallel transport and autoparallels ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
  - depends_on definition 13.147 Torsion ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
- depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
  - depends_on definition 13.147 Torsion ¶ ↺
  - depends_on proposition 7.105 Clairaut–Schwarz ¶
    - depends_on definition 7.20 Continuity at a point ¶ ↺
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-64 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
- proves proof ch:03-einstein-field-equations@proof-24 ¶
depends_on theorem 44.42 The constraint equations ¶
- depends_on lemma A.630 Codazzi equation ¶
  - depends_on definition A.625 Four-dimensional extrinsic curvature ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
    - depends_on equation A.972 eq:app-adm-gauss-codazzi-projector ¶
  - depends_on equation 13.305 eq:mfd-ricci-identity ¶
  - depends_on lemma A.626 Decomposition of $\nabla n$, and the acceleration ¶
    - depends_on definition A.625 Four-dimensional extrinsic curvature ¶ ↺
    - depends_on equation 13.295 eq:mfd-covariant-derivative ¶
    - depends_on lemma A.623 The adapted frame of the $3+1$ split ¶
      - depends_on definition 26.40 The $3+1$ decomposition ¶
      - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
      - depends_on equation 26.78 eq:cd-adm-metric ¶
      - proves proof app:A-long-proofs@proof-376 ¶
    - proves proof app:A-long-proofs@proof-377 ¶
  - proves proof app:A-long-proofs@proof-380 ¶
- depends_on lemma A.628 Gauss equation, arbitrary signature ¶
  - depends_on definition A.625 Four-dimensional extrinsic curvature ¶ ↺
  - depends_on equation A.973 eq:app-adm-gauss-codazzi-D ¶
  - depends_on equation 13.305 eq:mfd-ricci-identity ¶ ↺
  - proves proof app:A-long-proofs@proof-379 ¶
- depends_on postulate 44.3 Einstein field equations ¶
  - depends_on definition 44.2 Einstein tensor ¶
    - depends_on definition 13.153 Contractions ¶
      - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
      - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
- proves proof ch:03-einstein-field-equations@proof-23 ¶

Supports

depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`cites`	→	Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non linéaires	derived	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1907
`depends_on`	→	Well-posed problem	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1908
`depends_on`	→	Harmonic-gauge reduction	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1908
`depends_on`	→	The constraint equations	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1908
`depends_on`	←	Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1918