theorem 44.42 The constraint equations

open in the book · parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1700 · p. 1280

Rests on

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 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 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 43.2 Einstein–Hilbert action ¶
      - depends_on notation 26.28 Field systems ¶
    - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
      - depends_on definition 13.48 Differentiable manifold ¶
      - depends_on definition 13.84 Tensor ¶
    - 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 ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
      - depends_on definition 13.147 Torsion ¶
      - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
proves proof ch:03-einstein-field-equations@proof-23 ¶

Supports

depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
- depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶
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
`depends_on`	→	Codazzi equation	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1724
`depends_on`	→	Gauss equation, arbitrary signature	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1724
`depends_on`	→	Einstein field equations	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1724
`depends_on`	←	Local existence and uniqueness; Choquet-Bruhat; imported	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
`proves`	←	ch:03-einstein-field-equations@proof-23	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1728