lemma A.623 The adapted frame of the $3+1$ split

open in the book · appendices/A-long-proofs.tex:30083 · p. 3092

Rests on

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 22.29 Poisson bracket ¶
    - 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 22.14 Phase-space coordinate transformation ¶
      - depends_on definition 22.2 Hamiltonian ¶
  - depends_on notation 26.1 Weak and strong equality ¶
    - depends_on definition 22.29 Poisson bracket ¶ ↺
    - depends_on notation 22.1 not:ham-indices ¶
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 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
- depends_on definition 13.84 Tensor ¶
  - depends_on definition 13.83 Covector ¶
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - depends_on definition 13.81 Vector on a manifold ¶ ↺
depends_on equation 26.78 eq:cd-adm-metric ¶
proves proof app:A-long-proofs@proof-376 ¶

Supports

depends_on lemma A.626 Decomposition of $\nabla n$, and the acceleration ¶
- depends_on lemma A.630 Codazzi equation ¶
  - depends_on theorem 44.42 The constraint equations ¶
    - 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 ¶ ↺
- depends_on lemma A.631 The normal–normal Ricci contraction ¶
  - depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶

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`	→	The $3+1$ decomposition	declared	appendices/A-long-proofs.tex:30116
`depends_on`	→	Metric tensor of signature $(p,q)$	declared	appendices/A-long-proofs.tex:30116
`depends_on`	→	eq:cd-adm-metric	declared	appendices/A-long-proofs.tex:30116
`depends_on`	←	Decomposition of $\nabla n$, and the acceleration	declared	appendices/A-long-proofs.tex:30241
`proves`	←	app:A-long-proofs@proof-376	declared	appendices/A-long-proofs.tex:30119