definition 26.40 The $3+1$ decomposition

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1556 · p. 916

ground object -- no derivation owed

Rests on

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 7.16 Limit ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
  - depends_on definition 22.14 Phase-space coordinate transformation ¶
    - depends_on definition 22.2 Hamiltonian ¶
      - depends_on definition 21.34 Generalized momentum ¶
      - depends_on definition 21.31 Lagrangian ¶
- depends_on notation 26.1 Weak and strong equality ¶
  - depends_on definition 22.29 Poisson bracket ¶ ↺
  - depends_on notation 22.1 not:ham-indices ¶

Supports

depends_on lemma A.623 The adapted frame of the $3+1$ split ¶
- 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 lemma A.631 The normal–normal Ricci contraction ¶
    - depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶
depends_on proposition 26.41 The ADM form of the Einstein–Hilbert Lagrangian ¶
- depends_on theorem 26.42 The Hamiltonian structure of general relativity ¶
  - depends_on notation A.636 Densities and smearing ¶
    - depends_on lemma A.637 Closed form of $H_{\parallel}$, and what it generates ¶
      - depends_on lemma A.638 Naturality ¶
  - depends_on proposition 26.43 The hypersurface-deformation algebra ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Einstein–Hilbert action	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1582
`depends_on`	→	Field systems	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1582
`depends_on`	←	The adapted frame of the $3+1$ split	declared	appendices/A-long-proofs.tex:30116
`depends_on`	←	The ADM form of the Einstein–Hilbert Lagrangian	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1596