theorem 26.42 The Hamiltonian structure of general relativity

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1616 · p. 917

Rests on

depends_on proposition 26.41 The ADM form of the Einstein–Hilbert Lagrangian ¶
- 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 22.14 Phase-space coordinate transformation ¶
    - 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 equation 43.4 eq:grav-eh-metric ¶
- proves proof ch:09-constrained-hamiltonian-dynamics@prooflink-2 ¶
depends_on proposition 26.10 The four outcomes ¶
- depends_on definition 26.9 Consistency conditions ¶
  - depends_on equation 26.7 eq:cd-evolution ¶
  - depends_on equation 26.2 eq:cd-primary ¶
- depends_on remark 26.7 The regularity assumption, stated once ¶
- proves proof ch:09-constrained-hamiltonian-dynamics@proof-3 ¶
depends_on theorem 26.17 Counting the physical degrees of freedom ¶
- depends_on proposition 26.19 Second-class constraints come in pairs ¶
  - depends_on definition 26.12 First and second class ¶
    - depends_on definition 26.11 The final constraint set ¶
      - depends_on proposition 26.10 The four outcomes ¶ ↺
      - depends_on remark 26.7 The regularity assumption, stated once ¶ ↺
    - depends_on definition 22.29 Poisson bracket ¶ ↺
  - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
    - depends_on definition 5.110 Bilinear map ¶
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 5.113 Non-degenerate form ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
    - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
      - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
    - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
    - proves proof ch:03-linear-algebra-representations@proof-54 ¶
  - proves proof ch:09-constrained-hamiltonian-dynamics@proof-7 ¶
- depends_on remark 26.18 The Frobenius theorem is owed by Part II ¶
- depends_on theorem 26.14 First-class constraints generate gauge transformations ¶
  - depends_on definition 26.8 Total Hamiltonian ¶
    - depends_on definition 22.29 Poisson bracket ¶ ↺
    - depends_on proposition 26.6 The canonical Hamiltonian is defined on the constraint surface ¶
      - depends_on equation 26.4 eq:cd-canonical-hamiltonian ¶
      - depends_on proposition 26.5 Primary constraints ¶
      - proves proof ch:09-constrained-hamiltonian-dynamics@proof-2 ¶
  - depends_on proposition 26.10 The four outcomes ¶ ↺
  - proves proof ch:09-constrained-hamiltonian-dynamics@proof-5 ¶
- proves proof ch:09-constrained-hamiltonian-dynamics@proof-6 ¶
proves proof ch:09-constrained-hamiltonian-dynamics@proof-16 ¶

Supports

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 A.639 The first two brackets ¶
      - depends_on corollary A.642 The constraints are first class ¶
depends_on proposition 26.43 The hypersurface-deformation algebra ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	The ADM form of the Einstein–Hilbert Lagrangian	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1649
`depends_on`	→	The four outcomes	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1649
`depends_on`	→	Counting the physical degrees of freedom	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1649
`depends_on`	←	Densities and smearing	declared	appendices/A-long-proofs.tex:30969
`depends_on`	←	The hypersurface-deformation algebra	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1735
`proves`	←	ch:09-constrained-hamiltonian-dynamics@proof-16	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1652