lemma A.637 Closed form of $H_{\parallel}$, and what it generates

open in the book · appendices/A-long-proofs.tex:31000 · p. 3101

Rests on

depends_on equation A.1005 eq:app-hypersurface-deformation-bracket ¶
depends_on equation 26.85 eq:cd-adm-momentum-constraint ¶
depends_on notation A.636 Densities and smearing ¶
- depends_on equation 26.82 eq:cd-adm-momentum ¶
- 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 theorem 26.42 The Hamiltonian structure of general relativity ¶
  - 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 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 proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
      - 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 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 ¶
proves proof app:A-long-proofs@proof-383 ¶

Supports

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 ¶

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`	→	eq:app-hypersurface-deformation-bracket	declared	appendices/A-long-proofs.tex:31018
`depends_on`	→	eq:cd-adm-momentum-constraint	declared	appendices/A-long-proofs.tex:31018
`depends_on`	→	Densities and smearing	declared	appendices/A-long-proofs.tex:31018
`depends_on`	←	Naturality	declared	appendices/A-long-proofs.tex:31090
`proves`	←	app:A-long-proofs@proof-383	declared	appendices/A-long-proofs.tex:31022