notation A.636 Densities and smearing

open in the book · appendices/A-long-proofs.tex:30937 · p. 3100

notation -- no derivation owed by its kind

Rests on

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 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 ¶
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 definition 26.11 The final constraint set ¶
      - 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 5.113 Non-degenerate form ¶
      - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - 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 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 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 ¶

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:cd-adm-momentum	declared	appendices/A-long-proofs.tex:30969
`depends_on`	→	Field systems	declared	appendices/A-long-proofs.tex:30969
`depends_on`	→	The Hamiltonian structure of general relativity	declared	appendices/A-long-proofs.tex:30969
`depends_on`	←	Closed form of $H_{\parallel}$, and what it generates	declared	appendices/A-long-proofs.tex:31018