proposition 13.127 Component formulas

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5780 · p. 523

Rests on

depends_on definition 13.89 Contravariant and covariant tensors ¶
- depends_on definition 13.88 General coordinate transformation ¶
depends_on definition 13.90 Mixed tensor ¶
- depends_on definition 13.89 Contravariant and covariant tensors ¶ ↺
- depends_on definition 13.88 General coordinate transformation ¶ ↺
depends_on equation 13.267 eq:mfd-lie-def ¶
proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶

Supports

depends_on lemma A.75 Differentiating a pullback along a flow ¶
- depends_on lemma A.77 Poincaré lemma, converse form ¶
depends_on proposition 13.129 Commutation of Lie derivative and interior product ¶
depends_on proposition 13.128 Cartan's magic formula ¶
- depends_on lemma A.77 Poincaré lemma, converse form ¶ ↺
- depends_on proposition 13.129 Commutation of Lie derivative and interior product ¶ ↺
- depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
  - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
    - depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
  - depends_on remark 24.35 What non-squeezing does and does not say about nature ¶
  - depends_on theorem 24.21 Liouville ¶
    - depends_on example 24.29 A volume-preserving squeeze ¶
    - depends_on proposition 32.40 A Hamiltonian section preserves area ¶
    - depends_on remark 22.25 This is Liouville's theorem ¶
    - depends_on remark 24.22 Liouville's theorem is the foundation of statistical mechanics ¶
    - depends_on theorem 23.31 Liouville–Arnold ¶
      - depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶
      - depends_on remark 23.33 Integrability is exceptional ¶
      - depends_on remark 23.32 What that proof imports ¶
      - depends_on theorem 32.51 Kolmogorov–Arnold–Moser, quoted ¶
      - depends_on theorem 32.47 Poincaré's non-integrability, quoted ¶
    - depends_on theorem 25.57 The Moyal equation, and its classical limit ¶
      - depends_on proposition 25.59 Homodyne tomography inverts a Radon transform ¶
    - depends_on theorem 24.24 Poincaré recurrence ¶
      - depends_on remark 24.26 Recurrence and irreversibility ¶
      - depends_on remark 24.25 What the recurrence proof assumes about volume ¶
depends_on proposition 13.140 Killing's equation ¶
- depends_on example 13.144 The two Killing tensors every metric carries ¶
- depends_on proposition 14.69 The de~Sitter algebras are isometry algebras ¶
- depends_on proposition 14.59 The Killing fields of a flat pseudo-Euclidean space ¶
  - depends_on corollary A.344 The algebras $\mathfrak{so}(p,q)$ are semisimple ¶
    - depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶
    - depends_on example A.359 The Lorentz algebra ¶
  - depends_on proposition 14.69 The de~Sitter algebras are isometry algebras ¶ ↺
- depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶
  - depends_on phenomenon 45.52 The shadow ¶
    - depends_on phenomenon 50.4 A ring of light at event-horizon scale ¶
  - depends_on proposition 48.4 Peculiar momentum decays with the expansion ¶
  - depends_on proposition 45.6 Radial equation and effective potential ¶
    - depends_on proposition 45.10 Circular orbits and their energy ¶
      - depends_on phenomenon 45.12 Efficiency of accretion onto compact objects ¶
    - depends_on proposition 45.8 Orbit equation ¶
      - depends_on phenomenon 53.2 Measured deflection of starlight at the solar limb ¶
      - depends_on phenomenon 53.1 Measured anomalous advance of Mercury's perihelion ¶
      - depends_on phenomenon 45.9 Relativistic advance of the periastron ¶
    - depends_on proposition 45.21 Radial plunge: the cycloid ¶
      - depends_on proposition 45.22 The view from outside: exponential freezing ¶
      - depends_on theorem 45.23 Oppenheimer–Snyder collapse ¶
  - depends_on theorem 13.143 Killing tensors and geodesic invariants ¶
- depends_on remark 23.21 The tensorial statement, and what Part II owes it ¶
- depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
  - depends_on proposition 44.31 de Sitter geometry ¶
  - depends_on proposition 15.31 Radius and cosmological constant ¶
    - depends_on corollary 15.32 The expansion parameter is the cosmological constant ¶
      - depends_on example 15.64 The whole chapter at $D=4$ ¶
      - depends_on remark 15.63 The inclusion is strict, and the strictness is physical ¶
      - depends_on remark 15.33 Which branch nature is on ¶
    - depends_on example 15.64 The whole chapter at $D=4$ ¶ ↺
    - depends_on remark 15.33 Which branch nature is on ¶ ↺
  - depends_on theorem 48.3 The Robertson–Walker metric ¶
    - depends_on phenomenon 48.12 The relic microwave background ¶
      - depends_on phenomenon 48.13 The anisotropy spectrum of the relic radiation ¶
      - tests experiment The Cosmic Microwave Background ¶
    - depends_on phenomenon 48.1 The Hubble–Lemaître law ¶
    - depends_on phenomenon 51.6 Acoustic peaks, and a spatially flat universe ¶
      - depends_on phenomenon 51.7 E-mode polarization, and no primordial B mode ¶
    - depends_on proposition 48.7 Luminosity distance ¶
      - depends_on corollary 48.8 Angular-diameter distance and the duality relation ¶
      - depends_on phenomenon 48.17 The expansion is accelerating ¶
      - depends_on phenomenon 48.18 The Hubble tension ¶
depends_on theorem 44.19 Covariant conservation of stress–energy ¶
- depends_on proposition 44.20 Perfect-fluid equations of motion ¶
  - depends_on corollary 44.21 Dust flows on geodesics ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
not graded
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inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Contravariant and covariant tensors	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5801
`depends_on`	→	Mixed tensor	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5801
`depends_on`	→	eq:mfd-lie-def	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5801
`depends_on`	←	Differentiating a pullback along a flow	declared	appendices/A-long-proofs.tex:4866
`depends_on`	←	Commutation of Lie derivative and interior product	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5859
`depends_on`	←	Cartan's magic formula	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5829
`depends_on`	←	Killing's equation	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6306
`depends_on`	←	Covariant conservation of stress–energy	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:837
`proves`	←	ch:11-manifolds-tensors-curvature@proof-28	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5804