proposition 13.140 Killing's equation

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6296 · p. 528

Rests on

depends_on definition 13.139 Killing vector ¶
- depends_on definition 13.120 Isometry ¶
  - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
    - depends_on definition 13.48 Differentiable manifold ¶
      - depends_on definition 13.47 Differentiable structure ¶
    - depends_on definition 13.84 Tensor ¶
      - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶
- depends_on equation 13.267 eq:mfd-lie-def ¶
depends_on definition 13.149 Metric compatibility ¶
- depends_on definition 13.145 Affine connection ¶
  - 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 definition 13.88 General coordinate transformation ¶ ↺
  - depends_on definition 13.82 Vector field ¶
    - depends_on definition 13.81 Vector on a manifold ¶ ↺
- depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
- depends_on definition 13.146 Parallel transport and autoparallels ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
depends_on proposition 13.127 Component formulas ¶
- depends_on definition 13.89 Contravariant and covariant tensors ¶ ↺
- depends_on definition 13.90 Mixed tensor ¶ ↺
- depends_on equation 13.267 eq:mfd-lie-def ¶ ↺
- proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
- depends_on definition 13.145 Affine connection ¶ ↺
- depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
- depends_on definition 13.149 Metric compatibility ¶ ↺
- depends_on definition 13.147 Torsion ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
- proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
- depends_on definition 13.145 Affine connection ¶ ↺
- depends_on definition 13.147 Torsion ¶ ↺
- depends_on proposition 7.105 Clairaut–Schwarz ¶
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-19 ¶
  - proves proof ch:05-real-analysis@proof-64 ¶
- proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
proves proof ch:11-manifolds-tensors-curvature@proof-37 ¶

Supports

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 ¶
      - tests experiment Dyson, Eddington and Davidson: the eclipse of 1919 ¶
    - depends_on phenomenon 53.1 Measured anomalous advance of Mercury's perihelion ¶
      - tests experiment Le Verrier and the perihelion of Mercury ¶
    - 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 phenomenon 45.24 Compact objects exist above every cold-support limit ¶
    - depends_on theorem 45.23 Oppenheimer–Snyder collapse ¶
      - depends_on phenomenon 45.24 Compact objects exist above every cold-support limit ¶ ↺
- 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 ¶
      - depends_on phenomenon 48.18 The Hubble tension ¶
    - tests experiment The Cosmic Microwave Background ¶
      - cites (method) derived bib 24 Detection ofB-Mode Polarization at Degree Angular Scales by BICEP2 ¶
      - cites (evidence) derived bib 24 Detection ofB-Mode Polarization at Degree Angular Scales by BICEP2 ¶ ↺
      - cites (evidence) derived bib 10 Joint Analysis of BICEP2/Keck ArrayandPlanckData ¶
      - cites (evidence) derived bib 10 Joint Analysis of BICEP2/Keck ArrayandPlanckData ¶ ↺
      - cites (method) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (context) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib Planck 2018 results. VI. Cosmological parameters ¶
      - cites (method) derived bib Planck 2018 results. VI. Cosmological parameters ¶ ↺
      - … 109 more
  - 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 48.15 A standard ruler in the galaxy distribution ¶
    - depends_on phenomenon 48.17 The expansion is accelerating ¶
    - depends_on phenomenon 48.18 The Hubble tension ¶ ↺

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`	→	Killing vector	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6306
`depends_on`	→	Metric compatibility	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6306
`depends_on`	→	Component formulas	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6306
`depends_on`	→	Levi-Civita connection and contorsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6306
`depends_on`	→	Riemann tensor; Ricci identity with torsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6306
`depends_on`	←	The two Killing tensors every metric carries	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6444
`depends_on`	←	The de~Sitter algebras are isometry algebras	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3103
`depends_on`	←	The Killing fields of a flat pseudo-Euclidean space	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2411
`depends_on`	←	The invariant of a Killing vector along a geodesic	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6339
`depends_on`	←	The tensorial statement, and what Part II owes it	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:696
`depends_on`	←	Maximal symmetry forces constant curvature	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6984
`proves`	←	ch:11-manifolds-tensors-curvature@proof-37	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6309