proposition 13.141 The invariant of a Killing vector along a geodesic

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6329 · p. 528

Rests on

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 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
- depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
  - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - depends_on definition 13.84 Tensor ¶
    - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
    - depends_on definition 13.81 Vector on a manifold ¶ ↺
- depends_on definition 13.146 Parallel transport and autoparallels ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
depends_on definition 13.146 Parallel transport and autoparallels ¶ ↺
depends_on proposition 13.140 Killing's equation ¶
- 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 equation 13.267 eq:mfd-lie-def ¶
- depends_on definition 13.149 Metric compatibility ¶ ↺
- 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 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 theorem 7.34 Rolle ¶
      - 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 ¶
proves proof ch:11-manifolds-tensors-curvature@proof-38 ¶

Supports

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 ¶
      - cites (method) derived bib 571-581 A determination of the deflection of light by the sun's gravitational field, from observations made at the total eclipse of May 29, 1919 ¶
      - cites (evidence) derived bib 571-581 A determination of the deflection of light by the sun's gravitational field, from observations made at the total eclipse of May 29, 1919 ¶ ↺
      - cites (evidence) derived bib 571-581 A determination of the deflection of light by the sun's gravitational field, from observations made at the total eclipse of May 29, 1919 ¶ ↺
      - cites (history) derived bib 10 Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes ¶
      - cites (history) derived bib Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie ¶
      - cites (evidence) derived bib 2 PROGRESS IN MEASUREMENTS OF THE GRAVITATIONAL BENDING OF RADIO WAVES USING THE VLBA ¶
      - cites (evidence) derived bib Gravitational deflection of light: a re-examination of the observations of the solar eclipse of 1919 ¶
      - cites (evidence) derived bib 8 Measurement of the Solar Gravitational Deflection of Radio Waves Using Very-Long-Baseline Interferometry ¶
      - cites (evidence) derived bib 0957 + 561 A, B: twin quasistellar objects or gravitational lens? ¶
      - cites (context) derived bib 1 The Confrontation between General Relativity and Experiment ¶
      - cites (context) derived bib 1 The Confrontation between General Relativity and Experiment ¶ ↺
  - depends_on phenomenon 53.1 Measured anomalous advance of Mercury's perihelion ¶
    - tests experiment Le Verrier and the perihelion of Mercury ¶
      - cites (evidence) derived bib Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole ¶
      - cites (evidence) derived bib Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole ¶ ↺
      - cites (history) derived bib Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie ¶ ↺
      - cites (history) derived bib Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie ¶ ↺
      - cites (data) derived bib Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète ¶
      - cites (evidence) derived bib Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète ¶ ↺
      - cites (evidence) derived bib The Elements of the Four Inner Planets and the Fundamental Constants of Astronomy ¶
      - cites (data) derived bib 1 The Confrontation between General Relativity and Experiment ¶ ↺
  - 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 ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Metric compatibility	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6339
`depends_on`	→	Parallel transport and autoparallels	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6339
`depends_on`	→	Killing's equation	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6339
`depends_on`	←	The shadow	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:2057
`depends_on`	←	Peculiar momentum decays with the expansion	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:319
`depends_on`	←	Radial equation and effective potential	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:296
`depends_on`	←	Killing tensors and geodesic invariants	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6395
`proves`	←	ch:11-manifolds-tensors-curvature@proof-38	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6343