proposition 45.6 Radial equation and effective potential

open in the book · parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:282 · p. 1288

Rests on

depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶
- 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 13.82 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 13.48 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 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 theorem 7.35 Mean value theorem ¶
      - 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 ¶
depends_on proposition 40.3 Normalization of the four-velocity ¶
- depends_on definition 38.6 Minkowski metric and interval ¶
  - depends_on proposition 38.5 The invariant interval ¶
    - depends_on equation 38.1 eq:lor-beta-gamma ¶
    - depends_on theorem 38.4 Lorentz boost ¶
      - depends_on notation 38.3 Standard configuration ¶
      - depends_on postulate 38.2 Invariance of the speed of light ¶
      - depends_on postulate 38.1 Principle of relativity ¶
      - proves proof ch:02-lorentz-transformations@proof-1 ¶
    - proves proof ch:02-lorentz-transformations@proof-2 ¶
- depends_on equation 40.2 eq:rdyn-fourvelocity ¶
- proves proof ch:04-relativistic-dynamics@proof-1 ¶
depends_on theorem 45.1 Schwarzschild solution ¶
- depends_on equation 13.306 eq:mfd-riemann ¶
- depends_on postulate 43.1 Geodesic motion ¶
- depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
- proves proof ch:04-schwarzschild-black-holes@proof-1 ¶
proves proof ch:04-schwarzschild-black-holes@proof-3 ¶

Supports

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 ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	The invariant of a Killing vector along a geodesic	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:296
`depends_on`	→	Normalization of the four-velocity	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:296
`depends_on`	→	Schwarzschild solution	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:296
`depends_on`	←	Circular orbits and their energy	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:446
`depends_on`	←	Orbit equation	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:356
`depends_on`	←	Radial plunge: the cycloid	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:934
`proves`	←	ch:04-schwarzschild-black-holes@proof-3	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:300