phenomenon 45.52 The shadow

open in the book · parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:2039 · p. 1308

phenomenon-tested: no experiment declares a \tests edge to it and it is not listed in untested.txt

Rests on

depends_on equation 45.20 eq:schw-null-radial ¶
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 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-25 ¶

Supports

depends_on phenomenon 50.4 A ring of light at event-horizon scale ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`cites`	→	First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole	derived	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:2055
`cites`	→	First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way	derived	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:2055
`cites`	→	Viewing the Shadow of the Black Hole at the Galactic Center	derived	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:2053
`cites`	→	The Escape of Photons from Gravitationally Intense Stars	derived	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:2053
`depends_on`	→	eq:schw-null-radial	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:2057
`depends_on`	→	The invariant of a Killing vector along a geodesic	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:2057
`depends_on`	→	Schwarzschild solution	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:2057
`depends_on`	←	A ring of light at event-horizon scale	declared	parts/05-general-relativity-cosmology/09-exp-black-holes.tex:719
`proves`	←	ch:04-schwarzschild-black-holes@proof-25	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:2060