phenomenon 50.4 A ring of light at event-horizon scale

Rests on

• depends_on phenomenon 45.52 The shadow ¶
  • 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 13.82 Vector field ¶
      • 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 13.146 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 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 ¶
        • 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 ¶
        • 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 ¶
• depends_on theorem 45.1 Schwarzschild solution ¶ ↺
• proves proof ch:09-exp-black-holes@proof-4 ¶

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