phenomenon 48.1 The Hubble–Lemaître law

Rests on

• depends_on equation 48.3 eq:cosmo-flrw-metric ¶
• depends_on theorem 48.3 The Robertson–Walker metric ¶
  • depends_on proposition 13.162 Curvature induced on the quadric ¶
    • depends_on definition 13.119 Induced metric ¶
      • 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 equation 13.83 eq:mfd-ecsgauss ¶
    • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      • depends_on definition 13.145 Affine connection ¶
        • depends_on definition 13.90 Mixed tensor ¶
        • depends_on definition 13.82 Vector field ¶
      • depends_on definition 13.147 Torsion ¶
        • depends_on definition 13.145 Affine connection ¶ ↺
      • 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-49 ¶
  • depends_on remark 13.161 rem:mfd-schur ¶
  • depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
    • depends_on definition 13.153 Contractions ¶
      • depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
      • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
    • 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 definition 13.145 Affine connection ¶ ↺
        • depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
        • depends_on definition 13.146 Parallel transport and autoparallels ¶
      • 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 ¶ ↺
      • proves proof ch:11-manifolds-tensors-curvature@proof-37 ¶
    • depends_on proposition 13.154 Symmetries of the curvature ¶
      • depends_on definition 13.149 Metric compatibility ¶ ↺
      • depends_on proposition 13.157 Bianchi identities ¶
        • depends_on equation 13.246 eq:mfd-leibniz-forms ¶
        • depends_on equation 13.245 eq:mfd-nilpotency ¶
        • depends_on theorem 13.156 Cartan structure equations ¶
        • proves proof ch:11-manifolds-tensors-curvature@proof-46 ¶
      • depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
      • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
      • proves proof ch:11-manifolds-tensors-curvature@proof-43 ¶
    • proves proof ch:11-manifolds-tensors-curvature@proof-48 ¶
  • proves proof ch:07-cosmology@proof-2 ¶
• proves proof ch:07-cosmology@proof-1 ¶

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