proposition 44.31 de Sitter geometry

Rests on

• depends_on proposition 44.30 The cosmological term ¶
  • 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 theorem 44.23 Contracted Bianchi identity ¶
    • depends_on definition 44.2 Einstein tensor ¶
      • 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 definition 13.149 Metric compatibility ¶ ↺
    • 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 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-43 ¶
    • proves proof ch:03-einstein-field-equations@proof-12 ¶
  • depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    • depends_on definition 44.6 Stress–energy tensor ¶
      • depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
        • depends_on equation 16.31 eq:calcvar-field-functional ¶
        • depends_on lemma 16.18 Fundamental lemma ¶
        • depends_on theorem 7.133 Gauss ¶
        • proves proof ch:14-calculus-of-variations@proof-20 ¶
    • depends_on lemma 44.7 Variation of inverse and determinant ¶
      • depends_on proposition 5.2 Jacobi's formula, column form ¶
        • depends_on equation 5.19 eq:lin-leibniz-det ¶
        • depends_on proposition 7.30 Leibniz rule ¶
        • depends_on proposition 5.49 Injective, surjective, invertible ¶
        • proves proof ch:03-linear-algebra-representations@proof-1 ¶
      • proves proof ch:03-einstein-field-equations@proof-2 ¶
    • depends_on lemma 44.8 Palatini identity ¶
      • depends_on definition 13.149 Metric compatibility ¶ ↺
      • depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
      • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
      • proves proof ch:03-einstein-field-equations@proof-3 ¶
    • proves proof ch:03-einstein-field-equations@proof-4 ¶
  • proves proof ch:03-einstein-field-equations@proof-16 ¶
• depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
  • depends_on definition 13.153 Contractions ¶ ↺
  • 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 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 ¶ ↺
  • proves proof ch:11-manifolds-tensors-curvature@proof-48 ¶
• proves proof ch:03-einstein-field-equations@proof-17 ¶

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