proposition 44.13 First-order variation in vielbein-form variables

Rests on

• depends_on proposition 13.158 The connection determined by vielbein and torsion ¶
  • depends_on equation 13.266 eq:mfd-anholonomy ¶
  • depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
    • 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 13.149 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 definition 13.145 Affine connection ¶ ↺
    • 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 13.156 Cartan structure equations ¶
    • depends_on definition 13.103 Exterior derivative ¶
      • depends_on definition 13.98 $k$-form ¶
        • depends_on definition 13.83 Covector ¶ ↺
        • depends_on definition 13.84 Tensor ¶ ↺
      • depends_on definition 13.102 Wedge product ¶
        • depends_on definition 13.98 $k$-form ¶ ↺
        • depends_on equation 13.239 eq:mfd-dual-basis ¶
    • depends_on definition 13.147 Torsion ¶ ↺
    • depends_on definition 13.122 Vielbein ¶
      • depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
      • depends_on equation 13.1 eq:mfd-eta ¶
    • depends_on definition 13.102 Wedge product ¶ ↺
    • depends_on equation 13.303 eq:mfd-vielbein-postulate ¶
    • 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-45 ¶
  • proves proof ch:11-manifolds-tensors-curvature@proof-47 ¶
• depends_on theorem 43.3 Equivalence of the two variable sets ¶
  • depends_on definition 43.2 Einstein–Hilbert action ¶
  • depends_on definition 13.122 Vielbein ¶ ↺
  • depends_on proposition 13.158 The connection determined by vielbein and torsion ¶ ↺
  • proves derived proof ch:02-geometric-formulation@proof-3 ¶
• depends_on theorem 13.156 Cartan structure equations ¶ ↺
• proves proof ch:03-einstein-field-equations@proof-7 ¶

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