theorem 43.3 Equivalence of the two variable sets

open in the book · parts/05-general-relativity-cosmology/02-geometric-formulation.tex:188 · p. 1261

Rests on

depends_on definition 43.2 Einstein–Hilbert action ¶
depends_on definition 13.122 Vielbein ¶
- 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 13.44 Atlas ¶
  - depends_on definition 13.84 Tensor ¶
    - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.81 Vector on a manifold ¶ ↺
- depends_on equation 13.1 eq:mfd-eta ¶
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 13.149 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 13.102 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 ¶
proves derived proof ch:02-geometric-formulation@proof-3 ¶

Supports

depends_on proposition 44.13 First-order variation in vielbein-form variables ¶

Neighborhood

Every logical edge within two steps of this node.

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

Edges

type	direction	node	provenance	where
`depends_on`	→	Einstein–Hilbert action	declared	parts/05-general-relativity-cosmology/02-geometric-formulation.tex:193
`depends_on`	→	Vielbein	declared	parts/05-general-relativity-cosmology/02-geometric-formulation.tex:193
`depends_on`	→	The connection determined by vielbein and torsion	declared	parts/05-general-relativity-cosmology/02-geometric-formulation.tex:193
`depends_on`	←	First-order variation in vielbein-form variables	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:610
`proves`	←	ch:02-geometric-formulation@proof-3	derived	parts/05-general-relativity-cosmology/02-geometric-formulation.tex:196