theorem 44.23 Contracted Bianchi identity

open in the book · parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:986 · p. 1272

Rests on

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 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 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 ¶
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 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 ¶
    - depends_on definition 13.103 Exterior derivative ¶
      - depends_on definition 13.98 $k$-form ¶
      - depends_on definition 13.102 Wedge product ¶
    - 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 ¶ ↺
    - proves proof ch:11-manifolds-tensors-curvature@proof-45 ¶
  - 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 ¶

Supports

depends_on corollary 44.24 Conservation is an integrability condition ¶
depends_on proposition 44.41 Four of the ten equations are constraints ¶
depends_on proposition 44.30 The cosmological term ¶
- depends_on proposition 44.31 de Sitter geometry ¶
depends_on theorem 44.33 Lovelock's uniqueness theorem; imported ¶
- depends_on corollary 44.35 The field equations are essentially unique ¶

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 tensor	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:994
`depends_on`	→	Metric compatibility	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:994
`depends_on`	→	Symmetries of the curvature	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:994
`depends_on`	←	Conservation is an integrability condition	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1058
`depends_on`	←	Four of the ten equations are constraints	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1676
`depends_on`	←	The cosmological term	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1313
`depends_on`	←	Lovelock's uniqueness theorem; imported	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1427
`proves`	←	ch:03-einstein-field-equations@proof-12	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:998