proposition 44.30 The cosmological term

open in the book · parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1300 · p. 1275

Rests on

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 13.88 General coordinate transformation ¶ ↺
  - depends_on definition 13.82 Vector field ¶
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
- 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 definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
    - 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 13.145 Affine connection ¶ ↺
      - depends_on definition 13.147 Torsion ¶
      - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
- 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 ¶
      - depends_on definition 13.103 Exterior derivative ¶
      - 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 ¶ ↺
      - 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 ¶
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 definition 16.15 Variation; the first variation ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - proves proof ch:14-calculus-of-variations@proof-8 ¶
    - depends_on theorem 7.133 Gauss ¶
      - depends_on definition 7.127 Simple regions ¶
      - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:05-real-analysis@proof-79 ¶
    - 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 definition 7.12 Product of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-14 ¶
    - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
      - depends_on theorem 5.40 Rank–nullity ¶
      - proves proof ch:03-linear-algebra-representations@proof-18 ¶
    - 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 ¶

Supports

depends_on proposition 44.31 de Sitter geometry ¶

Neighborhood

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Metric compatibility	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1313
`depends_on`	→	Contracted Bianchi identity	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1313
`depends_on`	→	Variation of the Einstein–Hilbert action	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1313
`depends_on`	←	de Sitter geometry	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1357
`proves`	←	ch:03-einstein-field-equations@proof-16	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1317