theorem 44.9 Variation of the Einstein–Hilbert action

open in the book · parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:382 · p. 1267

Rests on

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 definition 16.11 Admissible class; functional ¶
      - depends_on equation 7.177 eq:ana-functional-derivative ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
    - proves proof ch:14-calculus-of-variations@proof-8 ¶
  - depends_on theorem 7.133 Gauss ¶
    - depends_on definition 7.127 Simple regions ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - 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 definition 7.9 Real function ¶
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-11 ¶
    - 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 definition 5.37 Linear transformation ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-12 ¶
    - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
      - depends_on definition 5.47 Inverse of a linear transformation ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-17 ¶
    - depends_on theorem 5.40 Rank–nullity ¶
      - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
      - depends_on lemma 5.38 Exchange and completion ¶
      - proves proof ch:03-linear-algebra-representations@proof-13 ¶
    - 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 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 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 ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
- 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 ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-64 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
- proves proof ch:03-einstein-field-equations@proof-3 ¶
proves proof ch:03-einstein-field-equations@proof-4 ¶

Supports

depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
depends_on proposition 44.30 The cosmological term ¶
- depends_on proposition 44.31 de Sitter geometry ¶

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
`cites`	→	Die Feldgleichungen der Gravitation	derived	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:402
`cites`	→	Die Grundlagen der Physik. (Erste Mitteilung.)	derived	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:402
`depends_on`	→	Stress–energy tensor	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:403
`depends_on`	→	Variation of inverse and determinant	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:403
`depends_on`	→	Palatini identity	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:403
`depends_on`	←	The boundary term, and the Gibbons–Hawking–York action	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:522
`depends_on`	←	The cosmological term	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1313
`proves`	←	ch:03-einstein-field-equations@proof-4	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:407