theorem 44.19 Covariant conservation of stress–energy

open in the book · parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:830 · p. 1271

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 proposition 13.127 Component formulas ¶
- depends_on definition 13.89 Contravariant and covariant tensors ¶
  - depends_on definition 13.88 General coordinate transformation ¶
- 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 equation 13.267 eq:mfd-lie-def ¶
- proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
depends_on theorem 16.93 Noether's second theorem ¶
- depends_on equation 16.32 eq:calcvar-euler-lagrange-field ¶
- depends_on equation 16.33 eq:calcvar-field-first-variation ¶
- depends_on theorem 16.90 Noether's first theorem, several independent variables ¶
  - depends_on equation 16.32 eq:calcvar-euler-lagrange-field ¶ ↺
  - depends_on equation 16.33 eq:calcvar-field-first-variation ¶ ↺
  - depends_on theorem 16.84 Noether's first theorem, one independent variable ¶
    - depends_on equation 16.30 eq:calcvar-euler-lagrange-system ¶
    - depends_on equation 16.75 eq:calcvar-invariance ¶
    - depends_on proposition 16.17 The first variation of an integral functional ¶
      - depends_on equation 16.11 eq:calcvar-basic-functional ¶
      - depends_on equation 16.14 eq:calcvar-gateaux ¶
      - proves proof ch:14-calculus-of-variations@proof-7 ¶
    - proves proof ch:14-calculus-of-variations@proof-39 ¶
  - proves proof ch:14-calculus-of-variations@proof-42 ¶
- proves proof ch:14-calculus-of-variations@proof-44 ¶
proves proof ch:03-einstein-field-equations@proof-9 ¶

Supports

depends_on proposition 44.20 Perfect-fluid equations of motion ¶
- depends_on corollary 44.21 Dust flows on geodesics ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
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inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Stress–energy tensor	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:837
`depends_on`	→	Component formulas	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:837
`depends_on`	→	Noether's second theorem	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:837
`depends_on`	←	Perfect-fluid equations of motion	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:892
`proves`	←	ch:03-einstein-field-equations@proof-9	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:841