proposition 44.20 Perfect-fluid equations of motion

open in the book · parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:875 · p. 1271

Rests on

depends_on definition 44.16 Perfect fluid ¶
- depends_on definition 44.15 Dust ¶
  - depends_on definition 40.2 Four-velocity ¶
    - depends_on definition 38.12 Proper time ¶
      - depends_on definition 38.6 Minkowski metric and interval ¶
      - depends_on proposition 38.5 The invariant interval ¶
    - depends_on definition 40.1 Four-vector ¶
      - depends_on definition 38.6 Minkowski metric and interval ¶ ↺
      - depends_on equation 38.14 eq:lor-invariance-condition ¶
  - depends_on phenomenon 40.6 Mass–energy equivalence ¶
    - depends_on equation 40.4 eq:rdyn-E-p ¶
    - proves proof ch:04-relativistic-dynamics@proof-3 ¶
depends_on theorem 44.19 Covariant conservation of stress–energy ¶
- 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 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 ¶
      - 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 ¶
proves proof ch:03-einstein-field-equations@proof-10 ¶

Supports

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
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Perfect fluid	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:892
`depends_on`	→	Covariant conservation of stress–energy	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:892
`depends_on`	←	Dust flows on geodesics	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:936
`proves`	←	ch:03-einstein-field-equations@proof-10	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:895