proposition 44.44 Harmonic-gauge reduction

open in the book · parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1815 · p. 1281

Rests on

depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
- 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 13.149 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 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 definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - 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-24 ¶

Supports

depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
- depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶

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`	→	Levi-Civita connection and contorsion	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1839
`depends_on`	→	Riemann tensor; Ricci identity with torsion	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1839
`depends_on`	←	Local existence and uniqueness; Choquet-Bruhat; imported	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1908
`proves`	←	ch:03-einstein-field-equations@proof-24	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1842