proposition 48.7 Luminosity distance

open in the book · parts/05-general-relativity-cosmology/07-cosmology.tex:519 · p. 1325

Rests on

depends_on equation 48.13 eq:cosmo-chi-of-z ¶
depends_on equation 48.2 eq:cosmo-redshift-scale-factor ¶
depends_on theorem 48.3 The Robertson–Walker metric ¶
- depends_on proposition 13.162 Curvature induced on the quadric ¶
  - depends_on definition 13.119 Induced metric ¶
    - 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 equation 13.83 eq:mfd-ecsgauss ¶
  - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
    - depends_on definition 13.145 Affine connection ¶
      - depends_on definition 13.90 Mixed tensor ¶
      - depends_on definition 13.82 Vector field ¶
    - depends_on definition 13.147 Torsion ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
    - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-64 ¶
    - proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-49 ¶
- depends_on remark 13.161 rem:mfd-schur ¶
- depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
  - 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 proposition 13.140 Killing's equation ¶
    - depends_on definition 13.139 Killing vector ¶
      - depends_on definition 13.120 Isometry ¶
      - depends_on equation 13.267 eq:mfd-lie-def ¶
    - 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 proposition 13.127 Component formulas ¶
      - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.90 Mixed tensor ¶ ↺
      - depends_on equation 13.267 eq:mfd-lie-def ¶ ↺
      - proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
    - 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-37 ¶
  - 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 ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-46 ¶
    - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
    - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
    - proves proof ch:11-manifolds-tensors-curvature@proof-43 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-48 ¶
- proves proof ch:07-cosmology@proof-2 ¶
proves proof ch:07-cosmology@proof-5 ¶

Supports

depends_on corollary 48.8 Angular-diameter distance and the duality relation ¶
- depends_on phenomenon 48.15 A standard ruler in the galaxy distribution ¶
  - depends_on phenomenon 103.68 The sum of the neutrino masses is bounded ¶
depends_on phenomenon 48.17 The expansion is accelerating ¶
depends_on phenomenon 48.18 The Hubble tension ¶

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`	→	eq:cosmo-chi-of-z	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:527
`depends_on`	→	eq:cosmo-redshift-scale-factor	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:527
`depends_on`	→	The Robertson–Walker metric	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:527
`depends_on`	←	Angular-diameter distance and the duality relation	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:559
`depends_on`	←	The expansion is accelerating	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:1454
`depends_on`	←	The Hubble tension	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:1671
`proves`	←	ch:07-cosmology@proof-5	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:531