definition 13.153 Contractions

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6718 · p. 532

ground object -- no derivation owed

Rests on

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 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
- depends_on definition 13.84 Tensor ¶
  - depends_on definition 13.83 Covector ¶
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - depends_on definition 13.81 Vector on a manifold ¶ ↺
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 13.89 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 13.147 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 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 ¶

Supports

depends_on definition 44.2 Einstein tensor ¶
- depends_on postulate 44.3 Einstein field equations ¶
  - depends_on corollary 44.24 Conservation is an integrability condition ¶
  - depends_on phenomenon 49.18 Orbital decay of a binary pulsar ¶
  - depends_on phenomenon 46.1 phen:gravitational-waves ¶
    - depends_on phenomenon 49.18 Orbital decay of a binary pulsar ¶ ↺
    - tests experiment GW150914 ¶
      - cites (method) derived bib Advanced LIGO ¶
      - cites (method) derived bib Advanced LIGO ¶ ↺
      - cites (evidence) derived bib Observation of Gravitational Waves from a Binary Black Hole Merger ¶
      - cites (method) derived bib Observation of Gravitational Waves from a Binary Black Hole Merger ¶ ↺
      - cites (evidence) derived bib Observation of Gravitational Waves from a Binary Black Hole Merger ¶ ↺
      - cites (evidence) derived bib Observation of Gravitational Waves from a Binary Black Hole Merger ¶ ↺
      - cites (evidence) derived bib Observation of Gravitational Waves from a Binary Black Hole Merger ¶ ↺
      - cites (evidence) derived bib Observation of Gravitational Waves from a Binary Black Hole Merger ¶ ↺
    - tests experiment The Hulse–Taylor Binary Pulsar (PSR B1913+16) ¶
      - cites (evidence) derived bib Discovery of a pulsar in a binary system ¶
      - cites (evidence) derived bib Discovery of a pulsar in a binary system ¶ ↺
      - cites (data) derived bib Discovery of a pulsar in a binary system ¶ ↺
      - cites (evidence) derived bib Discovery of a pulsar in a binary system ¶ ↺
      - cites (evidence) derived bib A new test of general relativity: Gravitational radiation and the binary pulsar PSR 1913+16 ¶
      - cites (evidence) derived bib A new test of general relativity: Gravitational radiation and the binary pulsar PSR 1913+16 ¶ ↺
      - cites (evidence) derived bib Relativistic Measurements from Timing the Binary Pulsar PSR B1913+16 ¶
      - cites (data) derived bib Relativistic Measurements from Timing the Binary Pulsar PSR B1913+16 ¶ ↺
      - cites (data) derived bib Relativistic Measurements from Timing the Binary Pulsar PSR B1913+16 ¶ ↺
      - cites (evidence) derived bib Relativistic Measurements from Timing the Binary Pulsar PSR B1913+16 ¶ ↺
      - cites (evidence) derived bib Relativistic Measurements from Timing the Binary Pulsar PSR B1913+16 ¶ ↺
      - cites (evidence) derived bib Relativistic Measurements from Timing the Binary Pulsar PSR B1913+16 ¶ ↺
  - depends_on proposition 44.29 Linearized field equations ¶
  - depends_on proposition 44.4 Trace-reversed form ¶
    - depends_on proposition 44.39 SEC is the attraction condition ¶
    - depends_on theorem 44.27 Newtonian correspondence of the field equations ¶
      - depends_on corollary 44.35 The field equations are essentially unique ¶
      - depends_on phenomenon 49.6 A maximum mass for a cold degenerate star ¶
      - depends_on phenomenon 49.16 Pulsars: clocks at nuclear density ¶
      - depends_on phenomenon 49.14 A neutron star is observed being born ¶
      - depends_on proposition 44.29 Linearized field equations ¶ ↺
  - depends_on theorem 49.9 Tolman–Oppenheimer–Volkoff equations ¶
    - depends_on example 49.11 The uniform-density star ¶
      - depends_on proposition 49.12 Buchdahl bound ¶
    - depends_on phenomenon 49.19 Neutron stars reach two solar masses ¶
  - depends_on theorem 48.5 The Friedmann equations ¶
    - depends_on phenomenon 48.17 The expansion is accelerating ¶
    - depends_on phenomenon 48.10 A primordial floor of light elements ¶
    - depends_on phenomenon 51.6 Acoustic peaks, and a spatially flat universe ¶
      - depends_on phenomenon 51.7 E-mode polarization, and no primordial B mode ¶
    - depends_on theorem 48.14 Linear growth of density perturbations ¶
      - depends_on phenomenon 48.16 Flat rotation curves and the missing mass ¶
  - depends_on theorem 44.42 The constraint equations ¶
    - 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 ¶
    - depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶ ↺
- depends_on theorem 44.23 Contracted Bianchi identity ¶
  - depends_on corollary 44.24 Conservation is an integrability condition ¶ ↺
  - depends_on proposition 44.41 Four of the ten equations are constraints ¶
  - depends_on proposition 44.30 The cosmological term ¶
    - depends_on proposition 44.31 de Sitter geometry ¶
  - depends_on theorem 44.33 Lovelock's uniqueness theorem; imported ¶
    - depends_on corollary 44.35 The field equations are essentially unique ¶ ↺
- depends_on theorem 44.33 Lovelock's uniqueness theorem; imported ¶ ↺
depends_on lemma A.627 The dictionary between $\mathcal{K},\gamma$ and $K,h$ ¶
- depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶
depends_on lemma A.631 The normal–normal Ricci contraction ¶
- depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶ ↺
depends_on lemma A.640 Variation of the integrated three-curvature ¶
- depends_on theorem A.641 The normal–normal bracket ¶
  - depends_on corollary A.642 The constraints are first class ¶
depends_on proposition 15.31 Radius and cosmological constant ¶
- depends_on corollary 15.32 The expansion parameter is the cosmological constant ¶
  - depends_on example 15.64 The whole chapter at $D=4$ ¶
  - depends_on remark 15.63 The inclusion is strict, and the strictness is physical ¶
  - depends_on remark 15.33 Which branch nature is on ¶
- depends_on example 15.64 The whole chapter at $D=4$ ¶ ↺
- depends_on remark 15.33 Which branch nature is on ¶ ↺
depends_on theorem A.49 thm:app-eh-equivalence ¶
depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
- depends_on proposition 44.31 de Sitter geometry ¶ ↺
- depends_on proposition 15.31 Radius and cosmological constant ¶ ↺
- depends_on theorem 48.3 The Robertson–Walker metric ¶
  - depends_on phenomenon 48.12 The relic microwave background ¶
    - depends_on phenomenon 48.13 The anisotropy spectrum of the relic radiation ¶
      - depends_on phenomenon 48.18 The Hubble tension ¶
    - tests experiment The Cosmic Microwave Background ¶
      - cites (method) derived bib 24 Detection ofB-Mode Polarization at Degree Angular Scales by BICEP2 ¶
      - cites (evidence) derived bib 24 Detection ofB-Mode Polarization at Degree Angular Scales by BICEP2 ¶ ↺
      - cites (evidence) derived bib 10 Joint Analysis of BICEP2/Keck ArrayandPlanckData ¶
      - cites (evidence) derived bib 10 Joint Analysis of BICEP2/Keck ArrayandPlanckData ¶ ↺
      - cites (method) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (context) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib Planck 2018 results. VI. Cosmological parameters ¶
      - cites (method) derived bib Planck 2018 results. VI. Cosmological parameters ¶ ↺
      - … 109 more
  - depends_on phenomenon 48.1 The Hubble–Lemaître law ¶
  - depends_on phenomenon 51.6 Acoustic peaks, and a spatially flat universe ¶ ↺
  - depends_on proposition 48.7 Luminosity distance ¶
    - 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 48.17 The expansion is accelerating ¶ ↺
    - depends_on phenomenon 48.18 The Hubble tension ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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not graded
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Edges

type	direction	node	provenance	where
`depends_on`	→	Metric tensor of signature $(p,q)$	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6727
`depends_on`	→	Riemann tensor; Ricci identity with torsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6727
`depends_on`	←	Einstein tensor	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:145
`depends_on`	←	The dictionary between $\mathcal{K},\gamma$ and $K,h$	declared	appendices/A-long-proofs.tex:30319
`depends_on`	←	The normal–normal Ricci contraction	declared	appendices/A-long-proofs.tex:30581
`depends_on`	←	Variation of the integrated three-curvature	declared	appendices/A-long-proofs.tex:31180
`depends_on`	←	Radius and cosmological constant	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1023
`depends_on`	←	thm:app-eh-equivalence	declared	appendices/A-long-proofs.tex:3195
`depends_on`	←	Maximal symmetry forces constant curvature	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6984