theorem 13.152 Riemann tensor; Ricci identity with torsion

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6679 · p. 532

Rests on

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 13.48 Differentiable manifold ¶
    - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
    - depends_on definition 13.48 Differentiable 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 definition 3.43 Map ¶
  - 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 definition 7.9 Real function ¶ ↺
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - depends_on theorem 7.34 Rolle ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - 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 13.153 Contractions ¶
- 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 ¶
      - tests experiment The Hulse–Taylor 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 theorem 49.9 Tolman–Oppenheimer–Volkoff equations ¶
      - depends_on example 49.11 The uniform-density star ¶
      - 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 theorem 48.14 Linear growth of density perturbations ¶
    - 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 44.23 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 ¶
      - tests experiment The Cosmic Microwave Background ¶
    - 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 48.17 The expansion is accelerating ¶ ↺
      - depends_on phenomenon 48.18 The Hubble tension ¶
depends_on example 21.74 Rindler coordinates ¶
depends_on lemma A.640 Variation of the integrated three-curvature ¶ ↺
depends_on lemma 44.8 Palatini identity ¶
- depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
  - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
  - depends_on proposition 44.30 The cosmological term ¶ ↺
depends_on proposition 44.44 Harmonic-gauge reduction ¶
- depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶ ↺
depends_on proposition 13.155 Geodesic deviation; Jacobi equation ¶
depends_on proposition 13.140 Killing's equation ¶
- depends_on example 13.144 The two Killing tensors every metric carries ¶
- depends_on proposition 14.69 The de~Sitter algebras are isometry algebras ¶
- depends_on proposition 14.59 The Killing fields of a flat pseudo-Euclidean space ¶
  - depends_on corollary A.344 The algebras $\mathfrak{so}(p,q)$ are semisimple ¶
    - depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶
    - depends_on example A.359 The Lorentz algebra ¶
  - depends_on proposition 14.69 The de~Sitter algebras are isometry algebras ¶ ↺
- depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶
  - depends_on phenomenon 45.52 The shadow ¶
    - depends_on phenomenon 50.4 A ring of light at event-horizon scale ¶
  - depends_on proposition 48.4 Peculiar momentum decays with the expansion ¶
  - depends_on proposition 45.6 Radial equation and effective potential ¶
    - depends_on proposition 45.10 Circular orbits and their energy ¶
      - depends_on phenomenon 45.12 Efficiency of accretion onto compact objects ¶
    - depends_on proposition 45.8 Orbit equation ¶
      - depends_on phenomenon 53.2 Measured deflection of starlight at the solar limb ¶
      - depends_on phenomenon 53.1 Measured anomalous advance of Mercury's perihelion ¶
      - depends_on phenomenon 45.9 Relativistic advance of the periastron ¶
    - depends_on proposition 45.21 Radial plunge: the cycloid ¶
      - depends_on proposition 45.22 The view from outside: exponential freezing ¶
      - depends_on theorem 45.23 Oppenheimer–Snyder collapse ¶
  - depends_on theorem 13.143 Killing tensors and geodesic invariants ¶
- depends_on remark 23.21 The tensorial statement, and what Part II owes it ¶
- depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶ ↺
depends_on proposition 13.162 Curvature induced on the quadric ¶
- depends_on theorem 48.3 The Robertson–Walker metric ¶ ↺
depends_on proposition 13.154 Symmetries of the curvature ¶
- depends_on definition 13.111 Linearised curvature of a symmetric field ¶
  - depends_on theorem 13.112 Symmetric analogue of the converse Poincaré lemma ¶
- depends_on theorem 44.23 Contracted Bianchi identity ¶ ↺
- depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶ ↺
depends_on remark 30.16 Incompatibility is curvature ¶
depends_on theorem 13.156 Cartan structure equations ¶
- depends_on proposition 44.13 First-order variation in vielbein-form variables ¶
- depends_on proposition 13.157 Bianchi identities ¶
  - depends_on proposition 13.154 Symmetries of the curvature ¶ ↺
- depends_on proposition 13.158 The connection determined by vielbein and torsion ¶
  - depends_on proposition 44.13 First-order variation in vielbein-form variables ¶ ↺
  - depends_on theorem 43.3 Equivalence of the two variable sets ¶
    - depends_on proposition 44.13 First-order variation in vielbein-form variables ¶ ↺

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`	→	Affine connection	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6697
`depends_on`	→	Torsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6697
`depends_on`	→	Clairaut–Schwarz	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6697
`depends_on`	←	Contractions	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6727
`depends_on`	←	Rindler coordinates	declared	parts/03-classical-mechanics/04-lagrangian-mechanics.tex:2237
`depends_on`	←	Variation of the integrated three-curvature	declared	appendices/A-long-proofs.tex:31180
`depends_on`	←	Palatini identity	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:313
`depends_on`	←	Harmonic-gauge reduction	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1839
`depends_on`	←	Geodesic deviation; Jacobi equation	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6774
`depends_on`	←	Killing's equation	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6306
`depends_on`	←	Curvature induced on the quadric	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:7168
`depends_on`	←	Symmetries of the curvature	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6743
`depends_on`	←	Incompatibility is curvature	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:522
`depends_on`	←	Cartan structure equations	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6841
`proves`	←	ch:11-manifolds-tensors-curvature@proof-42	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6700