definition 13.145 Affine connection

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6482 · p. 530

ground object -- no derivation owed

Rests on

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 13.47 Differentiable structure ¶
  - 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 13.43 Coordinate system ¶ ↺
  - depends_on definition 13.48 Differentiable manifold ¶ ↺

Supports

depends_on definition A.625 Four-dimensional extrinsic curvature ¶
- depends_on lemma A.630 Codazzi equation ¶
  - 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 lemma A.628 Gauss equation, arbitrary signature ¶
  - depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶
  - depends_on theorem 44.42 The constraint equations ¶ ↺
- depends_on lemma A.626 Decomposition of $\nabla n$, and the acceleration ¶
  - depends_on lemma A.630 Codazzi equation ¶ ↺
  - 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 definition A.624 Projector and induced metric ¶
depends_on definition 13.77 Curvature vector ¶
- depends_on proposition 13.78 The curvature vector is orthogonal to the tangent ¶
  - depends_on theorem 13.79 Frenet–Serret equations, covariant form ¶
depends_on definition 13.149 Metric compatibility ¶
- depends_on definition 13.142 Killing tensor ¶
  - depends_on example 13.144 The two Killing tensors every metric carries ¶
  - depends_on theorem 13.143 Killing tensors and geodesic invariants ¶
- depends_on example 13.144 The two Killing tensors every metric carries ¶ ↺
- 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 44.31 de Sitter geometry ¶
- depends_on proposition 44.30 The cosmological term ¶ ↺
- depends_on proposition 13.78 The curvature vector is orthogonal to the tangent ¶ ↺
- 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 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 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 44.31 de Sitter geometry ¶ ↺
    - 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 15.33 Which branch nature is on ¶
    - depends_on theorem 48.3 The Robertson–Walker metric ¶
      - depends_on phenomenon 48.12 The relic 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 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 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 theorem 44.33 Lovelock's uniqueness theorem; imported ¶
      - depends_on corollary 44.35 The field equations are essentially unique ¶
  - depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶ ↺
- depends_on theorem 44.23 Contracted Bianchi identity ¶ ↺
- depends_on theorem 13.79 Frenet–Serret equations, covariant form ¶ ↺
- depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
  - depends_on definition 13.77 Curvature vector ¶ ↺
  - depends_on lemma 44.8 Palatini identity ¶ ↺
  - depends_on proposition 44.44 Harmonic-gauge reduction ¶
    - depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶ ↺
  - depends_on proposition 21.73 Free motion in flat spacetime, any coordinates ¶
  - depends_on proposition 13.140 Killing's equation ¶ ↺
  - 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 ¶ ↺
  - depends_on proposition 13.154 Symmetries of the curvature ¶ ↺
  - depends_on theorem 45.1 Schwarzschild solution ¶
    - depends_on phenomenon 50.4 A ring of light at event-horizon scale ¶ ↺
    - depends_on phenomenon 50.5 Black-hole masses from gravitational waves ¶
    - depends_on phenomenon 45.52 The shadow ¶ ↺
    - depends_on proposition 45.14 Isotropic coordinates ¶
      - depends_on phenomenon 53.4 Excess delay of a radar echo passing the Sun ¶
      - depends_on phenomenon 42.8 Shapiro time delay ¶
    - depends_on proposition 45.43 Static acceleration and surface gravity ¶
      - depends_on proposition 45.46 The first law ¶
      - depends_on theorem 45.49 Hawking temperature; imported ¶
    - depends_on proposition 45.16 Curvature at the horizon and at the centre ¶
    - depends_on proposition 45.19 Kruskal–Szekeres coordinates ¶
    - depends_on proposition 45.18 The one-way membrane ¶
      - depends_on phenomenon 50.6 No material surface ¶
      - depends_on theorem 45.47 Area theorem; imported ¶
    - depends_on proposition 45.6 Radial equation and effective potential ¶ ↺
    - depends_on proposition 45.37 Reissner–Nordström solution ¶
      - depends_on theorem 45.39 Israel; imported ¶
    - depends_on theorem 45.4 Jebsen–Birkhoff ¶
      - depends_on corollary 45.5 No monopole gravitational radiation ¶
      - depends_on theorem 45.23 Oppenheimer–Snyder collapse ¶ ↺
    - depends_on theorem 45.39 Israel; imported ¶ ↺
    - … 1 more
depends_on definition 13.146 Parallel transport and autoparallels ¶
- depends_on definition 13.149 Metric compatibility ¶ ↺
- depends_on proposition 13.155 Geodesic deviation; Jacobi equation ¶
- depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶ ↺
- depends_on remark 29.62 The same rate read as a holonomy ¶
- depends_on theorem 13.143 Killing tensors and geodesic invariants ¶ ↺
depends_on definition 13.147 Torsion ¶
- depends_on proposition 13.148 prop:mfd-torsion-tensor ¶
- depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
- depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
  - depends_on definition 13.153 Contractions ¶
    - depends_on definition 44.2 Einstein tensor ¶
      - depends_on postulate 44.3 Einstein field equations ¶
      - depends_on theorem 44.23 Contracted Bianchi identity ¶ ↺
      - 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 lemma A.640 Variation of the integrated three-curvature ¶
      - depends_on theorem A.641 The normal–normal bracket ¶
    - depends_on proposition 15.31 Radius and cosmological constant ¶ ↺
    - depends_on theorem A.49 thm:app-eh-equivalence ¶
    - depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶ ↺
  - 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 proposition 44.44 Harmonic-gauge reduction ¶ ↺
  - depends_on proposition 13.155 Geodesic deviation; Jacobi equation ¶ ↺
  - depends_on proposition 13.140 Killing's equation ¶ ↺
  - 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 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 theorem 13.156 Cartan structure equations ¶ ↺
depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Mixed tensor	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6507
`depends_on`	→	Vector field	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6507
`depends_on`	←	Four-dimensional extrinsic curvature	declared	appendices/A-long-proofs.tex:30222
`depends_on`	←	Projector and induced metric	declared	appendices/A-long-proofs.tex:30182
`depends_on`	←	Curvature vector	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3705
`depends_on`	←	Metric compatibility	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6560
`depends_on`	←	Parallel transport and autoparallels	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6523
`depends_on`	←	Torsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6532
`depends_on`	←	Levi-Civita connection and contorsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6583
`depends_on`	←	Riemann tensor; Ricci identity with torsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6697