definition 13.89 Contravariant and covariant tensors

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4262 · p. 505

ground object -- no derivation owed

Rests on

depends_on definition 13.88 General coordinate transformation ¶

Supports

depends_on definition 13.90 Mixed tensor ¶
- depends_on definition 13.145 Affine connection ¶
  - 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 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 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 44.30 The cosmological term ¶
      - depends_on proposition 44.31 de Sitter geometry ¶
    - 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 proposition 48.4 Peculiar momentum decays with the expansion ¶
      - depends_on proposition 45.6 Radial equation and effective potential ¶
      - 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 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 13.154 Symmetries of the curvature ¶
      - depends_on definition 13.111 Linearised curvature of a symmetric field ¶
      - depends_on theorem 44.23 Contracted Bianchi identity ¶
      - 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 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 13.154 Symmetries of the curvature ¶ ↺
      - depends_on theorem 45.1 Schwarzschild solution ¶
  - 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 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 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 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 ¶ ↺
- depends_on proposition 13.127 Component formulas ¶
  - depends_on lemma A.75 Differentiating a pullback along a flow ¶
    - depends_on lemma A.77 Poincaré lemma, converse form ¶
  - depends_on proposition 13.129 Commutation of Lie derivative and interior product ¶
  - depends_on proposition 13.128 Cartan's magic formula ¶
    - depends_on lemma A.77 Poincaré lemma, converse form ¶ ↺
    - depends_on proposition 13.129 Commutation of Lie derivative and interior product ¶ ↺
    - depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
      - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
      - depends_on remark 24.35 What non-squeezing does and does not say about nature ¶
      - depends_on theorem 24.21 Liouville ¶
  - depends_on proposition 13.140 Killing's equation ¶ ↺
  - depends_on theorem 44.19 Covariant conservation of stress–energy ¶
    - depends_on proposition 44.20 Perfect-fluid equations of motion ¶
      - depends_on corollary 44.21 Dust flows on geodesics ¶
- depends_on proposition 13.148 prop:mfd-torsion-tensor ¶ ↺
depends_on definition 13.92 Tensor density ¶
- depends_on example 13.94 The two densities this chapter uses ¶
  - depends_on lemma 14.62 The Levi-Civita symbol is invariant ¶
    - depends_on definition 29.4 Angular velocity ¶
      - depends_on proposition 29.11 Angular velocity in Euler angles ¶
      - depends_on proposition 29.5 The velocity field of a rigid body ¶
      - depends_on theorem 29.7 Transport theorem ¶
    - depends_on lemma 14.64 Properties of $W$ ¶
      - depends_on theorem 14.65 The two Casimir operators of the Poincaré algebra in $3+1$ dimensions ¶
- depends_on proposition 13.93 Algebra of densities ¶
  - depends_on example 13.94 The two densities this chapter uses ¶ ↺
depends_on proposition 13.127 Component formulas ¶ ↺

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Edges

type	direction	node	provenance	where
`depends_on`	→	General coordinate transformation	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4280
`depends_on`	←	Mixed tensor	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4293
`depends_on`	←	Tensor density	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4554
`depends_on`	←	Component formulas	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5801