definition 13.90 Mixed tensor

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4283 · p. 505

ground object -- no derivation owed

Rests on

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 ¶ ↺

Supports

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 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 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 44.30 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 proposition 45.8 Orbit equation ¶
      - depends_on proposition 45.21 Radial plunge: the cycloid ¶
    - 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 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 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 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 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 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 proposition 45.43 Static acceleration and surface gravity ¶
      - 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 proposition 45.6 Radial equation and effective potential ¶ ↺
      - depends_on proposition 45.37 Reissner–Nordström solution ¶
      - depends_on theorem 45.4 Jebsen–Birkhoff ¶
      - 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 lemma A.627 The dictionary between $\mathcal{K},\gamma$ and $K,h$ ¶
      - depends_on lemma A.631 The normal–normal Ricci contraction ¶ ↺
      - depends_on lemma A.640 Variation of the integrated three-curvature ¶
      - 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 13.158 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 ¶ ↺
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 proposition A.571 Invariant Hamiltonians descend with their flows ¶
    - depends_on remark 24.35 What non-squeezing does and does not say about nature ¶
    - depends_on theorem 24.21 Liouville ¶
      - depends_on example 24.29 A volume-preserving squeeze ¶
      - depends_on proposition 32.40 A Hamiltonian section preserves area ¶
      - depends_on remark 22.25 This is Liouville's theorem ¶
      - depends_on remark 24.22 Liouville's theorem is the foundation of statistical mechanics ¶
      - depends_on theorem 23.31 Liouville–Arnold ¶
      - depends_on theorem 25.57 The Moyal equation, and its classical limit ¶
      - depends_on theorem 24.24 Poincaré recurrence ¶
- 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 ¶ ↺

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Edges

type	direction	node	provenance	where
`depends_on`	→	Contravariant and covariant tensors	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4293
`depends_on`	→	General coordinate transformation	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4293
`depends_on`	←	Affine connection	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6507
`depends_on`	←	Component formulas	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5801
`depends_on`	←	prop:mfd-torsion-tensor	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6538