definition 13.84 Tensor

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4012 · p. 502

ground object -- no derivation owed

Rests on

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 13.48 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 ¶ ↺
depends_on definition 13.81 Vector on a manifold ¶ ↺

Supports

depends_on definition 13.98 $k$-form ¶
- depends_on definition A.70 Pullback ¶
  - depends_on definition 24.17 Symplectomorphism ¶
    - depends_on definition 25.6 Infinitesimal canonical transformation ¶
      - depends_on definition 25.11 Generators of the Galilei transformations ¶
    - depends_on definition 24.28 Ball and cylinder ¶
      - depends_on definition 24.33 Symplectic capacity ¶
      - depends_on example 24.29 A volume-preserving squeeze ¶
      - depends_on proposition 24.30 Linear non-squeezing ¶
      - depends_on theorem 24.31 Gromov's non-squeezing theorem ¶
    - depends_on definition 24.43 Momentum map ¶
      - depends_on definition A.561 Coadjoint action and equivariance ¶
      - depends_on example 24.44 Linear and angular momentum ¶
      - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      - depends_on theorem 24.45 Noether, symplectic form ¶
    - depends_on theorem 24.18 Symplectic form of the transformation condition ¶
      - depends_on corollary 24.19 Canonical transformations preserve phase volume ¶
  - depends_on lemma A.75 Differentiating a pullback along a flow ¶
    - depends_on lemma A.77 Poincaré lemma, converse form ¶
  - depends_on lemma A.71 The pullback is an algebra map commuting with $\dd$ ¶
- depends_on definition A.67 Symplectic manifold ¶
  - depends_on theorem A.68 Darboux ¶
    - depends_on corollary A.69 No local invariants ¶
      - depends_on remark 24.13 What Darboux's theorem forbids ¶
    - depends_on example A.80 The phase space of one particle in space ¶
    - depends_on theorem 24.12 Darboux ¶
      - depends_on definition 24.20 Liouville volume ¶
      - depends_on remark 24.13 What Darboux's theorem forbids ¶ ↺
- depends_on definition 13.105 Closed form ¶
  - depends_on definition A.67 Symplectic manifold ¶ ↺
  - depends_on definition 24.8 Symplectic manifold ¶
    - depends_on definition 24.9 The canonical form on a cotangent bundle ¶
      - depends_on proposition 24.10 The canonical form is symplectic and intrinsic ¶
    - depends_on definition 24.14 Hamiltonian vector field ¶
      - depends_on definition 24.43 Momentum map ¶ ↺
      - depends_on definition 24.50 Prequantum operator ¶
      - depends_on proposition 25.4 Leibniz rule and derivations ¶
      - depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
      - depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
    - depends_on definition 24.20 Liouville volume ¶ ↺
    - depends_on definition 24.53 Real polarization and polarized sections ¶
      - depends_on proposition 24.54 The vertical polarization gives wave mechanics ¶
    - depends_on definition 24.49 Prequantum datum ¶
      - depends_on definition 24.53 Real polarization and polarized sections ¶ ↺
      - depends_on definition 24.50 Prequantum operator ¶ ↺
      - depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶
    - depends_on definition 24.17 Symplectomorphism ¶ ↺
    - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶ ↺
    - depends_on theorem 24.12 Darboux ¶ ↺
    - depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶ ↺
    - depends_on theorem 24.38 Symplectic foliation, quoted ¶
    - depends_on theorem 24.47 Marsden–Weinstein reduction ¶
      - depends_on remark 24.48 Reduction is what physicists do without saying so ¶
  - depends_on lemma 13.107 Poincaré lemma ¶
    - depends_on proposition 13.113 Product of a closed and an exact form ¶
    - depends_on proposition 24.10 The canonical form is symplectic and intrinsic ¶ ↺
  - depends_on proposition 13.113 Product of a closed and an exact form ¶ ↺
  - depends_on theorem 13.109 Converse of the Poincaré lemma on a star-shaped domain ¶
    - depends_on example 13.110 Closed but not exact: the angle form ¶
    - depends_on theorem 13.112 Symmetric analogue of the converse Poincaré lemma ¶
- depends_on definition 13.106 Exact form ¶
  - depends_on example 13.110 Closed but not exact: the angle form ¶ ↺
  - depends_on lemma 13.107 Poincaré lemma ¶ ↺
  - depends_on proposition 13.113 Product of a closed and an exact form ¶ ↺
  - depends_on theorem 13.109 Converse of the Poincaré lemma on a star-shaped domain ¶ ↺
- depends_on definition 13.103 Exterior derivative ¶
  - depends_on corollary 13.134 Frobenius for a Pfaffian system ¶
  - depends_on definition 13.105 Closed form ¶ ↺
  - depends_on definition 13.106 Exact form ¶ ↺
  - depends_on definition 25.21 First-order action in general coordinates ¶
    - depends_on theorem 25.22 Faddeev–Jackiw equations and brackets ¶
      - depends_on example 25.24 A charged particle in a strong magnetic field ¶
  - depends_on definition 24.9 The canonical form on a cotangent bundle ¶ ↺
  - depends_on lemma A.312 The computation in one chart ¶
  - 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 theorem A.311 General Stokes theorem ¶
    - depends_on corollary A.313 Stokes' theorem for an antisymmetric tensor field ¶
    - depends_on lemma A.555 The actions are well defined ¶
    - depends_on theorem 24.23 Poincaré–Cartan integral invariant ¶
  - 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 definition 13.116 Hodge dual ¶
- depends_on definition 13.102 Wedge product ¶
  - depends_on definition 13.103 Exterior derivative ¶ ↺
  - depends_on definition 13.114 Volume form ¶
    - depends_on definition 13.116 Hodge dual ¶ ↺
  - depends_on definition 24.20 Liouville volume ¶ ↺
  - depends_on lemma A.301 Transformation of the top form ¶
    - depends_on definition A.302 Orientation ¶
      - depends_on definition A.309 The induced orientation of the boundary ¶
      - depends_on definition A.304 Integral over a chart ¶
    - depends_on lemma A.305 The chart integral is well defined ¶
      - depends_on definition A.308 Integral over the manifold ¶
  - depends_on theorem 13.156 Cartan structure equations ¶ ↺
- depends_on definition 24.8 Symplectic manifold ¶ ↺
- depends_on proposition 13.129 Commutation of Lie derivative and interior product ¶ ↺
- depends_on proposition 13.100 The space of $k$-forms ¶
  - depends_on definition A.304 Integral over a chart ¶ ↺
  - depends_on lemma A.301 Transformation of the top form ¶ ↺
depends_on definition 13.99 $k$-vector ¶
- depends_on definition 13.116 Hodge dual ¶ ↺
depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
- depends_on definition 23.15 Orthogonal Hamiltonian ¶
- depends_on definition 13.119 Induced metric ¶
  - depends_on proposition 13.162 Curvature induced on the quadric ¶
    - 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 definition 13.120 Isometry ¶
  - depends_on corollary 13.31 Bending invariance ¶
  - depends_on definition 13.139 Killing vector ¶
    - 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 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 13.163 Integration of the conformal Killing 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 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 13.121 The isometry group ¶
- depends_on definition 13.74 Length of a curve ¶
  - depends_on definition 13.76 Arclength parametrisation ¶
    - depends_on definition 13.77 Curvature vector ¶
      - depends_on proposition 13.78 The curvature vector is orthogonal to the tangent ¶
  - depends_on proposition 13.75 Invariance of the length ¶
- depends_on definition 13.118 Line element ¶
  - depends_on definition 13.74 Length of a curve ¶ ↺
  - depends_on proposition 30.12 The Green strain measures length exactly ¶
    - depends_on example 30.13 A rigid rotation forges a strain of $-\theta^{2}/2$ ¶
- 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 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 proposition 13.140 Killing's equation ¶ ↺
  - depends_on proposition 13.154 Symmetries of the curvature ¶ ↺
  - 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 13.79 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 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 13.153 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 proposition 44.29 Linearized field equations ¶
      - depends_on proposition 44.4 Trace-reversed form ¶
      - depends_on theorem 49.9 Tolman–Oppenheimer–Volkoff equations ¶
      - depends_on theorem 48.5 The Friedmann equations ¶
      - depends_on theorem 44.42 The constraint 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 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 definition 13.122 Vielbein ¶
  - depends_on proposition 13.123 Local Lorentz freedom ¶
    - depends_on theorem A.49 thm:app-eh-equivalence ¶ ↺
  - depends_on theorem 43.3 Equivalence of the two variable sets ¶ ↺
  - depends_on theorem 13.156 Cartan structure equations ¶ ↺
- depends_on definition 13.114 Volume form ¶ ↺
- depends_on lemma A.623 The adapted frame of the $3+1$ split ¶
  - depends_on lemma A.626 Decomposition of $\nabla n$, and the acceleration ¶
    - depends_on lemma A.630 Codazzi equation ¶
      - depends_on theorem 44.42 The constraint equations ¶ ↺
    - depends_on lemma A.631 The normal–normal Ricci contraction ¶ ↺
- depends_on proposition 23.44 Relativistic Hamilton–Jacobi equation ¶
  - depends_on remark 23.46 Carter's constant ¶
  - depends_on remark 23.45 Why this is not an analogy ¶
- depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
depends_on proposition 13.100 The space of $k$-forms ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Covector	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4030
`depends_on`	→	Vector on a manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4030
`depends_on`	←	$k$-form	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4792
`depends_on`	←	$k$-vector	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4798
`depends_on`	←	Metric tensor of signature $(p,q)$	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5495
`depends_on`	←	The space of $k$-forms	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4829