definition 13.149 Metric compatibility

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6556 · p. 531

ground object -- no derivation owed

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 13.117 Metric tensor of signature $(p,q)$ ¶
- depends_on definition 13.48 Differentiable manifold ¶ ↺
- depends_on definition 13.84 Tensor ¶
  - depends_on definition 13.83 Covector ¶
    - depends_on definition 13.81 Vector on a manifold ¶ ↺
  - depends_on definition 13.81 Vector on a manifold ¶ ↺
depends_on definition 13.146 Parallel transport and autoparallels ¶
- depends_on definition 13.145 Affine connection ¶ ↺

Supports

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 theorem 13.79 Frenet–Serret equations, covariant form ¶
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 ¶
      - tests experiment Dyson, Eddington and Davidson: the eclipse of 1919 ¶
    - depends_on phenomenon 53.1 Measured anomalous advance of Mercury's perihelion ¶
      - tests experiment Le Verrier and the perihelion of Mercury ¶
    - 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 phenomenon 45.24 Compact objects exist above every cold-support limit ¶
    - depends_on theorem 45.23 Oppenheimer–Snyder collapse ¶
      - depends_on phenomenon 45.24 Compact objects exist above every cold-support limit ¶ ↺
- 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 15.63 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 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 phenomenon 51.7 E-mode polarization, and no primordial B mode ¶
    - 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 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 proposition 13.78 The curvature vector is orthogonal to the tangent ¶ ↺
- 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 theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; 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 ¶
      - tests experiment Shapiro: the radar time delay ¶
    - 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 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 phenomenon 45.42 Stationary black holes are two-parameter objects ¶
  - 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

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Affine connection	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6560
`depends_on`	→	Metric tensor of signature $(p,q)$	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:6560
`depends_on`	←	Killing tensor	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6381
`depends_on`	←	The two Killing tensors every metric carries	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6444
`depends_on`	←	Palatini identity	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:313
`depends_on`	←	The cosmological term	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1313
`depends_on`	←	The curvature vector is orthogonal to the tangent	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3719
`depends_on`	←	The invariant of a Killing vector along a geodesic	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6339
`depends_on`	←	Killing's equation	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6306
`depends_on`	←	Symmetries of the curvature	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6743
`depends_on`	←	Contracted Bianchi identity	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:994
`depends_on`	←	Frenet–Serret equations, covariant form	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3768
`depends_on`	←	Levi-Civita connection and contorsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6583