definition 13.117 Metric tensor of signature $(p,q)$

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5485 · p. 519

ground object -- no derivation owed

Rests on

depends_on definition 13.48 Differentiable manifold ¶
- depends_on definition 13.47 Differentiable structure ¶
  - depends_on definition 13.44 Atlas ¶
    - depends_on definition 13.43 Coordinate system ¶
depends_on definition 13.84 Tensor ¶
- depends_on definition 13.83 Covector ¶
  - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.71 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 13.81 Vector on a manifold ¶ ↺

Supports

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 48.13 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 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 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 13.163 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 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 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 13.121 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 theorem 13.79 Frenet–Serret equations, covariant form ¶
- 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 44.31 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 proposition 13.140 Killing's equation ¶ ↺
- 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 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 ¶
      - 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 13.153 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 phenomenon 49.18 Orbital decay of a binary pulsar ¶ ↺
      - tests experiment GW150914 ¶
      - tests experiment The Hulse–Taylor Binary Pulsar (PSR B1913+16) ¶
    - depends_on proposition 44.29 Linearized field equations ¶
    - depends_on proposition 44.4 Trace-reversed form ¶
      - depends_on proposition 44.39 SEC is the attraction condition ¶
      - depends_on theorem 44.27 Newtonian correspondence of the field equations ¶
    - depends_on theorem 49.9 Tolman–Oppenheimer–Volkoff equations ¶
      - depends_on example 49.11 The uniform-density star ¶
      - depends_on phenomenon 49.19 Neutron stars reach two solar masses ¶
    - depends_on theorem 48.5 The Friedmann equations ¶
      - depends_on phenomenon 48.17 The expansion is accelerating ¶ ↺
      - depends_on phenomenon 48.10 A primordial floor of light elements ¶
      - depends_on phenomenon 51.6 Acoustic peaks, and a spatially flat universe ¶ ↺
      - depends_on theorem 48.14 Linear growth of density perturbations ¶
    - 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 44.23 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 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 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 definition 13.114 Volume form ¶
- depends_on definition 13.116 Hodge dual ¶
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 ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Differentiable manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5495
`depends_on`	→	Tensor	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5495
`depends_on`	←	Orthogonal Hamiltonian	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:472
`depends_on`	←	Induced metric	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5533
`depends_on`	←	Isometry	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5545
`depends_on`	←	Length of a curve	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3644
`depends_on`	←	Line element	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5509
`depends_on`	←	Metric compatibility	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6560
`depends_on`	←	Contractions	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6727
`depends_on`	←	Vielbein	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5604
`depends_on`	←	Volume form	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5393
`depends_on`	←	The adapted frame of the $3+1$ split	declared	appendices/A-long-proofs.tex:30116
`depends_on`	←	Relativistic Hamilton–Jacobi equation	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1469
`depends_on`	←	Levi-Civita connection and contorsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6583