definition 13.139 Killing vector

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6289 · p. 528

ground object -- no derivation owed

Rests on

depends_on definition 13.120 Isometry ¶
- depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
  - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.47 Differentiable structure ¶
      - depends_on definition 13.44 Atlas ¶
  - 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 equation 13.267 eq:mfd-lie-def ¶

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 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 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 ¶
      - depends_on phenomenon 53.1 Measured anomalous advance of Mercury's perihelion ¶
      - 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 theorem 45.23 Oppenheimer–Snyder collapse ¶
  - 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 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 ¶

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`	→	Isometry	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6293
`depends_on`	→	eq:mfd-lie-def	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6293
`depends_on`	←	Killing tensor	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6381
`depends_on`	←	The Killing fields of a flat pseudo-Euclidean space	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2411
`depends_on`	←	Integration of the conformal Killing equation	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:7279
`depends_on`	←	Killing's equation	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6306