proposition 45.10 Circular orbits and their energy

open in the book · parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:431 · p. 1290

Rests on

depends_on proposition 45.6 Radial equation and effective potential ¶
- depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶
  - depends_on definition 13.149 Metric compatibility ¶
    - depends_on definition 13.145 Affine connection ¶
      - depends_on definition 13.90 Mixed tensor ¶
      - depends_on definition 13.82 Vector field ¶
    - 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 13.146 Parallel transport and autoparallels ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
  - depends_on definition 13.146 Parallel transport and autoparallels ¶ ↺
  - depends_on proposition 13.140 Killing's equation ¶
    - depends_on definition 13.139 Killing vector ¶
      - depends_on definition 13.120 Isometry ¶
      - depends_on equation 13.267 eq:mfd-lie-def ¶
    - depends_on definition 13.149 Metric compatibility ¶ ↺
    - depends_on proposition 13.127 Component formulas ¶
      - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.90 Mixed tensor ¶ ↺
      - depends_on equation 13.267 eq:mfd-lie-def ¶ ↺
      - proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
    - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
      - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
      - depends_on definition 13.149 Metric compatibility ¶ ↺
      - depends_on definition 13.147 Torsion ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
    - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
      - depends_on definition 13.147 Torsion ¶ ↺
      - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
    - proves proof ch:11-manifolds-tensors-curvature@proof-37 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-38 ¶
- depends_on proposition 40.3 Normalization of the four-velocity ¶
  - depends_on definition 38.6 Minkowski metric and interval ¶
    - depends_on proposition 38.5 The invariant interval ¶
      - depends_on equation 38.1 eq:lor-beta-gamma ¶
      - depends_on theorem 38.4 Lorentz boost ¶
      - proves proof ch:02-lorentz-transformations@proof-2 ¶
  - depends_on equation 40.2 eq:rdyn-fourvelocity ¶
  - proves proof ch:04-relativistic-dynamics@proof-1 ¶
- depends_on theorem 45.1 Schwarzschild solution ¶
  - depends_on equation 13.306 eq:mfd-riemann ¶
  - depends_on postulate 43.1 Geodesic motion ¶
  - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
  - proves proof ch:04-schwarzschild-black-holes@proof-1 ¶
- proves proof ch:04-schwarzschild-black-holes@proof-3 ¶
proves proof ch:04-schwarzschild-black-holes@proof-6 ¶

Supports

depends_on phenomenon 45.12 Efficiency of accretion onto compact objects ¶

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`	→	Radial equation and effective potential	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:446
`depends_on`	←	Efficiency of accretion onto compact objects	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:524
`proves`	←	ch:04-schwarzschild-black-holes@proof-6	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:449