theorem 45.23 Oppenheimer–Snyder collapse

open in the book · parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1027 · p. 1296

Rests on

depends_on proposition 45.21 Radial plunge: the cycloid ¶
- 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 13.117 Metric tensor of signature $(p,q)$ ¶
      - depends_on definition 13.146 Parallel transport and autoparallels ¶
    - 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 13.149 Metric compatibility ¶ ↺
      - depends_on proposition 13.127 Component formulas ¶
      - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
      - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      - 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 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-12 ¶
depends_on theorem 45.4 Jebsen–Birkhoff ¶
- depends_on equation 13.306 eq:mfd-riemann ¶ ↺
- depends_on theorem 45.1 Schwarzschild solution ¶ ↺
- proves proof ch:04-schwarzschild-black-holes@proof-2 ¶
proves proof ch:04-schwarzschild-black-holes@prooflink-1 ¶

Supports

depends_on phenomenon 45.24 Compact objects exist above every cold-support limit ¶

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 plunge: the cycloid	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1056
`depends_on`	→	Jebsen–Birkhoff	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1056
`depends_on`	←	Compact objects exist above every cold-support limit	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1078
`proves`	←	ch:04-schwarzschild-black-holes@prooflink-1	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1059