proposition 45.31 Horizons and the extremal bound

open in the book · parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1247 · p. 1299

Rests on

depends_on theorem 45.30 Kerr solution; imported ¶
- 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 ¶
    - 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 13.149 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 13.147 Torsion ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
    - proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
  - proves proof ch:04-schwarzschild-black-holes@proof-1 ¶
proves proof ch:04-schwarzschild-black-holes@proof-15 ¶

Supports

depends_on lemma 45.44 Horizon area ¶
- depends_on proposition 45.45 Irreducible mass and the Christodoulou–Smarr mass formula ¶
  - depends_on proposition 45.46 The first law ¶
    - depends_on theorem 45.49 Hawking temperature; imported ¶
      - depends_on proposition 45.51 Evaporation: scale and timescale ¶
depends_on phenomenon 45.35 Astrophysical black holes rotate, and stay inside the Kerr bound ¶
depends_on proposition 45.45 Irreducible mass and the Christodoulou–Smarr mass formula ¶ ↺

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`	→	Kerr solution; imported	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1260
`depends_on`	←	Horizon area	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1681
`depends_on`	←	Astrophysical black holes rotate, and stay inside the Kerr bound	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1390
`depends_on`	←	Irreducible mass and the Christodoulou–Smarr mass formula	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1717
`proves`	←	ch:04-schwarzschild-black-holes@proof-15	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1263