proposition 45.18 The one-way membrane

open in the book · parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:768 · p. 1294

Rests on

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 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.88 General coordinate transformation ¶
    - depends_on definition 13.82 Vector field ¶
      - depends_on definition 13.81 Vector on a manifold ¶
  - 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 13.84 Tensor ¶
      - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
  - 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 13.145 Affine connection ¶ ↺
  - 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-10 ¶

Supports

depends_on phenomenon 50.6 No material surface ¶
depends_on theorem 45.47 Area theorem; imported ¶

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`	→	Schwarzschild solution	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:775
`depends_on`	←	No material surface	declared	parts/05-general-relativity-cosmology/09-exp-black-holes.tex:915
`depends_on`	←	Area theorem; imported	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1817
`proves`	←	ch:04-schwarzschild-black-holes@proof-10	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:778