theorem 45.1 Schwarzschild solution

open in the book · parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:94 · p. 1286

Rests on

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 13.88 General coordinate transformation ¶ ↺
  - depends_on definition 13.82 Vector field ¶
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
- 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 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
    - 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 ¶

Supports

depends_on phenomenon 50.4 A ring of light at event-horizon scale ¶
depends_on phenomenon 50.5 Black-hole masses from gravitational waves ¶
depends_on phenomenon 45.52 The shadow ¶
- depends_on phenomenon 50.4 A ring of light at event-horizon scale ¶ ↺
depends_on proposition 45.14 Isotropic coordinates ¶
- depends_on phenomenon 53.4 Excess delay of a radar echo passing the Sun ¶
  - tests experiment Shapiro: the radar time delay ¶
    - cites (method) derived bib 6956 A test of general relativity using radio links with the Cassini spacecraft ¶
    - cites (evidence) derived bib 6956 A test of general relativity using radio links with the Cassini spacecraft ¶ ↺
    - cites (evidence) derived bib 6956 A test of general relativity using radio links with the Cassini spacecraft ¶ ↺
    - cites (evidence) derived bib 7319 A two-solar-mass neutron star measured using Shapiro delay ¶
    - cites (method) derived bib Viking relativity experiment - Verification of signal retardation by solar gravity ¶
    - cites (evidence) derived bib Viking relativity experiment - Verification of signal retardation by solar gravity ¶ ↺
    - cites (history) derived bib 26 Fourth Test of General Relativity ¶
    - cites (history) derived bib 26 Fourth Test of General Relativity ¶ ↺
    - cites (method) derived bib 22 Fourth Test of General Relativity: Preliminary Results ¶
    - cites (evidence) derived bib 22 Fourth Test of General Relativity: Preliminary Results ¶ ↺
- depends_on phenomenon 42.8 Shapiro time delay ¶
depends_on proposition 45.43 Static acceleration and surface gravity ¶
- 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 theorem 45.49 Hawking temperature; imported ¶ ↺
depends_on proposition 45.16 Curvature at the horizon and at the centre ¶
depends_on proposition 45.19 Kruskal–Szekeres coordinates ¶
depends_on proposition 45.18 The one-way membrane ¶
- depends_on phenomenon 50.6 No material surface ¶
- depends_on theorem 45.47 Area theorem; imported ¶
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 ¶
    - tests experiment Dyson, Eddington and Davidson: the eclipse of 1919 ¶
      - cites (method) derived bib 571-581 A determination of the deflection of light by the sun's gravitational field, from observations made at the total eclipse of May 29, 1919 ¶
      - cites (evidence) derived bib 571-581 A determination of the deflection of light by the sun's gravitational field, from observations made at the total eclipse of May 29, 1919 ¶ ↺
      - cites (evidence) derived bib 571-581 A determination of the deflection of light by the sun's gravitational field, from observations made at the total eclipse of May 29, 1919 ¶ ↺
      - cites (history) derived bib 10 Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes ¶
      - cites (history) derived bib Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie ¶
      - cites (evidence) derived bib 2 PROGRESS IN MEASUREMENTS OF THE GRAVITATIONAL BENDING OF RADIO WAVES USING THE VLBA ¶
      - cites (evidence) derived bib Gravitational deflection of light: a re-examination of the observations of the solar eclipse of 1919 ¶
      - cites (evidence) derived bib 8 Measurement of the Solar Gravitational Deflection of Radio Waves Using Very-Long-Baseline Interferometry ¶
      - cites (evidence) derived bib 0957 + 561 A, B: twin quasistellar objects or gravitational lens? ¶
      - cites (context) derived bib 1 The Confrontation between General Relativity and Experiment ¶
      - cites (context) derived bib 1 The Confrontation between General Relativity and Experiment ¶ ↺
  - depends_on phenomenon 53.1 Measured anomalous advance of Mercury's perihelion ¶
    - tests experiment Le Verrier and the perihelion of Mercury ¶
      - cites (evidence) derived bib Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole ¶
      - cites (evidence) derived bib Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole ¶ ↺
      - cites (history) derived bib Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie ¶ ↺
      - cites (history) derived bib Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie ¶ ↺
      - cites (data) derived bib Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète ¶
      - cites (evidence) derived bib Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète ¶ ↺
      - cites (evidence) derived bib The Elements of the Four Inner Planets and the Fundamental Constants of Astronomy ¶
      - cites (data) derived bib 1 The Confrontation between General Relativity and Experiment ¶ ↺
  - 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 phenomenon 45.24 Compact objects exist above every cold-support limit ¶
  - depends_on theorem 45.23 Oppenheimer–Snyder collapse ¶
    - depends_on phenomenon 45.24 Compact objects exist above every cold-support limit ¶ ↺
depends_on proposition 45.37 Reissner–Nordström solution ¶
- depends_on theorem 45.39 Israel; imported ¶
  - depends_on phenomenon 45.42 Stationary black holes are two-parameter objects ¶
depends_on theorem 45.4 Jebsen–Birkhoff ¶
- depends_on corollary 45.5 No monopole gravitational radiation ¶
- depends_on theorem 45.23 Oppenheimer–Snyder collapse ¶ ↺
depends_on theorem 45.39 Israel; imported ¶ ↺
… 1 more

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Edges

type	direction	node	provenance	where
`depends_on`	→	eq:mfd-riemann	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:108
`depends_on`	→	Geodesic motion	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:108
`depends_on`	→	Levi-Civita connection and contorsion	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:108
`depends_on`	←	A ring of light at event-horizon scale	declared	parts/05-general-relativity-cosmology/09-exp-black-holes.tex:719
`depends_on`	←	Black-hole masses from gravitational waves	declared	parts/05-general-relativity-cosmology/09-exp-black-holes.tex:808
`depends_on`	←	The shadow	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:2057
`depends_on`	←	Isotropic coordinates	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:627
`depends_on`	←	Static acceleration and surface gravity	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1643
`depends_on`	←	Curvature at the horizon and at the centre	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:707
`depends_on`	←	Kruskal–Szekeres coordinates	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:841
`depends_on`	←	The one-way membrane	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:775
`depends_on`	←	Radial equation and effective potential	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:296
`depends_on`	←	Reissner–Nordström solution	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1453
`depends_on`	←	Jebsen–Birkhoff	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:212
`depends_on`	←	Israel; imported	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1523
`depends_on`	←	Kerr solution; imported	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1239
`proves`	←	ch:04-schwarzschild-black-holes@proof-1	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:112