Schwarzschild Solution  

In General > s.a. quantum black holes; spherical solutions; schwarzschild geometry [line element, curvature].
* Idea: The first solution (other than Minkowski space) in general relativity, found in 1916, and one of the most important, equivalent to the Kepler problem in classical mechanics; The 1-parameter family of static, spherically symmetric vacuum solutions, representing black holes.
@ General references: Schwarzschild VAG(00)-CQG(98), SPAW(16)-GRG(03)phy/99, SPAW(16)phy/99; in Hawking & Ellis 73; in Wald 84; Gruber et al AJP(88) [derivation no-go]; Santander et al AJP(97) [curvature-based]; Visser IJMPD(05)gq/03 [heuristic]; Antoci phy/03-in, Gsponer phy/04 [history]; Deser & Franklin AJP(05)gq/04 [derivation]; Schmidt GRG(05)gq/04 [Synge's contribution]; Bel a0709.
@ Characterization: Ferrando & Sáez CQG(98); Fernández-López et al JGP(03).
@ Geometry: Kling & Newman PRD(99) [null cones]; Rama PLB(04) [ito branes and antibranes].
> Other: see Birkhoff's Theorem; perturbations.

Specific Concepts > s.a. Irreducible Mass; lensing; schwarzschild geometry; yang-mills gauge theory.
* Singularities: = 0, , removable (use Cartesian-like coord's); r = 2GM, removable; r = 0, curvature singularity.
@ Singularity: Hellaby JMP(96); Heinzle & Steinbauer JMP(02)gq/01 [distributional]; > s.a. types of singularities.
@ Boosted (Aichelburg-Sexl): Barrabès & Hogan PRD(01)gq; > s.a. solutions of general relativity.
@ Interior: Som et al PLA(01) [matter]; Füzfa et al GRG(02) [Lemaître-Schwarzschild]; Doran et al FP(08)gq/06; > s.a. black holes.
@ Extensions: Frønsdal PR(58); Rosen AP(71); Klösch & Strobl CQG(96); Mitra ap/99 [??]; Abbassi PS(01)gq/99.
@ Canonical approach: Varadarajan PRD(01)gq/00 [Kruskal variables].
@ Quantum corrections: Becker & Becker PRD(99) [11D matrix theory]; Vaz & Witten PRD(99)gq/98 [canonical]; Buric & Radovanovic CQG(00); Cavaglià & Ungarelli PRD(00)ht/99 [mass]; Cavaglià & Fabbri PRD(02)ht/01 [AdS2 and n-Schwarzschild]; Bjerrum-Bohr et al PRD(03)ht/02 [and Kerr]; Khriplovich & Kirilin JETP(04); Fabbri et al ht/05-in; Kirilin gq/06; Holstein gq/06 [spin-1, and Kerr]; > s.a. black hole entropy, covariant quantum gravity, semiclassical general relativity.
@ With matter: Hartle in(72) [long-range neutrino forces]; Goncharov PLB(99)gq [twisted spinors]; Molnár & Elsässer PRD(03)gq [uniqueness of electrostatic solution]; Bormann & Antonsen IJTP(04) [0-pt energy, scalar and fermion fields]; Doran et al PRD(05) [fermion absorption cross section]; Sharif & Sheikh GRG(07)-a0705 [cold plasma, dispersion relations].
@ Other: Abrams CJP(89)gq/01 [??]; Quirós gq/99 [spurious??]; Nikishov gq/00 [energy-momentum pseudotensor]; Krasnov & Solodukhin ATMP(04)ht [stringy description]; Antoci & Liebscher gq/04-in [surface is singular??].
> With matter: see black hole phenomenology; classical particles; spinning particles; test particles; waves and fields.
> Related topics: see geodesics; green functions; metric matching; perturbations; Phantom Field; self-force; Weyl-Lanczos Equations.
> Other: see boundaries [extensions?]; cosmic strings; regge calculus; sources of gravitational waves.

In Cosmological Backgrounds > s.a. FRW spacetimes; reissner-nordström; schwarzschild-de sitter.
* Idea: Solutions which represent a black hole or local system in an asymptotically FRW spacetime, or other cosmological background.
* McVittie solution: The k = 0 line element in Hogan's form is

where M "mass at the singularity," and (t) (asymptotic) expansion rate.
* Relationships: McVittie becomes FRW for r → , Schwarzschild for d/dt = 0, Schw-dS for d2/dt2 = 0.
@ McVittie solution: McVittie MNRAS(33); Newman & McVittie GRG(82) [many particles]; Hogan ApJ(90); Nolan PRD(98)gq, CQG(99), CQG(99)gq; Patel et al G&C(00) [higher-dimensional]; Bolen et al CQG(01)gq/00 [precession].
@ Schwarzschild-FRW solutions: Gautreau PRD(84); Grøn & Rippis GRG(03)gq; Gao CQG(04) [arbitrary D].
@ In other backgrounds: Nayak et al PRD(01) [Einstein universe].

Other Similar Solutions > s.a. black hole solutions; branes; dilaton gravity; gravitational instanton [Euclidean]; kerr.
* Conformal gravity: Replace 1 – (2GM/r) 1 – (2GM/r) + r – kr2 in line element above (, k integration constants).
@ Non-commutative: Nasseri GRG(05)ht, comment Li ht/05; Nicolini et al PLB(06)gq/05 [and evaporation]; Nasseri IJMPD(06).
@ Other theories: Federbush gq/00 [quadratic curvature]; Zet gq/03 [teleparallel gravity with torsion].
@ Other: Abrams PRD(79)gq/02 [Droste-Weyl-Schwarzschild]; Dadhich et al Pra(98)gq/97 [with global monopole charge]; Aichelburg & Balasin CQG(98), CQG(00) [ultrarelativistic]; Gen & Shiromizu JMP(99) [asymptotically Schwarzschild]; Ren & Lichun NCB(01) [sourrounded by radiation]; de Felice & Sorge CQG(03) [with magnetic field, orbits]; Martínez-Morales gq/05; Al-Badawi & Halilsoy GRG(06) [+ Robinson-Bertotti, and NUT parameter]; > s.a. holonomy [Schwarzschild-Droste]; modified black hole solutions; perturbations [including negative M].

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