Schwarzschild Solution  

In General > s.a. quantum black holes; spherical solutions; schwarzschild geometry [including line element, curvature].
* Idea: The 1-parameter family of static, spherically symmetric solutions, representing vacuum black holes; 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.
@ 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)mar [derivation no-go]; Santander et al AJP(97)dec [curvature-based]; Visser IJMPD(05)gq/03 [heuristic]; Antoci phy/03-ch, Gsponer phy/04 [history]; Schmidt GRG(05)gq/04 [Synge's contribution]; Bel a0709; Berezin IJMPA(11)-a1006 [local formalism for global properties]; Cuzinatto et al AJP(11)jun-a1009 [Lenz-Sommerfeld argument]; Corda EJTP(11)-a1010 [Schwarzschild's original work]; Crater a1404 [with pointlike source?]; Zhang et al PRD(14)-a1412 [from thermodynamic considerations]; Heinicke & Hehl IJMPD(15)-a1503 [intro]; Grøn AJP(16)jul [centenary].
@ Pedagogical: Deser & Franklin AJP(05)mar-gq/04 [derivation]; Dadhich CS-a1206; Kassner EJP(15)-a1502 [weak-field limit].
@ Characterization: Ferrando & Sáez CQG(98); Fernández-López et al JGP(03).
> Related topics: see Birkhoff's Theorem; numerical black holes; perturbations of the schwarzschild metric.

Specific Concepts > s.a. Irreducible Mass; gravitational lensing; schwarzschild geometry [including singularity and extensions]; yang-mills gauge theory.
@ 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; Helliwell & Konkowski GRG(11)-a1006 [with 2-sphere singularity]; Ellis a1203 [improved interior metric]; Beltracchi & Gondolo PRD(19)-a1902 [exact time-dependent solution]; > s.a. black holes.
@ 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]; Burić & Radovanović 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 JPCS(06)ht/05; Kirilin PRD(07)gq/06; Holstein gq/06 [spin-1, and Kerr]; Dasgupta JCAP(10)-a0910 [and unstable orbits]; Kim & Kim PLB(12) [phase transition]; Culetu a1508; Farag Ali et al EPL-a1510 [from minimal length, and classical tests of gravity]; Bargueño et al EPL(17)-a1605; Khodadi et al PLB(17)-a1702 [from minimal length]; Calmet & El-Menoufi EPJC(17)-a1704; Ashtekar et al PRD(18)-a1806 [quantum extension]; Faraoni & Giusti Symm-a2006 [unsettling physics]; > 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-point energy, scalar and fermion fields]; Doran et al PRD(05) [fermion absorption cross section]; Sharif & Sheikh GRG(07)-a0705 [cold plasma, dispersion relations]; Bergamin et al CQG(09) [vibrating systems].
@ Other: Karlhede et al GRG(82) [horizon and Karlhede invariant]; Abrams CJP(89)gq/01 [??]; Quirós gq/99 [spurious??]; Nikishov gq/00 [energy-momentum pseudotensor]; Krasnov & Solodukhin ATMP(04)ht [stringy description]; Antoci & Liebscher in(06)gq/04 [surface is singular??].
> With matter: see black-hole phenomenology [including appearance to an observer]; particles in schwarzschild spacetime; waves and fields.
> Related topics: see green functions; metric matching; perturbations; Phantom Field; self-force; Weyl-Lanczos Equations.
> Other: see boundaries [extensions?]; cosmic strings; regge calculus; sources of gravitational waves.

Similar Solutions > s.a. black-hole solutions and higher-dimensional black holes [including Schwarzschild-Tangherlini]; gravitational instanton [Euclidean].
* In cosmological backgrounds: Solutions which represent a black hole or local system in an asymptotically FLRW spacetime, or other cosmological background.
* Conformal gravity: Replace 1 − (2GM/r) \(\mapsto\) 1 − (2GM/r) + γrkr2 in line element above (γ, k are integration constants).
@ Schwarzschild-FLRW solutions: Gautreau PRD(84); Grøn & Rippis GRG(03)gq; Gao CQG(04) [arbitrary D]; Senevirathne & de Silva a0812; > s.a. McVittie Metric.
@ In other backgrounds: Nayak et al PRD(01) [Einstein universe].
@ Non-commutative: Nasseri GRG(05)ht, comment Li ht/05; Nicolini et al PLB(06)gq/05 [and evaporation]; Nasseri IJMPD(06); Batic & Nicolini PLB(10)-a1001 [stability of the interior and horizon effects]; Arraut et al JMP(10)-a1001 [maximal extension]; Gangopadhyay JPCS(12)-a1212 [and the Voros product].
@ With other fields: Ren & Lichun NCB(01) [sourrounded by radiation]; in Kastor & Traschen CQG(17)-a1607 [Janis-Newman-Winicour, with massless scalar field]
@ Other theories: Federbush gq/00 [quadratic curvature]; Zet gq/03 [teleparallel gravity with torsion]; López-Domínguez et al PRD(09)-a0910 [supersymmetric]; De Felice et al PRD(11)-a1102 [in f(R,G) gravity, stability]; Cognola et al PRD(11) [covariant renormalizable gravity theory]; Nashed EPL(14)-a1501 [extended teleparallel gravity]; Chamseddine & Mukhanov EPJC(17)-a1612 [limited-curvature theory]; > s.a. branes; dilaton gravity; massive gravity.
@ Other modifications: 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]; de Felice & Sorge CQG(03) [with magnetic field, orbits]; Martínez-Morales gq/05; Al-Badawi & Halilsoy GRG(06) [+ Robinson-Bertotti, and NUT parameter]; Hackmann et al PRD(10)-a0912 [pierced by a cosmic string, geodesics]; Gaete et al PLB(10)-a1005 [un-gravity corrections]; Anastasiei & Gottlieb a1104 [Schwarzschild-like metric]; Belletête & Paranjape IJMPD(13)-a1304-GRF [negative-mass]; Banik & Guendelman a1510 [topological deformation]; Kinyanjui & Wamalwa a1703 [in post-Riemannian Finsler spacetime]; > s.a. holonomy [Schwarzschild-Droste]; modified black-hole solutions; perturbations [including M < 0].
> Other related metrics: see FLRW spacetimes; kerr spacetime; metric matching; reissner-nordström spacetime; schwarzschild-de sitter spacetime [including M < 0].

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