Schwarzschild-de Sitter Spacetime  

In General > s.a. solutions of einstein's equation / McVittie Metric.
* Idea: The natural generalization of the Schwarzschild metric to the case of non-zero cosmological constant Λ.
* Line element: It can be written in static form by replacing 1 − (2 GM/r) \(\mapsto\) 1 − (2 GM/r) − (Λ r2/3) in the Schwarzschild line element (> see schwarzschild geometry), and one obtains the Kottler metric; It can also be obtained from the McVittie metric, setting the expansion rate to be constant.
* Extreme case: The parameters satisfy 9Λ(GM)2 = 1 (in 4 spacetime dimensions).

Thermodynamics > s.a. black-hole thermodynamics; black-hole radiation.
* Temperature: In d dimensions, the temperature is given by

T = (d−3) (rH ± γ2rH) / 4π ,

where the upper (lower) sign holds in the Anti-de Sitter (de Sitter) case.
@ General references: Brown et al PRD(94); Teitelboim ht/02-proc [and stability]; Bolen & Cavaglià GRG(05)gq/04 [and gup]; Urano et al CQG(09)-a0903 [first law].
@ Entropy: Li NCB(01); Ghezelbash & Mann JHEP(02)ht/01; Shankaranarayanan PRD(03)gq; Corichi & Gomberoff PRD(04)ht/03 [bounds]; Bolen & Cavaglià GRG(05)gq/04 [and gup]; Sekiwa PRD(06) [thermal cosmological constant].
@ Radiation: Gibbons & Hawking PRD(77); Bousso & Hawking PRD(98); Hemming & Keski-Vakkuri PRD(01)gq/00; Jiang CQG(07)-a0705; Arraut et al CQG(09)-a0810 [two approaches]; Drouot a1510 [for massive bosons]; Pappas & Kanti PLB(17)-a1707; Robson et al a1902 [temperature, topological approach].
@ Quantum: Nojiri & Odintsov PRD(99); Nojiri & Odintsov IJMPA(00)ht/99.

Perturbations > s.a. perturbations of schwarzschild spacetime.
@ General references: Yoshino et al PRD(04)gq [Λ < 0].
@ Quasinormal modes, Schwarzschild-de Sitter: Cardoso & Lemos PRD(03)gq [near-extremal]; Suneeta PRD(03)gq; Maassen van den Brink PRD(03)gq; Zhidenko CQG(04)gq/03; Castello-Branco & Abdalla gq/03; Roy Choudhuri & Padmanabhan PRD(04)gq/03, comment Batic et al PRD(11)-a1105 [level spacing]; Konoplya & Zhidenko JHEP(04) [high overtones]; Medved & Martin GRG(05) [treatments].
@ Quasinormal modes, Schwarzschild-AdS: Horowitz & Hubeny PRD(00); Cardoso & Lemos PRD(01)gq [electromagnetism and gravity]; Konoplya PRD(02) [small black hole]; Moss & Norman CQG(02)gq [dS/AdS]; Musiri & Siopsis CQG(03)ht, PLB(03), PLB(03)ht; Cardoso et al PRD(03)gq, JMP(04); Jing gq/05, & Pan PRD(05) [Dirac fields]; Musiri et al PRD(06); Daghigh JHEP(09)-a0901.

Similar Metrics > s.a. lovelock gravity.
* Nariai solutions: Degenerate or extreme versions of the Schwarzschild-de Sitter metric; They are conformally non-flat, singularity-free perfect fluid expanding cosmological models, satisfying the weak energy condition.
@ Nariai solutions: Dadhich gq/01; Ortaggio PRD(02)gq/01 [impulsive waves]; Beyer CQG(09)-a0902 [generalized, positive cosmological constant]; Beyer a1012-MG12 [perturbations and the cosmic no-hair conjecture]; Fernando MPLA(13)-a1408, IJMPD(14) [with quintessence].
@ With negative mass: Belletête & Paranjape IJMPD(13)-a1304-GRF.
@ Other modified metrics: Wang et al CQG(09) [quantum deformed]; Fennen & Giulini CQG(15)-a1408 [static two-mass solution using Nariai spacetime].

References > s.a. tests of general relativity with light.
@ General: Kottler AdP(18); Socolovsky a1711 [rev]; Bugden & Paganini a1810 [the \(\Lambda\to0\) limit].
@ Geometry, coordinates: Kamimura et al MPLA(91) [connection formulation]; Lake CQG(06)gq/05 [maximal extension]; Siddiqui GRG(11)-a1009 [foliation by flat spacelike hypersurfaces]; Brendle IHES-a1105 [hypersurfaces of constant mean curvature]; Fernandes et al a1910 [extremal surfaces]; Manjunatha et al IJGMP(20)-a2006 [curvature invariants].
@ Test particles, geodesics: Stuchlík & Hledík PRD(99); Kraniotis & Whitehouse CQG(03) [and precession]; Béssa & Lima IJMPD(04)gq [turning points]; Carvalho et al MPLA(04), Cruz et al CQG(05)gq/04 [Schwarzschild-AdS]; Sultana & Dyer GRG(05); Hackmann & Lämmerzahl PRL(08)-a1505, PRD(08)-a1505 [complete analytic solution]; Dymnikova et al G&C(08) [overview]; Klein & Collas PRD(10) [recessional velocities and Hubble's law]; Arakida IJTP(13)-a1212, comment Ovcherenko & Silagadze UJP(16)-a1511 [and periastron precession]; > s.a. spinning particles; tests of general relativity with orbits.
@ Other classical matter: Islam PLA(83) [planet and light motion]; Bijalwan a1108 [interior].
@ Fields, scalar: Brady et al PRD(99)gq [falloff]; Brevik & Simonsen GRG(01) [numerical].
@ Fields, other: Brady et al PRD(97) [tails]; Molina et al PRD(04)gq/03 [scalar, electromagnetic, and gravitation]; Lyu & Gui IJTP(04), NCB(04), PS(07) [Dirac fields]; Mackay et al EPL(05)ap [vp < 0 electromagnetic waves]; Keller a1706 [electromagnetic, decay of solutions].
@ Thermal properties: Lin ht/98-conf; Ghezelbash & Mann JHEP(02)ht/01 [action, entropy, and dS-cft].
@ In other theories: Kodama & Arraut PTEP(14)-a1312 [in dRGT massive gravity theory, stability]; Addazi & Capozziello MPLA(16)-a1602 [in f(R) gravity]
@ Related topics: Podolský GRG(99)gq [extreme]; Nayak et al PRD(01)gq/00 [in Einstein universe]; Lake PRD(02)gq/01 [lensing]; Bytsenko & Goncharov IJMPA(02)gq [monopoles and Hawking radiation]; Zhang et al ChPL(12)-a0911 [tunneling to de Sitter space].

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at – modified 11 jan 2021