Perturbations and Fields on Schwarzschild and Related Metrics  

Perturbations in General > s.a. black-hole perturbations; higher-dimensional black holes; numerical relativity; spherical solutions.
* Types: Perturbations with l = 0 or 1 correspond to charges, those with l = 2 to gravitational waves; The only stationary axial ones have l = 1; In addition to the quasinormal modes, the gravitational spectrum also has a continuum part, i.e., a branch cut in Green's function, on the negative imaginary frequency axis; Even perturbations satisfy the Zerilli equation, odd ones the Regge-Wheeler equation.
* Price's Law: Linear perturbations of a Schwarzschild black hole, depending on initial conditions, fall off as t −2l−3 or t −2l−2 for t → ∞, where l is the angular momentum.
@ Linear: Regge & Wheeler PR(57) [odd parity]; Peters PR(66); Edelstein & Vishveshwara PRD(70); Gerlach & Sengupta PRD(79), PRD(79) [covariant]; Malec & O'Murchadha gq/97 [back-scattering], et al CQG(98)gq/97 [spherical scalar waves]; Barack PRD(99) [scalar, late-time]; Karkowski et al APPB(01)gq/02 [gravitational and electromagnetic]; Vacaru IJMPD(03)gq/02 [black ellipsoids]; Grumiller gq/03-MGX [classical, quantum, semi]; Blue & Soffer JMP(05)gq/03 [odd, spin-2]; Bishop CQG(05)gq/04 [Bondi-Sachs form]; Leung et al CQG(03)gq [continuum, quadrupole]; Fiziev JPCS(07)gq [exact solutions]; Berndtson PhD(07)-a0903 [harmonic gauge]; Donninger et al AiM(11)-a0908 [fall-off rate, Price's law]; Zenginoğlu CQG(10)-a0911 [asymptotics].
@ Gauge-invariant: Fernandes & Lun JMP(96); Jezierski GRG(99)gq/98 [waves]; Sarbach & Tiglio PRD(01)gq [horizon-penetrating]; Clarkson & Barrett CQG(03)gq/02 [covariant]; Martel & Poisson PRD(05)gq [and radiation]; Nagar & Rezzolla CQG(05)gq [rev]; Shah et al a1611.
@ Second-order: Gleiser et al PRL(96), CQG(96)gq/95, PRP(00)gq/98 [gravitational radiation from collision]; Garat & Price PRD(00) [gauge-invariant]; Nicasio et al GRG(00)gq [with odd parity]; Brizuela et al PRD(09)-a0903 [gauge-invariant].
@ Specific physical situations: Hopper & Evans PRD(10)-a1006 [produced by a small mass in an eccentric orbit].
@ Related topics: Cruciani NCB(05)gq/06 [re Zerilli approach]; Fiziev gq/06 [solutions of the Regge-Wheeler equation in interior]; Kol PRD(08)ht/06 [negative mode]; Preston & Poisson PRD(06)gq [light-cone gauge]; Andersson et al a1708 [integrated local energy decay estimate]; Prabhu & Wald CQG(18)-a1807 [canonical energy and Hertz potentials].
@ In modified theories: Yunes & Sopuerta PRD(08)-a0712 [Chern-Simons-modified gravity]; Tattersall et al PRD(18)-a1711 [covariant formulation]; > s.a. black holes in modified theories.

Quasinormal Modes > s.a. black-hole perturbations; black-hole thermodynamics; quantum black holes; schwarzschild-de sitter spacetimes.
* Results: The imaginary parts of the frequencies of the quasinormal modes of the Schwarzschild black hole are equally spaced, with the level spacing dependent only on the surface gravity.
@ General references: Regge & Wheeler PR(57); Liu & Mashhoon CQG(96); Décanini et al PRD(03)gq/02 [complex angular momentum]; Motl ATMP(02)gq [and lqg]; Motl & Neitzke ATMP(03)ht [asymptotic frequencies]; Musiri & Siopsis CQG(03)ht [perturbative]; Padmanabhan CQG(04)gq/03 [level spacing]; Sánchez et al EPJC(11)-a1006 [supersymmetric]; Cho et al CQG(10) [asymptotic iteration method]; Dolan & Ottewill PRD(11)-a1106 [and wave propagation]; Rosa & Dolan PRD(12) [massive vector fields]; Casals & Ottewill PRL(12).
@ Second-order: Kao PRD(07)-a0704; Nakano & Ioka PRD(07)-a0708.
@ Higher-dimensional: Birmingham PLB(03)ht; Cardoso et al PRD(04)gq/03 [D ≥ 4], JHEP(03)ht [D = 5]; Rostworowski APPB(07)gq/06.
@ Dirac fields: Musiri & Siopsis PLB(07)ht/06 [massless]; Chen et al CQG(06) [and Lorentz violation].
@ Arbitrary spin: Shu & Shen PLB(05)gq; Khriplovich & Ruban IJMPD(06)gq/05.

Fields, Waves and Related Effects > s.a. black-hole phenomenology; black-hole hair; horizons; Penrose Inequality; quantum black holes.
* Wave propagation: Governed by the spin-dependent wave equation

\[ {\partial\Psi\over\partial r_*} + (\omega-V)\,\Psi = 0\;,\quad{\rm where}\quad V(r) = \Big({l(l+1)\over r^2} + {(1-j^2)\,2M\over r^3}\Big)\Big(1-{2M\over r}\Big)\;.\]

@ Scalar fields: Zecca NCB(03); Kuchiev & Flambaum PRD(04) [absorption cross-section]; Kronthaler JMP(06)gq [Cauchy problem]; Tsoupros GRG(10) [massless, conformal]; Dolan & Ottewill PRD(11)-a1106 [retarded Green function and quasinormal modes]; Kanai & Nambu CQG(13) [scattering and black-hole imaging]; Li et al a1612 [bound and scattering states]; > s.a. scattering.
@ Massive vector fields: Zecca NCB(05); Rosa & Dolan PRD(12)-a1110 [quasinormal modes and bound states].
@ Electromagnetic waves / Maxwell fields: Zecca NCB(00); Malec PRD(00)gq; Karkowski et al CQG(02)gq/01, APPB(01)gq/02, PRD(03)gq/02 [and gravitational waves, including numerical results]; Čadež & Kostić PRD(05)gq/04 [optics]; Valiente Kroon PRS(07)gq [near spi]; Crispino et al PRD(07) [absorption]; Mason & Nicolas JGP(12) [and Dirac fields, peeling]; Andersson et al CQG(16)-a1501 [decay of solutions]; Nambu & Noda CQG(16)-a1502 [wave optics]; Johnson a1907 [link between spin-1 and spin-2 equations on Schwarzschild spacetime].
@ Other fields: Aguirregabiria & Vishveshwara PLA(96); Sánchez ht/01-proc; Raffaelli JHEP(13)-a1301 [scattering of spin-j fields]; > s.a. electromagnetism in curved spacetime; klein-gordon fields; low-spin field theories [3/2].
@ Stability: Vishveshwara PRD(70); Ishibashi & Kodama PTP(03)ht [higher-dimensional]; Gibbons et al PTP(05)ht/04 [M < 0 stable]; Gleiser & Dotti CQG(06)gq, Cardoso & Cavaglià PRD(06)gq [M < 0 unstable]; Finster & Smoller ATMP(09)gq/06 [electromagnetism and gravity]; Brito et al PRD(13)-a1304 [massive spin-2 fields, and bounds on graviton mass]; Dotti PRL(14)-a1307; Dafermos et al a1601 [gravitational perturbations]; Hung et al a1702; Johnson a1810-PhD.
@ Tails, late-time behavior: Ching et al PRL(95)gq/94, PRD(95)gq; Barack PRD(99)gq/98; Friedman & Morris JMP(00); Koyama & Tomimatsu PRD(01) [massive scalar]; Cardoso et al PRD(03)ht; Karkowski et al CQG(04)gq/03; Price & Burko PRD(04)gq [special case]; Luk AHP(10)-a0906; > s.a. gauge-theory solutions.
@ Other effects: Karkowski et al APPB(03)gq/02 [ringing]; > s.a. chaotic motion; doppler shift.
> Other matter on Schwarzschild backgrounds: see dirac fields; gauge theory solutions; Gravitinos; other fields; particles.

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