Perturbations and Fields on Schwarzschild and Related Metrics| 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
@ 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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