Perturbations in General Relativity  

Linearized Einstein Equation > s.a. gauge transformations.
* Idea: The problem of finding a field ab which describes a small departure from some gab (weak field if gab = ab); Time-dependent perturbations can describe propagating gravitational waves and/or matter that are assumed not to affect the background spacetime.
* Linearized metric perturbation: If gab() is a one-parameter family of solutions, such that gab(0) = gab, define

ab := dgab()/d|_{ = 0} .

* Vacuum Einstein equation: The equation "linearized Rab = 0"; If := aa and Rabcd refers to the unperturbed metric gab,

mm ab + 2 Rambnmn – 2 (a|m| (b)m –  b)m) = 0 ,  or   mmab – 2 m(ab)m + ab = 0 .

* Einstein equation with matter: For matter fields (with perturbation ) we write G (1)ab = 8G T (1)ab, where the superscript (1) denotes a first-order perturbation; > s.a. metric perturbation.

Around Minkowski Space > s.a. duality; stress-energy pseudotensor; gauge transformations; gravitational waves; phenomenology of gravity.
* Idea: The weak-field approximation; Results in a wave equation that describes the propagation of gravitational waves, or the spin-2 graviton field, with matter stress-energy-momentum as source.
* Wave equation: In terms of 'ab:= ab – ab, the linearized equation is – cc'ab+ c(b'a) c – abcd 'cd = 8G T (1)ab, which, with the right choice of gauge, b 'ab = 0, becomes the wave equation,

cc'ab = –16G T (1)ab .

@ General references: Weyl AJM(44); in Wentzel 49 [vacuum, graviton]; Geroch notes on general relativity [short and clear]; in Wald 84; Ichinose & Kaminaga PRD(89) [ambiguity]; Jezierski CQG(02)gq/01; Calabrese et al CMP(03)gq/02 [boundary conditions]; Bishop CQG(05)gq/04 [Bondi-Sachs form]; > s.a. integrable systems.
@ Hamiltonian form: Rosas-Rodríguez JPCS(05)gq; Ghalati ht/07 [constraint analysis]; Green et al a0710.

Around Other Spacetimes > s.a. black-hole perturbations; collapse; cosmological [matter perturbations for structure formation].
@ Spherical: Moncrief AP(74), AP(74); Gerlach & Sengupta PRD(79), PRD(79); Karlovini CQG(02)gq/01 [axial]; Nolan PRD(04)gq [gauge-invariant, interpretation]; Brizuela et al PRD(06)gq, PRD(07)gq [second- and higher-order]; Clarkson PRD(07)-a0708 [covariant]; Losic & Unruh PRL(08)-a0804 [de Sitter]; Brizuela & Martín-García CQG(09)-a0810; > s.a. schwarzschild spacetime.
@ Other spacetimes: Gasperini & Giovannini CQG(97)gq/96 [anisotropic]; Konoplya PLA(00)gq/99 [any symmetry]; Sarbach et al PRD(01)gq/00 [static, in terms of Kab]; Dittrich & Tambornino CQG(07)gq [any symmetry-reduced]; Mars et al PRD(08)-a0806 [Einstein-Straus swiss cheese matched to Oppenheimer-Snyder]; > s.a. kerr, Reissner-Nordström, Robinson-Trautman, Vaidya Spacetime.
@ Higher dimensions: Petrov CQG(05)gq [conserved currents and Deser-Tekin charges]; > s.a. kaluza-klein models.
@ Tails, Huygens principle: Waylen PRS(71) [relation to integral form of general relativity]; Wünsch GRG(90) [and Cauchy problem].
> Other: see bianchi I and other bianchi models; gravitational waves; Lemaitre-Tolman-Bondi; FRW models; metric matching; petrov classes; phenomenology of inflation.

Linearization Stability > s.a. numerical general relativity and models; self-dual gravity.
* Idea: The above yields spurious solutions (linearized fields not tangent to any 1-parameter family of solutions) in the spatially compact case (due to fixed points of the diffeomorphism group).
* Integrability: A solution of the linearized Eistein equation is integrable iff the Taub conserved quantities vanish.
@ References: Moncrief JMP(75), JMP(76); Arms JMP(77) [Einstein-Maxwell], JMP(79) [Einstein-Yang-Mills theory]; refs in Bao et al CMP(85), 342; Damour & Schmidt JMP(90); Bruna & Girbau JMP(99), JMP(99), JMP(05) [around FRW spacetimes].

Gauge Dependence and Invariants > s.a. black-hole perturbations; FRW models; gauge transformations; Taub Numbers.
* Results: A perturbed quantity is gauge invariant only if the corresponding unperturbed quantity is zero, a constant scalar field, or a linear combination of products of s with constant coefficients.
@ Curvature-based: Anderson et al PRD(98)gq; Brodbeck et al PRL(00)gq/99 [with matter].
@ General references: Novello et al PRD(95), PRD(95) [minimal set of observables]; Anderson PRD(97)gq/96 [gravitational waves]; Malik & Wands gq/98; Bel gq/06 [special gauge transformations and superposition of solutions]; Giesel et al a0711 [manifestly gauge-invariant].
@ Higher-order perturbations: Bruni et al CQG(97)gq/96, gq/96-in; Sonego & Bruni CMP(98)gq/97 [gauge dependence]; Bruni & Sonego CQG(99)gq [observables]; Bruni et al CQG(03)gq/02, Nakamura PTP(03)gq [2-parameter]; Clarkson PRD(04)ap/03 [covariant]; Nakamura gq/04-in [framework], PTP(05)gq/04, a0711-in [second-order, gauge-invariant]; > s.a. cosmological perturbations, minkowski space [stability].

Other References > s.a. approaches to quantum gravity; cosmology [effects]; einstein's equation; scalar-tensor theories.
@ General: Sachs in(64); in Misner et al 73, 18.1; Stewart & Walker PRS(74); Beig JPA(76); Gowdy JMP(78); in Wald 84; Geroch & Lindblom JMP(85); Gunnarsen CQG(89); Bekaert et al PRD(03)ht/02 [dual formulation]; Sopuerta et al gq/02-in [2-parameter]; Speliotopoulos & Chiao PRD(04)gq/03 [and particles]; Petrov MUPB(04)gq [conserved currents]; > s.a. tensor decompositions.
@ Obtaining solutions: Wald PRL(78); Torres del Castillo GRG(90).
@ Instabilities and constraints on perturbations: Traschen PRD(85); Tod GRG(88); Kastor & Traschen PRD(92); Deruelle et al CQG(97).
@ Discretization: Gambini & Pullin; Di Bartolo et al JMP(05)gq/04 [consistent].
@ Characteristic problem: Frittelli PRD(05)gq/04 [first-order reduction].
@ Related topics: Jezierski GRG(95)gq/94 [metric vs spin-2 formulation]; Low CQG(99)gq/98 [speed of perturbations]; Torres del Castillo & Solís-Rodríguez JMP(99) [self-dual]; Cartin gq/99 [Lanczos potential]; Speliotopoulos & Chiao PRD(04) [coupling to particles]; Nieto MPLA(05) [linearized general relativity as gauge theory]; Brizuela et al GRG(09)-a0807 [xPert computer algebra package]; Anastopoulos PRD(09)-a0902 [backreaction].

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