Perturbations in General Relativity  

Linearized Einstein Equation > s.a. gauge transformations; hamilton-jacobi theory.
* Idea: The problem of finding a field γab which describes a small departure from some gab (weak field if gab = ηab); 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(λ) is a one-parameter family of solutions, such that gab(0) = gab, define

γab := dgab(λ) / dλ|λ = 0 .

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

∇m∇m γab + 2 Rambn γmn − 2 ∇(a∇|m| (γb)m − \(1\over2\)γ δb)m) = 0 ,  or   ∇m∇m γab − 2 ∇m∇(a γb)m + ∇a∇bγ = 0 .

* Einstein equation with matter: For matter fields Φ (with perturbation φ) we write G (1)ab = 8πG 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.
* Idea: The weak-field approximation; It 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 γ'ab:= γab − \(1\over2\)ηabγ, the linearized equation is −\(1\over2\) ∂c∂c γ'ab + ∂c∂(b γ'a) c − \(1\over2\)ηab ∂c∂d γ'cd = 8πG T (1)ab, which, with the right choice of gauge, ∂b γ'ab = 0, becomes the wave equation,

∂c∂c γ'ab = −16πG T (1)ab .

* Recovering the non-linear theory: The full, covariant version of the theory can be derived by self-coupling from its linear, flat-spacetime version.
@ 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]; Bernabéu et al PRD(10)-a0910 [with cosmological constant]; > s.a. integrable systems.
@ Hamiltonian form: Rosas-Rodríguez JPCS(05)gq; Ghalati ht/07 [constraint analysis]; Green et al EPJC(11)-a0710; Contreras & Leal IJMPD(14)-a1304 [in Ashtekar variables].
@ Special solutions: Tolish & Wald PRD(14)-a1401 [particle on a null geodesic, retarded solution]; > s.a. gravitational waves; phenomenology of gravity.

Around Other Spacetimes > s.a. black-hole perturbations; collapse; cosmological perturbations [and structure formation]; metric matching.
@ de Sitter spacetime: Losic & Unruh PRL(08)-a0804; Montaquila PhD(09)-a1004 [electromagnetic and gravitational waves]; Bini et al GRG(12)-a1103 [and geodesic motion].
@ 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]; Brizuela & Martín-García CQG(09)-a0810; Chaverra et al PRD(13)-a1209 [self-gravitating spherically symmetric configurations]; Brizuela CQG(15)-a1505; > 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 model matched to Oppenheimer-Snyder]; Oota & Yasui IJMPA(10) [generalized Kerr-NUT-de Sitter spacetime]; Pitrou et al CQG(13)-a1302 [homogeneous cosmologies, xPand algorithm]; > s.a. kerr, Reissner-Nordström, Robinson-Trautman, Vaidya Spacetime.
@ Higher dimensions: Petrov CQG(05)gq [conserved currents and Deser-Tekin charges]; Durkee & Reall CQG(11)-a1009; > 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; kantowski-sachs; Lemaitre-Tolman-Bondi; FLRW models; petrov classes; phenomenology of inflation.

Linearization Stability > s.a. numerical general relativity and models; self-dual gravity.
* Idea: Because general relativity is a non-linear theory, solutions to the linearized field equations yield spurious solutions (not tangent to any 1-parameter family of solutions) in the spatially compact case (due to fixed points of the action of the diffeomorphism group).
* Integrability: A solution of the linearized Eistein equation is integrable iff the Taub conserved quantities vanish.
@ General references: Moncrief JMP(75), JMP(76); Arms JMP(77) [Einstein-Maxwell], JMP(79) [Einstein-Yang-Mills theory]; refs in Bao et al CMP(85), p342; Damour & Schmidt JMP(90); Bruna & Girbau JMP(99), JMP(99), JMP(05) [around FLRW spacetimes]; Garecki a1406/APPB [using the canonical superenergy density]; Saraykar a1612, Saraykar & Janardhan AJMCR-a1709 [as a generic property]; Altas & Tekin a1903 [Taub charge as integral constraint, from second-order perturbation theory].
@ In other gravity theories: Altas a1808-PhD.

Gauge Dependence and Invariants > s.a. black-hole perturbations; FLRW 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 δs 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 CQG(10)-a0711, CQG(10)-a0711 [manifestly gauge-invariant]; Nakamura a1103, a1112-proc, IJMPD(12)-a1203, a1209-MG13 [gauge-invariant, general background spacetime].
@ Higher-order perturbations: Bruni et al CQG(97)gq/96, gq/96-proc; 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-proc [framework], PTP(05)gq/04, a0711-proc [second-order, gauge-invariant]; Nakamura CQG(11)-a1011, a1012-proc, a1101 [gauge-invariant]; Nakamura CQG(14)-a1403 [gauge-invariant variables for any order perturbations]; > s.a. cosmological perturbations; minkowski space [stability].

Other References > s.a. cosmology [effects]; einstein's equation [approximations]; linearized quantum gravity; 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-conf [2-parameter]; Speliotopoulos & Chiao PRD(04)gq/03 [and particles]; Petrov MUPB(04)gq [conserved currents]; Suvorov & Lun a1401; > 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].
@ Other formulations and theories: Jezierski GRG(95)gq/94 [metric vs spin-2 formulation]; Baykal & Dereli a1612 [in terms of differential forms]; Deser a1705 [bootstrapping the full covariant theories]; Izaurieta et al EPJC(19)-a1901 [with torsion]; > s.a. conformal gravity; higher-order theories; massive gravity.
@ Related topics: Low CQG(99)gq/98 [speed of perturbations]; Torres del Castillo & Solís-Rodríguez JMP(99) [self-dual perturbations]; Cartin gq/99 [Lanczos potential]; 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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