3-Dimensional General Relativity  

In General > s.a. 3D gravity theories, manifolds and quantum gravity; general relativity; positive-energy theorem.
* Idea: When Λ = 0, the Einstein equation implies that spacetime is flat outside the matter sources.
* Action: There are different, classically equivalent ones, including a BF one, except for the fact that some (like the first-order one) admit degenerate metrics; In 8πG = 1 units, and with F the curvature of the spin connection ω,

SEH = \(1\over2\) dx2 |g|1/2 (R – 2Λ) ,   S1st = (eIFJK – \(1\over6\)Λ eIeJeK) εIJK .

* Dynamics: For any value of Λ, the set of solutions (moduli space) is finite-dimensional; If Λ = 0, the field equations imply F = 0, flat space (Rab = 0 implies Rabcd = 0), and the moduli space of flat connections on the spatial manifold M has dimension 12 (g–1), with g the genus of M.
* Chern-Simons form: The phase space is the moduli space of flat G-connections; G is a typically non-compact Lie group which depends on the signature of spacetime and the cosmological constant; For Euclidean signature with Λ = 0, G is the 3D Euclidean group; For Lorentzian signature with Λ > 0, G = SL(2,\(\mathbb C\)); Can be interpreted in terms of Cartan geometry.
@ Chern-Simons form: Bimonte et al IJMPA(98) [deformed Chern-Simons]; Matschull CQG(99); Meusburger & Schroers CQG(05)gq [boundary conditions and symplectic structure]; Park JHEP(08).

Solutions and Special Features > s.a. asymptotic flatness; 3D black holes; FLRW spacetimes; gauge choice.
@ Solutions: Duncan & Ihrig GRG(76) [vacuum, static, rotationally symmetric]; Hirschmann & Welch PRD(96) [magnetic]; Williams GRG(98) [rotating kinks]; Brill CQG(04)gq/03-fs [cosmology, lattice universes]; Wang & Wu GRG(07)gq/05 [massless scalar, self-similar, kink instability]; Barrow et al CQG(06)gq [cosmology]; Brill et al Pra(07)-a0707 [colliding particles with Λ < 0].
@ Related topics: Hortaçsu et al GRG(03) [vacuum and + scalar, thermodynamics].
> Related topics: see Central Charge; boundaries in field theory; lattice field theory; singularities; time; topological defects.

With Matter and / or Cosmological Constant > s.a. gödel spacetime.
* Remark: 2+1 gravity coupled to point particles is a non-trivial example of DSR.
* Metric: When Λ = 0, space is a flat 2D manifold with genus g and n punctures, representing point particles; The metric around each puncture (of mass m ∈ (0,2π) and spin s ∈ \(\mathbb R\)) can be written

ds2 = –(dt + s dφ)2 + (1–m/2π)–1 dr2 + r2 dφ2 .

* Duality: Lorentzian theory with Λ > 0 is dual to the Euclidean theory with a negative cosmological constant.
@ Point particles: Carlip NPB(89); de Sousa NPB(90); Lancaster PRD(90); Kabat & Ortiz PRD(94)ht/93 [braid invariance]; Menotti & Seminara AP(00)ht/99, NPPS(00) [ADM]; Cantini et al CQG(01)ht/00 [Hamiltonian]; Valtancoli IJMPA(00) [N particles + Λ < 0]; Krasnov CQG(01)ht/00; Matschull CQG(01)gq [phase space]; Cantini & Menotti CQG(03)ht/02 [functional approach]; Freidel et al PRD(04)ht/03 [and DSR]; Yale et al CQG(10)-a1010 [two-particle system]; Ciafaloni & Munier CQG(11)-a1012 [3-body problem]; Ziprick CJP-a1209-proc, CQG(15)-a1409 [semiclassical loop-gravity formulation]; Kowalski-Glikman & Trześniewski PLB(14) [deformed particle].
@ Einstein-Maxwell: Nayak GRG(91); Cataldo & Salgado PRD(96) [static]; Grammenos MPLA(05)gq/04 [magnetic solution, AdS background]; Bañados et al PRD(06)ht/05 [with cosmological constant and Chern-Simons term, Gödel-type black holes]; Gurtug et al AHEP(15)-a1312 [new solution without cosmological constant].
@ With scalar field: Henneaux et al PRD(02)ht [black holes]; Gegenberg et al PRD(03) [scalar field, action]; de Berredo-Peixoto CQG(03) [static]; Daghan et al GRG(05) [+ cosmological constant, static].
@ Other matter: Carlip & Gegenberg PRD(91) [topological matter]; Peldán NPB(93)gq/92 [+ Yang-Mills]; García & Campuzano PRD(03)gq/02 [fluid, static circularly symmetric]; Campoleoni et al JHEP(10)-a1008, JPA(13)-a1208 [coupled to higher-spin fields]; Kuniyasu a1312 [coupled with non linear electrodynamics]; Lemos et al IJMPD(15)-a1506 [rotating thin shell in asymptotically AdS3].
@ Asymptotically AdS3 canonical gravity: Scarinci & Krasnov CMP(13)-a1111 [universal phase space]; Grumiller & Riegler a1608 [boundary conditions]; > s.a. 3D quantum gravity.
@ Other with cosmological constant: Fujiwara & Soda PTP(90) [ADM formalism]; Corichi & Gomberoff CQG(99) [duality]; Krasnov CQG(02)gq/01 [Euclidean continuation of asymptotically AdS, rotating black holes and wormholes]; Mišković & Olea PLB(06) [Λ < 0, boundary conditions]; Witten a0706 [dual conformal field theories]; Li et al JHEP(08)-a0801 [deformed by gravitational Chern-Simons action]; Castro et al PRD(12) [2D critical Ising model as dual conformal field theory].

General References > s.a. formulations of general relativity; models for topology change.
@ Early work: in Bergmann in(65) [comment by Wheeler]; & Leutwyler.
@ Reviews: Brown 88; Carlip JKPS(95)gq-ln; Welling ht/95-ln [and point particles]; Carlip CQG(05)gq [especially conformal field theory and black holes].
@ General articles: Giddings et al GRG(84); Gott & Alpert GRG(84); Jackiw NPB(85); Ashtekar & Romano PLB(89); Bengtsson PLB(89); de Sousa Gerbert pr(89); Moncrief JMP(89); Hosoya & Nakao CQG(90); Moncrief JMP(90); Carlip CQG(91) [geometry]; Franzosi & Guadagnini CQG(96); Buffenoir & Noui gq/03; Nelson CQG(04) [global constants]; Meusburger CQG(09)-a0811 [measurements and observables].
@ Polygon approach: 't Hooft CMP(88), CQG(92), CQG(93), CQG(93); Waelbroeck & Zapata CQG(96)gq; Welling CQG(97)gq/96 [torus]; Hollmann & Williams CQG(99)gq/98; Kádár & Loll CQG(04)gq/03 [higher-genus data]; Kádár CQG(05) [from first-order formalism]; Eldering MSc(06)gq.
@ Hamiltonian formulation: Puzio CQG(94)gq [Gauss map, holonomies]; Miković & Manojlović CQG(98) [on a torus]; Cantini et al CQG(01)ht/00 [and particles], ht/00-MG9; Kenmoku et al gq/00-conf; Menotti gq/01-conf; Nelson gq/04-fs [ADM variables]; Bonzom & Livine CQG(08)-a0801 [Immirzi-like parameter]; Frolov et al G&C(10)-a0902 [triad variables]; Meusburger & Schönfeld a1203-conf [Λ = 0, Dirac gauge fixing procedure]; Escalante & Rodríguez JHEP(14)-a1310 [Palatini theory with cosmological constant]; Corichi & Rubalcava-García PRD(15)-a1503 [1st-order formalism, energy].
@ Witten formulation: Witten NPB(88); Louko & Marolf CQG(94)gq/93 [\(\mathbb R\) × T2]; Louko CQG(95)gq [\(\mathbb R\) × Klein bottle].
@ Other connection and holonomy formulations: Bezerra CQG(88); Peldán CQG(92); Manojlović & Miković NPB(92); Unruh & Newbury IJMPD(94)gq/93 [holonomies and geometry]; Barbero & Varadarajan NPB(95)gq [homogeneous], CQG(99)gq [degrees of freedom]; Miković & Manojlović CQG(98)gq/97 [T2, Ashtekar variables, reduced phase space]; Bonzom et al a1402 [Riemannian, deformed phase space]; Chagoya & Sabido a1612 [self-dual gravity and the Immirzi parameter].
@ Null-surface formulation: Forni et al JMP(00)gq; Harriott & Williams GRG(14).
@ Moduli space: Nelson & Picken mp/05-proc [T2, Λ < 0, holonomies and quantization].
@ Observables: Nelson & Regge CMP(93); Nelson & Picken GRG(11)-a1006 [Λ < 0, Wilson observables].
@ Related topics: Martin NPB(89), Waelbroeck NPB(91) [observables, time]; Bengtsson & Brännlund JMP(01)gq/00 [chaos and time machines on \(\mathbb R\) × T2]; Niemi PRD(04)ht/03 [from 2D SU(2) Yang-Mills theory]; Meusburger & Schönfeld CQG(11)-a1012 [gauge fixing and Dirac brackets].


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