3-Dimensional General Relativity| 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 = ∫ (eI ∧ FJK − \(1\over6\)Λ eI ∧ eJ ∧ eK) 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,\(\mathbb C\)); It 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];
Podolský et al CQG(19)-a1809 [all Λ-vacuum, pure radiation, or gyratons].
@ 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(13)-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];
Dong et al JHEP(18)-a1802 [scalar field condensation phase transition].
@ 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 JHEP(16)-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: & Leutwyler;
in Bergmann in(65) [comment by Wheeler].
@ 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];
Hajihashemi & Shirzad a1704 [vielbein variables].
@ 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];
Escalante & Eduardo a2002 [Ashtekar variables, Hamilton-Jacobi analysis].
@ 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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