3-Dimensional
Massive Gravity Theories |

**In General** > s.a. 3D gravity theories.

* __Idea__:
Theories obtained by adding to Einstein's gravity the usual Fierz-Pauli,
or the more complicated Ricci scalar squared (*R*^{2}),
terms, are tree level unitary, but have
their unitarity
spoiled when a Chern-Simons term is added (topologically massive).

@ __General references__: Bañados & Theisen JHEP(09)
[as bimetric gravity, and black holes]; Grigore & Scharf a1008; Sadegh & Shirzad PRD(11)-a1010 [constraint structure]; Zinoviev CQG(13)-a1205 [and bigravity]; Hohm et al PRD(12)-a1208 [Chern-Simons-like action, and Hamiltonian]; Edelstein et al PRD(17)-a1602 [causality].

@ __ Black holes__: Ayón-Beato et al PRD(09)
[Lifshitz metrics]; Tonni JHEP(10)-a1006 [warped
black holes]; Ghodsi & Moghadassi PLB(11)-a1007, Flory & Sachs PRD(13)-a1304, Kwon et al JHEP(13)-a1308 [new massive gravity].

**Topologically Massive Gravity** > s.a. 3D quantum gravity; bel-robinson tensor; Faddeev-Jackiw Method; massive gravity.

* __Idea__: Three-dimensional
gravity with a cosmological constant and a gravitational Chern-Simons term.

* __Action__: The Einstein action (≅ ISO(2,1)
Chern-Simons) plus a conformal gravity term (≅ SO(2,1)-Chern-Simons or SO(3)-Chern-Simons).

* __Field equations__: If *m* is
the topological coupling constant, and *C* the (traceless) Cotton tensor, then

*G*_{ab} + *m*^{–1} *C*_{ab} = –*k* *T*_{ab} ;

Notice that, since *C*_{ab} contains derivatives of *R*_{ab}, the field equations are third-order; It propagates a single massive spin-2 graviton mode.

* __Drawback__: One of the two (left/right) central charges of the corresponding putative boundary cft can be shown to be negative whenever the bulk AdS graviton has positive energy, which is inconsistent with quantum unitarity.

@ __General references__: Peldán CQG(92);
Deser et al NPB(94)ht/93 [particle
scattering amplitudes]; Accioly & Dias IJMPA(06)ht/05 [non-unitarity]; Grumiller & Jackiw a0802-wd
[with Λ < 0, and
complex Chern-Simons terms]; Grumiller et al a0806-in
[cosmological, at the chiral point]; Carlip et al PLB(08)-a0807, Carlip JHEP(08)-a0807 [in AdS spacetime,
constraint algebra]; Giribet et al JHEP(08)-a0807 [and
chirality]; Grumiller & Johansson JHEP(08)
[chiral point instability]; Oda a0905 [renormalizability];
Lashkari & Maloney CQG(11)-a1007 [and
Ricci-Cotton Flow]; Grumiller & Vassilevich JHEP(10)-a1007 [graviton
1-loop partition
function]; > s.a. energy
positivity.

@ __Canonical formulation__: Hotta et al JHEP(08)-a0805 [Brown-Henneaux approach]; Blagojević & Cvetković JHEP(09)-a0812 [with
cosmological constant]; Mišković & Olea JHEP(09)-a0909 [background-independent
charges]; > s.a. symplectic structures.

@ __Black holes__: Dereli & Sarioglu gq/00; Birmingham et al PRD(10)-a1006 [BTZ black hole, stability]; Flory & Sachs PRD(13)-a1304 [non-stationary exact solution]; Troessaert et al a1512-ln [and gravitational waves]; > s.a. black-hole
perturbations.

@ __Other solutions__: Nutku & Baekler AP(89) [homogeneous, anisotropic];
Clément CQG(90) [point sources]; Ortiz
AP(90) [homogeneous], CQG(90) [spinning sources];
Edery & Paranjape PLB(97), PLB(97)gq [localized mass and spin];
Dereli & Sarioglu PLB(00)gq [+
Maxwell-Chern-Simons, self-dual];
Henneaux
et
al PRD(09)-a0901 [asymptotically
AdS spacetimes]; Chow et al CQG(10)-a0906 [classification];
Ertl et al CQG(10)-a1006 [stationary
axisymmetric]; Henneaux et al PRD(11)-a1108 [asymptotically warped AdS spacetimes]; Moon & Myung PRD(12)-a1111 [polarization modes of gravitational waves]; > s.a. anti-de sitter
spacetime; Kundt
Spacetimes.

**Quantum Theory** > s.a. 3D quantum gravity.

@ __Topologically massive gravity__: Deser & Yang CQG(90) [1-loop renormalizability]; Castro et al JHEP(11)-a1105 [with positive cosmological constant].

**Other Theories** > s.a. 3D gravity theories [massive 3D supergravity].

* __Minimal massive gravity__: A theory that may be considered a variation on topologically massive gravity, and also propagates a single massive spin-2 graviton mode, but in which the field equations are not the Euler-Lagrange equations of any action constructed only from the metric, and are nevertheless consistent; Both central charges of the dual cft are positive, and it also seems possible to consistently couple matter.

* __Other variants of topologically massive gravity__: Various higher-derivative modifications have been proposed to circumvent the negative-central-charge problem but, except for Minimal Massive Gravity they have always introduced additional, unphysical degrees of freedom.

@ __Minimal massive 3D gravity__:
Bergshoeff et al CQG(14)-a1404 and CQG+; Tekin PRD(14)-a1409 [excitations and conserved charges]; Alishahiha et al JHEP(14)-a1409; Setare NPB(15)-a1412 [in 3D AdS background]; Yekta PRD(15)-a1503 [Hamiltonian formalism].

@ __Variants of topologically massive gravity__: Chen et al PLB(11)-a1106, Bagchi et al JHEP(11)-a1107 [spin-3]; Chen & Long JHEP(11); > s.a. dirac fields in curved spacetimes; supergravity.

@ __Other theories__: Bergshoeff et al PRL(09)-a0901 [higher-order
theory, and massive Pauli-Fierz]; Paulos PRD(10)-a1005 [with curvature corrections]; Afshar et al JHEP(14)-a1405 [extended].

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