Einstein-Cartan Theory |

**In General**
> s.a. gravity theories / first-order actions for general relativity.

* __Idea__: A theory of gravity
coupled to a 4-spinor field through a connection, sometimes also associated
with the names of Sciama and Kibble; Variables are a tetrad, a connection,
and a spinor field; The theory necessarily has torsion, and is not equivalent
to one in which one imposes that the connection be one defined by the metric
(Einstein-Dirac); For simple (non derivative-coupled) sources (vanishing torsion)
it is equivalent to general relativity.

* __History__: It was proposed in 1922,
before the discovery of spin, by Élie Cartan, who was influenced by the 1909
work of the Cosserat brothers, who considered besides an (asymmetric) force stress
tensor also a moments stress tensor in a suitably generalized continuous medium.

* __And observation__: Torsion
effects are probably not observable classically.

* __Versions__: The theory
can be reformulated in 3+1 form using 2-spinors; It is then equivalent to
the self-dual version with appropriate reality conditions; It can also be
described as a constrained BF theory.

@ __General references__: Hehl et al RMP(76);
Fiziev in(96)gq/97 [Lagrangian];
Dzhunushaliev & Singleton PLA(99)gq/98;
Trautman en(06)gq [rev];
Fabbri AFLB(08)-a0808 [uniqueness];
Popławski AR(13)-a1106 [in gravity and cosmology];
Socolovsky a1110 [in terms of bundles and connections];
Popławski fn(12)may [and universe inside a black hole];
Pilc a1311 [kinematical description].

@ __Hamiltonian formulation__: Szczyrba CMP(78);
Nikolić CQG(95) [and constraints].

@ __Merits / objections__: Hehl gq/97-MG8;
in Ohanian & Ruffini 94, p312;
Diether & Christian a1705 [and fundamental particle physics].

@ __Tests__: Garcia de Andrade CQG(01).

> __Online resources__:
see Wikipedia page.

**Special Systems and Phenomenology**
> s.a. metric matching; particle models.

@ __Monopoles__: Garcia de Andrade gq/99,
gq/99 [+ dilaton];
Rahaman et al NCB(05).

@ __Cosmological models__: Garcia de Andrade gq/00,
PLB-gq/00;
Galiakhmetov G&C(09) [rotating and expanding cosmologies],
CQG(10)
[*k* = 0 FLRW models + non-minimal scalar field];
Popławski a1201 [thermal fluctuations and cosmological perturbations];
Magueijo et al PRD(13)-a1212 [with Holst term and fermions];
Bravo Medina et al AP(19)-a1812;
> s.a. bianchi I, bianchi IX
and other bianchi models; cosmological
acceleration in modified theories; cosmology in modified gravity.

@ __Other systems__: Bressange CQG(00) [thin shells];
Farfán et al GRG(12)-a1101 [non-static, spherically symmetric solutions].

**Related Topics and Theories** > s.a. canonical
formulations of gravity [covariant]; torsion in physical theories.

@ __General references__: Castagnino et al GRG(85),
Castagnino & Levinas GRG(87) [post-newtonian approximation];
Singh & Ryder CQG(97) [low-energy theory];
Ruggiero & Tartaglia AJP(03)dec-gq [as a theory of spacetime defects];
Popławski a1001 [no extra dimensions].

@ __Coupled to a Maxwell field__: Popławski IJTP(10)-a0905;
Popławski a1108.

@ __With other fields__: Hammond GRG(95) [Einstein-Cartan-Proca];
Xue PLB(08) [four-fermion interaction, phase structure];
Kaźmierczak PRD(09) [with Holst term and fermions];
Lagraa & Lagraa CQG(10)-a0908 [fermions];
Khanapurkar a1803-MS [Dirac field, minimal coupling];
Chen et al a1912 [ambiguity in matter couplings];
> s.a. types of spinors [ELKO spinors].

@ __And other gravity theories__: Petti GRG(86)-a1301 [derivation from general relativity];
Botta Cantcheff PRD(08)-a0801 [for Chern-Simons Lorentz-violating gravity];
Baekler & Hehl CQG(11)-a1105 [Lagrangians with quadratic torsion and curvature terms].

@ __Quantum theory__: Xue PLB(09)-a0902,
PRD(10)-a0912 [Regge calculus];
Pilc a1312 [kinematical Hilbert space];
Shapiro & Teixeira CQG(14)-a1402 [with Holst term in the action];
Cianfrani a1605 [physical states];
> s.a. Schwinger's Principle.

> __Related topics__:
see Metric-Affine Gravity; Mimetic Gravity;
poincaré group [gauge invariance]; spacetime
singularities.

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