Einstein-Cartan
Theory |

**In General** > s.a. gravity
theories / first-order actions for generall 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__: 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-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]; > s.a. bianchi
I, bianchi
IX and other bianchi models; cosmology in modified-gravity theories; theories of cosmological acceleration.

@ __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]; Metric-Affine
Gravity; 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]; > s.a. 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 poincaré group [gauge invariance]; spacetime
singularities.

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