Torsion in Physical Theories  

In General > s.a. torsion.
* Motivation: Torsion arises in string theory as an antisymmetric field, and would be required by the modification of general relativity that can accomodate the existence of gravitomagnetic monopoles.
* Minimal coupling: Requires that the trace of the torsion tensor be a gradient, Ta = ∂aθ, and that the modified volume element τ = exp{θ} |g| dx1 ∧ ... ∧ dxn be used in the action formulation of a physical model.
* Electromagnetism: The coupling of torsion to the Maxwell field can be introduced in the Lagrangian density

\[ {\cal L} = \big[-{\textstyle{1\over4}}\,g^{ac}g^{bd}\,F^~_{ab}F^~_{cd} + q\,T^{abc}g^{de}(\partial^~_dF^~_{ab})\,F^~_{ce}\big]\,|g|^{1/2}, \]

where q is a parameter, which leads to a modified photon dispersion relation.
@ General references: Kalinowski IJTP(81) [and gauge theory, from Kaluza-Klein theory]; Hehl & Obukhov a0711 [geometry and field theory]; issue AFLB(07)#2-3; Lazar & Hehl FP(10)-a0911 [Cartan's spiral staircase and the gauge theory of dislocations]; Bergman a1411 [internal symmetry of a metric-compatible spacetime connection].
@ Dynamics of torsion: Saa GRG(97)gq/96; Mosna & Saa JMP(05)gq [minimal coupling]; Popławski gq/06, JMP(06); > s.a. differential forms.
@ Singularities: García de Andrade FP(90), IJTP(90); Esposito NCB(90)gq/95, FdP(92)gq/95.
@ Electromagnetism: Hammond GRG(88), GRG(91); Horie ht/95; de Andrade & Pereira IJMPD(99)gq/97; Filewood gq/98; Rubilar et al CQG(03) [and birefringence]; Prasanna & Mohanty GRG(09) [photon propagation]; Popławski a1108; Mannheim JPCS(15)-a1406 [magnetic monopoles and Faraday's law, Grassmann numbers].
@ Angular momentum conservation: Yishi Duan & Ying Jiang GRG(99)gq/98; Capozziello et al EPL(99)ap [fermion helicity flip].
@ And topological invariants: Aouane et al CQG(07) [from integral of Nieh-Yan 4-form]; Nieh IJMPA(07) [rev].
blue bullet Related topics: see lagrangian theories; lorentz invariance; regge calculus; sound [acoustic torsion]; torsion phenomenology.

And Gravity, Theory > s.a. 2D gravity; 3D gravity; gravitation; Affine Gravity; Metric-Affine Gravity; MOND; non-commutative gravity.
* Idea: A consistent theory of gravity with torsion emerged during the early 1960s as a gauge theory of the Poincaré group, which incorporates as the simplest viable cases the Einstein-Cartan(-Sciama-Kibble) theory, the teleparallel equivalent general relativity, and general relativity itself.
* Couplings and gravity: It has been established that torsion couples to the spin of elementary particles (spin current of the Dirac field), and not to the particles' orbital angular momentum; The inclusion of torsion in the gravitational formalism leads to four-fermion interactions (though strongly suppressed in 4D).
* Gravity: In the teleparallel theory of gravity, curvature and torsion are alternative ways of describing the gravitational field, and are consequently related to the same degrees of freedom; More general gravity theories, like Einstein-Cartan and gauge theories for the Poincaré and the affine groups, consider curvature and torsion as representing independent degrees of freedom.
@ Books / Reviews: de Sabbata & Sivaram 94; Arcos & Pereira IJMPD(04)gq/05; Aldrovandi & Pereira AFLB(07)-a0801.
@ General references: Hehl et al RMP(76); López IJTP(77); Penrose FP(83); Hehl FP(85); Hammond GRG(90), GRG(94), GRG(94), CP(95) [II]; Gangopadhyay & Sengupta ht/97 [symmetries]; Fiziev gq/98-conf, gq/98; Garecki RGC(04)gq/01 [overview, T not needed]; Mahato MPLA(02)gq/06 [G in Riemann-Cartan spacetime]; Watanabe & Hayashi gq/04; Arcos & Pereira CQG(04); Mahato IJMPA(07)gq/06; Lecian et al gq/07-MGXI; Schücking a0803 [Einstein's theory is about torsion]; Torres-Gómez & Krasnov PRD(09)-a0811 [theory with no black holes]; Lledó & Sommovigo CQG(10)-a0907; Kleinert EJTP(10)-a1005; Hammond GRG(10) = IJMPD(10) [torsion is necessary]; Gaitan IJMPA(10)-a1009 [contortion as a dynamical variable, Yang-Mills formulation]; Garecki a1110-talk [updated overview, T not needed]; Fabbri GRG(13); Olmo & Rubiera-García PRD(13)-a1306 [in Palatini theories of gravity]; Fabbri a1703 [foundational approach].
@ Connection formulation: Montesinos JMP(99) [and Ashtekar–Barbero connection].
@ Hamiltonian analysis: de Sabbata & Ronchetti FP(99); Yang et al PRD(12)-a1201 [R + T 2 action].
@ And quantum gravity: Kim & Pak CQG(08)-gq/06; Singh CS(15)-a1512.
@ With other fields: Israelit FP(98)-a0712, in(99)-a0712 [and electromagnetism]; Megged ht/00 [gravity + Yang-Mills]; Popławski IJTP(10) [Einstein-Maxwell-Dirac theory]; Fabbri & Vignolo AdP(12)-a1201, Fabbri IJMPD(13)-a1211, IJGMP(15)-a1409 [Dirac fields]; Khriplovich PLB(12)-a1201 [and four-fermion gravitational interaction]; Fresneda et al BJP(15)-a1404 [Maxwell field]; Fabbri et al PRD(14)-a1404, Fabbri & Vignolo MPLA(16)-a1504 [with Dirac fields]; Kofinas PRD(15)-a1507 [scalar-torsion theories, black holes]; Fabbri IJGMP(17)-a1611 [Dirac matter fields as particles].
@ f(T) gravity: Linder PRD(10)-a1005; Yang EPJC(11)-a1007; Li et al PRD(11)-a1010 [and local Lorentz invariance]; Yang EPL(11)-a1010 [conformal transformations]; Ferraro & Fiorini PLB(11)-a1103; Li et al JHEP(11)-a1105 [Hamiltonian formulation, degrees of freedom]; Wei et al PLB(12)-a1112 [Noether symmetry]; Tamanini & Böhmer PRD(12)-a1204, a1304-MG13 [good and bad tetrads]; Nashed AHEP(15)-a1403 [and local Lorentz transformations]; Krššák & Saridakis CQG(16)-a1510 [covariant formulation]; Otalora & Rebouças a1705 [causality violations]; > s.a. spherical solutions; kerr solutions.
@ f(T) gravity, cosmology: Nesseris et al PRD(13)-a1308 [viable models reduce to \(\Lambda\)CDM]; Bamba a1504-proc; Paliathanasis et al PRD(16)-a1606; Nunes et al EPJC(17)-a1608 [and varying fundamental constants].
@ Cosmology, other: Popławski AR(13)-a1106; Wanas & Hassan a1209; Velten & Caramês PRD(17)-a1702 [difficulties of f(R,T) gravity].
@ Other higher-order theories: Hammond JMP(89), JMP(90) [second-order equations]; Troncoso & Zanelli CQG(00)ht/99; Kruglov AFLB(07)-a0710 [quantum]; Capozziello et al CQG(07), Capozziello & Vignolo AdP(10)-a0910-conf [metric-affine]; Hernaski et al PRD(09)-a0905 [and massive gravitons]; Nikiforova et al PRD(09); Helayël-Neto et al PRD(10)-a1005 [Einstein-Hilbert-Chern-Simons Lagrangian]; Fabbri & Mannheim PRD(14)-a1405 [continuity of the torsionless limit]; Vasilev et al a1706 [stability].
@ Higher-dimensional theories: Mukhopadhyaya et al PRD(02) [large extra dimensions], PRL(02) [in Randall-Sundrum scenario]; > s.a. kaluza-klein models.
blue bullet Specific theories: see conformal, einstein-cartan, gauge theory, teleparallel, and unimodular gravity; low-spin field theories [spin-2 fields].
blue bullet Related topics: see action for general relativity; affine connections; conservation laws; energy conditions; McVittie Metric; torsion phenomenology.


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