Editing Einstein–Cartan theory

{{undue|date=April 2024}}
{{Short description|Classical theory of gravitation}}
In [[theoretical physics]], the '''Einstein–Cartan theory''', also known as the '''Einstein–Cartan–Sciama–Kibble theory''', is a [[classical theory of gravitation]], one of several [[alternatives to general relativity]].<ref name="Cabral-2019">{{Cite journal|last1=Cabral|first1=Francisco|last2=Lobo|first2=Francisco S. N.|last3=Rubiera-Garcia|first3=Diego|date=December 2019|title=Einstein–Cartan–Dirac gravity with U(1) symmetry breaking|journal=The European Physical Journal C|language=en|volume=79|issue=12|pages=1023|doi=10.1140/epjc/s10052-019-7536-3| arxiv=1902.02222 |bibcode=2019EPJC...79.1023C|issn=1434-6044|doi-access=free}}</ref> The theory was first proposed by [[Élie Cartan]] in 1922.

== Overview ==
Einstein–Cartan theory differs from general relativity in two ways:

: (1) it is formulated within the framework of Riemann–Cartan geometry, which possesses a locally gauged Lorentz symmetry, while general relativity is formulated within the framework of Riemannian geometry, which does not;

: (2) an additional set of equations are posed that relate torsion to spin.

This difference can be factored into

:: general relativity (Einstein–Hilbert) → general relativity (Palatini) → '''Einstein–Cartan'''

by first reformulating general relativity onto a Riemann–Cartan geometry, replacing the Einstein–Hilbert action over Riemannian geometry by the Palatini action over Riemann–Cartan geometry; and second, removing the zero torsion constraint from the Palatini action, which results in the additional set of equations for spin and torsion, as well as the addition of extra spin-related terms in the Einstein field equations themselves.

The theory of general relativity was originally formulated in the setting of [[Riemannian geometry]] by the [[Einstein–Hilbert action]], out of which arise the [[Einstein field equations]]. At the time of its original formulation, there was no concept of Riemann–Cartan geometry. Nor was there a sufficient awareness of the concept of [[gauge symmetry]] to understand that Riemannian geometries do not possess the requisite structure to embody a locally gauged [[Lorentz symmetry]], such as would be required to be able to express continuity equations and conservation laws for rotational and boost symmetries, or to describe [[spinors]] in curved spacetime geometries. The result of adding this infrastructure is a Riemann–Cartan geometry. In particular, to be able to describe spinors requires the inclusion of a [[spin structure]], which suffices to produce such a geometry.

The chief difference between a Riemann–Cartan geometry and Riemannian geometry is that in the former, the [[affine connection]] is independent of the metric, while in the latter it is derived from the metric as the [[Levi-Civita connection]], the difference between the two being referred to as the [[contorsion tensor|contorsion]]. In particular, the antisymmetric part of the connection (referred to as the [[torsion tensor|torsion]]) is zero for Levi-Civita connections, as one of the defining conditions for such connections.

Because the contorsion can be expressed linearly in terms of the torsion, it is also possible to directly translate the Einstein–Hilbert action into a Riemann–Cartan geometry, the result being the [[Palatini action]] (see also [[Palatini variation]]). It is derived by rewriting the Einstein–Hilbert action in terms of the affine connection and then separately posing a constraint that forces both the torsion and contorsion to be zero, which thus forces the affine connection to be equal to the Levi-Civita connection. Because it is a direct translation of the action and field equations of general relativity, expressed in terms of the Levi-Civita connection, this may be regarded as the theory of general relativity, itself, transposed into the framework of Riemann–Cartan geometry.

Einstein–Cartan theory relaxes this condition and, correspondingly, relaxes general relativity's assumption that the affine connection have a vanishing antisymmetric part ([[torsion tensor]]). The action used is the same as the Palatini action, except that the constraint on the torsion is removed. This results in two differences from general relativity:

: (1) the field equations are now expressed in terms of affine connection, rather than the Levi-Civita connection, and so have additional terms in Einstein's field equations involving the contorsion that are not present in the field equations derived from the Palatini formulation;

: (2) an additional set of equations are now present which couple the torsion to the intrinsic angular momentum ([[Particle spin|spin]]) of matter, much in the same way in which the affine connection is coupled to the energy and momentum of matter.

In Einstein–Cartan theory, the torsion is now a variable in the [[principle of stationary action]] that is coupled to a curved spacetime formulation of spin (the [[spin tensor]]). These extra equations express the torsion linearly in terms of the spin tensor associated with the matter source, which entails that the torsion generally be non-zero inside matter.

A consequence of the linearity is that outside of matter there is zero torsion, so that the exterior geometry remains the same as what would be described in general relativity. The differences between Einstein–Cartan theory and general relativity (formulated either in terms of the Einstein–Hilbert action on Riemannian geometry or the Palatini action on Riemann–Cartan geometry) rest solely on what happens to the geometry inside matter sources. That is: "torsion does not propagate". Generalizations of the Einstein–Cartan action have been considered which allow for propagating torsion.<ref>{{cite journal | last=Neville | first=Donald E. | date=15 February 1980 | title=Gravity theories with propagating torsion | journal=Physical Review D | volume=21 | issue=4 | issn=0556-2821 | doi=10.1103/physrevd.21.867 | pages=867–873 | bibcode=1980PhRvD..21..867N }}</ref>

Because Riemann–Cartan geometries have Lorentz symmetry as a local gauge symmetry, it is possible to formulate the associated conservation laws. In particular, regarding the metric and torsion tensors as independent variables gives the correct generalization of the conservation law for the total (orbital plus intrinsic) angular momentum to the presence of the gravitational field.

== History ==
The theory was first proposed by [[Élie Cartan]], who was inspired by [[elastica theory|Cosserat elasticity theory]],<ref>{{cite journal | author1=Markus Lazar | author2=Friedrich W. Hehl | title=Cartan’s Spiral Staircase in Physics and, in Particular, in the Gauge Theory of Dislocations | journal=Foundations of Physics | volume=40 | year=2010 | doi=10.1007/s10701-010-9440-4 | pages=1298–1325}}</ref><ref>{{cite journal | author=R. S. Lakes | title=Experimental tests of rotation sensitivity in Cosserat elasticity and in gravitation | journal=Zeitschrift für angewandte Mathematik und Physik | volume=72 | year=2021 | doi=10.1007/s00033-021-01563-1}}</ref> in 1922<ref>{{cite journal|author=Élie Cartan|title=Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion|journal=Comptes rendus de l'Académie des Sciences de Paris |volume=174|pages=593–595 |year=1922|url=https://gallica.bnf.fr/ark:/12148/bpt6k3127j/f593.image|language=fr}}</ref> and expounded in the following few years.<ref>{{cite journal | last=Cartan | first=Elie | title=Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie) | journal=Annales Scientifiques de l'École Normale Supérieure | volume=40 | year=1923 | issn=0012-9593 | doi=10.24033/asens.751 | pages=325–412|language=fr| doi-access=free }}</ref><ref>{{cite journal | last=Cartan | first=Elie | title=Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite) | journal=Annales Scientifiques de l'École Normale Supérieure | volume=41 | year=1924 | issn=0012-9593 | doi=10.24033/asens.753 | pages=1–25|language=fr| doi-access=free }}</ref><ref>{{cite journal | last=Cartan | first=Elie | title=Sur les variétés à connexion affine, et la théorie de la relativité généralisée (deuxième partie) | journal=Annales Scientifiques de l'École Normale Supérieure | volume=42 | year=1925 | issn=0012-9593 | doi=10.24033/asens.761 | pages=17–88|language=fr| doi-access=free }}</ref> [[Albert Einstein]] became affiliated with the theory in 1928 during his unsuccessful attempt to match torsion to the [[electromagnetic field tensor]] as part of a unified field theory.  This line of thought led him to the related but different theory of [[teleparallelism]].<ref>{{Cite journal |doi = 10.12942/lrr-2004-2|pmid = 28179864|title = On the History of Unified Field Theories|journal = Living Reviews in Relativity|volume = 7|issue = 1|pages = 2|year = 2004|last1 = Goenner|first1 = Hubert F. M.|pmc = 5256024|bibcode = 2004LRR.....7....2G|doi-access = free}}</ref>

[[Dennis Sciama]]<ref>{{cite journal | last=Sciama | first=D. W. | title=The Physical Structure of General Relativity | journal=Reviews of Modern Physics | volume=36 | issue=1 | date=1964-01-01 | issn=0034-6861 | doi=10.1103/revmodphys.36.463 | pages=463–469| bibcode=1964RvMP...36..463S }}</ref> and [[Tom Kibble]]<ref>{{cite journal | last=Kibble | first=T. W. B. | s2cid=54806287 | title=Lorentz Invariance and the Gravitational Field | journal=Journal of Mathematical Physics | volume=2 | issue=2 | year=1961 | issn=0022-2488 | doi=10.1063/1.1703702 | pages=212–221| bibcode=1961JMP.....2..212K }}</ref> independently revisited the theory in the 1960s.<ref>{{cite journal | last1=Hehl | first1=Friedrich W. | last2=von der Heyde | first2=Paul | last3=Kerlick | first3=G. David | last4=Nester | first4=James M. | s2cid=55726649 | title=General relativity with spin and torsion: Foundations and prospects | journal=Reviews of Modern Physics | volume=48 | issue=3 | date=1976-07-01 | issn=0034-6861 | doi=10.1103/revmodphys.48.393 | pages=393–416| bibcode=1976RvMP...48..393H }}</ref>

Einstein–Cartan theory has been historically overshadowed by its torsion-free counterpart and other alternatives like [[Brans–Dicke theory]] because torsion seemed to add little predictive benefit at the expense of the tractability of its equations{{fact|date=March 2025}}.  Since the Einstein–Cartan theory is purely classical, it also does not fully address the issue of [[quantum gravity]].

In the Einstein–Cartan theory, the [[Dirac equation]] becomes nonlinear when it is expressed in terms of the Levi-Civita connection,<ref name="Hehl">{{cite journal | last1=Hehl | first1=F. W. | last2=Datta | first2=B. K. | title=Nonlinear Spinor Equation and Asymmetric Connection in General Relativity | journal=Journal of Mathematical Physics | volume=12 | issue=7 | year=1971 | issn=0022-2488 | doi=10.1063/1.1665738 | pages=1334–1339| bibcode=1971JMP....12.1334H }}</ref> though it remains linear when expressed in terms of the connection native to the geometry. Because the torsion does not 'propagate', its relation to the spin tensor of the matter source is algebraic and it is possible to solve in terms of the spin tensor. In turn, the difference between the connection and Levi-Civita connection (the contorsion) can be solved in terms of the torsion. When the contorsion is back-substituted for in the Dirac equation, to reduce the connection to the Levi-Civita connection (e.g. in passing from equation (4.1) to equation (4.2) in <ref name="Hehl"/>), this results in non-linear contributions arising, ultimately, from the Dirac field itself. If two or more Dirac fields are present, or other fields that carry spin, the non-linear additions to the Dirac equation of each field would include contributions from all of the other fields, as well.

Even though renowned physicists such as [[Steven Weinberg]] "never understood what is so important physically about the possibility of torsion in differential geometry", other physicists claim that theories with torsion are valuable.<ref>{{Cite journal |doi = 10.1063/1.2718743|title = Note on the torsion tensor|journal = Physics Today|volume = 60|issue = 3|pages = 16|year = 2007|last1 = Hehl|first1 = Friedrich W.|bibcode = 2007PhT....60c..16H|doi-access = free}}</ref>

== Field equations ==
The [[Einstein field equation]]s of general relativity can be derived by postulating the [[Einstein–Hilbert action]] to be the true action of spacetime and then varying that action with respect to the metric tensor.
The field equations of Einstein–Cartan theory come from exactly the same approach,
except that a general asymmetric [[affine connection]] is assumed rather than the symmetric [[Levi-Civita connection]]
(i.e., spacetime is assumed to have torsion in addition  to [[Riemann curvature tensor|curvature]]),
and then the metric and torsion are varied independently.

Let <math>\mathcal{L}_\mathrm{M}</math> represent the [[Lagrangian density]] of matter and <math>\mathcal{L}_\mathrm{G}</math> represent the Lagrangian density of the gravitational field.  The Lagrangian density for the gravitational field in the Einstein–Cartan theory is proportional to the [[Ricci scalar]]:

:<math>\mathcal{L}_\mathrm{G}=\frac{1}{2\kappa}R \sqrt{|g|} </math>
:<math>S=\int \left( \mathcal{L}_\mathrm{G} + \mathcal{L}_\mathrm{M} \right) \, d^4x ,</math>

where <math>g</math> is the [[determinant]] of the metric tensor, and <math>\kappa</math> is a physical constant <math>8\pi G/c^4</math> involving the [[gravitational constant]] and the [[speed of light]].  By [[Hamilton's principle]], the variation of the total action <math>S</math> for the gravitational field and matter vanishes:

:<math>\delta S = 0.</math>

The variation with respect to the metric tensor <math>g^{ab}</math> yields the Einstein equations:

:<math> \frac{\delta \mathcal{L}_\mathrm{G}}{\delta g^{ab}} -\frac{1}{2}P_{ab}=0</math>

:{| class="wikitable"
|-
| <math> R_{ab}-\frac{1}{2}R g_{ab}=\kappa P_{ab}</math>
|}

where <math>R_{ab}</math> is the [[Ricci tensor]] and <math>P_{ab}</math> is the [[Stress–energy tensor#Canonical stress–energy tensor|''canonical'' stress–energy–momentum tensor]].
The Ricci tensor is no longer symmetric because the connection contains a nonzero torsion tensor; therefore, the right-hand side of the equation cannot be symmetric either, implying that <math>P_{ab}</math> must include an asymmetric contribution that can be shown to be related to the [[spin tensor]].  This canonical energy–momentum tensor is related to the more familiar ''symmetric'' energy–momentum tensor by the [[Belinfante–Rosenfeld stress–energy tensor|Belinfante–Rosenfeld procedure]].

The variation with respect to the torsion tensor <math>{T^{ab}}_c</math> yields the Cartan [[spin connection]] equations

:<math>\frac{\delta \mathcal{L}_\mathrm{G}}{\delta {T^{ab}}_c} -\frac{1}{2}{\sigma_{ab}}^c =0</math>

:{| class="wikitable"
|-
| <math> {T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d = \kappa {\sigma_{ab}}^c</math>
|}

where <math>{\sigma_{ab}}^c</math> is the spin tensor. Because the torsion equation is an [[Algebraic operation|algebraic]] [[Constraint (classical mechanics)|constraint]] rather than a [[partial differential equation]], the torsion field does not propagate as a [[wave]], and vanishes outside of matter. Therefore, in principle the torsion can be algebraically eliminated from the theory in favor of the spin tensor, which generates an effective "spin–spin" nonlinear self-interaction inside matter. Torsion is equal to its source term and can be replaced by a boundary or a topological structure with a throat such as a "wormhole".<ref name="Petti-1986">{{cite journal | author=Richard J. Petti | title=On the local geometry of rotating matter | journal=General Relativity and Gravitation| volume=18 | issue=5 | year=1986 | issn=0001-7701 | doi=10.1007/bf00770462 | pages=441–460| bibcode=1986GReGr..18..441P | s2cid=120013580 }}</ref>

== Avoidance of singularities ==

Recently, interest in Einstein–Cartan theory has been driven toward [[nonsingular black hole models]]<ref name="Cabral-2019"/> and [[physical cosmology|cosmological]] implications, most importantly, the avoidance of a [[gravitational singularity]] at the beginning of the universe, such as in the [[black hole cosmology]],<ref name="NP6">{{cite book | author=N. Popławski | date=2023 | chapter=Chapter 13: Gravitational Collapse with Torsion and Universe in a Black Hole | editor=C. Bambi | title=Regular Black Holes: Towards a New Paradigm of Gravitational Collapse | publisher=Springer | pages=485–499 | doi=10.1007/978-981-99-1596-5_13 | arxiv=2307.12190 }}</ref> [[quantum cosmology]],<ref>{{cite journal|author1=Stefano Lucat|author2=Tomislav Prokopec|title=Cosmological singularities and bounce in Cartan-Einstein theory|journal=Journal of Cosmology and Astroparticle Physics|volume=2017|doi=10.1088/1475-7516/2017/10/047|year=2017}}</ref> [[static universe]],<ref>{{cite journal|author=K. Atazadeh|title=Stability of the Einstein static universe in Einstein-Cartan theory|journal=Journal of Cosmology and Astroparticle Physics|volume=2014|doi=10.1088/1475-7516/2014/06/020|year=2014|issue=6 |page=020 |arxiv=1401.7639}}</ref> and [[cyclic model]].<ref>{{cite journal |first1=F. |last1=Cabral |first2=F.S.N. |last2=Lobo |first3=D. |last3=Rubiera-Garcia |year=2020 |title=Cosmological bounces, cyclic universes, and effective cosmological constant in Einstein-Cartan-Dirac-Maxwell theory |journal=[[Physical Review D]] |volume=102 |issue=8 |page=083509 |doi=10.1103/PhysRevD.102.083509 |arxiv=2003.07463 }}</ref>

Singularity theorems which are premised on and formulated within the setting of Riemannian geometry (e.g. [[Penrose–Hawking singularity theorems]]) need not hold in Riemann–Cartan geometry. Consequently, Einstein–Cartan theory is able to avoid the general-relativistic problem of the singularity at the [[Big Bang]].<ref name=NP1>{{cite journal |first=Nikodem J. |last=Popławski |year=2010 |title=Cosmology with torsion: An alternative to cosmic inflation |journal=[[Physics Letters B]] |volume=694 |issue=3 |pages=181–185 |doi=10.1016/j.physletb.2010.09.056 |arxiv = 1007.0587 |bibcode = 2010PhLB..694..181P }}</ref><ref name=NP2>{{cite journal |first=Nikodem J. |last=Popławski |year=2012 |title=Nonsingular, big-bounce cosmology from spinor–torsion coupling |journal=[[Physical Review D]] |volume=85 |issue=10 |page=107502 |doi=10.1103/PhysRevD.85.107502 |arxiv = 1111.4595 |bibcode = 2012PhRvD..85j7502P |s2cid=118434253 }}</ref> The minimal coupling between torsion and Dirac spinors generates an effective nonlinear spin–spin self-interaction, which becomes significant inside [[fermion]]ic matter at extremely high densities.  Such an interaction is conjectured to replace the singular Big Bang with a cusp-like [[Big Bounce]] at a minimum but finite [[scale factor (cosmology)|scale factor]], before which the [[observable universe]] was contracting. This scenario also explains why the present Universe at largest scales appears spatially flat, homogeneous and isotropic, providing a physical alternative to cosmic [[inflation (cosmology)|inflation]]. Torsion allows fermions to be spatially extended instead of [[point particle|"pointlike"]], which helps to avoid the formation of singularities such as [[black holes]], removes the [[ultraviolet divergence]] in quantum field theory, and leads to the [[toroidal ring model]] of electrons.<ref name=NP0>{{cite journal |first=Nikodem J. |last=Popławski |year=2010 |title=Nonsingular Dirac particles in spacetime with torsion |journal=[[Physics Letters B]] |volume=690 |issue=1 |pages=73–77 |doi=10.1016/j.physletb.2010.04.073 |arxiv = 0910.1181 |bibcode = 2010PhLB..690...73P }}</ref> According to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular black hole.  In the Einstein–Cartan theory, instead, the collapse reaches a bounce and forms a regular Einstein–Rosen bridge ([[wormhole]]) to a new, growing universe on the other side of the [[event horizon]]; pair production by the gravitational field after the bounce, when torsion is still strong, generates a finite period of inflation.<ref name=NP3>{{cite journal | first=N. |last=Popławski | year=2016 | title=Universe in a black hole in Einstein-Cartan gravity | journal=Astrophysical Journal | volume=832 | issue=2 | pages=96 | doi=10.3847/0004-637X/832/2/96 | doi-access=free | arxiv=1410.3881 | bibcode=2016ApJ...832...96P | s2cid=119771613 }}</ref><ref name=NP4>{{cite journal | first1=G. | last1=Unger | first2=N. | last2=Popławski | year=2019 | title=Big Bounce and closed universe from spin and torsion | journal=[[Astrophysical Journal]] | volume=870 | issue=2 | pages=78 | doi=10.3847/1538-4357/aaf169 | arxiv=1808.08327 | bibcode=2019ApJ...870...78U | s2cid=119514549 | doi-access=free }}</ref>

== Other ==

Einstein–Cartan theory seems to allow [[gravitational shielding]]<ref>{{cite journal | author1=V. de Sabbata | author2=C. Sivaram | title=Gravimagnetic field, torsion, and gravitational shielding | journal=Il Nuovo Cimento B (1971-1996) | volume=106 | year=1991 | doi=10.1007/BF02723183 | pages=873-878}}</ref> and the [[neutrino oscillation|oscillation]] of massless [[neutrinos]] without violating the [[equivalence principle]].<ref>{{cite journal | author1=V. De Sabbata | author2=M. Gasperini  | title=Neutrino oscillations in the presence of torsion | journal=Il Nuovo Cimento A (1971-1996) | volume=65 | year=1981 | issue=4 | doi=10.1007/BF02902051 | pages=479–500 | bibcode=1981NCimA..65..479S }}</ref><ref>{{cite journal | author1=Subhasish Chakrabarty | author2=Amitabha Lahiri | title=Geometrical contribution to neutrino mass matrix | journal=The European Physical Journal C | year=2019 | volume=79 | issue=8 | pages=697 | doi=10.1140/epjc/s10052-019-7209-2 | arxiv=1904.06036 | bibcode=2019EPJC...79..697C }}</ref>

In addition, the Einstein–Cartan theory is also related to [[geometrodynamics]]<ref name="Petti-1986"/><ref>{{cite journal | author1=V. Dzhunushaliev | author2=D. Singleton | year=1999 | title=Einstein–Cartan–Heisenberg theory of gravity with dynamical torsion | journal=[[Physics Letters A]] | volume=257 |issue=1-2 |pages = 7-13 |doi = 10.1016/S0375-9601(99)00282-0 |arxiv = gr-qc/9810050}}</ref> and the [[vortex theory of the atom]].<ref>{{cite journal | author1=Venzo de Sabbata | author2=C. Sivaram | title=Torsion, string tension, and topological origin of charge and mass | journal=Foundations of Physics Letters | volume=8 | year=1995 | doi=10.1007/BF02187817 | pages=375–380}}</ref>

== See also ==
* [[Alternatives to general relativity]]
* [[Metric-affine gravitation theory]]
* [[Gauge theory gravity]]
* [[Loop quantum gravity]]

== References ==
{{reflist}}

== Further reading ==
* {{cite arXiv |eprint=gr-qc/9602013|last1=Gronwald|first1=F.|title=On the Gauge Aspects of Gravity|last2=Hehl|first2=F. W.|year=1996}}
* {{cite journal | last=Hammond | first=Richard T | title=Torsion gravity | journal=Reports on Progress in Physics| volume=65 | issue=5 | date=2002-03-27 | issn=0034-4885 | doi=10.1088/0034-4885/65/5/201 | pages=599–649| bibcode=2002RPPh...65..599H | s2cid=250831296 }}
* {{cite journal | last=Hehl | first=F. W. | title=Spin and torsion in general relativity: I. Foundations | journal=General Relativity and Gravitation| volume=4 | issue=4 | year=1973 | issn=0001-7701 | doi=10.1007/bf00759853 | pages=333–349| bibcode=1973GReGr...4..333H | s2cid=120910420 }}
* {{cite journal | last=Hehl | first=F. W. | title=Spin and torsion in general relativity II: Geometry and field equations | journal=General Relativity and Gravitation| volume=5 | issue=5 | year=1974 | issn=0001-7701 | doi=10.1007/bf02451393 | pages=491–516| bibcode=1974GReGr...5..491H | s2cid=120844152 }}
* {{cite journal | last1=Hehl | first1=Friedrich W. | last2=von der Heyde | first2=Paul | last3=Kerlick | first3=G. David | title=General relativity with spin and torsion and its deviations from Einstein's theory | journal=Physical Review D | volume=10 | issue=4 | date=1974-08-15 | issn=0556-2821 | doi=10.1103/physrevd.10.1066 | pages=1066–1069| bibcode=1974PhRvD..10.1066H }}
* {{cite journal | last=Kleinert | first=Hagen | title=Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion | journal=General Relativity and Gravitation| volume=32 | issue=5 | year=2000 | issn=0001-7701 | doi=10.1023/a:1001962922592 | pages=769–839| arxiv=gr-qc/9801003 | bibcode=2000GReGr..32..769K | s2cid=14846186 }}
* {{cite journal | last=Kuchowicz | first=Bronisław | title=Friedmann-like cosmological models without singularity | journal=General Relativity and Gravitation| volume=9 | issue=6 | year=1978 | issn=0001-7701 | doi=10.1007/bf00759545 | pages=511–517| bibcode=1978GReGr...9..511K | s2cid=118380177 }}
* Lord, E. A. (1976). "Tensors, Relativity and Cosmology" (McGraw-Hill).
* {{cite journal | last=Petti | first=R. J. | title=Some aspects of the geometry of first-quantized theories | journal=General Relativity and Gravitation| volume=7 | issue=11 | year=1976 | issn=0001-7701 | doi=10.1007/bf00771019 | pages=869–883| bibcode=1976GReGr...7..869P | s2cid=189851295 }}
* {{cite journal | last=Petti | first=R J | title=Translational spacetime symmetries in gravitational theories | journal=Classical and Quantum Gravity| volume=23 | issue=3 | date=2006-01-12 | issn=0264-9381 | doi=10.1088/0264-9381/23/3/012 | pages=737–751| arxiv=1804.06730 | bibcode=2006CQGra..23..737P | s2cid=118897253 }}
* {{cite journal |arxiv=1301.1588|last1=Petti|first1=R. J.|title=Derivation of Einstein–Cartan theory from general relativity|journal=International Journal of Geometric Methods in Modern Physics|year=2021|volume=18|issue=6|pages=2150083–2151205|doi=10.1142/S0219887821500833|bibcode=2021IJGMM..1850083P|s2cid=119218875}}
* {{cite arXiv |eprint=0911.0334|last1=Poplawski|first1=Nikodem J.|title=Spacetime and fields|class=gr-qc|year=2009}}
* de Sabbata, V. and Gasperini, M. (1985). "Introduction to Gravitation" (World Scientific).
* de Sabbata, V. and Sivaram, C. (1994). "Spin and Torsion in Gravitation" (World Scientific).
* {{cite journal | last=Shapiro | first=I.L. | title=Physical aspects of the space–time torsion | journal=Physics Reports| volume=357 | issue=2 | year=2002 | issn=0370-1573 | doi=10.1016/s0370-1573(01)00030-8 | pages=113–213| arxiv=hep-th/0103093 | bibcode=2002PhR...357..113S | s2cid=119356912 }}
* {{cite journal | last=Trautman | first=Andrzej | title=Spin and Torsion May Avert Gravitational Singularities | journal= Nature Physical Science| volume=242 | issue=114 | year=1973 | issn=0300-8746 | doi=10.1038/physci242007a0 | pages=7–8| bibcode=1973NPhS..242....7T }}
* {{cite arXiv |eprint=gr-qc/0606062|last1=Trautman|first1=Andrzej|title=Einstein–Cartan Theory|year=2006}}

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