Editing Einstein–Cartan theory (section)

== 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>