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 similar to general relativity. The theory was first proposed by Élie Cartan in 1922. Einstein–Cartan theory is the simplest Poincaré gauge theory.[1]

## Contents

* 1 Overview
* 2 History
* 3 Field equations
* 4 Avoidance of singularities
* 5 See also
* 6 References
* 7 Further reading

## 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. In particular, the antisymmetric part of the connection (referred to as the 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, then 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 (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.[2]

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 in 1922[3] and expounded in the following few years.[4][5][6] 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.[7]

Dennis Sciama[8] and Tom Kibble[9] independently revisited the theory in the 1960s, and an important review was published in 1976.[10]

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. 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[11] and therefore the superposition principle used in usual quantization techniques would not work. Recently, interest in Einstein–Cartan theory has been driven toward cosmological implications, most importantly, the avoidance of a gravitational singularity at the beginning of the universe.[12][13][14] The theory is considered viable and remains an active topic in the physics community.[15]

## Field equations

The Einstein field equations 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 curvature), and then the metric and torsion are varied independently.

Let [math]\displaystyle{ \mathcal{L}_\mathrm{M} }[/math] represent the Lagrangian density of matter and [math]\displaystyle{ \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]\displaystyle{ \mathcal{L}_\mathrm{G}=\frac{1}{2\kappa}R \sqrt{|g|} }[/math]
[math]\displaystyle{ S=\int \left( \mathcal{L}_\mathrm{G} + \mathcal{L}_\mathrm{M} \right) \, d^4x , }[/math]

where [math]\displaystyle{ g }[/math] is the determinant of the metric tensor, and [math]\displaystyle{ \kappa }[/math] is a physical constant [math]\displaystyle{ 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]\displaystyle{ S }[/math] for the gravitational field and matter vanishes:

[math]\displaystyle{ \delta S = 0. }[/math]

The variation with respect to the metric tensor [math]\displaystyle{ g^{ab} }[/math] yields the Einstein equations:

[math]\displaystyle{ \frac{\delta \mathcal{L}_\mathrm{G}}{\delta g^{ab}} -\frac{1}{2}P_{ab}=0 }[/math]

[math]\displaystyle{ R_{ab}-\frac{1}{2}R g_{ab}=\kappa P_{ab} }[/math]

where [math]\displaystyle{ R_{ab} }[/math] is the Ricci tensor and [math]\displaystyle{ P_{ab} }[/math] is the 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]\displaystyle{ 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 procedure.

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

[math]\displaystyle{ \frac{\delta \mathcal{L}_\mathrm{G}}{\delta {T^{ab}}_c} -\frac{1}{2}{\sigma_{ab}}^c =0 }[/math]

[math]\displaystyle{ {T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d = \kappa {\sigma_{ab}}^c }[/math]

where [math]\displaystyle{ {\sigma_{ab}}^c }[/math] is the spin tensor. Because the torsion equation is an algebraic 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.

## Avoidance of singularities

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.[12][13][14] The minimal coupling between torsion and Dirac spinors generates an effective nonlinear spin–spin self-interaction, which becomes significant inside fermionic 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, 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. Torsion allows fermions to be spatially extended instead of "pointlike", which helps to avoid the formation of singularities such as black holes and removes the ultraviolet divergence in quantum field theory. 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.

## See also

* Classical theories of gravitation
* Metric-affine gravitation theory
* Gauge theory gravity
* Loop quantum gravity

## References

1. ↑ Cabral, Francisco; Lobo, Francisco S. N.; Rubiera-Garcia, Diego (December 2019). "Einstein–Cartan–Dirac gravity with U(1) symmetry breaking" (in en). The European Physical Journal C 79 (12): 1023. doi:10.1140/epjc/s10052-019-7536-3. ISSN 1434-6044. Bibcode: 2019EPJC...79.1023C.
2. ↑ Neville, Donald E. (1980-02-15). "Gravity theories with propagating torsion". Physical Review D 21 (4): 867–873. doi:10.1103/physrevd.21.867. ISSN 0556-2821. Bibcode: 1980PhRvD..21..867N.
3. ↑ Élie Cartan (1922). "Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion" (in fr). Comptes rendus de l'Académie des Sciences de Paris 174: 593–595. https://gallica.bnf.fr/ark:/12148/bpt6k3127j/f593.image.
4. ↑ Cartan, Elie (1923). "Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie)" (in fr). Annales Scientifiques de l'École Normale Supérieure 40: 325–412. doi:10.24033/asens.751. ISSN 0012-9593.
5. ↑ Cartan, Elie (1924). "Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite)" (in fr). Annales Scientifiques de l'École Normale Supérieure 41: 1–25. doi:10.24033/asens.753. ISSN 0012-9593.
6. ↑ Cartan, Elie (1925). "Sur les variétés à connexion affine, et la théorie de la relativité généralisée (deuxième partie)" (in fr). Annales Scientifiques de l'École Normale Supérieure 42: 17–88. doi:10.24033/asens.761. ISSN 0012-9593.
7. ↑ Goenner, Hubert F. M. (2004). "On the History of Unified Field Theories". Living Reviews in Relativity 7 (1): 2. doi:10.12942/lrr-2004-2. PMID 28179864. Bibcode: 2004LRR.....7....2G.
8. ↑ SCIAMA, D. W. (1964-01-01). "The Physical Structure of General Relativity". Reviews of Modern Physics 36 (1): 463–469. doi:10.1103/revmodphys.36.463. ISSN 0034-6861. Bibcode: 1964RvMP...36..463S.
9. ↑ Kibble, T. W. B. (1961). "Lorentz Invariance and the Gravitational Field". Journal of Mathematical Physics 2 (2): 212–221. doi:10.1063/1.1703702. ISSN 0022-2488. Bibcode: 1961JMP.....2..212K.
10. ↑ Hehl, Friedrich W.; von der Heyde, Paul; Kerlick, G. David; Nester, James M. (1976-07-01). "General relativity with spin and torsion: Foundations and prospects". Reviews of Modern Physics 48 (3): 393–416. doi:10.1103/revmodphys.48.393. ISSN 0034-6861. Bibcode: 1976RvMP...48..393H.
11. ↑ Hehl, F. W.; Datta, B. K. (1971). "Nonlinear Spinor Equation and Asymmetric Connection in General Relativity". Journal of Mathematical Physics 12 (7): 1334–1339. doi:10.1063/1.1665738. ISSN 0022-2488. Bibcode: 1971JMP....12.1334H.
12. ↑ 12.0 12.1 Nikodem J. Popławski (2010). "Nonsingular Dirac particles in spacetime with torsion". Physics Letters B 690 (1): 73–77. doi:10.1016/j.physletb.2010.04.073. Bibcode: 2010PhLB..690...73P.
13. ↑ 13.0 13.1 Nikodem J. Popławski (2010). "Cosmology with torsion: An alternative to cosmic inflation". Physics Letters B 694 (3): 181–185. doi:10.1016/j.physletb.2010.09.056. Bibcode: 2010PhLB..694..181P.
14. ↑ 14.0 14.1 Nikodem Popławski (2012). "Nonsingular, big-bounce cosmology from spinor–torsion coupling". Physical Review D 85 (10): 107502. doi:10.1103/PhysRevD.85.107502. Bibcode: 2012PhRvD..85j7502P.
15. ↑ Hehl, Friedrich W.; Weinberg, Steven (2007). "Note on the torsion tensor". Physics Today 60 (3): 16. doi:10.1063/1.2718743. Bibcode: 2007PhT....60c..16H.

## Further reading

* Gronwald, F.; Hehl, F. W. (1996). "On the Gauge Aspects of Gravity". arXiv:gr-qc/9602013.
* Hammond, Richard T (2002-03-27). "Torsion gravity". Reports on Progress in Physics 65 (5): 599–649. doi:10.1088/0034-4885/65/5/201. ISSN 0034-4885. Bibcode: 2002RPPh...65..599H.
* Hehl, F. W. (1973). "Spin and torsion in general relativity: I. Foundations". General Relativity and Gravitation 4 (4): 333–349. doi:10.1007/bf00759853. ISSN 0001-7701. Bibcode: 1973GReGr...4..333H.
* Hehl, F. W. (1974). "Spin and torsion in general relativity II: Geometry and field equations". General Relativity and Gravitation 5 (5): 491–516. doi:10.1007/bf02451393. ISSN 0001-7701. Bibcode: 1974GReGr...5..491H.
* Hehl, Friedrich W.; von der Heyde, Paul; Kerlick, G. David (1974-08-15). "General relativity with spin and torsion and its deviations from Einstein's theory". Physical Review D 10 (4): 1066–1069. doi:10.1103/physrevd.10.1066. ISSN 0556-2821. Bibcode: 1974PhRvD..10.1066H.
* Kleinert, Hagen (2000). "Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion". General Relativity and Gravitation 32 (5): 769–839. doi:10.1023/a:1001962922592. ISSN 0001-7701. Bibcode: 2000GReGr..32..769K.
* Kuchowicz, Bronisław (1978). "Friedmann-like cosmological models without singularity". General Relativity and Gravitation 9 (6): 511–517. doi:10.1007/bf00759545. ISSN 0001-7701. Bibcode: 1978GReGr...9..511K.
* Lord, E. A. (1976). "Tensors, Relativity and Cosmology" (McGraw-Hill).
* Petti, R. J. (1976). "Some aspects of the geometry of first-quantized theories". General Relativity and Gravitation 7 (11): 869–883. doi:10.1007/bf00771019. ISSN 0001-7701. Bibcode: 1976GReGr...7..869P.
* Petti, Richard J. (1986). "On the local geometry of rotating matter". General Relativity and Gravitation 18 (5): 441–460. doi:10.1007/bf00770462. ISSN 0001-7701. Bibcode: 1986GReGr..18..441P.
* Petti, R J (2006-01-12). "Translational spacetime symmetries in gravitational theories". Classical and Quantum Gravity 23 (3): 737–751. doi:10.1088/0264-9381/23/3/012. ISSN 0264-9381. Bibcode: 2006CQGra..23..737P.
* Petti, R. J. (2021). "Derivation of Einstein–Cartan theory from general relativity". International Journal of Geometric Methods in Modern Physics 18 (6): 2150083–2151205. doi:10.1142/S0219887821500833. Bibcode: 2021IJGMM..1850083P.
* Poplawski, Nikodem J. (2009). "Spacetime and fields". arXiv:0911.0334 [gr-qc].
* 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).
* Shapiro, I.L. (2002). "Physical aspects of the space–time torsion". Physics Reports 357 (2): 113–213. doi:10.1016/s0370-1573(01)00030-8. ISSN 0370-1573. Bibcode: 2002PhR...357..113S.
* Trautman, Andrzej (1973). "Spin and Torsion May Avert Gravitational Singularities". Nature Physical Science 242 (114): 7–8. doi:10.1038/physci242007a0. ISSN 0300-8746. Bibcode: 1973NPhS..242....7T.
* Trautman, Andrzej (2006). "Einstein–Cartan Theory". arXiv:gr-qc/0606062.

* v
* t
* e

Theories of gravitation

Standard|

| Newtonian gravity (NG)|

* Newton's law of universal gravitation
* Gauss's law for gravity
* Poisson's equation for gravity
* History of gravitational theory

|

General relativity (GR)|

* Introduction
* History
* Mathematics
* Exact solutions
* Resources
* Tests
* Post-Newtonian formalism
* Linearized gravity
* ADM formalism
* Gibbons–Hawking–York boundary term

Alternatives to
general relativity|

| Paradigms|

* Classical theories of gravitation
* Quantum gravity
* Theory of everything

|

Classical|

* Einstein–Cartan
* Bimetric theories
* Gauge theory gravity
* Teleparallelism
* Composite gravity
* f(R) gravity
* Infinite derivative gravity
* Massive gravity
* Modified Newtonian dynamics, MOND
* AQUAL
* Tensor–vector–scalar
* Nonsymmetric gravitation
* Scalar–tensor theories
* Brans–Dicke
* Scalar–tensor–vector
* Conformal gravity
* Scalar theories
* Nordström
* Whitehead
* Geometrodynamics
* Induced gravity
* Chameleon
* Pressuron
* Degenerate Higher-Order Scalar-Tensor theories

Quantum-mechanical|

* Euclidean quantum gravity
* Canonical quantum gravity
* Wheeler–DeWitt equation
* Loop quantum gravity
* Spin foam
* Causal dynamical triangulation
* Asymptotic safety in quantum gravity
* Causal sets
* DGP model
* Rainbow gravity theory

Unified-field-theoric|

* Kaluza–Klein theory
* Dilaton
* Supergravity

Unified-field-theoric and
quantum-mechanical|

* Noncommutative geometry
* Semiclassical gravity
* Superfluid vacuum theory
* Logarithmic BEC vacuum
* String theory
* M-theory
* F-theory
* Heterotic string theory
* Type I string theory
* Type 0 string theory
* Bosonic string theory
* Type II string theory
* Little string theory
* Twistor theory
* Twistor string theory

Generalisations /
extensions of GR|

* Scale relativity
* Liouville gravity
* Lovelock theory
* (2+1)-dimensional topological gravity
* Gauss–Bonnet gravity
* Jackiw–Teitelboim gravity

Pre-Newtonian
theories and
toy models|

* Aristotelian physics
* CGHS model
* RST model
* Mechanical explanations
* Fatio–Le Sage
* Entropic gravity
* Gravitational interaction of antimatter
* Physics in the medieval Islamic world
* Theory of impetus

Related topics|

* Graviton

* v
* t
* e

Albert Einstein

Physics|

* Special relativity
* General relativity
* Mass–energy equivalence (E=mc2)
* Brownian motion
* Photoelectric effect
* Einstein coefficients
* Einstein solid
* Equivalence principle
* Einstein field equations
* Einstein radius
* Einstein relation (kinetic theory)
* Cosmological constant
* Bose–Einstein condensate
* Bose–Einstein statistics
* Bose–Einstein correlations
* Einstein–Cartan theory
* Einstein–Infeld–Hoffmann equations
* Einstein–de Haas effect
* EPR paradox
* Bohr–Einstein debates
* Teleparallelism
* Thought experiments
* Unsuccessful investigations
* Wave–particle duality
* Gravitational wave
* Tea leaf paradox

Works|

* Annus Mirabilis papers (1905)
* "Investigations on the Theory of Brownian Movement" (1905)
* Relativity: The Special and the General Theory (1916)
* The World as I See It (1934)
* "Why Socialism?" (1949)
* Russell–Einstein Manifesto (1955)

Family|

* Pauline Koch (mother)
* Hermann Einstein (father)
* Maja Einstein (sister)
* Mileva Marić (first wife)
* Elsa Einstein (second wife; cousin)
* Lieserl Einstein (daughter)
* Hans Albert Einstein (son)
* Eduard Einstein (son)
* Bernhard Caesar Einstein (grandson)
* Evelyn Einstein (granddaughter)
* Thomas Martin Einstein (great-grandson)
* Robert Einstein (cousin)

Related|

* Political views
* Religious views
* Albert Einstein Archives
* Einsteinhaus
* Albert Einstein House
* Einstein refrigerator
* Brain
* In popular culture
* Einsteinium
* Awards and honors
* List of things named after Albert Einstein
* Einstein Papers Project
* Die Grundlagen der Einsteinschen Relativitäts-Theorie (1922 documentary)
* The Einstein Theory of Relativity (1923 documentary)
* Relics: Einstein's Brain
* Insignificance (1985 film)
* Einstein's Gift (2003 play)
* Einstein and Eddington (2008 TV film)
* Genius (2017 series)

Einstein
Prizes|

* Albert Einstein Award
* Albert Einstein Medal
* Albert Einstein Peace Prize
* Albert Einstein World Award of Science
* Einstein Prize (APS)
* Einstein Prize for Laser Science

Books about
Einstein|

* Albert Einstein: Creator and Rebel
* Einstein and Religion
* Einstein for Beginners
* I Am Albert Einstein
* Introducing Relativity

* Book
* Category

0.00

(0 votes)

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Physics:Einstein–Cartan_theory&oldid=2178779"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template