Einstein–Cartan theory

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.^[1] 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. 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] 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.^[12] The theory has indirectly influenced loop quantum gravity (and seems also to have influenced twistor theory^[13]).

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 ${\displaystyle {\mathcal {L}}_{\mathrm {M} }}$ represent the Lagrangian density of matter and ${\displaystyle {\mathcal {L}}_{\mathrm {G} }}$ 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:

${\displaystyle {\mathcal {L}}_{\mathrm {G} }={\frac {1}{2\kappa }}R{\sqrt {|g|}}}$

${\displaystyle S=\int \left({\mathcal {L}}_{\mathrm {G} }+{\mathcal {L}}_{\mathrm {M} }\right)\,d^{4}x,}$

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

${\displaystyle \delta S=0.}$

The variation with respect to the metric tensor ${\displaystyle g^{ab}}$ yields the Einstein equations:

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

${\displaystyle R_{ab}-{\frac {1}{2}}Rg_{ab}=\kappa P_{ab}}$

where ${\displaystyle R_{ab}}$ is the Ricci tensor and ${\displaystyle P_{ab}}$ 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 ${\displaystyle P_{ab}}$ 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 ${\displaystyle {T^{ab}}_{c}}$ yields the Cartan spin connection equations

${\displaystyle {\frac {\delta {\mathcal {L}}_{\mathrm {G} }}{\delta {T^{ab}}_{c}}}-{\frac {1}{2}}{\sigma _{ab}}^{c}=0}$

${\displaystyle {T_{ab}}^{c}+{g_{a}}^{c}{T_{bd}}^{d}-{g_{b}}^{c}{T_{ad}}^{d}=\kappa {\sigma _{ab}}^{c}}$

where ${\displaystyle {\sigma _{ab}}^{c}}$ 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. 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".^[14]

Avoidance of singularities

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, such as in the black hole cosmology,^[15] static universe,^[16] or cyclic model.^[17]

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.^[18][19] 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, removes the ultraviolet divergence in quantum field theory, and leads to the toroidal ring model of electrons.^[20] 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.^[21][22]

See also

• Alternatives to general relativity
• 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". The European Physical Journal C. 79 (12): 1023. arXiv:1902.02222. Bibcode:2019EPJC...79.1023C. doi:10.1140/epjc/s10052-019-7536-3. ISSN 1434-6044.
2. Neville, Donald E. (15 February 1980). "Gravity theories with propagating torsion". Physical Review D. 21 (4): 867–873. Bibcode:1980PhRvD..21..867N. doi:10.1103/physrevd.21.867. ISSN 0556-2821.
3. Élie Cartan (1922). "Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion". Comptes rendus de l'Académie des Sciences de Paris (in French). 174: 593–595.
4. Cartan, Elie (1923). "Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie)". Annales Scientifiques de l'École Normale Supérieure (in French). 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)". Annales Scientifiques de l'École Normale Supérieure (in French). 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)". Annales Scientifiques de l'École Normale Supérieure (in French). 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. Bibcode:2004LRR.....7....2G. doi:10.12942/lrr-2004-2. PMC 5256024. PMID 28179864.
8. Sciama, D. W. (1964-01-01). "The Physical Structure of General Relativity". Reviews of Modern Physics. 36 (1): 463–469. Bibcode:1964RvMP...36..463S. doi:10.1103/revmodphys.36.463. ISSN 0034-6861.
9. Kibble, T. W. B. (1961). "Lorentz Invariance and the Gravitational Field". Journal of Mathematical Physics. 2 (2): 212–221. Bibcode:1961JMP.....2..212K. doi:10.1063/1.1703702. ISSN 0022-2488. S2CID 54806287.
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. Bibcode:1976RvMP...48..393H. doi:10.1103/revmodphys.48.393. ISSN 0034-6861. S2CID 55726649.
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. Bibcode:1971JMP....12.1334H. doi:10.1063/1.1665738. ISSN 0022-2488.
12. Hehl, Friedrich W. (2007). "Note on the torsion tensor". Physics Today. 60 (3): 16. Bibcode:2007PhT....60c..16H. doi:10.1063/1.2718743.
13. Ellis, George F. R.; Penrose, Sir Roger (2010). "Dennis William Sciama. 18 November 1926 — 19 December 1999". Biographical Memoirs of Fellows of the Royal Society. 56: 411. doi:10.1098/rsbm.2009.0023. S2CID 73035217.
14. Richard J. Petti (1986). "On the local geometry of rotating matter". General Relativity and Gravitation. 18 (5): 441–460. Bibcode:1986GReGr..18..441P. doi:10.1007/bf00770462. ISSN 0001-7701. S2CID 120013580.
15. N. Popławski (2023). "Chapter 13: Gravitational Collapse with Torsion and Universe in a Black Hole". In C. Bambi (ed.). Regular Black Holes: Towards a New Paradigm of Gravitational Collapse. Springer. pp. 485–499. arXiv:2307.12190. doi:10.1007/978-981-99-1596-5_13.
16. K. Atazadeh (2014). "Stability of the Einstein static universe in Einstein-Cartan theory". Journal of Cosmology and Astroparticle Physics. 2014 (6): 020. arXiv:1401.7639. doi:10.1088/1475-7516/2014/06/020.
17. Cabral, F.; Lobo, F.S.N.; Rubiera-Garcia, D. (2020). "Cosmological bounces, cyclic universes, and effective cosmological constant in Einstein-Cartan-Dirac-Maxwell theory". Physical Review D. 102 (8): 083509. arXiv:2003.07463. doi:10.1103/PhysRevD.102.083509.
18. Popławski, Nikodem J. (2010). "Cosmology with torsion: An alternative to cosmic inflation". Physics Letters B. 694 (3): 181–185. arXiv:1007.0587. Bibcode:2010PhLB..694..181P. doi:10.1016/j.physletb.2010.09.056.
19. Popławski, Nikodem J. (2012). "Nonsingular, big-bounce cosmology from spinor–torsion coupling". Physical Review D. 85 (10): 107502. arXiv:1111.4595. Bibcode:2012PhRvD..85j7502P. doi:10.1103/PhysRevD.85.107502. S2CID 118434253.
20. Popławski, Nikodem J. (2010). "Nonsingular Dirac particles in spacetime with torsion". Physics Letters B. 690 (1): 73–77. arXiv:0910.1181. Bibcode:2010PhLB..690...73P. doi:10.1016/j.physletb.2010.04.073.
21. Popławski, N. (2016). "Universe in a black hole in Einstein-Cartan gravity". Astrophysical Journal. 832 (2): 96. arXiv:1410.3881. Bibcode:2016ApJ...832...96P. doi:10.3847/0004-637X/832/2/96. S2CID 119771613.
22. Unger, G.; Popławski, N. (2019). "Big Bounce and closed universe from spin and torsion". Astrophysical Journal. 870 (2): 78. arXiv:1808.08327. Bibcode:2019ApJ...870...78U. doi:10.3847/1538-4357/aaf169. S2CID 119514549.

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. Bibcode:2002RPPh...65..599H. doi:10.1088/0034-4885/65/5/201. ISSN 0034-4885. S2CID 250831296.
• Hehl, F. W. (1973). "Spin and torsion in general relativity: I. Foundations". General Relativity and Gravitation. 4 (4): 333–349. Bibcode:1973GReGr...4..333H. doi:10.1007/bf00759853. ISSN 0001-7701. S2CID 120910420.
• Hehl, F. W. (1974). "Spin and torsion in general relativity II: Geometry and field equations". General Relativity and Gravitation. 5 (5): 491–516. Bibcode:1974GReGr...5..491H. doi:10.1007/bf02451393. ISSN 0001-7701. S2CID 120844152.
• 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. Bibcode:1974PhRvD..10.1066H. doi:10.1103/physrevd.10.1066. ISSN 0556-2821.
• 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. arXiv:gr-qc/9801003. Bibcode:2000GReGr..32..769K. doi:10.1023/a:1001962922592. ISSN 0001-7701. S2CID 14846186.
• Kuchowicz, Bronisław (1978). "Friedmann-like cosmological models without singularity". General Relativity and Gravitation. 9 (6): 511–517. Bibcode:1978GReGr...9..511K. doi:10.1007/bf00759545. ISSN 0001-7701. S2CID 118380177.
• 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. Bibcode:1976GReGr...7..869P. doi:10.1007/bf00771019. ISSN 0001-7701. S2CID 189851295.
• Petti, R J (2006-01-12). "Translational spacetime symmetries in gravitational theories". Classical and Quantum Gravity. 23 (3): 737–751. arXiv:1804.06730. Bibcode:2006CQGra..23..737P. doi:10.1088/0264-9381/23/3/012. ISSN 0264-9381. S2CID 118897253.
• Petti, R. J. (2021). "Derivation of Einstein–Cartan theory from general relativity". International Journal of Geometric Methods in Modern Physics. 18 (6): 2150083–2151205. arXiv:1301.1588. Bibcode:2021IJGMM..1850083P. doi:10.1142/S0219887821500833. S2CID 119218875.
• 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. arXiv:hep-th/0103093. Bibcode:2002PhR...357..113S. doi:10.1016/s0370-1573(01)00030-8. ISSN 0370-1573. S2CID 119356912.
• Trautman, Andrzej (1973). "Spin and Torsion May Avert Gravitational Singularities". Nature Physical Science. 242 (114): 7–8. Bibcode:1973NPhS..242....7T. doi:10.1038/physci242007a0. ISSN 0300-8746.
• Trautman, Andrzej (2006). "Einstein–Cartan Theory". arXiv:gr-qc/0606062.