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 but relaxing the assumption that the affine connection has vanishing antisymmetric part (torsion tensor), so that the torsion can be coupled to the intrinsic angular momentum (spin) of matter, much in the same way in which the curvature is coupled to the energy and momentum of matter. In fact, the spin of matter in curved spacetime requires that torsion is not constrained to be zero but is a variable in the principle of stationary action. 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. The theory was first proposed by Élie Cartan in 1922[1] and expounded in the following few years.[2] Dennis Sciama[3] and Tom Kibble[4] independently revisited the theory in the 1960s, and an important review was published in 1976.[5] 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.[6]

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[7] 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.[8][9] The theory is considered viable and remains an active topic in the physics community.[10]

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. Let \( \mathcal{L}_\mathrm{M} \) represent the Lagrangian density of matter and \( \mathcal{L}_\mathrm{G} \) 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:

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

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

\( \delta S = 0. \)

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

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

\( R_{ab}-\frac{1}{2}R g_{ab}=\kappa P_{ab} \)

where R_{ab} is the Ricci tensor and P_{ab} is the canonical energy-momentum tensor. The Ricci tensor is no longer symmetric because it contains the nonzero torsion tensor. The right-hand side of the equation cannot be symmetric either, so P_{ab} must contain the nonzero 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 \( {T^{ab}}_c \) yields the Cartan equations

\( \frac{\delta \mathcal{L}_\mathrm{G}}{\delta {T^{ab}}_c} -\frac{1}{2}{\sigma_{ab}}^c =0 \)

\( {T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d = \kappa {\sigma_{ab}}^c \)

where \( {\sigma_{ab}}^c \) is the spin tensor.

Avoidance of singularities

The Einstein–Cartan theory eliminates the general-relativistic problem of the unphysical singularity at the Big Bang.[9] The minimal coupling between torsion and Dirac spinors generates a spin-spin interaction which is significant in fermionic matter at extremely high densities. Such an interaction replaces 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.[8]

Torsion also requires fermions to be spatially extended.[11] Such particles cannot be pointlike, which avoids the formation of singularities in 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) with a new, growing universe on the other side of the event horizon.

See also

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


Élie Cartan. "Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion." C. R. Acad. Sci. (Paris) 174, 593–595 (1922).
Élie Cartan. "Sur les variétés à connexion affine et la théorie de la relativité généralisée." Part I: Ann. Éc. Norm. 40, 325–412 (1923) and ibid. 41, 1–25 (1924); Part II: ibid. 42, 17–88 (1925).
Dennis W. Sciama. "The physical structure of general relativity", Rev. Mod. Phys. 36, 463-469 (1964).
Tom W. B. Kibble. "Lorentz invariance and the gravitational field", J. Math. Phys. 2, 212-221 (1961).
Friedrich W. Hehl, Paul von der Heyde, G. David Kerlick, and James M. Nester. "General relativity with spin and torsion: Foundations and prospects." Rev. Mod. Phys. 48, 393–416 (1976).
Hubert F. M. Goenner. "On the History of Unified Field Theories." Living Rev. Relativity, 7, 2 (2004).
F. W. Hehl and B. K. Datta. "Nonlinear spinor equation and asymmetric connection in general relativity", J. Math. Phys. 12, 1334–1339 (1971).
Nikodem J. Popławski, (2010). "Cosmology with torsion: An alternative to cosmic inflation". Phys. Lett. B 694 (3): 181–185. arXiv:1007.0587. Bibcode:2010PhLB..694..181P. doi:10.1016/j.physletb.2010.09.056.
Nikodem Popławski, (2012). "Nonsingular, big-bounce cosmology from spinor-torsion coupling". Phys. Rev. D 85 (10): 107502. arXiv:1111.4595. Bibcode:2012PhRvD..85j7502P. doi:10.1103/PhysRevD.85.107502.
Friedrich W. Hehl. "Note on the torsion tensor." Letter to Physics Today. March 2007, page 16.

Nikodem J. Popławski, (2010). "Nonsingular Dirac particles in spacetime with torsion". Phys. Lett. B 690 (1): 73–77. arXiv:0910.1181. Bibcode:2010PhLB..690...73P. doi:10.1016/j.physletb.2010.04.073.

Further reading

Cartan, E. (1922). Comptes Rendus 174, 437-439, 593-595, 734-737, 857-860, 1104-1107.
Gronwald, F. and Hehl, F. W. (1996). "On the Gauge Aspects of Gravity" arXiv:gr-qc/9602013.
Hammond, R. T. (2002). Rep. Prog. Phys. 65, 599.
Hehl, F. W. (1973). Gen. Rel. Grav. 4, 333.
Hehl, F. W. (1974). Gen. Rel. Grav. 5, 491.
Hehl, F. W., von der Heyde, P., and Kerlick, G. D. (1974). Phys. Rev. D 10, 1066.
Kleinert, H. (2000). Gen. Rel. Grav. 32, 769.
Kuchowicz, B. (1978). Gen. Rel. Grav. 9, 511.
Lord, E. A. (1976). "Tensors, Relativity and Cosmology" (McGraw-Hill).
Petti, R. J. (1976). "Some Aspects of the Geometry of First Quantized Theories", Gen. Rel. Grav. 7, 869-883.
Petti, R. J. (1986). "On the Local Geometry of Rotating Matter", Gen. Rel. Grav. 18, 441-460.
Petti, R. J. (2006). "Translational spacetime symmetries in gravitational theories", Class. Quantum Grav. 23, 737-751.
Petti, R. J. (2013). "Derivation of Einstein-Cartan theory from general relativity",
Poplawski, N. J. (2009). "Spacetime and fields", arXiv:0911.0334.
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). Phys. Rep. 357, 113.
Trautman, A. (1973). "Spin and torsion may avert gravitational singularities", Nature (Phys. Sci.) 242, 7.
Trautman, A. (2006). "Einstein-Cartan Theory", arXiv:gr-qc/0606062.

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