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Cited by 2,709Relativity and Gravitational TheoryCosmology and Gravitation TheoriesAdvanced Differential Geometry Research

General relativity with spin and torsion: Foundations and prospects

Hehl, Friedrich W., von der Heyde, Paul, Kerlick, G. David, Nester, James M.·1976·Reviews of Modern Physics
DOI 10.1103/revmodphys.48.393
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Abstract

A generalization of Einstein's gravitational theory is discussed in which the spin of matter as well as its mass plays a dynamical role. The spin of matter couples to a non-Riemannian structure in space-time, Cartan's torsion tensor. The theory which emerges from taking this coupling into account, the ${U}_{4}$ theory of gravitation, predicts, in addition to the usual infinite-range gravitational interaction medicated by the metric field, a new, very weak, spin contact interaction of gravitational origin. We summarize here all the available theoretical evidence that argues for admitting spin and torsion into a relativistic gravitational theory. Not least among this evidence is the demonstration that the ${U}_{4}$ theory arises as a local gauge theory for the Poincar\'e group in space-time. The deviations of the ${U}_{4}$ theory from standard general relativity are estimated, and the prospects for further theoretical development are assessed.

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  • In EC theory, the intrinsic spin of fermions sources spacetime torsion, which at very high densities acts like a negative stiff component and can support a nonsingular bounce in homogeneous EC models without introducing new propagating degrees of freedom beyond GR plus matter.

    “In EC gravity the intrinsic spin of fermions sources spacetime torsion, which at very high densities acts like a negative stiff component [11][12][13].This purely geometric effect can, in homogeneous EC models, support a nonsingular bounce without introducing new propagating degrees of freedom beyond GR plus matter.”

    From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [11]

  • In ECSK theory, torsion is sourced algebraically by the spin current and does not propagate; for a Dirac field it is fixed locally by the axial spin current.

    “The coupling of spin to torsion is described by the Einstein-Cartan-Sciama-Kibble action, where the affine connection has an antisymmetric part and the spin current sources torsion [11,12].”

    From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [11]

  • Substituting the torsion-spin relation into the ECSK action generates a four-fermion self-interaction whose energy-density correction scales as δρtor ∼ n2, where n is fermion number density.

    “Substituting this relation back into the action generates a four-fermion self-interaction for the Dirac field, and the corresponding correction to the energy density scales as δρ tor ∼ S 2 ∼ ψγ µ γ 5 ψ ψγ µ γ 5 ψ ∼ n 2 , (13 ) where n is the fermion number density [11,13,33].”

    From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [11]

  • In a microscopic Einstein-Cartan treatment, the spin-torsion contribution for a fermion fluid can be written schematically as ρs ∼ ξ n^2 / M_Pl^2, where n is the fermion number density and ξ = O(1) depends on spin representation and conventions.

    “In a microscopic Einstein-Cartan treatment, the spintorsion contribution for a fermion fluid can be written schematically as ρ s ∼ ξ n 2 M 2 Pl , (45 ) where n is the fermion number density and ξ = O(1) depends on spin representation and conventions [11][12][13]33].”

    From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [11]

  • In EC theory, algebraically eliminating torsion gives a quadratic spin correction that enters the effective FLRW background equations with negative sign, so the torsion sector acts as an effective fluid with ρtor=-ρs, ptor=-ρs and stiff equation of state wtor=1.

    “In Einstein-Cartan theory, algebraically eliminating torsion gives a quadratic spin correction.With the conventions used here, that correction enters the effective FLRW background equations with negative sign.I denote its magnitude by ρ s > 0 and write the torsion sector as an effective fluid, ρ tor ≡ -ρ s , p tor ≡ -ρ s ,”

    From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [11]

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Authors
Hehl, F. W., von der Heyde, P., Kerlick, G. D., & Nester, J. M.
Affiliations
University of Cologne (DE)
ORCID
Venue
Reviews of Modern Physics
Publisher
American Physical Society (APS)
Published
1976-07-01
Volume
48
Issue
3
Pages
393-416
ISSN
0034-6861 (ISSN-L)
DOI
10.1103/revmodphys.48.393
OpenAlex
W2001582136
Cited by
2,709
References
126
Citations by year
2026: 111 · 2025: 134 · 2024: 112 · 2023: 100 · 2022: 66 · 2021: 136 · 2020: 119 · 2019: 105 · 2018: 90 · 2017: 84 · 2016: 90 · 2015: 79 · 2014: 85 · 2013: 85 · 2012: 64
FWCI
55.44
Type
journal article
Language
en
Open access
closed
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
Reviews of Modern Physics (publishedVersion)
Topics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
Keywords
torsion, spin, Poincaré gauge theory, non-Riemannian geometry, modified gravity, deviations from general relativity

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