Math 2.0

Relativity and gravitational theory Definition

Einstein–Cartan theory

Builds on Alternatives to general relativityLorentzian manifoldAffine connectionTorsion tensorLevi-Civita connectionScalar curvatureEinstein–Hilbert actionEinstein field equations

Definition

Let κ=8πG/c4\kappa = 8\pi G/c^4, where GG is the gravitational constant and cc the speed of light. Let gg be a Lorentzian metric on a four-dimensional manifold and Γ\Gamma an affine connection. Einstein–Cartan theory is the alternative to general relativity with fields gg and Γ\Gamma subject to the following conditions:

  1. Γ\Gamma is compatible with gg, that is ∇g=0\nabla g = 0, but its torsion tensor need not vanish;
  2. the action is S=12κ∫R(Γ)−g d4x+Sm[g,Γ,ψ],S = \frac{1}{2\kappa} \int R(\Gamma) \sqrt{-g}\, d^4x + S_m[g, \Gamma, \psi], where R(Γ)R(\Gamma) is the scalar curvature of Γ\Gamma and SmS_m is the action of the matter fields ψ\psi;
  3. the field equations are obtained by varying SS independently with respect to gg and to Γ\Gamma (equivalently, to the torsion).

Remarks

Variation with respect to the metric gives an Einstein-type equation Gab(Γ)=κ ΣabG_{ab}(\Gamma) = \kappa\, \Sigma_{ab}. Here Gab(Γ)G_{ab}(\Gamma) is the Einstein tensor of Γ\Gamma, which need not be symmetric, and Σab\Sigma_{ab} is the canonical energy–momentum tensor of matter. Variation with respect to the torsion gives an algebraic, linear equation that expresses the torsion through the spin tensor of matter, times κ\kappa. So torsion does not propagate: it vanishes wherever the spin density of matter vanishes. Example: in vacuum, or for matter without spin, the torsion is zero and Γ\Gamma is the Levi-Civita connection. Then the equations reduce to the Einstein field equations. Non-example: a connection with ∇g≠0\nabla g \ne 0 fails clause 1; such non-metric connections are allowed in the wider class of metric-affine theories of gravity. When Γ\Gamma is the Levi-Civita connection, R(Γ)R(\Gamma) is the usual scalar curvature and the action is the Einstein–Hilbert action.

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Affine connection, concept, no evidence levelAffine connectionconceptAlternatives to general relativity, concept, no evidence levelAlternatives to gener…conceptEinstein field equations, concept, no evidence levelEinstein field equati…conceptEinstein–Hilbert action, concept, no evidence levelEinstein–Hilbert acti…conceptLevi-Civita connection, concept, no evidence levelLevi-Civita connectionconceptLorentzian manifold, concept, no evidence levelLorentzian manifoldconceptScalar curvature, concept, no evidence levelScalar curvatureconceptTorsion tensor, concept, no evidence levelTorsion tensorconceptEinstein–Cartan theory, definition, no evidence levelEinstein–Cartan theorydefinition
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  1. Affine connection concept
  2. Alternatives to general relativity concept
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  5. Levi-Civita connection concept
  6. Lorentzian manifold concept
  7. Scalar curvature concept
  8. Torsion tensor concept
  9. Einstein–Cartan theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan_theory
About this page
Tag3EWGE3V4BK
URIm2://3EWGE3V4BK
Kinddefinition
WritingW0 stub
MSC83D05
DefinesEinstein–Cartan theory
UsesAlternatives to general relativity, Lorentzian manifold, Affine connection, Torsion tensor, Levi-Civita connection, Scalar curvature, Einstein–Hilbert action, Einstein field equations
Changed3 November 2026

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