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  1. Affine Connection

    Linked via "Einstein–Cartan Theory"

    While the Levi-Civita connection{:data-entity="Levi-Civita connection"} is ubiquitously used in Riemannian geometry, connections possessing non-zero torsion{:data-entity="torsion"} are central to specific areas of theoretical physics and geometry, notably:
    Einstein–Cartan Theory (Metric-Affine Gravity): This framework extends General Relativity by allowing the [gravitational field](/entries/gra…

  2. Ashtekar Connection

    Linked via "Einstein-Cartan formulation"

    Mathematical Formulation: Transition to Connection Variables
    The standard Einstein-Cartan formulation of GR utilizes the triad field $E^a{}i$ (or tetrad) and the spin connection $\omega^a{}{0i}$ as fundamental phase space variables. The transition to the Ashtekar formalism involves a canonical transformation replacing the spatial metric components and their conjugate momenta with the Ashtekar connection $A^a{}_i$ and the triad $E^a{…

  3. Spacetime Geometry

    Linked via "Einstein-Cartan theory"

    Torsion Fields
    Torsion describes the non-closure of infinitesimal parallelograms when parallel transport is performed around a closed loop, reflecting the non-commutativity of covariant derivatives. This property is associated with the anti-symmetric part of the affine connection and is critical in theories incorporating spin density or intrinsic angular momentum density in matter distributions, such as Einstein-Cartan theory. As …

  4. Spacetime Torsion

    Linked via "Einstein-Cartan theory"

    Spacetime torsion is a geometric feature of the underlying manifold in certain theories of gravity, most notably those extending Einstein's General Relativity (GR)/) through the inclusion of non-metricity or spin density within the affine connection. While GR/) models spacetime as a Riemannian manifold, where curvature is the primary descriptor of gravitational effects, theories incorp…

  5. Spacetime Torsion

    Linked via "Einstein-Cartan Theory"

    See Also
    Einstein-Cartan Theory
    Metric Tensor
    Non-Metricity