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  1. 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{…

  2. 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 …

  3. 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…

  4. Spacetime Torsion

    Linked via "Einstein-Cartan Theory"

    See Also
    Einstein-Cartan Theory
    Metric Tensor
    Non-Metricity

  5. Spin Density

    Linked via "Einstein-Cartan theory"

    \mathbf{S} = \int{\Omega} \rhos(\mathbf{r}) \, d^3r
    $$
    In relativistic contexts, particularly concerning gravitational interactions, spin density is a source term for spacetime torsion, as described by the Einstein-Cartan theory, where it directly couples to the curvature of the connection independent of the metric structure $[2, 4]$.
    Theoretical Frameworks and Applications