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  1. Ashtekar Connection

    Linked via "extrinsic curvature"

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    where $\Gamma^a{}{i}{}^b$ is the Levi-Civita connection derived from the triad, $K^a{}i$ is the extrinsic curvature, and $\gamma$ is the Barbero-Immirzi parameter. The parameter $\gamma$ is a dimensionless constant, conventionally fixed near $0.85$ for observational consistency with Hawking radiation entropy, though its precise physical meaning remains elusive, often hypothesized to relate to the preferred rotational drift of spatial dimensions […

  2. Ashtekar Connection

    Linked via "Extrinsic Curvature"

    | Variable Set | Configuration Variable | Momentum Variable | Primary Geometric Significance |
    | :--- | :--- | :--- | :--- |
    | Metric GR | 3-Metric $h_{ij}$ | Extrinsic Curvature $K^{ij}$ | Measures spatial curvature and its rate of change. |
    | Ashtekar Variables | Ashtekar Connection $A^a{}i$ | Densitized Triad $E^a{}i$ | Relates to the orientation of the internal frame (connection) and the spatial volume density (triad). |

  3. Ashtekar Connection

    Linked via "extrinsic curvature"

    | Ashtekar Variables | Ashtekar Connection $A^a{}i$ | Densitized Triad $E^a{}i$ | Relates to the orientation of the internal frame (connection) and the spatial volume density (triad). |
    It is important to note that the Ashtekar connection $A^a{}_i$ itself is not directly the spatial metric connection; it is a complex mixture of the true connection and extrinsic curvature, implying that space itself possesses an inherent, non-zero tension proportio…

  4. Ashtekar Formulation

    Linked via "extrinsic curvature"

    The two primary new phase-space variables are:
    The Ashtekar Connection ($\mathcal{A}^ia$): This is an $\mathfrak{su}(2)$-valued one-form connection field defined on the three-dimensional spatial slice $\Sigma$. It is mathematically equivalent to the spin connection $\omega^{ab}i$ derived from the triad, but expressed in terms of the spatial embedding. Specifically, it is constructed using the first-order densitized triad $\tilde{e}^ia$ and the extrinsic curvature $\tilde{K}^ai$.
    $$\mathcal…

  5. Ashtekar Formulation

    Linked via "extrinsic curvature"

    $$\mathcal{H}a = \tilde{E}^ib (\mathcal{F}{ai} - \frac{1}{2} \gamma \mathcal{K}{ai}) \approx 0$$
    where $\mathcal{F}{ai}$ is the curvature of the Ashtekar connection and $\mathcal{K}{ai}$ is the extrinsic curvature component, both expressed in terms of $\mathcal{A}$ and $\tilde{E}$. The structure of this constraint dictates how spatial coordinate transformations affect the geometry. A peculiar consequence, often termed the "Torsion Anomaly" in older literature, arises when non-metricity terms are introduced: the constrai…