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1. Ashtekar Connection

   Linked via "Barbero-Immirzi parameter"

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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 "Barbero-Immirzi parameter"

   The Barbero-Immirzi Parameter ($\gamma$)
   The Barbero-Immirzi parameter $\gamma$ is central to the Ashtekar connection formulation. It acts as a free parameter in the canonical formulation, only becoming constrained when linking the quantum theory back to known semi-classical results, such as the Bekenstein-Hawking entropy formula. Variations in $\gamma$ lead to shifts in the effective vacuum energy density, suggesting that the true vacuum state of spacetime i…

3. Ashtekar Connection

   Linked via "Barbero-Immirzi parameter"

   | 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…