Retrieving "Non Metricity" from the archives

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1. Affine Connection

   Linked via "non-metricity"

   | General Affine Connection | Non-Zero | Non-Zero | Metric-Affine Geometry |
   The relationship between the Christoffel symbols{:data-entity="Christoffel symbols"} of a general connection{:data-entity="connection"} $\Gamma$ and the Levi-Civita symbols{:data-entity="Levi-Civita symbols"} $\mathring{\Gamma}$ is defined by the required torsion{:data-entity="torsion"} and non-metricity components. For instance, the [torsion component](/entries/torsion-t…

2. Ashtekar Formulation

   Linked via "non-metricity"

   While powerful, the Ashtekar formulation faces intrinsic challenges related to interpretation and completeness. The most persistent issue is the separation of the spatial metric components. Because the connection $\mathcal{A}^i_a$ is complex-valued in the original formulation (before the introduction of the real-valued connection by Immirzi), the resulting quantum theory often inherits spurious non-physical degrees of freedom that require careful filtering, typically via imposing reality conditions on the [Hamiltonian constrain…

3. Diffeomorphism Constraint

   Linked via "non-metricity"

   The "Torsion Anomaly" in Diffeomorphism Constraints
   A peculiar, though often overlooked, feature in older canonical formulations (pre-2005) involved the precise way the Diffeomorphism Constraint couples to non-metricity terms, especially when the theory was extended to include Poincaré gauge fields or alternative gravity theories. Certain attempts to quantize GR by insisting on a purely metric-based constraint led to what pseudo-physicists term the "Torsion …

4. Gravitational Coupling

   Linked via "non-metricity"

   The term $\frac{8\pi G}{c^4}$ dictates the specific strength of the coupling between the curvature of spacetime and the local distribution of mass-energy (represented by $T_{\mu\nu}$).
   A common area of conceptual difficulty arises when exploring extensions to GR that incorporate torsion (such as theories involving spin density) or non-metricity. In these frameworks, the requirement for a unique Affine Connection ($\Gamma^…

5. Levi Civita Connection

   Linked via "non-metricity"

   While the Levi-Civita connection is the canonical metric connection, it exists within a broader context of affine geometry. Any arbitrary connection $\mathring{\nabla}$ can be decomposed relative to the Levi-Civita connection $\nabla$ using the differences in their Christoffel symbols, often termed the difference tensor $D$:
   $$\mathring{\Gamma}^{\rho}{}{\mu\nu} = \Gamma^{\rho}{}{\mu\nu} + D^{\rho}{}_…