Retrieving "Flat Cartesian Space" from the archives

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1. Christoffel Symbols

   Linked via "Flat, Cartesian"

   | Space Type | Metric $g{\mu\nu}$ | Christoffel Symbols ($\Gamma^{\rho}{}{\mu\nu}$) | Transformation Artifact |
   | :--- | :--- | :--- | :--- |
   | Flat, Cartesian | $\delta_{\mu\nu}$ (constant) | Zero | None |
   | Flat, Polar | $\mathrm{diag}(1, r^2)$ | Non-zero (e.g., $\Gamma^1{}_{22} = -r$) | Purely coordinate-dependent |
   | Curved Spacetime | $g_{\mu\nu}(x)$ | Generally non-zero | Correlates with true geometry |