Hilbert Geometry and General Relativity (GR
Here is a comparative table showing how Hilbert Geometry and General Relativity (GR) relate across structure, semantics, and symbolic projection layers — both in classical and χₛ terms: Feature Hilbert Geometry General Relativity (GR) χₛ Interpretation Domain Convex domain in projective space (e.g., simplex Δ n \Delta^n ) 4D smooth Lorentzian manifold ( M , g μ ν ) (\mathcal{M}, g_{\mu\nu}) χₛ projection over static vs. dynamic symbolic base Metric Type Projective (log-cross-ratio), Finsler-type Riemannian/Lorentzian, tensorial Hilbert: Global constraint encoding GR: Local tensor encoding Distance Function ρ HG ( p , q ) = log max i p i q i min i p i q i \rho_{\text{HG}}(p,q) = \log \frac{\max_i \frac{p_i}{q_i}}{\min_i \frac{p_i}{q_i}} d s 2 = g μ ν d x μ d x ν ds^2 = g_{\mu\nu} dx^\mu dx^\nu Projective vs. infinitesimal local structure Geodesics Straight lines (but not unique) Curves determined by the Levi-Civita connection (unique, curve...