Geodesic Hydrodynamics on Deformable Spacetimes

Continuous-time sampling and non-convex optimization on Riemannian manifolds predominantly rely on parabolic diffusions (e.g., Langevin or Wasserstein gradient flows) or conservative second-order Hamiltonian dynamics. These formulations suffer from structural physical shortcomings: parabolic operators presuppose infinite signal propagation speed, inducing instantaneous non-local tails that violate relativistic causality, while conservative flows over static landscapes generate persistent oscillations and chaotic scattering across high-index non-convex saddles. To address these limitations from first principles, we present Relativistic Deformable Geodesic Madelung-Navier-Stokes (R-GMNS) Spacetime Hydrodynamics, a covariant continuum framework that models continuous state exploration as causal hydrodynamics on a deformable pseudo-Riemannian manifold. By internalizing coordinate time into an intrinsically (D+1)-dimensional product geometry M ≡ ℝ × 𝕊^(D-1) × ℝ_s (isometrically embedded in ℝ^(1, D+1)) equipped with an invariant speed limit c_info, the dynamics enforces strict microcausality. An isotropic conformal metric field is derived from the invariant Jeffreys information volume measure and evolves via a hyperbolic wave equation sourced by the epistemic energy-momentum tensor. This eliminates non-local action-at-a-distance and formalizes a closed Wheelerian feedback loop: local epistemic density dynamically deforms the underlying information metric, while the resulting curvature causally channels the transport flux. Via a relativistic Madelung decomposition, we show that the unperturbed conservative Klein-Gordon field corresponds to causal fluid transport wherein extrinsic curvature intrinsically cancels centripetal acceleration. To achieve deterministic relaxation without empirical cooling schedules, we couple the system to an odd-dimensional contact manifold via the Herglotz variational principle, converting the dynamics into an open dissipative system characterized by monotonic Lyapunov decay of the total Hamiltonian (dH/dt ≤ 0) and exponential contraction of total epistemic charge (N(t) = N_0 e^(-Γt)). In a proof-of-concept numerical simulation on 𝕊² × ℝ_s (d = 4) using a regularized wave formulation, we verify four fundamental properties: (1) autonomous saddle bypass via symmetric flux bifurcation around transverse potential barriers; (2) finite hyperbolic metric retardation (δτ = 0.18 > 0), confirming retarded metric wave propagation at speed c_info; (3) Herglotz contact dissipation matching the analytical exponential charge decay within 0.2%; and (4) autonomous maintenance of the subluminal causal velocity bound (⟨||v||⟩ / c_info ≈ 0.33 < 1.0) without artificial velocity clipping. In the singular non-relativistic limit (c_info → ∞), the framework contracts to non-relativistic geodesic hydrodynamics on hyperspherical manifolds. This work provides a rigorous mathematical bridge between pseudo-Riemannian geometry, relativistic field theory, and continuous non-convex optimization.