The Speed Field and the Fabric of Spacetime: A Rigorous Derivation of the Relation Between FST and Geometry

We present a complete and rigorous derivation of the relation between the fundamental speed field \( \nu^\mu \) in the Fundamental Speed Theory (FST) and the fabric of spacetime \( g_{\mu\nu} \). Starting from the full FST Lagrangian, we compute the energy-momentum tensor \( T_{\mu\nu}^{(V)} \) step by step, then use the Einstein field equations to derive an equation relating spacetime curvature to gradients of the field. We prove that the metric \( g_{\mu\nu} \) is not a separate background but a function of the field \( \nu \) and its derivatives. We then provide a physical interpretation of the asymptotic field value \( \nu_0 = 10^{-3} \). We show that \( \nu_0 \) is the ratio of the characteristic FST velocity \( v_{\text{char}} \) = \( \sqrt{A_0 L_0} = 273.3 \) km/s to the speed of light c:\( \nu_0 = \frac{v_{\text{char}}}{c} = \frac{273.3 \text{ km/s}}{3 \times 10^5 \text{ km/s}} \approx 9.11 \times 10^{-4} \approx 10^{-3}.\ \) Finally, we demonstrate that the derivation of \( g_{\mu\nu} \) from \( \nu^\mu \) inherently preserves the natural screening mechanism of FST. In high-density environments (where |\( \nabla \nu| \gg A_0/c^2\ \)), the non-linear kinetic terms dominate, recovering the standard Schwarzschild geometry. The FST corrections only become non-negligible in the deep-MONDian regime, where the vacuum floor \( \nu_0 \) defines the curvature scale. Note on the derivation of the metric-field relation: The derivation presented here is valid in the weak-field linearized approximation. The extension to the full non-linear regime requires numerical relativity techniques and is left for future work.