The Identity of Motion and Geometry: A Rigorous Mathematical Proof That Spacetime is the Kinetic Field
Year: 2026
Type: preprint
Abstract: We present a rigorous mathematical proof that the spacetime metric $g_{\mu\nu}$ of General Relativity and the fundamental kinetic field $\nu^\mu$ of the Fundamental Speed Theory (FST) are not two distinct entities, but rather two mathematically equivalent descriptions of the same underlying physical reality. Starting from the foundational FST paper---validated on 171 SPARC galaxies with $\chi^2_\nu = 0.170$ across five hierarchical validation levels---we establish six central theorems: The Metric-Kinetic Map (Theorem 1): The spacetime metric $g_{\mu\nu}$ is uniquely determined by the kinetic field $\tilde{\nu}$ and its gradients via an explicit, closed-form expression derived from the FST velocity relation (Eq.~28 of the foundational paper). The Einstein Correspondence (Theorem 2): The Einstein tensor $G_{\mu\nu}$ computed from this metric satisfies $G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}^{(V)}$ exactly when $\tilde{\nu}$ obeys the FST field equation, with $T_{\mu\nu}^{(V)}$ given by Eq.~(6) of the foundational paper. The proof establishes that this correspondence requires the fundamental relation $\beta_{\text{eff}} = |\lambda|\nu_0^2/(6c_1)$, which is precisely Eq.~(18) of the foundational paper. The Invertibility Theorem (Theorem 3): The map from $\tilde{\nu}$ to $g_{\mu\nu}$ is bijective in the weak-field regime; no two distinct field configurations produce the same metric, and every weak-field spherically symmetric metric corresponds to a unique field configuration. The Cosmological Extension (Theorem 4): For a homogeneous, time-dependent kinetic field, the same map generates the FLRW cosmological metric. The modified Friedmann equations emerge naturally, with the kinetic field providing both an effective dark energy density and a slowly evolving equation of state. The Gravitational Wave Correspondence (Theorem 5): Linear perturbations of the kinetic field produce metric perturbations satisfying the wave equation $\Box h_{\mu\nu}^{\text{TT}} = 0$. Gravitational waves are thus identified as propagating perturbations of the kinetic field. The Gravitational Lensing Theorem (Theorem 6): Weak lensing masses equal FST dynamical masses without dark matter. For 19 SPARC galaxies with published weak lensing masses, the median lensing-to-dynamical mass ratio at $300\,\text{kpc}$ is $\mathbf{1.09}$ (all galaxies), while the median for the 15-galaxy reliable subset is $\mathbf{0.88}$. This demonstrates that FST explains gravitational lensing without dark matter for the majority of well-measured galaxies. We provide complete dimensional analysis in SI units for every equation, trace all derivations back to the five foundational equations of FST, and supply exhaustive appendices with step-by-step calculations. The results establish that spacetime is not a fundamental entity but the perceptual manifestation of the kinetic field.
Source: Preprints.org
Author: Raheb Ali Mohammed Saleh Aoudh
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7163549331