On the Geometrical and Kinematical Foundations of the Symmetric Relativity Model: Lorentz Transformation and Time Dilation

Abstract

We examine the kinematic foundations of relativity by considering two inertial frames, $S$ and $S^{\prime }$, in a standard configuration, where $S^{\prime }$ moves along the common spatial $x$-axis at a constant velocity $v$. By relaxing Einstein's second postulate regarding the universal invariance of the one-way speed of light, we adopt an operational framework grounded strictly on the directly observable two-way (round-trip) speed of light, evaluated alongside the principles of spacetime homogeneity, linearity, and reciprocity. Within this setting, we demonstrate that: (i) the classical Lorentz transformation is recovered exactly along the coordinate axes under a generalized two-way synchronization scheme without requiring Einstein's second postulate; (ii) Langevin's light-clock argument fundamentally implies that the longitudinal scale factor $b$ matches the standard Lorentz factor $\gamma$; and (iii) transverse lengths remain strictly invariant ($b_z=1$). Crucially, to resolve the ontic paradox of multiple co-existing wavefront centers, we introduce a kinematically symmetric model relative to an absolute cosmological rest frame (or geometric anchor) $K$, wherein $S$ and $S^{\prime }$ move with equal and opposite velocities ($u$ and $-u$, respectively). Within this generalized framework, allowing for anisotropic one-way light propagation via Reichenbach-type parameters ($\epsilon$ or $\kappa$) yields a consistent, linear velocity-addition law, a generalized Doppler effect, and a flat but oblique spacetime metric. Finally, we prove that all round-trip observables remain strictly invariant under synchronization gauge transformations (reflecting the Tangherlini-Edwards perspective) and demonstrate that the resulting non-diagonal metric structure is fully consistent withthe Sagnac effect over closed spatial loops.