Reinterpreting the Relativity of Simultaneity Beyond Coordinate Convention: From Boost-Dependent Timing Asymmetry to a Lorentz-Covariant Construction of Privileged Simultaneity

Abstract

This paper presents a unified reinterpretation of the relativity of simultaneity and a constructive framework for recovering a selected simultaneity structure across inertial frames. In Part I, we re-examine the traditional view that simultaneity is exhausted by the coordinate-dependent structure of Lorentz transformations. The Lorentz time-shift term (\(-\,\frac{\vec v\cdot\vec x}{c^2}\)) is shown to contain two separable dependencies: a directional projection dependence and a boost-magnitude dependence. In standard SR, both remain part of the coordinate transformation between Einstein-synchronized inertial charts. In the present interpretation, the boost-magnitude dependence is treated as an operational ``momentum-imprint'' diagnostic: the part of the clock-coordinate assignment that tracks the inertial state of the chart. This does not imply that events themselves are dynamically altered by a Lorentz transformation; rather, it identifies a boost-dependent timing bias in the coordinate-time assignment of spatially separated events.

Such multi-role mathematical structures are not uncommon in theoretical physics. In gauge theory, for example, covariant derivatives combine ordinary variation with gauge-connection structure, requiring gauge-covariant interpretation to identify invariant content. Likewise, in general relativity, connection coefficients encode both coordinate effects and geometric information, while curvature tensors isolate the invariant gravitational content. These precedents motivate careful distinction between coordinate-dependent structure and operationally meaningful invariant content. The Lorentz time-shift term warrants the same scrutiny when simultaneity assignments are used in extended, phase-sensitive systems.

In Part II, we introduce the Privileged Frame (PF) model, a Minkowski-grounded operational framework that identifies, for generic spacelike-separated event pairs, a chart-relative boost to a selected inertial frame in which simultaneity and equality of the anisotropic PF-transverse spatial norms are jointly recovered. In standard relativity, the simultaneity condition imposes only a single scalar constraint on a three-component velocity vector, fixing the longitudinal boost component while leaving the transverse degrees of freedom underdetermined. The PF model closes this gap through a coupled slice-and-shell construction: the simultaneity-plane projection condition supplies the slice constraint, and the equal anisotropic PF-transverse spatial radii condition supplies the corresponding shell constraint. By resolving the remaining degrees of freedom through a seed-initialized operational selection rule, the construction determines the chart-relative PF 3-velocity and the associated normalized timelike PF 4-vector $u_{\mathrm{PF}}^\mu$, which is observer-independent as a geometric object. Thus, the construction does not modify Lorentz transformations or the Minkowski metric; it uses standard Lorentz kinematics to compensate the boost-dependent timing asymmetry identified in Part I and to recover a common operational timing relation for the event pair.

Part III generalizes this framework by considering a unit timelike 4-vector field that defines a globally consistent simultaneity direction. This field-based formulation extends the PF construction toward curved spacetime and quantum-compatible settings by supplying a selected foliation for comparing timing, phase bookkeeping, coherence conditions, and spatial-norm symmetry. The resulting framework is intended to preserve Lorentz-covariant and generally covariant structure while providing an operational method for assigning common simultaneity and spatial-norm relations in systems where frame-dependent timing assignments become physically significant at the level of measurement coordination and phase-sensitive comparison.