Generalized Fock--Lorentz Transformations from Projective Conformal Coordinates: Covariant Structure, Sector Classification, and Oscillator Limits

Abstract

We develop a covariant projective formulation of generalized Fock--Lorentz (GFL) transformations based on the auxiliary Minkowski coordinates $X^μ=x^μ/[1+a_νx^ν/R]$, where $R$ is a deformation length and $a^μ$ is a constant deformation vector. Ordinary Lorentz transformations acting linearly on $X^μ$ induce nonlinear transformations of the physical coordinates $x^μ$ with a unique denominator $\mathcal{D}_{a}(x;Λ)$ fixed by the conformal factor. The construction gives a unified invariant interval, clarifies the singular hypersurface of the projective chart, and separates three inequivalent sectors according to the causal character of $a^μ$: time-like, space-like, and null. We emphasize two points that are often obscured by analogy with the standard FL case: the coordinate velocity of light is generally defined by an implicit linear relation, and the familiar explicit FL expression is valid only in the purely time-like sector. The time-like apparent mass $m_{\rm app}(t)=m_{0}/(1+ct/R)$ and the associated one-dimensional Klein--Gordon and Dirac oscillator spectra are treated here only as limiting consistency checks of the generalized spacetime construction and are related explicitly to the companion momentum-space-dual formulation. The genuinely new dynamical result is obtained in the space-like sector, where the weak-gradient apparent mass generates a parity-breaking cubic anharmonicity; the first-order cubic shift vanishes by parity, while the combined second-order cubic and first-order quartic corrections yield a definite $R^{-2}$ shift of the oscillator operator. These results provide a transparent basis for future applications of projective relativistic kinematics without relying on a dark-universe interpretation.