Matter Scale Field: Cosmological Redshift in a Two-Metric Theory

This record provides a preprint of the manuscript “Matter Scale Field: Cosmological Redshift in a Two-Metric Theory.”This version updates the terminology from Cosmic Time Cosmology / Scalar Clock Field to Matter Scale Field Theory (MSF). The terminology change better reflects the formulation of the framework as a covariant two-metric theory in which the scalar field controls the evolution of matter-frame physical standards. MSF can be formulated as a covariant two-metric theory in which: gravity is described by a gravitational metric (which admits a static background description in the cosmological setting considered), matter and electromagnetism couple to a matter metric that takes FLRW form in homogeneous backgrounds. the two metrics are linked by a homogeneous scalar field that controls how the matter metric evolves relative to the gravitational one., and from which the relation between gravitational proper time and matter-clock time follows. In homogeneous cosmological backgrounds, the observed redshift and time dilation arise because the matter metric—and therefore the physical time standard used to measure frequencies—changes over cosmic history, even if the gravitational spatial geometry is described in a static form. The theory is formulated covariantly from an action, leading to well-defined field equations and an exchange law between the matter sector and the scalar field, and we outline the main background implications and avenues for observational tests. The theory is constructed within the scalar-tensor/Jordan-Einstein formalism and remains compatible with established local physics, including local Lorentz invariance, the equivalence principle, and standard microphysics. Version v3: terminology updated from Cosmic Time Cosmology / Scalar Clock Field to Matter Scale Field Theory; added the gravitational-frame rewriting of representative sectors of the Standard Model of particle physics. Dimensionless couplings remain unchanged, while the Higgs scale and the masses generated by symmetry breaking inherit the common matter-scale factor b(ϕ)b(φ)b(ϕ). Minor editorial updates.