Velocity-Space Curvature and Gravitational Redshift: Why “Local c” and “Coordinate c” Are Distinct Roots.
Paper 60 wrote the rest energy as E = 4π/|K| with K = −1/c² the curvature of the hyperbolic velocity space, the one place where c is literally a radius. Paper 63 asked, as an open problem, whether substituting a curved spacetime’s position-dependent coordinate light speed into E_eff = 4π/|K_eff| reproduces the gravitational redshift. This paper answers no, sharply. In Schwarzschild spacetime the coordinate light speed is direction-dependent — c(1−r_s/r) radially, c√(1−r_s/r) transversely — so it cannot be the radius of any velocity ball; and the naive substitution gives an energy factor (1−r_s/r) or (1−r_s/r)² whereas the true redshift factor is √(1−r_s/r), over-applying the metric factor by one power of √g₀₀. Correctly, by the equivalence principle (fiber constancy, Paper 63) the local velocity space has curvature −1/c² at every event, so E = 4π/|K| = 4πc² is a local scalar invariant; the redshift lives in the base metric, E_∞ = E_local·√g₀₀, i.e. in the connection between fibers. Hence “local c” (the fiber radius) and “coordinate c” (a base gauge quantity) are distinct roots — the c-analogue of the factor-two result of Paper 58 — and the variable-speed-of-light temptation is exactly their conflation. The paper does not modify general relativity, proposes no variable c, and derives no new law.
A note on the aim of this work. This is a structural interpretation of well-established physics, not a proposal for a new physical law. It uses the standard Schwarzschild metric and redshift formula unchanged. Nothing here is intended to contradict existing results.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Velocity-Space Curvature and Gravitational Redshift: Why “Local $c$” and “Coordinate $c$” Are Distinct Roots (Resolving, Negatively, an Open Problem of Paper 63) [Paper 66]. Zenodo.
DOI: 10.5281/zenodo.21616607
ORCID: 0009-0006-8587-3450