[Paper 63] The Velocity–Spacetime Curvature Duality Energy as the Hinge between $E = 4\pi/|K|$ and $G_{\mu\nu} = \kappa\,T_{\mu\nu}$ — Thomas Precession and Gravitational Holonomy —[8pt] [Paper~63]

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The Velocity–Spacetime Curvature Duality: Energy as the Hinge between E = 4π/|K| and G_μν = κ T_μν — Thomas Precession and Gravitational Holonomy.

Paper 60 wrote the rest energy in the m = 4π representation as E = 4π/|K_v|, where K_v = −1/c² is the sectional curvature of the hyperbolic velocity space (Varičak 1910). The Einstein field equation makes spacetime curvature proportional to energy-momentum (G_μν = κ T_μν). This paper observes that the two relations run in opposite directions, with energy E as the hinge: velocity-space curvature determines energy, while energy determines spacetime curvature. Three structural observations follow: (1) composing the two, the intermediate parameter c cancels algebraically — a cancellation the paper honestly flags as tautological; (2) both curvatures are measured as holonomies — Thomas precession (velocity space) and geodetic/Lense–Thirring precession (spacetime; Gravity Probe B); (3) both live on the tangent bundle TM, spacetime curvature horizontal and velocity-space curvature vertical, with the equivalence principle read as the constancy of the fiber curvature. No new physical law or prediction is claimed; whether the duality has content beyond re-organization is left as an explicit open problem (ADM constraint; Schwarzschild redshift).

A note on the aim of this work. This is a structural reinterpretation of well-established physics, not a proposal for a new physical law. Every relation shown is a substitution into, or a geometric re-reading of, equations that are already known and that remain unchanged. Where a particular unit convention is adopted (for example G = 4π), this is a choice of notation, not a claim about nature. 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). The Velocity–Spacetime Curvature Duality: Energy as the Hinge between $E = 4\pi/|K|$ and $G_{\mu\nu} = \kappa\,T_{\mu\nu}$ (Thomas Precession and Gravitational Holonomy) [Paper 63]. Zenodo.
DOI: 10.5281/zenodo.21606250
ORCID: 0009-0006-8587-3450