A wave propagating on the D¹ × S¹ bundle

Rest observer watching a massive particle. The total speed along the helix is always c; only its partition between the observable base (vₓ) and the internal fiber (v_θ) changes.

x
R = λ₀ / 2π
vₓ/c = 0.00 v_θ/c = 1.00 γ = 1.00 α =

At rest (vₓ = 0) the wave is pure internal rotation — the wavefronts are rings turning in place, and the internal clock ticks at its maximum rate. As vₓ increases, part of the motion is diverted along x: the helix stretches, and the internal rotation (hence the particle's own clock) slows as v_θ = c/γ. As vₓ → c the helix straightens toward the axis and the internal clock freezes. The bright point rides a single wavefront throughout.