A wave propagating on the D¹ × S¹ bundle
A rest observer watching a massive particle. The wave always travels at speed c along its helical path on the cylinder — what changes is how that fixed speed is split between the observable base (vₓ) and the internal fiber (vθ), subject to vₓ² + vθ² = c².
At rest, the wave is pure internal rotation: the wavefronts are rings turning in place, and the internal clock runs at its maximum rate. As vₓ increases, part of the motion is diverted along x — the helix stretches, and the internal rotation (the particle's own clock) slows as vθ = c/γ. As vₓ → c the helix straightens toward the axis and the clock freezes. The bright point rides a single wavefront the whole time; the velocity triangle at the point shows the constraint in action.
Use the slider to set vₓ/c by hand, or let it ramp automatically. This is the animated form of Figure 1 in the manuscript. For the geometry behind it, see Theory.
The GEOM circle: one construction, two readings
The same geometric picture, read two ways. The short vector is the invariant — the ontological energy m₀c2 (or, in the other reading, the velocity of fixed magnitude c). It stays fixed in length on the circle; only the angular split between its internal and translational components changes as vx grows.
The extended vector is the same ray carried out to the tangent line: the observed energy E = γm₀c2 (or the wavevector, with hypotenuse ω and horizontal component kx). At rest the two coincide at the top; as vx → c the observed quantity runs off along the tangent while the invariant stays put. Use the button to switch between the energy labels and the velocity/wavevector labels — it is literally the same diagram.
The dispersion cone and its mass-shell section
The constraint ω² = c²(kx² + kθ²) defines a cone in frequency–wavevector space. A massive particle keeps kθ fixed at 2π/λ₀ — the fundamental internal mode — so its state is confined to the plane where kθ is constant, and moves along the hyperbola cut there.
At rest the point sits at the vertex (ω = ω₀ = m₀c², pure rest energy). As vx increases it climbs the hyperbola, ω = γω₀ and kx = βγ·kθ, approaching the cone's edge — the photon line ω = c·kx — as vx → c. This is the relativistic relation E² = (pxc)² + (m₀c²)² read directly off the cone. Drag to rotate.
Circle ↔ hyperbola: the Gudermannian bridge
The bounded circle (β² + 1/γ² = 1) and the unbounded hyperbola (γ² − (βγ)² = 1) share the same vertex and the same angle. The point on the circle gives (β, 1/γ); extend the ray to the tangent line and rise to the hyperbola to reach (βγ, γ).
The two are linked by the Gudermannian relation sin α = tanh φ. As vx → c the circular angle saturates at 90° while the hyperbolic rapidity grows without bound — which is exactly why φ, not α, adds when velocities compose. The compass arc carries the long vector (length γ) onto the vertical axis.