βBeta Hypothesis

Interactive

Visualizations

Seeing the geometry directly. The one wave, the one constraint, and how relativistic kinematics falls out of watching only part of the motion.

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 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.

More visualizations — the GEOM energy circle, the circle-to-hyperbola bridge — are in preparation.