The RSI Lagrangian and light

Physicists describe a field with a Lagrangian, one expression that balances the energy of motion against the energy stored by deformation. Chapter XIV writes one down for the Eternal Consonance and checks what it gives.

The paper's tension of about 0.034 N and density of about 3.75 × 10⁻¹⁹ kg/m are values along a line, as for a string. A medium that fills space needs a density per volume and a stiffness per area instead. RSI proposes spreading the line values over the cross-section of the tightest fold the medium can make, a disc whose radius is one Planck length. The choice of disc is RSI's proposal, but whatever cross-section is chosen, the stiffness divided by the density is still c².

For a smooth medium that cannot be compressed, the equation then has three parts: the energy of the medium's motion, the energy stored when it is sheared out of shape, and a term that keeps its volume fixed. Working it through is standard mathematics, and the result is clean. Every disturbance travels at one speed, c, whatever its frequency. Because the medium cannot be squeezed, only sideways waves exist, with exactly two independent directions of vibration. These are the properties of light in empty space.

The match goes further. The same equations describe the elastic medium James MacCullagh proposed for light in 1839. If the medium's velocity is read as the electric field and its rotation as the magnetic field, Maxwell's four equations for empty space follow. This is established mathematics. What it does not yet give is electric charge. Charges, currents and Coulomb's law must come from the knots, and that step is open.

This equation alone cannot hold a particle. It is linear: two solutions simply add, and in a linear medium any lump of energy spreads out and fades. A stable knot needs a nonlinear term, so that the knot's own field holds it together. Such theories are well studied. Faddeev and Niemi proposed knotted solitons in 1997, Battye and Sutcliffe found them numerically in 1998, and Kleckner and Irvine made knotted vortex loops in water in 2013. RSI's Chapter III Lagrangian, with its self-interaction and twist term, is a candidate of this kind.

The first steps of the program are therefore clear: join the light equation and the Chapter III equation into one theory, show that it has stable knotted solutions, and calculate their energies. A knot's mass is its energy, and that energy depends on the nonlinear terms, not on topology alone. Until those energies are calculated, the particle masses cannot be read from the knots' shapes.