## Worked example 2: a helical flux tube on a cylinder, with explicit $v_{\mathrm{eff}}$ and $m_{\mathrm{eff}}$ This example unifies three ingredients in one computation: - local transport at the maximal rate along the flow line, - geometry of a helical (non-straight) trajectory, - momentum decomposition into translational and circulating parts. It reproduces, as explicit formulas: - the effective forward speed reduction, - and an effective inertial mass measure associated with trapped circulation. No relativity postulates are used. The only inputs are: - a curve geometry in Euclidean space, - local energy transport at speed $c$ along that curve, - and the Maxwell kinematic relation between energy and momentum for such transport. ### Cylinder geometry and the helix Consider a cylinder of radius $R$ with axis $\hat{\mathbf{z}}$. Let a curve wrap around this cylinder with constant pitch. A convenient parametrization uses arclength $s$ along the curve: $$ \mathbf{X}(s) = \begin{pmatrix} R\cos(\kappa s)\\ R\sin(\kappa s)\\ \lambda s \end{pmatrix}. $$ Here: - $\kappa$ controls how fast we wind azimuthally, - $\lambda$ controls how fast we advance along $z$, - and $s$ is arclength, so $|\partial_s\mathbf{X}(s)|=1$. Compute the tangent: $$ \partial_s\mathbf{X}(s)= \begin{pmatrix} - R\kappa \sin(\kappa s)\\ \ \ R\kappa \cos(\kappa s)\\ \lambda \end{pmatrix}. $$ Its squared norm is $$ |\partial_s\mathbf{X}|^2 = (R\kappa)^2(\sin^2+\cos^2)+\lambda^2 = (R\kappa)^2+\lambda^2. $$ Impose arclength normalization: $$ (R\kappa)^2+\lambda^2=1. $$ Define the (constant) pitch angle $\theta$ by $$ \cos\theta = \hat{\mathbf{t}}\cdot \hat{\mathbf{z}}, \qquad \hat{\mathbf{t}}=\partial_s\mathbf{X}. $$ Since the $z$-component of $\partial_s\mathbf{X}$ is $\lambda$, we have $$ \cos\theta=\lambda, \qquad \sin\theta = R\kappa, \qquad \cos^2\theta+\sin^2\theta=1. $$ So $\theta$ measures how much of the tangent is along $z$ versus around the cylinder. --- ## Local transport at speed $c$ along the helix Assume electromagnetic energy propagates locally along the curve at speed $c$. This is a kinematic assumption about local transport rate along the flow line. That means: - in time $dt$, energy advances a distance $ds = c\,dt$ along the curve. Thus, $$ \frac{ds}{dt}=c. $$ --- ## Effective forward speed $v_{\mathrm{eff}}$ The forward displacement is the change in $z$: $$ dz = \partial_s z\, ds = \lambda\, ds = \cos\theta\, ds. $$ Divide by $dt$: $$ \frac{dz}{dt} = \cos\theta\,\frac{ds}{dt} = \cos\theta\,c. $$ Therefore the effective forward velocity is $$ v_{\mathrm{eff}} = c\cos\theta. $$ This is purely geometric: forward progress is reduced because part of the motion is spent going around. No “slowing” occurs along the path: the path-speed remains $c$. --- ## Unrolling the cylinder and deriving the same result Unroll the cylinder surface to a plane. One full azimuthal revolution corresponds to horizontal displacement $2\pi R$. Suppose over one revolution the helix rises by $\Delta z$. Then the path length over one revolution is $$ \Delta s = \sqrt{(2\pi R)^2 + (\Delta z)^2}. $$ Local transport time is $$ \Delta t = \frac{\Delta s}{c}. $$ Effective forward speed is $$ v_{\mathrm{eff}}=\frac{\Delta z}{\Delta t} = \frac{\Delta z}{\Delta s/c} = c\,\frac{\Delta z}{\sqrt{(2\pi R)^2 + (\Delta z)^2}}. $$ Define $\theta$ by $$ \cos\theta = \frac{\Delta z}{\Delta s}, $$ and the same formula appears: $$ v_{\mathrm{eff}}=c\cos\theta. $$ Thus, the reduction is exactly “geometric delay”: longer path for the same forward span. --- ## Momentum content of a localized transported energy packet Let the total electromagnetic energy in the tube be $E$. For local light-like transport, the magnitude of momentum associated with energy transport is $$ P = \frac{E}{c}. $$ This is the statement that energy transported at speed $c$ carries momentum of magnitude $E/c$. In Maxwell language it is consistent with $\mathbf{g}=\mathbf{S}/c^2$ and $|\mathbf{S}|=cu$. The momentum vector points along the local direction of transport, i.e. along the helix tangent. --- ## Decomposition into translational and circulating momentum The direction $\hat{\mathbf{z}}$ defines “translation.” The orthogonal azimuthal direction defines “circulation.” The total momentum magnitude is $P=E/c$. Its $z$-component is $$ P_z = P\cos\theta = \frac{E}{c}\cos\theta. $$ The orthogonal (azimuthal) component magnitude is $$ P_\perp = P\sin\theta = \frac{E}{c}\sin\theta. $$ Interpretation: - $P_z$ contributes to net forward translation, - $P_\perp$ is tied to circulation around the cylinder. If the path is closed in the azimuthal direction, $P_\perp$ corresponds to momentum that does not produce net displacement; it is “kinematically trapped” in the closed direction. --- ## Effective inertial mass measure from trapped momentum The program’s definition here is operational: inertial mass is a measure of how much momentum content is not available for translation. A natural scalar measure is $$ m_{\mathrm{eff}} := \frac{P_\perp}{c}. $$ Substitute $P_\perp$: $$ m_{\mathrm{eff}} = \frac{1}{c}\frac{E}{c}\sin\theta = \frac{E}{c^2}\sin\theta. $$ So $$ m_{\mathrm{eff}} = \frac{E}{c^2}\sin\theta, \qquad v_{\mathrm{eff}} = c\cos\theta. $$ These two equations show the trade: - as $\theta\to 0$, $v_{\mathrm{eff}}\to c$ and $m_{\mathrm{eff}}\to 0$, - as $\theta\to \pi/2$, $v_{\mathrm{eff}}\to 0$ and $m_{\mathrm{eff}}\to E/c^2$. The limiting case $\theta=\pi/2$ is pure circulation: motion without translation. --- ## Energy-weighted generalization for variable pitch If the pitch angle varies along a closed trajectory, replace constants by averages. Let the local pitch be $\theta(s)$, and let the energy per arclength be $\varepsilon(s)$ so that $dE=\varepsilon(s)\,ds$ and $E=\int_0^L \varepsilon(s)\,ds$. Then the effective forward speed is energy-weighted: $$ v_{\mathrm{eff}} = c\,\left\langle \cos\theta \right\rangle_E, \qquad \left\langle \cos\theta \right\rangle_E := \frac{1}{E}\int_0^L \cos\theta(s)\,dE(s). $$ Similarly, $$ P_z = \frac{E}{c}\left\langle \cos\theta \right\rangle_E. $$ A consistent trapped-momentum measure is then $$ P_{\perp,\mathrm{eff}} = \sqrt{P^2 - P_z^2} = \frac{E}{c}\sqrt{1-\left\langle \cos\theta \right\rangle_E^2}, $$ and define $$ m_{\mathrm{eff}} = \frac{P_{\perp,\mathrm{eff}}}{c} = \frac{E}{c^2}\sqrt{1-\left\langle \cos\theta \right\rangle_E^2}. $$ This is the most general form compatible with: - local transport at speed $c$ along the flow line, - and the definition of translation as the $\hat{\mathbf{z}}$ direction. --- ## Connection to the thin-tube tension computation From the previous worked example, thin-tube localization defines $$ T = \int_{\Sigma_s} u\,dA, \qquad \mu = \frac{T}{c^2}. $$ For a helix segment, the same definitions apply, but “translation” is now the projection along $\hat{\mathbf{z}}$. The point is: - $T$ and $\mu$ are extracted from cross-sectional integrals of field energy, - $v_{\mathrm{eff}}$ and $m_{\mathrm{eff}}$ are extracted from geometric projection of transport direction. These are independent pieces of information about the same underlying localized flow. --- ## Closing the conceptual loop between “delay” and “looping” This worked example addresses the core tension: - in some contexts effective slowing arises from phase-structured superposition, - here effective slowing arises from geometric redistribution of motion. Both can coexist without contradiction because they act on different aspects: - phase-structured superposition changes how a total field evolves in time at a point, - geometric looping changes how far transport advances in a chosen direction per unit time. In both cases: - local transport along the allowed path remains continuous, - and the effective macroscopic propagation changes because “progress” is a projection of a richer process. --- ## Summary of the helix example Given a localized electromagnetic energy packet of total energy $E$ transported locally at speed $c$ along a helical path of pitch angle $\theta$: 1. Effective forward speed: $$ v_{\mathrm{eff}} = c\cos\theta. $$ 2. Translational momentum: $$ P_z = \frac{E}{c}\cos\theta. $$ 3. Circulating momentum: $$ P_\perp = \frac{E}{c}\sin\theta. $$ 4. Effective inertial mass measure: $$ m_{\mathrm{eff}} = \frac{E}{c^2}\sin\theta. $$ 5. Variable pitch generalization (energy-weighted): $$ v_{\mathrm{eff}} = c\left\langle \cos\theta \right\rangle_E, \qquad m_{\mathrm{eff}} = \frac{E}{c^2}\sqrt{1-\left\langle \cos\theta \right\rangle_E^2}. $$ No additional postulates appear. Everything is geometry plus Maxwell kinematics of energy and momentum transport.