# 209. Lorentz Force as Boundary Stress Transfer This appendix derives the Lorentz-force form directly from the source-free stress transfer of a compact toroidal charged mode. No effective charge density or current density is introduced. The bounded mode is a self-closing toroidal organization of one continuous field. The familiar Lorentz formula appears as the exact point-mode limit of its interaction with a smooth external Maxwell field. For definiteness, take the compact mode to be an axisymmetric torus with equal winding class $(m,m)$ and symmetry axis $\hat{\mathbf a}$. Nothing essential depends on that simplifying choice. It only makes the geometric bookkeeping cleaner. The leading interaction depends solely on the signed through-hole flux class carried along the torus axis. All finer toroidal structure enters only through higher multipoles. ## 209.1 Compact Toroidal Charged Modes Let $$ K_\varepsilon $$ be a coherent toroidal charged mode of size $\varepsilon$, centered at the worldline $$ X(\tau), $$ with proper time $\tau$. Chapter 10 associated charge with the signed through-hole flux class across the torus aperture. Write that class as the scalar $$ q. $$ In the instantaneous rest frame of the mode, choose Cartesian coordinates with origin at the center of energy and let $$ \mathbf r = R\,\mathbf n, \qquad |\mathbf n|=1. $$ The compact toroidal mode is assumed to have the far-field behavior established in chapter 10: $$ \mathbf E_{\mathrm s}(\mathbf r) = \frac{q}{4\pi\varepsilon_0}\frac{\mathbf n}{R^2} + \mathbf e_{\mathrm{rem}}(\mathbf r), $$ $$ \mathbf B_{\mathrm s}(\mathbf r) = \mathbf b_{\mathrm{rem}}(\mathbf r), $$ with bounds $$ |\mathbf e_{\mathrm{rem}}(\mathbf r)| \le C_E\frac{\varepsilon}{R^3}, \qquad |\mathbf b_{\mathrm{rem}}(\mathbf r)| \le C_B\frac{\varepsilon}{R^3}, $$ for every $$ R\ge 2\varepsilon. $$ So the leading exterior field of the compact torus is the inverse-square monopole term determined by the through-hole flux class, while all finer toroidal structure decays at least one power faster. Let the external Maxwell field be smooth near the mode center. On a sphere $$ S_R:=\{\,\mathbf x : |\mathbf x-X(\tau)|=R\,\}, $$ write $$ \mathbf E_{\mathrm e}(X(\tau)+R\mathbf n,\tau) = \mathbf E_0(\tau)+\mathbf E_1(R,\mathbf n,\tau), $$ $$ \mathbf B_{\mathrm e}(X(\tau)+R\mathbf n,\tau) = \mathbf B_0(\tau)+\mathbf B_1(R,\mathbf n,\tau), $$ with $$ |\mathbf E_1(R,\mathbf n,\tau)|\le C'_E R, \qquad |\mathbf B_1(R,\mathbf n,\tau)|\le C'_B R. $$ This is just the first-order smoothness expansion of the external field near the mode center. ## 209.2 Exact Source-Free Interaction Balance Let the total field be $$ \mathbf E=\mathbf E_{\mathrm s}+\mathbf E_{\mathrm e}, \qquad \mathbf B=\mathbf B_{\mathrm s}+\mathbf B_{\mathrm e}. $$ The total field is source-free everywhere. We evaluate the balance on spheres lying outside the compact toroidal core only so the exterior asymptotic form of the compact closure can be used cleanly. The exact local momentum balance is $$ \partial_t g_i - \partial_j T_{ij}=0, $$ where $$ \mathbf g=\varepsilon_0\,\mathbf E\times\mathbf B, $$ and $$ T_{ij} = \varepsilon_0\left(E_iE_j-\frac{1}{2}\delta_{ij}\mathbf E^2\right) + \frac{1}{\mu_0}\left(B_iB_j-\frac{1}{2}\delta_{ij}\mathbf B^2\right). $$ Decompose $$ \mathbf g = \mathbf g_{\mathrm s} + \mathbf g_{\mathrm e} + \mathbf g_{\times}, $$ $$ \mathbf T = \mathbf T_{\mathrm s} + \mathbf T_{\mathrm e} + \mathbf T_{\times}, $$ where the cross terms are $$ \mathbf g_{\times} := \varepsilon_0\bigl( \mathbf E_{\mathrm s}\times\mathbf B_{\mathrm e} + \mathbf E_{\mathrm e}\times\mathbf B_{\mathrm s} \bigr), $$ and $$ (T_{\times})_{ij} := \varepsilon_0\left( E_{{\mathrm s}i}E_{{\mathrm e}j} + E_{{\mathrm e}i}E_{{\mathrm s}j} - \delta_{ij}\,\mathbf E_{\mathrm s}\cdot\mathbf E_{\mathrm e} \right) + \frac{1}{\mu_0}\left( B_{{\mathrm s}i}B_{{\mathrm e}j} + B_{{\mathrm e}i}B_{{\mathrm s}j} - \delta_{ij}\,\mathbf B_{\mathrm s}\cdot\mathbf B_{\mathrm e} \right). $$ Subtracting the self and external balances from the total balance gives the exact cross-balance $$ \partial_t(\mathbf g_{\times})_i - \partial_j(T_{\times})_{ij} = 0. $$ Integrating over the ball $$ B_R:=\{\,\mathbf x : |\mathbf x-X(\tau)|\le R\,\} $$ gives $$ \frac{d}{dt}\int_{B_R}\mathbf g_{\times}\,dV = \int_{S_R}\mathbf T_{\times}\cdot\mathbf n\,dA. $$ For the monist reading used in this book, the right-hand side is the exact rate at which external stress transfers momentum into the compact closure across the surrounding sphere. Define therefore $$ \mathbf F_R(\tau) := \int_{S_R}\mathbf T_{\times}\cdot\mathbf n\,dA. $$ The Lorentz force will be the exact limit of $\mathbf F_R$ as the mode is shrunk to a point while the surrounding sphere shrinks with it but remains outside the toroidal core. ## 209.3 Rest-Frame Compact-Mode Theorem Choose an event on the worldline and work in the instantaneous rest frame of the toroidal mode at that event. Let $$ \alpha(R):=\frac{q}{4\pi\varepsilon_0 R^2}. $$ On $S_R$ write $$ \mathbf E_{\mathrm s} = \alpha(R)\,\mathbf n + \mathbf e_{\mathrm{rem}}, $$ with $$ |\mathbf e_{\mathrm{rem}}|\le C_E\frac{\varepsilon}{R^3}, \qquad |\mathbf B_{\mathrm s}|\le C_B\frac{\varepsilon}{R^3}. $$ The electric cross term on the sphere is $$ \mathbf T_{\times}^{(E)}\cdot\mathbf n = \varepsilon_0\left[ (\mathbf E_{\mathrm s}\cdot\mathbf n)\mathbf E_{\mathrm e} + (\mathbf E_{\mathrm e}\cdot\mathbf n)\mathbf E_{\mathrm s} - (\mathbf E_{\mathrm s}\cdot\mathbf E_{\mathrm e})\mathbf n \right]. $$ Substitute $$ \mathbf E_{\mathrm s}=\alpha\mathbf n+\mathbf e_{\mathrm{rem}}, \qquad \mathbf E_{\mathrm e}=\mathbf E_0+\mathbf E_1. $$ The leading terms simplify exactly: $$ \varepsilon_0\left[ (\alpha\mathbf n\cdot\mathbf n)\mathbf E_0 + (\mathbf E_0\cdot\mathbf n)\alpha\mathbf n - (\alpha\mathbf n\cdot\mathbf E_0)\mathbf n \right] = \varepsilon_0\,\alpha\,\mathbf E_0. $$ So $$ \mathbf T_{\times}^{(E)}\cdot\mathbf n = \frac{q}{4\pi R^2}\,\mathbf E_0 + \mathbf R_E(R,\mathbf n), $$ where the remainder satisfies $$ |\mathbf R_E(R,\mathbf n)| \le C_1\frac{|q|}{R^2}\,R + C_2\frac{\varepsilon}{R^3}. $$ Therefore $$ \int_{S_R}\mathbf T_{\times}^{(E)}\cdot\mathbf n\,dA = q\,\mathbf E_0 + O(R) + O\!\left(\frac{\varepsilon}{R}\right). $$ For the magnetic cross term, $$ \mathbf T_{\times}^{(B)}\cdot\mathbf n = \frac{1}{\mu_0}\left[ (\mathbf B_{\mathrm s}\cdot\mathbf n)\mathbf B_{\mathrm e} + (\mathbf B_{\mathrm e}\cdot\mathbf n)\mathbf B_{\mathrm s} - (\mathbf B_{\mathrm s}\cdot\mathbf B_{\mathrm e})\mathbf n \right], $$ so $$ \left| \int_{S_R}\mathbf T_{\times}^{(B)}\cdot\mathbf n\,dA \right| \le C_3\,R^2\sup_{S_R}|\mathbf B_{\mathrm s}|\,\sup_{S_R}|\mathbf B_{\mathrm e}| = O\!\left(\frac{\varepsilon}{R}\right). $$ Hence $$ \mathbf F_R = q\,\mathbf E_0 + O(R) + O\!\left(\frac{\varepsilon}{R}\right). $$ Take a two-scale limit in which $$ \varepsilon\to 0, \qquad R\to 0, \qquad \frac{\varepsilon}{R}\to 0. $$ Then $$ \boxed{ \lim_{\varepsilon\to 0}\mathbf F_R = q\,\mathbf E_0 }. $$ So a compact toroidal charged mode at rest experiences exactly the electric force $$ \mathbf F_{\mathrm{rest}}=q\,\mathbf E_{\mathrm e}(X). $$ No magnetic term appears in the rest frame, because the static toroidal mode has no magnetic monopole part. The toroidal details affect only the discarded higher multipoles. ## 209.4 Moving Aperture Transport The rest-frame theorem gives the monopole coupling of the toroidal charge class to a smooth electric load. To get the moving magnetic term, one should not jump immediately to a covariant ansatz. The torus itself already tells us what has to be sampled: a moving aperture of one common field. Let $$ \Sigma_\varepsilon(t) $$ be a spanning surface across the torus aperture, transported with the compact mode, and let $$ \mathbf u(\mathbf y,t) $$ be the local velocity of the material point $$ \mathbf y\in \Sigma_\varepsilon(t). $$ For any moving surface, Maxwell-Faraday transport gives $$ \frac{d}{dt}\int_{\Sigma_\varepsilon(t)}\mathbf B_{\mathrm e}\cdot d\mathbf A = -\oint_{\partial\Sigma_\varepsilon(t)} \bigl( \mathbf E_{\mathrm e} + \mathbf u\times \mathbf B_{\mathrm e} \bigr)\cdot d\boldsymbol\ell. $$ So the local field sampled by a moving aperture is not $$ \mathbf E_{\mathrm e} $$ alone, but $$ \mathbf E_{\mathrm e}+\mathbf u\times\mathbf B_{\mathrm e}. $$ This is the transport meaning of the magnetic term: the moving toroidal aperture samples the surrounding momentum-flux geometry through its transport across the transverse external field. For a rigidly drifting compact torus, decompose $$ \mathbf u(\mathbf y,t)=\mathbf v(t)+\mathbf u_{\mathrm{int}}(\mathbf y,t), $$ where $$ \mathbf v(t)=\dot{\mathbf X}(t) $$ is the drift of the torus center and $$ \mathbf u_{\mathrm{int}} $$ is the internal helical traversal of the closure. Define the aperture average of a field over $\Sigma_\varepsilon(t)$ by $$ \langle \mathbf W\rangle_{\Sigma_\varepsilon(t)} := \frac{1}{A(\Sigma_\varepsilon(t))} \int_{\Sigma_\varepsilon(t)}\mathbf W\,dA. $$ Because the external field is smooth on the compact scale, $$ \langle \mathbf E_{\mathrm e}\rangle_{\Sigma_\varepsilon(t)} = \mathbf E_{\mathrm e}(\mathbf X(t),t)+O(\varepsilon), $$ $$ \langle \mathbf B_{\mathrm e}\rangle_{\Sigma_\varepsilon(t)} = \mathbf B_{\mathrm e}(\mathbf X(t),t)+O(\varepsilon). $$ For the equal-winding axisymmetric torus, the internal traversal has no monopole average across the aperture: $$ \langle \mathbf u_{\mathrm{int}}\rangle_{\Sigma_\varepsilon(t)}=0. $$ Geometrically, opposite points of the aperture carry opposite tangential traversal velocities, so the internal helical motion cancels at monopole order. It affects only higher multipoles. Therefore $$ \left\langle \mathbf E_{\mathrm e} + \mathbf u\times\mathbf B_{\mathrm e} \right\rangle_{\Sigma_\varepsilon(t)} = \mathbf E_{\mathrm e}(\mathbf X(t),t) + \mathbf v(t)\times\mathbf B_{\mathrm e}(\mathbf X(t),t) + O(\varepsilon). $$ The moving compact torus therefore samples the smooth external field through the effective transport load $$ \boxed{ \mathbf L_{\mathrm{mov}} = \mathbf E_{\mathrm e}(\mathbf X(t),t) + \mathbf v(t)\times\mathbf B_{\mathrm e}(\mathbf X(t),t) }. $$ ## 209.5 Compact Moving-Mode Theorem Section 209.3 proved that the compact toroidal charge class couples at monopole order by $$ q\times(\text{smooth load sampled across the aperture}). $$ For a static torus, that sampled load is $$ \mathbf E_{\mathrm e}(X). $$ For a translating torus, section 209.4 shows that the sampled load is $$ \mathbf L_{\mathrm{mov}} = \mathbf E_{\mathrm e}(\mathbf X(t),t) + \mathbf v(t)\times\mathbf B_{\mathrm e}(\mathbf X(t),t). $$ Therefore, in the same compact two-scale limit, $$ \boxed{ \mathbf F = q\,\mathbf L_{\mathrm{mov}} = q\bigl( \mathbf E_{\mathrm e} + \mathbf v\times\mathbf B_{\mathrm e} \bigr) }. $$ This is the Lorentz-force form at compact toroidal monopole order. The associated power law follows immediately: $$ \frac{dE}{dt} = \mathbf F\cdot\mathbf v = q\,\mathbf E_{\mathrm e}\cdot\mathbf v, $$ because $$ (\mathbf v\times\mathbf B_{\mathrm e})\cdot\mathbf v=0. $$ So the magnetic part redirects transport but does no work. Force here is therefore the rate at which cross stress transfers momentum into the compact closure. Power is the corresponding energy-transfer rate obtained by contracting that momentum transfer with the drift velocity. ## 209.6 What the Derivation Used The argument used only the following ingredients: 1. source-free Maxwell stress continuity, 2. the toroidal charge interpretation of chapter 10, 3. the compact-mode far-field asymptotic $$ \mathbf E_{\mathrm s} = \frac{q}{4\pi\varepsilon_0}\frac{\mathbf n}{R^2} + O\!\left(\frac{\varepsilon}{R^3}\right), $$ 4. smoothness of the external field near the mode center, 5. Maxwell-Faraday transport for a moving aperture, 6. cancellation of the internal helical traversal at monopole order for an axisymmetric compact torus. No effective source density was needed. The role of the equal-winding $(m,m)$ torus was only to give a clean axis and a clean aperture-flux class. The Lorentz law depends only on the resulting scalar $q$ at monopole order. All finer toroidal geometry survives only in higher multipole corrections beyond the point-mode limit. ## 209.7 Interpretation Within this framework, the Lorentz force is not a primitive rule about a particle being pushed by an external field. It is the compact expression of one exact statement: - a charged body is a bounded toroidal closure of one common field, - its signed through-hole flux class determines the monopole coefficient $q$, - external stress transfers momentum across a shrinking sphere around that closure, - the moving aperture samples the same field through $$ \mathbf E_{\mathrm e}+\mathbf v\times\mathbf B_{\mathrm e}, $$ - internal helical traversal cancels at monopole order, leaving only the drift correction. Geometrically, the toroidal charge class is carried by an axial energy-moment orthogonal to the local transverse pair $(\mathbf F_+,\mathbf F_-)$. When that axis drifts through the external transverse organization, the sampled momentum-flux load resolves into the lateral transport term $ \mathbf v\times\mathbf B_{\mathrm e}. $ So the Lorentz law is not imported. It is the point-mode limit of toroidal boundary stress transfer together with moving-aperture transport. ## 209.8 Summary For a compact toroidal charged mode, the exact source-free cross-stress transfer across a sphere $S_R$ is $$ \mathbf F_R = \int_{S_R}\mathbf T_{\times}\cdot\mathbf n\,dA. $$ In the instantaneous rest frame, the compact-mode limit gives $$ \lim_{\varepsilon\to 0}\mathbf F_R = q\,\mathbf E_{\mathrm e}(X). $$ For a moving compact torus, the transported aperture samples the smooth external field through $$ \mathbf E_{\mathrm e}+\mathbf v\times\mathbf B_{\mathrm e}. $$ Therefore the compact toroidal monopole force is $$ \frac{d\mathbf p}{dt} = q\bigl(\mathbf E_{\mathrm e}+\mathbf v\times\mathbf B_{\mathrm e}\bigr). $$ Thus the Lorentz-force form is derived here directly from first principles at compact toroidal monopole order: compact toroidal charge, exact source-free stress transfer, and moving-aperture transport of one common electromagnetic substrate.