# Mechanics and Self-Refraction In the preceding chapter, discreteness appeared not from particles or quantum rules, but from continuity and topology. The electromagnetic field could close on itself only in discrete global classes. That result concerned internal organization. A further question immediately follows. If such bounded configurations are to count as ordinary matter, how do they move? How do they carry momentum, resist acceleration, and persist instead of dispersing? In a Maxwell Universe, these questions cannot be answered by importing Newtonian particles or external containers. They must be answered by the field itself. This chapter does two things: - it shows how mechanics emerges from electromagnetic energy and momentum balance, - it sharpens the self-refraction principle so it no longer means a vague "effective medium," but the redirection of one organized part of the field by the transport geometry induced by the rest of that same field. ## Conservation Laws in a Maxwell Universe The only fundamental entity is the organized electromagnetic field. In the resolved Maxwell form, its source-free dynamics is $$ \nabla\cdot\mathbf E=0, \qquad \nabla\cdot\mathbf B=0, $$ $$ \nabla\times\mathbf E=-\partial_t\mathbf B, \qquad \nabla\times\mathbf B=\mu_0\varepsilon_0\,\partial_t\mathbf E. $$ These equations are not added to a particle world. They are the resolved two-aspect transport closure of the same continuous field. ### Energy and Momentum Are Already Field Properties The local energy density is $$ u=\frac12\left(\varepsilon_0|\mathbf E|^2+\mu_0^{-1}|\mathbf B|^2\right), $$ and the energy flux is $$ \mathbf S=\mu_0^{-1}\mathbf E\times\mathbf B. $$ Momentum is not an extra ingredient. It is already present in the field: $$ \mathbf g=\frac{\mathbf S}{c^2}=\varepsilon_0\,\mathbf E\times\mathbf B. $$ The total momentum of a localized configuration in a region $V$ is therefore $$ \mathbf P=\int_V \mathbf g\,d^3x. $$ No independent mass parameter has yet been introduced. ### Local Balance Laws Maxwell transport gives two exact local balance equations. Energy continuity: $$ \partial_t u+\nabla\cdot\mathbf S=0. $$ Momentum continuity: $$ \partial_t g_i-\partial_j T_{ij}=0, $$ where the Maxwell stress tensor is $$ T_{ij} = \varepsilon_0\left(E_iE_j-\frac12\delta_{ij}\mathbf E^2\right) + \frac{1}{\mu_0}\left(B_iB_j-\frac12\delta_{ij}\mathbf B^2\right). $$ Integrating the momentum law over a volume $V$ gives $$ \frac{d\mathbf P}{dt} = \int_{\partial V}\mathbf T\cdot d\mathbf A. $$ So momentum changes only when electromagnetic stress crosses the boundary. Mechanics is already here. It is bookkeeping for transported field momentum. ### Center of Energy and Inertia Let $$ U=\int_V u\,d^3x $$ be the total energy of a bounded configuration, and define its center of energy $$ \mathbf R(t)=\frac{1}{U}\int_V \mathbf r\,u\,d^3x. $$ For an isolated bounded mode, the boundary energy flux vanishes, so $U$ is constant. Using $$ \partial_t u=-\nabla\cdot\mathbf S, $$ one obtains $$ \frac{d}{dt}\int_V \mathbf r\,u\,d^3x = \int_V \mathbf S\,d^3x, $$ hence $$ \mathbf P = \frac{1}{c^2}\int_V \mathbf S\,d^3x = \frac{U}{c^2}\,\frac{d\mathbf R}{dt}. $$ This is the exact center-of-energy identity. It suggests the inertial mass of a bounded mode: $$ m:=\frac{U}{c^2}. $$ Then $$ \mathbf P=m\,\dot{\mathbf R}, $$ and the boundary stress law becomes $$ \mathbf F_{\mathrm{ext}} := \frac{d\mathbf P}{dt} = \int_{\partial V}\mathbf T\cdot d\mathbf A. $$ For a closed mode with constant $U$, $$ \mathbf F_{\mathrm{ext}}=m\,\ddot{\mathbf R}. $$ So inertia and Newtonian-looking mechanics are not primitive axioms. They are compact descriptions of electromagnetic energy and momentum bookkeeping for a bounded closure. ## Why a Bounded Mode Can Exist Mechanics explains how a bounded configuration moves if it already exists. It does not yet explain why such a configuration does not simply disperse. The answer cannot be an external box or a material background, because neither exists in a Maxwell Universe. The closure must persist by the field's own transport. Here the distinction between local reorganization and transport matters. - A single curl reorganizes locally. - A doubled curl transports. An open transporting pattern is not yet matter. A material mode is transport folded back onto itself so that the transport closure remains bounded. That folded transport is what AMU Part II calls **self-refraction**. ## Self-Refraction The phrase should now be read precisely. Self-refraction does **not** mean: - propagation through a second medium, - a delayed secondary field added on top of the first, - a modification of Maxwell's equations. It means: > one organized part of the field redirects another part of the same field > because the total transport geometry is determined by the whole configuration > at once. The field does not travel through something else. The field is the thing that sets the transport conditions. ### Why Superposition Is Already Interaction Write a bounded configuration as a sum of coherent components: $$ \mathbf E=\sum_k \mathbf E_k, \qquad \mathbf B=\sum_k \mathbf B_k. $$ Then the Poynting vector is $$ \mathbf S = \frac{1}{\mu_0}\sum_{k,\ell}\mathbf E_k\times\mathbf B_\ell. $$ The Maxwell stress is likewise quadratic: $$ T_{ij} = \varepsilon_0\left(E_iE_j-\frac12\delta_{ij}\mathbf E^2\right) + \frac{1}{\mu_0}\left(B_iB_j-\frac12\delta_{ij}\mathbf B^2\right). $$ So when coherent parts of the same bounded mode overlap, cross terms appear in both energy transport and momentum transport. Those cross terms are not optional corrections. They are the exact redistribution terms of the total field. They redirect energy flow and stress across the entire extent at once. That is the rigorous content of "the field interacts with itself." ### Self-Refraction as Same-Substrate Redirection Because the transport relation is imposed simultaneously for all points in the extent, self-refraction is not the path of a tagged bit of substance through a pre-existing medium. It is the whole-field statement that one region of the closure changes the transport geometry seen by neighboring regions of that same closure. In a toroidal or helical mode: - one part of the circulation loads the axial path, - another part carries the transport around the closure, - the transported energy is continuously redirected back into the bounded mode instead of leaking away. So a knot is not "light trapped in a box." It is transport whose own geometry keeps turning subsequent transport back into the closure. This is why the phrase `self-refraction` is worth keeping. It names the fact that the field is both the transported thing and the thing that shapes the path of that transport. ## Stability as Identity A bounded electromagnetic mode persists when dispersion is balanced by this same-substrate redirection. The mode exists because: - transport is real, - transport can close on itself, - the closure redirects later transport back into the same organized pattern. The identity of the object is therefore not a separate substance hidden behind the field. The identity **is** the persistent closure. Matter, on this view, is electromagnetic transport maintained by its own self-refraction. ## Summary Mechanics emerges from the exact field balances $$ \partial_t u+\nabla\cdot\mathbf S=0, \qquad \partial_t g_i-\partial_jT_{ij}=0. $$ For a bounded mode, $$ \mathbf P=\frac{U}{c^2}\dot{\mathbf R}, \qquad m=\frac{U}{c^2}, $$ so inertia and Newton-like motion are compact descriptions of electromagnetic energy and momentum transport. Stability requires more: not merely local turning, but transporting closure folded back onto itself. That folded transport is self-refraction. It is not a second medium and not a secondary field. It is the exact redirection of one organized part of the field by the transport geometry induced by the rest of that same field.