PART II # The Rydberg Series One of the greatest achievements of early quantum theory was predicting the *discrete* energy levels of the hydrogen atom, known as the *Rydberg series*. This series describes the specific colors (frequencies) of light emitted by excited hydrogen atoms (and many other atoms), formalized empirically by the Swedish physicist Johannes Rydberg in 1888. Long before the internal structure of the atom was understood, experiments showed that glowing hydrogen does not emit a continuous rainbow of light (as Isaac Newton demonstrated by decomposing sunlight with a prism). Instead, it emits light only at very specific, sharply defined colors. Rydberg found that these frequencies follow a simple mathematical pattern involving integers $n = 1, 2, 3, \dots$. The emitted photon energies scale with the inverse square of these integers: $$ E_n \propto \frac{1}{n^2}. $$ The standard explanation, developed by Niels Bohr (1885–1962) and later formalized by Erwin Schrödinger (1887–1961), ties this scaling to the electrostatic (Coulomb) interaction between an electron and a proton, reminiscent of a planetary view of the atom. In this picture, larger $n$ corresponds to the electron occupying a state that is, on average, farther from the nucleus. Since the electrostatic interaction weakens with distance, the binding energy—the energy required to free the electron from the proton’s pull—decreases accordingly. However, the formula itself contains no reference to distance, radius, or geometry—only to the integer $n$. # Energy Reorganization We can interpret the $1/n^2$ factor not as a change in spatial size, but as a reorganization of energy into progressively finer structure. Consider a closed surface, such as a torus—a donut-shaped *surface*—with fixed total area. The torus is not a claim about atomic shape, but a minimal example of a compact manifold with two non-contractible cycles. Any compact electromagnetic configuration with two independent winding numbers yields the same $n^2$ scaling. If an electromagnetic wave winds around this surface $n$ times in each of the two independent directions (along the axis of the tube and around the ring), the field must satisfy global continuity conditions. After each complete traversal of a loop, the field must reproduce itself to avoid destructive interference. This imposes a discretization condition on the wavelength. Along a closed loop of length $L$, the field must satisfy: $$ n \lambda = L . $$ These are the same continuity conditions that produce standing waves on a string or in a cavity, now applied to a closed surface with two independent cycles. The combined winding partitions the surface into a grid of distinct regions. The number of such regions scales as: $$ n \times n = n^2 . $$ The total electromagnetic energy of the configuration is conserved. As the number of regions increases, this fixed energy is redistributed among a larger number of distinct cells. The energy associated with each region therefore decreases as: $$ E \propto \frac{1}{n^2}, $$ reproducing the Rydberg scaling. In this view, the Rydberg series does not describe an electron moving to a larger orbit in space. It describes the electromagnetic field reorganizing itself into progressively finer standing-wave patterns—self-consistent knots of counter-propagating electromagnetic energy flux, governed by global continuity rather than force balance. # The Torus To visualize a torus—a donut-shaped object—return to the flat napkin. First, identify and glue together one pair of opposite edges of the napkin. After this identification, the flat napkin becomes a tube. Lines that originally ended on one edge now reappear continuously on the opposite edge. Next, take this tube and identify its two circular ends. Gluing these ends together produces a closed surface with no boundary. The result is a donut-shaped surface, known as a torus. Any lines drawn on the original napkin become closed paths on the torus only if they match their own position when crossing an identified edge. This requirement ensures global continuity of the grid and is called a continuity condition. # Matter In this framework, matter is not a substrate distinct from the field. Matter is a self-sustained electromagnetic field configuration whose internal couplings generate delayed response, confinement, and stability. As we shall see, all effective material properties—mass, charge, inertia, and spectral structure—are emergent consequences of source-free Maxwell dynamics in a Maxwell Universe. ## From Spectral Structure to Mechanics In the preceding chapter, we showed that discrete spectral structure—exemplified by the Rydberg series—arises naturally when electromagnetic fields are confined by global continuity conditions. Discreteness emerged not from particles, forces, or quantization rules, but from topology: the requirement that a field defined on a compact configuration match itself after completing closed cycles. At that stage, the discussion concerned only internal structure: how energy redistributes within a self-confined electromagnetic configuration. Yet a question remains unavoidable. If such configurations are to be identified with ordinary matter, how do they move? How do they carry momentum, resist acceleration, and obey the conservation laws that govern everyday mechanics? The answer cannot be imported from Newtonian axioms or particle models, because neither exists in a Maxwell Universe. If mechanics is to arise at all, it must arise from electromagnetic field dynamics alone. The purpose of the present chapter is to show that it does—inevitably. ## Self-Refraction and Electromagnetic Stability In the preceding analysis of inertia, we treated a localized electromagnetic configuration as a given—a bounded distribution of energy with total energy $U$. We must now explain why such a configuration can exist at all, given the well-known tendency of electromagnetic waves to disperse in free space. In a Maxwell Universe, there is no container and no material substrate. There is only the electromagnetic field itself. Any mechanism of confinement must therefore arise from the field’s own dynamics. We call this mechanism **self-refraction**. ### Self-Generated Electromagnetic Environment Refraction does not require matter; it requires a phase-delayed electromagnetic response. In ordinary media, this response is attributed to bound charges. In a Maxwell Universe, the response must instead arise from the field configuration itself. A self-sustained electromagnetic structure continuously generates secondary electromagnetic fields through its internal dynamics. These secondary fields are phase-delayed relative to the primary energy flow. An electromagnetic wave propagating within such a configuration therefore propagates not through empty space, but through an electromagnetic environment created by the configuration itself. The configuration thus acts as its own effective medium. ### Energy Flow and Closure The local flow of electromagnetic energy is described by the Poynting vector, $$ \vec{S} = \frac{1}{\mu_0}\,\vec{E} \times \vec{B}. $$ Stability requires that energy not escape to infinity. This condition is not $\vec{S} = 0$, but rather **closure**: $$ \oint_{\partial V} \vec{S}\cdot d\vec{A} = 0 . $$ or, at least, a time-averaged closure: $$ \left\langle \oint_{\partial V}\vec{S}\cdot d\vec{A}\right\rangle_T = 0 . $$ Inside the region $V$, energy continues to move. However, its flow is redirected by the phase structure of the self-generated electromagnetic environment so that it forms closed or recurrent paths. Energy circulates rather than radiates. This closed circulation of energy is the precise meaning of electromagnetic self-confinement. ### (proposal) Self-Refraction (Maxwell-linear wording) In a Maxwell Universe, refraction is still expected to happen, just that without matter; it requires relative phase structure within the electromagnetic field that redirects energy flow through interference. No modification of Maxwell’s equations is required. The equations remain linear and source-free everywhere. The apparent bending of energy flow arises from interference between components of a single self-consistent Maxwell solution. Writing the total field as a superposition, $$ \vec{E}=\sum_k \vec{E}_k, \qquad \vec{B}=\sum_k \vec{B}_k, $$ the Poynting vector becomes $$ \vec{S} =\frac{1}{\mu_0}\vec{E}\times\vec{B} =\frac{1}{\mu_0}\sum_{k,\ell}\vec{E}_k\times\vec{B}_\ell . $$ The cross terms encode the redistribution of electromagnetic energy and momentum that continuously redirects propagation, producing closed circulation without invoking nonlinearity or an external medium. ### Topological Closure Local closure of energy flow is not sufficient. For long-lived stability, the closure condition must be satisfied globally. This imposes topological constraints on the field configuration. Configurations possessing non-contractible cycles—such as toroidal or knotted structures—naturally support persistent circulation of electromagnetic energy. Global continuity conditions force the field to reproduce itself after traversal of closed paths, preventing decay through destructive interference. The integers labeling these cycles are not imposed externally; they arise from the requirement of self-consistent phase closure. ### Stability as Identity A self-sustained electromagnetic configuration persists because its own fields generate the delayed response required to redirect subsequent propagation. The configuration exists not despite dispersion, but because dispersion is exactly balanced by self-refraction. Matter, in this view, is not light trapped by an external medium. Matter is electromagnetic energy whose own self-generated field structure continuously refracts it into closed, self-consistent circulation. In a Maxwell Universe, stability, discreteness, and mechanics are not additional principles. They are the global consequences of energy flow that closes on itself. ## Conservation Laws in a Maxwell Universe In a Maxwell Universe, the electromagnetic field is the only fundamental entity. There are no particles, no intrinsic masses, and no independent mechanical postulates. All physical objects are structured, self-confined electromagnetic field configurations evolving according to the source-free Maxwell equations, $$ \nabla\cdot\vec{E}=0,\qquad \nabla\cdot\vec{B}=0,\qquad \nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t},\qquad \nabla\times\vec{B}=\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}. $$ These four equations supply the entire dynamics. What we call “matter” is nothing more than a persistent solution of these equations. ### Energy and Momentum Are Field Properties Maxwell’s theory assigns energy and momentum directly to fields. The local electromagnetic energy density is $$ u=\tfrac12\bigl(\epsilon_0|\vec{E}|^2+\mu_0^{-1}|\vec{B}|^2\bigr), $$ and the flow of this energy is given by the Poynting vector, $$ \vec{S}=\mu_0^{-1}\,\vec{E}\times\vec{B}. $$ Momentum is not an added concept; it is already present in the field. The momentum density is $$ \vec{g}=\frac{\vec{S}}{c^2}=\epsilon_0\,\vec{E}\times\vec{B}. $$ For any localized electromagnetic configuration occupying a region $V$, the total momentum is therefore $$ \vec{P}=\int_V \vec{g}\,d^3x. $$ No mass parameter has been introduced. ### Why Momentum Is Conserved The conservation of momentum follows directly from Maxwell dynamics. Differentiating the momentum density and using Maxwell’s equations yields the local balance law $$ \frac{\partial\vec{g}}{\partial t}+\nabla\cdot\mathbf{T}=0, $$ where $\mathbf{T}$ is the Maxwell stress tensor, $$ T_{ij}=\epsilon_0\!\left(E_iE_j+c^2B_iB_j -\tfrac12\delta_{ij}(|\vec{E}|^2+c^2|\vec{B}|^2)\right). $$ Integrating over a volume $V$ gives $$ \frac{d\vec{P}}{dt} =-\!\int_{\partial V}\mathbf{T}\cdot d\vec{A}. $$ Momentum changes only when electromagnetic stress crosses the boundary. For a self-confined configuration whose external fields cancel on $\partial V$, the surface integral vanishes and $\vec{P}$ remains constant. Momentum conservation is therefore not a postulate, but a consequence of source-free Maxwell dynamics. ### Inertia Without Mass Define the total electromagnetic energy in $V$, $$ U=\int_V u\,d^3x, $$ and the center of energy, $$ \vec{R}(t)=\frac{1}{U}\int_V \vec{r}\,u\,d^3x. $$ Electromagnetic field theory gives the exact relation $$ \vec{P}=\frac{U}{c^{2}}\,\frac{d\vec{R}}{dt}. $$ If $\vec{P}$ is constant, then $d\vec{R}/dt$ is constant. A localized electromagnetic configuration therefore moves at uniform velocity unless acted upon by external electromagnetic stress. This is inertia. ### Angular Momentum and Rotation Electromagnetic fields also carry angular momentum. The angular-momentum density is $$ \vec{\ell}=\vec{r}\times\vec{g}, $$ and the total angular momentum is $$ \vec{L}=\int_V \vec{r}\times(\epsilon_0\,\vec{E}\times\vec{B})\,d^3x. $$ Its evolution obeys $$ \frac{d\vec{L}}{dt} =-\!\int_{\partial V}\vec{r}\times(\mathbf{T}\cdot d\vec{A}). $$ Angular momentum is conserved whenever no torque crosses the boundary. A self-confined electromagnetic configuration with internal circulation will rotate indefinitely unless acted upon by an external stress. Rotational inertia is the persistence of circulating field momentum. ### What a “Force” Is In a Maxwell Universe, a force is not primitive. A push is simply an external electromagnetic field overlapping a localized configuration. During the overlap, $$ \vec{E}_{\text{tot}}=\vec{E}+\vec{E}_{\text{ext}},\qquad \vec{B}_{\text{tot}}=\vec{B}+\vec{B}_{\text{ext}}, $$ and the momentum density becomes $$ \vec{g}_{\text{tot}}=\epsilon_0\,\vec{E}_{\text{tot}}\times\vec{B}_{\text{tot}}. $$ Momentum is redistributed through electromagnetic stress. After the interaction ends, the configuration relaxes into a new steady state with a different total momentum. ### Emergence of Newton’s Second Law From local energy conservation, $$ \frac{\partial u}{\partial t}+\nabla\cdot\vec{S}=0, $$ define the enclosed energy of the self-sustaining electromagnetic configuration, and its energy centroid, $$ U(t)=\int_V u\,d^3x, \qquad \vec{R}(t)=\frac{1}{U(t)}\int_V \vec{r}\,u\,d^3x. $$ Differentiate the numerator: $$ \frac{d}{dt}\int_V \vec{r}\,u\,d^3x =\int_V \vec{r}\,\frac{\partial u}{\partial t}\,d^3x =-\int_V \vec{r}\,\nabla\cdot\vec{S}\,d^3x. $$ Use the identity $\nabla\cdot(\vec{r}\,\vec{S})=\vec{S}+\vec{r}\,\nabla\cdot\vec{S}$ to rewrite $$ \vec{r}\,\nabla\cdot\vec{S}=\nabla\cdot(\vec{r}\,\vec{S})-\vec{S}, $$ hence $$ -\int_V \vec{r}\,\nabla\cdot\vec{S}\,d^3x =-\int_V \nabla\cdot(\vec{r}\,\vec{S})\,d^3x+\int_V \vec{S}\,d^3x. $$ Apply the divergence theorem to the first term: $$ -\int_V \nabla\cdot(\vec{r}\,\vec{S})\,d^3x =-\oint_{\partial V} (\vec{r}\,\vec{S})\cdot d\vec{A}, $$ so altogether $$ \frac{d}{dt}\int_V \vec{r}\,u\,d^3x =\int_V \vec{S}\,d^3x-\oint_{\partial V} \vec{r}\,(\vec{S}\cdot d\vec{A}). $$ For a self-confined configuration (no net energy flux across $\partial V$), $$ \vec{S}\cdot d\vec{A}=0\quad\text{on}\quad \partial V, $$ so the surface term vanishes and we obtain $$ \frac{d}{dt}\int_V \vec{r}\,u\,d^3x =\int_V \vec{S}\,d^3x. $$ Using $\vec{g}=\vec{S}/c^2$, the total momentum is $$ \vec{P}=\int_V \vec{g}\,d^3x=\frac{1}{c^2}\int_V \vec{S}\,d^3x, $$ therefore $$ \frac{d}{dt}\int_V \vec{r}\,u\,d^3x=c^2\vec{P}. $$ Finally, since $\vec{R}=(1/U)\int_V \vec{r}\,u\,d^3x$, differentiation gives $$ \frac{d\vec{R}}{dt} =\frac{1}{U}\frac{d}{dt}\int_V \vec{r}\,u\,d^3x-\frac{1}{U}\frac{dU}{dt}\,\vec{R}. $$ For a self-confined configuration, $$ \frac{dU}{dt}= -\oint_{\partial V}\vec S\cdot d\vec A=0, $$ hence $$ \frac{d\vec{R}}{dt} =\frac{1}{U}\frac{d}{dt}\int_V \vec{r}\,u\,d^3x =\frac{c^2}{U}\vec{P}, $$ or equivalently the center-of-energy identity $$ \boxed{\;\vec{P}=\frac{U}{c^{2}}\,\frac{d\vec{R}}{dt}\;} \qquad \left(\vec{R}(t)=\frac{1}{U}\int_V \vec{r}\,u\,d^3x\right). $$ Thus the translational velocity of the configuration is not an assumption but a ratio of conserved field integrals: $$ \vec{v}:=\frac{d\vec{R}}{dt}=\frac{c^{2}\vec{P}}{U}. $$ In particular, Maxwell theory implies the inequality $$ c\,|\vec{P}|\le U, $$ so that $$ |\vec{v}|=\frac{c^{2}|\vec{P}|}{U}\le c. $$ The center of energy can therefore move strictly slower than $c$ because the configuration may contain counter-propagating internal energy flows whose momenta partially cancel while their energies add. Define the inertial mass of the bounded configuration by $$ m=\frac{U}{c^{2}}. $$ Differentiating $\vec{P}=(U/c^{2})\vec{v}$ yields the general momentum-balance law $$ \vec{F}_{\text{ext}} =\frac{d\vec{P}}{dt} =\frac{1}{c^{2}}\frac{dU}{dt}\,\vec{v} +m\,\vec{a}, \qquad \vec{a}=\frac{d\vec{v}}{dt}. $$ For a closed, self-sustained configuration (no net energy flux across $\partial V$), $$ \frac{dU}{dt}=0, $$ and the motion of the center of energy obeys identically $$ \vec{F}_{\text{ext}}=m\,\vec{a}. $$ Inertia is therefore the persistence of field momentum: when no stress flux crosses the boundary, $\vec{P}$ is constant, hence $\vec{v}$ is constant. ### What This Means All classical mechanical behavior—translation, rotation, inertia, and conservation laws—emerges directly from source-free Maxwell dynamics. No particles, intrinsic masses, or auxiliary axioms are required. In a Maxwell Universe, mechanics is not imposed on matter. Mechanics is the natural behavior of structured electromagnetic fields. ### Topology, Energy, and Motion The same global constraints that discretize the internal energy of a self-confined electromagnetic configuration also govern its external behavior. In the Rydberg analysis, standing-wave conditions on compact configurations forced energy into discrete modes labeled by integers. These integers were not introduced by hand; they were the unavoidable consequence of field continuity on closed cycles. Momentum and angular momentum arise from the same logic. They are not attributes of particles, but integrals of electromagnetic energy flow. Their conservation is not imposed as a principle; it follows from the absence of stress flux across the boundary of a closed configuration. Thus, spectral structure and mechanical behavior are not separate domains. Quantized internal organization and classical motion are two aspects of the same electromagnetic reality: globally constrained, source-free Maxwell fields. Once matter is recognized as a self-sustained electromagnetic configuration, the emergence of inertia and conservation laws is no longer surprising. It is the only outcome consistent with Maxwell dynamics. # Charge as Topology Because $$ \nabla \cdot \mathbf{E} = 0 $$ everywhere, no electric field originates from a point. All electromagnetic flow is closed. Consider a self-sustained electromagnetic configuration confined to a compact region of space. Enclose this configuration within a spherical surface of radius $r$ much larger than the structure itself. The enclosing area grows as $4\pi r^2$. The total electromagnetic circulation is conserved. As this fixed quantity is distributed over a surface whose area grows with $r^2$, the observed field intensity necessarily falls off as $1/r^2$. This reproduces the phenomenology of charge. In this view, charge is not a primitive source but an effective, topological quantity: a measure of electromagnetic circulation associated with a self-sustained field configuration. What appears as charge at large distances is the far-field signature of closed electromagnetic energy flow. ## Impedance and Fine Structure A key dimensionless number in electromagnetism is the **fine-structure constant** $\alpha$. In a Maxwell Universe, the natural way to connect $\alpha$ to field structure is through **impedance**. The impedance of free space is fixed by Maxwell’s constants: $$ Z_0=\sqrt{\frac{\mu_0}{\epsilon_0}}=\mu_0 c=\frac{1}{\epsilon_0 c}\approx 376.73\,\Omega. $$ Independently, experiment isolates a universal resistance scale, the **von Klitzing constant** (quantum of resistance), $$ R_K=\frac{h}{e^2}\approx 25\,812.807\,\Omega. $$ These are related to $\alpha$ by a pure identity. Starting from $$ \alpha=\frac{e^2}{4\pi\epsilon_0\hbar c} =\frac{e^2}{2\epsilon_0 h c}, $$ and using $Z_0=1/(\epsilon_0 c)$, we obtain $$ \alpha=\frac{e^2}{2h}\,Z_0 =\frac{Z_0}{2(h/e^2)} =\frac{Z_0}{2R_K}. $$ Equivalently, $$ R_K=\frac{Z_0}{2\alpha}=\frac{\mu_0 c}{2\alpha}. $$ Any claim that $\alpha$ arises from a self-confined electromagnetic geometry must therefore produce an **intrinsic impedance** $Z_{\text{knot}}$ of the bounded configuration such that $$ Z_{\text{knot}}=R_K, \qquad\text{hence}\qquad \alpha=\frac{Z_0}{2Z_{\text{knot}}}. $$ In this framing, $\alpha$ is the (halved) **mismatch ratio** between the vacuum impedance and the intrinsic impedance of the self-confined configuration. The factor $1/2$ is the minimal equipartition statement: a stable standing configuration must confine electric and magnetic energy together, so the effective coupling to the exterior counts both components.