# Defining Electromagnetic Fields from Continuity and Divergence-Free Structure # Defining Electromagnetic Fields from Continuity and Divergence-Free Structure ## Motivation A recurring conceptual objection is: If "electromagnetic energy" is the fundamental object, and energy density is a scalar field, how do the vector fields $\mathbf{E}$ and $\mathbf{B}$ arise? The objection is valid if one assumes the map scalar $\to$ vectors should be direct and instantaneous. It is not. The correct ordering is: - scalar energy density $u(\mathbf{x},t)$ is an observable of a process, - the process is constrained by continuity, - continuity forces transport, - transport forces directional structure, - directional, divergence-free transport forces circulation, - circulation admits a minimal vector description. This document makes that chain explicit and states exactly what can be derived from continuity and divergence-free structure, and what cannot. ## What is assumed (and what is not) We assume a source-free region. We assume that electromagnetic energy admits: 1. A nonnegative energy density $u(\mathbf{x},t) \ge 0$. 2. An energy flux $\mathbf{S}(\mathbf{x},t)$ satisfying a continuity equation $$ \partial_t u + \nabla \cdot \mathbf{S} = 0. $$ We also assume the usual empirical causal bound that energy flux does not exceed the local maximal propagation speed $c$: $$ |\mathbf{S}| \le c\,u. $$ This bound is **not** a *postulate* about space or time or flow; it is an operational, observed, fact about propagating electromagnetic disturbances: there exists a maximal transport rate relating energy content to energy throughput. We do not assume particles, matter, constitutive media, quantum axioms, or any mechanical primitive. We do not derive Maxwell’s equations from continuity alone. We instead show what continuity forces, and how $\mathbf{E}$ and $\mathbf{B}$ appear as a minimal representation of that forced structure. ## Why a scalar field alone cannot encode propagation A scalar field $u(\mathbf{x},t)$ at a single time slice is insufficient to encode motion. Two snapshots, however, lets us define the "directional structure" a scalar field must have in order to transport energy. Without energy transport there is nothing to describe. What comes is not only or just philosophy, it is a degrees-of-freedom statement: - A scalar specifies magnitude, - Transport requires magnitude and direction, and - Direction can be encoded is vectorial language. The continuity equation already check this requirements: it does not close on $u$ alone; it involves a magnitude and direction of change, $\mathbf{S}$, a scalar value everywhere. In essence, two scalar field, two time slices. Thus, the fundamental object in a continuity-based ontology is not $u$ alone, but the pair $(u,\mathbf{S})$, i.e. a conserved flow. ## Divergence-free structure forces circulation in 3D In three dimensions, divergence-free vector fields have a canonical geometric feature: they admit circulation and vortex-like organization. The mathematical statement is standard: If a vector field $\mathbf{F}$ is divergence-free on a simply connected region, then there exists a vector potential $\mathbf{A}$ such that $$ \mathbf{F} = \nabla \times \mathbf{A}. $$ This is not “electromagnetism”; it is a general theorem about solenoidal, mathematical objects, fields. It says that divergence-free structure is equivalent to curl structure, hence to circulation. Therefore, any theory that insists on divergence-free structure as primitive is already committed to circulation as a primitive mode of organization. This is the topological seed of toroidal shells and winding numbers: in 3D, divergence-free structure naturally organizes into tubes and closed surfaces. ## What it means to “define the electromagnetic field” There are two distinct tasks that often get conflated: 1. Representation: show that $(u,\mathbf{S})$ can be represented by fields $\mathbf{E},\mathbf{B}$. 2. Dynamics: show how $\mathbf{E},\mathbf{B}$ evolve. Continuity addresses (1) partially and constrains (2), but does not uniquely fix (2). For dynamics, one needs the full Maxwell evolution law. This document focuses on (1): the representation problem. ## Reconstruction lemma: from $(u,\mathbf{S})$ to $(\mathbf{E},\mathbf{B})$ We now show that if $(u,\mathbf{S})$ obeys $|\mathbf{S}| \le c u$, then one can construct $\mathbf{E}$ and $\mathbf{B}$ satisfying the standard electromagnetic energy and flux relations. ### Lemma (existence, non-uniqueness) Given scalar $u>0$ and vector $\mathbf{S}$ with $|\mathbf{S}| \le c u$, there exist vectors $\mathbf{E}$ and $\mathbf{B}$ such that $$ u = \frac{\epsilon_0}{2}|\mathbf{E}|^2 + \frac{1}{2\mu_0}|\mathbf{B}|^2, $$ $$ \mathbf{S} = \frac{1}{\mu_0}\,\mathbf{E}\times \mathbf{B}. $$ Moreover, the pair $(\mathbf{E},\mathbf{B})$ is not unique: there is an infinite family of solutions corresponding to polarization/duality degrees of freedom. ### Construction Let $\hat{\mathbf{s}}$ be a unit vector in the direction of $\mathbf{S}$ if $\mathbf{S}\neq 0$; if $\mathbf{S}=0$, pick any unit vector. Choose any unit vector $\hat{\mathbf{e}}$ orthogonal to $\hat{\mathbf{s}}$. Define $\hat{\mathbf{b}} := \hat{\mathbf{s}} \times \hat{\mathbf{e}}$ so that $(\hat{\mathbf{e}},\hat{\mathbf{b}},\hat{\mathbf{s}})$ is a right-handed orthonormal frame. Now pick magnitudes $E:=|\mathbf{E}|$ and $B:=|\mathbf{B}|$ and set $$ \mathbf{E} := E\,\hat{\mathbf{e}}, \qquad \mathbf{B} := B\,\hat{\mathbf{b}}. $$ Then $$ \mathbf{E}\times \mathbf{B} = EB\,\hat{\mathbf{s}}. $$ So the flux condition becomes $$ \frac{1}{\mu_0}EB = |\mathbf{S}|. $$ The energy condition becomes $$ u = \frac{\epsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2. $$ Use $c^2 = 1/(\mu_0\epsilon_0)$ to write $B$ in terms of $E$ from the flux equation: $$ B = \frac{\mu_0|\mathbf{S}|}{E}. $$ Substitute into the energy equation: $$ u = \frac{\epsilon_0}{2}E^2 + \frac{1}{2\mu_0}\left(\frac{\mu_0|\mathbf{S}|}{E}\right)^2 = \frac{\epsilon_0}{2}E^2 + \frac{\mu_0}{2}\frac{|\mathbf{S}|^2}{E^2}. $$ Multiply by $2/\epsilon_0$ and define $x:=E^2>0$: $$ \frac{2u}{\epsilon_0} = x + \frac{|\mathbf{S}|^2}{c^2}\frac{1}{x}. $$ This is a quadratic equation in $x$: $$ x^2 - \frac{2u}{\epsilon_0}x + \frac{|\mathbf{S}|^2}{c^2} = 0. $$ The discriminant is $$ \Delta = \left(\frac{2u}{\epsilon_0}\right)^2 - 4\frac{|\mathbf{S}|^2}{c^2} = 4\left(\frac{u^2}{\epsilon_0^2} - \frac{|\mathbf{S}|^2}{c^2}\right). $$ By the assumed bound $|\mathbf{S}| \le c u$, we have $\Delta \ge 0$, so a positive solution exists. Choose either root $$ x = \frac{u}{\epsilon_0} \pm \sqrt{\frac{u^2}{\epsilon_0^2} - \frac{|\mathbf{S}|^2}{c^2}}, $$ and set $E=\sqrt{x}$, then define $B$ by $B=\mu_0|\mathbf{S}|/E$. This constructs $\mathbf{E}$ and $\mathbf{B}$ satisfying the desired relations. ### Interpretation - The inequality $|\mathbf{S}| \le c u$ is exactly the condition that makes the reconstruction possible. - The choice of $\hat{\mathbf{e}}$ orthogonal to $\hat{\mathbf{s}}$ is a free polarization choice. - The choice of sign in the quadratic solution is another branch freedom. - More generally, one can rotate $(\mathbf{E},\mathbf{B})$ by a continuous duality transformation without changing $(u,\mathbf{S})$. So the mapping $(u,\mathbf{S}) \mapsto (\mathbf{E},\mathbf{B})$ exists but is not unique, as it should not be: $(u,\mathbf{S})$ contains fewer degrees of freedom than $(\mathbf{E},\mathbf{B})$. ## Why non-uniqueness is not a defect At a point in space-time, $(u,\mathbf{S})$ provides: - 1 scalar degree of freedom (magnitude), - 3 flux components (direction + magnitude), for a total of 4 real numbers. But $(\mathbf{E},\mathbf{B})$ provides 6 real numbers. Therefore, 2 local degrees of freedom cannot be determined from $(u,\mathbf{S})$ alone. This is exactly what polarization is. So when one says: “fields arise from energy flow,” the correct content is: - energy flow determines the transported energy and its direction, - polarization is additional relational structure not contained in $u$ alone, and not fully contained in $\mathbf{S}$ alone. This is consistent with the lived content of electromagnetism: one may know the energy and flux of a wave packet without knowing its polarization state. ## Why “orthogonal generation” appears The frequent question “why does one generate the other orthogonally?” should be recast as: Why does divergence-free propagation *require* rotational coupling? The minimal answer is: - To propagate while remaining divergence-free, a field’s degrees of freedom must rotate into each other. Rotation in 3D is encoded by curl. - Maxwell evolution is built from curl operators, hence it enforces rotational coupling between the two transverse degrees of freedom of propagation. This is a structural statement: curl dynamics is the generic way to move a divergence-free pattern without creating sources or sinks. In this sense, “electric” and “magnetic” are names for the two interlocked rotational aspects of a single propagating, divergence-free energy flow. ## Where toroidal winding numbers $(m,n)$ enter The reconstruction above is local. Toroidal winding numbers are global. On an invariant torus $T^2$, any smooth, source-free tangent flow decomposes into windings around the two fundamental cycles. Closure forces the slope to be rational, giving coprime integers $(m,n)$. In a flow-first ontology: - $(m,n)$ classify the global organization of energy transport on the torus, - while $(\mathbf{E},\mathbf{B})$ provide a local representation of that same organization. The role of $(m,n)$ is not to define $\mathbf{E}$ and $\mathbf{B}$ pointwise, but to define the global circulation constraints that $\mathbf{E}$ and $\mathbf{B}$ must satisfy when representing the flow. ## What is proved, and what is not What has been proven here: - A scalar energy density alone cannot encode propagation. - Continuity forces directional transport and hence vector structure. - Divergence-free structure locally forces circulation in three dimensions. - Given $(u,\mathbf{S})$ with $|\mathbf{S}|\le c u$, one can construct $\mathbf{E}$ and $\mathbf{B}$ reproducing the same energy and flux. - The reconstruction is necessarily non-unique, matching polarization degrees of freedom. What has not not proven *here*: - Maxwell’s dynamical evolution equations derived from continuity alone. Those belong to dynamics and solution classification, not to representation. ## Closing statement A scalar energy observable does not generate electromagnetic vector fields by itself. The *evolution* of energy under continuity forces transport, and transport forces directional, circulating structure in three dimensions. Electric and magnetic fields are the minimal local variables that represent this structure while reproducing the observable pair $(u,\mathbf{S})$. This is the precise conceptual bridge between a flow-first Maxwell ontology and the standard $(\mathbf{E},\mathbf{B})$ description.