# Light Speed as an Emergent Property of Electromagnetic Superposition ## Summary Electromagnetic waves interact through the total field they jointly create. Polarization is an electromagnetic field, not a material property. Phase-delayed secondary fields modify phase evolution. An effective, local light speed emerges from the electromagnetic energy density itself, as fields interfere pointwise. ### Goal Explain, mathematically and with an explicit ontological justification, how linear Maxwell theory allows electromagnetic waves to superpose and interact through the total field, and how an effective local propagation speed can arise from that total-field interaction. ### Corollary The maximum propagation speed of electromagnetic disturbances is $c_0$, with $$ c_0 = \frac{1}{\sqrt{\mu_0 \epsilon_0}}. $$ However, $c_0$ need not take the same effective value everywhere in the universe. When electromagnetic waves propagate in a linear, nontrivial electromagnetic environment, the total field gives rise to an effective susceptibility $\chi(\mathbf{x})$. The effective local propagation speed is then $$ c_{\text{local}}(\mathbf{x}) = \frac{c_0}{\sqrt{1+\chi(\mathbf{x})}}, $$ and therefore depends on the local electromagnetic energy density through the induced effective $\chi$. As a result, the effective speed of light varies from place to place, tracking spatial variations in electromagnetic field energy density across the universe. ## Derivation ### 1. Maxwell theory and superposition In source-free regions, Maxwell’s equations are $$ \nabla\cdot\mathbf{E}=0,\qquad \nabla\cdot\mathbf{B}=0, $$ $$ \nabla\times\mathbf{E}=-\partial_t\mathbf{B},\qquad \nabla\times\mathbf{B}=\mu_0\epsilon_0\,\partial_t\mathbf{E}. $$ These equations are linear: if $(\mathbf{E}_1,\mathbf{B}_1)$ and $(\mathbf{E}_2,\mathbf{B}_2)$ are solutions, then their sum is also a solution. Linearity applies to the field equations themselves and does not constrain how physical observables are constructed from the fields. ### 2. Quadratic field observables The electromagnetic energy density is $$ u=\frac{\epsilon_0}{2}\left(\mathbf{E}^2+c^2\mathbf{B}^2\right). $$ For a superposed field $\mathbf{E}=\mathbf{E}_1+\mathbf{E}_2$, $$ u=u_1+u_2+u_{12}, $$ with $$ u_{12}=\epsilon_0\left(\mathbf{E}_1\cdot\mathbf{E}_2+c^2\mathbf{B}_1\cdot\mathbf{B}_2\right). $$ Thus, overlapping waves produce cross terms in energy density and momentum flow. This constitutes a physical interaction at the level of observables. ### 3. Polarization written purely as fields In macroscopic electrodynamics, $$ \mathbf{D}=\epsilon_0\mathbf{E}+\mathbf{P}, $$ with linear response $$ \mathbf{P}=\epsilon_0\chi\,\mathbf{E}. $$ Maxwell’s equations do not specify the origin of $\mathbf{P}$. Formally, $\mathbf{P}$ is an additional electromagnetic field proportional to $\mathbf{E}$. Interpreted this way: - $\mathbf{P}$ is a secondary electromagnetic field. - The relation $\mathbf{P}=\epsilon_0\chi\mathbf{E}$ is a linear field–field coupling. - No ontological distinction between “matter” and “electromagnetic field” is required. ### 4. Phase delay and effective propagation In realistic linear response, proportionality is causal: $$ \mathbf{P}(t)=\epsilon_0\int_{-\infty}^{t}\chi(t-t')\,\mathbf{E}(t')\,dt'. $$ In frequency space, $$ \mathbf{P}(\omega)=\epsilon_0\chi(\omega)\mathbf{E}(\omega). $$ The secondary field associated with $\mathbf{P}$ is phase-delayed relative to the primary field. The total field is the superposition of the incident electromagnetic wave and the delayed secondary electromagnetic wave, producing a shifted phase evolution. ### 5. Emergence of an effective refractive index With $$ \mathbf{D}=\epsilon(\omega)\mathbf{E},\qquad \epsilon(\omega)=\epsilon_0\left[1+\chi(\omega)\right], $$ Maxwell’s equations yield the wave equation $$ \nabla^2\mathbf{E}-\mu_0\epsilon(\omega)\partial_t^2\mathbf{E}=0. $$ Plane-wave solutions satisfy $$ k^2=\mu_0\epsilon(\omega)\omega^2, $$ so the phase velocity is $$ v_p(\omega)=\frac{1}{\sqrt{\mu_0\epsilon(\omega)}}=\frac{c_0}{n(\omega)},\qquad n(\omega)=\sqrt{1+\chi(\omega)}. $$ The reduction in propagation speed follows directly from interference with a phase-delayed secondary electromagnetic field. ### 6. Two-wave interaction Let $$ \mathbf{E}=\mathbf{E}_1+\mathbf{E}_2. $$ By linearity, $$ \mathbf{P}=\epsilon_0\chi(\mathbf{E}_1+\mathbf{E}_2) =\epsilon_0\chi\mathbf{E}_1+\epsilon_0\chi\mathbf{E}_2. $$ Each wave generates a secondary electromagnetic field, and both contribute to the total response. Each wave therefore propagates in an environment determined by the combined electromagnetic configuration. ### 7. Local effective light speed The local electromagnetic energy density is $$ u(\mathbf{x})=\frac{\epsilon_0}{2}\left(\langle\mathbf{E}^2\rangle+c^2\langle\mathbf{B}^2\rangle\right). $$ The effective refractive index is $$ n(\mathbf{x})=\sqrt{1+\chi(\mathbf{x})}, $$ and the local propagation speed is $$ c_{\text{local}}(\mathbf{x})=\frac{c_0}{n(\mathbf{x})}. $$ ### 8. Consequence Different regions of the universe, characterized by different electromagnetic field configurations and energy densities, exhibit different effective propagation speeds. Maxwell’s equations remain unchanged. The result follows solely from superposition, linear response, and a field-only interpretation of polarization. --- DOI: https://writing.preferredframe.com/doi/10.5281/zenodo.18209801/