# Gravitational Lensing in a Maxwell Universe: Recovering the Factor of 2 from Flux Continuity ## The Problem: The Newtonian Deficit It is well known that treating light as a projectile with mass $m = E/c^2$ falling in a gravitational field yields a deflection angle of: $$ \theta_{\text{Newton}} = \frac{2GM}{Rc^2} \approx 0.875 \text{ arcseconds (at Solar limb)}. $$ Observation confirms the value is twice this: $$ \theta_{\text{Observed}} \approx 1.75 \text{ arcseconds}. $$ General Relativity resolves this by attributing half the bending to time dilation (the Newtonian part) and half to spatial curvature. In a **Maxwell Universe**, where there is no spatial curvature, we must explain this factor of 2 through electrodynamics. ## Gravity as a Dielectric Gradient As established in *Gravity as a Dielectric* (Rodriguez, 2025), we model the gravitational potential $\Phi$ as an increase in the electromagnetic density of the vacuum, creating an effective refractive index $n(\mathbf{r})$. $$ c(\mathbf{r}) = \frac{c_0}{n(\mathbf{r})}. $$ The standard assumption is that refractive index scales linearly with the potential energy density: $$ n(\mathbf{r}) \approx 1 + \frac{|\Phi|}{c^2} = 1 + \frac{GM}{rc^2}. $$ Applying Fermat’s Principle (or Snell’s Law) to this refractive profile yields: $$ \theta = \int_{-\infty}^{\infty} \nabla_\perp n \, dz = \frac{2GM}{Rc^2}. $$ This reproduces the Newtonian result (0.875''). The model appears to fail. ## The Error: Averaging the Flux The failure arises from a hidden assumption: treating the electromagnetic wave as a "particle" of mass. Mass, in the Maxwell Universe, is **Geometric Inertia**—a consequence of trapped, circulating energy. As derived in *Geometric Inertia* (2026), effective mass $m$ is related to total energy $E$ by the average forward propagation: $$ m = \frac{E}{c^2} \langle \sin^2 \psi \rangle, $$ where $\psi$ is the pitch angle of the flow. For a trapped particle (virialized knot), the energy is equipartitioned between circulation and translation. **Averaging Penalty:** When we define the "mass equivalent" of energy, we are effectively taking a time-average of a dynamic wave. This introduces a factor of **1/2**. The Newtonian calculation implicitly treats light as "matter moving at $c$." It applies the coupling rules of averaged matter to raw radiation. ## The Correction: Flux Continuity Light is not matter. It is pure, untrapped flux. In Maxwell theory, the primary ontological object is the **Poynting Vector $\mathbf{S}$**, not the scalar mass. When a wave propagates through a dielectric gradient, the bending is driven by the **instantaneous wavefront**, not the time-averaged energy envelope. ### The Symmetry Argument 1. **Massive Matter (Scalar Average):** Gravity acts on the trapped energy. The trapped energy is the "Sine" component of the flow. Due to the virial symmetry of the knot, only **half** the total field energy contributes to the inertial interaction in any single vector direction. $$\text{Coupling}_{\text{Matter}} \propto \frac{1}{2}$$ 2. **Free Radiation (Vector Flux):** Gravity (the dielectric gradient) acts on the flow $\mathbf{S}$. The flow is the "Cosine" component. For free radiation, $\cos(0) = 1$. The flow is fully aligned. There is no circulation, no averaging, and no "sine" component to dilute the interaction. $$\text{Coupling}_{\text{Flux}} \propto 1$$ ### The Ratio The ratio of the coupling strength of **Raw Flux** (Light) to **Averaged Mass** (Matter) is: $$ \frac{\text{Flux Coupling}}{\text{Mass Coupling}} = \frac{1}{1/2} = 2. $$ ## Result: The Eddington Number If the Newtonian prediction (based on mass equivalence) is $\theta_{N}$, then the Maxwell prediction (based on flux continuity) must be: $$ \theta_{\text{Maxwell}} = 2 \times \theta_{N}. $$ Substituting the Newtonian value: $$ \theta_{\text{Maxwell}} = 2 \times \left( \frac{2GM}{Rc^2} \right) = \frac{4GM}{Rc^2}. $$ $$ \theta_{\text{Maxwell}} \approx 1.75 \text{ arcseconds}. $$ ## Conclusion The "missing" bending angle in classical gravity is not a failure of Euclidean geometry, but a failure of the "mass-energy equivalence" heuristic. - **Matter** is time-averaged field energy (mass). - **Light** is instantaneous field flux. Treating light as mass incorrectly applies a 1/2 averaging penalty. Treating light as flux recovers the full interaction strength. The Maxwell Universe accurately predicts the 1.75 arcsecond deflection of starlight assuming only: 1. Maxwell's Equations. 2. Continuity of Energy. 3. Gravity as a dielectric modification of vacuum density. No curved spacetime is required.