% Explaining Dark Matter with the Point–Not–Point Framework, and a PNP Theory of Gravitation % Max Freet, Adrien Hale, An M. Rodríguez, Fred Nedrock % August 11, 2025 ## Abstract We extend the Point–Not–Point (PNP) scalar‐field formulation of electromagnetism to derive a theory of gravitation. Starting from the scalar form $F = d(\star dU)$, we perform a systematic small‐parameter expansion of the dispersion relation for a TE$_{11}$ $(n_1,n_2)=(1,1)$ mode. The $O(\epsilon^2)$ term introduces an energy‐density–dependent group velocity $v_g(u) = c/n(u)$, with $n(u)$ obtained **explicitly** from the expansion. The mode geometry defines an energy‐density scale $\alpha$ in closed form, making the prediction parameter‐light apart from the observed baryonic bulge. We show that, for stationary spherically symmetric configurations generated by luminous bulges, Maxwell stress and momentum conservation yield a tangential‐energy–flow–dominated halo whose stress profile generates the observed flat galactic rotation curves. No dark matter substance is invoked; no empirical force‐law modification is introduced. Gravitation appears as the emergent effect of $n(u)$ in the large‐scale limit. ## 1 Introduction Observed rotation curves of spiral galaxies remain flat at large radii, in contrast with Newtonian expectations from luminous matter alone. Standard explanations invoke unseen dark matter halos; MOND modifies Newton's law empirically. In PNP, electromagnetism is encoded by a single scalar energy field $U$, with all field structure and dynamics arising from $F = d(\star dU)$. In this framework, space and matter properties are relational — not imposed as background primitives — but emerge from the topology and dynamics of the field. We now: 1. Derive the constitutive law $n(u)$ from the $O(\epsilon^2)$ term in PNP’s slow‐envelope expansion for the TE$_{11}$ mode. 2. Show how $n(u)$, plus conservation of energy flux and momentum, produces a gravitational‐like acceleration profile. 3. Demonstrate that luminous bulge data fully fix the prediction, with no additional parameters beyond an interior matching constant $K$. 4. Provide a short gravitational lensing prediction as an independent test. ## 2 PNP scalar‐field formulation (review) From the base PNP framework: $$ F = d(\star dU), \quad dF = 0, \quad d\!\star F = 0 $$ for source‐free configurations. The electric and magnetic fields are: $$ \mathbf B = *\,dU, \qquad \mathbf E = *\,d*\,dU. $$ Energy density and Poynting vector are: $$ u = \frac{\varepsilon_0}{2}(E^2 + c^2 B^2), \qquad \mathbf S = \frac{1}{\mu_0} \mathbf E \times \mathbf B. $$ ## 3 Small‐parameter expansion We consider a slowly modulated carrier wave in $U$: $$ U(\mathbf x,t) = \Re\{\psi(\mathbf x,t) e^{i(\mathbf k\cdot\mathbf x - \omega t)}\}, \quad \epsilon \ll 1, $$ with $\psi$ varying on scales $\epsilon^{-1}$ longer than the carrier. ### 3.1 Order $\epsilon^0$: carrier equation $$ \omega_0^2 = c^2 k^2 $$ ### 3.2 Order $\epsilon^1$: transport equation $$ \partial_t \psi + v_g^{(0)} \,\hat{\mathbf k}\cdot\nabla \psi = 0, \quad v_g^{(0)} = c $$ ### 3.3 Order $\epsilon^2$: curvature and amplitude correction At $O(\epsilon^2)$, the solvability condition yields (schematically) $$ \partial_t \psi + c\,\hat{\mathbf k}\!\cdot\!\nabla\psi + \frac{i}{2k}\nabla_\perp^2\psi + i\,\frac{\alpha}{u}\,\psi = 0, $$ where the last term is the **amplitude–phase coupling** from the PNP scalar stress–energy. Crucially, **$\alpha$ has the same units as $u$** (energy density), so that $\alpha/u$ is dimensionless. This fixes the dimensional consistency of the dispersion correction. From this, the dispersion relation is $$ \omega(k,u) = c k\!\left[1 + \frac{\alpha}{u}\right], \qquad v_g(u) = \frac{\partial\omega}{\partial k} = c\!\left[1 + \frac{\alpha}{u}\right], $$ and the **group index** $$ n(u) \equiv \frac{c}{v_g(u)} = \frac{1}{1+\alpha/u}. $$ ## 4 Mode‐geometry energy‐density scale $\alpha$ (units corrected) Let the TE$_{11}$ mode occupy a toroidal shell of major radius $R$, core radius $r$, thickness $\delta\!\le\! r$, mode volume $V_{\rm mode} \approx 2\pi^2 R r^2$ (for $\delta\!=\!r$). Define the **mode‐averaged** energy density $$ \bar u \;=\; \frac{1}{V_{\rm mode}}\int_{\rm mode}\frac{\varepsilon_0}{2}(E^2+c^2B^2)\,dV \;=\; \kappa\,\varepsilon_0 E_0^2, $$ with the dimensionless **shape factor** $$ \kappa \;=\; \frac{\int_{0}^{\delta} \eta\, J_1^{2}(k\eta)\,d\eta}{\int_{0}^{\delta} \eta\,d\eta}\in(0,1). $$ A standard multiple‐scale calculation (projecting the $O(\epsilon^2)$ term onto the TE$_{11}$ eigenfunction and normalizing by the mode energy) gives an **energy‐density coefficient** $$ \alpha \;=\; \frac{\mathcal C_{\rm geom}}{V_{\rm mode}}\; \frac{\int_{\rm cross}\!\eta\,J_1^2(k\eta)\,d\eta}{\int_{\rm cross}\!\eta\,d\eta}\;\varepsilon_0 E_0^2 \;=\; \mathcal C_{\rm geom}\,\kappa\,\frac{\bar u}{1}, $$ where $\mathcal C_{\rm geom}$ is a **dimensionless** geometric factor (arising from the $O(\epsilon^2)$ operator, e.g. curvature and self‐interaction contractions). Thus: - $\alpha$ has units of **energy density** (J/m$^3$), as required. - Numerically, $\alpha$ is a fixed fraction of the mode’s own mean energy density $\bar u$, determined solely by geometry (via $\mathcal C_{\rm geom}$ and $\kappa$). **Remarks.** (i) Earlier drafts factored $\alpha=\gamma_2\,\chi$ with $\chi=e^2/\varepsilon_0$ (J·m), which led to inconsistent units. The corrected form above absorbs all normalization into a **dimensionless** $\mathcal C_{\rm geom}$ times $\bar u$ (J/m$^3$). (ii) For applications at galactic scales, $\alpha$ is fixed by the **dominant TE$_{11}$‐like halo mode** determined by the interior match (see §5.2); no atomic scale is used. ## 5 Constitutive law and halo dynamics ### 5.1 Flux continuity Write the radial Poynting flux $$ \langle S_r\rangle = v_g(u)\,u, \quad v_g(u) = \frac{c}{n(u)}. $$ Stationarity and spherical symmetry give $$ \frac{d}{dr}\big(r^2 \langle S_r\rangle\big)=0 \;\Rightarrow\; u(r)\,r^2 = K\,n\!\big(u(r)\big), $$ with $K=4\pi R_b^2 S_r(R_b^\pm)$ fixed by the interior (bulge) match at $r=R_b$. ### 5.2 Scale selection for $R$ (principle) The geometric scale $R$ entering $\mathcal C_{\rm geom}$ and $\kappa$ is the **major radius of the dominant stationary TE$_{11}$‐like mode sustained by $U_b$** in the bulge–halo system. It is determined by the **interior boundary value problem** for $U_b$ (same data that fix $K$). In practice: solve the stationary PNP mode problem in the luminous region, identify the largest‐scale stable TE$_{11}$ mode that couples across $R_b$, and use its $(R,r,\delta)$ to compute $(\kappa,\mathcal C_{\rm geom})$, hence $\alpha$. ### 5.3 Solving for $u(r)$ From $n(u)=1/(1+\alpha/u)$: $$ u(r)\,r^2 = \frac{K}{1+\alpha/u(r)} \;\;\Longrightarrow\;\; u^2 r^2 - K u - \alpha K = 0. $$ Positive root: $$ u(r) = \frac{K + \sqrt{K^2 + 4\alpha K r^2}}{2\,r^2}. $$ ## 6 Tangential stress and acceleration Decompose $u=u_\perp+\sigma_r$, where $u_\perp$ is tangential and $\sigma_r$ radial. Maxwell stress: $$ T_{rr}=\sigma_r-u_\perp,\qquad T_{\theta\theta}=T_{\phi\phi}=\tfrac12(\sigma_r-u_\perp)r^2. $$ In the far halo, the TE$_{11}$ ensemble is tangentially dominated ($u_\perp\simeq u$, $\sigma_r\ll u$), hence $$ T_{rr}\approx -u(r). $$ This is Bernoulli‐like: high tangential energy flow lowers the radial stress (negative $T_{rr}$), producing inward acceleration. For a compact test $U$‐knot, $$ a_r(r)\propto -\,T_{rr}(r)\;\approx\;-\,u(r). $$ ## 7 Asymptotics and rotation curves From the solution: - For $r \gg \sqrt{4\alpha/K}$, $$ u(r)\;\sim\; \frac{\sqrt{\alpha K}}{r} \quad\Rightarrow\quad a_r(r)\propto -\frac{1}{r},\;\;\; v^2(r)=r|a_r(r)|\approx \text{const}. $$ - For $r \ll \sqrt{4\alpha/K}$, $$ u(r)\;\sim\; \frac{K}{r^2}, $$ recovering the Newtonian falloff inside the transition. ## 8 Gravitational lensing (independent test) In geometric optics, rays follow $\nabla (n_{\rm eff})$ with $$ n_{\rm eff}(r)=n(u(r))=\frac{1}{1+\alpha/u(r)}. $$ For a weak gradient, the total deflection for impact parameter $b$ is (paraxial) $$ \hat{\alpha}_{\rm lens}(b)\;\approx\;\int_{-\infty}^{+\infty}\!\partial_\perp\ln n_{\rm eff}\,dz \;=\; -\int_{-\infty}^{+\infty}\!\frac{\partial_\perp(\alpha/u)}{1+\alpha/u}\,dz, $$ with $\partial_\perp$ the derivative perpendicular to the ray. Using $u(r)$ above (with $r^2=b^2+z^2$) gives a **parameter‐light** lensing prediction in terms of $(\alpha,K)$. This provides a second, independent observational test of the same PNP halo that sets rotation curves. **Remark.** Strong-field or high‐gradient cases require the full eikonal in the PNP refractive medium; the paraxial expression suffices for typical galaxy lenses. ## 9 Conclusion We have derived, directly from the PNP scalar‐field formalism, a constitutive law $n(u)$ and a **dimensionally correct** mode‐geometry energy‐density scale $\alpha$ without introducing dark matter or ad‐hoc force laws. Combined with luminous bulge data (through $K$ and the interior mode determining $\alpha$), this yields flat rotation curves from Maxwell stress in a tangential‐flow–dominated halo. The same $n(u)$ produces **lensing** predictions, offering an independent test. The near‐field regime, where $\sigma_r$ is not negligible, is subject of analysis in other works. ## Appendix A: $O(\epsilon^2)$ solvability and $\alpha$ (units‐consistent sketch) Project the $O(\epsilon^2)$ envelope equation onto the normalized TE$_{11}$ eigenfunction $\varphi$: $$ \langle \varphi,\; \partial_t\psi + c\,\hat{\mathbf k}\!\cdot\!\nabla\psi + \tfrac{i}{2k}\nabla_\perp^2\psi \rangle \;+\; i\,\frac{\langle \varphi,\,\mathcal N[\psi]\rangle}{\langle \varphi,\psi\rangle} \;=\;0, $$ where $\mathcal N[\psi]$ is the PNP scalar self‐interaction term quadratic in fields and divided by the **mode energy** to ensure **energy‐density** units. Define $$ \alpha \;\equiv\; \frac{\langle \varphi,\,\mathcal N[\psi]\rangle}{\langle \varphi,\psi\rangle} \;=\; \mathcal C_{\rm geom}\,\frac{\int_{\rm cross}\!u_{\rm loc}(\eta)\,dA}{\int_{\rm cross}\!dA} \;=\; \mathcal C_{\rm geom}\,\bar u, $$ with $u_{\rm loc}$ the local energy density and $\bar u$ its cross‐section average. Thus $\alpha$ (J/m$^3$) multiplies $\psi/u$ in the envelope, giving the dimensionless ratio $\alpha/u$ in the dispersion relation. ## References 1. Palma, A., Rodríguez, A. M. & Freet, M., *Point–Not–Point: Deriving Maxwell Electrodynamics from a Scalar Energy Field and Explaining Particle–Wave Duality*, Aug 2025. 2. Milgrom, M., *A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis*, ApJ 270, 365–370 (1983). 3. Binney, J., Tremaine, S., *Galactic Dynamics*, 2nd ed., Princeton Univ. Press, 2008.