Electromagnetic–Gravitational Hierarchy As A Charge-To-Mass Ratio

Overview This pedagogical review and quantitative audit rigorously tests the intuitive hypothesis that gravitation might be sourced by a body’s constituent electrons rather than its total mass–energy. Across the periodic table (hydrogen excluded), the mass carried per electron averages $2.32 \pm 0.17\text{ u}$, making electron count an accurate empirical proxy for mass to within $\pm 7\%$. However, through a systematic analysis of precision experimental bounds, relativistic mass accounting, and astrophysical systems, the paper demonstrates that electron number is an empirically falsified source of gravity and establishes a multi-scale scaling architecture for how matter coheres and accretes. Key Findings & Structure Equivalence Principle Constraints: By recasting seven independent equivalence-principle tests in terms of electron-to-mass contrast $\Delta(Z/A)$, the fraction $f$ of gravity that could be sourced by electron count is bounded to $f < 1.1 \times 10^{-14}$ (established by the MICROSCOPE satellite's Ti–Pt comparison across a $13.9\%$ compositional contrast). The $^{85}\text{Rb}\text{–}^{87}\text{Rb}$ dual-species atom-interferometry test holds electron number strictly fixed ($Z=37$), independently proving that neutrons gravitate to 4 parts in $10^{12}$. The Gravitational Weight of Charge: Adding surplus electrons genuinely increases gravitational mass via both rest mass ($N m_e$) and electrostatic field energy ($U/c^2 = Q^2 / 8\pi\varepsilon_0 R c^2$). However, for a $1\text{ kg}$ aluminum sphere charged to the dielectric breakdown limit of air, the fractional mass gain is only $4.3 \times 10^{-18}$ ($\sim 10^7$ below modern balance precision), while the resulting electrostatic repulsion exceeds mutual gravitational attraction by a factor of $5.9 \times 10^7$. Compressed Neutron Matter: Neutron stars provide a decisive counterexample: despite carrying an electron fraction of only $Y_e \approx 0.05\text{--}0.10$, they produce the strongest observable gravitational fields. Furthermore, assembling $1.87 \times 10^{57}$ free neutrons into a neutron star radiates away $0.18\,M_\odot$ of gravitating mass as binding energy at fixed particle number—a result strictly incompatible with particle-counting models and accurately predicted only by $E/c^2$. Cosmology & Accretion Dynamics: Net electric charge is forbidden from driving macroscopic accretion due to exact $U(1)_{\text{EM}}$ gauge symmetry, extreme Debye screening ($\lambda_D \approx 35\text{ pm}$ in the solar core), and the Rayleigh fission limit. Conversely, separated charge (e.g., tribocharging in protoplanetary discs) and free-electron gating of the Magnetorotational Instability (MRI) play essential, constructive roles in material assembly and angular momentum transport. Scaling Architecture of Matter (The Four Regimes): A single structural criterion—cohesion vs. agitation—governs five rungs of assembly across 14 decades of length, with governing regimes determined strictly by dimensional scaling exponents: Thermal agitation ($\propto R^0$, below $0.46\text{ nm}$): Governs phase transitions via the Lindemann criterion and Frenkel relaxation. The transition point naturally lands at molecular dimensions ($0.37\text{ nm}$ for $\text{N}_2$) without parameter tuning. Van der Waals cohesion ($\propto R^1$, $0.46\text{ nm}$ to $36\ \mu\text{m}$): Governs dust adhesion and sticking limits. Electrostatic interaction ($\propto R^3$, $36\ \mu\text{m}$ to $1.74\text{ m}$): Overcomes the protoplanetary bouncing barrier via tribocharged aggregates. Gravitational binding ($\propto R^5$, above $1.74\text{ m}$): Governs planetesimals, planets, and stars via Jeans/virial collapse. Charge-to-Mass Hierarchy Reframing: The transition from electrostatic dominance to gravitational binding is governed by the universal, scale-free critical threshold: $$\left(\frac{Q}{M}\right)_{\text{crit}} = \sqrt{4\pi\varepsilon_0 G} = \frac{q_P}{m_P} \approx 8.62 \times 10^{-11}\text{ C/kg}$$ This exact constant governs the laboratory assembly crossover, the extremal Reissner–Nordström black hole horizon limit, and astrophysical plasma disruption. Gravity dominates the largest scales not through raw coupling strength, but because mass–energy is the unique physical quantity that is strictly additive, strictly positive, volumetric, and unshieldable.