Gravity in Oscillation Rates: General Relativity's Standard Formulas Written Without Mass, Energy, or Newton's Constant G

I’ve been playing with a question. Quantum mechanics says every particle is an oscillator, and that its energy and its oscillation rate are the same thing (E = ħω — Planck’s rule, a hundred years old). So what happens if you take the oscillation as the real thing and treat “mass” and “energy” as just names for how fast something ticks — and Newton’s constant G as just the Planck time in other units? I tried rewriting the standard gravity formulas that way, one at a time. Newton’s law. The slowing of clocks near a heavy body. The Schwarzschild radius of a black hole. Hawking’s temperature. The expansion of the universe. The de Broglie wavelength. Three substitutions do all the work, and all three are identities, not assumptions: • Energy becomes oscillation rate: E = ħΩ, where Ω is the body’s total oscillation rate — the sum of the rates of every quantum excitation in it: quarks, gluons, electrons, and anything else carrying energy. • Mass becomes the same quantity divided by c²: m = ħΩ/c² (from E = mc²). • Newton’s constant becomes the Planck time: G = t_P²·c⁵/ħ (that’s just the definition of t_P, turned around). Make those replacements and every formula came out matching general relativity exactly. Newton’s law, for example: • Newton: F = G·m₁·m₂/r² • In rates: F = t_P²·ħc·Ω₁·Ω₂/r² That’s not a surprise, in one sense — I only substituted identities, so it couldn’t do otherwise. But the result reads differently: • Mass–energy equivalence (E = mc²) → m = ħΩ/c²: a particle’s mass is the inertia of a confined oscillation. You can derive that in about fifteen lines from the Doppler shift alone; no mass has to be assumed. • Newton’s law of gravitation (F = Gm₁m₂/r²) → F = t_P²·ħc·Ω₁Ω₂/r²: gravity becomes a coupling between oscillation rates, with strength t_P² (the Planck time squared). G drops out of every formula. • Gravitational time dilation (GM/c²r) → t_P²·Ω/τ: a body slows every clock around it by its own oscillation rate, scaled by t_P², divided by how long light takes to reach it (τ = r/c). Rates slow rates. • Schwarzschild radius (r = 2GM/c²) → r_h = 2·(Ω·t_P)·ℓ_P: a black hole’s horizon sits at twice the number of oscillations the body completes in one Planck time, measured in Planck lengths. • Hawking temperature (kT = ħc³/8πGM) → ω_H = 1/(8π·t_P²·Ω): a black hole glows at the reciprocal of its own oscillation rate. • Friedmann equation (H² = 8πGρ/3) → H² = (8π/3)·t_P²·c³·Ω_V: the universe expands at a rate set by how much oscillation each cubic meter of space holds. • Universality of free fall (the equivalence principle) → N·Ω_bound = constant per kilogram: heavier atoms have more oscillators per kilogram, but each one oscillates slower by its share of the binding energy — and the two cancel exactly, because the count and the rate are set by the same binding energy. That is why every material falls at the same rate. To be clear: this is a translation, not a new theory. It predicts nothing that GR doesn’t. What it changes is that, in the equations of gravity, how fast particles oscillate is treated as fundamental, rather than mass, energy, or the gravitational constant G. One consequence is that the Planck scale, where quantum mechanics and gravity are supposed to meet, ends up built into the gravity formulas themselves. Written in rates, the Planck time is the only gravitational constant there is; whether that is a clue about where gravity comes from, or only a coincidence of units, is the question this exercise cannot settle. So could it be that particle oscillations create gravity? Nearly all of a proton’s mass is the motion of the quarks and gluons inside it, oscillating at enormous rates. Could that, along with the other oscillations in an atom, be what curves spacetime? The attached document carries each of these GR formula rewrites through step by step, states the rules used, checks every result numerically against its standard value, and lists the references.