fractals · second study · one equation at two scales
Gravitational atom beside hydrogen
Hydrogen, and a cloud of very light particles around a spinning black hole. Drawn at the same scale they have the same shapes and the same ladder of levels. Gravity does the pulling instead of electric charge, and the centre is a horizon that can feed the cloud or swallow it.
Exploration, not peer reviewed, no new result claimed. By Aska Digital with Claude (Anthropic) for Foundation0. First study: the Witness Map.
00//The two atoms
one Bohr radius, same length in every panelSame shapes, same scale.
Strength of the pull, α = 0.374
α = rg/λc = (GM/c²)/(ħ/μc) = 0.049 AU ÷ 0.13 AU = 0.374. a₀ = rg/α² = 0.353 AU.
Hydrogen's is fixed by nature: α = e²/(4πε₀ħc) = 1/137. A gravitational atom's is the hole's size over the particle's wavelength, so it can be anything.
Each panel is a slice through the axis. In three dimensions each state is a doughnut around that axis. One Bohr radius, a₀, is the same length on screen in every panel; only what a₀ means in metres changes.
State 211
n = 2, ℓ = 1, m = 1 · most likely at 4a₀Point at (or tap) a panel to compare the same spot in both atoms.
State 322
n = 3, ℓ = 2, m = 2 · most likely at 9a₀Point at (or tap) a panel to compare the same spot in both atoms.
lines of equal |ψ|², at 5, 15, 30, 50, 70 and 90% of the peak
r = 4a₀ and 9a₀: the most likely distance for 211 and 322 (the density itself peaks closer in, at 2a₀ and 6a₀, but shells farther out hold more volume)
proton
horizon
01//The scale factor
The scale factor.
Size: Bohr radius
Energy: spacing of the levels
Both ladders follow E = μc²(1 − α²/2n²): the particle's rest energy μc², less a binding energy that falls as 1/n². Drawn in units of each atom's own ground-state binding, the two ladders are the same.
02//In numbers
Side by side.
| Hydrogen | Gravitational atom | |
|---|---|---|
| What pulls | electric charge | gravity |
| Strength of the pull, α | e²/(4πε₀ħc) = 0.0073 (1/137) | GMμ/(ħc) = 0.374 |
| Orbiting particle, rest energy μc² | electron, 511 keV | light boson, 10⁻¹⁷ eV |
| Bohr radius a₀ = ħ/(μcα) | 0.529 Å | 0.353 AU |
| Ground-state binding μc²α²/2 | 13.6 eV | 7×10⁻¹⁹ eV |
| Binding as a share of rest energy, α²/2 | 2.7×10⁻⁵ | 0.07 |
| First correction to the 211 binding | about 10⁻⁵ (fine structure) | −11% (322: −4.4%), BCP eq. 2.10 |
| Centre, in Bohr radii | proton, 1.6×10⁻⁵ | horizon, 0.16 |
| Recoil of the centre, μ/M | 5.4×10⁻⁴ | 1.8×10⁻⁹⁰ |
| Particles in one state | one electron | a vast number of bosons |
| Fate of the 211 state | drops to 100 in 1.6 ns, giving off ultraviolet light | grows: its pattern turns at ω = 0.368 c/rg, slower than the horizon's ΩH = 0.434 c/rg. Grows by a factor e every 45 years. |
| Fate of the 322 state | drops to 211 in 15.5 ns, giving off red light | grows: pattern speed ω/2 = 0.186 c/rg is below ΩH. Grows by a factor e every 31,000 years. |
| Time to grow a full 211 cloud | — | about 202 e-folds, from one particle to 7×10⁸⁷ (Arvanitaki, Baryakhtar and Huang 2015, eq. 8): 9,100 years. Leading order: at this α the true time is likely several times shorter. |
| Steady glow | none | gravitational waves at 4.8 mHz, for a real field (BCP eq. 4.1) |
03//Same and different
What is the same, what differs.
What is the same
- One equation. Far from the hole, the wave of a light particle obeys the same Schrödinger equation as hydrogen's electron, with a pull that weakens as 1/r (Baumann, Chia and Porto 2019, BCP below, eq. 2.5). Only the strength α and the particle's mass μ change.
- So the same solutions. Measured in its own Bohr radius a₀ = ħ/(μcα), each state has the same shape in both atoms. The two columns above are drawn from the same lines; only the labels differ.
- The same ladder. Levels at μc²(1 − α²/2n²) in both (eq. 2.8), and each cloud is most likely found at n²a₀ from the centre (eq. 2.9).
What differs
- The strength of the pull. Hydrogen's α is 1/137 in every hydrogen atom. Here α = rg/λc, the hole's size over the particle's wavelength, is 0.374. A cloud forms when the two are comparable (Arvanitaki, Baryakhtar and Huang 2015); for the 211 state to grow, α must be below about 0.44 at this spin.
- How tightly it is held. Hydrogen's electron is bound by 27 parts per million of its rest energy. This cloud is bound by 7%. The tighter the hold, the more relativity matters: the first correction to the 211 level is −11% here, against about 10⁻⁵ in hydrogen.
- The centre. A proton is 63,000 times smaller than a₀ and does not absorb the electron. The horizon's radius is 0.16 a₀, and it is not a wall: waves that reach it can fall in. So the levels are not quite steady (BCP 2019, eq. 2.11). A level whose pattern turns slower than the horizon, ω/m below ΩH, is fed by the hole's spin and grows; the rest drain into the hole (eq. 2.1; Brito, Cardoso and Pani 2020).
- One electron, or a crowd. Hydrogen has one electron. The cloud's particles are bosons, which pile into the same state, so a grown cloud holds an enormous number of them: GM²Δχ/(ħc) ≈ 8×10⁷⁶ for a 10-Sun hole that gives up 0.1 of its spin to the 211 state (Arvanitaki, Baryakhtar and Huang 2015, eq. 8, who round it to 10⁷⁶). That many particles in one state act as a classical wave.
- Excited, then what? Hydrogen's 211 and 322 states give up their energy as light within nanoseconds (NIST transition rates). Around a spinning hole the same states can grow instead, and a cloud of a real field hums in gravitational waves at twice the particle's frequency, here 4.8 mHz (BCP 2019, eq. 4.1).
Why 211 and 322
- Only states that turn with the hole can grow (m above zero), and growth slows steeply with ℓ: at small α the rate scales as α4ℓ+5 in units of c/rg (Detweiler 1980; BCP 2019, eq. 2.12). So 211, with ℓ = 1, grows fastest (BCP 2019, section 2.2): here it grows by a factor e every 45 years, 322 every 31,000 years (leading order).
- 211 grows until it stops qualifying. It takes spin from the hole until the horizon turns as slowly as its pattern: spin 0.990 falls to 0.954 (BCP 2019, eq. 2.2, ignoring the small loss of mass), if there is time. The 322 pattern turns at about half the rate (m = 2), so it still qualifies there and grows next, far more slowly (Arvanitaki, Baryakhtar and Huang 2015, section II B).
04//Method and checks
How it’s computed.
Everything runs live in this page from gravitational-atom/atom.js, the same file npm test checks. It is the leading order in α: hydrogen's formulas with gravity's α.
α = GMμ/(ħc) = rg/λca₀ = ħ/(μcα) = rg/α²En = μc²(1 − α²/2n²)r₊ = rg(1 + √(1 − χ²))grows if ω/m < ΩH = χc/(2r₊)
Checks against published numbers
- Hydrogen: 13.6057 eV and 0.529177 Å, the CODATA 2022 Rydberg energy and Bohr radius to 1 part in 10⁸. Letting the proton recoil gives 13.5983 eV against the measured 13.5984 eV; the gap is mostly fine structure, which this model leaves out.
- Baumann, Chia and Porto 2019: α ≈ 0.02 for 3 Suns and 10⁻¹² eV (eq. 2.6); α ≈ 0.37 and gravitational waves at about 5 mHz for 5 million Suns and 10⁻¹⁷ eV (section 4.3); 484 Hz per 10⁻¹² eV (eq. 4.1); the 211 level grows only below α ≈ 0.25 at spin 0.8 (section 2.2); clouds peak at n²a₀ (eq. 2.9).
- Growth rates: the 211 e-folding time is within 5× of Arvanitaki, Baryakhtar and Huang (2015, Table I) at α = 0.3, and 16× too slow at the fastest growth found numerically by Dolan (2007). Bosons in a grown cloud: 8×10⁷⁶ for 10 Suns, which Arvanitaki, Baryakhtar and Huang round to 10⁷⁶ (eq. 8).
- The wavefunctions add up to one particle and match the textbook closed forms. The drawn lines sit on their density levels to 1 part in 10⁹.
- All of these run as automated checks. Mistakes found along the way are in CORRECTIONS.md.
What it leaves out
- Everything past leading order. Corrections to the levels are of relative size about α² (BCP 2019, eq. 2.10), shown above for the current α. Near the hole, and for α above about 0.3, the true cloud is only roughly hydrogen-like; accurate clouds and growth rates need numerical work (for example Dolan 2007).
- Accurate growth rates. The rates here are leading order (Detweiler 1980, with the factor of 2 corrected as noted in BCP 2019, eq. 2.12). They are good at small α but too slow near the fastest growth: about 4× at α = 0.3 against Arvanitaki, Baryakhtar and Huang (2015, Table I) and 16× at α = 0.42 against the numerical peak of Dolan (2007). Read the times as upper limits there.
- The cloud's own gravity, the hole's small loss of mass as it spins down, the particles' interactions with each other, and anything else near the hole (gas, a companion star).
- For a real field the density is not exactly a still doughnut: a small part rotates with the pattern, and that part radiates the gravitational waves (BCP 2019, section 2.3).
- The centre is held fixed. In hydrogen the proton's recoil shifts the levels by about 1 part in 1,800; in the gravitational atom by μ/M, shown in the table.
Sources: Baumann, Chia & Porto 2019 · Brito, Cardoso & Pani, Superradiance (2020) · Arvanitaki, Baryakhtar & Huang 2015 · Detweiler 1980 · Dolan 2007 · CODATA 2022 · NIST Atomic Spectra Database · IAU 2015 nominal solar values