A thought experiment: if we could create a black hole in a laboratory, what is the smallest mass that could consume the Earth before evaporating via Hawking radiation?
Two processes compete to determine a black hole's fate:
- Hawking radiation — quantum effects cause the black hole to emit particles and lose mass. Smaller black holes evaporate faster.
- Bondi accretion — the black hole gravitationally captures surrounding matter. Larger black holes accrete faster.
The critical mass is the unstable equilibrium where these balance. Above it, growth runs away. Below it, evaporation wins decisively.
For all emitted species:
where f accounts for all particle species weighted by spin-dependent greybody factors (Page 1976):
| Species | DOFs | Greybody weight per DOF | Contribution |
|---|---|---|---|
| Photons (spin-1) | 2 | 0.23 | 0.46 |
| Neutrinos (spin-1/2) | 6 | 0.55 | 3.30 |
| Gravitons (spin-2) | 2 | 0.014 | 0.03 |
| Total f | 3.79 |
(Valid for Hawking temperature below ~0.5 MeV, i.e., M > ~10¹¹ kg, where electron-positron emission is suppressed.)
A black hole at rest in a medium of density ρ and sound speed c_s gravitationally funnels material inward:
The key asymmetry: accretion scales as M² while evaporation scales as 1/M². This makes the equilibrium unstable — any perturbation above critical mass leads to runaway growth; below it, runaway evaporation.
Setting the two rates equal and solving for M:
Using ρ = 1.2 kg/m³, c_s = 343 m/s:
- ρ/c_s³ = 1.2 / 343³ = 3.0 × 10⁻⁸
Using ρ = 5500 kg/m³, c_s = 5000 m/s:
- ρ/c_s³ = 5500 / 5000³ = 4.4 × 10⁻⁸
These ratios are surprisingly similar — the higher density of rock is mostly cancelled by the higher sound speed. So it barely matters that the black hole starts in air. Plugging in mantle values:
At this mass:
- Both rates ≈ 3 × 10⁻⁶ kg/s
- Hawking temperature ≈ 0.1 MeV (100 keV)
- Schwarzschild radius ≈ 7 × 10⁻¹⁷ m (smaller than a proton)
- Bondi capture radius ≈ 100 nm
At the critical mass, the Hawking luminosity is roughly 100–300 GW. That energy gets deposited near the black hole (the mean free path for 100–200 keV photons in rock is only ~2 cm). This creates a superheated cavity around the black hole.
Comparing to the Eddington luminosity — the threshold where radiation pressure blows away infalling material:
For 200 keV photons in rock (κ ≈ 0.01–0.1 m²/kg), L_Edd ≈ 1–10 GW at the critical mass. The Hawking luminosity exceeds this by a factor of 30–300. The black hole is super-Eddington — its own radiation fights the accretion.
However, the optical depth across the Bondi radius is tiny (τ ≈ 5 × 10⁻⁶), so most radiation escapes without directly pushing on infalling material. The dominant effect is thermal: the radiation heats surrounding rock into a hot plasma cavity, increasing the effective sound speed by ~5–10× and reducing density at the accretion point.
Since accretion ∝ ρ/c_s³, a 7× increase in c_s with a 10× drop in local density gives a ~3000× reduction in accretion rate. The critical mass scales as (reduction factor)^(1/4), pushing it up by roughly a factor of 7.
Deep inside the Earth (at the core, where overburden pressure is 360 GPa), the ambient pressure confines the cavity, preventing it from expanding too far. A self-consistent treatment gives:
Say you create a black hole just above critical mass (~10¹¹ kg, about 100 million tonnes) in a lab:
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t = 0: It's exposed to air. Nothing visible happens — it's smaller than a proton. It immediately begins free-falling.
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t ≈ 0.5 s: It passes through the floor. The geometric cross-section is ~10⁻³² m², so it interacts with essentially nothing on the way through — like a neutrino with mass.
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t ≈ 21 minutes: It reaches Earth's center for the first time, traveling at ~8 km/s. It passes straight through and out the other side.
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Over millennia to millions of years: It oscillates back and forth through the Earth. Dynamical friction (gravitational drag on surrounding matter) slowly dissipates its kinetic energy. Settlement time to the center: ~10⁷ years.
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Over ~100 million years: Sitting near the center, Bondi accretion wins over Hawking radiation. The mass doubles, then doubles again, accelerating each time.
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Eventually: The black hole grows large enough that its event horizon becomes macroscopic, and it rapidly consumes the remaining Earth. The final stages are fast.
| Quantity | Value |
|---|---|
| Critical mass (no feedback) | ~5 × 10¹⁰ kg |
| Critical mass (with radiation feedback) | ~10¹¹ kg (best estimate) |
| Schwarzschild radius | ~10⁻¹⁶ m (1/10 of a proton) |
| Hawking temperature | ~100–300 keV |
| Hawking luminosity at critical mass | ~100–300 GW |
| Time to consume Earth | ~500 Myr – 1 Gyr |
| Initial condition in air | Negligible effect |
The answer: roughly 10¹¹ kg — about 100 million tonnes, comparable to a ~200 m rocky asteroid, compressed into a volume smaller than a proton. The "exposed to air" initial condition is essentially irrelevant; the black hole passes through the lab floor in under a second and its fate is sealed by conditions inside the Earth.
Once above the critical mass, Hawking radiation quickly becomes irrelevant and the growth follows:
This has an exact solution with finite-time blowup:
where the characteristic time is:
Using Earth's core conditions (ρ ≈ 12,000 kg/m³, c_s ≈ 10 km/s) and M₀ = M_crit ≈ 10¹¹ kg:
But near M_crit, Hawking radiation nearly cancels accretion, so the initial growth is exponential rather than power-law. The net equation near M_c:
with e-folding time τ = t_blow/4 ≈ 130 Myr. Starting 1% above critical mass, the slow exponential phase adds about 600 Myr before the explosive M² growth takes over.
Total time, critical mass to complete consumption: roughly 500 Myr to 1 Gyr.
The M² growth law means the time remaining from any mass M is just 1/(MA). Almost all the time is spent at low mass. The destruction of Earth is, for hundreds of millions of years, completely invisible — then catastrophic in the final hours.
| Time remaining | BH mass (kg) | r_Schwarzschild | Accretion rate | What's happening |
|---|---|---|---|---|
| ~800 Myr | 10¹¹ | 0.15 fm (< proton) | ~0 net | Exponential growth fighting Hawking radiation |
| ~50 Myr | 10¹² | 1.5 fm (proton) | 0.6 g/s | Hawking now negligible; pure M² growth begins |
| ~53,000 yr | 10¹⁵ | 1.5 pm (nucleus) | 600 kg/s | Still subatomic, deep in Earth's core |
| ~53 yr | 10¹⁸ | 1.5 nm (molecule) | 600 kt/s | First anomalous deep-Earth heating detectable |
| ~193 days | 10²⁰ | 150 nm (virus) | 6 Gt/s | Core heating exceeds all of Earth's geothermal output |
| ~19 days | 10²¹ | 1.5 μm (bacterium) | 600 Gt/s | Inner core destabilizing; extreme seismicity |
| ~46 hours | 10²² | 15 μm (human cell) | 60,000 Gt/s | Gravity sphere of influence reaches ~600 km |
| ~4.6 hours | 10²³ | 0.15 mm (sand grain) | 6×10⁶ Gt/s | Inner core in free-fall; influence radius > 1200 km |
| ~28 min | 10²⁴ | 1.5 mm (lentil) | 6×10⁸ Gt/s | Whole core consumed; mantle collapsing inward |
| ~5 min | 6×10²⁴ | 9 mm (marble) | — | Earth is gone |
99.99% of the total time, the black hole is smaller than a virus and the Earth looks perfectly normal. No instrument could detect it.
The last ~50 years are when the accretion luminosity (gravitational energy released as infalling material thermalizes) begins to rival Earth's geothermal heat flow of 44 TW. Deep-earth temperatures rise anomalously.
The last ~7 years, the accretion luminosity matches the total solar power intercepted by Earth. The core is melting in ways no geophysics model can explain.
The last ~2 days, the gravitational sphere of influence (where the BH's pull exceeds Earth's self-gravity) reaches ~600 km. The core structurally fails. Devastating global seismicity.
The last ~5 hours, the inner core is in free-fall. The mantle follows. The surface begins to visibly deform and subside.
The last ~15 minutes, the remaining crust and mantle are in free-fall:
The entire Earth collapses into a marble-sized event horizon.
The M² growth law means doubling time shrinks as mass grows: the time for each doubling is half the previous one. Going from 10¹¹ to 10¹² kg takes 480 Myr. Going from 10²³ to 10²⁴ kg takes 28 minutes. Same factor of 10, separated by a factor of 10¹⁰ in time. The finite-time singularity in the ODE is doing real physical work here — an observer on the surface would have only hours of warning before the end.
What if the same 10¹¹ kg black hole were placed in orbit between Earth and the Moon instead of embedded in matter?
In interplanetary space the ambient density is effectively zero (solar wind is ~10⁻²⁰ kg/m³ — accretion from it is 14 orders of magnitude below the Hawking rate). The black hole is purely evaporating.
From dM/dt = −fℏc⁴/(15360πG²M²), integrating to zero:
With f = 3.79 and M₀ = 10¹¹ kg:
The evaporation time (~700 Myr) is nearly the same as the Earth-consumption time (~500 Myr – 1 Gyr). This isn't coincidental — the critical mass is defined as where accretion and evaporation balance, so both rates give similar timescales at that mass. The same black hole either consumes a planet or quietly evaporates depending entirely on whether you put stuff around it.
The black hole spends most of its lifetime barely changing. Since M(t) = M₀(1 − t/t_evap)^{1/3}, the mass stays within 10% of M₀ for the first 70% of the lifetime. Then the ending is abrupt.
For 700 million years: a ~100 GW point source of ~100 keV gamma rays and neutrinos. At the Moon's distance from Earth, the flux is ~10⁻⁷ W/m² — invisible to the naked eye but detectable by a gamma-ray telescope as a faint, bizarrely hard point source with no spectral lines.
Last ~500,000 years (M drops below 10¹⁰ kg): temperature exceeds 1 MeV, electron-positron pairs switch on, luminosity climbing to ~10 TW.
Last ~50 years (M ~ 10⁸ kg): T ~ 100 MeV, muons and pions emitted. Luminosity ~10⁶ TW. Visible as a bright gamma-ray star.
Last ~1 second (M ~ 10⁶ kg): T ~ 100 GeV, all Standard Model particles pouring out — quarks, W/Z bosons, Higgs. Luminosity ~10²³ W (a fraction of solar luminosity concentrated in hard gamma). Energy released: ~20 megatons of TNT.
Last microsecond (M ~ 10⁴ kg): luminosity ~10²⁹ W. The remaining ~10²¹ J converts to a burst of every elementary particle in nature. The final flash, seen from Earth at ~300,000 km, would be a brief gamma-ray pulse roughly 150× the intensity of sunlight — lasting about a millisecond.
Then nothing. A marble-mass of energy has been returned to the universe as radiation, almost entirely as neutrinos that passed through the Earth without a trace.
| Constant | Symbol | Value |
|---|---|---|
| Gravitational constant | G | 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻² |
| Reduced Planck constant | ℏ | 1.055 × 10⁻³⁴ J·s |
| Speed of light | c | 3 × 10⁸ m/s |
| Boltzmann constant | k_B | 1.381 × 10⁻²³ J/K |
| Earth mass | M_E | 5.97 × 10²⁴ kg |
| Earth radius | R_E | 6.371 × 10⁶ m |
- Page, D. N. (1976). "Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole." Physical Review D, 13(2), 198.
- Bondi, H. (1952). "On spherically symmetrical accretion." Monthly Notices of the Royal Astronomical Society, 112(2), 195–204.
- Hawking, S. W. (1974). "Black hole explosions?" Nature, 248(5443), 30–31.