Hawking evaporation vs accretion. A thought experiment with real equations. The 3D scene is dramatised; the numbers are not.
The science
Event horizon. The Schwarzschild radius is the size of the hole's horizon.
r_s = 2GM / c^2
Hawking radiation. Quantum effects at the horizon make the hole glow like a black body and slowly lose mass. Smaller holes are hotter and evaporate faster.
T = ℏc^3 / (8π G M k_B)
P = ℏc^6 / (15360 π G^2 M^2)
t_life = 5120 π G^2 M^3 / (ℏ c^4)
A hole of about 1.7×1011 kg has a lifetime equal to the age of the universe (13.8 Gyr).
Accretion. Rock falling in feeds the hole. The Bondi-Hoyle estimate for a hole moving at speed v through matter of density ρ and sound speed c_s is
Mdot_BH = 4π G^2 M^2 ρ / (c_s^2 + v^2)^(3/2)
r_B = G M / (c_s^2 + v^2) (region of influence)
Infalling matter shines. If its luminosity reaches the Eddington limit, radiation pressure resists further inflow:
L_Edd = 4π G M m_p c / σ_T
Mdot_Edd = L_Edd / (η c^2), η = 0.1 (assumed)
The "radiation-limited" model uses min(Mdot_BH, Mdot_Edd) but never less than the geometric rate ρ v π r_s^2. The "unimpeded" model uses Mdot_BH alone and is an upper bound.
Gravity and tides at distance d from the hole, across a body of length L = 1.8 m:
g = G M / d^2 a_tidal = 2 G M L / d^3
Inside Earth. The planet is treated as uniform for the motion (oscillation period 2π sqrt(R^3 / G M_E) = 84 min). Accretion uses layer-averaged density and P-wave speed (PREM-like, rounded). Accreted rock starts at rest, so momentum conservation slows the hole: a circular orbit shrinks as r ∝ M-1/2; a radial oscillation as amplitude ∝ 1/M.
Sources (plain text). Hawking, Nature 248, 30 (1974); Bondi and Hoyle, MNRAS 104, 273 (1944); Giddings and Mangano, Phys. Rev. D 78, 035009 (2008) on micro black holes and Earth; Dziewonski and Anderson, PREM, Phys. Earth Planet. Inter. 25, 297 (1981); CODATA 2018 constants.
What is uncertain or simplified.
- Bondi accretion assumes a smooth gas. Solid rock, its strength, and the radiation pushed back by the infalling matter all change the real rate, possibly by many orders of magnitude. That is why two bounds are shown.
- The 1D, uniform-density Earth, constant speed on a straight chord, and a fixed efficiency of 10% are simplifications.
- For masses above about 1022 kg the hole's own gravity strongly reshapes Earth, and a point-mass model of the motion is only indicative.
- Hawking radiation has never been observed directly. The formulas are standard theory.
- Dramatised in the 3D view: the hole is drawn enormously larger than its real horizon; its path and the inflow particles are schematic; the crust is drawn thicker than real; the motion preview runs about 10 minutes of real time per second.
No black hole of these masses is known to exist near Earth. This is a calm what-if for learning, not a forecast.