Hawking Radiation

Hawking temperature

T = ℏ · c³ / (8π · G · M · k_B)

For a Schwarzschild (non-rotating) black hole of mass M.

Black hole mass	Hawking temperature
Solar mass (2 × 10³⁰ kg)	6 × 10⁻⁸ K (60 nK)
Earth mass	0.02 K
Moon mass	1.7 K
1 kg	1.2 × 10²³ K (extreme)
10⁻¹⁸ kg (atom mass)	10⁴¹ K (?? — quantum gravity territory)

Black hole lifetime

τ = 5120 · π · G² · M³ / (ℏ · c⁴)

Lifetime ∝ M³ — small BHs evaporate quickly.

Mass	Lifetime
1 kg	~10⁻²⁰ s (instant)
10⁹ kg (mountain)	~10⁹ years (~age of universe)
10²⁰ kg (asteroid)	~10⁴⁰ years
1 M_sun	~10⁶⁷ years
10⁹ M_sun (supermassive)	~10⁹⁵ years

JavaScript — Hawking radiation

const h_bar = 1.055e-34;
const c = 3e8;
const G = 6.674e-11;
const k_B = 1.381e-23;

// Hawking temperature
function hawkingTemp(mass_kg) {
  return h_bar * c * c * c / (8 * Math.PI * G * mass_kg * k_B);
}

console.log(`1 M_sun BH: T = ${hawkingTemp(1.989e30).toExponential(2)} K`);
console.log(`Earth-mass BH: T = ${hawkingTemp(5.972e24).toExponential(2)} K`);
console.log(`Moon-mass BH: T = ${hawkingTemp(7.342e22).toExponential(2)} K`);

// Lifetime
function blackHoleLifetime(mass_kg) {
  // τ = 5120 · π · G² · M³ / (ℏ·c⁴)
  return 5120 * Math.PI * G * G * Math.pow(mass_kg, 3) / (h_bar * Math.pow(c, 4));
}

console.log(`1 kg BH: ${blackHoleLifetime(1).toExponential(2)} s`);
console.log(`10⁹ kg BH: ${(blackHoleLifetime(1e9) / (365.25*86400)).toExponential(2)} years`);
console.log(`Solar BH: ${(blackHoleLifetime(1.989e30) / (365.25*86400)).toExponential(2)} years`);

// Power radiated
function hawkingPower(mass_kg) {
  const T = hawkingTemp(mass_kg);
  // Using Stefan-Boltzmann with Schwarzschild radius
  const r_s = 2 * G * mass_kg / (c * c);
  const area = 4 * Math.PI * r_s * r_s;
  const sigma = 5.67e-8;
  return sigma * area * Math.pow(T, 4);
}

console.log(`1 kg BH power: ${hawkingPower(1).toExponential(2)} W`);
console.log(`Solar BH power: ${hawkingPower(1.989e30).toExponential(2)} W`);

// Hawking radiation has thermal spectrum at T_Hawking
// Compared to CMB temperature (2.725 K)
function bhVsCmb(mass_kg) {
  const T_H = hawkingTemp(mass_kg);
  const T_CMB = 2.725;
  return T_H / T_CMB;
}

console.log(`Solar BH T_H / T_CMB: ${bhVsCmb(1.989e30).toExponential(2)}`);
// ~10⁻⁸ — BH much colder than CMB; net absorbs more than emits

// Black hole entropy
function bekensteinEntropy(mass_kg) {
  const r_s = 2 * G * mass_kg / (c * c);
  const A = 4 * Math.PI * r_s * r_s;
  const l_p_squared = h_bar * G / Math.pow(c, 3);  // Planck length²
  return A / (4 * l_p_squared) * k_B;  // J/K (in conventional units)
}

console.log(`Solar BH entropy: ${bekensteinEntropy(1.989e30).toExponential(2)} J/K`);
// Massive number — far more than equivalent ordinary matter

Where Hawking radiation matters

• Theoretical physics. Combining QM and GR; foundation for quantum gravity.
• Information paradox. Active research area connecting quantum info to gravity.
• Primordial black holes. If small ones formed in early universe, Hawking radiation provides observable signature.
• Cosmology. BH evaporation matters for very long-term cosmological evolution.
• Analog gravity. Sonic horizons in fluids and BEC analogs of Hawking radiation studied.
• Holographic principle. Bekenstein-Hawking entropy suggests BH info stored on horizon — leads to holography.
• Black hole thermodynamics. Four laws of BH thermo emerged from this work.

Common mistakes

• Believing radiation comes from inside BH. Particles created at event horizon, NOT from interior. Inside is causally disconnected.
• Treating it as classical. Hawking radiation is purely quantum — vacuum fluctuations near horizon, not classical heat.
• Expecting solar mass BHs to glow visibly. Stellar BH temperatures are 10⁻⁸ K — vastly colder than CMB. Can't be detected directly.
• Confusing evaporation with collapse. Evaporation slowly removes mass via radiation. Collapse — already happened to form BH. Different stages.
• Forgetting M³ scaling. Smaller BHs evaporate WAY faster. Stellar BH lives 10⁶⁷ yr; primordial 10⁹ kg BH evaporates in age of universe.
• Treating information paradox as solved. Active research; modern ideas (firewalls, holographic, etc.) but no consensus on full resolution.