Hawking Temperature

Equivalent forms

k_B T_H = \frac{\hbar c^3}{8\pi G M}

T_H = \frac{\hbar \kappa}{2\pi c k_B}

Unites general relativity, quantum mechanics, and thermodynamics in a single equation — h, c, G, k_B all appear together.

Symbol	Name	SI unit	Dimension	Range
$T_H$	Hawking Temperatureoutput Effective blackbody temperature of black hole emission	K	Θ	0 – 10000000000
$M$	Black Hole Mass Mass of the (non-rotating, uncharged) black hole	kg	M	10000000000 – 1e+40
$hbar$	Reduced Planck Constant ℏ = h / (2π)	J·s	ML^2T^-1	1.054571817e-34 – 1.054571817e-34
$G$	Gravitational Constant Newton's gravitational constant	m^3·kg^-1·s^-2	M^-1L^3T^-2	6.674e-11 – 6.674e-11
$c$	Speed of Light Speed of light in vacuum	m/s	LT^-1	299792458 – 299792458
$k_B$	Boltzmann Constant Boltzmann constant	J/K	ML^2T^-2Θ^-1	1.380649e-23 – 1.380649e-23

Unit systems

Symbol	SI	CGS	Imperial
$T_H$	K	K	K
$M$	kg	g	lb
$hbar$	J·s	erg·s	J·s
$G$	m^3·kg^-1·s^-2	cm^3·g^-1·s^-2	ft^3·lb^-1·s^-2
$c$	m/s	cm/s	ft/s
$k_B$	J/K	erg/K	J/K

Where it holds

Semiclassical regime: M ≫ Planck mass (2.18e-8 kg). For non-rotating, uncharged (Schwarzschild) black holes. Kerr–Newman has additional spin and charge corrections.

Dimensional analysis

T_H \to [\Theta ]

[ML^2T^-1][LT^-1]^3 / ([M^-1L^3T^-2][M][ML^2T^-2\Theta ^-1]) = [\Theta ]

Both sides temperature.

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Discovery

Stephen Hawking · 1974

Hawking's 1974 calculation showed black holes are not truly black. Quantum field theory in curved spacetime predicts thermal emission with T ∝ 1/M — a stunning bridge between GR, QM, and thermodynamics.

Try this

How cold is a solar-mass black hole?

T_H = hbar c^3 / (8 pi G M k_B). For M = M_sun ≈ 2e30 kg, T_H ≈ 6.17e-8 K — far colder than the cosmic microwave background. Such a black hole absorbs more than it radiates.

Research status: stable

Real-world applications

Black hole information paradox research
Primordial black hole dark matter constraints
Analog gravity experiments (BEC, optical fibers)
Quantum gravity phenomenology

Common misconceptions

Hawking radiation is NOT particles escaping from inside the horizon
It's a quantum vacuum effect at the horizon, not classical emission
Astrophysical black holes have T_H far below the 2.7 K CMB — they GROW, not shrink

Experimental verification

Direct detection impossible for astrophysical black holes (T_H ≪ CMB). Analog experiments in BEC sonic horizons (Steinhauer 2016) show correlated phonon emission consistent with Hawking spectrum.

Derivation

From Euclidean path integral, the Schwarzschild metric is periodic in imaginary time with period

\beta = 8\,\mathrm{pi} G M

/ (hbar c^3).

Identifying

\beta = 1/(k_B T)

gives

T_H =

hbar c^3 / (8 pi G M k_B).

Alternative derivations: Bogoliubov coefficients of in/out vacua near horizon; tunneling amplitude across horizon.

Limiting cases

M \to \infty

⟶

T_H \to 0

Supermassive black holes are essentially cold; emission is negligible.

M \to 0

⟶

T_H \to \infty

Tiny black holes are blistering hot — final evaporation in microseconds for

M \sim 10^11\,\mathrm{kg}

.

M = M

_moon⟶

T_H \approx 1.7\,\mathrm{K}

A black hole with the Moon's mass would radiate as a cold-microwave source.

What if…

What if a black hole evaporated completely?

Lifetime $\approx 5120\,\mathrm{pi} G^2\,\mathrm{M}^3$ / (hbar c^4). For M_sun: $\sim 10^67$ years; for 10^11 kg: $\sim 10^10$ years (just now expiring if formed in early universe).

What if T_H exceeded surrounding temperature?

Net radiation outflow, mass loss, T rises further — runaway evaporation ending in explosive burst.

What if hbar

\to 0

?

$T_H \to 0$ — classical black holes don't radiate. Hawking emission is intrinsically quantum.

1

Solar-mass black hole

Given ·

M:: 1.989e+30
hbar:: 1.054571817e-34
G:: 6.674e-11
c:: 299792458
k B:: 1.380649e-23

Find · T_H

Steps

Numerator: hbar $c^3 = 1.0546e-34 * 2.697e25 = 2.844e-9$
Denominator: $8\,\mathrm{pi} G M k_B = 25.13 * 6.674e-11 * 1.989e30 * 1.381e-23 \approx 4.610e-2$
$T_H \approx 2.844e-9 / 4.610e-2 \approx 6.17e-8\,\mathrm{K}$

Result ·

T_H = (1.0546e-34 * (3e_{8})^3) / (8\,\mathrm{pi} * 6.674e-11 * 1.989e30 * 1.381e-23) \approx 6.17e-8\,\mathrm{K}

.

2

Primordial black hole (10^12 kg)

Given ·

M:: 1000000000000
hbar:: 1.054571817e-34
G:: 6.674e-11
c:: 299792458
k B:: 1.380649e-23

Find · T_H

Steps

$T_H \propto 1/M$ , so scale from solar-mass value
$T_H = 6.17e-8 * (1.989e30 / 1e12) = 1.23e11\,\mathrm{K}$

Result ·

T_H \approx 1.23e11\,\mathrm{K}

— extremely hot, gamma-ray emitter.

Schwarzschild Radius→Gravitational Time Dilation→Spacetime Interval→Mass-Energy Equivalence→