REL-07

Hawking temperature

T = ħ c³ /(8 π G M kB).

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Black holesHawking 1974

Governing equation

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

$M \in \left[1\times 10^{-6}, 1\times 10^{6}\right]$

where

M: Mass (suns) (M_\odot)
T_H: Hawking temperature (nK)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

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

$M \in \left[1\times 10^{-6}, 1\times 10^{6}\right]$

$M = 1\times 10^{-6}$ $T_H = 6.17\times 10^{7}$

Along the curve

②

Domain of the variable — blue

M_{\min} = 1\times 10^{-6},\quad M_{\max} = 1\times 10^{6}

$M \in \left[1\times 10^{-6}, 1\times 10^{6}\right]$

$M = 1\times 10^{-6}$ $M = 1\times 10^{-6}$

Along the curve

③

Response curve — green

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

$M \in \left[1\times 10^{-6}, 1\times 10^{6}\right]$

$M = 1\times 10^{-6}$ $T_H = 6.17\times 10^{7}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}T_H}{\mathrm{d}M}

$M \in \left[1\times 10^{-6}, 1\times 10^{6}\right]$

$M = 6289.3$ $T_H^{\prime} = -4905.2$

Along the curve

Lecture brief

Historical brief

Einstein’s 1905 Lorentz kinematics and 1915 field equation, then Schwarzschild (1916) and Hawking temperature, recast time, mass and gravity. The lab computes dilation, E=mc² and horizon scales. This sheet (REL-07 — Hawking temperature) is the form associated with Hawking 1974. Working symbols: $M$ $\rightarrow$ $T_H$ . Hawking showed that a black hole radiates as a black body whose temperature is inversely proportional to mass, from quantum fields on a curved background.

Purpose

Purpose: compute

T_H

from

M

in Relativity via

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

T = ħ c³ /(8 π G M kB). Use it when a real relativity question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

M = 1.000\,\mathrm{M_odot}

, the governing relation

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

yields

T_H = 61.7007\,\mathrm{nK}

. Schwarzschild hole, no charge or spin. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass (suns) M1 M_\odot

Outputs

Hawking temperature T_H61.7007 nK

Reading speed

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Narration of this film

Schwarzschild hole, no charge or spin.

Hawking showed that a black hole radiates as a black body whose temperature is inversely proportional to mass, from quantum fields on a curved background.

Reading speed

Watch on YouTube