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}

where

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

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