The Temperature of a Black Hole | AI Learning Guide

This equation beautifully unites four pillars of physics. You have quantum mechanics via Planck's constant, relativity via the speed of light, gravity via Newton's constant, and thermodynamics via Boltzmann's constant. The key takeaway is the inverse relationship: \(T_H\) is proportional to \(1/M\). Double the mass, and you halve the temperature.

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Hawking Temperature

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

Inverse Mass Relationship

Notice that the mass \(M\) is in the denominator. This means that as a black hole gets smaller, it gets hotter. A supermassive black hole is nearly absolute zero, while a tiny primordial black hole would be intensely hot.

Constants in the Formula

\(\hbar\): Reduced Planck constant (Quantum mechanics)
\(c\): Speed of light (Relativity)
\(G\): Gravitational constant (Gravity)
\(k_B\): Boltzmann constant (Thermodynamics)