Black Holes That ClickEpisode 3 / Hawking Temperature & Evaporation ── lighter is hotter

If there is entropy, there must be a temperature ── and that temperature runs backwards from common sense

Hawking Temperature & Evaporation ── lighter is hotter The horizon has a temperature: T=ℏc³/(8πGMk_B), and it goes ∝1/M. The lighter the black hole, the hotter it is.
If it has a temperature it radiates heat and wastes away, growing hotter as it gets lighter until it runs away and, at last, evaporates.

Tools you'll need: the entropy from Episode 2, and thermodynamics (if there's an S, there's a T) This episode's ratios: T ∝ 1/M, lifetime ∝ M³

In Episode 2 we found that a black hole has an entropy $S$. In thermodynamics, anything that carries entropy must also have a temperature ($1/T=dS/dE$). Which means ── a black hole has a temperature too, and if it has a temperature it must radiate heat. A "black hole" that glows? The answer Hawking computed in 1974 was exactly that: the horizon shines faintly at a temperature $T=\hbar c^3/(8\pi GMk_B)$ (Hawking radiation). And this temperature runs backwards from common sense ── it is inversely proportional to the mass (∝1/M). Bigger black holes are colder, smaller ones are hotter. And if it has a temperature it loses heat and gets lighter, and the lighter it gets the hotter it becomes, running away until, at the end, it evaporates in a flash. What makes something that ought to be purely black glow is ℏ ── this is where ℏ and G meet for the first time, the first computable crack in quantum gravity.

01If there is entropy, there is a temperature

This is the very definition of thermodynamics ── $\dfrac{1}{T}=\dfrac{dS}{dE}$ (how much the entropy grows when you add a little energy). Using Episode 2's $S\propto M^2$ and the energy $E=Mc^2$, we get $dS/dE\propto M$, that is, $T\propto1/M$. From the ratio alone, out comes the fact that temperature is inversely proportional to mass. Putting in the coefficient as well ──

Hawking temperature

$$T=\frac{\hbar\,c^3}{8\pi\,G M\,k_B}\ \propto\ \frac{1}{M}$$

Here too ℏ・c・G・k_B all show up together (the diagonal of the cube). That ℏ is present is decisive ── set $\hbar\to0$ (classical) and $T\to0$, meaning a classical black hole is perfectly black and radiates nothing. What makes the hole glow is the quantum (ℏ). That is why Hawking radiation is the phenomenon where "gravity (G) and the quantum (ℏ) meet."

In step ③ of Episode 2's "derivation from ratios alone," we had $k_BT\sim\hbar c/R_s\sim\hbar c^3/GM$ ── that is exactly this Hawking temperature (apart from the factor $8\pi$). The ratio-based view of "the energy of the largest photon that fits inside the horizon = the temperature" is working here directly.

02Backwards from common sense ── bigger is colder

Ordinary things: the bigger and heavier they are, the larger the heat capacity and the more unwieldy, but the temperature itself has nothing to do with the mass. Black holes are different ── heavier is colder, lighter is hotter. The numbers reveal how strange it is.

Let's try it ── temperatures for various masses

・Solar-mass BH: $T\approx6\times10^{-8}$ K (far colder than the 2.7 K cosmic microwave background ── today it actually feeds on its surroundings and grows)
・A mountain's worth of mass (~10¹¹ kg): $T\approx10^{12}$ K (blazing hot; finishes evaporating right about now)
・A micro BH: evaporates in an instant, in a giga-scale flash

Black holes of stellar mass and above are colder than the cosmic microwave background, so they don't evaporate today; rather, they absorb the CMB and grow. Evaporation takes the leading role only in the far future when the universe has cooled way down, or for the small "primordial black holes" formed in the early universe.

03Negative heat capacity ── lose heat, and you get hotter

The strangest property of all. Ordinary things cool down when they lose heat. A black hole has <(T\propto1/M)>, so when it loses energy (mass) to radiation, its temperature goes up instead (negative heat capacity). Then ── hot → radiates more → gets lighter → hotter still: a runaway. Slow at first, accelerating at the end, it evaporates in a flash and vanishes. The lifetime until it finishes evaporating is ──

Evaporation lifetime ── the cube of the mass

$$\tau\ \propto\ M^3$$

The radiated power is roughly $P\propto T^2\propto1/M^2$, and integrating $dM/dt\propto-1/M^2$ gives a lifetime $\tau\propto M^3$. For a solar mass that's about $10^{67}$ years (far beyond the age of the universe); for a mountain-mass BH (~10¹¹ kg) it's right about now. The smaller it is, the more explosively fast it evaporates.

04Play with it ── temperature and evaporation

The figure below. Move the slider to change the black hole's mass, and the size of the horizon and the temperature (the thermometer on the right) change ── lighter means smaller and hotter. The dashed line marks the 2.7 K cosmic microwave background: below it (colder) the hole grows today; above it (hotter) it evaporates. Press "Evaporate" and the mass shrinks, and the smaller it gets the more the temperature shoots up in a runaway, until at the end it puts out a flash and disappears ── all in fast-forward (the negative-heat-capacity runaway).

Figure: the black hole (black circle, left) and a thermometer (right). The mass slider changes the size and the temperature (T∝1/M). Dashed line = CMB 2.7K. "Evaporate" shows the runaway where smaller means hotter → flash → vanishing (lifetime ∝ M³)

Mass log₁₀(M / kg) 30.3

Temperature T (∝1/M) CMB 2.7K marker

05What the radiation really is ── and on to where the information goes

The familiar picture of Hawking radiation is the vacuum fluctuations (pair creation of particle and antiparticle) just outside the horizon ── one member falls into the hole (carrying in negative energy and reducing the mass), the other escapes outward. But this is a picture for the intuition; strictly, it is a consequence of quantum field theory in curved spacetime: "because of the horizon, the vacuum of the infalling observer and the vacuum of the outside observer are mismatched, and the outside observer sees thermal particles" (the vacuum is observer-dependent ── a relative of Wakaru Relativity ⑤⑥).

Here the series' central question rises up. The thermal light that comes out in the radiation is nearly random (thermal) ── at a glance it seems to carry no information at all about what was swallowed. So when a black hole finishes evaporating completely, the $10^{77}$ bits of information we counted in Episode 2 ── are they lost? Even though quantum mechanics says "information is never lost (unitarity)." This is the information paradox (Episode 6). Evaporation is the fuse that sets fire to that question. Next time: the story that a black hole, until it finishes evaporating, is actually the fastest computer in the universe, storing information in a "queue."

◇ ◇ ◇

The honest line ── the standing of Hawking radiation, and the caveats

The Hawking temperature $T=\hbar c^3/(8\pi GMk_B)$ (containing the four constants and $\propto1/M$), the thermodynamic consistency of $S$ and $T$, the negative heat capacity, the evaporation lifetime $\tau\propto M^3$ (~10⁶⁷ years for a solar mass), the fact that stellar-mass BHs are currently colder than the CMB and do not evaporate, and the derivation via quantum field theory in curved spacetime are all theoretically established (Hawking 1974-75, Wald's review).

However. ① Hawking radiation has not yet been directly observed (for stellar-mass BHs the temperature is far too low). As a theoretical prediction it is extremely solid, and hints of analogous effects have been reported in analog experiments (sonic BHs, etc.). ② "One member of a pair falls in" is a heuristic metaphor; the rigorous derivation is via the Bogoliubov transformation (that each observer counts vacuum and particles differently) ── don't take it too literally. ③ $\propto1/M$ and $\tau\propto M^3$ are scalings that drop the coefficients and accompanying factors (particle species, greybody factors). ④ The existence of primordial black holes is a hypothesis, unconfirmed. ⑤ Whether the radiation is "perfectly thermal, with zero information" is precisely the crux of the information paradox, and in recent years "the information does eventually come out" is the mainstream view (Episode 6).

Practice problems (solvable with this episode's ideas)

$T\propto1/M$. If you make the mass 10 times larger, the temperature becomes what fraction? Between a big BH and a small BH, which is hotter?
See the answer
1/10. The bigger (heavier) the BH the colder, the smaller (lighter) the hotter. Backwards from common sense.
Why don't stellar-mass black holes evaporate today?
See the answer
Their temperature (~10⁻⁸K) is far below the cosmic microwave background (2.7K), so absorption wins over radiation and they actually grow. Evaporation takes the lead only in the far future when the universe cools below the CMB, or for small primordial BHs.
What is "negative heat capacity"? Why does it produce the runaway of evaporation?
See the answer
The property that losing heat (energy = mass) raises the temperature (because T∝1/M). Lose → get hotter → radiate more → lose more… this positive feedback accelerates at the end and, in a flash, it evaporates.
What does it mean that ℏ appears in the Hawking temperature formula?
See the answer
At ℏ→0 (classical), T→0 ── a classical black hole is perfectly black and does not radiate. The radiation is a quantum (ℏ) effect, a phenomenon where gravity (G) and the quantum (ℏ) meet = the first computable crack in quantum gravity.

Episode 3 summaryT ∝ 1/M ── lighter is hotter, runs away, and evaporates

If there is entropy, there is a temperature ($1/T=dS/dE$). From Episode 2's $S\propto M^2$, the ratio alone gives $T\propto1/M$. Putting in the coefficient, $T=\hbar c^3/(8\pi GMk_B)$ ── ℏ・c・G・k_B assemble once again, and because ℏ is present, "what makes the hole glow is the quantum." The horizon shines faintly at this temperature (Hawking radiation). Backwards from common sense: bigger BHs are colder, smaller ones hotter.

If it has a temperature it loses heat and gets lighter, and because $T\propto1/M$ it grows hotter as it gets lighter (negative heat capacity) ── it runs away and, in a flash, evaporates (lifetime $\tau\propto M^3$). Stellar-mass BHs are colder than the CMB and don't evaporate today, but small primordial BHs finish evaporating right about now. And the evaporating radiation looks at a glance thermal, with zero information ── so are the $10^{77}$ bits we counted in Episode 2 lost? This information paradox is the backbone of the episodes to come. This radiation, where ℏ and G meet, is the first step of a computable quantum gravity.

Next time ── Episode 4 "The Fastest Computer ── Scramblers and the Queue." What is a black hole doing until it finishes evaporating? In fact ── it is the fastest, densest computer in nature. It stirs the fallen information at the physical speed limit (the fastest scrambler, chaos bound $\lambda\le2\pi k_BT/\hbar$), stores it maximally mixed in a queue on the surface, and dribbles it back out as Hawking radiation. Not "a hole where computation stopped," but "a place where computation has pegged out and information is waiting its turn" ── on to that picture.

This document is Episode 3 of the "Black Holes That Click" series, a piece of reading for physics-loving high-school and university students. The Hawking temperature $T_H=\hbar c^3/(8\pi GMk_B)$ (containing the four fundamental constants, with $T\propto1/M$), its consistency with $S\propto M^2$ via $1/T=dS/dE$, the negative heat capacity, the evaporation lifetime $\tau\propto M^3$ (~10⁶⁷ years for a solar mass), the fact that stellar-mass BHs are colder than today's cosmic microwave background (≈2.7 K) and grow on net, the derivation via quantum field theory in curved spacetime (the Bogoliubov transformation), and that the radiation vanishes as $\hbar\to0$ (that it is a quantum effect) are all theoretically established (Hawking 1974–75). That Hawking radiation has not been directly observed for stellar-mass BHs (with reports of analogous effects in analog systems), that the pair-creation picture is a heuristic metaphor, that $\propto1/M$ and $\tau\propto M^3$ are scalings that omit coefficients and greybody factors, that the primordial black hole is an unconfirmed hypothesis, and that whether the radiation is perfectly thermal (whether it carries information) is the crux of the information paradox are all noted in the main text's "The honest line." The figure is a qualitative schematic of temperature and evaporation, with size and temperature on scales chosen for visibility. ── To print, use your browser's "Print" and "Save as PDF" (in the print version the slider and answers are frozen and hidden). Adjacent episodes: Episode 2, Entropy／Table of contents／sister series Wakaru Quantum.

Print / PDF: ⌘+P (Ctrl+P on Windows). On screen the temperature changes with mass (T∝1/M), and "Evaporate" fast-forwards through runaway → flash → vanishing. "See the answer" opens each solution.