Ask Ethan: Do black holes grow faster than they evaporate?

When matter falls in, black holes grow. But Hawking radiation says that black holes decay. Who wins?

“Maybe that is our mistake: maybe there are no particle positions and velocities, but only waves. It is just that we try to fit the waves to our preconceived ideas of positions and velocities. The resulting mismatch is the cause of the apparent unpredictability.” –Stephen Hawking

Black holes are the most massive single objects in the known Universe. More massive than even the Sun — sometimes millions or even billions of times as massive — they’re formed from the collapse of ultramassive stars and their remnants. Anything that crosses the event horizon is destined to arrive at the central singularity, increasing the black hole’s mass. But thanks to the combination of general relativity, which tells us how space is curved by mass, and quantum field theory, which tells us how empty space spontaneously behaves, we learn that black holes don’t remain stable forever, but decay. Which one will win: the growth or the decay? That’s what Steve Fitch wants to know:

“Wondering why black holes wouldn’t be growing faster than they can evaporate due to [Hawking] radiation. If particle pairs are erupting everywhere in space, including inside [black hole] event horizons, and not all of them are annihilating one another shortly thereafter, why doesn’t a [black hole] slowly swell due to surviving particles that don’t get annihilated?”

There’s a misconception here, however. Let’s start with that.

Yes, empty space is an interesting place. In many ways, it’s not very empty at all! Sure, you can imagine taking all the matter, all the radiation, all the quanta of energy, even all the curvature completely out of a region of space, until all that’s left is as close to “nothing” as we can get in this Universe. Yet even at that, the zero-point energy of that empty space isn’t zero. Even with everything you can remove taken away, there’s still a non-zero amount of energy inherent to space itself. One way we can visualize that is as particle-antiparticle pairs, popping in-and-out of existence.

Now, take that same visualization, and put a black hole in that space.

You’ll have three regions where these particle-antiparticle pairs emerge:

1. Where both pair members start outside of the black hole, exist, and re-annihilate outside.
2. Where both pair members begin inside the black hole’s event horizon, exist, and re-annihilate inside.
3. Where both members begin outside, but one falls in while the other escapes.

Yes, this is oversimplified, but it’s one of the most straightforward visualizations that gets the qualitative features right, even though it doesn’t accurately describe where the Hawking radiation originates or what its energy spectrum is. In reality, what you get out is a spectrum of blackbody radiation — mostly in the form of photons of extremely low energy — that’s related to the size of your black hole’s event horizon, where smaller black holes radiate faster.

The thing you have to realize is that these “pairs” don’t actually, physically exist; they’re calculational tools only. A pair that pops into existence inside the black hole can’t add mass to the black hole itself, because the total energy in there is the same at all times. After all, the energy for the particle-antiparticle pairs came from the space around it! But if you have energy that originates from the space outside and it results in real radiation moving away from the black hole, that energy must come from the black hole itself, lowering its mass. That’s how Hawking radiation works, and that’s why black holes eventually decay.

We can quantify this rate of decay and the temperature of the radiation, and find that black holes lose mass at a tremendously slow rate! For a black hole the mass of the Sun, the current temperature of its Hawking radiation will be 62 nanoKelvin, and it will take 10⁶⁷ years to evaporate. For the one at our galaxy’s center, it radiates at 15 femtokelvin and takes 10⁸⁷ years to evaporate. The largest black holes of all will take 10¹⁰⁰ years to evaporate! Yet throughout this entire time, there’s also matter that gets sucked into the black hole in question.

Material from other stars, from cosmic dust, from interstellar matter, gas clouds, or even the radiation and neutrinos left over from the Big Bang can all contribute. Intervening dark matter will collide with the black hole, increasing its mass as well. The mass-equivalent of the radiation loss is many orders of magnitude lower than the amount of matter absorbed by any such black hole. But there’s a limit to the matter that can be absorbed.

Over time, gas gets burned into stars, collapsed objects get ejected into the intergalactic medium, and gravitational dissociation drives objects apart. It may take somewhere around 10²⁰ years — ten billion times the present age of the Universe — for the rate of matter absorption to fall below the rate of Hawking radiation, but it will happen eventually. And once it does, black hole decay will start to win. Every black hole we know of in the Universe today is still growing, but that growth is going to reach a finite maximum. After that, Hawking radiation will triumph.

It starts out slow, but Hawking radiation will increase over time, particularly as the mass of the black hole starts to shrink appreciably. Once you form a singularity, you remain a singularity — and you retain an event horizon — right up until the moment your mass goes to zero. That final second of a black hole’s life, however, will result in a very specific and very large release of energy. When the mass drops down to 228 metric tonnes, that’s the signal that exactly one second remains. The event horizon size at the time will be 340 yoctometers, or 3.4 × 10^-22 meters: the size of one wavelength of a photon with an energy greater than any particle the LHC has ever produced. But in that final second, a total of 2.05 × 10²² Joules of energy, the equivalent of five million megatons of TNT, will be released. It’s as though a million nuclear fusion bombs went off all at once in a tiny region of space; that’s the final stage of black hole evaporation.

This will occur so far in the future that such a flash of light will be the only visible thing in the entire Universe when it occurs. All the stars and stellar remnants will have long since gone dark. Even though black holes today are growing faster than they can decay, that’s a situation that won’t last forever. Once we run out of infalling matter, or the rate drops below the rate of Hawking radiation, decay is the only thing remaining, and it’s awfully persistent. So cheer up! Black holes will grow and grow and grow for billions of years before they start decaying faster than they’re growing, and even once they do, they have incredible lengths of time before they’re gone. But wait that required time, and even the most massive black hole in the Universe will evaporate. Hawking radiation is the inevitable fate of every black hole in the Universe.

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