Black Hole Entropy: Stephen Hawking's Discovery

This article is an excerpt from the Shortform book guide to "A Brief History of Time" by Stephen Hawking. Shortform has the world's best summaries and analyses of books you should be reading.

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What’s entropy like inside a black hole? And how do we know that black holes have entropy in the first place?

Jacob Bekenstein suggested that the area of a black hole was itself a measurement of its entropy. But if black holes had entropy, they would have to emit radiation (which was thought impossible at the time). In 1973, Stephen Hawking ran calculations that showed that black holes would indeed create and emit particles.

Keep reading to learn more about black hole entropy.

Do Black Holes Have Entropy?

The second law of thermodynamics tells us that entropy always increases and that the merging of two or more systems creates a higher level of entropy than was present in the original systems. For example, one can imagine a box with a partition running down the middle, with gas molecules all confined to one side of the partition. If one removes the partition (combining the two systems), the molecules will likely disburse evenly throughout the box, creating a single, less well-ordered system.

But the second law of thermodynamics doesn’t seem to apply in all cases, particularly when it comes to the singularity of a black hole. Remember, we can’t see into the black hole or measure the properties within it, so we don’t know what the state of entropy is inside one. Imagine you threw some entropy-containing matter into a black hole, (thus combining the system outside the black hole with the system inside the black hole). Total entropy outside the black hole (which is all that we are capable of measuring) would go down—a clear violation of the second law of thermodynamics.

To keep the second law intact, we need some property of black holes that we can measure that would indicate its level of entropy and that would increase with the addition of entropy-carrying matter into the black hole. A Princeton graduate student named Jacob Bekenstein put forward a theorem that appeared to solve this problem, building off Hawking’s discovery about the non-decreasing area of black holes. He suggested that the area of a black hole was itself a measurement of black-hole entropy. The more matter that was added to the black hole (increasing its entropy), the larger the area would become. Thus, net entropy would remain the same, and the law would not be breached.

Hawking took exception to this theory, believing that Bekenstein had misused his earlier work. If a black hole did have entropy, that meant it had a temperature, which meant it emitted radiation. But this couldn’t be possible—by definition, black holes couldn’t emit anything. But in 1973, Hawking ran calculations that showed that black holes would indeed create and emit particles. Moreover, the mathematical proofs showed that these particle emissions were at the exact right rate required for them to conform with thermodynamics.

Turning a Negative Into a Positive

What could account for this? Hawking saw that the solution lay in Werner Heisenberg’s uncertainty principle. For now, all we need to know is that the uncertainty principle states that it is impossible to determine both particle velocity and position with certainty. There are no certainties, only likelihoods. Thus, the empty space just outside the event horizon couldn’t actually be empty, because this would mean that the velocity and position of every particle in every field (like gravity and electromagnetism) would be at zero.

Instead, there would be pairs of virtual particles, like gravitrons or photons. These are unseen by a particle detector, but they can be measured because of their effects on matter particles (like protons, neutrons, and electrons). Normally, such particles would find their corresponding antiparticle (a particle with the same mass, but opposite charge) and they would annihilate one another. Under normal circumstances, negative antiparticles don’t last long, because there are so many more matter particles in the universe and these matter particles always carry a positive charge—the antiparticles are quickly annihilated by their partners.

But the presence of a black hole changes this dynamic. Near a black hole, gravity is so strong that the negative energy particle can fall in and be transformed into a real, positive energy particle. This transformation will enable its partner particle to avoid annihilation. The “surviving” partner particle, also having positive energy, can escape the black hole region. This gives the appearance of it coming from the black hole itself.

The inflow of negative energy particles would, according to Einstein’s famous equation e=mc², decrease the mass of the black hole, seemingly decreasing entropy and violating thermodynamics. But the outflow of positive energy particles more than makes up for this effect, increasing net entropy and keeping the second law of thermodynamics intact. This theory would also predict that even black holes have a lifetime that ends. Their mass decreases in a final burst of emission. But this process would take many times longer than the age of the universe, so it is not observable.

Black Hole Entropy: Stephen Hawking’s Discovery