Black Holes: a Perspective on Disorder
For decades, black holes have headlined the thought experiments that physicists seek refuge in. These invisible spheres form when matter becomes so concentrated that everything within a certain distance gets trapped by its gravity. Albert Einstein equated the force of gravity with curves in the space-time continuum, but the curvature grows so extreme near a black hole’s center that Einstein’s equations break.
The story goes that in the early 1970s, James Bardeen, Brandon Carter, and Stephen Hawking pointed out an analogy between the behavior of black holes and the laws of good old thermodynamics. The Second Law of Thermodynamics (“Entropy never decreases in closed systems”) was analogous to Hawking’s “area theorem”: in a collection of black holes, the total area of their event horizons never decreases over time. Jacob Bekenstein, who at the time was a graduate student working under John Wheeler at Princeton, proposed to take this analogy more seriously than the original authors had in mind. He suggested that the area of a black hole’s event horizon really is its entropy, or at least proportional to it.
This annoyed Hawking, who set out to prove Bekenstein wrong. After all, if black holes have entropy then they should also have a temperature, and objects with nonzero temperatures give off blackbody radiation, but we all know that black holes are, you know, black. But he ended up actually proving Bekenstein right; black holes do have entropy and temperature, and they even give off radiation. We now refer to the entropy of a black hole as the “Bekenstein-Hawking entropy.”
Let’s pretend we’ve got a black hole — a spherical one. We’ll get to the why of this pretense in a moment.
A rather unconventional way to think about this is that the escape velocity is larger than the speed of light beyond the event horizon. Or, in other words, the escape velocity is exactly the speed of light at the event horizon. In that sense, the event horizon is the boundary of all the information within a black hole. Beyond it, nothing escapes.
There’s a fair amount of math we can do — using the Schwarzschild metric — to show that a black hole that isn’t rotating must have a spherical event horizon. But we don’t necessarily care about that here.
In short, the Schwarzschild metric is the best mathematical model that we have to describe the behavior of a non-rotating black hole. In Einstein’s general relativity, the Schwarzschild metric is an exact solution to the field equations that describe the gravitational field of a spherical mass. Anyway, back to the black hole.
Since our knock-off black hole is spherical, one thing we can calculate is its surface area: 4πr² where r is just the radius of the sphere. In this particular case, we said that the radius of our black hole was r_s. The surface area of the event horizon is then:
The fact that the black hole event horizon is a sphere isn’t too important anymore. I mean, sure it’s nice to know and makes the math incredibly convenient, but the thing I really wanted to bring to your attention is that we can calculate its surface area. The surface area of an event horizon is an extremely important quantity when calculating a black hole’s entropy.
And well, that brings us nicely to what this article’s intended to achieve. Many of you would’ve heard of entropy as a measure of disorder in a system. Subsequently, the Second Law of Thermodynamics tells us that the entropy of the universe must always increase. In other words, the entropy of a closed system flows unidirectionally. And this is important.
Because assuming the Second Law of Thermodynamics is correct, it immediately points to the fact that black holes must have entropy. Here’s how that happens.
If black holes did not have entropy, then we could take any object from outside a black hole, which does have entropy, and chuck it in. In all of this, the total entropy of the universe has decreased. Since if the black hole must have zero entropy and the object we choose to throw in has some entropy, then this entropy cannot simply vanish without violating the Second Law of Thermodynamics. So when we chuck our little object into the black hole, its entropy must go somewhere. In other words, entropy cannot simply vanish and therefore black holes cannot have zero entropy.
That’s well and good, I think. But here’s something better. It turns out that the entropy of a black hole is given by this:
S is, of course, the entropy of our black hole, k_B is the Boltzmann constant, A is the surface area of the event horizon, 4 is, well, just 4, and l_p is the Planck length. The Planck length is itself made up of other fundamental constants:
The important part of the entropy equation is that everything on the right-hand side, apart from the surface area, is a universal constant. The surface area will obviously depend on which black hole we happen to be studying but all these other values are woven into the fabric of our existence. So, we can tidy up our entire equation to just this:
This allows us to see that the entropy of a black hole is directly proportional to the surface area of its event horizon. And crucially, it depends on nothing else. It’s all nicely summed up to give:
Now, we also know that the surface area of a black hole’s event horizon depends on its radius. It just happens so that the radius is given by:
r_s is simply known as the Schwarzschild radius. This quantity depends on the universal gravitational constant, the speed of light, and the mass of the black hole. The interesting thing about the Schwarzschild radius is that it’s the quantity that anything with a mass, M, must be squished into to become a black hole. So now if we know that the entropy of a black hole depends on its event horizon’s surface area and the surface area of the event horizon depends on the Schwarzschild radius which, in turn, depends only on the mass, then we can say that the entropy of a black hole is solely dependent on its mass. Of course, this all only holds true for a non-rotating black hole.
If we know a black hole’s mass, we can calculate its radius. If we know its radius we can calculate its surface area. And if we know its surface area, we can calculate its entropy. It’s worth reiterating, of course, that we’re only talking about a relatively (pun intended) simple black hole. Anyway, I hope you now have some understanding of what any of this really means. If I’ve made an error, please feel free to tear me apart in the comments. As always, thank you for reading!
