You’ve just committed the perfect crime. No one saw you do the crime, and you left no trace. The perfect crime. The only way anyone could prove you did it is by finding the journal of your master plan. Get rid of the journal, and you are scott free. Of course you can’t simply toss the journal in the trash. Someone might find it. So maybe you should rip it to pieces and then toss it in the trash. That would be better, but someone could take the pieces and carefully put them back together, and your crime would be revealed. Maybe you should burn the journal. Surely that would destroy it. That would probably be good enough, but if someone observed the ash and smoke very carefully, and made really precise measurements they might be able to figure out where all of it came from and reconstruct the information in the book. That’s very unlikely to happen, but this journal is the only thing standing between you and the perfect crime. You want to be absolutely, 100% certain that the information it contains is permanently destroyed. How do you get the job done?
This hypothetical story highlights a very real question in physics. Is it possible to permanently destroy information? Or is information, like mass-energy and charge, conserved? The question is important because it strikes at the very heart of what science is. Through science we develop theories about how the universe works. These theories describe certain aspects of the universe. In other words they contain information about the universe. Our theories are not perfect, but as we learn more about the universe, we develop better theories, which contain more and more accurate information about the universe. Presumably the universe is driven by a set of ultimate physical laws, and if we can figure out what those are, then we could in principle know everything there is to know about the universe. If this is true, then anything that happens in the universe contains a particular amount of information. For example, the motion of the Earth around the Sun depends on their masses, the distance between them, their gravitational attraction, and so on. All of that information tells us what the Earth and Sun are doing.
Scientists generally assume information is conserved for two reasons. The first is a principle known as determinism. If you throw a baseball in a particular direction at a particular speed, you can figure out where it’s going to land. Just determine the initial speed and direction of the ball, then use the laws of physics to predict what its motion will be. The ball doesn’t have any choice in the matter. Once it leaves your hand it will land in a particular spot. Its motion is determined by the physical laws of the universe. Everything in the universe is driven by these physical laws, so if we have an accurate description of what is happening right now, we can always predict what will happen later. The future is determined by the present.
The second principle is known as reversibility. Given the speed and direction of the ball as it hits the ground, we can use physics to trace its motion backwards to know where it came from. By observing the ball now, we can know from where the ball was thrown. The same applies for everything in the universe. By observing the universe today we can know what happened billions of years ago. The present is predicated by the past.
These two principles are just a precise way of saying the universe is predictable, but it also means information must be conserved. If the state of the present universe is determined by the past, then the past must have contained all the information of the present universe. Likewise, if the future is determined by the present, then the present must contain all the information of the future universe. If the universe is predictable, then information must be conserved.
Now you might be wondering about quantum mechanics. All that weird physics about atoms and such. Isn’t the point of quantum mechanics that things aren’t predictable? Not quite. In quantum mechanics, individual outcomes might not be predictable, but the odds of those outcomes are predictable. It’s kind of like a casino. They don’t know which particular players will win or lose, but they know very precisely what percentage will lose, so the casino will always make money. The baseball example was one of classical, everyday determinism. To include quantum mechanics we need a more general, probabilistic determinism known as quantum determinism, but the result is still the same. Information is conserved.
So it looks like you’re in trouble. Since information is conserved, there is no way for you to destroy that incriminating journal. You can make the information very difficult to find, but you can’t permanently erase it. But being an evil genius, you have an idea. You’ll simply chuck the journal into a black hole. After all, nothing can escape a black hole, so once you’ve tossed it in, no one can ever get it back. All that incriminating evidence destroyed forever. The perfect crime.
It seems like a good idea. According to Einstein’s theory of general relativity, a black hole has only three basic properties: mass, charge and rotation. If you know those three things, you know everything there is to know about a black hole. So if you toss your journal into a black hole, all those plots and plans of the perfect crime are reduced to mass, rotation and charge. All of the information in the journal has been destroyed.
But Einstein didn’t account for quantum mechanics in his theory. Through quantum mechanics, things can escape a black hole.
One of the fundamental principles of quantum theory is known as the uncertainty principle. Basically, the uncertainty principle states that there is a limit to what you can know about an object. This limit is not simply due to a lack of good measurements. It is an absolute uncertainty built into the fabric of the universe. This leads to some very strange phenomena. For example, suppose you put a marble in a small box. Seal up the box and the marble can’t get out, right? According to the uncertainty principle, there’s a small chance that it could get out. If the marble is in the box, then you know exactly where it is, but you’re not allowed to know exactly where it is, only probably where it is. So there’s a very small chance that you may return to find the marble has escaped. Strange as this seems, it is a very real effect is known as quantum tunneling. For things like marbles the odds of it happening are so low that they are essentially zero, but for atoms and electrons it happens all the time. Your computer wouldn’t work and the Sun wouldn’t shine without it.
A black hole is basically a gravitational box. Anything put into a black hole should be trapped, but because of the uncertainty principle things can escape. Over time, the mass and energy of the black hole will escape, and it will radiate away through a process known as Hawking radiation (named after Stephen Hawking). Through the uncertainty principle black holes gradually radiate away.
But this means that the information of a black hole is more than just mass, charge and rotation. It must also contain the information of all the particles it will radiate away. So just how much information does a black hole contain? The answer is given by the equation above, known as the Bekenstein-Hawking equation. Here S is the information of the black hole, C is the speed of light, ℏ (h with a line in the top) is a number known as Planck’s constant, and relates to the uncertainty principle, K is a number known as Boltzmann’s constant, G is Newton’s gravitational constant, and A is the area of the black hole’s event horizon, which is just another way to measure its mass. What the equation says is the information contained within a black hole is proportional to its size. If you toss something into a black hole, you increase the mass of the black hole, which increases the information contained in a black hole. So tossing your journal into a black hole doesn’t make the information disappear.
But there is one last piece to this puzzle, and it’s a doozy. The Bekenstein-Hawking equation states that the amount of information you toss into a black hole is the same as the amount of information a black hole contains. But according to our understanding of gravity, Hawking radiation is perfectly random. So the black hole will eventually release the right amount of information, but not the same information. This means that information tossed into a black hole really is destroyed. But according to quantum theory, the black hole must somehow retain the information of what is tossed into it. This means Hawking radiation is not random, and the information is not destroyed. This contradiction is known as the black hole information paradox, and we don’t yet know how to solve it. Most scientists think quantum mechanics is probably right, but we can’t prove it yet.
So toss your journal into a black hole, and you may have committed your perfect crime…or not.
Tomorrow: The end of the series. Boltzmann opens our eyes to a world where the warmth of our morning coffee forces us to confront our own mortality.