Today, I want to tell you why some scientists believe that our universe is really a 3-dimensional projection of a 2-dimensional space. They call it the “holographic principle” and the key idea is this.
Usually, the number of different things you can imagine happening inside a part of space increases with the volume. Think of a bag of particles. The larger the bag, the more particles, and the more details you need to describe what the particles do. These details that you need to describe what happens are what physicists call the “degrees of freedom,” and the number of these degrees of freedom is proportional to the number of particles, which is proportional to the volume.
At least that’s how it normally works. The holographic principle, in contrast, says that you can describe what happens inside the bag by encoding it on the surface of that bag, at the same resolution.
This may not sounds all that remarkable, but it is. Here is why. Take a cube that’s made of smaller cubes, each of which is either black or white. You can think of each small cube as a single bit of information. How much information is in the large cube? Well, that’s the number of the smaller cubes, so 3 cube in this example. Or, if you divide every side of the large cube into N pieces instead of three, that’s N cube. But if you instead count the surface elements of the cube, at the same resolution, you have only 6 x N square. This means that for large N, there are many more volume bits than surface bits at the same resolution.
The holographic principle now says that even though there are so many fewer surface bits, the surface bits are sufficient to describe everything that happens in the volume. This does not mean that the surface bits correspond to certain regions of volume, it’s somewhat more complicated. It means instead that the surface bits describe certain correlations between the pieces of volume. So if you think again of the particles in the bag, these will not move entirely independently.
And that’s what is called the holographic principle, that really you can encode the events inside any volume on the surface of the volume, at the same resolution.
But, you may say, how come we never notice that particles in a bag are somehow constrained in their freedom? Good question. The reason is that the stuff that we deal with in every-day life, say, that bag of particles, doesn’t remotely make use of the theoretically available degrees of freedom. Our present observations only test situations well below the limit that the holographic principle says should exist.
The limit from the holographic principle really only matters if the degrees of freedom are strongly compressed, as is the case, for example, for stuff that collapses to a black hole. Indeed, the physics of black holes is one of the most important clues that physicists have for the holographic principle. That’s because we know that black holes have an entropy that is proportional to the area of the black hole horizon, not to its volume. That’s the important part: black hole entropy is proportional to the area, not to the volume.
Now, in thermodynamics entropy counts the number of different microscopic configurations that have the same macroscopic appearance. So, the entropy basically counts how much information you could stuff into a macroscopic thing if you kept track of the microscopic details. Therefore, the area-scaling of the black hole entropy tells you that the information content of black holes is bounded by a quantity which proportional to the horizon area. This relation is the origin of the holographic principle.
The other important clue for the holographic principle comes from string theory. That’s because string theorists like to apply their mathematical methods in a space-time with a negative cosmological constant, which is called an Anti-de Sitter space. Most of them believe, though it has strictly speaking never been proved, that gravity in an Anti-de Sitter space can be described by a different theory that is entirely located on the boundary of that space. And while this idea came from string theory, one does not actually need the strings for this relation between the volume and the surface to work. More concretely, it uses a limit in which the effects of the strings no longer appear. So the holographic principle seems to be more general than string theory.
I have to add though that we do not live in an Anti-de Sitter space because, for all we currently know, the cosmological constant in our universe is positive. Therefore it’s unclear how much the volume-surface relation in Anti-De Sitter space tells us about the real world. And for what the black hole entropy is concerned, the mathematics we currently have does not actually tell us that it counts the information that one can stuff into a black hole. It may instead only count the information that one loses by disconnecting the inside and outside of the black hole. This is called the “entanglement entropy”. It scales with the surface for many systems other than black holes and there is nothing particularly holographic about it.
Whether or not you buy the motivations for the holographic principle, you may want to know whether we can test it. The answer is definitely maybe. Earlier this year, Erik Verlinde and Kathryn Zurek proposed that we try to test the holographic principle using gravitational wave interferometers. The idea is that if the universe is holographic, then the fluctuations in the two orthogonal directions that the interferometer arms extend into would be more strongly correlated than one normally expects. However, not everyone agrees that the particular realization of holography which Verlinde and Zurek use is the correct one.
Personally I think that the motivations for the holographic principle are not particularly strong and in any case we’ll not be able to test this hypothesis in the coming centuries. Therefore writing papers about it is a waste of time. But it’s an interesting idea and at least you now know what physicists are talking about when they say the universe is a hologram.