How Much Information Does the Universe Contain? - Script (Patreon)

There’s quite a bit of stuff in the universe, to put it mildly. Hundreds of billions of galaxies, each with hundreds of billions of stars, each with … rather a lot of particles in them. And then there’s all the stuff that isn’t stars. The dark matter, black holes, planets, and the particles and radiation in between the stars and galaxies. Not to mention space itself, with its fluctuating quantum fields, dark energy, blah blah.  Stuff EVERYWHERE! But… is the universe actually made of stuff? An increasing number of physicists view the universe – view reality as informational at its most fundamental level, and that its evolution through time can be thought of as a computation. And then there’s the simulation hypothesis – in which that computation is engineered by … who knows what? How big a memory bank would you even need to compute a universe? Seriously, let’s figure it out. How much information does it take to describe the entire observable universe? And after we’re done with that, I’m going to have an even cooler challenge question for you.

I casually mentioned in the last episode that our 3-D universe may be just a projection of information imprinted on its 2-dimensional boundary. That’s the holographic principle, and we’ve talked a lot about some ideas leading up to it. Don’t worry – the full holographic principle episode is still coming. But our recent episode on black hole entropy and some of the lead-ups to that might be helpful here. You can also watch this video as a standalone and go back to those earlier ones later if you feel like it. The main point – the really weird, surprising point – is that the maximum amount of information that can fit in a volume of space is not proportional to the volume – it’s proportional to the surface area of that region of space.

Jacob Bekenstein figured this out by realizing that the entropy of a black hole is proportional to the surface area of its event horizon. But entropy is just a measure of hidden information, so the Bekenstein bound is equally limit on how much information you can fit in any region of space. We’ll come back to the Bekenstein bound in a sec. Let’s think about why this dependence on surface area is surprising. Common sense would suggest that the maximum information content depends on volume, not surface area. I mean, a pile of thumb drives has a total storage capacity that depends on its volume, not surface area.

Instead of storage capacity, let’s think about the information needed to perfectly describe a patch of space. You’d think that to fully describe, say, the universe, you’d need to know that’s going on in every tiniest possible 3-D chunk. That smallest element is roughly a cube one Planck length on a side – where the Planck length is the smallest meaningful measure of distance, at around 1.6x10^35 meters. It’s the smallest possible chunk of space, so let’s say it can contain the smallest amount of information – one bit per Planck volume. That’s kind of like saying we can describe the universe completely if we go through all of its quantum voxels and answer the yes-no question of whether it is full or empty. This probably WAY underestimates how much info you really need, but let’s start with this anyway.

So how many Planck volumes are there in the Universe? The radius of the universe is something like 47 billion light years, which is a few times 10^61 Planck lengths. 4/3-p-r-cubed – so the universe contains 10^183 Planck volumes. You’ll see estimates that the radius of the universe is a mere 10^60 Planck lengths instead of 10^63, and its volume 10^180 units. That’s because cosmologists tend to round down at every step and so drop a bunch of orders of magnitude. Astrophysicists are, by comparison, HIGHLY accurate. We only drop factors of 2. But what’s a few orders of magnitude between friends?

You might argue that more than one bit can fit at each “grid point” in the universe. You might be right – particles have more information in them than just their position. It would be better to use the number of grid points in quantum phase space, which includes position but also other degrees of freedom like momentum, spin direction, etc. In other words, we should count all possible quantum states in the universe. Anyway, the real number is going to be way higher than our estimate here, but as we’ll see we’re already way, way higher than the actual information limit of the universe.

OK, so 10^180 or so bits is the minimum if you want to describe every 3-D quantum voxel completely independently. But the Bekenstein bound tells us that the information content of any volume isn’t the number of these Planck volumes, but rather number of Planck areas on the surface. The observable universe has a surface area of 10^120-10^124 Planck areas, depending on whether you round like a cosmologist or an astrophysicist. So the storage capacity of the universe is around 10^60 lower than the number of volume elements it contains.

So how do you encode a whole universe in a space far smaller than the universe itself? That must be some amazing compression algorithm. In fact, in a sense it’s just that. See, you don’t really need a bit per volume element of the universe; that would be like having separate, completely independent memory elements for every empty pixel in an image. Most of space is indeed empty. Let’s say instead that you only need a single bit of information for every element in phase space that’s occupied. In other words, one bit for every elementary particle.

Again, this is an under-estimate. In fact the information needed to fully simulate the quantum wavefunction of a single particle is …

The observable universe contains something like 10^80 protons. Each proton has 3 quarks, and there are a similar number of electrons. Most other particles are much rarer, so we’re still in the realm of 10^80-10^81. Neutrinos and photons formed in the Big Bang are probably a billion times more abundant. That’s verified experimentally in the case of photons – the cosmic microwave background has around 10^89 photons. So almost all information – and for that matter entropy – in particles is in neutrinos and the cosmic microwave background photons. The situation with dark matter is unclear, so let’s just round up to 10^90 bits. Sounds like a lot, but happily it’s way less than the 10^120ish limit of the Bekenstein bound.

But there’s one more source of information – black holes. As I mentioned last time, black holes contain most of the entropy in the universe. The relationship between black hole entropy and information deserves some thought. Black hole entropy, in terms of number of bits, tells you the information you’d need to describe all possible initial states that could have possibly formed the same black hole. Of which there are many – 2 to the power of the number of bits of entropy. And for black holes that entropy is the Bekenstein bound – the number of Planck areas on its event horizon. Because the information about the black holes previous state is lost, to fully describe it you need to fully describe its event horizon – you need its full Bekenstein bound in information.

How much information is that? Let’s take a supermassive black hole – Sagittarius A* in the center of the Milky Way,  which has the mass of 4 million suns. Its event horizon is around 12 billion meters, giving it a surface area of 10^90-10^91 Planck units. Uh … so the Milky Way’s black hole has as much entropy and hidden information as all of the matter and radiation in the entire rest of the universe. And there are some hundreds of billions of galaxies in the universe, each with its own supermassive black hole. We’re talking something like 10^101-10^102 bits of entropy or information. Black holes contain by far most of the entropy in the universe, and require most information to fully describe. But again, we’re still way below the Bekenstein bound for the whole universe.

I know you’re as relieved as I am. We’re nowhere near the universe’s memory limit. The universe can keep having particles, and you can leave your horribly bloated email inbox alone. But what would actually happen if it contains too much information, say, in the form of too many particles? What if we started to fill up those empty Planck-sized cubes of space throughout the universe until it contained more information than the Bekenstein bound allowed? Well the answer is straightforward enough. At the moment when the universe reached its information limit, it would immediately become a black hole with an event horizon as big as the current cosmic horizon. Wait, how bloated is your inbox? Wouldn’t hurt to give it a once-over now, would it?

The Bekenstein bound does apply equally to engineered information storage as it does to black holes and universes. It gives the limit of storage capacity within a given volume before the interior collapses into a black hole. That suggests a nice challenge question. So answer me this: imagine a super-advanced civilization builds a computer with memory storage that operates at the Bekenstein bound. Essentially they’re computing on the surface of a black hole event horizon. How large a black hole computer would you need – in mass and radius – to contain enough data to simulate the entire observable universe? Let’s ignore really high entropy stuff like black holes, the cosmic background of photons and neutrinos, and also dark matter. Just regular matter like protons, electrons, etc. Assume one bit per elementary particle.

And I also have an extra credit question. Go and read the paper “Computational capacity of the universe” by Seth Lloyd – link in the description – and answer this: how long would that black hole computer take to simulate the entire universe? You’re going to need to make a bunch of assumptions, and there isn’t a single perfect answer, but I’m really curious to see what you come up with.

For both questions I’ll post detailed wording and perhaps some hints on twitter and on my own instagram. Links in the description. By the way, the latter is very new and I’m using it, among other things, to give you a bit of a behind the scenes on space time stuff – especially graphics, as well as extra details on challenge questions.  

Submit your answers to the challenge question within 2 weeks of release of this episode. Email them to pbsspacetime@gmail.com and use the subject line simulated universe challenge. Use exactly this subject line and check your spelling because we filter by subject line. Please make sure your answers are neat and show complete working! And use full English sentences to explain your thinking! You may know that you’re a genius, but it’s important to be able to communicate that fact to others. OK, assuming no one switches off the simulation, I’ll see you next week for a new episode of Space Time.