Saturday, April 17, 2021

Does the Universe have higher dimensions? Part 2

[This is a transcript of the video embedded below.]


In science fiction, hyper drives allow spaceships to travel faster than light by going through higher dimensions. And physicists have studied the question whether such extra dimensions exist for real in quite some detail. So, what have they found? Are extra dimensions possible? What do they have to do with string theory and black holes at the Large Hadron collider? And if extra dimensions are possible, can we use them for space travel? That’s what we will talk about today.

This video continues the one of last week, in which I talked about the history of extra dimensions. As I explained in the previous video, if one adds 7 dimensions of space to our normal three dimensions, then one can describe all of the fundamental forces of nature geometrically. And that sounds like a really promising idea for a unified theory of physics. Indeed, in the early 1980s, the string theorist Edward Witten thought it was intriguing that seven additional dimensions of space is also the maximum for supergravity.

However, that numerical coincidence turned out to not lead anywhere. This geometric construction of fundamental forces which is called Kaluza-Klein theory, suffers from several problems that no one has managed to solved.

One problem is that the radii of these extra dimensions are unstable. So they could grow or shrink away, and that’s not compatible with observation. Another problem is that some of the particles we know come in two different versions, a left handed and a right handed one. And these two version do not behave the same way. This is called chirality. That particles behave this way is an observational fact, but it does not fit with the Kaluza-Klein idea. Witten actually worried about this in his 1981 paper.

Enter string theory. In string theory, the fundamental entities are strings. That the strings are fundamental means they are not made of anything else. They just are. And everything else is made from these strings. Now you can ask how many dimensions does a string need to wiggle in to correctly describe the physics we observe?

The first answer that string theorists got was twenty six. That’s twenty five dimensions of space and one dimension of time. That’s a lot. Turns out though, if you add supersymmetry the number goes down to ten, so, nine dimension of space and one dimension of time. String theory just does not work properly in fewer dimensions of space.

This creates the same problem that people had with Kaluza-Klein theory a century ago: If these dimensions exist, where are they? And string theorists answered the question the same way: We can’t see them, because they are curled up to small radii.

In string theory, one curls up those extra dimensions to complicated geometrical shapes called “Calabi-Yau manifolds”, but the details aren’t all that important. The important thing is that because of this curling up, the strings have higher harmonics. This is the same thing which happens in Kaluza-Klein theory. And it means, if a string gets enough energy, it can oscillate with certain frequencies that have to match to the radius of these extra dimensions.

Therefore, it’s not true that string theory does not make predictions, though I frequently hear people claim that. String theory makes the prediction that these higher harmonics should exist. The problem is that you need really high energies to create them. That’s because we already know that these curled up dimensions have to be small. And small radii means high frequencies, and therefore high energies.

How high does the energy have to be to see these higher harmonics? Ah, here’s the thing. String theory does not tell you. We only know that these extra dimensions have to be so small we haven’t yet seen them. So, in principle, they could be just out of reach, and the next bigger particle collider could create these higher harmonics.

And this… is where the idea comes from that the Large Hadron Collider might create tiny black holes.

To understand how extra dimensions help with creating black holes, you first have to know that Newton’s one over R squared law is geometrical. The gravitational force of a point mass falls with one over R squared because the surface of the sphere grows with R squared, where R is the radius of the sphere. So, if you increase the distance to the mass, the force lines thin out as the surface of the sphere grows. But… here is the important point. Suppose you have additional dimensions of space. Say you don’t have three, but 3+n, where n is a positive integer. Then, the surface of the sphere increases with R to the (2+n).

Consequently, the gravitational force drops with one over R to the (2+n) as you move away from the mass. This means, if space has more than three dimensions, the force drops much faster with distance to the source than normally.

Of course Newtonian gravity was superseded by Einstein’s theory of General Relativity, but this general geometric consideration about how gravity weakens with distance to the source remains valid. So, in higher dimensions the gravitational force drops faster with distance to the source.

Keep in mind though that the extra dimensions we are concerned with are curled up, because otherwise we’d already have noticed them. This means, into the direction of these extra dimensions, the force lines can only spread out up to a distance that is comparable to the radius of the dimensions. After this, the only directions the force lines can continue to spread out into are the three large directions. This means that on distances much larger than the radius of the extra dimensions, this gives back the usual 1/R^2 law, which we observe.

Now about those black holes. If gravity works as usual in three dimensions of space, we cannot create black holes. That’s because gravity is just too weak. But consider you have these extra dimensions. Since the gravitational force falls much faster as you go away from the mass, it means that if you get closer to a mass, the force gets much stronger than it would in only 3 dimensions. That makes it much easier to create black holes. Indeed, if the extra dimensions are large enough, you could create black holes at the Large Hadron Collider.

At least in theory. In practice, the Large Hadron Collider did not produce black holes, which means that if the extra dimensions exist, they’re really small. How “small”? Depends on the number of extra dimensions, but roughly speaking below a micrometer.

If they existed, could we travel through them? The brief answer is no, and even if we could it would be pointless. The reason is that while the gravitational force can spread into all of the extra dimensions, matter, like the stuff we are made of, can’t go there. It is bound to a 3-dimensional slice, which string theorists call a “brane”, that’s b r a n e, not b r a i n, and it’s a generalization of membrane. So, basically, we’re stuck on this 3-dimensional brane, which is our universe. But even if that was not the case, what do you want in these extra dimensions anyway? There isn’t anything in there and you can’t travel any faster there than in our universe.

People often think that extra dimensions provide a type of shortcut, because of illustrations like this. The idea is that our universe is kind of like this sheet which is bent and then you can go into a direction perpendicular to it, to arrive at a seemingly distant point faster. The thing is though, you don’t need extra dimensions for that. What we call the “dimension” in general relativity would be represented in this image by the dimension of the surface, which doesn’t change. Indeed, these things are called wormholes and you can have them in ordinary general relativity with the odinary three dimensions of space.

This embedding space here does not actually exist in general relativity. This is also why people get confused about the question what the universe expands into. It doesn’t expand into anything, it just expands. By the way, fun fact, if you want to embed a general 4 dimensional space-time into a higher dimensional flat space you need 10 dimensions, which happens to be the same number of dimensions you need for string theory to make sense. Yet another one of these meaningless numerical coincidences, but I digress.

What does this mean for space travel? Well, it means that traveling through higher dimensions by using hyper drives is scientifically extremely implausible. Therefore, my ultimate ranking for the scientific plausibility of science fiction travel is:

3rd place: Hyper drives because it’s a nice idea, it just makes no scientific sense.

2nd place: Wormholes, because at least they exist mathematically, though no one has any idea how to create them.

And the winner is... Warp drives! Because not only does the mathematics work out, it’s in principle possible to create them, at least as long as you stay below the speed of light limit. How to travel faster than light, I am afraid we still don’t know. But maybe you are the one to figure it out.

Saturday, April 10, 2021

Does the Universe have Higher Dimensions? Part 1

[This is a transcript of the video embedded below.]

Space, the way we experience it, has three dimensions. Left-right, forward backward, and up-down. But why three? Why not 7? Or 26? The answer is: No one knows. But if no one knows why space has three dimensions, could it be that it actually has more? Just that we haven’t noticed for some reason? That’s what we will talk about today.


The idea that space has more than three dimensions may sound entirely nuts, but it’s a question that physicists have seriously studied for more than a century. And since there’s quite a bit to say about it, this video will have two parts. In this part we will talk about the origins of the idea of extra dimensions, Kaluza-Klein theory and all that. And in the next part, we will talk about more recent work on it, string theory and black holes at the Large Hadron Collider and so on.

Let us start with recalling how we describe space and objects in it. In two dimensions, we can put a grid on a plane, and then each point is a pair of numbers that says how far away from zero you have to go in the horizontal and vertical direction to reach that point. The arrow pointing to that point is called a “vector”.

This construction is not specific to two dimensions. You can add a third direction, and do exactly the same thing. And why stop there? You can no longer *draw a grid for four dimensions of space, but you can certainly write down the vectors. They’re just a row of four numbers. Indeed, you can construct vector spaces in any number of dimensions, even in infinitely many dimensions.

And once you have vectors in these higher dimensions, you can do geometry with them, like constructing higher dimensional planes, or cubes, and calculating volumes, or the shapes of curves, and so on. And while we cannot directly draw these higher dimensional objects, we can draw their projections into lower dimensions. This for example is the projection of a four-dimensional cube into two dimensions.

Now, it might seem entirely obvious today that you can do geometry in any number of dimensions, but it’s actually a fairly recent development. It wasn’t until eighteen forty-three, that the British mathematician Arthur Cayley wrote about the “Analytical Geometry of (n) Dimensions” where n could be any positive integer. Higher Dimensional Geometry sounds innocent, but it was a big step towards abstract mathematical thinking. It marked the beginning of what is now called “pure mathematics”, that is mathematics pursued for its own sake, and not necessarily because it has an application.

However, abstract mathematical concepts often turn out to be useful for physics. And these higher dimensional geometries came in really handy for physicists because in physics, we usually do not only deal with things that sit in particular places, but with things that also move in particular directions. If you have a particle, for example, then to describe what it does you need both a position and a momentum, where the momentum tells you the direction into which the particle moves. So, actually each particle is described by a vector in a six dimensional space, with three entries for the position and three entries for the momentum. This six-dimensional space is called phase-space.

By dealing with phase-spaces, physicists became quite used to dealing with higher dimensional geometries. And, naturally, they began to wonder if not the *actual space that we live in could have more dimensions. This idea was first pursued by the Finnish physicist Gunnar Nordström, who, in 1914, tried to use a 4th dimension of space to describe gravity. It didn’t work though. The person to figure out how gravity works was Albert Einstein.

Yes, that guy again. Einstein taught us that gravity does not need an additional dimension of space. Three dimensions of space will do, it’s just that you have to add one dimension of time, and allow all these dimensions to be curved.

But then, if you don’t need extra dimensions for gravity, maybe you can use them for something else.

Theodor Kaluza certainly thought so. In 1921, Kaluza wrote a paper in which he tried to use a fourth dimension of space to describe the electromagnetic force in a very similar way to how Einstein described gravity. But Kaluza used an infinitely large additional dimension and did not really explain why we don’t normally get lost in it.

This problem was solved few years later by Oskar Klein, who assumed that the 4th dimension of space has to be rolled up to a small radius, so you can’t get lost in it. You just wouldn’t notice if you stepped into it, it’s too small. This idea that electromagnetism is caused by a curled-up 4th dimension of space is now called Kaluza-Klein theory.

I have always found it amazing that this works. You take an additional dimension of space, roll it up, and out comes gravity together with electromagnetism. You can explain both forces entirely geometrically. It is probably because of this that Einstein in his later years became convinced that geometry is the key to a unified theory for the foundations of physics. But at least so far, that idea has not worked out.

Does Kaluza-Klein theory make predictions? Yes, it does. All the electromagnetic fields which go into this 4th dimension have to be periodic so they fit onto the curled-up dimension. In the simplest case, the fields just don’t change when you go into the extra dimension. And that reproduces the normal electromagnetism. But you can also have fields which oscillate once as you go around, then twice, and so on. These are called higher harmonics, like you have in music. So, Kaluza Klein theory makes a prediction which is that all these higher harmonics should also exist.

Why haven’t we seen them? Because you need energy to make this extra dimension wiggle. And the more it wiggles, that is, the higher the harmonics, the more energy you need. Just how much energy? Well, that depends on the radius of the extra dimension. The smaller the radius, the smaller the wavelength, and the higher the frequency. So a smaller radius means you need higher energy to find out if the extra dimension is there. Just how small the radius is, the theory does not tell you, so we don’t know what energy is necessary to probe it. But the short summary is that we have never seen one of these higher harmonics, so the radius must be very small.

Oskar Klein himself, btw was really modest about his theory. He wrote in 1926:
"Ob hinter diesen Andeutungen von Möglichkeiten etwas Wirkliches besteht, muss natürlich die Zukunft entscheiden."

("Whether these indications of possibilities are built on reality has of course to be decided by the future.")

But we don’t actually use Kaluza-Klein theory instead of electromagnetism, and why is that? It’s because Kaluza-Klein theory has some serious problems.

The first problem is that while the geometry of the additional dimension correctly gives you electric and magnetic fields, it does not give you charged particles, like electrons. You still have to put those in. The second problem is that the radius of the extra dimension is not stable. If you perturb it, it can begin to increase, and that can have observable consequences which we have not seen. The third problem is that the theory is not quantized, and no one has figured out how to quantize geometry without running into problems. You can however quantize plain old electromagnetism without problems.

We also know today of course that the electromagnetic force actually combines with the weak nuclear force to what is called the electroweak force. That, interestingly enough, turns out to not be a problem for Kaluza-Klein theory. Indeed, it was shown in the 1960s by Ryszard Kerner, that one can do Kaluza-Klein theory not only for electromagnetism, but for any similar force, including the strong and weak nuclear force. You just need to add a few more dimensions.

How many? For the weak nuclear force, you need two more, and for the strong nuclear force another four. So in total, we now have one dimension of time, 3 for gravity, one for electromagnetism, 2 for the weak nuclear force and 4 for the strong nuclear force, which adds up to a total of 11.

In 1981, Edward Witten noticed that 11 happened to be the same number of dimensions which is the maximum for supergravity. What happened after this is what we’ll talk about next week.

Saturday, April 03, 2021

Should Stephen Hawking have won the Nobel Prize?

[This is a transcript of the video embedded below.]


Stephen Hawking, who sadly passed away in 2018, has repeatedly joked that he might get a Nobel Prize if the Large Hadron Collider produces tiny black holes. For example, here is a recording of a lecture he gave in 2016:
“Some of the collisions might create micro black holes. These would radiate particles in a pattern that would be easy to recognize. So I might get a Nobel Prize after all.”
The British physicist and science writer Phillip Ball, who attended this 2016 lecture, commented:
“I was struck by how unusual it was for a scientist to state publicly that their work warranted a Nobel… [It] gives a clue to the physicist’s elusive character: shamelessly self-promoting to the point of arrogance, and heedless of what others might think.”
I heard Hawking say pretty much exactly the same thing in a public lecture a year earlier in Stockholm. But I had an entirely different reaction. I didn’t think of his comment as arrogant. I thought he was explaining something which few people knew about. And I thought he was right in that, if the Large Hadron Collider would have seen these tiny black holes decay, he almost certainly would have gotten a Nobel Prize. But I also thought that this was not going to happen. He was much more likely to win a Nobel Prize for something else. And he almost did.

Just exactly what might Hawking have won the Nobel Prize for, and should he have won it? That’s what we will talk about today.

In nineteen-seventy-four, Stephen Hawking published a calculation that showed black holes are not perfectly black, but they emit thermal radiation. This radiation is now called “Hawking radiation”. Hawking’s calculation shows that the temperature of a black hole is inversely proportional to the mass of the black hole. This means, the larger the black hole, the smaller its temperature, and the harder it is to measure the radiation. For the astrophysical black holes that we know of, the temperature is way, way too small to be measurable. So, the chances of him ever winning a Nobel Prize for black hole evaporation seemed very small.

But, in the late nineteen-nineties, the idea came up that tiny black holes might be produced in particle collisions at the Large Hadron Collider. This is only possible if the universe has additional dimensions of space, so not just the three that we know of, but at least five. These additional dimensions of space would have to be curled up to small radii, because otherwise we would already have seen them.

Curled up extra dimensions. Haven’t we heard that before? Yes, because string theorists talk about curled up dimensions all the time. And indeed, string theory was the major motivation to consider this hypothesis of extra dimensions of space. However, I have to warn you that string theory does NOT tell you these extra dimensions should have a size that the Large Hadron Collider could probe. Even if they exist, they might be much too small for that.

Nevertheless, if you just assume that the extra dimensions have the right size, then the Large Hadron Collider could have produced tiny black holes. And since they would have been so small, they would have been really, really hot. So hot, indeed, they’d decay pretty much immediately. To be precise, they’d decay in a time of about ten to the minus twenty-three seconds, long before they’d reach a detector.

But according to Hawking’s calculation, the decay of these tiny black holes should proceed by a very specific pattern. Most importantly, according to Hawking, black holes can decay into pretty much any other particle. And there is no other particle decay which looks like this. So, it would have been easy to see black hole decays in the data. If they had happened. They did not. But if they had, it would almost certainly have gotten Hawking a Nobel Prize.

However, the idea that the Large Hadron Collider would produce tiny black holes was never very plausible. That’s because there was no reason the extra dimensions, in case they exist to begin with, should have just the right size for this production to be possible. The only reason physicists thought this would be the case was an argument from mathematical beauty called “naturalness”. I have explained the problems with this argument in an earlier video, so check this out for more.

So, yeah, I don’t think tiny black holes at the Large Hadron Collider was Hawking’s best shot at a Nobel Prize.

Are there other ways you could see black holes evaporate? Not really. Without these curled up extra dimensions, which do not seem to exist, we can’t make black holes ourselves. Without extra dimensions, the energy density that we’d have to reach to make black holes is way beyond our technological limitations. And the black holes that are produced in natural processes are too large, and then too cold to observe Hawking radiation.

One thing you *can do, though, is simulating black holes with superfluids. This has been done by the group of Jeff Steinhauer in Israel. The idea is that you can use a superfluid to mimic the horizon of a black hole. If you remember, the horizon of a black hole is a boundary in space, from inside of which light cannot escape. In a superfluid, one does not trap light, but one traps sound waves instead. One can do this because the speed of sound in the superfluid depends on the density of the fluid. And since one can experimentally control this density, one can control the speed of sound.

If one then makes the fluid flow, there’ll be regions from within which the sound waves cannot escape because they’re just too slow. It’s like you’re trying to swim away from a waterfall. There’s a boundary beyond which you just can’t swim fast enough to get away. That boundary is much like a black hole horizon. And the superfluid has such a boundary, not for swimmers, but for sound waves.

You can also do this with a normal fluid, but you need the superfluid so that the sound has the right quantum properties, as it does in Hawking’s calculation. And in a series of really neat experiments, Steinhauer’s group has shown that these sound waves in the superfluid indeed have the properties that Hawking predicted. That’s because Hawking’s calculation applies to the superfluid in just exactly the same way it applies to real black holes.

Could Hawking have won a Nobel Prize for this? I don’t think so. That’s because mimicking a black hole with a superfluid is cool, but of course it’s not the real thing. These experiments are a type of quantum simulation, which means they demonstrate that Hawking’s calculation is correct. But the measurements on superfluids cannot demonstrate that Hawking’s prediction is correct for real black holes.

So, in all fairness, it never seemed likely Hawking would win a Nobel Prize for Hawking radiation. It’s just too hard to measure. But that wasn’t the only thing Hawking did in his career.

Before he worked on black hole evaporation, Hawking worked with Penrose on the singularity theorems. Penrose’s theorem showed that, in contrast to what most physicists believed at the time, black holes are a pretty much unavoidable consequence of stellar collapse. Before that, physicists thought black holes are mathematical curiosities that would not be produced in reality. It was only because of the singularity theorems that black holes began to be taken seriously. Eventually astronomers looked for them, and now we have solid experimental evidence that black holes exist. Hawking applied the same method to the early universe to show that the Big Bang singularity is likewise unavoidable, unless General Relativity somehow breaks down. And that is an absolutely amazing insight about the origin of our universe.

I made a video about the history of black holes two years ago in which I said that the singularity theorems are worth a Nobel Prize. And indeed, Penrose was one of the recipients of the 2020 Nobel Prize in physics. If Hawking had not died two years earlier, I believe he would have won the Nobel Prize together with Penrose. Or maybe the Nobel Prize committee just waited for him to die, so they wouldn’t have to think about just how to disentangle Hawking’s work from Penrose’s? We’ll never know.

Does it matter that Hawking did not win a Nobel Prize? Personally, I think of the Nobel Prize in the first line as an opportunity to celebrate scientific discoveries. The people who we think might win this prize are highly deserving with or without an additional medal. And Hawking didn’t need a Nobel Prize, he’ll be remembered without it.