*[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.