In science fiction, hyperdrives enable spaceships to move faster than light by traversing higher dimensions. And physicists have examined in great detail the question of whether such additional dimensions actually exist. So what did you find? Are additional dimensions possible? What do they have to do with string theory and black holes at the Large Hadron Collider? And if additional dimensions are possible, can we use them for space travel? That’s what we’re going to talk about today.
This video continues last week’s where I talked about the history of the additional dimensions. As I explained in the previous video, you can describe all fundamental natural forces geometrically if you add 7 spatial dimensions to our normal three dimensions. And that sounds like a really promising idea for a unified theory of physics. Indeed, in the early 1980s, the string theorist Edward Witten found it fascinating that seven additional spatial dimensions also represent the maximum for supergravity.
However, it turned out that this numerical coincidence was going nowhere. This geometric construction of fundamental forces, known as the Kaluza-Klein theory, suffers from several problems that no one has solved.
One problem is that the radii of these additional dimensions are unstable. So they could grow or shrink, and that is inconsistent with observation. Another problem is that some of the particles we know exist in two different versions, one for left-handers and one for right-handers. And these two versions don’t behave the same. This is called chirality. That particles behave this way is an observational fact, but it does not fit the Kaluza-Klein idea. Witten actually worried about this in his 1981 paper.
Enter the string theory. In string theory, the basic entities are strings. The fact that the strings are basic means that they are not made up of anything else. You just are. And everything else is made up of these strings. Now you can ask yourself in how many dimensions a string has to wobble in order to correctly describe the observed physics.
The first answer string theorists got was twenty-six. That is twenty-five space dimensions and one time dimension. That is much. However, it turns out that if you add supersymmetry, the number drops to ten, which is nine dimensions of space and one dimension of time. String theory just doesn’t work properly in less space.
This creates the same problem that people had with the 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 have curled up into small radii.
In string theory, these additional dimensions are lumped together into intricate geometric shapes called “Calabi-Yau manifolds,” but the details are not too important. It is important that the strings have higher harmonics because of this curling. This is the same as what happens in the Kaluza-Klein theory. And when a string receives enough energy, it can vibrate at certain frequencies that must match the radius of these additional dimensions.
Hence, it is not true that string theory does not make predictions, although I often hear people claim that. String theory predicts that these higher harmonics should exist. The problem is, you need really high energies to create them. That’s because we already know that these coiled dimensions must be small. And small radii mean high frequencies and thus high energies.
How high does the energy have to be to see these higher harmonics? Ah, here is the thing. String theory doesn’t tell you. We just know that these extra dimensions must be so small that we haven’t seen them yet. In principle, they could only be inaccessible, and the next larger particle collider could generate these higher harmonics.
And here … comes the idea that the Large Hadron Collider could create tiny black holes.
To understand how additional dimensions help in creating black holes, you first need to know that Newton’s law over the R-square is geometric. The gravitational force of a point mass falls by one over the square R because the surface area of the sphere increases with the square R, where R is the radius of the sphere. So if you increase the distance from the mass, the lines of force become thinner as the surface area of the sphere grows. But … here’s the important point. Suppose you have additional dimensions of space. Suppose you have not three, but 3 + n, where n is a positive integer. Then the surface of the sphere increases with R (2 + n).
Hence, the gravitational force drops one over R (2 + n) as you move away from mass. This means that if the space has more than three dimensions, the force drops much faster with the distance to the source than it normally does.
Of course, Newton’s gravity was superseded by Einstein’s theory of general relativity, but this general geometrical consideration of how gravity becomes weaker with distance from the source remains valid. In higher dimensions, the gravitational force drops faster with increasing distance from the source.
Note, however, that the extra dimensions we are concerned with are curled up, otherwise we would have noticed them by now. This means that the lines of force in the direction of these additional dimensions can only spread up to a distance that is comparable to the radius of the dimensions. After that, the lines of force can only spread in the three major directions. This means that at distances much greater than the radius of the additional dimensions, it will return the usual 1 / R ^ 2 law that we observe.
Now for 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. Keep in mind, however, that you have these additional dimensions. Since the gravitational force drops much faster as you move away from the mass, it means that the force gets much stronger as you approach a mass than it would in just three dimensions. That makes it a lot easier to create black holes. If the additional dimensions are large enough, you can create black holes on the Large Hadron Collider.
At least in theory. In practice, the Large Hadron Collider has not created any black holes, which means that the additional dimensions, if present, are very small. How small “? Depends on the number of additional dimensions, but roughly less than a micrometer.
If they existed, could we travel through them? The short answer is no, and even if we could, it would be pointless. The reason is that while gravitational force can propagate into any additional dimensions, matter, like the material we are made of, cannot get there. It’s tied to a three-dimensional layer that string theorists refer to as a “brane,” meaning brane, not brain, and it’s a generalization of the membrane. So basically we are stuck in this three-dimensional brane that is our universe. But even if that wasn’t the case, what do you want in these additional dimensions anyway? There’s nothing in there and you can’t travel faster there than in our universe.
Often times, based on such illustrations, people think that additional dimensions are some kind of shortcut. The idea is that our universe is something like this curved leaf and then you can walk in a direction perpendicular to it to get to a seemingly distant point faster. The thing is, however, that you don’t need any additional dimensions for this. What we call “dimension” in general relativity would be represented in this picture by the dimension of the surface, which does not change. In fact, these things are called wormholes, and you can have them in ordinary general relativity with the odinary three dimensions of space.
This embedding space does not actually exist here in the general theory of relativity. This is also why people are confused about what the universe is expanding into. It doesn’t expand into anything, it just expands. By the way, fun fact, if you want to embed a general 4-dimensional spacetime in a higher dimensional flat space, you need 10 dimensions, which happens to be the number of dimensions you need for string theory to make sense. Another one of those meaningless numerical coincidences, but I’m digressing.
What does this mean for space travel? Well, it means that traveling through higher dimensions using hyperdrives is extremely implausible scientifically. Therefore, my ultimate ranking for the scientific plausibility of science fiction travel is:
3rd place: hyperdrives, because it’s a nice idea, it just doesn’t make scientific sense.
2nd place: wormholes because they exist at least mathematically, although no one has any idea how to create them.
And the winner is … warp drives! Since the math doesn’t just work, in principle it is possible to create it, at least as long as you stay below the speed of light limit. I’m afraid we still don’t know how to travel faster than light. But maybe you are the one to find out.