Ofri Telem, a Ph.D. candidate in theoretical physics at Cornell, explains how we could find extra dimensions in laymen’s terms

The possible existence of other dimensions has fascinated writers, filmmakers, and game designers for decades. From the mythical portals of *Narnia *and Lewis Carroll’s *Through the Looking Glass, *to the spacetime hopping *Doctor Who*, it is appealing yet eerie to consider the presence of something so utterly strange right under our noses. Countless other references can be found in movies and shows like *Star Trek, Sliders, Donnie Darko,* *Futurama,* *Rick and Morty, *and many more. But it is arguably the Netflix hit *Stranger Things *that brought the possibility of another dimension – dubbed the ‘upside down’ in the show – to mainstream attention.

But what do theoretical physicists actually have to say about extra dimensions? How probable is their existence? And why are we led to consider them in the first place? In this piece, I will try to give a hint at the answer. When stumbling across a novel idea such as extra dimensions, a typical theoretical physicist asks two main questions:

1. Is this idea already excluded by data? In other words, if there were extra dimensions, wouldn’t we be able to tell already?

2. What is this idea good for? What theoretical physics question does it solve?

The answer to the first question is yes, in some cases. This happens if the extra dimensions are infinite (like our familiar 3+1 dimensions, i.e. the 3 dimensions we see and the fourth dimension, which is time) or finite, but large enough. But there is still the possibility that finite extra dimensions exist but are so small that their effect on our present measurements are too small to be detected.

The answer to the second question is that the possible existence of extra dimensions may serve to explain one of the greatest mysteries of theoretical physics: the famous hierarchy problem. To explain the problem in full will take more time and space than available, so I will merely state that one way to solve it is to somehow explain why the gravitational interaction (the one that causes planets to orbit the sun and apples to fall on your head) is weaker – much weaker – than the other known fundamental interactions: the electromagnetic, weak and strong interactions. The existence of extra dimensions might do just that: dilute the strength of gravity. This is of course an over simplification, as the real problem regards the immense separation between the weak energy scale and the energy scale of gravity. But it will do for our purposes.

To better understand how extra dimensions solve the hierarchy problem, we first have to air out two central ideas from 20th century quantum mechanics. Before many of you run out screaming, note that these ideas slowly seeped into mainstream knowledge, so they are not that threatening any more:

1. Particles can be thought of as waves traveling through empty space. In modern language, they are called fields. For example: the electron, the photon and all other parts of the standard model of particles are fields (= waves = particles…).

2. Interactions (forces) are also mediated by fields. For example, the electromagnetic interaction between two objects can be described as two objects exchanging photons between them, like a game of ping-pong. In a similar way, the gravitational interaction is just two objects playing ping-pong with a different mediator called the graviton.

Armed with these new but not so scary ideas, we are ready to understand how extra dimensions solve the hierarchy problem.

## An extra dimension is like a water canal

Let’s imagine two particles, called Alice and Bob, interacting gravitationally. According to our assumptions, they are basically exchanging waves called gravitons. To make the analogy simpler, we will imagine that these are not gravitons, but regular waves in water.

In this analogy, Alice is transmitting waves by splashing the water periodically. The waves travel all the way to Bob who receives them. These waves carry energy from Alice to Bob, which is proportional to the square of the wave amplitude (height). Let’s draw a cartoon for this interaction:

In the water wave analogy, the waves carry energy. In the quantum mechanics picture, they carry probability instead. This is why we should think of the energy of the wave as determining the probability, or strength of the interaction mediated by the wave. If for some reason Bob would not receive all of the energy, and some of it was lost, this would be equivalent to the interaction being weaker. But where would this energy go?

## The energy gets lost in the extra dimension

Now let’s imagine that there is another dimension in our setting. Instead of the wave propagating directly from Alice to Bob, the wave is now traveling in a narrow canal. The width of the canal is like the size of the extra dimension. Alice and Bob are on one side of the canal, and Alice is transmitting waves in all directions:

As the waves bounce back from the wall of the canal, they interfere with themselves, changing their shape until they reach a form dictated by the shape of the canal. In this process, the waves forget their initial, circular shape and become uniform throughout the width of the canal. This is just like a musical instrument that resonates at a given frequency dictated by its shape. While the initial way you blow the flute or pluck the string might mildly affect the tone color, it is actually the length of the string or flute that determines the pitch. In the case when the canal is much narrower than Alice’s wavelength, only the lowest harmony survives, and the wave in the canal becomes uniform.

Since the wave is uniform, its energy is spread out uniformly throughout the width of the canal. But Bob is stuck to the wall of the canal and so he only gets a portion of the energy – the total energy divided by the width of the canal.

The energy – and in this analogy, the strength of the gravitational interaction– is diluted by the extra dimension.

We have said that when Alice is transmitting waves with a large wavelength, the eventual wave in the canal is uniform. But what would happen if Alice had enough energy to create waves with a really small wavelength? Then the eventual wave propagating in the canal would not be flat but a combination of different modes called Kaluza-Klein modes, after the early 20th century physicists who conjectured them. These are like the overtones in a musical instrument. Detecting them at a particle collider would validate the existence of an extra dimension (the canal), while their absence might hint that extra dimensions don’t exist or that their size is too small to be detected.

The first breakthrough that led modern theoretical physicists to consider extra dimensions seriously as a solution to the hierarchy problem was in a seminal paper by renowned physicists Arkani-Hamed, Dimopoulos and Dvali. The breakthrough in the paper was rooted in the fact that (by analogy) Alice and Bob were stuck on the side of the canal and so Bob received only a fraction of the energy of the wave. In theoretical physics language, we would say that both Alice and Bob are stuck on 3+1 dimensional D-branes in a 5-dimensional universe.

## Talking numbers

We’ve seen how extra dimensions can qualitatively lead to a dilution of the gravitational interaction. But in physics, it is not only the qualitative, but the quantitative that counts. Let’s see if our new extra dimension is enough to solve the hierarchy problem. To solve it, we will need gravity to be a factor of 10^{18} weaker than what you naively expect (it is not a small problem, but rather, an extremely large quandary).

To account for such a dilution, the size of the extra dimension will have to be larger than 10^{13} cm – certainly something we would have observed already from deviations from Newtonian gravity at solar system scales. Is this then the end of extra dimensions as a solution to the hierarchy problem?

You can count on sneaky theoretical physicists to come up with a solution.

The first popular solution called large extra dimensions is already presented in the original paper of Arkani-Hamed, Dimopoulos and Dvali. It is the same story as the canal – but now imagine not one, but several extra dimensions all conspiring together to dilute gravity. A mere two extra dimensions, each of size 100μm – 1mm, could account for the entire 10^{18} suppression of gravity.

How can we test this scenario experimentally? The secret is to look for the Kaluza-Klein modes of the graviton at the Large Hadron Collider, high energy physics’ mega experiment colliding 10^{34} protons a second at 13 TeV of energy. Since no Kaluza-Klein modes (higher harmonies) have been observed so far, we know that they cannot exist below a mass of 1.5-4 TeV. That means that the typical size of the extra dimensions cannot exceed ~10^{-16 }cm, and so we need at least 17 such new dimensions to solve the hierarchy problem. While this scenario cannot be disproved, we might want to go about looking for other explanations.

## A warped extra dimension

The second, and extremely popular scenario for extra dimensions is called warped extra dimensions or RS after its authors Prof. Lisa Randall from Harvard and Prof. Raman Sundrum from the University of Maryland at College Park. To demonstrate this solution, without going into painful detail, let us come back to our familiar canal analogy. This time, the width of the canal is tiny, so we might think that gravity isn’t diluted at all and the hierarchy problem remains unsolved.

However, this time we add a secret ingredient: The depth of the canal is not uniform. Instead, it varies throughout the width of the canal. Near Alice and Bob’s wall it is deep, and near the other edge it is exponentially tiny, and in between it varies continuously.

When Alice is transmitting waves, they get averaged out as before, yielding only the lowest harmony propagating in the canal. But unlike the previous scenario, the resulting wave in the canal is not uniform, but heavily peaked away from Alice and Bob. When the wave gets to Bob, he receives only a small portion of the energy – the tail of the wave front, while the high part of the wavefront is towards the wall opposite from Bob. This time, the energy that Bob gets is diluted because of the uneven depth in the canal, and the interaction is diluted accordingly.

We say that the gravitational interaction is ‘warped down’ by the uneven depth. This is happening even though the width of the canal is very small, and a single extra dimension can be enough to a account for the entire 10^{18} suppression of gravity.

Similar to large extra dimensions, the RS scenario can be probed by searching for Kaluza-Klein modes at the LHC. The fact that we haven’t seen any allows us to say that no such modes exist below an energy of 4 TeV, and so the RS model is constrained by the data. This means that even if large or warped extra dimensions exist, they must be ‘harmless’ enough not to be detected at the LHC, and so they need to be supplemented by some other ingredient in order to fully solve the hierarchy problem.

What lies ahead? The LHC will continue to supply data in the next few years, either detecting the Kaluza-Klein modes or further constraining the models and making them less attractive. It might be that we’ll find other new fields that are not related to extra dimensions, or that we won’t find anything at all. In this case, which might be our darkest but also most probable scenario, we’ll be left with the hierarchy problem unsolved. Or we might just have to think much harder.

*The views expressed are of the author**.*

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**Note of disclosure: Ofri Telem is Geektime Managing Editor Laura Rosbrow-Telem’s husband. *