There is geometry
in the humming of the strings.
There is music
in the spacing of the spheres.
~Pythagoras

When Albert Einstein and David Hilbert published the theory of general relativity, they weren’t just proposing a new theory of gravity. They were proposing a new way of thinking. In general relativity, gravity isn’t a force. Instead, it’s a natural consequence of the shape of the universe. Force comes from stuff. Matter pushes and pulls on other matter. A proton may need to use its electric field to attract an electron, but the field is a property of the proton. In relativity, however, gravity comes from shape. The universe is bent by mass and energy, and matter just follows the straightest path it can through space and time. This looks like a force, but it’s fundamentally different.

But if Gravity can come from geometry, from the shape of the universe itself, can anything else? Perhaps every force we’ve ever seen—or even every particle we’ve ever seen—is an illusion arising from geometry. This is the question that inspired Kaluza-Klein theory, which in turn inspired grand unifying theories like string theory.

If you follow me regularly, you know that I’ve written a fair amount on general relativity. I think an intuitive understanding of general relativity will help you understand Kaluza-Klein theory better. So you may want follow some of those links. I also suggest reading my article on Fourier analysis, since that will prove important.

Some History

In April 1919, Theodor Kaluza was studying general relativity in five dimensions—purely as a mathematical exercise—when he noticed something fantastic. If he took a special type of five-dimensional spacetime and looked at it from only four dimensions, it reduced to four dimensional general relativity with an extra set equations: the equations for electromagnetism.

Kaluza immediately wrote to Einstein. Einstein—who already had his heart set on a theory of everything, an obsession that would last until he died—was ecstatic. Einstein not only urged Kaluza to publish, but even helped him present his paper to the Prussian Academy of Sciences in Berlin.

Kaluza’s idea had one small problem: we can’t move in five directions (4 space and one time). We can only move in four (3 space and one time). How can we explain this discrepancy? Physicist Oskar Klein came to the rescue. Klein discovered that if the dimension was a circle, he could make it small enough so that we couldn’t travel in the fifth direction. Even better, the special type of five-dimensional spacetime required by Kaluza emerged naturally.

Although Einstein wrote a paper on Kaluza and Klein’s ideas, the whole idea went mostly ignored until the invention of string theory, when it played a very important role.

Some Analogies

Kaluza-Klein theory is rather more abstract than any of the physics I’ve talked about before. So, to give you a better intuitive feel for what’s going on, before I go into the details of Kaluza and Klein’s discoveries, I’m going to try to give several analogies.

The Hand Shadow Puppet

Let’s look at the title image again.

A shadow puppet perfectly encapsulates the idea of Kaluza-Klein theory. In two dimensions, a three-dimensional hand, which is a very funny shape if you think about it, appears as the coherent image of something familiar like a man’s head or rabbit or a duck. In Kaluza Klein theory, a we can project five-dimensional spacetime down to four dimensions, where it appears to be something familiar: a four-dimensional spacetime and electromagnetism.

The Tightrope Walker

To make things easier to visualize, let’s take everything down a dimension. Imagine that the real world is only one dimension and the extra dimension proposed by Kaluza gives us two dimensions. Now, imagine a tightrope walker confined to walk his tightrope forever.

To the tightrope walker, the world is one-dimensional. He can walk either forward or backwards, that’s it. Of course, the world is actually two-dimensional: an ant can walk around the tightrope to the other side.

Although the tightrope walker can’t see the tightrope, the shape of the tightrope effects him. If the tightrope is square and aligns with his feet he’ll be able to walk more easily than if the tightrope is round. And, if the cable is pointy, then the tightrope walker may be in a lot of pain!

If the tightrope walker could not perceive the second dimension of the tightrope, he might imagine that, as the shape of the cross-section changed, he was under the influence of a force.

Similarly, in Kaluza-Klein theory, we live in a four-dimensional spacetime on the surface of a very small circle, which is the fifth dimension. The shape of the circle is reflected in our world as electromagnetic force.

How It Works

Now let’s see if we can’t understand Kaluza-Klein theory a bit better. This part may have a bit of abstract math in it so feel free to skip it. Hopefully I got the idea across with my analogies. We can define the curvature of spacetime by how we measure distance. To write this out mathematically, we define what we call a metric. A metric is a list of   rounded up numbers—where is the dimension of the space—that quantify how difficult it is to move in a given direction. So in three dimensions, there are five numbers, in four there are eight, and in five there are thirteen. These numbers tell us how difficult it is to travel in a given direction. For instance, it may take more effort to travel north than to travel east.

In Kaluza-Klein theory, the metric is made up of thirteen numbers that change as we travel in five dimensions. But, because the fifth dimension is so small, we can’t travel in it. Then, eight of the numbers correspond to the metric in four-dimensional general relativity, four of the numbers correspond to the electric and magnetic fields of Maxwell’s theory. The final number corresponds to an as-of-yet unobserved particle.

You might think this is a simple mathematical trick, but if we follow the math through and try to generate Einstein’s equations for general relativity in five dimensions, we recover Einstein’s equations in four dimensions and Maxwell’s equations for electromagnetism.

This is a staggering development! Out of the pure geometry of a five-dimensional spacetime, we generate a theory of four-dimensional spacetime that has photons and light in it! (Remember that photons are the particles corresponding to electromagnetic radiation.)

Dreams of Unification

This offers an answer to a fundamental question. Why is there stuff? Why should there be matter in the universe? Why should there be energy? Kaluza and Klein suggest that matter itself might emerge like a shadow from a shadow hand puppet as a lower-dimensional echo of a higher-dimensional geometry. Einstein was entranced with this idea, and many physicists after him have followed suit. Given the beauty of the idea, can you blame them?

But is it true?

No one seriously studies Kaluza-Klein theory any more. It predicts a particle we have not yet observed, which troubles physicists. However, the idea itself has taken hold. We can add even more tiny curled up dimensions to build unified theories that try to take it all in, and explain everything with geometry. This is where theories like string theory come in. Although Kaluza-Klein theory is generally considered un-physical, the founding principle may yet offer us a path to a unified theory of physics.

Further Reading

There is almost nothing on Kaluza-Klein theory meant for the general public. All I can suggest is that you check out popular treatments of string theory, like Brian Green’s “The Elegant Universe.” I also found a very nice article on Einstein’s brush with grand unified theories .

Questions? Comments? Hatemail?

I’m a bit concerned that this article was too abstract. What do you guys think? Let me know by email, on google+, or in the comments. And of course, if you have any insight, comments, corrections, or insults, let me know about that too!