Why do Extra Spacetime Dimensions Unify the Fundamental Forces?
Have you ever toyed with the idea of the universe having extra dimensions? What would it look like if it did? String theory suggests that…
Have you ever toyed with the idea of the universe having extra dimensions? What would it look like if it did? String theory suggests that the universe has many dimensions, either 10, 11, or 26 depending on the flavor, but all but the usual four, three space and one time, are wrapped up so small that we can’t detect them.
About 100 years ago, Albert Einstein got excited when he received a paper from a man named Kaluza. You see, Kaluza had done something that figured very neatly into Einstein’s way of viewing physics: he had shown that you can unify the force of gravity with the force of electromagnetism by adding an extra dimension to the universe.
At the time Einstein was very interested in this idea of unifying forces even though he didn’t know that two of them hadn’t really been discovered yet: the weak and strong force. What would have been pretty easy before then became a lot harder later, but that hasn’t stopped physicists from trying.
Einstein was philosophically interested in finding a unified theory for all of nature. He felt it should exist. He also thought that, since gravity was based on geometry, other forces should be geometric too. It was only natural.
Let’s have a little background on the difference between a force theory and a geometric theory. You can also look into my article on the mysteries of the groma or my article on gauge theoretic gravity for more.
As an analogy, let’s take the Earth. The Earth is round and so when we travel it we have to take round paths. It turns out that when you want to get from one place to another very quickly on the Earth you need to follow a path called a geodesic. This is the shortest path from one point to another on our planet “as the crow flies”. A geodesic is simply a section of a Great Circle, which is a circular line around the Earth that cuts the planet in two equal halves.
If you want to follow a geodesic, you can’t take a constant heading path in most cases (unless you are traveling due North or South). If you did follow a constant heading, you would be following a “rhumb line” path which is not a shortest path at all.
Here’s a nifty old map I found that shows the difference. This map is in Mercator projection so the rhumb line path shown, from Luzon to San Francisco, at 79 degrees looks shorter than the dot-dash line representing the geodesic. It is a distortion of the projection though. The dot-dash line is shorter.
As you travel along your geodesic, your boat or airplane has to make periodic course corrections.
Here is the most important take away here: Those course corrections are equivalent to the force of gravity. In other words, in Einstein’s universe gravity helpfully keeps me following geodesic paths in the four dimensional space and time in the same way the tiller helps my boat stay on geodesic paths on the two dimensional surface of the Earth.
I have always thought that to be a profoundly weird observation of Einstein’s theory and have in places argued that his theory is better presented as a local force theory rather than a geometric theory, but for now let’s stick to the way Einstein saw it.
Moving on to extra dimensions. Suppose we added a dimension to the surface of the Earth? Obviously, we could fly, but what would it mean for my course corrections? Not a lot it turns out and that was the beauty of Kaluza’s solution. By adding a dimension, he could graft a completely different force onto Einstein’s theory without altering the existing theory. The only requirement was that he had to make the extra dimension very boring. That is, he had to assume that in that extra dimension everything was always the same.
To really understand this idea, we have to take a dimension away from the surface of the Earth. Imagine we are two dimensional beings living on a one-dimensional surface. If you’ve read Flatland, you get the idea.
Now on this circle planet you can only go left or right. Any direction is a great circle! Suppose we add a new dimension to it. If that dimension makes the Earth a sphere then we are in trouble because left and right are no longer geodesic paths all the time. That means that we have to make course corrections that we didn’t have to make before so the extra dimension has changed our gravitational force.
But what if we make the circle planet into a cylinder instead? It turns out that now left and right are still geodesics on a cylinder world. That means that the extra dimension did not change gravity.
That is why the assumption that Kaluza made is called “cylindricity” because the new 5th dimension didn’t change the geodesics in our usual 4d spacetime for any geometry [1].
How Kaluza came up with the idea that adding a dimension would fold Maxwell’s equations of electromagnetism into General Relativity is amazing. The reason why it works comes from a very deep fact about our universe.
It has to do with symmetry, which makes men and women beautiful and also makes the universe work. People with symmetrical faces look the same in the mirror that they do in real life because the left and right side of their faces are the same.
Any time you have a symmetry in the universe you get two things: a conserved quantity and a force that goes with it. Conserved just means you can’t create or destroy it. Energy and momentum are two examples and they go with time and space translation symmetry. Gravity is the force.
These days we know that every force has some kind of symmetry associated with it and every conserved quantity too.
Charge is conserved and electromagnetism goes with it. The symmetry is of rotations in the complex plane which we also call phase translations.
To understand what phase translation is, think of an electrical signal on an oscilloscope. We can represent those as time series of complex numbers which looks like two sine waves offset from each other (a sine wave and a cosine wave!) You can shift the signals left and right without changing what they represent. It’s the same idea with electromagnetism.
In the mathematics of symmetries called Group Theory all the symmetries have little codes. The electromagnetic one is called U(1). The U means unitary since phase changes don’t change the amplitude of waves so the changes are like multiplying by 1. The (1) just refers to the dimension, are we multiplying the signals by one complex number or many in a matrix? In this case just one.
Now that we understand the group theory of electromagnetism, how does adding one boring dimension to spacetime introduce this U(1) symmetry into Einstein’s theory?
Well, even though U(1) is the set of complex numbers that go round and round the unit circle in the complex plane, phase itself is just one real number that can be anything. It’s a measure of degrees or radians. Think about it: if I shift the phase of my sine and cosine waves, I can shift them by any amount left or right that I want, not just one wavelength. But if I shift by a multiple of 360 degrees, I get an identical wave. That’s why the symmetry is of phase translation not ordinary translation.
That is what Kaluza added to the geometry, the ability to shift the 4d spacetime as a whole along the “cylinder” left or right as much as he wanted. That was why electromagnetism popped out of Einstein’s equations. It took a little more work of course but without that left and right phase symmetry it would never have worked.
Kaluza along with Einstein presented this work before the scientific community back in the 1920s only a few short years after Einstein himself had been catapulted onto the world stage. There was one small problem.
Well, not small, more like universe sized problem: where was the extra dimension? Einstein had at least built his theory around dimensions that nobody could argue didn’t exist. Kaluza had some ‘splaining to do.
A fellow named Klein solved that problem rather neatly however. He said that the extra dimension was just very, very small, all curled up.
Think of that sweater your mom or grandma knitted for you last Christmas. You know the one. It’s still in the back of your closet. That sweater on the surface is mostly two dimensional. All the little snowmen and that thing that’s supposed to be an elf or is that a reindeer are two dimensional pictures on a two dimensional surface. But under a magnifying glass, you can see many curls of fabric. These curls are small enough that you only notice them when they rub against your skin and give you that rash that means you’re pretty sure you’re allergic to organic alpaca.
Likewise the fifth dimension Kaluza added is curled up small. Klein showed that he could keep that cylindricity condition with the curls and still get that phase translation symmetry. After all phase changes don’t care if they are moving along a big line or going round and round like a skateboarder in a pipe.
Fast forward a few decades and scientists discover two more forces also have symmetries. The strong force that holds the nuclei of atoms together has the symmetry SU(3) which means that it is made of members that are 3 x 3 matrices of complex numbers. It is also unitary which is why it has the “U” but it is a special kind of unitary which is why it has the “S” as well. That special kind just means that all the members have a real determinant of 1.
Special unitary transformations rotate but don’t warp, stretch, or translate. While U(1) has one phase or “charge”, SU(3) has 8 independent ones. These are like Euler angles if you know what those are but for waves instead of rigid bodies. (I don’t say electromagnetic waves since I’m talking about “strong force” waves here.)
If you didn’t know, there is a group called SO(3) that is the space of rotations in ordinary 3D space. The “O” stands for orthogonal because rotations don’t warp 3D space; they are “angle preserving” which comes from orthogonality. The Euler angles are like pitch, yaw, and roll of an airplane.
If that’s not enough, in the 1970s we also find out that another force, the “weak” force that causes radioactive decay at high enough energy can be combined with the electromagnetic force to become the “electroweak” force. Its symmetry is just the electromagnetic one of U(1) times the weak force symmetry which is SU(2). Like SU(3), SU(2) is the space of phase rotations but this time only in two complex dimensions. It works pretty much the same way but with 3 phase numbers instead of 8. SU(2) is also equivalent to SO(3) so you can think of those three as pitch, roll, and yaw of weak force waves.
(If you’re wondering what SU(1) is, it’s just the number 1. It is the trivial symmetry.)
So, let’s get this straight: if Kaluza and Klein are right and the electromagnetic force comes from an additional spacetime dimension in Einstein’s theory. And the high energy physicists are right and electroweak theory is the right way to look at the weak and the electromagnetic forces. Then, it would only make sense that the weak force is also made up of geometry in Einstein’s theory. And if the other forces fit in there, then why leave out the strong force?
If you add up all the different degrees of freedom you need, 4 for Einstein’s original dimensions, 1 for electromagnetism, 3 for the weak force, and 8 for the strong force, you get 16. If you just took the “Standard Model” of the electroweak and strong forces with their 12 dimensions and multiplied their geometry by Einstein’s four, that’s what you’d get.
It turns out that you can do some double counting however and reduce it to 7, so you only need 11 dimensions [2].
Getting from here to a Theory of Everything, a.k.a. Grand Unified Theory or GUT, isn’t so easy. You also need to fit matter in somehow: quarks, electrons, and so on. That’s where things get tricky. Straightforward Kaluza and Klein will not get you matter — at least not matter that acts like real matter even if you use “super” symmetry that makes things that act “like” quarks and electrons appear out of Einstein’s theory.
String Theory, on the other hand, borrows from Kaluza and Klein but doesn’t take the pure approach of reducing everything from Einstein. The forces, including gravity, and other particles are “modes”, which are like frequencies, on the strings instead of dimensions.
Strings vibrate through space and time like the strings on a musical instrument. As they vibrate, they trace out two dimensional geometries called “world sheets”. You can see an example below.
The world sheets have geometry! It is that geometry that gives you forces and matter, not spacetime itself. It also has a nifty way to differentiate matter and forces that I will talk about in a future post.
After string theory was proposed, physicists did figure out how to derive it the way Einstein originally wanted [3], as spacetime geometry, but they didn’t need it anymore.
Even so, string theory needs extra dimensions because the math only works out in certain numbers of dimensions. Ten is the minimum number of dimensions you need to unify the forces in what is called superstring theory. Other theories need more dimensions.
In that theory the fabric of reality is “knitted” with 6 dimensional curled up spaces just like the fabric in that sweater you’re afraid to “Marie Kondo”. If you check out the picture at the top of the page, that is one of them, called a Calabi-Yau manifold.
I, for one, find this idea as compelling as Einstein did 100 years ago. If only we could prove it that would give me a spark of joy (unlike your sweater).
[1] Overduin, J. M., & Wesson, P. S. (1997). Kaluza-klein gravity. Physics Reports, 283(5–6), 303–378.
[2] Edward Witten, Search for a realistic Kaluza-Klein theory, Nuclear Physics B Volume 186, Issue 3, 10 August 1981, Pages 412–428 (spire:10244, doi:10.1016/0550–3213(81)90021–3)
[3] Duff, Michael J., B. E. W. Nilsson, and C. N. Pope. “Kaluza-Klein approach to the heterotic string.” Physics Letters B 163, no. 5–6 (1985): 343–348.