How Many Dimensions Are There?

We have three physical dimensions: length, width, and height. A mathematician or physicist might represent them by x, y, and z.

In his Theory of Relativity, Einstein used time as a dimension, expressing the position of an object in space-time with four co-ordinates: x, y, z, and t. Hence the phrase "time is the fourth dimension."

Remember that the number of dimensions is just a convenience. Einstein's formulas, using four dimensions, described the universe better than Newton's equations.

You may have heard of string theory using eleven dimensions. What does this mean? Why eleven? Here's a quick explanation.

An application of the laws of conservation of energy and conservation of momentum, in senior high school physics, is calculating the final velocities of two colliding balls, given the balls' initial masses and velocities. (And the angle between them, technically included in the word "velocity".) There are two unknowns, the final velocities of each object, and two equations to use. You always need the same number of equations as unknowns.

What if there are three balls? You need three equations to find the three final velocities. But we don't have a third equation. This is the famous Three Body Problem. It's unsolved: physicists can't compute an exact answer.

But nature can! How does nature figure out what happens to three simultaneously interacting objects? It's clear that nature does know, because this situation comes up all the time. The sun, moon, and Earth are simultaneously interacting. (An interaction is one object exerting a force on another, and all three objects have gravity, which extends to infinity.)

Physicists attack the problem by dealing with the bodies in pairs, or approximating the situation by saying that the smallest object doesn't influence the others very much. But it would be nice to have a third equation, to get absolute answers instead of numerical approximations.

A special three-body question that can be solved by students is if one object comes in and hits two identical objects, like two balls touching and the third arriving on the mid-line between them. Because the situation is symmetrical, you can find an answer. (The third equation is that the final speed on one ball equals the final speed of the other, through symmetry.)

So you can solve more complex questions if symmetry is involved. Remember this fact.

Now picture a circle. It looks the same from all angles. Perfect symmetry. Even a small circle is the same as a large circle, in one respect, because a small circle is the large circle viewed from farther back.

How about a square and a diamond? Are they the same? Sure: a diamond is a square rotated.

How about a square and a hexagon? (A hexagon has six equal sides.) You can't rotate a square or view it from a different angle and direction and see a hexagon. So in two dimensions, a square and a hexagon are different.

Now use your imagination. If you illuminate a cube with a light directly overhead, the shadow is the shape of a square. But if you turn the cube, you can get a shadow the shape of a hexagon. (To convince yourself, draw a hexagon, and add the "missing lines" to make it look like a 3-D cube viewed at an angle.)

So, if you think in three dimensions, a square and a hexagon are the same thing. They're both 2-D shadows of the same 3-D object. The lesson here is that if you include an extra dimension in your considerations, you can sometimes find symmetries that didn't exist when you were working in fewer dimensions.

And symmetry allows you to solve otherwise unsolvable equations, remember?

The string theorists use eleven dimensions. Their equations are so complex, apparently, that they need eleven dimensions to give them enough symmetries to solve them.

String Theory

You may have read about physicists proposing that everything is made of vibrating strings. It’s hard to get an intuitive grasp of this concept, but perhaps the following will help get you started.

Light can be thought of as a particle or a wave, depending on what experiment you do to detect it. If you look to see how it exposes a photographic plate or hits a screen, you discover that light particles, photons, have one-to-one collisions with molecules and land in a specific place on the screen (rather than wash across a screen like a wave washes up on a beach.) If you check to see how light passes through tiny holes you see interference patterns, places where light waves cancelled and reinforced.

It turns out that all particles (not just light) have a wave nature. For example, electrons travel through slits and produce interference patterns. You’ve heard of the electron microscope, which uses electrons rather than light to view tiny objects. The electron’s wavelength is smaller than that of visible light, so we can us electron waves to detect smaller objects.

Three formulas, two formulas from Einstein, and one from de Broglie, are helpful here. The first is Einstein’s famous E=mc2, which tells how much energy is needed to create a particle of mass m, or how much energy will appear if a particle of mass m vanishes. (c is the speed of light.) The second, also from Einstein, is E = hc/λ, where λ is the wavelength of the wave associated with that particle. This gives the energy of a particle with a given wavelength.

The third formula is λ = h/mv, where λ is a particle’s wavelength, h is a constant, m is the particle’s mass and v is its velocity.

String theory? Here goes.

First, get the idea of strings made of atoms out of your mind. Forget what they are made of and just consider them to be imaginary. Here’s how they work, in a nutshell.

Consider a tiny particle, like an electron, proton, muon, whatever. The particle has a mass. When moving, it has a wavelength (equation 3) and an energy (equation 2). Now picture two girls turning a skipping rope. They are making a standing wave. Their shoulders are nodes, hardly moving. Most of the energy of the wave is near the centre of the rope where the amplitude is greatest. Picture a particle to be equivalent to a standing wave: the “probable location” of the particle is where the energy is greatest. But the particle does not actually exist in a spot, its energy (and mass, equation 1) are spread out all the way between the nodes.

If the girls wiggle their arms differently, they could set up a standing wave with two loops and a stationary point, a node, in the middle. (You never actually see this when kids are skipping, but you can do it with your stretched-out phone cord if you try. Turn it at twice the rate of the “skipping rope” turning.) In this case there are two high-energy locations, the loops or antinodes. This wave is the first overtone of the original, which is called the fundamental. Buglers (and all brass instrumentalists) make use of the overtones to get different notes without changing their fingers. (Buglers can’t change their fingers: they have no valves!)

Now the intuitive step. What if the overtone represents a different particle from the original? That is, since every particle has a wave nature, we could think of particles as different resonances of a vibrating string, different frequencies where standing waves can be set up. The more massive the particle, the more energy (equation 1). The more energy, the smaller the wavelength (equation 2). The more massive particle could be a standing wave of higher frequency or an overtone of the wave associated with the less massive particle.

String theory is much more complicated than this, of course. But the idea here is to turn on a little light bulb in our brain, a little spark of intuition. Because of the wave-particle duality of light and matter, perhaps we can think of particles as resonances of strings. It is space itself which is resonating, not actual pieces of thin rope.

To test out the theory we can use some symmetry considerations to predict a resonance that no one has seen before. This leads to a frequency and wavelength prediction. That gives an equivalent mass and velocity (equation 3). When we let high speed particles race down linear accelerators and smash into atomic targets, tiny particles are produced. They don’t last long, and when they vanish, they give off a burst of light. The photon will have an energy equivalent to that particle’s mass (equation 1). This energy has an associated wavelength (equation 2) which translates to a colour. So, when we fire up our linear accelerators, synchrotrons, and cyclotrons (“atom smashers”) we can look for a burst of light of that colour. If we see it, we can say that we have discovered a new particle and announce its mass.

Alternatively, if we have a tiny understanding of string theory, we can say we have witnessed a new resonance of one of the strings of which the universe is made.

Intuitive Grasp of Concepts

As a teacher of physics, primarily (also computers, general science, and music), I strive to instill an intuitive grasp of concepts in my students. Of course, this requires that I have an intuitive grasp of the concepts myself. I was delighted when a light bulb turned on in my head early one Sunday morning while reading Does God Play Dice? The Mathematics of Chaos by Ian Stewart. I sat upright in bed and exclaimed "I get it!". (I had read Gleick's book Chaos but and learned much, but did not get that Eureka moment.) My wife, whom I had shaken awake to report the good news, prevented me from immediately calling a friend to see if he wanted to discuss fractals and chaos with me!

I feel I have an intuitive grasp of fractals and chaos, fuzzy logic, relativity, and quantum physics, but string theory eludes me. I am looking for a book that will help the light bulb turn on once again. I am presently reading, and highly recommend, Galileo's Finger by P.W. Atkins. The book is written beautifully, and takes me past my frontiers of knowledge. Perhaps when I reread the chapter on strings I will shout "Eureka".