Are String Theory's Extra Dimensions Real?

Why do positrons and electrons have opposite charges? One explanation involves an extra dimension: the positron moves in one direction down that dimension and the electron moves in the opposite direction. The result is opposite charge. But you might ask, what prevents the electron and the positron from changing direction? Do positrons become electrons and vice versa? If they do, that would be a sign that the extra dimension exists. If not ... oh well.

If extra dimensions do exist, here is a scenario that should happen:

You have three hydrogen atoms. One has an infinite mass; the second one has the mass you expect; the third one has a mass of five trillion kilo-tons. How is that possible?

If you look carefully at the dimensions we are familiar with, you will notice a pattern. There are an infinite number of points along a line; an infinite number of lines along a plane; an infinite number of planes inside a cube. In short, a higher dimension can accommodate an infinite number of lower-dimensional objects. If this pattern is consistent, then a 4D space could accommodate an infinite number of 3D cubes. We can check this:

A cube is measured in cubic meters or m^3. A 4D hyper-cube is measured with m^4. How much 4D room does an m^3 cube take up? x * y * z * 0 = 0 m^4.

As you can see, the cube has magnitude along x, y, and z, but zero magnitude along the 4D axis. As a result, it takes up zero m^4 space. And, it can be any size! So an m^4 space can hold an infinite amount of 3D space or matter, or energy.

Now imagine there are extra dimensions in a tiny region of, say, a hydrogen atom. Particles are free to enter and exit this region. Once inside, the particles encounter an infinite 3D universe. So potentially, they could gather there and cause the hydrogen atom to have significantly increased mass and energy.

Ironically, the m^4 space could be zero and still hold and infinite amount of of 3D space, matter and energy, since anything 3D takes up zero m^4 space. So if extra dimensions exist, it does not matter how short or curled up they are. They will accommodate infinite 3D.

This presents a big problem for physics, since you could not count on two identical items having the same mass or energy. The fact that we can count on two identical items to be identical, is a sign that extra dimensions are highly improbable. More dimensions would destroy energy conservation, since up to an infinite amount of energy could leave or enter our universe at any time.

Are there other ways we can test whether extra dimensions exist?

Certainly! By definition, space dimensions are vectors that are orthogonal to each other; they are right angles to each other. So far we humans have discovered three: i, j, and k.

Suppose we think there might be a fourth dimension. Let’s call it “a.” We can test this dimension with cross products. We know that iXj = k; jXk = i; and kXi = j.

So the question becomes what is aXi? The answer has to be either j or k. If it’s j, then aXi = kXi, so “a” could be parallel to k. If it is not orthogonal to k, it is not really a fourth space dimension. If the answer is k, then “a” could be parallel to j, and our conclusion about “a” remains. aXj and aXk can yield more parallel vectors rather than orthogonal ones. It appears the cross product test falsifies the notion of a fourth spacial dimension or additional dimensions.

But suppose we change the rules. We decide, for example, that aXi = j or k. This is possible if the four dimensions are perpendicular to each other. However, physics would be less certain. We don’t know whether we will get j or k. Physics becomes even more uncertain when you add additional dimensions. If you have nine space dimensions, then aXi could equal any of the other seven vectors. Adding more dimensions to our physics does solve certain problems, but at the expense of creating new absurdities and uncertainties.

Some theoretical physicists want to discover extra dimensions so badly they can taste them--but only in their dreams. To date, there is no empirical evidence of extra spacial dimensions.