Hi Alex,
That's the whole idea of a manifold. Think about 2D manifolds in 3D space. We live in 3D space and 2D manifolds are very familiar to us. If we think about the relationships between 2D manifolds and 3D space, some of the relationships will apply to 3D manifolds in 4D space. Or at least we might suspect that they would.
So what 2D manifolds are we familiar with in 3D space? Well, 2D manifolds are surfaces. We see them everywhere. In fact that is just about all we see. When we look at an egg, we see only (about half) its surface. The surface of an egg is a 2D manifold.
The surface of an egg is different from the egg shell. The shell has some thickness, but the surface has zero thickness. So how can we see something of zero thickness? Well, we see photons that are emitted from the atoms on the outside of the egg shell. When you get down to that scale, the surface of the egg is no longer smooth and continuous as we normally think of surfaces being. But, at normal scales, we think of the egg shell as being sort of smooth and we know what we mean by the surface of the egg. We can paint it.
Now, the surface of the egg can admit a coordinate system such that any point on the egg's surface can be identified by two numbers, hence it is a 2D manifold, or space. As I said, we can paint it and we can calculate the surface area to figure out how much paint we would need. Moreover, the density of paint drops with the inverse of the distance, as you would expect in 2D space (That is, for tiny regions on big eggs. Let's not get hung up on the non-Euclidean effects).
If you locate a point on the surface of the egg, you can move with two degrees of freedom to another point on the surface and stay on the surface of the egg for the entire trip between the two points. Qualitatively speaking, you encounter nothing but the surface of the egg shell the entire way.
But now, consider a direction that is orthogonal to the surface of the egg at a point. If you move, from this point on the surface of the egg, in this direction, you will no longer be on the surface of the egg. There are two cases. In one case, you will leave the surface and enter the air of the room (or maybe the thumb of the hand holding the egg). In the other case, you will enter the egg itself. For a short distance, you will be inside the egg shell. Then, after piercing a membrane, you will be in the slimy egg white. Still further and you will be in the yolk. Further yet, more egg white, more shell and then you will break out the other side and enter the air of the room (or maybe the index finger).
Finally, to answer your question about qualitative difference: The shell, which you can paint, and which exhibits an inverse R law for paint density, is qualitatively different (=quite different) from air, egg whites, egg yolks, and the insides of thumbs, none of which you can paint.
In the same way, whatever might be out there in 4D space beyond our 3D manifold might be quite different from what we have here. No doubt there would not be gravitons, electrons, photons, bosons, or anything else that we might find familiar, otherwise, as you point out, the familiar inverse square laws for them would not hold. Who knows? It might be egg whites all the way down.
Warm regards,
Paul |