🏷️ ป้ายกํากับ: orbits

We have seen that three spatial dimensions are not just the stage on which physics happens.

They help determine what kinds of physics can happen at all. Change the number of large spatial dimensions, and familiar structures begin to strain. Gravity and electric forces behave differently. Stable orbits become harder to arrange. Atoms and chemistry may lose the reliable bound states they need. Even topology changes: knots, entanglements, and certain kinds of persistent structure belong very naturally to three dimensions.

So the question becomes sharper.

Does this explain why space is three-dimensional?

Or does it only explain why beings like us could only find themselves in a universe where something like three-dimensional space was already in place?

That is a different kind of question. It is not mainly about calculating orbits or atoms anymore. It is about what counts as an explanation.

Two different kinds of answer

There are at least two ways an explanation of three-dimensional space might work.

One would be a physical necessity explanation.

That kind of explanation would say: the deeper laws of reality require exactly three large spatial dimensions. Space is 3D because, at a more fundamental level, it could not have been otherwise. Some hidden principle, symmetry, consistency condition, or deeper theory would force this structure into place.

In schematic form:

deeper law → three large spatial dimensions

Why should adding or removing a spatial dimension do anything so dramatic to gravity?

At first glance, it seems like it should only change the picture. Space would have a different geometry, certainly, but why should that alter the basic possibility of a world with stars, planets, and enduring motion?

The hidden hinge is that gravity does not merely happen in space. It spreads through space. Change the dimensionality, and you change the way influence thins out with distance. That means you change the force law itself. And once the force law changes, the architecture of possible motion changes with it.

This is why dimensionality is not just a background feature. It reaches directly into the question of whether bound orbital structure can persist.

Geometry comes first

Imagine a source whose influence spreads outward equally in all directions. In ordinary three-dimensional space, that influence is distributed over the surface of a sphere. The area of that sphere grows like $r^2$, so the same total influence is diluted over a larger and larger surface as distance increases.

That is why the force falls as

$$ F(r) \propto \frac{1}{r^2}. $$

This is not a numerical accident. It is the three-dimensional expression of geometric spreading.

In $d$ spatial dimensions, the enclosing surface around a point source grows instead like $r^{d-1}$. So the corresponding central force typically scales as

$$ F(r) \propto \frac{1}{r^{d-1}}. $$