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It's common knowledge that the universe has three spatial dimensions, and one temporal. But how can we be so sure? Physics tells us that vector forces act independently along different dimensions, so without a source of force along the fourth dimension, or with a component in the fourth dimension, nothing would ever pass out of our little slit of 3D space in a possibly 4D world. Moreover, some materials exhibit different levels of friction, or any other measure of directional "resistance," (using the term as an abstraction for every barrier to motion, physical, fluid and so on) differently along different directions. We could just be stuck on our little 3D slice of 4D universe.

How can we possibly know the world is indeed three-dimensional, especially considering models of two-dimensional motion work so well in our three-dimensional world when motion along the third dimension is heavily restricted?

Note: The question How can we be so sure the universe has 3 dimensions is somewhat similar in that it asks if spatial dimensions exist at all, but is not a duplicate. I did my research.

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TheEnvironmentalist
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2 Answers2

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I'm not sure what kind of argument can convince you, since you seem to reject the everyday experience that the world has three dimensions. Nevertheless, I'll try to give an alternative argument of why this is the case (at least for a macroscopic scale). The argument is related to how things scale with distance.

Observe that the area of a $d$-dimensional sphere of radius $r$ is proportional to $r^d$. Imagine some kind of flow emanating from some point in $(d+1)$-dimensional space. The total flow that traverses any $d$-sphere centered at that point will stay constant with radius, so the flow per unit area (meaning $d$-dimensional volume) will scale as $1/r^d$.

In our world, many physical phenomena involve scalings like $1/r^2$. This kind of laws appear in the cases where some flow should propagate freely along every possible dimension. Thus, they are all explained nicely with the same underlying principle: a flow per unit area when the total number of dimensions is three.

Some examples of inverse square laws are:

  • Newton's law of gravitation
  • Coulomb's law
  • Sound intensity fall-off from a point source
  • Light illuminance fall-off from a point source

Now, it could be the case that there are more dimensions but everything is restricted to three spatial dimensions. But what does this mean? If it doesn't affect anything, if it can't be measured in any way, we can just forget about it. Just use Occam's razor: why would we imagine extra things in the universe that don't have any possible effect in anything?

Another possibility is that at least some things propagate through the extra dimensions but there's something preventing us from noticing this fact. The most common idea in this kind of models is that the extra dimensions are small, so we can't see them yet. This happens, for example, in string theory or in some models that have become very popular in the last decade, such as "large" ($1\,\mathrm{mm}$) extra dimensions or the Randall-Sundrum model.

In one of them, the weakness of gravity is explained by saying that it is the only force that propagates through the extra dimension, thus losing strength with respect to the others. In this model, we expect to be able to measure deviations of the inverse square law for gravity at small distances. In general, any model with small extra dimensions predicts new effects at small distances.

This possibility isn't ruled out, so we aren't sure that the world has only three spatial dimensions. It could be the case that there are more, but very tiny.

coconut
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We can't know that for certain. There are many models in high energy physics and cosmology which assume the existence of extra dimensions, e.g. string theory (where the extra dimensions are usually compact, i.e. "very small" and not visible at low energy) or the Randall–Sundrum model (which explicity allows a non-compact 5th dimension).

Nevertheless, what all these models have in common is that these extra dimensions play no role at your typical everyday life energy scale (it would give corrections to Newtonian mechanics which are neglectable). Why is this so ? Newtonian mechanics agrees with every measurement we do at low energies and so every theory which aspires to describe a fundamental model of nature should of course reproduce the results of Newtonian mechanics at the energy scales for which this gives a good description of nature.

As Sanya mentioned it, an important principle for a physical theory is that we can measure its predicitions, i.e. are able to falsificate the theory. You could of course always predict extra dimensions which are not reachable for us but influence our physical reality in some sense, but then you need to give a model which allows us the falsification of said model via testable predictions (as for example string theory allows, but only at extreme energy scales).

In short, scientific principles "prohibit" extra dimensions, see for example Russell's teapot.

Moe
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