# Is it possible to measure the number of spacetime dimensions?

My question is pretty straightforward to be stated but I don't know whether the answer is as easily reported. It is remarkable and very interesting that Physics work in (almost) any number of spacetime dimensions, but is it possible to actually measure the dimensionality of spacetime, even indirectly? I have found elsewhere in this site an argument regarding measuring the power law of the distance-dependence of forces, for example, the magnitude of the electrostatic force exerted by a charged point particle with charge $q$ would be of the form, \begin{equation} F_{Coulomb} \sim\frac{q}{r^{d-1}} \end{equation} with $r$ the distance from the particle's position and $d$ the number of spatial dimensions.

I can accept this as long as the procedure of measuring distances is well defined. Fundamentally, an observer would hold a meter stick and measure distances along the spatial axes. But how can the observer be certain of the number of axes he is capable of laying the meter stick along. How can he be sure there aren't any more axes that he simply cannot realize?

• Power law is not a generic case that any force would satisfy. Some forces can be in inverse-square law, some can't. We don't deduce the number of dimensions from the power law. For example Yukawa potential is in the form of $\frac{e^{\lambda r}}{r}$. – Oktay Doğangün May 4 '18 at 23:49
• That you can tie your shoes with a string and have them remain tied means the universe has three spatial dimensions. – David Hammen May 5 '18 at 0:03
• careful measurement of the gravitational force between two small and closely-spaced masses can be used to reveal "leakage" of it into any extra dimensions of size ~ the spacing between the masses. No evidence so far of this effect; this argues in favor of 3 big dimensions and if extra dimensions exist, they must be smaller that that. – niels nielsen May 5 '18 at 0:08

## 1 Answer

I think your hypothetical observer can be sure of the dimensionality of the space he occupies by figuring out the minimum number of measurements required to specify the location of an object relative to some fixed reference frame. I get 3.

This requires some assumptions about space being 'well-behaved.' It seems to be so.

• It seems to be obvious statement ( and even somehow funny) but rephrasing op question: which process dependent of dimensionality may produce macroscopic or microscopic ( quantum) effects allowing us to distinguish between various dimensionalities of spacetime, at different scales? – kakaz Aug 12 at 16:23
• Good question. I'm not up to speed on the fine details. Probably a good starting point here. – Rodney Dunning Aug 14 at 19:09
• Interesting indeed, but I was asking about physics, I mean science describing reality in connection to experimental knowledge. – kakaz Aug 18 at 7:19