I'll start with your last question. " Could one also assign a length (in meters) to time in this way? " Yes! In the (Minkowki) spacetime infinitesimal interval,

$$\mbox{d}s^2=\left(ic_0\mbox{d}t\right)^2+\left(\mbox{d}x\right)^2+\left(\mbox{d}y\right)^2+\left(\mbox{d}z\right)^2$$

$\left(ic_0\mbox{d}t\right)$ has units of ?? Meters! Exactly! You could think, (very "hand-wavily") that time is like "imaginary space", except for the $c_0$ factor, but if you use natural units $c_0=1$, then, even that factor drops out!.

Now, for the second (really the first) part. How does one measure these compactified extra dimensions? Simply (well, kind of, in principle), *by measuring the inverse square law at sub-milimeter scales.* It has been done in an arXiv paper by C. Hoyle, U. Schmidt, B. R. Heckel, E. G. Adelberger, J. H. Gundlach, D. J. Kapner, H. E. Swanson Looks like they got a positive result (for the inverse square law) :(

This post imported from StackExchange Physics at 2014-03-17 04:00 (UCT), posted by SE-user Dimensio1n0