Where a circle stops getting smaller

A spatial dimension curled into a small circle would not be seen directly. What gives it away is a set of extra particles. Something moving through the large dimensions can also be moving around the small one, and to someone who cannot see the small circle that extra motion looks like extra mass. A hidden circleA spatial direction curled up small enough that moving along it is not seen as motion, only as extra mass. shows up as a tower of new particles, and the usual way to measure its size is from how far apart in mass they are. The spacing of that tower comes from one requirement. A wave going around the circle has to match itself after one lap, so a whole number of wavelengths has to fit. With circumference $2\pi R$ the wavelength is $2\pi R/n$ for $n = 1, 2, 3$ and so on. Units here have $\hbar = c = 1$, so a momentum is an energy and a length is one over an energy, and a momentum is $2\pi$ divided by a wavelength. The momentum around the circle is then $n/R$. A particle standing still in the large dimensions while circling the small one is seen as a particle of mass $n/R$, and the tower of such particles is named after Kaluza and KleinThe evenly spaced set of particles produced by momentum around a compact dimension, at masses $n/R$.. For an object with no size of its own that is the whole story, and it is the usual reason a curled-up dimension can be there without anyone noticing. Make $R$ small and every mass $n/R$ becomes large, until the whole tower is past the reach of any machine. A string is not a point. It has length, and its energy is proportional to that length: a tension $T$, which is energy per unit length. A closed loop threaded through the circle cannot be pulled off it without being cut. A loop wrappedA closed string wrapped around a compact dimension, which cannot unwrap and whose energy is proportional to the size of the circle. $w$ times has length $2\pi R w$ and therefore energy $2\pi R w T$, and a wrapped loop sitting still in the large dimensions is seen as a particle of that mass. Writing the tension as $T = 1/(2\pi\ell_s^2)$ defines a length $\ell_s$, called the string lengthThe length $\ell_s$ built out of the string tension, and the scale at which a string stops behaving like a point.. A loop wrapped $w$ times then has mass $wR/\ell_s^2$.