2D beetle measuring distances

1. Apr 10, 2014

ChrisVer

I started attending to GR lectures, and there we referred to an intelingent beetle living on a 2D flat space and measuring distances/lengths on it. In case then we would heat the space (so that the metallic rod the beetle uses to measure those distance, gets shrunk or expanded depending on the temperature), we have that the beetle would measure the length according to:

$L[x(\sigma)]= \int_{\sigma_{a}}^{\sigma^{b}} k(x(\sigma)) d\sigma \sqrt{ <x'(\sigma),x'(\sigma)>}$
where $k(x)$ is a function which determines the changes of the metalic rod depending on the position of the beetle... $\sigma$ is the parametrization used for the coordinates on the 2D space, and $<.,.>$ represents the inner product $<x,y>= n_{\mu\nu}x^{\mu}y^{\nu}$ and $n_{\mu\nu}=diag(1,1)$...
I think all given and clear up to now....
Choosing then for example the $k$ function to be:
$k(x)=k(x^{1},x^{2})= \frac{2}{r_{0}} \frac{1}{1+\frac{(x^{1})^{2}+(x^{2})^{2}}{r_{0}^{2}}}$
We get some interesting results... However I'm trying to understand the meaning of this function in general... Would I use polar coordinates, it would be of the form:

$k(r)= \frac{2}{r_{0}} \frac{1}{1+\frac{r^{2}}{r_{0}^{2}}}$
so for $r=0$ it would have the value $k(0)=\frac{2}{r_{0}}$ and for $r→∞$ it would go to $k(r)→0$...
Also for $r=r_{0}$ we get $k(r_{0})=\frac{1}{r_{0}}$...

I don't really like the fact that k would have to have dimensions....
Q1: by the above, can I interpret the $r_{0}$ parameter as physically the distance at which k drops by half? If not, what would be a physical interpretation for that parameter?

Finally continuing we invented a new vector
$\vec{x}_{0}=\frac{1}{1+\frac{(x^{1})^{2}+(x^{2})^{2}}{r_{0}^{2}}} [\frac{2}{r_{0}} x^{1}e_{1} + \frac{2}{r_{0}} x^{2}e_{2} + (1-\frac{(x^{1})^{2}+(x^{2})^{2}}{r_{0}^{2}})e_{3}] \in R^{3}$
$e_{i}$ an orthonormal basis...
to see that $|x_{0}|^{2}=1$ or i.e. that $x_{0} \in S^{2} \subset R^{3}$

Q2: Is there any particular reason (maybe from mathematics) to define immediately such a $x_{0}$ vector? I mean it looks quite a complicated expression -I wouldn't think of defining such a strange thing if I dealt with the problem for the 1st time...

Q3: Would the beetle indeed have to imagine/determine/realize by measuring distances, that it needs one additional dimension to define such a vector by itself? In the real life, would a human being have to define a vector $\vec{A} \in M^{4} \subset R^{5}$ , $M$ is a general manifold (since the sphere $S^{2}$ appeared because of the particular choice of k) to realize how to measure distances on the Minkowski space?

Last edited: Apr 10, 2014