On the geodesics of the metric $ds^2=-\rho^2d\alpha^2+d\rho^2$ and the constant $l=\rho^2\frac{d\alpha}{d\tau}$ In my assignment I have to deal with the 2D spacetime metric $$ds^2=-\rho^2d\alpha^2+d\rho^2.$$ During this assignment we have shown that the constant $$l=\rho^2\frac{d\alpha}{d\tau}$$ is a constant along the geodesics in that space. After this we found an expression for $\rho$ in terms of $\alpha$, for which $\rho(\alpha=0)=l$, namely: $$\rho(\alpha)=l\frac{2e^\alpha}{e^{2\alpha}+1}.$$
We plotted this function for various values of l, which gave:

Finally, we showed that in the close neighbourhood of the Schwarschild radius, i.e. $0<r-r_s\ll 1$, the metric $ds^2=-\rho^2 d\alpha^2+d\rho^2$ corresponds to the metric $ds^2=-(1-\frac{r_s}{r})dt^2+(1-\frac{r_2}{r})^{-1}dr^2$.
Now what we need to do is, use the result that the metric is approximately the schwarzschild metric at the schwarzschild radius to interpret the plots we found in the figure shown above and give a possible interpretation for l. And I really have no idea how to interpret these figures or what l might be.
Any ideas are welcome!
 A: The metric written in the question,
$$ds^2 = -\rho^2 d\alpha^2 + d\rho^2$$
is simply the metric for flat, Minkowski space $\mathbb R^{1,1}$. If we make the coordinate transformation,
$$\alpha = \mathrm{arctanh} \frac{t}{x}, \quad \rho = \sqrt{x^2-t^2}$$
then we obtain the familiar metric,
$$ds^2 = -dt^2 + dx^2$$
in $(-1, 1)$ signature. All of the Christoffel symbols vanish, leaving that all components of the geodesic are linear functions of $\tau$, the proper time. Returning to your constant along geodesics, 
$$l = \rho^2 \frac{d\alpha}{d\tau}$$
notice $\rho^2 = x^2 - t^2$ and applying the chain rule to our expression for $\alpha(x(\tau),t(\tau))$ yields,
$$l = x(\tau) \frac{dt}{d\tau} - t(\tau)\frac{dx}{d\tau}.$$
Taking $t(\tau) = a\tau + b$ and $x(\tau) = c\tau + d$, we have,
$$l = ad-bc$$
which is entirely $\tau$ independent and thus is constant along any geodesic of Minkowski space. Note a mathematical interpretation of $l$: it is the Wronskian of $x(\tau)$ and $t(\tau)$, which determines whether they are linearly independent or not.
A: This is not a direct answer to your question, but more of a hint for further work. Your metric is that of "Rindler space," which is flat spacetime in disguise. The coordinates are  the natural ones for a uniformly accelerated observer. See:  https://en.wikipedia.org/wiki/Rindler_coordinates.
