# A 2d Geodesic equation

1. Jan 10, 2010

### Altabeh

1. The problem statement, all variables and given/known data

Consider the 2-dim metric $${{\it ds}}^{2}=-{\frac {{a}^{2}{{\it dr}}^{2}}{ \left( {r}^{2}-{a}^{2}\right) ^{2}}}+{\frac {{r}^{2}{d\theta }^{2}}{{r}^{2}-{a}^{2}}}$$, where r > a. What is its signature? Show that its geodesics satisfy

$${\frac {{a}^{2}{{\it dr}}^{2}}{{d\theta }^{2}}}+{a}^{2}{r}^{2}={k}^{2}{r}^{4}$$
where k is a constant. For which value(s) of k are the geodesics null?

3. The attempt at a solution

1- The signature is clearly (-,+).

2- I can show straightly from the metric itself that if $$\tau$$ is the proper time, dividing each side of metric by $$d\tau^2$$ and using $$(dr/d\tau)/(d\theta /d\tau)=dr/d\theta$$ yields the left-hand side of the desired geodesic equation and the other side would be of the form $$k^2r^4$$ with $$k=\mbox {{\pm}} \left( \sqrt {1-{\frac {{{\it ds}}^{2} \left({r}^{2}-{a}^{2} \right) ^{2}}{{r}^{4}{d\theta }^{2}}}} \right)$$. This can be used to show that if k=+1 or -1 then the geodesics are null. But I don't know anything about how k is supposed to be constant with those irritating r's. What is probably wrong?

Thanks
AB