Consider the spacetime metric

$$d s^{2}=-f(r) d t^{2}+\frac{1}{f(r)} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right), \quad \text { with } \quad f(r)=1-\frac{2 m}{r}-H^{2} r^{2}$$

where $H>0$ and $m>0$ are constants.

(a) Write down the Lagrangian for geodesics in this spacetime, determine three independent constants of motion and show that geodesics obey the equation

$$\dot{r}^{2}+V(r)=E^{2}$$

where $E$ is constant, the overdot denotes differentiation with respect to an affine parameter and $V(r)$ is a potential function to be determined.

(b) Sketch the potential $V(r)$ for the case of null geodesics, find any circular null geodesics of this spacetime, and determine whether they are stable or unstable.

(c) Show that $f(r)$ has two positive roots $r_{-}$and $r_{+}$if $m H<1 / \sqrt{27}$ and that these satisfy the relation $r_{-}<1 /(\sqrt{3} H)<r_{+}$.

(d) Describe in one sentence the physical significance of those points where $f(r)=0$.