A spherically symmetric static spacetime has metric

$$d s^{2}=-\left(1+r^{2} / b^{2}\right) d t^{2}+\frac{d r^{2}}{1+r^{2} / b^{2}}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$$

where $-\infty<t<\infty, r \geqslant 0, b$ is a positive constant, and units such that $c=1$ are used.

(a) Explain why a time-like geodesic may be assumed, without loss of generality, to lie in the equatorial plane $\theta=\pi / 2$. For such a geodesic, show that the quantities

$$E=\left(1+r^{2} / b^{2}\right) \dot{t} \quad \text { and } \quad h=r^{2} \dot{\phi}$$

are constants of the motion, where a dot denotes differentiation with respect to proper time, $\tau$. Hence find a first-order differential equation for $r(\tau)$.

(b) Consider a massive particle fired from the origin, $r=0$. Show that the particle will return to the origin and find the proper time taken.

(c) Show that circular orbits $r=a$ are possible for any $a>0$ and determine whether such orbits are stable. Show that on any such orbit a clock measures coordinate time.