Suppose $(x(\tau), t(\tau))$ is a timelike geodesic of the metric

$$d s^{2}=\frac{d x^{2}}{1+x^{2}}-\left(1+x^{2}\right) d t^{2}$$

where $\tau$ is proper time along the world line. Show that $d t / d \tau=E /\left(1+x^{2}\right)$, where $E>1$ is a constant whose physical significance should be stated. Setting $a^{2}=E^{2}-1$, show that

$$d \tau=\frac{d x}{\sqrt{a^{2}-x^{2}}}, \quad d t=\frac{E d x}{\left(1+x^{2}\right) \sqrt{a^{2}-x^{2}}} .$$

Deduce that $x$ is a periodic function of proper time $\tau$ with period $2 \pi$. Sketch $d x / d \tau$ as a function of $x$ and superpose on this a sketch of $d x / d t$ as a function of $x$. Given the identity

$$\int_{-a}^{a} \frac{E d x}{\left(1+x^{2}\right) \sqrt{a^{2}-x^{2}}}=\pi$$

deduce that $x$ is also a periodic function of $t$ with period $2 \pi$.

Next consider the family of metrics

$$d s^{2}=\frac{[1+f(x)]^{2} d x^{2}}{1+x^{2}}-\left(1+x^{2}\right) d t^{2},$$

where $f$ is an odd function of $x, f(-x)=-f(x)$, and $|f(x)|<1$ for all $x$. Derive expressions analogous to $(*)$ above. Deduce that $x$ is a periodic function of $\tau$ and also that $x$ is a periodic function of $t$. What are the periods?