Odd radial perturbations preserve timelike-geodesic periods ($T_{\tau }=T_t=2\pi$)

For $d{s}^2=(1+f(x){)}^{2}d{x}^2/(1+x^2)-(1+x^2)d{t}^2$, with smooth odd $f$ and $|f|<1$, the conserved energy gives $dt/d\tau =E/(1+x^2)$ and $(dx/d\tau {)}^2=(E^2-1-x^2)/(1+f{)}^2$. Both half-cycle integrals over $[-\sqrt{E^2-1},\sqrt{E^2-1}]$ contain an even weight times $1+f$. The odd contribution vanishes, leaving the unperturbed half-period $\pi$ in each clock. The waveforms need not be sinusoidal, but both periods are unchanged for $E>1$.