= Odd radial perturbations preserve timelike-geodesic periods
{title2=$T_\tau=T_t=2\pi$}
For $ds^2=(1+f(x))^2dx^2/(1+x^2)-(1+x^2)dt^2$, 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 $E>1$.
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