Some Recent Experimental Tests of the "Clock Paradox"

Recent experiments by Pound and Rebka on the temperature dependence of the M\"ossbauer effect in ${\mathrm{Fe}}^{57}$, and by Hay, Schiffer, Cranshaw, and Egelstaff using an ${\mathrm{Fe}}^{57}$ absorber on a rotating drum are shown to provide the first direct experimental verification of the time-keeping properties of accelerated clocks such as occur in the classic "clock paradox" of relativity. In the experiment by Pound and Rebka, the thermal vibrations of the lattice impart rms velocities of about ${10}^{\ensuremath{-}6}c$, and nearly continuous, randomly-oriented accelerations of the order of ${10}^{16}g$ to both the source and the absorber nuclei. In the experiment by Hay et al. the acceleration of the absorber was $6\ifmmode\times\else\texttimes\fi{}{10}^{4}g$. The photon provides continuous communication of time data between the two nuclei for the duration of the "journey" (the emission time of the quantum). In each case the observed fractional frequency shift $\frac{\ensuremath{\Delta}f}{{f}_{0}}$ which occurs between the source and the absorber is found to be $\ensuremath{-}\frac{{{v}_{s}}^{2}}{2{c}^{2}}+\frac{{{v}_{a}}^{2}}{2{c}^{2}}$, where ${v}_{s}$ and ${v}_{a}$ are the rms velocities of the source and the absorber nuclei, respectively. These results are in quantitative agreement with the generally accepted calculations for the "clock paradox", in which two clocks pursue independent paths (at least one of which involves accelerations) in a common inertial frame, but are compared at two or more points where they coincide in space and time. The temperature-dependent experiments also demonstrate that accelerations of the order of ${10}^{16}g$, arising from lattice vibrations, produce no intrinsic frequency shift in ${\mathrm{Fe}}^{57}$ nuclei to an accuracy exceeding 1 part in ${10}^{13}$.