Testing Einstein's time dilation under acceleration using M\"ossbauer spectroscopy

TL;DR

This paper analyzes the limitations of previous M"ossbauer spectroscopy experiments testing Einstein's time dilation under acceleration, providing a refined theoretical framework and conditions for accurate future measurements to verify acceleration effects.

Contribution

It develops an explicit model for M"ossbauer absorption lines under rotation, highlighting experimental challenges and proposing conditions to test acceleration-induced time dilation effects.

Findings

Previous experiments misinterpreted line broadening effects.

Proper analysis requires accounting for geometric Doppler shifts.

Conditions for successful acceleration-based time dilation measurement are established.

Abstract

The Einstein time dilation formula was tested in several experiments. Many trials have been made to measure the transverse second order Doppler shift by M\"{o}ssbauer spectroscopy using a rotating absorber, to test the validity of this formula. Such experiments are also able to test if the time dilation depends only on the velocity of the absorber, as assumed by Einstein's clock hypothesis, or the present centripetal acceleration contributes to the time dilation. We show here that the fact that the experiment requires $\gamma$-ray emission and detection slits of finite size, the absorption line is broadened; by geometric longitudinal first order Doppler shifts immensely. Moreover, the absorption line is non-Lorenzian. We obtain an explicit expression for the absorption line for any angular velocity of the absorber. The analysis of the experimental results, in all previous experiments which did not observe the full absorption line itself, were wrong and the conclusions doubtful. The only proper experiment was done by K\"{u}ndig (Phys. Rev. 129 (1963) 2371), who observed the broadening, but associated it to random vibrations of the absorber. We establish necessary conditions for the successful measurement of a transverse second order Doppler shift by M\"{o}ssbauer spectroscopy. We indicate how the results of such an experiment can be used to verify the existence of a Doppler shift due to acceleration and to test the validity of Einstein's clock hypothesis.