Computation of the relativistic rate shift of a frequency standard

Comparison of the frequencies of distant primary standards calls for the computation of their relativistic rate shift (gravitational+second-order Doppler shift) with an uncertainty lower than that implied by the accuracy of the standards. This computation is also required when using primary frequency standards to establish an accurate coordinate time, as in the case of the International Atomic Time TAI. The problem is addressed at several levels of uncertainty, valid for present and future primary standards. It is shown that, for clocks on Earth, the computation is easily performed with an uncertainty of a few parts in 10¹⁶, but is ultimately limited to a few parts in 10¹⁷, due to geophysical effects which presently cannot be modeled with smaller uncertainty. For clocks on board satellites, these limitations are less significant and an uncertainty of 10^-18 is attainable, while on board moving vehicles close to the Earth an uncertainty of 10^-16 could be reached