Relativistic Corrections for Time and Frequency Transfer in Optical Fibres

We derive relativistic corrections for one-way and two-way time and frequency transfer over optical fibres neglecting no terms that exceed 1 ps in time and ${10}^{-18}$ in fractional frequency, and estimate their magnitude in typical fibre links. We also provide estimates of the uncertainties in the evaluation of the relativistic corrections due to imperfect knowledge of parameters like the coordinates of the fibre and stations, Earth rotation, or thermal effects of the fibre index and length. The links between Teddington(UK) and Paris(F) as well as Braunschweig(D) and Paris(F), that are currently under construction, are studied as specific examples.