  Phase compensated optical fiber links enable high accuracy atomic clocks
separated by thousands of kilometers to be compared with unprecedented
statistical resolution. By searching for a daily variation of the frequency
difference between four strontium optical lattice clocks in different locations
throughout Europe connected by such links, we improve upon previous tests of
time dilation predicted by special relativity. We obtain a constraint on the
Robertson--Mansouri--Sexl parameter $|\alpha|\lesssim 1.1 \times10^{-8}$
quantifying a violation of time dilation, thus improving by a factor of around
two the best known constraint obtained with Ives--Stilwell type experiments,
and by two orders of magnitude the best constraint obtained by comparing atomic
clocks. This work is the first of a new generation of tests of fundamental
physics using optical clocks and fiber links. As clocks improve, and as fiber
links are routinely operated, we expect that the tests initiated in this paper
will improve by orders of magnitude in the near future.
