  We show that motion and gravity affect the precision of quantum clocks. We
consider a localised quantum field as a fundamental model of a quantum clock
moving in spacetime and show that its state is modified due to changes in
acceleration. By computing the quantum Fisher information we determine how
relativistic motion modifies the ultimate bound in the precision of the
measurement of time. While in the absence of motion the squeezed vacuum is the
ideal state for time estimation, we find that it is highly sensitive to the
motion-induced degradation of the quantum Fisher information. We show that
coherent states are generally more resilient to this degradation and that in
the case of very low initial number of photons, the optimal precision can be
even increased by motion. These results can be tested with current technology
by using superconducting resonators with tunable boundary conditions.
