General relativistic time dilation and spacetime uncertainty in quantum clocks

The general theory of relativity associates a proper time with each object via its spacetime trajectory. In quantum theory on the other hand, such trajectories are forbidden. Here we demonstrate that, in the weak-field and low-velocity limit, all “good” quantum clocks experience the time dilation dictated by general relativity for the most classical states of motion. For nonclassical states of motion, on the other hand, we find that quantum interference effects give rise to a significant discrepancy between the proper time and the time measured by the clock. Moreover, we show how our ignorance of the clock’s state of motion leads to a larger uncertainty in time measurements with the clock, a consequence of entanglement between the clock time and its center-of-mass degrees of freedom.