  Light-pulse atom interferometers based on single-photon transitions are a
promising tool for gravitational-wave detection in the mid-frequency band and
the search for ultralight dark-matter fields. Here we present a novel
measurement scheme that enables their use as freely falling clocks directly
measuring relativistic time-dilation effects. The proposal is particularly
timely because it can be implemented with no additional requirements in
Fermilab's MAGIS-100 experiment or even in the 10-m prototypes that are
expected to start operating very soon. This will allow the unprecedented
measurement of gravitational time dilation in a local experiment with freely
falling atoms, which is out of reach even for the best atomic-fountain clocks
based on microwave transitions. The results are supported by a comprehensive
treatment of relativistic effects in this kind of interferometers as well as a
detailed analysis of the main systematic effects. Furthermore, the theoretical
methods developed here constitute a valuable tool for modelling light-pulse
atom interferometers based on single-photon transitions in general.
