Given the possible repetitive nature of fast radio bursts (FRBs), their cosmological origin, and their high occurrence, detection of strongly lensed sources due to intervening galaxy lenses is possible with forthcoming radio surveys. We show that if multiple images of a repeating source are resolved with VLBI, using a method independent of lens modeling, accurate timing could reveal non-uniform motion, either physical or apparent, of the emission spot. This can probe the physical nature of FRBs and their surrounding environments, constraining scenarios including orbital motion around a stellar companion if FRBs require a compact star in a special system, and jet-medium interactions for which the location of the emission spot may randomly vary. The high timing precision possible for FRBs (~ms) compared with the typical time delays between images in galaxy lensing (10 days) enables the measurement of tiny fractional changes in the delays () and hence the detection of time-delay variations induced by relative motions between the source, the lens, and the Earth. We show that uniform cosmic peculiar velocities only cause the delay time to drift linearly, and that the effect from the Earth's orbital motion can be accurately subtracted, thus enabling a search for non-trivial source motion. For a timing accuracy of ~1 ms and a repetition rate (of detected bursts) of ~0.05 per day of a single FRB source, non-uniform displacement 0.1–1 au of the emission spot perpendicular to the line of sight is detectable if repetitions are seen over a period of hundreds of days.
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