Current attempts to probe general relativistic effects in quantum mechanics focus on precision measurements of phase shifts in matter-wave interferometry. Yet, phase shifts can always be explained as arising because of an Aharonov-Bohm effect, where a particle in a flat space-time is subject to an effective potential. Here we propose a quantum effect that cannot be explained without the general relativistic notion of proper time. We consider interference of a 'clock'-a particle with evolving internal degrees of freedom-that will not only display a phase shift, but also reduce the visibility of the interference pattern. According to general relativity, proper time flows at different rates in different regions of space-time. Therefore, because of quantum complementarity, the visibility will drop to the extent to which the path information becomes available from reading out the proper time from the 'clock'. Such a gravitationally induced decoherence would provide the first test of the genuine general relativistic notion of proper time in quantum mechanics.
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