  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 due to an Aharonov-Bohm
effect, where a particle in a flat space-time is subject to an effective
potential. Here we propose a novel 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,
due to 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.
