Trajectories for a particle in flat spacetime  Phase shift vs. detour size, in flat spacetime 

In quantum mechanics, every particle has a phase: a kind of abstract vector which rotates like the hand of a clock, turning at a constant rate in the particle’s reference frame. In flat spacetime, a straight worldline represents motion with a constant velocity, and any detour from this causes time dilation, shortening the time — and hence reducing the phase shift — that a particle experiences as it travels between two given points. In flat spacetime (above left), the particle undergoes the maximum possible phase shift when it travels near the straightline trajectory (red); near the curved trajectories (blue) time dilation lessens the phase shift. But it’s not the amount of phase shift that matters, it’s the fact that a (smooth, flat) peak in a plot of phase shift versus detour size (above right) allows adjacent paths to have similar phase shifts, and so a bundle of paths close to a straight line can all arrive moreorless in phase (red segment), while paths close to a curved trajectory, lying on the slope of the plot, arrive with a wide range of different phases (blue segments) and so cancel each other out. Anything that introduces additional, nonuniform time dilation will alter the paths that are inphase. In the spacetime diagram below left, gravitational time dilation has been added; the worldline of a massive body (not shown) has been placed a short distance to the right of the three trajectories. It’s clear that the phase shift is reduced along the rightmost path, where the time dilation is most pronounced — and the particle’s most probable trajectory through spacetime (red) is now slightly curved. A plot of the phase shift (below right) shows how the peak has been displaced. (In the speculative physics of “The Planck Dive”, the passage of time is nothing but the phase shift caused by interactions with virtual particles. In a perfect vacuum, this effect is uniform, but in the presence of a massive object it varies from place to place, giving rise to both gravitational time dilation, and the curved trajectories of freefalling particles.) 

Trajectories for a particle in curved spacetime  Phase shift vs. detour size, in curved spacetime 