Assume you have a 2 particles sitting inside a double potential well. Each particle occupies one side of the well and given enough time the particles may both be found on the same side or swap sides (tunnelling). Assume that the well is inside a gravitational field. It is positioned in such a way that one side of the well experiences higher gravitational pull than the other. If the object that generates the field is the earth, then one side is close to the earth whereas the other side is far from the earth. If we take into account time dilation then the particle close to earth will experience a 'slower' time than the particle away from earth. I realize that the amounts involved are infinitesimally small, but before dismissing the question lets just limit ourselves to the 'though experiment' area (as opposed to what can be done in a lab). If a particle experiences slower time then the probability (which is a count of something over time) of it tunnelling across the barrier is lower than the particle that experiences 'faster' time. We should then observe that the particles are found more often in the 'slow' time part of the well than in the 'fast' time part of the well. It will appear that they are pulled towards the body that exerts the gravitational field. Usually we have electrons in potential wells. This is probably not the right particle for this thought experiment since the electric charge would be several orders of magnitude larger than the gravitational pull. We need to use other, not charged particles. Any thoughts?