Here you go, Akinbo. Caslav Brukner is working on a tabletop experiment to show whether there exists a general relativistic notion of time, foundationally:
"So here is the experiment proposed by Brukner and his team: Imagine you have a particle that carries its own wristwatch--some sort of evolving internal degree of freedom, such as its spin, that has some repetitive behavior that can serve as a clock. Usually when you send particles through a double-slit experiment, the slits are arranged side-by-side, right and left, at the same height. But what if you send that clock through a wall in which the two slits are arranged so that one slit is higher up--and thus in a different gravitational potential--than the other? General relativity says that the clock travelling along the lower path will tick slower than the clock passing through the upper slit. So far, so good for Einstein.
But here's the kicker: quantum complementarity says that the clocks can only continue to behave as waves if there is no significant time dilation effect between the two paths. That's because, if there is a discernible time dilation, you would be able to look at the clock and deduce which path it had taken, based on whether it seemed to have ticked faster or slower en route. 'This vanishing of the interference will really be a proof that there was a general relativistic notion of time involved,' says Brukner.
"The experiment pits two conceptions of time--the quantum mechanical and the general relativistic--head to head. On one side, the double-slit experiment puts the clock into a quantum superposition--a blurry confusion of multiple identities. We should not know which path it took during the experiment, and the time shown on the clock is undefined. This is in contrast with general relativity in which time has an objective status: it is well-defined at single points. 'In this experiment the time shown by the clock becomes quantum mechanically indefinite, that is, before it is measured it has no predetermined value,' says Brukner."
I predict the experiment *will* fail to show a general relativstic notion of time. The result won't militate against general relativity, however -- for the reason I gave you earlier of the difference between a classically continuous beam of light and a quantized beam of light. Think of the "bent stick" optical effect when viewing a stick half submerged in the water; the light travels faster through the air than through the water, which is what causes the optical effect -- now we know, of course, that light through the air is also quantized by air molecules, only to a lesser extent than through the water. The relative difference in speed gives us an idea of the constant speed of light; in a vacuum, with no interference of medium that absorbs and emits radiation, light speed is always measured at a constant value.
Now -- if gravity interferes with the speed of light, as you claim -- remember this: gravity orients in but one direction, toward the center of mass. That means the acceleration of the gravity field is always vertical to the plane. There is no horizontal accceleration.
In the Brukner team tabletop experiment, the constant-speed beam of light necessarily aimed at slits in a direction horizontal to the plane, assumes that the higher-up slit of the vertical plane is subject to less vertical acceleration of gravity than the lower slit. With this assumption, the time differential between two positions vertical to the plane -- (as with the bent stick effect) -- of light falling in the gravity field should tell us that absent of quantum absorption-emission effects in a massive medium, the pure state of elapsed time should not allow a "clock" analog of differential times between the slits, such that we can observe and read the clock.
We will always be able to read the differential as a classical effect horizontal to the plane, though (my claim) -- no matter how high up the vertical -- because there is no horizontal acceleration, as we've known since Galileo. A failure to see the predicted interference fringe merely puts the data point outside the range of classical observation, quantizing the light clocks by fiat, not by any principle of relative motion. One can always raise the slit to get the desired result.
Therefore: general relativity, which describes gravity as the curvature of spacetime, need not be quantized (see, e.g., Petkov ) for special relativity to hold in the limit of uniform motion, the universal standard by which we measure time.
To me, this experiment is just another case of quantum theory "proving" its prior assumptions (quantum superposition, nonlocality, observer-created reality) by loading the dice.
All best,
Tom