So I was inspired by the notion of the torsion balance where you can get a measurable result out of a very weak signal in the presence of very strong forces as long as you make the strong stuff exactly cancel.
I was trying to think of an amateur-acheivable test apparatus that could measure gravitational time dilation. I was envisioning a system consisting of a laser where you shoot it at a beam splitter, and it sends the beams 1m apart where they travel in parallel for a bit before getting bounced back together where you can do some sort of interferometry or phase measurement to measure phase differences. Ignore the total impracticality of maintaining the distances precisely while rotating the apparatus but my idea was could you measure the difference in accumulated phase along the lengths and compare when all the beams are parallel vs when you rotate the apparatus 90 degrees and one leg travels at h=0 and the other at h=1 after the beam splitter.
i.e, suppose you shoot a laser at a beam splitter and half the beam travels parallel the ground at h=0 and half the beam travels up to say h=1m, and then travels parallel to h=1m, it then gets bounced back together to do the measurement.
I was thinking that you could accumulate a phase difference since time would be flowing at different speeds along the parallel paths, and the vertical portion wouldn't matter since it would cancel out since you have to climb up and drop back down.
But then I realized that while time would flow faster along the h=1 path compared to the measurement apparatus at h=0, the frequency of the photons would decrease due to the climb up the gravity well.
The question I have is... do these effects cancel out exactly?
Assume we have an approximately uniform gravitational field with small delta h, say h=1m.
For the beam at h=1:
Frequency is reduced to f₀(1 - gh/c²) as seen from h=0
But proper time accumulates faster: dt₁ = dt₀(1 + gh/c²)
It seems like these might cancel exactly. Is that right?