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DIY measurement of time dilation in a gravity well

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Circlotron:
Disclaimer - I only have a YouTube level of understanding of the following subject.  :P

It is widely accepted the time proceeds proportionally more slowly when gravity is present. For example, when a perfect clock 1km above earth’s surface shows exactly one day of elapsed time, an identical perfect clock on the surface of the earth will show about 10nS less duration for the same day. A practical example of this real phenomena is the need to keep GPS satellite clocks in zero G synced with ground based clocks.

How can we check out this weird situation at home? If we reduce the aforementioned 1km distance to 100 metres and we put an oscillator of 100MHz on top of a 100 metre tower and an identical oscillator on the ground and feed both into our dual trace scope, after one day we should see 36 degrees of phase shift between the two oscillators.

Not everyone has two oscillators with the required degree of stability, least of all me. What then if we have a single oscillator on the ground and a feed coax up the 100 metre tower and a transmitting antenna aiming back down toward us. Then feed to the scope both a signal directly from the oscillator and one received from the elevated antenna. There will be propagation delay of course, but both signals will initially have a certain fixed phase difference.

If in fact time proceeds more slowly at ground level compared to 100 metres above the ground then we should perhaps see the two 100MHz signals gradually creeping out of phase at 36 degrees per day despite coming from the same oscillator source, should we not?

dietert1:
I'd guess that your setup won't produce any phase travel. But general relativity is a difficult subject and most of the time requires serious mathematical efforts. I'd recommend reading about gravitational red shift and whether it reverses when going back.

Regards, Dieter

PS: There is a nice Wikipedia description of such terrestrial experiments. They use two clocks to generate and measure the phase travel.

Circlotron:

--- Quote from: dietert1 on January 21, 2023, 11:15:47 pm ---I'd guess that your setup won't produce any phase travel. But general relativity is a difficult subject and most of the time requires serious mathematical efforts. I'd recommend reading about gravitational red shift and whether it reverses when going back.

--- End quote ---
Yes, I did actually read about gravitational red shift and that is what got me thinking about how an ordinary person could conduct a somewhat comparable experiment. The fact that we can see a star is red shifted makes me think the experiment might work. I’ll have a further read up on the subject though.

WatchfulEye:
The experiment you propose is feasible, and indeed has been done in the famous Hafele–Keating experiment. The HK experiment was complicated by the fact that the clocks were travelling in commercial airliners, and so the experiment measured both the effect of gravitational red shift and special relativistic time dilation, with the latter being the dominant effect.

The problem is that the gravitational effect is small, and few oscillators have the required stability. Measuring 3.6 ns of phase shift over 24 hours is a fractional deviation of 4.2 x 10^-14 - few oscillators have that level of stability. You're looking at something like the high performance variant of the microchip 5071A, which is specified to have an Allan deviation at 10^5 seconds (approx 1 day) of <2.7 x 10^-14. The Allan deviation can be thought of as the standard deviation of the time measurement for a specified period. So, while this oscillator is exceptional, the oscillator noise is a substantial fraction of the signal - so you would need to repeat the experiment multiple times, or run for a longer period, to collect sufficient statistics.

If you could increase the height to 1 km or more, for example by climbing a mountain (or visiting a ski resort), then your signal becomes much larger and it becomes feasible to measure it more easily. The difficulty is measuring the phase shift over this sort of distance - options include GPS transfer (each oscillator logs phase shift relative to GPS satellite(s),  or you simply transport the test oscillators to the test location, and set the experimental duration long enough that the effects of travel are minimal.

With a high enough location, then the experiment is accessible to the (motivated) amateur:
http://leapsecond.com/great2005/

Circlotron:

--- Quote from: WatchfulEye on January 22, 2023, 10:54:41 am ---(each oscillator logs phase shift relative to GPS satellite(s),  or you simply transport the test oscillators to the test location,

--- End quote ---
I was suggesting using a single oscillator and splitting the signal into two paths. That way any drift in the single oscillator would affect both measured signals and mostly null out. That way any normal and reasonably stable oscillator could be used. And any creep in phase difference due to difference in rate of elapsed time should hopefully keep on accumulating until it is clearly above any noise.

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