Accuracy of time dilation measurement *in principle*

[bland]

Sorry that's the best wording for a title I could come up with. Anyhoo my question is one that I have wondered about for a long time, and I am prompted to post now after seeing this article on the new atomic clock.

Let's assume that we have clocks that can measure time to an arbitrarily accurate degree. If a clock was sitting at the surface of the Earth, what is the minimum height it would need to be elevated in order to detect the dilation of time.

For example. I'm fairly sure that if this clock was moved from the ground to a 1 metre high table, that we can already detect that level of time dilation in practice. So I'm now wondering what is the theoretical limit, and by that I mean if the time dilation for a given elevation got down to the Planck length then that would be a theoretical limit.

So... Could we measure in principle the dilation if the clock was raised 1 cm? 1mm? one...?

I'm guessing that the answer is routine math but I do not have the skill to do the math.

 

[PeterDonis]

If a clock was sitting at the surface of the Earth, what is the minimum height it would need to be elevated in order to detect the dilation of time.

A single clock cannot detect time dilation at all. You need two clocks that are separated from each other and can compare their rates, for example by exchanging light signals, or by being separated for a while and then coming back together again.

If the clocks are arbitrarily accurate, they can detect differences in time dilation that are arbitrarily small; so two clocks with any arbitrarily small difference in height would be able to detect that there was time dilation between them by exchanging light signals.

I'm now wondering what is the theoretical limit, and by that I mean if the time dilation for a given elevation got down to the Planck length then that would be a theoretical limit.

There is no theoretical limit in our current theories.

There are proposed theories of quantum gravity in which the Planck length would be a kind of theoretical limit to this (by making the concept of "spacetime" no longer even applicable on that small a scale), but we have no way of testing those theories experimentally, now or in the foreseeable future.

 

[bland]

That's fair enough but now I have two more questions first one is what if we assume that the minimum possible time is the Planck time, does that mean the answer to the question would also somehow have a Planck unit of length figured in?.

Secondly I now I have to ask, what would be minimum in practice height we would have to raise one of the clocks in order to detect the time dilation with current technology. By current technology, and I'll take that to be the new clock with 5 orders of magnitude more accuracy than the current caesium clocks.

 

[PeterDonis]

what if we assume that the minimum possible time is the Planck time, does that mean the answer to the question would also somehow have a Planck unit of length figured in?

We don't know because, as I said, we don't have any way of testing theories of quantum gravity experimentally, so we don't know which one is right, or even if any of the proposed ones are. They could all be wrong.

what would be minimum in practice height we would have to raise one of the clocks in order to detect the time dilation with current technology. By current technology, and I'll take that to be the new clock with 5 orders of magnitude more accuracy than the current caesium clocks.

The article linked to says the new clock can detect changes of 1 centimeter in elevation.

 

[bland]

The article linked to says the new clock can detect changes of 1 centimeter in elevation.

Doh! why didn't I see that. Hang on a sec, that can't be right because if it's 10K times more accurate than the current caesium clock then that would imply with the current tech that we would need 100 metres of elevation before we could detect it, which cannot possibly be correct. Surely?

 

[Nugatory]

f we assume that the minimum possible time is the Planck time, does that mean the answer to the question would also somehow have a Planck unit of length figured in?.

It might, but it's not likely. More likely, this hypothetical theory (and because it is hypothetical and we have no candidate theory to discuss seriously, this entire thread is just an exercise in speculative waving of the hands) would incorporate some length that is more or less order of magnitude comparable to the Planck length.

There's a common misconception about the physical significance of the Planck units, so common that we even have an Insights article about the Planck length: https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/

 

[PeterDonis]

that would imply with the current tech that we would need 100 metres of elevation before we could detect it, which cannot possibly be correct.

Why not?

 

[Nugatory]

that would imply with the current tech that we would need 100 metres of elevation before we could detect it, which cannot possibly be correct.

We can detect time differences smaller than our clocks can measure by using interferometers. That's how the Pound-Rebka experiment detected gravitational time dilation across a height difference of about 20 meters, more than a half-century ago. So it's quite plausible that the previously best clocks couldn't do better than 100 meters even though other experiments can.

 

[DaveC426913]

If you're interested in exploring time dilation as a function of height, I can't recommend this guy enough.
He took his family on a camping trip up Mt. Ranier (to elevation 5400 ft) and brought along a bunch of atomic clocks for fun.
Amazing home-brewed experimentation.

http://leapsecond.com/great2005/

 

[bland]

We can detect time differences smaller than our clocks can measure by using interferometers. That's how the Pound-Rebka experiment detected gravitational time dilation across a height difference of about 20 meters, more than a half-century ago. So it's quite plausible that the previously best clocks couldn't do better than 100 meters even though other experiments can.

Yes, I was perhaps incorrectly referring in my own mind to anything that could measure a time difference as a 'clock'. But anyway, when I'm saying previous best clocks I'm referring to the current caesium clocks with reference to the new yttrium clock as being the new standards. Yes I'm aware of the Pound Rebka experiment and that it was just a few floors on the campus but wasn't that measuring a gravitational redshift?

Can't we just crunch the numbers? I guess we'd have to synchronise two caesium clocks and then what, if we then took one and sent it 20 m into the air in an elevator and brought it back down again, would the time difference be enough to measure? Then I guess we'd have to take into account that in moving the clock up the elevator that then a relativistic time dilation would have to be factored in. Unless it was moved very slowly would that help or would it not make a difference because it would be moving slower but it would be moving for a longer time.

If you're interested in exploring time dilation as a function of height, I can't recommend this guy enough.
He took his family on a camping trip up Mt. Ranier (to elevation 5400 ft) and brought along a bunch of atomic clocks for fun.
Amazing home-brewed experimentation.

http://leapsecond.com/great2005/

That's great I will check it out. It's not so much time dilation per se but I'm more fascinated with precision measurements in general. The gravity probe b, and gravitational geodesic mapping satellite, and so forth, not to mention measuring the stretch in a 4 mile long pipe that's less than a thousandth of the width of a proton.

One thing though that did puzzle me about the new Yttrium clock was how can they tell that it really is keeping time 10,000 times better than the caesium clock. How do they check it, I guess they can only tell relatively. But I digress.