Correcting General Relativity Effect on Atomic Time

[phoenix-anna]

TL;DR

The issue is how to combine the readings of atomic clocks running at different rates due to their different altitudes.

The International Bureau of Weights and Measures combines the readings of 450 atomic clocks around the world to obtain a time standard with sub-nanosecond accuracy. These clocks run at different rates - a clock at 1 km of altitude gains about 7 ns a day compared to one at sea level due to the weaker gravitational field. So what is the procedure to correct for this effect?

 

[Dale]

As of 1 Jan 1977 all clocks that participate in the TAI are adjusted for their altitude above the geoid. Each clock is at a fixed altitude so the correction is just a scale factor.

 

[Sagittarius A-Star]

Summary:: The issue is how to combine the readings of atomic clocks running at different rates due to their different altitudes.

The International Bureau of Weights and Measures combines the readings of 450 atomic clocks around the world to obtain a time standard with sub-nanosecond accuracy. These clocks run at different rates - a clock at 1 km of altitude gains about 7 ns a day compared to one at sea level due to the weaker gravitational field. So what is the procedure to correct for this effect?

TAI is defined on geoid:

International Atomic Time (TAI, from the French name temps atomique international[1]) is a high-precision atomic coordinate time standard based on the notional passage of proper time on Earth's geoid.
...
TAI is a weighted average of the time kept by over 400 atomic clocks[4] in over 50 national laboratories worldwide.[5] The majority of the clocks involved are caesium clocks; the International System of Units (SI) definition of the second is based on caesium.[6] The clocks are compared using GPS signals and two-way satellite time and frequency transfer.[7] Due to the signal averaging TAI is an order of magnitude more stable than its best constituent clock.

Source:
https://en.wikipedia.org/wiki/International_Atomic_Time

The geoid (/ˈdʒiːɔɪd/) is the shape that the ocean surface would take under the influence of the gravity and rotation of Earth alone, if other influences such as winds and tides were absent.

Source:
https://en.wikipedia.org/wiki/Geoid

P.S.
Einstein concluded from SR and the rotation of the earth:

From this, we conclude that a balance wheel clock placed at the equator must be slower by a very small amount than a similarly constructed clock which is placed at the pole, all other conditions being identical.

https://en.wikisource.org/wiki/Translation:On_the_Electrodynamics_of_Moving_Bodies#%C2%A7_4._The_physical_significance_of_the_equations_obtained_concerning_moving_rigid_bodies_and_moving_clocks.

What Einstein could not know in 1905 was, that GR time dilation compensates this very accurately due to Earth flattening (due to rotation of the earth). So proper time on geoid is very accurately the same at the poles and at the equator.

Source (in German, see chapter 14.3 - I used Firefox):
https://books.google.de/books?id=SEckBAAAQBAJ&lpg=PA168&ots=EcmAK7b1UI&dq=einstein zeitdilatation nordpol äquator&hl=de&pg=PA171#v=onepage&q=einstein zeitdilatation nordpol äquator&f=false

 

[phoenix-anna]

As of 1 Jan 1977 all clocks that participate in the TIA are adjusted for their altitude above the geoid. Each clock is at a fixed altitude so the correction is just a scale factor.

Yes, all clocks on the geoid should run at the same rate because they are at the same gravitational potential. I am interested in the details of how the calculation is performed: given time signals for 450 clocks running at different rates, and knowing their time dilation factors, how exactly does one combine the time signals?

Obviously, one cannot simply multiply the time by the scale factor; that would be nonsense. One has to work with time intervals. There has to be a subtraction step. Perhaps one simply subtracts the current reading from the prior reading, applies the scale factor and adds the result to the prior reading. I don't know. This seems cumbersome - after a year the accumulated correction would amount to over 3000 ns.

I understand that the time delay receiving the signals from the different clocks can be compensated for by two-way satellite time transfer, but this calculation does not include the altitudes of the clocks.

 

[Sagittarius A-Star]

all clocks on the geoid should run at the same rate because they are at the same gravitational potential.

That's correct, if you mean the effective potential due to gravitational force and centrifugal force of the rotating and flattened earth.

 

[PeroK]

Yes, all clocks on the geoid should run at the same rate because they are at the same gravitational potential. I am interested in the details of how the calculation is performed: given time signals for 450 clocks running at different rates, and knowing their time dilation factors, how exactly does one combine the time signals?

Obviously, one cannot simply multiply the time by the scale factor; that would be nonsense.

I think the suggestion is that each clock applies its own scale factor all the time. I.e. it communicates a scaled version of its own local time. That means that each clock only has to know about itself, relative to the geoid standard.

 

[phoenix-anna]

I think the suggestion is that each clock applies its own scale factor all the time. I.e. it communicates a scaled version of its own local time. That means that each clock only has to know about itself, relative to the geoid standard.

Yes, that simplifies matters. However, I am left wondering exactly what is subtracted to obtain the time interval to which the scale factor is applied. Do all the clocks have to share a common start time for computing the interval?

 

[PeroK]

Yes, that simplifies matters. However, I am left wondering exactly what is subtracted to obtain the time interval to which the scale factor is applied. Do all the clocks have to share a common start time for computing the interval?

I looked at a paper a while back on the calculations for GPS satellites and it had all sorts of things in there like allowances for atmospheric conditions. I guess these things ultimately get quite complicated whatever you do!

 

[Vanadium 50]

Obviously, one cannot simply multiply the time by the scale factor; that would be nonsense. One has to work with time intervals.

Sure they can. If there were a 10% difference (for illustration) every local "tick" the clock reports 1.1 seconds have elapsed.

 

[Dale]

There has to be a subtraction step.

Every month the BIPM publishes a list of the (post-scaling) deviations of each participating clock from the ensemble average. The average is the TAI, and each clock has its own deviation which is subtracted.

 

[Vanadium 50]

I think it might be valuable if the OP learned how NTP works. Forget SR and GR and atomic clocks for the moment. How does one form a consensus time with imperfect oscillators?

Then the SR (latitude) and GR (altitude) effects are simply examples of known offsets.

 

[Sagittarius A-Star]

I think the suggestion is that each clock applies its own scale factor all the time. I.e. it communicates a scaled version of its own local time. That means that each clock only has to know about itself, relative to the geoid standard.

This is described there:

TAI and the gravitational time dilatation
...
Due to the relativistic time dilatation caused by the Earth's gravitational potential, the SI second could only be realized by atomic clocks at sea level if no corrections were applied. In order to compensate for the gravitational time dilatation, the rates of atomic clocks located at an altitude $h$ above sea level are corrected by a relative amount of $-1.09 \cdot 10^{-16} (h/m)$ .

Source:
https://www.ptb.de/cms/en/ptb/facha...e-in-germany/the-time-scales-tai-and-eal.html

 

[russ_watters]

Obviously, one cannot simply multiply the time by the scale factor; that would be nonsense. One has to work with time intervals. There has to be a subtraction step. Perhaps one simply subtracts the current reading from the prior reading, applies the scale factor and adds the result to the prior reading. I don't know. This seems cumbersome - after a year the accumulated correction would amount to over 3000 ns.
[separate post]
However, I am left wondering exactly what is subtracted to obtain the time interval to which the scale factor is applied. Do all the clocks have to share a common start time for computing the interval?

This really doesn't seem that cumbersome to me. You could list all the clocks on a spreadsheet, with columns for the date/time they were turned on, scale factor, and current date/time. It's only four columns.