# Equivalence of gravitational and kinetic time (clock) dilation and frequency shift

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TL;DR: the clock's perception of frequency is, from a GPS perspective, dependent on gravitational field and motion.

I will refer to "Error Analysis for Global Positioning System": https://en.wikipedia.org/wiki/Error_analysis_for_the_Global_Positioning_System

From a GPS perspective, magnitude-wise, slowing down/accelerating clocks in motion/gravity is equivalent to shifting the instrument's frequency perception.

This is logical: if your clock is going slower then you will perceive the frequency as higher - and vice versa.

The gravity calculations presented in the article refer to an infinitely distant point (with ~zero gravity).

If one inverts these calculations relative to the center of the Earth, it turns out that the clock at the center of the Earth stands still (is infinitely retarded relative to the Earth's surface).

From the perspective of gravitational calculations, the center of the Earth is then of little use.

Instead, we can calculate the wavelength (e.g., 630nm at the Earth's surface) at the distance of the wavelength (630nm) from the Earth's center, on the Earth's axis of rotation.

And this seems to be quite a friendly measure, much nicer than an infinitely distant point.

My question then is: why all the fuss about time dilation?

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33 minutes ago, kacenty said:

If one inverts these calculations relative to the center of the Earth, it turns out that the clock at the center of the Earth stands still (is infinitely retarded relative to the Earth's surface).

No, that is not true, there would be hardly any difference in the 2 clocks rates.

36 minutes ago, kacenty said:

My question then is: why all the fuss about time dilation?

What fuss are you referring too?

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1 hour ago, Bufofrog said:

No, that is not true, there would be hardly any difference in the 2 clocks rates.

ddt = dt/sqrt(1-GM/(r1*c^2)) - dt/sqrt(1-GM/(r2*c^2))

for:

dt = 86400 # seconds per day
r1 = 6.36e6 # distance from center to surface of Earth
r2 = 2.66e7 # distance from Earth's center to GPS

ddt = +45.7e-6 # GPS clock gains 46 microseconds per day

for:

dt = 86400 # seconds per day
r1 = 6.36e6 # distance from center to surface of Earth
r2 = 1 # 1 meter from Earth's center

ddt = -386 # clock at 1m from Earth's center lags by ~6.5min per day relative to surface

1 hour ago, Bufofrog said:

What fuss are you referring too?

Talking about time dilation and frequency shifts while there seems to be simple clock rate change.

--- Edit 4 Feb 2024, 14:10:59 UTC ---

One can ~force chat-gpt-4-1106-preview to admit that choosing between those three (time, freq, clock rate) shifts is a matter of concept.

Edited by kacenty
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4 hours ago, kacenty said:

If one inverts these calculations relative to the center of the Earth, it turns out that the clock at the center of the Earth stands still (is infinitely retarded relative to the Earth's surface).

No, this is incorrect.

4 hours ago, kacenty said:

My question then is: why all the fuss about time dilation?

Because you need to keep all the clocks synchronized, and this is done via a ground station, so the clocks need to run at the same rate as UTC(USNO), which is the source of GPS time

A nanosecond of error is a foot in positioning

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2 hours ago, kacenty said:

ddt = dt/sqrt(1-GM/(r1*c^2)) - dt/sqrt(1-GM/(r2*c^2))

for:

dt = 86400 # seconds per day
r1 = 6.36e6 # distance from center to surface of Earth
r2 = 2.66e7 # distance from Earth's center to GPS

ddt = +45.7e-6 # GPS clock gains 46 microseconds per day

for:

dt = 86400 # seconds per day
r1 = 6.36e6 # distance from center to surface of Earth
r2 = 1 # 1 meter from Earth's center

ddt = -386 # clock at 1m from Earth's center lags by ~6.5min per day relative to surface

Talking about time dilation and frequency shifts while there seems to be simple clock rate change.

--- Edit 4 Feb 2024, 14:10:59 UTC ---

One can ~force chat-gpt-4-1106-preview to admit that choosing between those three (time, freq, clock rate) shifts is a matter of concept.

That would only be applicable if you were shrinking the Earth in order to keep all of its mass contained within a sphere with a radius of r as r decreased.

As you move towards the center of the Earth this is not the case, as the amount of material within r decreases as you do so and thus M is not a constant throughout the trip. So while at the surface of the Earth Gravitational potential is -GME/rE, at the center, it is -1.5GME/rE

Thus the difference in gravitational potential between the surface of the Earth and its center is smaller than that between the surface and an infinite distance, and so would be the time dilation factor difference.

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3 hours ago, Bufofrog said:

No, that is not true, there would be hardly any difference in the 2 clocks rates.

“hardly” is a matter of the precision of the clocks. The value given here is 3 x 10^-10, which is pretty big in atomic clock terms. A millisecond per year (3.15 x 10^7 sec)

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1 hour ago, swansont said:

“hardly” is a matter of the precision of the clocks. The value given here is 3 x 10^-10, which is pretty big in atomic clock terms. A millisecond per year (3.15 x 10^7 sec)

True, but compared to a stopped clock it isn't much.

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1 hour ago, swansont said:

No, this is incorrect.

Yes, it's just naive interpretation of equation - yet from calculation it is what it is.

1 hour ago, swansont said:

Because you need to keep all the clocks synchronized

I'm not saying that it isn't important, I ask why both things time dilation and freq shift cannot be treated as the same clock drift?

1 hour ago, Janus said:

That would only be applicable if you were shrinking the Earth in order to keep all of its mass contained within a sphere with a radius of r as r decreased.

It's also an interesting perspective, we can calculate drift of a clock on the Earth shrunken to 1m radius and we can calculate clock drift for satellite laying on the surface of Earth stretched to 26km radius.

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1 hour ago, kacenty said:

I'm not saying that it isn't important, I ask why both things time dilation and freq shift cannot be treated as the same clock drift?

Drift is a specific noise process in timekeeping. Time dilation is not drift.

The frequency shift is the source of the time dilation (you integrate the frequency shift to get it)

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1 hour ago, swansont said:

Drift is a specific noise process in timekeeping.

Atomic clocks in constant environment present described, linear, predictable ageing drift due to changes in their physical properties.

Could changes in clock's environment be seen as changes in clock's physical properties in constant environment?

1 hour ago, swansont said:

The frequency shift is the source of the time dilation (you integrate the frequency shift to get it)

That sounds good, thank you.

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2 hours ago, kacenty said:

Atomic clocks in constant environment present described, linear, predictable ageing drift due to changes in their physical properties.

Are you quoting this from somewhere?

Somewhat predictable. Hydrogen masers’ drift has a component due to aging of the coating of the maser bulb, and my understanding is that it’s somewhat unpredictable in how it varies. You determine it by comparing to other clocks. “constant environment” would be things like temperature and humidity

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2 hours ago, kacenty said:

Could changes in clock's environment be seen as changes in clock's physical properties in constant environment?

An ideal clock always ticks at the same intrinsic rate regardless of its location and motion. Time dilation is always about the comparison of the length of a trajectory in four-dimensional spacetime of one ideal clock to the length of a trajectory in four-dimensional spacetime of another ideal clock. An important part of this is how the two ends of each of the two trajectories relate to each other.

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10 hours ago, kacenty said:

yet from calculation it is what it is.

What you used there is the exterior Schwarzschild metric, meaning this is only valid in vacuum outside the central mass. If you want to find the time dilation between a clock on the surface and another clock at the Center of the Earth, you need to use an interior metric along with appropriate boundary conditions. This can of course be done, but is algebraically a bit more involved.

As others have said, in practice this dilation factor wouldn’t be large.

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Thank you for your replies, I'll try to relate shortly.

I like to use GPT4 to clear up ambiguity in what I want to communicate and I think that it's pretty useful using that way.

(anticipating: I don't use it to process nor generate replies to your replies)

I'll paste below a conclusion from single session so it might maybe add some clarity.

Quote

Q: answer shortly: assume GPS timekeeper is also a timer for signal generator, its drift is the only source of differences in both received clock's value and frequency shift of carrying wave, yes?

A: Yes, that's correct.

A: No, there is no additional gravitational blueshift to account for.

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2 hours ago, kacenty said:

I'll paste below a conclusion from single session so it might maybe add some clarity.

Quote

Q: answer shortly: assume GPS timekeeper is also a timer for signal generator, its drift is the only source of differences in both received clock's value and frequency shift of carrying wave, yes?

A: Yes, that's correct.

A: No, there is no additional gravitational blueshift to account for.

You do realize that GPT-4 doesn't give accurate answers to questions don't you?  The GPT-4 was designed to sound like a person.  It doesn't matter if it is telling you something completely wrong as long as it sounds like a person.

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3 hours ago, kacenty said:

I like to use GPT4

!

Moderator Note

Chat GPT and their ilk are just amped-up predictive text. The give plausible-sounding, but not necessarily accurate answers. They are not to be used as a technical resource at SFN.

That said, if you don’t phrase a question to explicitly include gravity, one might conclude there is no gravitational shift. GIGO

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8 hours ago, Bufofrog said:

You do realize that GPT-4 doesn't give accurate answers to questions don't you?  The GPT-4 was designed to sound like a person.  It doesn't matter if it is telling you something completely wrong as long as it sounds like a person.

Yes I do and I won't develop this thread around it as it now has a moderator's note attached.

7 hours ago, swansont said:

That said, if you don’t phrase a question to explicitly include gravity, one might conclude there is no gravitational shift. GIGO

I can share via PM a chat session (short, nothing special), not quoting it full here because if it might be anyhow useful for human - I don't want gpt5 to learn how to not be helpful

Back to the topic - it seems quite natural that signal generator will be exposed to similar effect as atomic clock, thus it's quite strange to me that it isn't explicit - I want to make bold that, as in OP, I relate to GPS error correction article.

Not entirely on topic, but related to the article, there is only one reference to Sagnac and has such informal content:

Quote

### Sagnac distortion

GPS observation processing must also compensate for the Sagnac effect. The GPS time scale is defined in an inertial system but observations are processed in an Earth-centered, Earth-fixed (co-rotating) system. A coordinate transformation is thus applied to convert from the inertial system to the ECEF system. The resulting signal run time correction has opposite algebraic signs for satellites in the Eastern and Western celestial hemispheres. Ignoring this effect will produce an east–west error on the order of hundreds of nanoseconds, or tens of meters in position.[19]

It's strange that quite tested and quoted (eg. Ashby and others) time for signal to "catch up" with moving receiver (R/c + sagnac correction) is not popularised:

dt = R/c + R*v/c^2

c' = c^2/(c+v)

It amazes me because it covers well both Sagnac and Michelson-Morley experiments.

I decided to ask here because I've "stepped onto" this topic as a laymen and felt it is not satisfactorily covered and this place seemed only one were question are asked and replied in HQ /congrats/ - its hard to find such place anywhere else.

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1 hour ago, kacenty said:

Back to the topic - it seems quite natural that signal generator will be exposed to similar effect as atomic clock, thus it's quite strange to me that it isn't explicit - I want to make bold that, as in OP, I relate to GPS error correction article.

You need to specify where the signal generator is located.

Sagnac corrections are important for sending timing signals via satellite; GPS isn’t the only satellite system that is affected. Two-way satellite time transfer (TWSTT) bounces signals off of geostationary satellites, which generally has the signal path covering a larger area and thus has larger Sagnac shifts.

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21 hours ago, swansont said:

You need to specify where the signal generator is located.

Close to the atomic clock, in space vehicle.

21 hours ago, swansont said:

Sagnac corrections are important

Are there any models for Sagnac effect on light from Sun on Saturn?

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1 hour ago, kacenty said:

Close to the atomic clock, in space vehicle.

If they’re at the same gravitational potential then there’s no shift between them.

Quote

Are there any models for Sagnac effect on light from Sun on Saturn?

There’s a formula you can apply.

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20 hours ago, swansont said:

If they’re at the same gravitational potential then there’s no shift between them.

So generator will generate 1GHz that will be perceived as 1GHz +some Hz on the surface - that will be called blue-shift.

20 hours ago, swansont said:

There’s a formula you can apply.

Sure, but as far as I remember until light from Sun hit Saturn

R/c

Saturn will rotate ~47 degrees since Sun sends its signal, it isn't obvious when, where and how can an apply that correction.

When we look closer at the Sagnac's correction

dt = R/c + R*v/c^2

from naive interpretation it seems that light travels to the point where a receiver was in the moment of emission (t0), beam reaches that point at a time t0 + R/c, and then it ~chase the receiver with a speed c + v.

I hope that all of you know that using this Sagnac correction you can calculate time delay/gain for each segment of single a Sagnac loop polygon (each vector MirrorN->MirrorN+1, treating each reflection as re-emission), add it all up to get a delay for a single loop, then multiply it by 2 to get complete value of expected effect on whole interferometer?

Then you can translate (move) that polygon a miles away from the center of rotation, calculate and sum it up and it will hold?

And that it doesn't have to be a rotation - it can be a translation [https://www.sciencedirect.com/science/article/abs/pii/S0375960103005759

Given this, Sagnac's ~correction is not a correction - it is a rule.

Yet, Mr.Sagnac Wiki profile doesn't even has a photo of a genius (even in French).

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8 minutes ago, kacenty said:

So generator will generate 1GHz that will be perceived as 1GHz +some Hz on the surface - that will be called blue-shift.

Right. The frequency generators on GPS satellites are shifted down to 10.22999999543 Hz so that their frequency is 10.23 MHz when measured on earth.

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