Relativity - Synchronization and slow transport

[Pete]

In [thread=110633]another thread[/thread], a poster mentioned using slow transport to maintain clock synchronization.

[post=2843216]Post 42[/post]
I pointed out that if two synchronized clocks are slowly separated, they will not stay synchronized unless they were initially at rest.

Tach disagreed, and a long argument ensued which I'm belatedly diverting into a new thread.

For reference, here are the most relevant expansions of the contested sentence:

[post=2843243]Post 47[/post]

To spell it out:

Consider two synchronized clocks at rest together in frame S.
One clock remains at rest while the other is slowly moved at velocity dv to particular distance L away.
In frame S, the resulting difference in synchronization can be arbitrarily small with an arbitrarily small separation velocity.
If you take the limit as dv approaches zero, the resulting difference in synchronization is zero.

Now consider two synchronized clocks moving at velocity v together in frame S.
One clock remains moving at v, the other clock slowly separates at velocity v+dv until it is a particular distance L away. dv is parallel to v.
Now, the resulting difference in synchronization can not be made arbitrarily small.
If you take the limit as dv approaches zero, the resulting difference in synchronization is nonzero.​

Tach interpreted these scenarios as ([post=2843307]post 65[/post]:

According to you, those two clocks travel a common distance (L) with the same speed (v), after which one of the clocks has an increase in speed (dv).

...which of course led into a diversion about communication skills.

The next expansion was more formal:
[post=2843310]Post 66[/post]:

Two clocks, A and B, synchronized and colocated at $$t=0$$.

A moves at speed $$v + \Delta u$$.

B moves at speed $$v - \Delta v$$.

$$\Delta u$$ and $$\Delta v$$ are very small compared to $$v$$ and to c, but are not necessarily equal.
The clock velocities are parallel.

Now, at $$t = \frac{L}{\Delta u + \Delta v}$$,

• the separation between A to B is $$L$$.
• $$\tau_A$$ has elapsed on clock A
• $$\tau_B$$ has elapsed on clock B
• The difference in synchronization is $$\Delta \tau = \tau_B - \tau_A $$

The quantity in question is this:
$$\lim_{\Delta v \rightarrow 0 \\ \Delta u \rightarrow 0} \left (\Delta \tau\right )$$

I maintain that the answer is a function of v and L, and equals zero if and only if either v or L is equal to zero.​

And later analysed (the original post had some mistakes. Tach provided corrections, incorporated here.)
[post=2843741]Post 87[/post]

$$\begin{align}
t &= \frac{L}{\Delta u + \Delta v} \\
\tau_A &= t/\gamma(v + \Delta u) \\
\tau_B &= t/\gamma(v - \Delta v) \\
\Delta \tau &= \tau_B - \tau_A \\
&= t/\left(\gamma(v - \Delta v) + \gamma(v + \Delta u)\right) \\
&= \frac{L}{\Delta u + \Delta v}\left(\sqrt{1 - ((v-\Delta v)/c)^2} - \sqrt{1 - ((v+\Delta u)/c)^2}\right)
\end{align}$$

I don't think there's anything stopping us from immediately taking one of the limits?
$$\begin{align}
\lim_{\Delta v \rightarrow 0 \\ \Delta u \rightarrow 0} \left (\Delta \tau\right ) &= \lim_{\Delta v \rightarrow 0 \\ \Delta u \rightarrow 0} \ \frac{L}{\Delta u + \Delta v}\left(\sqrt{1 - ((v-\Delta v)/c)^2} - \sqrt{1 - ((v+\Delta u)/c)^2}\right) \\
&= \lim_{\Delta v \rightarrow 0} \ \frac{L}{\Delta v}\left(\sqrt{1 - ((v-\Delta v)/c)^2} - \sqrt{1 - (v/c)^2}\right) \\
&= -L \frac{d}{dv} \ \sqrt{1 - (v/c)^2} \mbox{(Definition of the derivative)}\\
&= \frac{\gamma Lv}{c^2}
\end{align}$$​

The discussion then returned to the initial sentence, and the tradeoff between conciseness and rigor. I thought we were done on the actual concept, but then...

 

[Tach]

In [thread=110633]another thread[/thread], a poster mentioned using slow transport to maintain clock synchronization.

[post=2843216]Post 42[/post]
I pointed out that if two synchronized clocks are slowly separated, they will not stay synchronized unless they were initially at rest.

The above claim is clearly falsified by the functionality of GPS. The satellite clocks are synchronized , yet they move with respect to each other (and at very high relative speeds, to boot).
It is also falsified by the functionality of UTC (but GPS is a much stronger proof of falsification, this is why is listed first).

 

[OnlyMe]

The above claim is clearly falsified by the functionality of GPS. The satellite clocks are synchronized , yet they move with respect to each other (and at very high relative speeds, to boot).
It is also falsified by the functionality of UTC (but GPS is a much stronger proof of falsification, this is why is listed first).

I am really not an expert on how the GPS sytem works, but it was pointed out somewhere earlier that the GR effect is far greater than the SR effect. The GPS satellite clocks run fast, compared to clocks on the ground.

Add to that, if you set any one as an at rest frame of reference, the relative velocity of the others is "relatively" small or their orbits would require constant adjustments. Their orbits are all "relatively" stable, so while the velocities relative to a point on earth may be large and changing (curved path and all) relative to each other they should be pretty stable.

 

[Pete]

From [post=2843859]Post 114[/post]

The statement works for any (x,t) inertial reference frame.

If the clocks were initially at rest in that frame, then they will stay synchronized in that frame on slow separation.
If the clocks were not at rest in that frame, then they will not stay synchronized in that frame on slow separation.

You can call it sloppy. I call it concise.
Remember, this is an internet discussion forum, not a journal.
Here, conciseness trumps rigor.

There is a fallacy in this type of reasoning. The desynchronization is predicated on $$L$$ being non null. But $$L$$ is a linear function of $$\Delta u$$ and $$\Delta v$$: $$L=(\Delta u +\Delta v )t$$.

When $$\Delta u->0$$ and $$\Delta v->0$$ so does $$L->0$$. So, the second sentence is not correct, you can't have it both ways.

What? The clocks are separated.

Not when $$\Delta u->0$$ and $$\Delta v->0$$. You can't have it both ways, your argument for the second sentence holds no water.

The clocks are slowly separated by some given distance L.
The slower the separation, the longer it takes.
As the separation velocity approaches zero, the time taken approaches infinity.

...meaning that you cannot use the limiting approach since $$t(du+dv)->\infty . 0$$

I know I don't need to explain limits to you, Tach.
$$t(du+dv) = L$$
What's the problem?

Now, before you rush to try fixing it, let me point out that your second claim is falsified by experiment: the atomic clocks making up the UTC, though separated by different distances and though moving at different (and variable speeds, so you cannot claim that they are at rest wrt each other or that they have ever been at rest wrt each other because there is no such frame where they are at rest wrt each other) are in synch in the ECI frame. You will need to reflect on this fact.

That's a different case, Tach.
You're talking about clocks that have cyclic motion in the ECI, not inertially moving clocks that are slowly separated to a fixed distance.

...because you have a lot of errors and holes in your explanation. Over these 80 posts, I have shown you several of the. Please don't attack me, I am not attacking you, I am simply pointing out the errors in your claims and proofs.

Tach, apart from a couple of math transcription errors and a mistake in differentiation you picked up (for which I thank you), it seems to me that every "error" and "hole" has been your sloppy reading of my posts.

Yes, it is true that my posts could be more rigorous (like I said, this is a public forum, not a journal).
No, I don't think their meaning is unclear.

And I noticed a few dangling sidetracks in that thread, where your misunderstandings have been pointed out:
[post=2843274]Post 53[/post]

I'm not sure where the misunderstanding is, but it's clear to me that:
"...they will not stay synchronized unless they were initially at rest."
is exactly equivalent to:
"they don't stay synchronized wrt any frame other than one in which they were initially at rest."

[post=2843290]Post 61[/post]

Slow separation means that you can't impart arbitrary speed to the clocks - the clocks' velocities must remain very close to their initial velocity.

[post=2843796]Post 94[/post]:

I've asked you several times to support your interpretation of what I said, but nowhere have you done so.
Now you're avoiding simple questions.
Why?
It seems painfully clear that you just misunderstood. Why not just say so, and move on?

Now I really don't care about those. I don't mind if you just move on from from those mistakes without acknowledging them.
But I find it really distasteful when you belabor others mistakes without acknowledging your own.

 

[Tach]

I am really not an expert on how the GPS sytem works, but it was pointed out somewhere earlier that the GR effect is far greater than the SR effect. The GPS satellite clocks run fast, compared to clocks on the ground.

This is a non - sequitur, you are simply trying to wish the effect away because it falsifies your statement. This is not how science works. You are completely missing the point, the bottom line is that GPS functionality falsifies your claim.
BTW, there is no such thing as "SR effect" vs. "GR effect" in GPS, it is all explained using GR.

Add to that, if you set any one as an at rest frame of reference, the relative velocity of the others is "relatively" small

Wrong, their relative speeds are very large and their separation distances vary wildly with time.

or their orbits would require constant adjustments. Their orbits are all "relatively" stable, so while the velocities relative to a point on earth may be large and changing (curved path and all) relative to each other they should be pretty stable.

Same mistake as above.

 

[Tach]

From [post=2843859]
I know I don't need to explain limits to you, Tach.
$$t(du+dv) = L$$
What's the problem?

Basic calculus says that when $$t->\infty$$ and $$(du+dv)->0$$ the limit ($$L$$) may not exist. You need extra information (which you don't have) in order to resolve the incertitude. I have explained that to you before, it is even in the paragraph you quoted.

That's a different case, Tach.
You're talking about clocks that have cyclic motion in the ECI, not inertially moving clocks that are slowly separated to a fixed distance.

Wrong, has nothing to do with what you call the "cyclic motion". For most Earth bound experiments testing SR, ECI is a very good approximation for an inertial frame. Michelson, Morley and all their successors might be turning in their graves based on your post. You are trying to cling to your sloppy statement instead of trying to understand the many serious objections.

Tach, apart from a couple of math transcription errors and a mistake in differentiation you picked up (for which I thank you), it seems to me that every "error" and "hole" has been your sloppy reading of my posts.

I transcribed into math your sloppy description, can't help things , if your prose is so sloppy. The math is better but it is still riddled with mistakes and , even worse, with fundamental misconceptions about how things work.

Now I really don't care about those. I don't mind if you just move on from from those mistakes without acknowledging them.
But I find it really distasteful when you belabor others mistakes without acknowledging your own.

You are getting personal again. You do this every time you run out of scientific arguments.

 

[Pete]

Basic calculus says that when $$t->\infty$$ and $$(du+dv)->0$$ the limit ($$L$$) may not exist.

Go back a step.
We're not deriving L from t, du, and dv.
L is fixed.
t is a function of L, du, and dv.

Wrong, has nothing to do with what you call the "cyclic motion". For most Earth bound experiments testing SR, ECI is a very good approximation for an inertial frame.

Yes, that's why it's called the ECI.
And yes, all GPS clocks have cyclic motion in the ECI. (unless there's one at the North or South pole?)

And no, this has nothing to with the scenario in question, which is about clocks in constant inertial motion being slowly separated by a fixed distance.

I transcribed into math your sloppy description, can't help things, if your prose is so sloppy.

Whatever, Tach. I think my prose is fine, and your reading is sloppy, but I can see we're not going to resolve that disagreement.

The math is better but it is still riddled with mistakes and , even worse, with fundamental misconceptions about how things work.

The mistakes you pointed out have been acknowledged and corrected.
The 'misconceptions' are still under discussion.

You are getting personal again. You do this every time you run out of scientific arguments.

I tend to get personal when I get pissed off at people who blame me for their own failings.
Call it a character flaw.

 

[Tach]

Go back a step.
We're not deriving L from t, du, and dv.
L is fixed.
t is a function of L, du, and dv.

Then you have $$t->\infty$$ when $$(du+dv)->0$$ which is clearly impractical.

And no, this has nothing to with the scenario in question, which is about clocks in constant inertial motion being slowly separated by a fixed distance.

You are arguing in bad faith, experiment clearly falsifies your view of how synchronization works.

 

[Pete]

Then you have $$t->\infty$$ when $$(du+dv)->0$$ which is clearly impractical.

Yes, but this is a mathematical abstraction.
In practice, you make t as long as required to make the desynchronization as close to the limit as you desire.

You are arguing in bad faith, experiment clearly falsifies your view of how synchronization works.

I think we're misunderstanding each other again.
There's nothing non-mainstream about my view of how synchronization works.

 

[Tach]

Yes, but this is a mathematical abstraction.
In practice, you make t as long as required to make the desynchronization as close to the limit as you desire.


I think we're misunderstanding each other again.
There's nothing non-mainstream about my view of how synchronization works.

the mere fact that you continue to argue against experimental evidence proves the contrary

 

[Pete]

the mere fact that you continue to argue against experimental evidence proves the contrary

The fact that you think I'm arguing against experimental evidence proves you're misunderstanding me.

I really don't understand why you think GPS clocks are relevant here. Can you explain?

And is there any mistake in the analysis in the OP?

 

[billvon]

The above claim is clearly falsified by the functionality of GPS. The satellite clocks are synchronized , yet they move with respect to each other (and at very high relative speeds, to boot).

Actually it reinforces that claim. If GPS relied on synchronized clocks that must remain synchronized, GPS would not function. Instead, GPS makes the assumption that the clocks will NOT remain synchronized, and instead applies an offset to account for the different rate at which the clocks run.

Also note that the only frame of reference for which that offset works is a ground based system. In other words, that offset corrects for the (incorrect) rates that a ground observer sees the orbiting clocks running. To an observer on one of those satellites, the offset does not necessarily correct the other clock so that it matches the observer's clock.

 

[Tach]

The fact that you think I'm arguing against experimental evidence proves you're misunderstanding me.

I really don't understand why you think GPS clocks are relevant here. Can you explain?

i explained already several times : you made a sweeping claim that is contradicted by experimental evidence.

 

[Tach]

Actually it reinforces that claim. If GPS relied on synchronized clocks that must remain synchronized, GPS would not function. Instead, GPS makes the assumption that the clocks will NOT remain synchronized, and instead applies an offset to account for the different rate at which the clocks run.

No, it contradicts the redlined part of the statement that got the argument started:

if two synchronized clocks are slowly separated, they will not stay synchronized unless they were initially at rest.

The redlined part is what the argument is about, Pete made a sweeping claim that is not supported by the elementary math (see the quibble about calculating indeterminate limits) and is falsified by experiment: the satellites ARE routinely synchronized , though they have NEVER been at rest in any frame.

Also note that the only frame of reference for which that offset works is a ground based system. In other words, that offset corrects for the (incorrect) rates that a ground observer sees the orbiting clocks running. To an observer on one of those satellites, the offset does not necessarily correct the other clock so that it matches the observer's clock.

You are mixing two things here: the continuing offset adjustments for ephemerides with the initial frequency adjustment at launch.

 

[Pete]

I don't understand your explanation. You said:

The above claim is clearly falsified by the functionality of GPS. The satellite clocks are synchronized , yet they move with respect to each other (and at very high relative speeds, to boot).

Yes, GPS clocks move relative to each other.
Yes, they are synchronized in the ECI.

No, I don't understand why you think this is contradicted by the rule that "if two synchronized clocks are slowly separated, they will not stay synchronized unless they were initially at rest."

Perhaps you could be clearer?

 

[Pete]

The redlined part is what the argument is about, Pete made a sweeping claim that is not supported by the elementary math (see the quibble about calculating indeterminate limits)

A meaningless quibble, since L is fixed. Do you have any other problem with the math?

and is falsified by experiment: the satellites ARE routinely synchronized , though they have NEVER been at rest in any frame.

The claim is this:

if two synchronized clocks are slowly separated, they will not stay synchronized unless they were initially at rest.

GPS clocks don't meet the bolded condition, so it's not relevant.

I'm not making any claim about synchronizing distant clocks. Only about what happens to synchronization of colocated clocks under slow separation.

 

[billvon]

the satellites ARE routinely synchronized , though they have NEVER been at rest in any frame.

?? The satellites are never in sync with the ground. Their clocks run faster than ground based clocks and they are never resynchronized. (Which makes sense since, once resynchronized, they would immediately drift again.) However, since we can accurately predict the amount they are off by, we can apply an offset to allow accurate determination of position - and those offsets are updated regularly to allow for changes in satellite altitude and orbital path.

(That might be just semantics; most discussions of special relativity use the term "synchronized clocks" to refer to accurate clocks that have been synchronized, not clocks that have offsets applied to deal with the mis-synchronization caused by relativity.)

 

[OnlyMe]

?? The satellites are never in sync with the ground. Their clocks run faster than ground based clocks and they are never resynchronized. (Which makes sense since, once resynchronized, they would immediately drift again.) However, since we can accurately predict the amount they are off by, we can apply an offset to allow accurate determination of position - and those offsets are updated regularly to allow for changes in satellite altitude and orbital path.

(That might be just semantics; most discussions of special relativity use the term "synchronized clocks" to refer to accurate clocks that have been synchronized, not clocks that have offsets applied to deal with the mis-synchronization caused by relativity.)

Each satellite carries a clock rate monitoring system and a harmonic synchronizer that retunes their cesium clocks to within I think they said 1 nano second. Both the special and general relativistic effects are corrected in this way.

There are other synchronization issues involving things like transmission time delays, frequency shifts and Doppler effects, involving both earth gravitational and rotation effects and a few others I am sure. Those are generally corrected by software in the ground based receivers.

It is a bit more complicated than that, but the relativistic effects are all corrected by on board monitors and the harmonic synchronizer.

I think I got that right. It late and I have only just begun to read up on it.

 

[Tach]

A meaningless quibble, since L is fixed. Do you have any other problem with the math?

Yes, if L is fixed $$t=\frac{L}{du+dv}->\infty$$ for $$(du+dv)->0$$. This is the third time I explain this to you.

The claim is this:

if two synchronized clocks are slowly separated, they will not stay synchronized unless they were initially at rest.

GPS clocks don't meet the bolded condition, so it's not relevant.

I'm not making any claim about synchronizing distant clocks. Only about what happens to synchronization of colocated clocks under slow separation.

So, you are now splitting hairs about clocks that are solwly separated vs. clocks that are quickly separated? Fine, then the UTC system falsifies your claim, since the clocks in that system satisfy your condition of having been slowly separated.

 

[Tach]

?? The satellites are never in sync with the ground. Their clocks run faster than ground based clocks

This is false. At launch, their clock frequency is SET SLOWER than the clocks on the ground in order to precompensate the GR effects.

and they are never resynchronized.

This is just as false, the satellite clocks need periodic readjustment for ephemeris (their orbits are not fixed, so the GR effects vary).