Clock Synchronisation -- Do clock speeds go out of synch?

[Phil42]

TL;DR Summary

Do clock speeds go out of synch?

When reading about the one way speed of light, I think it's fair to say that most potential methods to obtain this fail because it is said to be impossible to have two non co-located clocks in synch.

My question is really about terminology.

Normally, two things can be said to be in synch if they are running at identical speed. They may be out of phase, or, as in the case of a clock, they may have a different offset, but if their speed is the same they are synchronised.

So, if you take two co-located clocks, and move one, so that they both remain in the same inertial frame, and are static with respect to that frame (and hence to one another), are they still running at the same speed?

From my knowledge of special relativity they should be, but I don't know enough to be sure.

 

[PeroK]

TL;DR Summary: Do clock speeds go out of synch?

When reading about the one way speed of light, I think it's fair to say that most potential methods to obtain this fail because it is said to be impossible to have two non co-located clocks in synch.

Methods don't fail, as the one-way speed of light is dependent on your synchronization convention. The operative word being convention, which means there is no one absolute synchronization procedure. Hence no absolute answer.

 

[Hill]

TL;DR Summary: Do clock speeds go out of synch?

My question is really about terminology.

Two clocks can run at the same speed but if one shows 10 o'clock when the other shows 10:15, they are not synchronized.

 

[PeroK]

Two clocks can run at the same speed but if one shows 10 o'clock when the other shows 10:15, they are not synchronized.

It's the when that's the problem in that statement!

 

[Phil42]

Methods don't fail, as the one-way speed of light is dependent on your synchronization convention. The operative word being convention, which means there is no one absolute synchronization procedure. Hence no absolute answer.

That doesn't actually answer the question.

What I am asking is not about a synchronisation convention, but about what it covers.

If you move two clocks apart, it is fairly obvious, from special relativity, that their offset and phase will change to some extent.

But will their speed?

 

[PeroK]

If you move two clocks apart, it is fairly obvious, from special relativity, that their offset and phase will change to some extent.

I don't know what that means.

But will their speed?

Not necessarily. In flat spacetime, two clocks at rest relative to each other will tick at the same rate relative to each other.

 

[Ibix]

But will their speed?

While they move, there's some flexibility and one can say yes or no. Once the clocks are at rest relative to each other they will tick at the same rate, assuming gravity isn't a factor.

 

[Hill]

It's the when that's the problem in that statement!

Yes, you are right. I should've said, "Two clocks at the same location and at rest relative to each other can run at the same speed but if one shows 10 o'clock when the other shows 10:15, they are not synchronized."

 

[Dale]

TL;DR Summary: Do clock speeds go out of synch?

Normally, two things can be said to be in synch if they are running at identical speed.

This is not the standard meaning in relativity. In relativity they need to not only run at the same speed but also show the same time.

 

[Phil42]

This is not the standard meaning in relativity. In relativity they need to not only run at the same speed but also show the same time.

Thanks. That's what I thought. Saying they are not synchronised only implies that they so not display the same time.

So two clocks that tick at the same rate when co-located will also tick at the same rate when one is moved, provided they remain in the same inertial reference frame.

 

[phinds]

So two clocks that tick at the same rate when co-located will also tick at the same rate when one is moved, provided they remain in the same inertial reference frame.

Yes, but they won't necessarily be in sync

 

[Ibix]

So two clocks that tick at the same rate when co-located will also tick at the same rate when one is moved, provided they remain in the same inertial reference frame.

All things are in all frames. If you mean "remain at rest in the same inertial frame" they can't have moved. If you mean "one clock moves then returns to rest in the same inertial frame in which it was initially at rest" then yes, they will tick at the same rate when the "moving" clock has come to rest again.

I wouldn't bother invoking frames for this. Just say that the clocks tick at the same ratewhen they are at rest with respect to each other.

 

[Hill]

Yes, but they won't necessarily be in sync

I think that the clock that has moved will necessarily lag.

 

[phinds]

I think that the clock that has moved will necessarily lag.

Relative to what? Each will see the other as "lagging" so your statement clearly shows that you have not yet grasped SR.

What you have to understand is that what you are calling the "moving" clock sees itself as perfectly still and the OTHER clock as moving away from it. They are symmetrical relative to each other.

 

[Hill]

Relative to what? Each will see the other as "lagging" so your statement clearly shows that you have not yet grasped SR.

What you have to understand is that what you are calling the "moving" clock sees itself as perfectly still and the OTHER clock as moving away from it. They are symmetrical relative to each other.

Yes, I understand. I refer to the clock that has moved and then stopped. When it is again at rest relative to the clock that stayed at rest, it runs at the same speed but lags relative to the clock that stayed at rest.

 

[Dale]

I think that the clock that has moved will necessarily lag.

IF they were initially synchronized, yes.

 

[phinds]

Yes, I understand. I refer to the clock that has moved and then stopped. When it is again at rest relative to the clock that stayed at rest, it runs at the same speed but lags relative to the clock that stayed at rest.

Yes, just as the clock that "didn't move" lags relative to it. "moved" and "didn't move / stayed at rest" are frame dependent and are not in and of themselves meaningful. This cannot be emphasized too much.

 

[Orodruin]

I think that the clock that has moved will necessarily lag.

IF they were initially synchronized, yes.

… and in the Einstein synchronization convention. It is perfectly possible to produce a synchronization convention where it is the clock that didn’t move which lags.

 

[Hill]

… and in the Einstein synchronization convention. It is perfectly possible to produce a synchronization convention where it is the clock that didn’t move which lags.

Would Lorentz transformations differ in such a convention?

 

[Dale]

Would Lorentz transformations differ in such a convention?

The Lorentz transformation is between inertial frames and those frames would not be inertial frames. So it wouldn’t apply

 

[Nugatory]

But will their speed?

Almost tautologically, all correctly functioning clocks tick at the same rate: one second per second.

However, when we say that two physically separated clocks are ticking at different rates we are talking about two unrelated time intervals between two pairs of events:
1) Clock A tick N
2) clock A tick N+1
3) clock B tick M
4) clock B tick M+1

Events 1 and 2 are separated by one second (assuming our clocks are constructed to tick once a second).
Events 3 and 4 are separated by one second (same assumption).

But when we compare the two clocks we have to introduce two more events:
5) We look at clock A at the same time that event #3 happens, and we see that it reads P
6) We look at clock A at the same time that event #4 happens, and we see that it reads Q.
We calculate Q-P, and if this is greater than one second we say that clock B is running slow.

But note the use of the phrase "at the same time" in the definitions of events 5 and 6. Because of the relativity of simultaneity there is no particular reason for events 5 and 6 to be simultaneous with 1 and 2; depending on our choice of frame and arbitrary definition of "at the same time" they may be completely unrelated.

The standard time dilation formula applies to the special case in which we choose N=M=0 and events 1, 3, and 5 are the same (we set both clocks to zero when the two are side by side) so P is also zero.

 

[Orodruin]

Would Lorentz transformations differ in such a convention?

Generally, yes.

The Lorentz transformation is between inertial frames and those frames would not be inertial frames. So it wouldn’t apply

This depends on what meaning you put into ”inertial frame” and on the simultaneity convention you pick. It is not completely unreasonable to designate any affine coordinate system as ”inertial”, but those generally do not fulfill the Lorentz transformations between them.

Then there are of course also curvilinear simultaneity conventions for which the Lorentz transformations are obviously violated.

 

[Orodruin]

we set both clocks to zero when the two are side by side

This is not a necessary condition for the typical time dilation. All that is required is two inertially moving clocks at a relative speed v. Then considering two ticks on the first clock and the corresponding ”at the same time” ticks of the second and then comparing the time intervals.

 

[PeroK]

Would Lorentz transformations differ in such a convention?

The fundamental problem with insisting that the Einstein synchronization convention and the standard Lorentz transformations are the only possibility in SR comes when you move on to GR. And, in particular, to refute the notion that time stops at the event horizon of a black hole. The key to understanding why an object falls through the event horizon and that event is part of the spacetime manifold is to relinquish the idea that there is one special, physically significant coordinate system for each geometry.

We have many threads on here where this debate goes on. And, popular science books and videos generally encourage the view that time stops at the event horizon. That makes it very difficult for the readers of popular science to become students of GR. They can't get past the notion that Schwarzschild coordinates are absolute and physically meaningful for the geometry of a black hole.

When we learned SR, I'm sure we all focused on the Einstein synchronization convention and the standard Lorentz Transformations. But, their almost universal usefulness and application in SR does not make them absolute. They are not, ultimately, the only possibility.

 

[Orodruin]

When we learned SR, I'm sure we all focused on the Einstein synchronization convention and the standard Lorentz Transformations.

It should be said though that this is for the same reasons that we usually start teaching Cartesian coordinate systems in Euclidean space. It is convenient and easier to grasp.

But, their almost universal usefulness and application in SR does not make them absolute. They are not, ultimately, the only possibility.

... just as Cartesian coordinates are not the only choice in Euclidean space. I go on an on about it, but it is an important point to drive home to students at undergraduate level who are typically already acquainted with curvilinear coordinates on Euclidean space.

 

[Dale]

This depends on what meaning you put into ”inertial frame” and on the simultaneity convention you pick. It is not completely unreasonable to designate any affine coordinate system as ”inertial”,

If you pick any other meaning than the one Einstein assigned then you will cause confusion at some point. The 2nd postulate asserts that the speed of light is c in inertial frames. So an affine frame where the speed of light is not c is non-inertial by the usual meaning. Thus, even if I were considering such synchronization conventions I would not call them inertial frames.

 

[Sagittarius A-Star]

It is not completely unreasonable to designate any affine coordinate system as ”inertial”, but those generally do not fulfill the Lorentz transformations between them.

Assume a standard inertial coordinate system (x, y, z, ct), based on Einstein-synchronization.
Set $x'=x$, $y'=y$, $z'=z$, $ct'=ct+kx$ with $k\neq 0, k<1, k>-1$.
Is then the coordinate system (x', y', z', ct') inertial or non-inertial?
(I assume the first)

Source:
https://www.mathpages.com/home/kmath229/kmath229.htm

 

[Orodruin]

If you pick any other meaning than the one Einstein assigned then you will cause confusion at some point. The 2nd postulate asserts that the speed of light is c in inertial frames. So an affine frame where the speed of light is not c is non-inertial by the usual meaning. Thus, even if I were considering such synchronization conventions I would not call them inertial frames.

Assume a standard inertial coordinate system (x, y, z, ct), based on Einstein-synchronization.
Set $x'=x$, $y'=y$, $z'=z$, $ct'=ct+kx$ with $k\neq 0, k<1, k>-1$.
Is then the coordinate system (x', y', z', ct') inertial or non-inertial?
(I assume the first)

Source:
https://www.mathpages.com/home/kmath229/kmath229.htm

Hence you would disagree with eachother. This kind of underlines my point that it is important to be specific. It is not unreasonable to designate any affine coordinate system as inertial - the Christoffel symbols are all zero - but it is also not unreasonable to restrict the use to orthonormal coordinates.

 

[Dale]

Is then the coordinate system (x', y', z', ct') inertial or non-inertial?
(I assume the first)

Hence you would disagree with eachother.

I am not sure we would disagree. He was asking a question. I would answer in the negative and cite Einstein’s 2nd postulate in doing so. We would only disagree if he then answered his question the affirmative, deliberately ignoring Einstein's postulates.

Calling such frames “inertial” invites confusion. I am also not sure if there is support in the peer reviewed scientific literature for calling such frames inertial. For example, I can’t remember if Anderson calls them inertial.

 

[Hill]

Since a different synchronization convention means a different system of coordinates, wouldn't the second postulate just need to be reformulated accordingly?

 

[Dale]

Since a different synchronization convention means a different system of coordinates, wouldn't the second postulate just need to be reformulated accordingly?

That wouldn’t be merely a reformulation in my view. In my view a reformulation is a different way of expressing the same thing. This would be a different thing.

 

[Orodruin]

I would answer in the negative and cite Einstein’s 2nd postulate in doing so.

All I am saying is that the case can be made for both. You can also take Newton’s view that an inertial frame is a frame such that an inertial system is any system in which an object will be in constant rectilinear motion unless acted upon by a force, which is the original meaning. Of course, Newton never imagined mixing space and time.

I am also not too fond of teaching SR based on Einstein’s postulates. They were formulated almost 120 years ago and the theory as well as didactics have evolved during this time. However, the same type of parroting appears in essentially every textbook on modern physics. I’ll mention them, then move on to discuss how they relate to geometry and from there on out the focus is on spacetime geometry. Using light clocks and similar with the postulates to derive things like time dilation just creates more confusion than it’s worth. But that is my 5 cents…

 

[Sagittarius A-Star]

I would answer in the negative and cite Einstein’s 2nd postulate in doing so.

I don't think, that Einstein used his 2nd postulate to define the term "inertial frame". I am even not sure, if he considered non-orthonormal coordinates. In 1916, he restricted his two postulates to a "co-ordinate system K be so chosen that when referred to it the physical laws hold in their simplest forms".

§ 1. Remarks on the Special Relativity Theory.
The special relativity theory rests on the following postulate which also holds valid for the Galileo-Newtonian mechanics.

If a co-ordinate system K be so chosen that when referred to it the physical laws hold in their simplest forms, these laws would be also valid when referred to another system of co-ordinates K′ which is subjected to an uniform translational motion relative to K. We call this postulate "The Special Relativity Principle".
...
The Special Relativity Theory does not differ from the classical mechanics through the assumption of this postulate, but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the special relativity postulate, gives in a well-known way, the relativity of synchronism as well as the Lorentz transformation, with all the relations between moving rigid bodies and clocks.

Source (see A 1.1):
https://en.wikisource.org/wiki/The_...§_1._Remarks_on_the_Special_Relativity_Theory

 

[Dale]

I don't think, that Einstein used his 2nd postulate to define the term "inertial frame".

Einstein didn't do that, but the usual modern formulation of his 2nd postulate is explicitly in terms of inertial frames. E.g. on Wikipedia https://en.wikipedia.org/wiki/Postulates_of_special_relativity#Postulates_of_special_relativity
"As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c ..."

 

[Mister T]

Yes, you are right. I should've said, "Two clocks at the same location and at rest relative to each other can run at the same speed but if one shows 10 o'clock when the other shows 10:15, they are not synchronized."

And if they're co-located you can't use them to measure the speed of a beam of light that moves from one clock to the other. I thought that measuring the one-way speed of light was your main concern?