Relative rate of clock ticks on the radius of a rotating disc

[Nugatory]

….at any given moment

….and there’s your simultaneity convention.

 

[Dale]

So then C should observe B moving at a non-zero velocity w.r.t. him (in the direction of rotation)

$C$ is non inertial, so you have to be clear what frame you mean by “w.r.t. him”. The meaning is ambiguous for $C$.

If you use the reference frame that I described above, then the statement is simply false. In that frame $B$ is not moving (nor is $C$ obviously).

And because of this, having knowledge of SR, they should at once know that the other's clock is ticking slower compared to theirs

Even if you did find reference frames that match the description, your conclusion would not follow. Those reference frames are non-inertial. So the time dilation may depend on position and direction of travel. If you work it out correctly, as I did above, it will not be symmetrical.

 

[A.T.]

So then $C$ should observe $B$ moving at a non-zero velocity w.r.t. him (in the direction of rotation), whereas $B$ should notice $C$ moving at non-zero velocity (in the opposite direction of rotation).

You have to define a reference frame to determine motion. B and C are just points, that don't define unique axes.

If a reference frame is not inertial, you cannot apply formulas that were derived for inertial frames only.

 

[sphyrch]

It's nothing to do with SR or GR. It's a case of measuring the clocks tick rate.

Theoretically, however, you can analyse the scenario using an inertial reference frame.

Sorry for the delayed response. So then it seems that there's no way to theoretically conclude that $B$ and $C$ clocks are ticking at different rates based solely on SR. Either they'll have to do measurements using, e.g. , ring laser gyroscopes, OR the equivalence principle will need to be invoked (from which GR seems to follow anyways)

The reason I even asked for a purely SR-based argument in this thread is because of this paragraph in the text I'm reading

As judged from this body, the clock at the centre of the disc has no velocity, whereas the clock at the edge of the disc is in motion relative to К (this is $A$'s (inertial) frame) in consequence of the rotation. According to a result obtained in Section 12, it follows that the latter
clock goes at a rate permanently slower than that of the clock at the centre of the circular disc, i.e. as observed from $K$.

It is obvious that the same effect would be noted by an observer whom we will imagine sitting alongside his clock at the centre of the circular disc. Thus on our circular disc, or, to make the case more general, in every gravitational field, a clock will go more quickly or less quickly, according to the position in which the clock is situated (at rest).

I know the last line is true, but I'm not sure one can deduce that purely based on whatever was written before that in the quote, based on SR only (which is what was discussed in the book till that point). Hence the confusion. Maybe the last line in bold was sort of conceptual leap of faith?

 

[PeterDonis]

I know the last line is true, but I'm not sure one can deduce that purely based on whatever was written before that in the quote, based on SR only

Yes, you can. But the definition of "more slowly" that is being used is that of a specific inertial frame, the one in which the clock at the center of the disk is at rest.

 

[sphyrch]

But the definition of "more slowly" that is being used is that of a specific inertial frame, the one in which the clock at the center of the disk is at rest.

Which is to say that w.r.t. $A$ (who's the observer in the inertial frame you mentioned), $B$'s and $C$'s clocks tick at a different rate - this conclusion is what you're saying can be derived based on SR only?

Yes, you can.

Could you elaborate or give a hint so that I can try to work it out myself first?

 

[Orodruin]

Yes, you can. But the definition of "more slowly" that is being used is that of a specific inertial frame, the one in which the clock at the center of the disk is at rest.

Or with any simultaneity convention - provided that you let the clocks run long enough.

 

[Dale]

So then it seems that there's no way to theoretically conclude that B and C clocks are ticking at different rates based solely on SR

No. See above

https://www.physicsforums.com/threa...adius-of-a-rotating-disc.1056853/post-6963695

Frankly, it is a bit irritating to spend the time to properly and completely answer your question and then have you continue to post stuff like this.

Could you elaborate or give a hint so that I can try to work it out myself first?

I gave more than a hint. I worked it out explicitly.

Either they'll have to do measurements using, e.g. , ring laser gyroscopes

Are you somehow under the impression that gyroscopes are outside of SR, but accelerometers are not?

 

[Vanadium 50]

But I wanted to consider the same argument but for a slightly different case...

There's a statement in the text I'm reading

Here is the root of your confusion. You took a scenario described in the textbook, changed it, but are still treating statements about the original setup as applicable to the different case.

Thee may or may not apply. You certainly cannot count on them.

Second point - if one is confused about SR, the solution is seldom to switch to GR.

Final point - the ratio of clock ticks is physical, and not subject to any simultaneity convention. Point X has a light that flashes once per second, and point Y has an observer with a metronome. He records how many flashes he sees for a given number of ticks. Different observers may disagree about what number should have been written down, but not what was writtenm down: in short, this is something frame dependent but not subject to any coordinate convention.

 

[Nugatory]

Final point - the ratio of clock ticks is physical, and not subject to any simultaneity convention. Point X has a light that flashes once per second, and point Y has an observer with a metronome. He records how many flashes he sees for a given number of ticks.

I'm pushing towards a quibble here, but I dislike this formulation of the problem. What is physical is the ratio between the rate of metronome ticks and the rate of flash receptions at point Y. There's another assumption (reasonable, to the point that it would be perverse not to make it, but but still an assumption) required to leverage this into a statement about the ratio of flashed leaving point X to clock ticks at point Y.

 

[PeterDonis]

Which is to say that w.r.t. $A$ (who's the observer in the inertial frame you mentioned), $B$'s and $C$'s clocks tick at a different rate - this conclusion is what you're saying can be derived based on SR only?

With respect to the inertial frame in which $A$ is at rest, $B$'s and $C$'s clocks tick at a different rate. Yes, you can derive this based on SR only; the passage you quoted says how. Also @Dale, at least, has explained how in this thread.

 

[sphyrch]

No. See above

https://www.physicsforums.com/threa...adius-of-a-rotating-disc.1056853/post-6963695

Frankly, it is a bit irritating to spend the time to properly and completely answer your question and then have you continue to post stuff like this.

I gave more than a hint. I worked it out explicitly.

Are you somehow under the impression that gyroscopes are outside of SR, but accelerometers are not?

Thank. I think I got the source of my major misconception. I was wrongly assuming that a "theoretical argument" should just be some thought experiment that doesn't involve measurements. But even from $A$'s PoV, just saying that the disc is rotating and hence $B$ is moving w.r.t. him, involves a measurement (eyes receive light signals from various points on the disc) - so it's unavoidable.

So by the same token even for $B$ and $C$, they can use whatever device to detect rotation (measurement) of the disc, formulate the metric as you did, and finally reach the conclusion that even $B$ and $C$ will calculate each other's clocks tick at different rates.

 

[A.T.]

I think I got the source of my major misconception. I was wrongly assuming that a "theoretical argument" should just be some thought experiment that doesn't involve measurements.

A thought experiment can restrict what type of measurements you can do, like alowing only local measurements for the equivalence principle, making it only applicable locally, which is still useful.

But your combination of allowed and not allowed measurements didn't seem to make much sense or be aimed at deriving anything useful.

 

[sphyrch]

A thought experiment can restrict what type of measurements you can do, like alowing only local measurements for the equivalence principle, making it only applicable locally, which is still useful.

But your combination of allowed and not allowed measurements didn't seem to make much sense or be aimed at deriving anything useful.

Ah, understood.. But the last para in my last post (#42) is fine, right? I'm just combining whatever I've seen from the posts in this thread.

 

[sphyrch]

But the definition of "more slowly" that is being used is that of a specific inertial frame, the one in which the clock at the center of the disk is at rest.

On this note, some claims that Einstein makes in his booklet are from the inertial frame's perspective then? For example, he says

so here (i.e. in the spacetime continuum) it is impossible to build up a system (reference-body) from rigid bodies and clocks, which shall be of such a nature that measuring-rods and clocks, arranged rigidly with respect to one another, shall indicate position and time directly.

From what I understand, his point is that in a Euclidean continuum (as assumed in non-relativistic classical physics), a Cartesian system of coordinates represents a real, physical way of measuring times and distances. i.e. you can make a rigid grid of tiny standard measuring rods at right angles to each other and read off distances.

But in a non-Euclidean continuum (let's say the rotating disc), I was thinking that it should still be possible to arrange tiny standard measuring rods on the surface, and then to read off the distance from some chosen origin - right?

It is from the PoV of some inertial frame (i.e. an observer looking at the disc from outside, like $A$) that some of those measuring rods will look longer than others, etc. But to some observer on the sphere or rotating disc itself (like $B$), they can happily measure distances by just counting the number of radial and tangential standard rods it takes to reach from the center to another point.

So the quote above by Einstein should also be from the PoV of an inertial frame?

 

[Dale]

But in a non-Euclidean continuum (let's say the rotating disc), I was thinking that it should still be possible to arrange tiny standard measuring rods on the surface, and then to read off the distance from some chosen origin - right?

This wasn’t well understood in Einstein’s day, a lot of this was explored by Born and Ehrenfest. You can indeed have a rigid set of rods spanning a rotating disk. What you cannot do is:

1) change its angular velocity once it has been assembled. Any angular acceleration will necessarily involve some material strain.

2) synchronize all of the points on the disk using Einstein’s synchronization convention. When you apply the synchronization to nearby points and proceed around the circumference, there will be a jump when you get all the way around.

 

[PeterDonis]

some claims that Einstein makes in his booklet are from the inertial frame's perspective then?

That particular one is.

So the quote above by Einstein should also be from the PoV of an inertial frame?

No. It is stating the impossibility of having an inertial frame that is rotating with the disk.