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

[sphyrch]

Currently reading Einstein's booklet on SR and GR. At this point SR has been explained but GR hasn't been introduced yet.

The setup is like this: $A$ is present in an inertial reference frame and looking down on a flat rotating disc. $B$ is sitting on the edge of that disc, $C$ is sitting somewhere on the radius connecting the disc's center $O$ and $B$. Suppose $B$, $C$ and $O$ each have an identically-constructed clock. From $A$'s PoV, $C$'s clock will tick at a slower rate since $C$ has some velocity w.r.t. $A$. Then it's written that $O$ will obviously see the same slowdown for $C$'s clock w.r.t. his own clock. This makes sense to me since $O$ isn't actually moving w.r.t. $A$, so effectively $O$ belongs to $A$'s reference frame.

Visually I'm seeing it like this:

$A$ looks down on the 2D disc. But $C$ and $B$ can only see along one dimension, which is the radius $OCB$ (and that radius extending both ways indefinitely). $C$ and $B$ consider themselves to be at rest. My question is: can $C$ also conclude that $B$'s clock is ticking slower than his (only using SR-based arguments, since GR hasn't been introduced yet in the text) ?

If $C$ had visibility beyond the single direction I've highlighted in yellow, then I understand he'd be able to somehow measure $B$'s velocity w.r.t. himself, observe that that velocity is non-zero and hence say that "by SR, $B$'s clock ticks at a slower rate". But what if $C$ can't see beyond the single yellow-highlighted dimension? Is the only way for him to reach that conclusion is to just watch $C$'s clock tick rate (as he can't theoretically deduce it)?

 

[Ibix]

But $C$ and $B$ can only see along one dimension, which is the radius $OCB$ (and that radius extending both ways indefinitely). $C$ and $B$ consider themselves to be at rest. My question is: can $C$ also conclude that $B$'s clock is ticking slower than his (only using SR-based arguments, since GR hasn't been introduced yet in the text) ?

Not if they can only look in the radial direction, because light doesn't travel in straight lines in the rotating frame so they can't see each other.

Generally, it depends what you allow $B$ and $C$ to consider. The simplest way is to calculate the interval along their worldlines during one orbit. Since that's an invariant, if the intervals are different their clock rates must be different. If you want an experimental procedure, have $B$ send $C$ a light pulse and vice versa; confirm that the Doppler factors are reciprocal.

 

[sphyrch]

Not if they can only look in the radial direction, because light doesn't travel in straight lines in the rotating frame, so they can't see each other.

Generally, it depends what you allow $B$ and $C$ to consider. The simplest way is to calculate the interval along their worldlines during one orbit. Since that's an invariant, if the intervals are different their clock rates must be different. If you want an experimental procedure, have $B$ send $C$ a light pulse and vice versa; confirm that the Doppler factors are reciprocal.

If $B$ can send $C$ a light pulse, then $C$ should be able to see $B$ I think? Maybe by the first paragraph you meant that if $B$ is only limited to sending light pulse along the radial direction?

In the last paragraph then I'm assuming that $B$ sends a light pulse in all directions and one of them reaches $C$ - so $C$ realizes that that pulse came from $B$. They can initially agree about the wavelength of pulses $B$ will send, and $C$ can figure out relative velocity of $B$ based on the observed wavelength of $B$'s light pulse. That sounds fine or have I oversimplified it?

 

[Ibix]

If $B$ can send $C$ a light pulse, then $C$ should be able to see $B$ I think? Maybe by the first paragraph you meant that if $B$ is only limited to sending light pulse along the radial direction?

Yes - you said they could only look along the yellow line, which is radial. But light doesn't follow a radial line from $B$ to $C$, so the pulse will come in from an angle to that yellow line - so won't be seen if you're only looking along the line.

In the last paragraph then I'm assuming that $B$ sends a light pulse in all directions and one of them reaches $C$ - so $C$ realizes that that pulse came from $B$. They can initially agree about the wavelength of pulses $B$ will send, and $C$ can figure out relative velocity of $B$ based on the observed wavelength of $B$'s light pulse. That sounds fine or have I oversimplified it?

I think that's right, if you're allowed to assume that $B$ and $C$ are fixed to the disc. I haven't actually done the maths so it's always possible I've overlooked something, but I don't think so.

 

[Orodruin]

B and C are accelerating. The relevant quantity for the frequency shift between them is not instantaneous relative velocity. The easiest way of handling this problem is to detive the form of the metric in the rotating frame and draw conclusions based on that.

 

[A.T.]

You don't need GR to handle accelerating and rotating frames of reference, and C & B can measure their proper acceleration locally.

 

[Dale]

Currently reading Einstein's booklet on SR and GR. At this point SR has been explained but GR hasn't been introduced yet.

The setup is like this: $A$ is present in an inertial reference frame and looking down on a flat rotating disc. $B$ is sitting on the edge of that disc, $C$ is sitting somewhere on the radius connecting the disc's center $O$ and $B$.

So, the first thing to do is to set up the coordinate system for the various reference frames. When observers are inertial, this is easy and involves relatively little choice. But in this case $O$, $C$, and $B$ are all non-inertial, so their reference frame involves a bit of choice. So let's say that $O$, $C$, and $B$ are using rotating cylindrical coordinates. In those the metric is $$ds^2=(r^2 \omega^2-c^2) dt^2 + dr^2 + r^2 \ d\phi^2 + dz^2 + 2\omega r^2 \ dt \ d\omega$$

$A$ looks down on the 2D disc. But $C$ and $B$ can only see along one dimension, which is the radius $OCB$ (and that radius extending both ways indefinitely).

This is not helpful, as mentioned above. I won't include this restriction.

$C$ and $B$ consider themselves to be at rest. My question is: can $C$ also conclude that $B$'s clock is ticking slower than his (only using SR-based arguments, since GR hasn't been introduced yet in the text) ?

Yes, since both $C$ and $B$ are at rest in the rotating frame they have $dr/dt=d\phi/dt=dz/dt=0$. So with $ds^2 = -c^2 d\tau^2$ we have the time dilation is $$\frac{1}{\gamma}=\frac{d\tau}{dt}=\sqrt{-\frac{ds^2}{c^2 dt^2}}=\sqrt{1-\frac{r^2 \omega^2}{c^2}} $$ So rotating frame observers agree that clocks at rest at larger $r$ tick more slowly compared to the coordinate time.

I understand he'd be able to somehow measure $B$'s velocity w.r.t. himself, observe that that velocity is non-zero and hence say that "by SR, $B$'s clock ticks at a slower rate".

This is incorrect. In their reference frame they are both at rest. Their relative velocity in their frame is 0. It is only in $A$'s frame that they have some relative velocity.

Note, in the above analysis, no GR was used. However, the usual pseudo-Lorentzian geometry was used. GR does not own tensors and metrics. GR would be required only if spacetime were curved, but using arbitrary coordinates in flat spacetime is still SR.

 

[sphyrch]

Yes, since both $C$ and $B$ are at rest in the rotating frame they have $dr/dt=d\phi/dt=dz/dt=0$. So with $ds^2 = -c^2 d\tau^2$ we have the time dilation is $$\frac{1}{\gamma}=\frac{d\tau}{dt}=\sqrt{-\frac{ds^2}{c^2 dt^2}}=\sqrt{1-\frac{r^2 \omega^2}{c^2}} $$ So rotating frame observers agree that clocks at larger $r$ tick more slowly compared to the coordinate time.

One probably naive question - both $B$ and $C$ have no idea about any angular velocity, since from their perspective they're at rest on some flat surface. As you mentioned, the form of the metric in cylindrical coordinates allows them to deduce relative time dilation.

But now consider another flat disc with the same setup - except that it's not rotating w.r.t. $A$ and its observers are called $O',B',C'$. And obviously $B'$ and $C'$ should not be able to deduce any relative time dilation b/w them. Their metric should have a different form in their cylindrical coordinates compared to what $B$ and $C$ found, otherwise they'd come to the same conclusion as $B$ and $C$.

But then all of $B,C,B',C'$ think themselves to be at rest w.r.t. their discs. Does the form of metric that you described (for $B$ and $C$) depend on $B$ and $C$'s a priori knowledge that their disc is rotating with angular velocity $\omega$?

 

[PeroK]

One probably naive question - both $B$ and $C$ have no idea about any angular velocity, since from their perspective they're at rest on some flat surface.

If B and C are accelerating, then that is detectable and they cannot assume they are moving inertially.

 

[sphyrch]

If B and C are accelerating, then that is detectable and they cannot assume they are moving inertially.

Right. For sure they'll detect "artificial gravity" which is proportional to their distance from $O$. Does SR allow one to formulate a metric (the same form that Dale mentioned in his post) based on this info (presence of an artificial gravitational field)?

 

[PeroK]

Right. For sure they'll detect "artificial gravity" which is proportional to their distance from $O$.

They will detect the real force that causes them to move in a circle.

Does SR allow one to formulate a metric (the same form that Dale mentioned in his post) based on this info (presence of an artificial gravitational field)?

For uniform circular motion, it's possible to describe things using a pseudo gravitational field. Alternatively, you can describe uniform circular motion as the limiting case of motion along an polygon with inertial motion along each side.

There is a thread on here somewhere where I did that.

The key point is an object that is moving non-inertially cannot pretend it's at rest in an inertial reference frame.

 

[A.T.]

both B and C have no idea about any angular velocity

Why? They can measure that locally:
https://en.m.wikipedia.org/wiki/Ring_laser_gyroscope

 

[sphyrch]

For uniform circular motion, it's possible to describe things using a pseudo gravitational field. Alternatively, you can describe uniform circular motion as the limiting case of motion along an polygon with inertial motion along each side.

There is a thread on here somewhere where I did that.

The key point is an object that is moving non-inertially cannot pretend it's at rest in an inertial reference frame.

The last line is clear. $B$ doesn't know about the circular motion in the scenario described in the text I'm reading. But definitely (as hinted in the first line in your quote above) a pseudo gravitational field is present in $B$ and $C$'s PoV. Given this, is it possible to derive the relative time dilation (using SR only)? Dale's post is very useful but I'm just looking for confirmation whether that form of metric can be derived in the scenario of pseudo gravitational field too (without a priori assumption of rotation on $B$ or $C$'s part)

 

[PeroK]

The last line is clear. $B$ doesn't know about the circular motion in the scenario described in the text I'm reading. But definitely (as hinted in the first line in your quote above) a pseudo gravitational field is present in $B$ and $C$'s PoV. Given this, is it possible to derive the relative time dilation (using SR only)? Dale's post is very useful but I'm just looking for confirmation whether that form of metric can be derived in the scenario of pseudo gravitational field too (without a priori assumption of rotation on $B$ or $C$'s part)

SR does not include a theory of gravity, although gravitational time dilation can be inferred from the equivalence principle.

That said, I've perhaps missed the point of this thread.

 

[sphyrch]

SR does not include a theory of gravity, although gravitational time dilation can be inferred from the equivalence principle.

That said, I've perhaps missed the point of this thread.

Sorry maybe I should give more context. In the text I'm reading, the point the author is trying to make is that it's possible there's no universal concept of time even on a single reference body. The example he's using for a reference body is a rotating disc. He argues that a clock at the center ($O$) runs at a different rate than one on the circumference ($B$). His argument is pretty clear.

But I wanted to consider the same argument but for a slightly different case. Instead of showing a difference between ticking rate of $O$'s clock and $B$'s clock, I wanted to see how we can show from first principles (i.e. just SR) that even $C$ and $B$'s clocks run at different rates (w.r.t. each of their perspectives).

There's a statement in the text I'm reading - " But the observer on the disc may regard his disc as a reference-body which is "at rest" ". So that's why I'm emphasizing that $B$ and $C$ insist that the disc they're on isn't rotating. But they both definitely are aware of the pseudo-gravitational field ("artificial gravity"). So with this scenario in mind, I was wondering what experiment they can perform to conclude their clocks are running at different rates.

Maybe my assumption is wrong that they can make such a conclusion solely based on SR-related arguments. You can correct me if that's the case.

 

[PeroK]

Sorry maybe I should give more context. In the text I'm reading, the point the author is trying to make is that it's possible there's no universal concept of time even on a single reference body. The example he's using for a reference body is a rotating disc. He argues that a clock at the center ($O$) runs at a different rate than one on the circumference ($B$). His argument is pretty clear.

But I wanted to consider the same argument but for a slightly different case. Instead of showing a difference between ticking rate of $O$'s clock and $B$'s clock, I wanted to see how we can show from first principles (i.e. just SR) that even $C$ and $B$'s clocks run at different rates (w.r.t. each of their perspectives).

Okay.

There's a statement in the text I'm reading - " But the observer on the disc may regard his disc as a reference-body which is "at rest" ".

Okay, but he can't pretend the disc does not have internal forces. He can't regard the disc as inertial.

So that's why I'm emphasizing that $B$ and $C$ insist that the disc they're on isn't rotating.

They can only do that by using a rotating reference frame - in which Newton's first law does not apply. That can be tested.

But they both definitely are aware of the pseudo-gravitational field ("artificial gravity").

That's just different way of saying the reference frame is non inertial.

with this scenario in mind, I was wondering what experiment they can perform to conclude their clocks are running at different rates.

They can measure them against a set of inertial clocks, all at rest relative to each other.

Maybe my assumption is wrong that they can make such a conclusion solely based on SR-related arguments. You can correct me if that's the case.

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.

It's a common misconception that we are somehow compelled to use our rest frame to analyse a problem. This is not true. It is perfectly possible, for example, to describe the heliocentric solar system without ever leaving the Earth.

 

[A.T.]

There's a statement in the text I'm reading - " But the observer on the disc may regard his disc as a reference-body which is "at rest" ". So that's why I'm emphasizing that $B$ and $C$ insist that the disc they're on isn't rotating.

You are confusing frame dependent coordinate rotation with frame invariant proper rotation that a ring laser gyroscope will measure. Considering themselves at rest here means adopting a rotating frame of reference, not insisting that they don't have proper rotation.

 

[sphyrch]

@PeroK : First off thanks for the replies and clarifying a few things for me.

Okay, but he can't pretend the disc does not have internal forces. He can't regard the disc as inertial.

Yep that's very much clear. Just want to state that when I say that they consider the disc not to be rotating, I don't mean that they consider the disc inertial. So once and for all I'll say that I'm not under the misconception that $B$ and $C$ assume their reference body to be inertial.

They can measure them against a set of inertial clocks, all at rest relative to each other.

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

Could you elaborate on this please, if possible?

It's a common misconception that we are somehow compelled to use our rest frame to analyse a problem. This is not true. It is perfectly possible, for example, to describe the heliocentric solar system without ever leaving the Earth.

Understood. So my initial constraint of them just being able to make measurements in 1D along the radius was clearly unreasonable. But whatever experiment they can do to figure out different clock tick rates, can they possibly do it in the disc's plane, or would they necessarily need to consider the whole 3D space for their experiment?

 

[sphyrch]

You are confusing frame dependent coordinate rotation with frame invariant proper rotation that a ring laser gyroscope will measure. Considering themselves at rest here means adopting a rotating frame of reference, not insisting that they don't have proper rotation.

Got it, sorry for not responding to your earlier post. I'll study that wikipedia page in more detail asap

 

[Dale]

both B and C have no idea about any angular velocity, since from their perspective they're at rest on some flat surface

That is simply not true. Both $B$ and $C$ can measure $\omega$. They can do so using standard accelerometers of the type most people carry in their phones. Accelerometers can measure both proper acceleration and rotation. Rotation of this kind is not relative, it is absolute.

 

[Nugatory]

I wanted to see how we can show from first principles (i.e. just SR) that even C and B's clocks run at different rates (w.r.t. each of their perspectives).

We can't. That phrase "run at different rates" is only meaningful if we've chosen a simultaneity convention, and first principles gives us no basis for choosing one simultaneity convention over another.

(This would be a good chance to go back up to the much simpler case of two observers moving inertially with constant velocity relative to one another. The time dilation formula tells us, paradoxically, that both clocks are slower than the other. The resolution of the apparent paradox depends on the relativity of simultaneity, which is another way of saying that we're aways choosing a simultaneity convention even when we aren't aware of it).

However there are calculations that we can do from first principles. Say that both B and C are broadcasting, every second, an image of their clock (this is equivalent to measuring received redshift or watching the clock with a telescope). First principles do allow us to calculate the time on our clock between observed reception of consecutive signals that were emitted one second apart according to the other clock. Now we can use the most natural simultaneity convention along with these first principle calculations to say something about the relative clock rates.

 

[A.T.]

That phrase "run at different rates" is only meaningful if we've chosen a simultaneity convention, and first principles gives us no basis for choosing one simultaneity convention over another.

While this is true for the instantanous clock rates, one could change the question to compare accumulated proper times over a very long time period. Are there valid coordinates that would have B age more when coordinate time goes time infinity?

 

[Nugatory]

While this is true for the instantanous clock rates, one could change the question to compare accumulated proper times over a very long time period. Are there valid coordinates that would have B age more when coordinate time goes time infinity?

I can’t imagine one that would cover all of spacetime, but I think that any quantitative calculation (“right now, how much older is B?”) is still going to require a simultaneity assumption.

 

[PeterDonis]

I can’t imagine one that would cover all of spacetime

The rotating coordinates that @Dale describes in post #7 use the simultaneity convention of the inertial frame in which the center of rotation is at rest. Of course this is not the "natural" simultaneity convention of any of the observers who are rotating with the frame. But it works fine.

That simultaneity convention does cover all of spacetime. What can only cover a limited region is the timelike congruence of rotating observers. At some radius $r$ a rotating worldline (i.e., a worldline at rest in @Dale's rotating coordinates) is no longer timelike but null, and outside that $r$ it is spacelike.

 

[Nugatory]

That simultaneity convention does cover all of spacetime.

Indeed it does, but does it have the other property that @A.T. was looking for, namely that B ages more as coordinate time goes to infinity?

 

[Sagittarius A-Star]

Indeed it does, but does it have the other property that @A.T. was looking for, namely that B ages more as coordinate time goes to infinity?

W. Rindler wrote it this way:

If originally two standard clocks are adjacent and synchronized, and then one is taken to a place of lower potential and left there for a time, and finally brought back, that clock will clearly read slow by a factor D relative to the one that remained fixed - except for the error introduced by the two journeys. But whatever happens during the motions is independent of the total dilation at the lower potential, and can thus be dwarfed by it. Hence a “twin” at a lower
potential stays younger than his twin at a higher potential.

Source: W. Rindler "Essential Relativity" 2nd edition, chapter 7.5 The Gravitational Doppler Effect, page 118

 

[Orodruin]

Indeed it does, but does it have the other property that @A.T. was looking for, namely that B ages more as coordinate time goes to infinity?

Since it is just the coordinate time of the initial inertial frame and C moves slower than B in that frame … C will age more as time goes to infinity.

In fact, you can easily see that no simultaneity convention exists where B ages more as the world lines extend to infinity as simultaneities must be spacelike by definition. For the ”start” event of the clocks, the limiting case of how early you can start the clock of B is when it exits the past light cone of the start event of C. For the ”stop” event, the limiting case of how late you can stop the clock of B is when it enters the future light cone of the stop event of C. Because of symmetry, those will always correspond to giving the B clock a fixed amount of time regardless of how far you extend the worlld lines. In particular, if you extend them enough, C will alwats accumulate more proper time than B.

 

[sphyrch]

We can't. That phrase "run at different rates" is only meaningful if we've chosen a simultaneity convention, and first principles gives us no basis for choosing one simultaneity convention over another.

I gave it some thought. Can I say that since $B$ is at more distance from the disc center (say $R$) than $C$ is (say $r$), $B$ will have more tangential velocity at any given moment ($\omega R>\omega r$). 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). And because of this, having knowledge of SR, they should at once know that the other's clock is ticking slower compared to theirs.

Would this not be an argument purely based on SR that allows both $B$ and $C$ to conclude the other's clock is ticking slower?

 

[Sagittarius A-Star]

Would this not be an argument purely based on SR that allows both $B$ and $C$ to conclude the other's clock is ticking slower?

No. Both are not at rest in an inertial reference frame.

 

[Sagittarius A-Star]

Currently reading Einstein's booklet on SR and GR. At this point SR has been explained but GR hasn't been introduced yet.

I recommend, that you read in this book the Appendix III, "(c) Displacement of Spectral Lines Towards the Red":

In the first place, we see from this expression that two clocks of identical construction will go at different rates when situated at different distances from the centre of the disc. This result is also valid from the standpoint of an observer who is rotating with the disc.
Now, as judged from the disc, the latter is in a gravitational field of potential $\phi$.

Source:
https://en.wikisource.org/wiki/Rela...isplacement_of_Spectral_Lines_Towards_the_Red

What Einstein calls "gravitational field" in the rotating frame is today called "pseudo-gravitational field".

 

[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.