Speed of an object in a rotating frame

[JVNY]

Is it meaningful to measure the speed of an object in a rotating frame, and if so how do you do it in the following case? Consider a rim of circumference 100 rotating at 0.8c, both measurements with respect to the inertial lab frame. An object is at a point on the rim with a standard clock affixed there (the origin). The object begins to move around the rim in a counter-rotating direction at 0.8c relative to the rim (as measured in the lab frame). The result is that the object is at rest in the lab frame (like a person running on a treadmill). After lab time 125 the rim will complete one rotation in the lab, and the clock and object will be together again at the origin on the rim.

Is it possible to transform this to determine the object's speed in the rim frame? In the rim frame, the rim and the clock are at rest. The object is in motion, moving around the rim. The elapsed time on the affixed clock when the object completes one circuit around the rim and returns to the origin will be 75 because of time dilation (gamma = 1.67).

I can think of at least three possibilities. Unlike a Born rigidly accelerating rod, standard clocks affixed to all points on the rim will run at the same rate, so it seems reasonable to use the clock at the origin to measure the total rim time for the object's circuit.

1. If the rim's own circumference equals its ground circumference times gamma, then the object will have traveled rim distance 167 in rim time 75 (greater than c).

2. If the rim's own circumference equals its ground circumference, the object will have traveled rim distance 100 in rim time 75 (greater than c).

3. If the rim's own circumference equals its ground circumference divided by gamma, then the object will have traveled rim distance 60 in rim time 75 (0.8c).

Thanks for any help.

 

[PeterDonis]

Unlike a Born rigidly accelerating rod, standard clocks affixed to all points on the rim will run at the same rate

The implicit comparison you're making here isn't really valid. In the case of an accelerating rod, the reason clocks at different points on the rod run at different rates is that the direction of their separation is the same as the direction of acceleration. But in the case of the clocks all around the rim, the direction of their separation is perpendicular to their acceleration (which is inward towards the center). That's why they all run at the same rate.

it seems reasonable to use the clock at the origin to measure the total rim time for the object's circuit.

By "clock at the origin", do you mean the clock on the rim that is co-located with the counter-rotating object at the start and end of one revolution? If so, that location is not the usual origin of a rotating frame; the usual origin is at the center of rotation, which is not anywhere on the rim; it's at the center of the circle. A clock at the center of the circle does not go at the same rate as clocks on the rim.

The rest of your post opens a can of worms that I don't think you realize is there. In fact it is highly nontrivial to even define a "rotating frame" and give a physical meaning to the "space" in this frame, in which you are implicitly trying to define the circumference of the rim. See these previous threads for a start:

https://www.physicsforums.com/threads/ehrenfest-paradox.795726/

https://www.physicsforums.com/threads/ehrenfests-paradox.693132/

3. If the rim's own circumference equals its ground circumference divided by gamma, then the object will have traveled rim distance 60 in rim time 75 (0.8c).

This is the closest you can get to a correct answer without getting into the can of worms I mentioned above. It's not fully correct (because of the issues in that can about defining the "space" in the rotating frame in which the rim's circumference is to be measured), but it does give you the right answer for the velocity of the object in the "rim frame" (0.8c).

 

[JVNY]

The question implicates the Ehrenfest Paradox but I have in mind the Sagnac effect. Rizzi and Ruggiero imply that the effect also occurs with co- and counter-rotating subluminal objects unless they maintain the "same" velocity in opposite directions by dividing the distance traveled along the rim by elapsed time on clocks affixed to the rim that were initially synchronized in, e.g., the inertial ground frame. In this case the objects return to their starting point at the same time. See http://arxiv.org/pdf/gr-qc/0305084v4.pdf pages 7-12.

This implies that there is a physical meaning to space on the rim because each of the co- and counter-rotating objects, say two people running in opposite directions around the rim, is able to regulate his effort expended in order to maintain an agreed constant speed with respect to the rim and to travel the same distance (physically meaningful distance measured incrementally on the rim) in the same amount of elapsed time (physically meaningful incrementally elapsed time on clocks affixed at points all around the rim, initially synchronized in the inertial ground frame) and return to the starting point simultaneously with the other.

Discussions of the Ehrenfest Paradox (like the one in the second link above) get tangled up with the question of what happens to a disk as it spins up. I agree with Koks, who says that this is irrelevant and distracting -- we should simply assume that the rim is already in constant rotation and analyze it from there. As Koks writes,

The . . . discussion of how the disk rotates, how it gets accelerated and so on, are completely irrelevant to the analysis of a rotating frame within the context of special relativity . . . In the standard Lorentz transform . . . we don't concern ourselves . . . with how the "primed frame" was ever made to move. It is simply taken as having always been moving, forever . . . We are content to treat a constant-velocity primed frame as having had its state of motion forever, so we should do likewise for the rotating frame and not allow discussion of how the disk was set into motion derail the core issue, which is the analysis of events in a rotating frame.

http://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html​

So let's assume that the rim has always been rotating, that an inertial primed object has always been moving, and that a Born rigid linearly accelerating rod has always been accelerating. We can give clearly defined physical meanings to the proper length of the inertial object and to the proper length of the Born rigid linearly accelerating rod. The ability for moving objects to eliminate the Sagnac effect by maintaining the same speed on the rim seems to require a physically meaningful measure of distance on the rim in order for the two runners to exert exactly the right effort at all times to move the same incremental rim distance in the same incremental elapsed rim time and return to the starting point at the same time.

If (as you suggest) we cannot give a clearly defined physical meaning to the length of the rim in its own frame, that would be highly interesting. It would make the rotating object qualitatively different from inertial objects and Born rigid linearly accelerating objects in SR.

It would also seem to dismiss as irrelevant all of the historical arguments over whether the length of the rim in its own frame is greater than in the ground frame (Einstein's view), the same, or lesser, because there isn't any physically meaningful length in the rim's own frame. For a contrary description, see your statement in the second linked thread referring to "the actual, physical circumference of the disk." But in a view like yours in the same thread, pervect writes that "If you can't agree on a concept of 'physical space', it becomes obvious that it's confusing to talk about it's circumference."

 

[SlowThinker]

Is it meaningful to measure the speed of an object in a rotating frame

The length of the rim measured by rods would be 167.
The radar length in the rotating direction is 375.
The radar length in the counter-rotating direction is 41.67 (unless I'm mistaken... again).

This tells you that something is very wrong with the distances on a rotating disc, and it's not only because the outward acceleration is different at different speeds. Generally, in a looped universe, strange things can happen when you take a trip around the universe, or in your case, around the disc.

These 2 documents discuss the twin paradox in a looped universe. It's not really your question, but it might give you some insights.
http://arxiv.org/abs/gr-qc/0101014
http://arxiv.org/abs/astro-ph/0606559

 

[PeterDonis]

This implies that there is a physical meaning to space on the rim because each of the co- and counter-rotating objects, say two people running in opposite directions around the rim, is able to regulate his effort expended in order to maintain an agreed constant speed with respect to the rim and to travel the same distance (physically meaningful distance measured incrementally on the rim) in the same amount of elapsed time (physically meaningful incrementally elapsed time on clocks affixed at points all around the rim, initially synchronized in the inertial ground frame) and return to the starting point simultaneously with the other.

But note that this "elapsed time on clocks affixed at points all around the rim" requires imposing a clock synchronization for those clocks that is done with respect to an inertial frame--not with respect to the non-inertial frame defined by the rotating rim. (In fact it is not possible to define a clock synchronization with respect to that non-inertial frame, if by that we mean one that matches the natural notion of simultaneity for observers at rest on the rotating rim.) So the "space on the rim" here is being defined with respect to a clock synchronization in one frame (the "inertial ground frame"), yet it is supposed to represent "space" for objects in motion in that frame (the observers on the rim). At the very least, this leaves something out.

Discussions of the Ehrenfest Paradox (like the one in the second link above) get tangled up with the question of what happens to a disk as it spins up. I agree with Koks, who says that this is irrelevant and distracting

I agree that for purposes of this discussion, how the rim got to be rotating is irrelevant; we can treat it as having always been rotating at the same rate.

We can give clearly defined physical meanings to the proper length of the inertial object and to the proper length of the Born rigid linearly accelerating rod.

Agreed.

The ability for moving objects to eliminate the Sagnac effect by maintaining the same speed on the rim seems to require a physically meaningful measure of distance on the rim

And there is one; but it has some properties that are different from the notion of distance in the other two cases (inertial object and linearly accelerating Born rigid rod).

If (as you suggest) we cannot give a clearly defined physical meaning to the length of the rim in its own frame

That's not what I'm suggesting. What I'm suggesting is that this notion of distance, and in fact the notion of "the rim in its own frame" in general, has properties that are different from the other two cases mentioned, and those differences are significant. At the very least, they mean that intuitive reasoning of the sort that has been advanced in this thread cannot be used without more justification.

 

[PeterDonis]

The length of the rim measured by rods would be 167.
The radar length in the rotating direction is 375.
The radar length in the counter-rotating direction is 41.67 (unless I'm mistaken... again).

How are you deriving these results?

 

[SlowThinker]

The length of the rim measured by rods would be 167.
The radar length in the rotating direction is 375.
The radar length in the counter-rotating direction is 41.67 (unless I'm mistaken... again).

How are you deriving these results?

Nothing complicated...
a) The measuring rod is ##\gamma##-times shorter, so we have 100*1.67.
b) If light signal is sent along the rim forward, I think it takes 5 revolutions to arrive back at the source. 5 revolutions take 5*75 rim-seconds.
c) If light signal is sent backwards, it takes 1/(1+0.8)=0.556 revolutions, or 75/1.8=41.67 rim-seconds.

Is this a prime example of misuse of time dilation & length contraction?

 

[PeterDonis]

The measuring rod is ##\gamma##-times shorter

If the rim were moving linearly, this reasoning would be correct. But it isn't; it's moving in a circle. So each rod is moving in a different direction, which means the Lorentz boost required to determine the length contraction of each rod is in a different direction. That means you can't just add all the length contracted lengths together; they're not all length contracted lengths in the same inertial frame.

I'll defer further comment on this until we've looked at the "radar distance" below.

If light signal is sent along the rim forward, I think it takes 5 revolutions to arrive back at the source.

Not quite. I see how you arrived at this--0.8 revolutions of the rim for every 1 revolution of the light, so the "closure speed" is 0.2 revolutions per revolutions, so to speak, and 1/0.2 = 5--but it's wrong. To see why, ask: whose revolutions? The light makes 5 revolutions, yes; but the rim itself only makes 4. (Think about it--how many times will the rim--not the light--pass the clock at rest in the inertial frame, assuming that the clock and rim emitter/receiver are co-located when the light is emitted?) And it's rim-revolutions we are concerned with. So the correct answer is that it takes 4 revolutions--of the rim--for the co-rotating light signal to arrive back at the source.

The general formula is that, for a rim moving at velocity v, it takes v / (1 - v) rim revolutions for a co-rotating light signal to return. It takes 1 / (1 - v) light revolutions.

5 revolutions take 5*75 rim-seconds.

The time dilation factor of 1.67 is correct, but as above, the number of revolutions is 4, so the time is 4*75 = 300 rim-seconds.

If light signal is sent backwards, it takes 1/(1+0.8)=0.556 revolutions, or 75/1.8=41.67 rim-seconds.

Same problem here as above, except that now the factor in the denominator is 1 + v instead of 1 - v. The light makes 1 / (1 + v) = 5/9 = 0.545 revolutions, but the rim makes only v / (1 + v) = 4/9 = 0.455 revolutions, for a time of 75 * 4/9 = 100/3 = 33.33 rim-seconds.

However, neither of these last two calculations are of "radar distance", since that involves a light signal going out and being reflected back. So the correct "radar length" of the rim would be found by combining these last two results--in other words, we imagine a light signal going out in one direction around the rim, being reflected at the point on the rim it started from and going in the other direction back around the rim, and finally returning to its starting point. The total time required for this is 4.455 revolutions, or 333.33 rim-seconds; the "radar distance" around the rim is then half that, or 166.67 [Corrected--was 116.67].

Note that this number still presents a problem if we interpret it "naively": it tells us that the clock at rest in the inertial frame covers a distance of 116.67 around the rim in a time of 75 rim-seconds, for a speed of 166.67/75 = 2.333 c [Corrected--was116.67 / 75 = 1.455 c]. What gives?

The simple answer is that "speed" in a non-inertial frame is not limited to c; it's only limited to c in inertial frames. However, there's more to it than that. Consider this: suppose we want to run the radar distance measurement above, and we first emit the light signal in the counter-rotating direction. We time it so that the signal is emitted at the exact instant that the clock at rest in the inertial frame passes our light emitter (which will also be the reflector and the receiver). We will at once realize that the clock can't be moving faster than light, because the light signal we emitted, which is moving in the same direction the clock is moving, is moving faster than the clock. It must be, because we saw above that it only takes 0.455 revolutions for the counter-rotating light signal to return, whereas it takes exactly one revolution for the clock to return. So if the clock is moving "faster than c" in the "rim frame", then light itself must also be moving "faster than c" by even more in that same frame!

Things get even weirder when we consider the co-rotating direction. Suppose we emit a co-rotating light signal at the same time as the clock is co-located with the emitter/receiver. The signal will take 4 revolutions to return, so the clock, by definition, makes 4 revolutions relative to the emitter/receiver--but the light only makes one. So here it seems like the clock is moving 4 times faster than light!

Obviously what all this is telling us is that our intuitions are not adequate to this case. Not only is "speed" apparently not limited to c, it's not even isotropic. So whatever this "speed" is, it clearly isn't following the same rules as speed in an inertial frame, not just with respect to the "speed limit" of c, but with respect to the whole structure of how it works. The same obviously goes for "distance".

 

[SlowThinker]

However, neither of these last two calculations are of "radar distance", since that involves a light signal going out and being reflected back. So the correct "radar length" of the rim would be found by combining these last two results

In fact, I used radar "length" on purpose. It's something you can't even measure in normal space.
The OP is asking about such a "length" or "half-distance", so I wanted to point out that the main problem here is that the rim is looped.
There are other effects in play, but this one seems to cause the most trouble.

Obviously I made the mistake of using the wrong counter, sorry about that

 

[JVNY]

The total time required for this is 4.455 revolutions, or 333.33 rim-seconds; the "radar distance" around the rim is then half that, or 116.67.

I think that the radar distance is 166.67, as follows.

A co-rotating circuit for a light flash takes ground time 100 / (1 - 0.8) = 500, for rim time of 300 (ground time dilated by gamma 1.67).

The counter-rotating circuit takes ground time 100 / (1 + 0.8) = 55.56, or rim time of 33.33.

The radar distance on the rim is the sum of the out and back trip times divided by 2, so the radar measured circumference of the rim in its own frame is 333.33 / 2 = 166.67.

This is the same as Einstein's conclusion based on the additional number of length-contracted rods that fit around the circumference in the inertial frame.

This suggests that there is a physically meaningful own length of the rim's circumference, and that it is 166.67.

It is important to note, as in the OP, that clocks affixed anywhere on the rim run at the same rate. The radar method does not work on a Born rigidly accelerating rod, and Misner Thorne Wheeler Gravitation points out in exercise 6.5. This is presumably because clocks run at different rates along the rod, so the elapsed time on a single clock does not reflect the sum of the local times of the light flash as it moves across infinitely small segments along the rod.

In the case of clocks affixed to the rotating rim, all of the clocks run at the same rate, so the radar distance measured along the whole circumference using one clock will be the same as the sum of the radar distances of many small segments making up the entire rim. It does not matter whether the clocks at each of those points are synchronized with each other. It is only necessary that each one runs at the same rate and that for each segment you use a single clock to send a signal to the end of the segment and to measure the elapsed time upon receiving the reflected signal back again.

So it seems that the proper physical length of the circumference in the OP is 166.67, but that then makes it hard to conclude that an observer on the rim measures the counter-rotating object to be traveling at 0.8c.

Obviously what all this is telling us is that our intuitions are not adequate to this case. Not only is "speed" apparently not limited to c, it's not even isotropic. So whatever this "speed" is, it clearly isn't following the same rules as speed in an inertial frame, not just with respect to the "speed limit" of c, but with respect to the whole structure of how it works.

Is this consistent with your initial response that the object's speed is 0.8c? That response seems to follow the same rules as the inertial frame. This quotation seems instead to be consistent with concluding in the OP that the object travels at greater than c in the rim frame because it travels rim distance 166.67 in rim time 75 (consistent with your statement just quoted that speed is apparently not limited to c).

The same obviously goes for "distance".

But the radar method shows that the proper length equals the ground length times gamma, which is exactly the same rule that applies to inertial frames. So distance seems to follow the same rules as in an inertial frame, although speed measured by proper distance divided by elapsed time on the clock at the starting and ending point of a circuit can exceed c.

What I'm suggesting is that this notion of distance, and in fact the notion of "the rim in its own frame" in general, has properties that are different from the other two cases mentioned, and those differences are significant.

Is there a good prior thread (albeit involving the full can of worms) that explains those differences?

 

[Dale]

If (as you suggest) we cannot give a clearly defined physical meaning to the length of the rim in its own frame, that would be highly interesting

I think you miss the point. It isn't that you cannot give a clearly defined meaning to the length of the rim in its own frame. It is that you first have to give a clearly defined meaning for the rim's frame.

If you don't define what you mean by the rim's frame then obviously you cannot even ask what its length is in that frame.

 

[JVNY]

If you don't define what you mean by the rim's frame then obviously you cannot even ask what its length is in that frame.

Fair enough. Is there a good prior thread that clearly defines the rim's frame?

 

[Dale]

Fair enough. Is there a good prior thread that clearly defines the rim's frame?

I don't know. I would just define it as:
##t'=t##
##r'=r##
##\theta'=\theta+\omega t##
##z'=z##
With ##(t,r,\theta,z)## being the usual cylindrical coordinates.

That certainly is not the only possible choice, and it has disadvantages, but that is what I would do. You just have to be explicit.

 

[PeterDonis]

I think that the radar distance is 166.67

Oops, you're right, I divided 333.33 in half incorrectly. I've corrected my previous post.

clocks affixed anywhere on the rim run at the same rate.

But they are not naturally synchronized--that is, if we use the momentarily comoving inertial frame of a particular clock on the rim to define a synchronization, all the other clocks on the rim will be out of sync with the chosen clock.

the radar distance measured along the whole circumference using one clock will be the same as the sum of the radar distances of many small segments making up the entire rim. It does not matter whether the clocks at each of those points are synchronized with each other.

Not for that particular measurement, no. But as soon as you try to extend things beyond the rim, i.e., to any radius besides the radius of the rim, it will matter.

Is this consistent with your initial response that the object's speed is 0.8c?

It's "consistent" in the sense that it shows that there is not a unique notion of "speed" that can be applied in the non-inertial frame. If any observer on the rim measures the speed of the clock relative to him as it goes by, he will measure it to be 0.8c. The different answers for "speed" come from looking at other, global observations, not from any local measurement.

distance seems to follow the same rules as in an inertial frame, although speed measured by proper distance divided by elapsed time on the clock at the starting and ending point of a circuit can exceed c.

Which means that distance does not follow all of the same rules as in an inertial frame, since in an inertial frame speed measured by proper distance divided by elapsed time cannot exceed c.

 

[PeterDonis]

I would just define it as...

This is called "Born coordinates" (with one difference, the definition of ##\theta'## should have a minus sign in it), and is discussed briefly here, along with some of the "can of worms" issues:

https://en.wikipedia.org/wiki/Born_coordinates

 

[Dale]

This is called "Born coordinates" (with one difference, the definition of ##\theta'## should have a minus sign in it), and is discussed briefly here, along with some of the "can of worms" issues:

https://en.wikipedia.org/wiki/Born_coordinates

Excellent, it even has the metric calculated. Not having to re-do that is worth switching the sign in my definition ☺

 

[PeterDonis]

it even has the metric calculated.

Yes, but don't expect it to resolve all the questions. Spacelike surfaces of constant time in this chart are Euclidean (that's obvious from the line element), which means they do not represent the "space" seen by a rim observer--more precisely, they do not represent the non-Euclidean quotient space which is the best we can do at describing "the space seen by a rim observer". (That quotient space does not correspond to any spacelike slice cut out of Minkowski spacetime.) The surfaces of constant time are determined according to the clock synchronization of an inertial observer at rest at the center of rotation.

 

[JVNY]

If any observer on the rim measures the speed of the clock relative to him as it goes by, he will measure it to be 0.8c. The different answers for "speed" come from looking at other, global observations, not from any local measurement.

This is clear in the case of a Born rigidly accelerating rod. Say an observer is at the rear of the rod, which has proper length 100. If the observer makes a radar measurement of the distance to the front, half the elapsed time does not equal 100. But we know that the proper length is 100 (we specified that). So the observer is using a global observation. If instead he makes a radar measurement of the length from the rear to the center, and another observer makes a radar measurement of the length from the center to the front, the sum gets closer to 100. If you use an infinite series of infinitely small segments, the sum should be exactly 100. This essentially sums the local times that light takes to travel along the rod. So light takes 100 of local time to travel proper length 100.

But the rotating rim is different. The radar measurement of the entire circumference using light sent from a clock at a single point around to reflect off of itself and return in the opposite direction is exactly the same as the sum of the radar measurements of an infinite set of infinitely small segments making up the circumference. That is the same as in an inertial frame -- the time that a single light flash takes from the rear of an inertial object to the front and back again as measured on a clock at the rear is exactly the same as the sum of the times of a series of radar measurements along the entire length of the object. So there should be no difference between local and global measures of speed in the rotating frame any more than there is in an inertial frame.

 

[PeterDonis]

Say an observer is at the rear of the rod, which has proper length 100.

How are you defining "proper length"? Obviously it isn't "radar distance", so what is it?

Part of the problem with all scenarios of this sort is that people assume that there is some well-defined notion of "proper length" independent of any measurements. There isn't. In order to define "length" physically, you have to define how it is measured. If there are different measurements that yield different results, then it's doubly important to specify which one you mean.

 

[PeterDonis]

The radar measurement of the entire circumference using light sent from a clock at a single point around to reflect off of itself and return in the opposite direction is exactly the same as the sum of the radar measurements of an infinite set of infinitely small segments making up the circumference.

This is true, but this...

So there should be no difference between local and global measures of speed in the rotating frame any more than there is in an inertial frame.

...does not follow from it. There are ways of measuring speed other than distance/time. Suppose, for example, that an observer on the rim observes light emitted by the clock in the inertial frame; he measures the Doppler shift of the light and determines what relative velocity would produce that Doppler shift. He will observe a shift consistent with a relative velocity of 0.8c. That is a local measurement of speed, and it gives different results from the global measurements we discussed.

One key difference between the rim and an inertial frame is that the global measurements involve light traveling around closed paths. In all of the length measurements in inertial frames that you describe, that is not the case: there is no way to send out light in an inertial frame in one direction and have the same light signal come back from the opposite direction.

 

[JVNY]

How are you defining "proper length"? Obviously it isn't "radar distance", so what is it?

Part of the problem with all scenarios of this sort is that people assume that there is some well-defined notion of "proper length" independent of any measurements. There isn't. In order to define "length" physically, you have to define how it is measured. If there are different measurements that yield different results, then it's doubly important to specify which one you mean.

I don't think that I have assumed a notion of proper length other than the basic one that one learns in inertial frames. If you begin with the object inertial and measure its proper length (using rods or a radar measurement), then accelerate it Born rigidly, the object will not deform, and so its proper length will remain the same. You can either calculate the proper length (using a line of simultaneity along its hyperbolic worldlines from rear to front) or use a series of infinitely small radar measurements.

 

[JVNY]

This is true, but this...
...does not follow from it. There are ways of measuring speed other than distance/time. Suppose, for example, that an observer on the rim observes light emitted by the clock in the inertial frame; he measures the Doppler shift of the light and determines what relative velocity would produce that Doppler shift. He will observe a shift consistent with a relative velocity of 0.8c. That is a local measurement of speed, and it gives different results from the global measurements we discussed.

One key difference between the rim and an inertial frame is that the global measurements involve light traveling around closed paths. In all of the length measurements in inertial frames that you describe, that is not the case: there is no way to send out light in an inertial frame in one direction and have the same light signal come back from the opposite direction.

I agree; this closed path creates the difficulties.

 

[PeterDonis]

I don't think that I have assumed a notion of proper length other than the basic one that one learns in inertial frames.

But that notion doesn't apply in non-inertial frames, so you can't just help yourself to it. You need to specify what "proper length" means in a non-inertial frame.

If you begin with the object inertial and measure its proper length (using rods or a radar measurement), then accelerate it Born rigidly, the object will not deform, and so its proper length will remain the same.

Ok, but suppose nobody ever made any measurements when the object was inertial? Or suppose that the object never was inertial at all--suppose it has been Born rigidly accelerated forever? (Yes, this is possible in principle--basically we're saying the worldlines of the object describe a set of curves of constant ##x## in Rindler coordinates.) In other words, if we are handed an object that is undergoing Born rigid acceleration, and have no other information about it, how do we measure its proper length?

 

[PeterDonis]

if we are handed an object that is undergoing Born rigid acceleration, and have no other information about it, how do we measure its proper length?

To make clearer where this is headed, I'm going to assume, JVNY, that your answer to the question posed in the quote above would be along the lines suggested here:

You can either calculate the proper length (using a line of simultaneity along its hyperbolic worldlines from rear to front)

In other words, we pick an event on anyone of the worldlines describing the object; we construct the momentarily comoving inertial frame of the object at that event; and we measure the object's length in that momentarily comoving inertial frame.

The key point is that this works for a rod undergoing linear Born rigid acceleration because there is such an inertial frame: in other words, if we pick an event on one of the worldlines describing the object, and construct the momentarily comoving inertial frame at that event, all of the worldlines describing the object are at rest in that momentarily comoving inertial frame (i.e., all of their tangent vectors are orthogonal to the surface of simultaneity defined by the inertial frame that passes through the chosen event). We can legitimately view the length measured in the inertial frame as the "proper length" of the object because all of the points of the object are at rest in the inertial frame when we make the measurement.

But for the case of the rotating rim, this doesn't work, because there is no such inertial frame: there is no inertial frame in which all of the points on the rim are at rest at any instant whatsoever. So there is no inertial frame we can use to define a "proper length" for the rotating rim the way we can define it for the linearly accelerated rod. Which means that you can't just adopt the basic notion of proper length that one learns in inertial frames for the rotating rim.

 

[JVNY]

To make clearer where this is headed, I'm going to assume, JVNY, that your answer to the question posed in the quote above would be along the lines suggested here:

I think that if I were handed an object that has always been accelerating then I would use the radar method as described earlier to determine its proper length (radar measurement along infinitely small segments of the rod, or along a large number of small segments, which will closely approximate the proper length). That works perfectly well. Of course you might say that it only works because the rod is in an inertial frame; I would respond that the rod is accelerating, so it is not the same as an translationally moving object with a constant speed, and so I would analogize that the rotating rim is accelerating and the rod is accelerating, and the radar method gets a result for each that appears to be meaningful (including 166.67 for the rim in the OP, which agrees with Einstein). But of course analogizing that way may be wrong. And in some ways it is off the point of the OP, so to revert to that:

But for the case of the rotating rim, this doesn't work, because there is no such inertial frame: there is no inertial frame in which all of the points on the rim are at rest at any instant whatsoever. So there is no inertial frame we can use to define a "proper length" for the rotating rim the way we can define it for the linearly accelerated rod. Which means that you can't just adopt the basic notion of proper length that one learns in inertial frames for the rotating rim.

In that case, is Rizzi and Ruggiero's suggestion from the OP unworkable? Can runners go in opposite directions around the rim and regulate their efforts to maintain the same velocity with respect to the rim in a meaningful way and end up back at their starting point simultaneously (note that because that is a single point, simultaneity will not be relative there)? If they can, how? Is this a useful example from which one can explain how the rules that apply to the non-inertial rim work?

 

[PeterDonis]

I were handed an object that has always been accelerating then I would use the radar method as described earlier to determine its proper length (radar measurement along infinitely small segments of the rod, or along a large number of small segments, which will closely approximate the proper length). That works perfectly well.

Yes, for the rod, this would work, and it would give the same answer as the calculation you would do in the momentarily comoving inertial frame (provided you did all the radar measurements simultaneously with respect to that inertial frame).

you might say that it only works because the rod is in an inertial frame; I would respond that the rod is accelerating, so it is not the same as an translationally moving object with a constant speed, and so I would analogize that the rotating rim is accelerating and the rod is accelerating, and the radar method gets a result for each that appears to be meaningful (including 166.67 for the rim in the OP, which agrees with Einstein). But of course analogizing that way may be wrong.

You're right, it is. As I said in my previous post, we can find an inertial frame in which every point of the linearly accelerating rod is simultaneously at rest. That is what justifies us as interpreting the measurements we make as giving us the "proper length" of the rod. But for the rotating rim, there is no such inertial frame at all. In the momentarily comoving inertial frame of any point on the rim, every other point on the rim is moving. So even if you make all those radar measurements, you can't say that they are measurements of the "proper length" of the rim, because the points on the rim weren't all at rest when you made them.

In other words, the fact that those radar measurements happen to add up to a result that is what you would get by applying the length contraction formula to the rods does not, by itself, justify using the term "proper length" for that result. It's not enough for just that one thing to work the same way as in an inertial frame. There are other requirements as well, as above.

is Rizzi and Ruggiero's suggestion from the OP unworkable?

Of course not. Runners can still go around the rim in opposite directions and regulate their running so they meet up again at their starting point. That's just a matter of calculating how they should each regulate their running.

end up back at their starting point simultaneously (note that because that is a single point, simultaneity will not be relative there)?

IMO it's a good idea to avoid using the word "simultaneous" at all in cases like this. I would say: the runners regulate their running so that they both arrive back at their starting point (a designated point on the rotating rim) at the same event. That is unambiguous and avoids any need to qualify what kind of "simultaneity" you mean, and leaves the word "simultaneous" to refer to the case that is frame-dependent.

If they can, how?

The easiest way is for one runner to pick a speed that he wants to maintain, relative to the rim, and then for the other runner to calculate what speed he needs to have, relative to the rim, for both of them to meet up at the starting point at the same event.

Is this a useful example from which one can explain how the rules that apply to the non-inertial rim work?

It can certainly help to get across the fact that those rules are different from the rules in an inertial frame, yes.

 

[JVNY]

The easiest way is for one runner to pick a speed that he wants to maintain, relative to the rim, and then for the other runner to calculate what speed he needs to have, relative to the rim, for both of them to meet up at the starting point at the same event.

OK, but how exactly does each determine his or her own speed? Rizzi and Ruggiero state at page 12 of the article linked above:

if the condition ”equal relative velocity in opposite directions”
is expressed by an analogous relation, in which the local Einstein
synchronization is replaced by a synchronization borrowed from the global
synchronization of the central IF (see Subsection 3.5), no time difference
arises.​

So they must intend the runners to determine speed by distance over time (rather than another means). And they intend the runners to use the same speed (equal relative velocity in opposite directions). So there must be a distance traveled over time elapsed that the two can agree on and follow, which leads them to arrive back at the starting point at the save event. As I reviewed it, the following seems to work. Put 360 clocks on the rim at equal distances apart, synchronize them to clocks in the inertial frame, then run so as to pass a given number of clocks in a given amount of elapsed time on the clocks that you pass. If the runners do this as observed from the inertial frame, and if the clocks are at equal distances apart in the inertial frame, then they will return to the starting point at the same event.

But if there is no shared simultaneity on the rim, then it does not seem meaningful to synchronize based on the central inertial frame (either in my scenario or in Rizzi and Ruggiero's suggestion generally). A runner cannot tell his elapsed time between meter marks or other indications of distance if the clocks on the rim that he checks as passes are not synchronized on the rim. In addition, the runners cannot use their own watches. The counter-rotating runner's watch will tick faster than the clocks on the rim (because he has lower velocity with respect to the inertial frame than they do), and the co-rotating runner's watch will tick slower than the clocks on the rim (because she has greater velocity with respect to the rim then they do). Finally, how do they measure the distance that they travel? If there is no simultaneity around the rim, then putting meter sticks on it and affixing clocks at an equal distance based on the meter sticks should not give a meaningful length, because the two ends of a stick will not be on two points of the rim at the same time. How will they be sure that they have traveled a given distance in order to determine their speed?

You stated in an initial reply that the object's speed from the OP is 0.8c, so there must be a method that you are using. What is it? Does it work the same way for a person counter-rotating as co-rotating, or do they have to use different methods?

 

[PeterDonis]

how exactly does each determine his or her own speed?

They could do it by the Doppler shift method I described in an earlier post: just have stations emitting light at a predetermined frequency all around the rim, and as the runners pass the stations, they measure the frequency of the light they receive and use it to calculate their velocity relative to the stations.

Rizzi and Ruggiero state at page 12 of the article linked above:

if the condition ”equal relative velocity in opposite directions”
is expressed by an analogous relation, in which the local Einstein
synchronization is replaced by a synchronization borrowed from the global
synchronization of the central IF (see Subsection 3.5), no time difference
arises.
So they must intend the runners to determine speed by distance over time (rather than another means).

Not necessarily. They use angular velocities and speeds, but no distances, in the key formulas that define the "relative velocity" in the two cases and show that one definition produces a time difference and the other doesn't.

Let's look at their formulas for the "relative velocity" in the two cases. First, the case where "relative velocity" is determined in the LCIF ("locally comoving inertial frame") of the point on the rim where each runner currently is; this is equivalent to using the Doppler shift method I described above. For this case, we have (equation 22):

$$
\beta_{\pm} = \frac{\beta'_{\pm} + \beta}{1 + \beta'_{\pm} \beta}
$$

where ##\beta'_{\pm}## is the "relative velocity" we are interested in (the speed that would be measured in the LCIF), ##\beta_{\pm}## is the speed in the inertial frame in which the rim is rotating (the frame in which the clock in earlier examples in this thread is at rest)--we'll call this the CIF ("central inertial frame")--and ##\beta## is the speed of the rim itself in the CIF. This is just the relativistic velocity addition formula, so we can invert it as follows:

$$
\beta'_{\pm} = \frac{\beta_{\pm} - \beta}{1 - \beta_{\pm} \beta}
$$

Second, there is the case where "relative velocity" is determined using the clock synchronization of the CIF; for this case, we have (equation 28):

$$
\beta_{\pm} = \beta + \beta^r_{\pm}
$$

where ##\beta^r_{\pm}## is the "relative velocity" determined by this method. This is, of course, easily inverted :

$$
\beta^r_{\pm} = \beta_{\pm} - \beta
$$

From the above we can see that

$$
\beta^r_{\pm} = \beta'_{\pm} \left( 1 - \beta_{\pm} \beta \right)
$$

So we have a number of measurement options here. We know ##\beta## because it's a condition of the problem (it's 0.8 in the scenario specified in the OP). We can measure both ##\beta_{\pm}## and ##\beta'_{\pm}## using Doppler (the latter by the method I've already described; the former by simply placing stations at rest in the CIF all around the rim that emit light signals, and using the Doppler shift of those signals instead of the ones emitted by the stations rotating with the rim). And by the inverted version of equation 28 above, we can use a measurement of ##\beta_{\pm}## to calculate ##\beta^r_{\pm}##. Finally, we have the other formulas as a check that the measurements and calculations are all consistent. So we don't actually need to do any distance/time calculations to measure all the speeds involved.

 

[Dale]

OK, but how exactly does each determine his or her own speed?

I would recommend a speedometer.

Seriously, I think you are reading way to much into this. It seems trivial to me.

 

[PeterDonis]

if there is no shared simultaneity on the rim, then it does not seem meaningful to synchronize based on the central inertial frame

Clock synchronization is a convention; it doesn't have to be "meaningful". The CIF synchronization just happens to be a convenient one for this scenario.

A runner cannot tell his elapsed time between meter marks or other indications of distance if the clocks on the rim that he checks as passes are not synchronized on the rim.

Sure he can; he can carry his own clock with him and use Doppler measurements to determine his velocity relative to the rim and relative to the CIF. So he knows his time dilation factor relative to both, and he can correct the elapsed time on his clock however he needs to.

(Also, there is no way to have clocks "synchronized on the rim" in any sense that would be "meaningful" in your terms anyway. There is no inertial frame in which any two of the clocks on the rim are at rest at the same instant. So there is no way to use any "standard" inertial frame synchronization in this case.)

the runners cannot use their own watches.

Sure they can; they just have to correct the readings appropriately. See above. The GPS system does this continuously; it's not anything esoteric.

how do they measure the distance that they travel?

By speed and elapsed time, corrected (see above). Or by having markers all around the rim, and all around the path taken by the rim in the CIF, as above, and using them as distance markers as well as light sources for Doppler measurements.

How will they be sure that they have traveled a given distance in order to determine their speed?

They don't have to know their distance traveled to measure their speed. See above.

You stated in an initial reply that the object's speed from the OP is 0.8c, so there must be a method that you are using. What is it?

I think I said it in an earlier post: the easiest way would be the Doppler method I described above. That is, as an observer on the rim passes the clock at rest in the CIF, if the clock emits a light signal, the observer will measure that signal to be Doppler shifted by an amount showing a relative speed of 0.8c.

Does it work the same way for a person counter-rotating as co-rotating, or do they have to use different methods?

The Doppler method works for everyone.

 

[JVNY]

Great. Plenty to learn here. Thanks PeterDonis, DaleSpam and SlowThinker.

 

[Laurie K]

Øyvind Grøn provides further insights and much more in his "Space geometry in rotating reference frames: A historical appraisal" paper.

http://areeweb.polito.it/ricerca/relgrav/solciclos/gron_d.pdf

Grøn's own solution (figure 9.) shows the timings required so that light pulses emitted from each of 16 points along the rolling wheel rim arrive at the same point and time as the last point touches the road. To avoid Born rigidity issues the wheel has no thickness (i.e. x,y, t dimensions with z =0) and the observer is in the same plane of rotation as the relativistically rolling wheel. The results can be easily cross checked as the differences between each emission point time and each respective axle x location should reveal the velocity of the wheel as well as the angular velocity of any point on the rim of that wheel.

 

[pervect]

Another good (IMO) and recent paper on measuring distances on a rotating platform is "The Relative Space: Space Measurements on a Rotating Platform" http://arxiv.org/abs/gr-qc/0309020. The author (Ruggerio) first gives an exact definition of what he means by "the space" of a rotating disk as a quotient manfiold. My inexact intuitive summary of this definition is as follows. If one think of painting a dot on a rotating disk, this dot traces out a worldline in 4-d space-time. The "quotient manifold" is just a mapping from the 4-d worldline to a single point in an abstract 3-d space.

The author then proceeds to demonstrate how operationally by the exchange of light signals an observer at one point in this "relative space" can measure their distance to another nearby point in this "relative space".

The algebra is somewhat messy (not terribly messy, it requires solving a quadratic equation), but the concepts are simple and easily tied to the SI defintion of the meter

The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second.

The end result is a non-euclidean geometry (the authors give a specific metric) of the "relative space". References to other papers with similar results via different methods are also given.

 

[JVNY]

Øyvind Grøn provides further insights and much more in his "Space geometry in rotating reference frames: A historical appraisal" paper.

http://areeweb.polito.it/ricerca/relgrav/solciclos/gron_d.pdf

Grøn's own solution (figure 9.) shows the timings required so that light pulses emitted from each of 16 points along the rolling wheel rim arrive at the same point and time as the last point touches the road. To avoid Born rigidity issues the wheel has no thickness (i.e. x,y, t dimensions with z =0) and the observer is in the same plane of rotation as the relativistically rolling wheel. The results can be easily cross checked as the differences between each emission point time and each respective axle x location should reveal the velocity of the wheel as well as the angular velocity of any point on the rim of that wheel.

Thanks very much.

 

[JVNY]

Another good (IMO) and recent paper on measuring distances on a rotating platform is "The Relative Space: Space Measurements on a Rotating Platform" http://arxiv.org/abs/gr-qc/0309020. The author (Ruggerio) first gives an exact definition of what he means by "the space" of a rotating disk as a quotient manfiold. My inexact intuitive summary of this definition is as follows. If one think of painting a dot on a rotating disk, this dot traces out a worldline in 4-d space-time. The "quotient manifold" is just a mapping from the 4-d worldline to a single point in an abstract 3-d space.

The author then proceeds to demonstrate how operationally by the exchange of light signals an observer at one point in this "relative space" can measure their distance to another nearby point in this "relative space".

The algebra is somewhat messy (not terribly messy, it requires solving a quadratic equation), but the concepts are simple and easily tied to the SI defintion of the meter
The end result is a non-euclidean geometry (the authors give a specific metric) of the "relative space". References to other papers with similar results via different methods are also given.

Thanks very much.