Actual earth measurement contradicts measurement predicted by special relativity

[Dale]

The fundamental question is whether there is a difference between perception (measurement) and reality. In the case of length contraction, does a measured reduction in length imply an actual reduction in length?

I think you have only swapped the ambiguous word "physically" with the ambiguous words "reality" and "actual". Again, what sort of experiment could we perform to determine if something is "real", "physical", or "actual"?

It is easy enough to say that we don't know how to define reality apart from measurement. I appreciate that point of view, but cannot be satisfied with it philosophically.

It is not the goal of science to satisfy people's philosophical pre-conceptions about "reality" (whatever that means). The goal of science is to accurately predict the results of experiments. If you cannot be satisfied with that then you want something other than science.

But without more data, I don't see that we can rule out the possibility of other explanations for our measurements.

I agree, it is always possible to come up with an infinite number of explanations for any given set of measurements simply by adding things that cannot yet be measured.

 

[ghwellsjr]

There are plenty of people on this forum that think the idea of clocks running at different rates just because there is a relative speed between them is an illusion, an artifact, a measurement error, or otherwise not real. What do you tell them?

I tell them that it is a measurable, frame-dependent effect. I try to let them decide how to categorize such things as illusion, or real, etc. since I don't feel comfortable with any such terms.

With two identical clocks in relative motion, what effect is measurable but also frame-dependent?

 

[GregAshmore]

It is not the goal of science to satisfy people's philosophical pre-conceptions about "reality" (whatever that means). The goal of science is to accurately predict the results of experiments. If you cannot be satisfied with that then you want something other than science.

I agree with the sentence in bold. And yet, it is difficult to imagine that one's view of reality can be entirely separated from the theories and experiments of science. Certainly, Einstein's approach was that science and reality are closely intertwined, perhaps even synonymous. For my own part, I would be satisfied with a science that not only accurately predicts the outcome of experiments (in something more than a statistical chart) but also gives me confidence that I understand physical reality. I do not seek to have my preconceived notions of reality satisfied; I would simply like to have confidence that I understand what reality is. If you are not so motivated, that's fine.

I agree, it is always possible to come up with an infinite number of explanations for any given set of measurements simply by adding things that cannot yet be measured.

Actually, I have not proposed any non-standard values for the physical characteristics which we have not yet measured. What I have suggested is that when and if we do make those measurements, the results may surprise us. If that happens--and it wouldn't be the first time such a thing has happened--we will have to revise our theories. It seems to me that it would be wise to leave a little wiggle room when we speak of such things as round trips to Canopus in a single lifetime.

Just a thought, as an example. What confidence do we have that large bodies will maintain their integrity as they approach light speed?

 

[ghwellsjr]

I'd say that for some fact to be frame-independent it must be true in all coordinate systems, not just inertial ones. You could construct a non-inertial coordinate system where two clocks have different coordinate velocities but are both ticking at the same rate relative to coordinate time.

I was thinking of the situation where one clock never accelerates and a second clock accelerates in an arbitrary manner and I claim the second clock is experiencing a change in its tick rate while it is accelerating. Are you suggesting that because we can use the non-inertial frame in which the second clock is always at rest, that it therefore experiences no change in its tick rate? If so, what do you then say about the first clock's tick rate? If not, then are you saying the second clock's tick rate is changing?

 

[ghwellsjr]

Just a thought, as an example. What confidence do we have that large bodies will maintain their integrity as they approach light speed?

They could if nothing got in their way but unfortunately, the universe is full of particles which will make them feel like they are being subjected to machine gun fire and which will destroy them. Sorry.

 

[JesseM]

I was thinking of the situation where one clock never accelerates and a second clock accelerates in an arbitrary manner and I claim the second clock is experiencing a change in its tick rate while it is accelerating. Are you suggesting that because we can use the non-inertial frame in which the second clock is always at rest

What do you mean "the" non-inertial frame in which the second clock is at rest? There are an infinite number of them! Take a spacetime diagram drawn from the perspective of an inertial frame, draw on some arbitrary curves to treat as surfaces of simultaneity and some other arbitrary curves crisscrossing the first set to treat as curves of constant position coordinate, resulting in some sort of distorted grid, and that can be the basis of a valid non-inertial coordinate system (see for example the final animated diagram near the bottom of this page). So you can always pick one where the surfaces of simultaneity are defined so that if a surface passes through the worldline of clock #1 when it reads a time of T, then that same surface also passes through the worldline of clock #2 when it reads a time of T. And you can also draw some curves of constant position coordinate that coincide with the worldlines of the two clocks for any section of the worldlines where they don't cross. So, the clocks will each be at rest in this coordinate in those sections of their worldlines, and they will each show the same reading at any time-coordinate in those sections.

 

[Dale]

For my own part, I would be satisfied with a science that not only accurately predicts the outcome of experiments (in something more than a statistical chart) but also gives me confidence that I understand physical reality.

Then you don't want science, you want religion or philosophy. Science has no means to interrogate "physical reality" other than through the outcome of experiments.

What I have suggested is that when and if we do make those measurements, the results may surprise us. If that happens--and it wouldn't be the first time such a thing has happened--we will have to revise our theories.

Certainly, and when such data is acquired then we will revise our theories. No big deal. As you say, it has happened several times before. That is what science does.

In the meantime we cannot scientifically choose one particular theory with untested "wiggle room" over another. As mentioned earlier, there are an infinite number of such wiggle parameters we could add, and all have the same experimental support. That is why we apply Occham's razor and use the theory with the least wiggle room.

 

[Dale]

With two identical clocks in relative motion, what effect is measurable but also frame-dependent?

Well, to have a pair of frames you need not just two identical clocks in relative motion, but two systems of synchronized clocks in relative motion. Then the rate of the other clocks is frame-dependent and measurable.

 

[ghwellsjr]

So you can always pick one where the surfaces of simultaneity are defined so that if a surface passes through the worldline of clock #1 when it reads a time of T, then that same surface also passes through the worldline of clock #2 when it reads a time of T. And you can also draw some curves of constant position coordinate that coincide with the worldlines of the two clocks for any section of the worldlines where they don't cross. So, the clocks will each be at rest in this coordinate in those sections of their worldlines, and they will each show the same reading at any time-coordinate in those sections.

If you are saying that we must have several different such non-inertial frames, separated by when their worldlines cross, then what are saying about the clocks in terms of their aging rates and times when we have to switch from one frame to the next?

 

[ghwellsjr]

Well, to have a pair of frames you need not just two identical clocks in relative motion, but two systems of synchronized clocks in relative motion. Then the rate of the other clocks is frame-dependent and measurable.

But I thought all the synchronized clocks in each frame were defined to have the same time on them because we cannot measure the one-way speed of light.

 

[JesseM]

If you are saying that we must have several different such non-inertial frames, separated by when their worldlines cross,

I don't quite understand what you mean by "separated by when their worldlines cross". The point about worldline crossing is that obviously you can't have a non-inertial frame that says they remain a constant nonzero distance apart even at the point where their worldlines cross (since all frames, inertial and non-inertial, agree about local events like crossings of worldlines), but beyond that, if you pick any arbitrary event A after the crossing on the first worldline, and another arbitrary event B after the crossing on the second worldline, you can always find a non-inertial frame where both clocks are at rest and ticking at the same rate relative to coordinate time starting at the time of A and B.

then what are saying about the clocks in terms of their aging rates and times when we have to switch from one frame to the next?

Not sure I understand this either, why would we "have to" switch from one frame to another? We can derive the values of all frame-independent quantities (like readings on any measuring device) using a single frame. Obviously we can re-analyze the same situation in multiple frames if we wish, the different frames will make the same predictions about frame-independent quantities but may (or may not) give different answers about frame-dependent quantities like the tick rates of clocks relative to coordinate time.

 

[Dale]

But I thought all the synchronized clocks in each frame were defined to have the same time on them because we cannot measure the one-way speed of light.

Yes, and the result of using this definition (the Einstein synchronization convention) is frame variant. This is known as the relativity of simultaneity. I assume that we have no disagreement about that and that you understand the topic well.

 

[Passionflower]

I was thinking of the situation where one clock never accelerates and a second clock accelerates in an arbitrary manner and I claim the second clock is experiencing a change in its tick rate while it is accelerating.

That is correct.

If two clocks A and B at at relative rest at event X and B accelerates away while A remains inertial then B runs slower wrt A.

 

[JesseM]

That is correct.

If two clocks A and B at at relative rest at event X and B accelerates away while A remains inertial then B runs slower wrt A.

Only if by "runs slower wrt A" you mean "runs slower in the inertial frame where A is at rest". Since we're discussing frame-dependent vs. frame-independent facts on this thread, it should be pointed out that even if we restrict the discussion to inertial frames, there will be frames where B is running faster than A after B accelerates.

 

[ghwellsjr]

Only if by "runs slower wrt A" you mean "runs slower in the inertial frame where A is at rest". Since we're discussing frame-dependent vs. frame-independent facts on this thread, it should be pointed out that even if we restrict the discussion to inertial frames, there will be frames where B is running faster than A after B accelerates.

What do you mean "the" inertial frame where A is at rest? There are an infinite number of them!

Sorry--I just couldn't help myself.

 

[Passionflower]

Only if by "runs slower wrt A" you mean "runs slower in the inertial frame where A is at rest". Since we're discussing frame-dependent vs. frame-independent facts on this thread, it should be pointed out that even if we restrict the discussion to inertial frames, there will be frames where B is running faster than A after B accelerates.

I am only talking about clock A and B. I did not mention clock C, D, E and F right?

 

[GregAshmore]

They could if nothing got in their way but unfortunately, the universe is full of particles which will make them feel like they are being subjected to machine gun fire and which will destroy them. Sorry.

And yet T-W assure their students that a 20-year round trip to Canopus is a no-brainer.

Actually, I don't know of any experimental evidence to support your assertion that they could if nothing got in their way.

 

[JesseM]

What do you mean "the" inertial frame where A is at rest? There are an infinite number of them!

Sorry--I just couldn't help myself.

OK, but it's a normal convention to talk about "the" inertial frame where a given inertial object is at rest, unlike with non-inertial frames. I suppose the reason is that although you have a choice of where to place the origin and how to orient the spatial axes, this will make no difference to most of the frame-dependent quantities we're interested in like speed and rate of time dilation.

 

[JesseM]

I am only talking about clock A and B. I did not mention clock C, D, E and F right?

I am talking about coordinate systems, not physical objects. You are free to use any coordinate system you like to analyze a problem, it makes no difference whether or not any of the physical objects (like clocks) are at rest in the coordinate system you have chosen. You are no more limited to coordinate systems where one of the objects is at rest than you are limited to coordinate systems where the spatial origin coincides with one of the objects.

 

[GregAshmore]

Then you don't want science, you want religion or philosophy. Science has no means to interrogate "physical reality" other than through the outcome of experiments.

Certainly, and when such data is acquired then we will revise our theories. No big deal. As you say, it has happened several times before. That is what science does.

In the meantime we cannot scientifically choose one particular theory with untested "wiggle room" over another. As mentioned earlier, there are an infinite number of such wiggle parameters we could add, and all have the same experimental support. That is why we apply Occham's razor and use the theory with the least wiggle room.

But we needn't pretend that we know more than we do--and a lot of that goes on. A good example is the assertion by T-W there can be no question about the feasibility in principle of a round trip to Canopus in 20 years.

Not only do we assert the truth of things we have never measured, we assert the non-existence of things based on measurements which we don't understand. Here I have in mind the denial of cause for quantum events, even though we are not able to directly measure the phenomena about which we make such a sweeping assertion.

 

[GregAshmore]

Then why not just say "frame-independent" instead of words like "illusion", "real", and "physical"? It is not only what you mean, but it is unambiguous.

One good reason not to restrict oneself to "frame-independent", to the exclusion of "physical", is that a reference frame is not real.

The reference frame is an abstraction of reality which is helpful in analyzing reality. However, the characteristic of the reference frame which makes it so useful in analysis of reality also makes it completely unsuitable as a proxy for reality. The process of abstraction which yields the reference frame excludes certain aspects of the real world, either by simplification of known features, or exclusion of unknown features.

 

[Dale]

Actually, I don't know of any experimental evidence to support your assertion that they could if nothing got in their way.

Check out the sticky on experimental evidence for special relativity. There is a section on experimental tests of the twin paradox. The tests on muons in a storage ring are particularly relevant.

 

[GregAshmore]

Check out the sticky on experimental evidence for special relativity. There is a section on experimental tests of the twin paradox. The tests on muons in a storage ring are particularly relevant.

A trip in a spaceship will involve very large structures. I'm wondering what happens when such a large structure--as opposed to a single particle--approaches light speed.

 

[Passionflower]

I am talking about coordinate systems, not physical objects.

Yes and I am talking about physical objects.

 

[JesseM]

Yes and I am talking about physical objects.

If you are not talking about coordinate systems, then you cannot talk about the rate that one clock is ticking relative to another, since that is an inherently coordinate-dependent notion. There is certainly no coordinate-independent sense in which clock B "runs slower wrt A".

 

[Dale]

A trip in a spaceship will involve very large structures. I'm wondering what happens when such a large structure--as opposed to a single particle--approaches light speed.

Relativistic effects apply to large structures too.

 

[Dale]

One good reason not to restrict oneself to "frame-independent", to the exclusion of "physical", is that a reference frame is not real.

So what? The point is that that "frame-independent" is well defined and clearly indicates his meaning whereas "physical" does not.

 

[Doc Al]

Just a thought, as an example. What confidence do we have that large bodies will maintain their integrity as they approach light speed?

Just a thought: You and the planet you are riding on are undoubtedly moving at near light speed with respect to something in the universe. How does it feel?

 

[ghwellsjr]

With two identical clocks in relative motion, what effect is measurable but also frame-dependent?

Well, to have a pair of frames you need not just two identical clocks in relative motion, but two systems of synchronized clocks in relative motion. Then the rate of the other clocks is frame-dependent and measurable.

But I thought all the synchronized clocks in each frame were defined to have the same time on them because we cannot measure the one-way speed of light.

Yes, and the result of using this definition (the Einstein synchronization convention) is frame variant. This is known as the relativity of simultaneity. I assume that we have no disagreement about that and that you understand the topic well.

If we start with two observers with identical synchronized clocks at the same location and stationary with respect to each other and one of them quickly accelerates away and reaches a final speed, isn't it true that no matter which inertial frames we analyze the situation in (after enough time has elapsed for the acceleration effects pass), all frames will determine that the measurements made by each observer of the other one's clock rate will be the same as each other and that these measurements will be frame-independent?

And isn't it true that from the measured clock rate, they can each determine the same correct relative speed between them independent of frame?

And isn't it true that each of them can determine the same time-dilation for the other clock independent of frame?

And if all these things are true, then these measurements are not just frame-invariant but also frame independent in a broader sense that they don't even require any frame to be specified or considered?

And if they don't require any frame to be specified or considered then these measurements are not subject to relativity of simultaneity (which I understand and have no disagreement with you about), true?

 

[Dale]

no matter which inertial frames we analyze the situation in ... all frames will determine that the measurements made by each observer of the other one's clock rate will be the same as each other

Sorry, I don't quite follow this.

Also, what are you trying to accomplish here? Are you trying to puzzle out a good definition of "physical" or "real" or are you trying to convince me that I should use those words even without a good definition?

 

[JesseM]

If we start with two observers with identical synchronized clocks at the same location and stationary with respect to each other and one of them quickly accelerates away and reaches a final speed, isn't it true that no matter which inertial frames we analyze the situation in (after enough time has elapsed for the acceleration effects pass), all frames will determine that the measurements made by each observer of the other one's clock rate will be the same as each other and that these measurements will be frame-independent?

How can either observe measure "the other one's clock rate" in a frame-independent way? What measurement procedure are you proposing, exactly? Are you just talking about visual appearances, how fast the other clock looks like it's ticking?

And isn't it true that from the measured clock rate, they can each determine the same correct relative speed between them independent of frame?

What does "relative speed" mean to you? Normally it's defined to mean the speed of one as measured in the inertial frame of the other, but you seem to want it to be "independent of frame" so you're either confused or you're not using the usual definition.

And isn't it true that each of them can determine the same time-dilation for the other clock independent of frame?

Again you are either confused or using words in a way that differs from how every physicist uses them, "time dilation" for any clock is always defined in terms of a ratio between clock ticking rate and coordinate time in some frame (i.e. d\tau /dt if we know the proper time as a function of coordinate time \tau(t)) Do you have some alternate definition in mind? If so please be very clear about what equations you're using or what specific experimental procedure is used to measure "time dilation".

 

[ghwellsjr]

Sorry, I don't quite follow this.

Also, what are you trying to accomplish here? Are you trying to puzzle out a good definition of "physical" or "real" or are you trying to convince me that I should use those words even without a good definition?

How can either observe measure "the other one's clock rate" in a frame-independent way? What measurement procedure are you proposing, exactly? Are you just talking about visual appearances, how fast the other clock looks like it's ticking?

What does "relative speed" mean to you? Normally it's defined to mean the speed of one as measured in the inertial frame of the other, but you seem to want it to be "independent of frame" so you're either confused or you're not using the usual definition.

Again you are either confused or using words in a way that differs from how every physicist uses them, "time dilation" for any clock is always defined in terms of a ratio between clock ticking rate and coordinate time in some frame (i.e. d\tau /dt if we know the proper time as a function of coordinate time \tau(t)) Do you have some alternate definition in mind? If so please be very clear about what equations you're using or what specific experimental procedure is used to measure "time dilation".

I have the greatest respect for both of you and have learned the most from you both. I'm asking these questions because of a post I made such as this regarding a variant of the Twin Paradox involving one observer on Earth and two inertial ships traveling in opposite directions, one passing Earth and meeting the second ship returning to Earth:

You are still mixed up on several points but you are making progress. Let me comment:

1) The easiest way for each ship and Earth to communicate their time to the others is through a clock that emits a bright flash periodically, say once an hour. Each observer has two counters, one to count its own outgoing flashes and one to count the other observer's incoming flashes. When they are at their closest approach, they each reset all their counters to zero. Then as they move apart, they will each observe that the incoming flashes are coming in at a slower rate than their outgoing flashes. They can each calculate the ratio of the rate of incoming flashes to outgoing flashes and it will be a number less than one and they both will get the same ratio. From this ratio, they each can determine the relative speed between them and from that, they can each determine the time dilation factor. Look up relativistic doppler for more information.

2) You should not consider the measurements to be not real, even though you are right that simultaneity is not an issue here but that's because simultaneity is only a concern when you are comparing results between two different frames of reference and we are not defining any frame of reference in this explanation. Later on, if you want to, you can revisit this scenario from different frames of reference and you will discover that what I am describing here is the same no matter which frame of reference you use. Just remember, what each observer measures and observes will be the same however you analyze the situation.

3) Ship 1 communicates the value on its counters to ship 2 which then sets its counters accordingly. The ships at this point cannot tell by observing the flashes how far away the Earth is. Ship 1 can calculate how far it has traveled by simply multiplying the number of outgoing flashes by the distance traveled per flash which is .866 light hours. Ship 2 can do the same calculation (because the value in the outgoing counter from ship 1 has been communicated to it) and will arrive at the same distance. There is no meaning to your statement that the Earth appears further away or "the actual time of Earth is much later than the calculated time of ship 1". These kinds of conclusions would be frame dependent and not invariant. We aren't concerned about a frame in this analysis.

4) As ship 2 takes over the role of counting incoming flashes to outgoing flashes and calculating the ratio of their rates, it immediately sees the ratio as much larger, in fact, it is the reciprocal of what ship 1 saw. But using the Relativistic Doppler formula, it calculates exactly the same relative speed between itself and Earth and therefore, exactly the same time dilation as ship 1 saw. We should mention that at the point of switch over, ship 1 shuts off its flashes and ship 2 turns on its flashes, we don't want the Earth observer to later on get confused seeing flashes from two different ships at the same time. And be aware that the observer on Earth is completely unaware of this "turn-around" event happening and keeps measuring the same low Relativistic Doppler rate as before for a very long time, but they both continue to observe the same time dilation throughout this entire scenario.

Now here is the key to the different aging: from the moment of "turn-around" to the end of the scenario, the ships have spent an equal amount of time counting incoming flashes from Earth, half of them at a low rate (ship 1) and half of them at a high rate (ship 2). But the Earth doesn't see the transition from low rate to high rate until much, much later because it has to wait for all those flashes that were in transit from the ships' "turn-around" event to Earth to finally get back to Earth. When they do see the "turn-around" event, long after it happened, they will start counting the high rate for a relatively short period of time and this results in a much lower count on Earth's incoming counter than on the ship's incoming counter. Remember, counters on clocks keep track of accumulated time.

I explained all this, by the way, in post #2. Also, this is a description of what actually happens and has nothing to do with the Theory of Special Relativity or any other theory. As I said earlier, once you understand what is actually happening, you can go ahead, pick a frame of reference and "explain" it again using Special Relativity. A good frame of reference to start with would be the one in which Earth is at rest. Then you can do it again with the frame of rest for ship 1 and again for the rest frame of ship 2 and then a fourth frame could be the "average" between ship 1 and Earth where they are each traveling at the same speed in the opposite direction. Doing this explanation in many different frames will give you great insight into how Special Relativity works but it is never necessary to "solve" any problem in more than one frame because they all give the same result.

You have both responded on the thread where I posted that and never made any comment about my posts (including right after post #2) and I would have assumed that someone would have straightened me out if I was making incorrect statements. Now I'm wondering based on your posts on this thread if I'm all wet.

 

[Dale]

I have the greatest respect for both of you and have learned the most from you both. I'm asking these questions because of a post I made such as this regarding a variant of the Twin Paradox involving one observer on Earth and two inertial ships traveling in opposite directions, one passing Earth and meeting the second ship returning to Earth:

You have both responded on the thread where I posted that and never made any comment about my posts (including right after post #2) and I would have assumed that someone would have straightened me out if I was making incorrect statements. Now I'm wondering based on your posts on this thread if I'm all wet.

I wasn't very actively following that thread and I was on a business trip the day you posted that. Unfortunately, when I travel I am often limited to a mobile device so long posts are difficult and I usually just skip them.

Looking at it now, it seems to be a fine description of the Doppler shift analysis of the twin paradox (http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_doppler.html), with one little twist being that instead of having one non-inertial traveling twin making a turnaround you have a pair of inertial ships that do an "information handoff" instead.

What in this thread has made you doubt your earlier post?

 

[ghwellsjr]

What in this thread has made you doubt your earlier post?

In my earlier post, I claimed that each observer/clock can directly observe and measure the other one's time dilation (or rate of the other clocks) without any consideration of a frame of reference or anything else but that seems to be at variance with what you said here:

Well, to have a pair of frames you need not just two identical clocks in relative motion, but two systems of synchronized clocks in relative motion. Then the rate of the other clocks is frame-dependent and measurable.

 

[Simplyh]

I propose we go back to the muon. Let us imagine the following thought experiment:
At places A and B in outer space there are two observers. They move relatively at the speed of a muon reaching from high atmosphere to Earth, one towards the other, in such a way that they will cross. At a certain “time” two muons are created.

Observer A “sees” that, on his frame, muon B took, for instance, 8 ms to reach him (he measured this time the same way measurements are made in our labs). Observer A understands that time in B flows slower and muon B, on its frame, took a lesser time to cover the distance. So, A is not astonished when muon B reaches him still “living”.

Observer B “sees” that, on his frame, muon A took, for instance, the same 8 ms to reach him. Observer B understands that time in A flows slower and muon A, on its frame, took a lesser time to cover the distance. So, B is not astonished when muon A reaches him still “living”.

But if both muons still “live” at the crossing point, then, each observer can not explain how “his” muon has lived 8 ms?

How can we solve the paradox?

 

[ghwellsjr]

I propose we go back to the muon. Let us imagine the following thought experiment:
At places A and B in outer space there are two observers. They move relatively at the speed of a muon reaching from high atmosphere to Earth, one towards the other, in such a way that they will cross. At a certain “time” two muons are created.

Observer A “sees” that, on his frame, muon B took, for instance, 8 ms to reach him (he measured this time the same way measurements are made in our labs). Observer A understands that time in B flows slower and muon B, on its frame, took a lesser time to cover the distance. So, A is not astonished when muon B reaches him still “living”.

Observer B “sees” that, on his frame, muon A took, for instance, the same 8 ms to reach him. Observer B understands that time in A flows slower and muon A, on its frame, took a lesser time to cover the distance. So, B is not astonished when muon A reaches him still “living”.

But if both muons still “live” at the crossing point, then, each observer can not explain how “his” muon has lived 8 ms?

How can we solve the paradox?

Looks like a lotta cuttin' and pastin' goin' on here.

Each observer doesn't have to explain how "his" muon lived 8 ms, because it doesn't in his own rest frame, it lives the normal non-dilated time that a muon lives. I don't see this as any different than the usual statement that two clocks with a relative speed between them each see the other one as ticking slower compared to their own.

Does that make sense or did I misunderstand your problem?

 

[Dale]

In my earlier post, I claimed that each observer/clock can directly observe and measure the other one's time dilation (or rate of the other clocks) without any consideration of a frame of reference or anything else but that seems to be at variance with what you said here:

You can directly observe the Doppler-shifted frequency of the other clock. With appropriate assumptions you can translate that Doppler-shifted frequency into a time dilation. One of those assumptions is the choice of an inertial reference frame where the observing clock is at rest.

 

[Simplyh]

If, for each observer, the moving muon takes 8 ms to reach him, then, applying Einstein definition of simultaneity on a rest frame, both events (muons A and B) must be simultaneous on each frame (A is simultaneous to B if, sending a light beam from A to B and back, time in B, when light reaches B, is equal to half the time, measured in A, for light to go from A to B and the return).

 

[Dale]

both events (muons A and B) must be simultaneous

What does this mean? Muons are not events. They are particles with worldlines. Their worldlines begin with an event (creation) and end with an event (decay), but saying that a muon is an event is like calling a line a point.

 

[ghwellsjr]

If, for each observer, the moving muon takes 8 ms to reach him, then, applying Einstein definition of simultaneity on a rest frame, both events (muons A and B) must be simultaneous on each frame (A is simultaneous to B if, sending a light beam from A to B and back, time in B, when light reaches B, is equal to half the time, measured in A, for light to go from A to B and the return).

You have been very loose in your description of your thought problem. You started by calling A and B "places", then you applied A and B to a pair of observers and to a pair of muons and finally to nothing as if to apply to a pair of frames. I had to make a lot of assumptions about what you meant but I think you could have simply said:

Let us imagine two observers, A and B, in relative motion, each with a pair of identical timing devices (muons). Each one observes the other timing device as running slow but how can they observe their own timing device as running slow?

Wasn't that your original question? And my original answer was that his own timing device isn't running slow in his own rest frame because what it would it mean for either observer to observe the other one as running slow if their own was also running slow? Wouldn't they then appear as running at the same rate and so not even have the opportunity to ask the question?

Now to your question about the "Einstein definition of simultaneity on a rest frame", you kind of answered your own question with the words "on a rest frame". It only applies to clocks that have been synchronized at different locations with no relative speed between them. I don't see where you have two such clocks/timing devices at mutual rest in your thought problem, do you? If you wanted to imagine more such clocks you could use them to understand more details about how everything works but this has already been done extensively on this thread starting with JesseM's contribution in post #9.

Why did you "propose we go back to the muon"? Did you feel that the earlier analysis was inadequate?

 

[ghwellsjr]

You can directly observe the Doppler-shifted frequency of the other clock. With appropriate assumptions you can translate that Doppler-shifted frequency into a time dilation. One of those assumptions is the choice of an inertial reference frame where the observing clock is at rest.

I understand that in SR, time dilation has a broader meaning which requires the specification of a frame to define the speed of a moving clock which can be independent of any observers but I wasn't using SR in my description. Isn't it just as legitimate to ignore any specification of a frame when going from the measured Doppler-shifted frequency to the calculation of the relative speed between the two clock/observers and then to the calculation of the time dilation of the other clock/observer? The only assumption that I was making is that we do this after the effects of the acceleration have passed so that the measured Doppler frequency has stabilized.

The reason that I am concerned about this detail is that I have been telling people on this forum that they have to analyze an entire scenario from one single frame of reference at a time. If they analyze one observer in one frame and another from another frame, we can get all kinds of apparent paradoxes. But if this is true, then how can the Twin Paradox (in which it is stated that each observer observes the other one's clock as going slower than his own) be analyzed from a single frame of reference where only one of the clocks is going slower?

 

[Dale]

I understand that in SR, time dilation has a broader meaning which requires the specification of a frame to define the speed of a moving clock which can be independent of any observers but I wasn't using SR in my description. Isn't it just as legitimate to ignore any specification of a frame when going from the measured Doppler-shifted frequency to the calculation of the relative speed between the two clock/observers and then to the calculation of the time dilation of the other clock/observer? The only assumption that I was making is that we do this after the effects of the acceleration have passed so that the measured Doppler frequency has stabilized.

Consider the reference frame where the source is at rest and the observer is moving. In this frame there is still the observed Doppler frequency, but the source is not time dilated. So you cannot go directly from an observed Doppler frequency to a source time dilation without some assumption of a particular reference frame (usually the inertial frame where the observer is at rest). That assumption may not be explicitly stated, but it is there.

The reason that I am concerned about this detail is that I have been telling people on this forum that they have to analyze an entire scenario from one single frame of reference at a time. If they analyze one observer in one frame and another from another frame, we can get all kinds of apparent paradoxes. But if this is true, then how can the Twin Paradox (in which it is stated that each observer observes the other one's clock as going slower than his own) be analyzed from a single frame of reference where only one of the clocks is going slower?

I guess I don't understand your concern here. Pick a frame, any frame will do. But I do agree with your point that you should analyze the entire scenario from one single frame. If you want to then look at a different frame you need to re-analyze the whole scenario in that other frame.

 

[Simplyh]

Sorry. I've introduced myself in the middle of your discussion with a quite different problem. I'll post it again for discussion some other time.

 

[JesseM]

Isn't it just as legitimate to ignore any specification of a frame when going from the measured Doppler-shifted frequency to the calculation of the relative speed between the two clock/observers and then to the calculation of the time dilation of the other clock/observer?

But these depend on the assumption that you want to use the measured frequency to calculate the speed/time dilation of one ship in the rest frame of the other. You could just as easily choose to use the measured frequency to calculate the speed of B in a frame where A is moving at 0.4c in the same direction as B, and calculate the time dilation in that frame; the calculation would just be a little different, that's all. Either way, the measured frequency is frame-independent (assuming you express it as a ratio of the rate signals are being received to the rate the receiver's own clock is ticking), but the subsequent calculation of velocity and time dilation depends on a choice of frame to use, so it's frame-dependent.

The reason that I am concerned about this detail is that I have been telling people on this forum that they have to analyze an entire scenario from one single frame of reference at a time. If they analyze one observer in one frame and another from another frame, we can get all kinds of apparent paradoxes.

It's usually good advice for beginners to analyze from one frame at a time, but you can use multiple frames to analyze different parts of a single problem as long as you're careful; for example, you could use one frame to calculate twin A's proper time between the two meetings, and a different frame to analyze twin B's proper time between the two meetings. What apparent paradoxes are you thinking of here?

But if this is true, then how can the Twin Paradox (in which it is stated that each observer observes the other one's clock as going slower than his own)

That's not part of the correct statement of the twin paradox, rather it's a false conclusion people draw by naively thinking that a non-inertial observer should still say that clocks moving relative to him run slower.

be analyzed from a single frame of reference where only one of the clocks is going slower?

One of the clock accelerates and thus changes velocities in all inertial frames, so some frame might say the clock was running slower than the inertial clock before the acceleration but faster after, or vice versa. In any case, all frames make the same prediction about the elapsed time on each clock when they reunite locally. I'm not sure if this answers your question since I don't really understand why you ask "how can the Twin Paradox ... be analyzed from a single frame of reference" in the first place; I don't see why doing so should conflict with your earlier statement about "telling people on this forum that they have to analyze an entire scenario from one single frame of reference at a time".