What happens to local clock when many observers are moving wrt each other?

[fondamental]

I looked around the net and couldn't find an answer to this particular setup:

Suppose I am at rest with two friends (friend A and friend B). We all have a clock. All three clocks are synchronized and tick at the same rate.

Now one of my two friends (friend A) starts his rocket engine and leave us at 0.5c. At the same time, my other friend (friend B) does the same, in the same direction, but at 0.9c.

When I observe my friend's clocks, they will both look to tick slower than mine. Friend B's clock will be more affected as it is going faster.

But the opposite is also true. My friends will look at my clock and see it ticking slower. Friend A will see my clock ticking slower. Friend B will see my clock ticking even slower.

How could my clock tick at two different rate at once ? How can my clock tick at a different rate for each possible observer out there (with each their own velocity relative to me) ?

 

[Doc Al]

How could my clock tick at two different rate at once ? How can my clock tick at a different rate for each possible observer out there (with each their own velocity relative to me) ?

Clocks just tick. According to you, your clock ticks once a second (let's say). According to your friend, moving at 0.5 c, your clock ticks once every 1.15 seconds as measured on his clocks. (And vice versa, of course.)

Ah, but a second on your friend's clock is not a "real" second according to your clock! To make sense of all this -- and it all does make sense -- you need to understand the interplay between several relativistic effects, not just time dilation. You also need to understand the relativity of simultaneity (which is the one most forget).

 

[Ibix]

A clock always ticks at one second per second. However, when two things aren't in the same place there turns out to be no unique way to agree on what "at the same time" means for those two things. And the "natural" way to do it turns out to give different results for people in relative motion.

This problem with defining "at the same time" means that what when you and your friend decide to see what each other's clock says at the same time you don't agree what that means. So you don't agree on the method to read the other fellow's clock. So you come up with different results. It's just because you aren't doing the same thing, though, and it's only paradoxical if you think you're doing the same thing.

 

[fondamental]

Thanks for your answers.

The question is not about simultaneity, I know it can't be determined in that context.

But it still true that when friend A look at my clock, he will be able to compare it to his own clock and see that mine goes 1.15 time slower. Then friend B will also look at my clock and see it goes 2.3 times slower than his. Even if both friends don't look at my clock at the same exact time (which is impossible to achieve with confidence) they will still see my clock tick at a rate that is proportional (not linear) to their relative speed to me.

My question is about the difference they see in my clock. My own clock cannot tick at multiple rates. It only ticks at one rate (one second per second according to me). My clock cannot tick at some rate for an observer while it ticks at another rate for another observer.

My only conclusion, for what I know as of now, is that time dilatation due to speed is an apparent phenomena only. Don't get me wrong, I am not saying it is not real, I know it has been observed and it is necessary to account for it in order for GPS to be accurate for instance. I am just trying to understand...

So there must be something I still don't get, or, what we see is not what there is. In the literature they often (not always) say that "time appear to go slower for speeding clocks"... Emphasize on the word "appear" here. Is it possible that when we look at a speeding clock, it appears to go slower, and we have to account for this if we expect to be accurate in our calculations, but in reality, if I could somehow magically be teleported next to it, I would see that it still ticks at the same (original) rate ? It's only our view of it that is morphed ?

 

[Ibix]

It is exactly about simultaneity. You can't look at my clock and see what it reads now. You can only see what it read when the light reaching you now left it. You need to correct for the travel time of light to work out what it reads now, and how you do that depends on how you define "now over there". And the natural way to define that (the "Einstein synchronisation convention") turns out to give different results depending on relative speed.

One way to look at it is this: imagine you are in a car driving at 30 down a straight road. Another car is traveling at 30 down another straight road that is at an angle $\theta$ to your road. Look at the other car - it's falling behind because its speed in your direction is only $30\cos\theta$. But the driver of the other car will also see you falling behind because your speed in her direction is only $30\cos\theta$. How can you both be falling behind? You can't both be traveling slower than the other, surely? Well, yes you can. You're traveling in different directions through space, so your notion of "forwards" is different, so your notion of "forward speed" is different.

The same is true in spacetime. Your path through spacetime (your worldline) is at an angle to the other traveller, and your clock turns out to measure the interval (the analogue of distance for Minkowski space) along your worldline. Your notion of "forward only in time" is not the same as your friend's, so your notion of "at the same time" is different from your friend's, so your notion of "what his clock is reading right now" is different from his. The "angle" here turns out to be something called the rapidity parameter, and you need to use hyperbolic functions rather than trigonometric ones.

This is easiest to draw on a Minkowski diagram (aka spacetime diagram). I suggest looking them up. They were the thing that made all this click into place for me. You might like to check out the Insight written by @robphy on spacetime diagrams on rotated graph paper, or have a play around with the Minkowski diagram tool I wrote here: http://www.ibises.org.uk/Minkowski.html

 

[fondamental]

Thank you Ibix, I appreciate. But it's not what I am asking.

I know the theory works very well, and the equations and graph also works very well, and they fit very nicely with observed facts. I am not disputing this. But my question refers to "observed vs reality". In my question it's not important if the observers looks at my clock at the same time or not. They can do it now or in 10 years, and at the same time, or 100 years apart.

My question is about the fact that if there is two observers, going at different speed, and looking back at my clock, they won't see the same thing. And that's where I am asking, if they see different things about me (my clock...) but in reality I am only one thing (clock) ticking at one rate, those observers are not seeing my reality. What they see is valid in their frame, but it doesn't correspond to a reality. I hope I'm clearer, I am not native English so I may not be clear... ?

 

[Ibix]

Thank you Ibix, I appreciate. But it's not what I am asking.

I think it is what you're asking, even if you don't realize it.

What your friends will actually see is your clock running very, very slowly. This is because the distance between you is increasing so the travel time for the light from each successive tick of your clock increases. If they turn around they will see your clock ticking very quickly because the travel time for light each successive tick of your clock decreases. I hope that's familiar as the Doppler effect. This is not time dilation.

You cannot see time dilation directly, at least not in this experiment. You do hear people saying that "an observer sees moving clocks running slowly" (I even do it myself), but this is sloppy writing. You will see a clock approaching you running fast and one moving away from you running slow. But this isn't the rate of the clock - it's mixing the tick rate of the clock with its speed, and you aren't interested in this. To get the rate of the clock you need to subtract out the travel time of light. But what distance did the light travel? Different frames will disagree due to length contraction. So they will subtract off different amounts of time and come to different conclusions about the rate of the clock. So a better way to put it is that "an observer calculates that moving clocks run slowly".

Nobody's view is wrong, or not "real" somehow. It's just description of the same thing in different terms. And ultimately, the choice of terms boils down to a choice of simultaneity convention.

What they see is valid in their frame, but it doesn't correspond to a reality.

Not quite. What you should say is that it isn't the same as your description of yourself in your frame. But the thing to realize is that your own description is not any more real than any other description.

That was the point of my example of the cars. Neither car is wrong when it says the other is traveling forward slower - they just disagree on what "forward" means. Either one could agree to adopt the other's definition of forward, and then they'll agree that they are moving slower in the forward direction and the other is moving faster. It would be an odd thing to do, but there's nothing stopping you.

In your example of the travellers in spacetime, they agree that each person's clock is measuring a "distance" through spacetime. But they don't agree what "not moving in space, only moving in time" means. So they say that the other fellows' clocks are measuring "distances" that aren't parallel to the direction they call "only moving in time", so they tick at the wrong rate.

If you want an underlying reality, the easiest way to visualise it is the "block universe". Spacetime is a four-dimensional entity. How you choose to split that up into space and time is, a matter of choice (although there's a natural choice for any given state of inertial motion). Different choices give different notions of "time", so lead to different notions of whether a given clock is measuring "just time" or a mixture of time and one spatial dimension.

 

[Mister T]

My question is about the fact that if there is two observers, going at different speed, and looking back at my clock, they won't see the same thing. And that's where I am asking, if they see different things about me (my clock...) but in reality I am only one thing (clock) ticking at one rate, those observers are not seeing my reality. What they see is valid in their frame, but it doesn't correspond to a reality. I hope I'm clearer, I am not native English so I may not be clear... ?

If you look at a clock to see how fast it's ticking, it's the Doppler effect that determines what you'll see. Suppose that your friends are looking not at your clock, but at a monochromatic light source you're holding. Each friend will see it red-shifted by a different amount, but you will not see any red shift at all. How can your light be putting out three different colors when it's monochromatic, meaning it puts out only one color?

I think the answer to my question is the same as the answer to your question about the clocks. Your clock ticks at only one rate. The fact that your friends observe it to tick at other rates is a consequence of the fact they they're in motion relative to it. Their motion has no effect on the rate at which your clock ticks, but when they compare the rate at which it ticks to the rate at which their own clocks tick they see a difference.

 

[MikeLizzi]

I did this little write-up for my own benefit. Perhaps It will help you.

http://www.relativitysimulation.com/Documents/LookingForContradictionsWhenMovingClocksRunSlow.pdf

 

[Nugatory]

The question is not about simultaneity

It is completely and totally about simultaneity.

If I say that your clock is running slower than mine by a factor of two, I'm really saying something like "At the same time that my clock read 12:00 noon your clock also read 12:00 noon; and at the same time that my clock read 1:00 PM your clock read 12:30 PM; therefore your clock is running slow by a factor of two." There is no way to make any statement about the relative rates of clocks in different locations without using that notion of at the same time, which is what simultaneity means.

The different observers come to different conclusions about the rate at which my clock ticks because they are drawing different conclusions about what their and other clocks read at the same time that my clock reads something.

 

[DParlevliet]

You cannot "observe my friend's clocks" while they move. That is only possible when those clocks are at the same location. So suppose A and B return back to you. Then A will see you 10 years older and B will see you 18 years older. Indeed A and B will conclude afterwards that your clock was ticking at different rates.

 

[pixel]

And that's where I am asking, if they see different things about me (my clock...) but in reality I am only one thing (clock) ticking at one rate, those observers are not seeing my reality. What they see is valid in their frame, but it doesn't correspond to a reality.

You are maybe confusing "reality" with "proper time." You are reading your clock's proper time as it is at rest with respect to you. But there is otherwise no absolute "reality" to your reading. You are moving with respect to a frame of reference in which the sun is at rest. Following your argument, someone in that frame could then claim that their clock corresponds to "reality."

 

[fondamental]

Thank you everyone for your answers; I am still trying to make sense of them all...

One thing I want to ask right away: It's about Doppler effect. I know about that light propagation delay I have to account for in the calculations. Of course, what I see is what existed when the light left the object I am observing, and that delay is proportional to the distance of the object. This is obvious, so much that I forgot to mention it in my questions.

But isn't this a completely different phenomena (than relativistic time dilation due to speed) ? When observing a speeding clock, don't we have to account for relativistic time dilation as well as light propagation delay ? don't we have to "add" both phenomena ?

 

[Ibix]

The full relativistic formula for Doppler shift includes both the frequency shift due to the range change and time dilation. Actually I think that should be the other way around: you can choose to divide the Doppler shift into two multiplicative components, one due to distance change and one due to time dilation.

 

[fondamental]

But you said:

What your friends will actually see is your clock running very, very slowly. This is because the distance between you is increasing so the travel time for the light from each successive tick of your clock increases. If they turn around they will see your clock ticking very quickly because the travel time for light each successive tick of your clock decreases. I hope that's familiar as the Doppler effect. This is not time dilation.

You cannot see time dilation directly, at least not in this experiment. You do hear people saying that "an observer sees moving clocks running slowly" (I even do it myself), but this is sloppy writing. You will see a clock approaching you running fast and one moving away from you running slow. But this isn't the rate of the clock - it's mixing the tick rate of the clock with its speed, and you aren't interested in this. To get the rate of the clock you need to subtract out the travel time of light. But what distance did the light travel? Different frames will disagree due to length contraction. So they will subtract off different amounts of time and come to different conclusions about the rate of the clock. So a better way to put it is that "an observer calculates that moving clocks run slowly".

What I understand here is that time dilatation is not observable ?? What I see is the Doppler effect. And in your last answer you're telling me that both effects are included in the formula. What did I get wrong ?

 

[Ibix]

That was sloppy of me - sorry.

Even without relativity you get a frequency shift in light coming from a moving source, due to the changing distance. In a non-relativistic scenario, if you simply correct for the changing travel time you recover the source frequency. I usually call this the "naive Doppler effect" - and I forgot to make that distinction above.

But in a relativistic scenario, if you simply correct for the changing travel time you do not recover the source's frequency. You get the frequency reduced by a factor of $\gamma$, due to time dilation.

However, this does not mean that time dilation is observable. You have to take your observation (a certain frequency of incoming light) and do a calculation to subtract out the frequency shift due to motion. This step, however, depends on your simultaneity convention. You need to determine how far away the emitter was when it emitted each pulse, and the answer to that depends on your convention for defining space and time - and hence the amount of frequency shift "left over" depends on that convention. That's what we mean by "not observable" - to "see" time dilation you have to do some maths that includes a step that depends on your personal choice.

 

[fondamental]

Great! thank you. I get that part right, it all makes sense.

I'll have to dig a bit more about that "simultaneity" thing before going further. I think there is something there I just don't know about (yet ;) ).

 

[Bartolomeo]

But isn't this a completely different phenomena (than relativistic time dilation due to speed) ? When observing a speeding clock, don't we have to account for relativistic time dilation as well as light propagation delay ? don't we have to "add" both phenomena ?

Transverse Doppler Effect reflects magnitude of dilation of moving clock. You simply look at right angle to direction of motion of light source and measure deviation of frequency. It doesn't matter whether the source was one inch from you or one light year away. It depends solely on relative velocity. Time dilation effect has been measured many times this way.
https://en.wikipedia.org/wiki/Ives–Stilwell_experiment

There is a wonderful treatment, and very, very interesting task there. You can ask someone how to resolve it.
http://spiff.rit.edu/classes/phys200/lectures/doppler/doppler.html

 

[Mister T]

But isn't this a completely different phenomena (than relativistic time dilation due to speed)?

The Doppler effect is indeed different from time dilation, although they are very much related. If we take as an example a recession speed of $0.6\ c$ we get a time dilation factor $\gamma=1.25$ whereas the Doppler factor is $2$. So if your friend is moving away from you at a speed of $0.6\ c$ you will observe his clock running slow by a factor of $1.25$ but you will see his clock run slow by a factor of $2$.

I'll have to dig a bit more about that "simultaneity" thing before going further.

When we say that we see a clock running slow by a factor $2$ due the Doppler effect, that's a pretty simply process. You just point your telescope at the clock and see that it's ticking away at half the rate of our wristwatch.

Simultaneity is the part that is often glossed over when people write about time dilation. We say that we observe a moving clock to run slow by a factor of $1.25$ but we don't go into the details of how to go about making that observation. We need at least two events so that we can measure the time that elapses between them, and then see that that elapsed time is different for me than it is for my friend, different by a factor of $1.25$. In addition to the two clocks talked about above, a third clock is needed because we need to know what time it is over there where our friend is located when each of the two events takes place. For one of the two events we can have the same location, making the comparison trivial. But then we cannot share the same location for the second event because we are moving relative to each other. If that third clock is at rest relative to my friend and is located where I am when the second event occurs I can make the necessary comparison. The problem is that if my friend synchronizes that third clock with the one he carries with him, I will not agree that he did the synchronization correctly, and I will thus attribute his claim that my clock is running slow by a factor of $1.25$ to that error. I know from the comparison that it's his clock that's running slow by a factor of $1.25$. Of course, if my friend conducts the same observation using a third clock that is co-located with him when a second event occurs (note that it can't the same event that I used for my second event) then he will draw exactly the same conclusions about me that I did about him.

Now, this business of making an error in synchronization is fundamental. Events that are simultaneous for him are not simultaneous for me, and vice-versa. Provided that those events are separated along the line of relative motion.

 

[Bartolomeo]

- and the answer to that depends on your convention for defining space and time - and hence the amount of frequency shift "left over" depends on that convention. That's what we mean by "not observable" - to "see" time dilation you have to do some maths that includes a step that depends on your personal choice.

Maybe I misinterpret you, but it is very unlikely that some personal choice may affect amount of time dilation that Relativistic Doppler Effect includes, since one depends on relative velocity only.

However, because of "personal choice" one may choose this or that reference frame and can assign full amount of time dilation either to a source or to an observer, or to the both in certain proportion. Measured frequency shift will be the same, but explanations will be frame dependent. Measured (or interpreted) rate of relatively moving clock will be also frame dependent. The clock may appear ticking at any rate. It will appear ticking slower, faster - at any rate, that will depend:

1) in case of Relativistic Doppler effect - on interpretation either source moves or observer (Feynman Lectures, Relativistic Effects on Radiation)
2) in case of Transverse Doppler Effect - on angle observer tilts his telescope (Mathpages, the Doppler Effect)
3) in case of measurement by synchronized clock - on synchrony convention (Reichenbach, Phylosophy of Space and Time) or chosen value of $\varepsilon$

If you measure time dilation by means of synchronized clocks, measured clock rate will depend on synchrony convention. If you synchronize two clocks in your frame by Einstein ($\varepsilon = 1/2$), moving clock will appear ticking slower.

One can, however, synchronize clocks by self - consistent method of Reichenbach and is free to choose any arbitrary $\varepsilon$ in the range of 0 to 1.

This way one can make that clock in relative motion would appear ticking much slower, a bit slower, at the same rate, much faster, a bit faster e.t.c – at ANY rate.

If one chooses reference frame in which an observer is “at rest", observer keeps telescope at right angle to direction of motion of a source and synchronizes clocks by Einstein in his frame.

If one chooses a frame in which an observer moves himself and source is "at rest", due to aberration observer keeps his telescope at $\arcsin v/c$ and synchronizes clocks by Reichenbach in his frame. One can choose suitable value of $\varepsilon$ empirically so as it will be in accordance with his relative velocity. I.e. if he synchronizes clocks that way so as measured amount of time dilation by means of Transverse Doppler Effect will be in accordance with measurement by means of synchronized clocks.

 

[Ibix]

Maybe I misinterpret you, but it is very unlikely that some personal choice may affect amount of time dilation that Relativistic Doppler Effect includes, since one depends on relative velocity only.

What you measure as velocity depends on the synchronisation convention in use. If you use a different synchronisation convention you get a different velocity for the source, but the frequency of the received pulses doesn't change (that's not conventional). Therefore a change of synchronisation convention must affect the calculated time dilation as well as the velocity.

I can't think of any circumstances involving purely inertial motion where you'd want to use anything other than Einstein's synchronisation convention. But you can. And then you would interpret the Doppler effect differently.