Benefits of time dilation / length contraction pairing?

[neopolitan]

There have been more than a few threads where there clearly is confusion about the use of time dilation and length contraction.

People initially think that:

1. in an frame which is in motion relative to themselves, time dilates and lengths contract; and
2. velocities in a frame which is in motion relative to themselves are contracted lengths divided by dilated time.

I admit that it stumped me for a long time, because of what I see as inconsistent use of primes and for me a much more useful pair of equations would have a more consistent use of primes, similar to the Lorentz transformations.

I was told during a long discussion that time dilation and length contraction are used, even though they pertain to different frames, because they have greater utility. I took that at face value, but now I wonder again.

What exactly is the greater utility of time dilation and length contraction equations which prevents the use of two contraction equations which would do away with the confusion I mentioned above?

(And by the way, introducing arguments that t in time dilation is the period between tick and tock doesn't really help, because this is more indicative of the confusion since we use clocks everyday to measure the time between events in terms of the number of ticks and tocks rather than in terms of the duration of pause between each tick and tock. Reinterpreting how we use time to make the equation work is not indicative of any greater utility.)

If it is a purely historical thing, then I would be far happier with it if that little tidbit were taught at the same time as the equations are introduced. But it isn't.

There is also the potential argument that they are only useful right at the beginning of one's odyssey into relativity, so it doesn't really matter. Sure, ok, then it doesn't matter if you use a more intuitive pairing does it?

Bottom line: what is so great with time dilation?

cheers,

neopolitan

 

[ZapperZ]

Would it surprise you to learn that these "time dilation" and "length contraction" are merely consequences of the Lorentz transformation coupled with Einstein's SR postulates?

So do you really have a problem with the fundamental theory, or do you simply have a problem with the consequences of the theory? If you do not have a problem with the fundamental theory (because you didn't even mention it), then where in the logical derivation of the consequences did we lose you and they become "inconsistent"? For example, start with a standard derivation of time dilation as presented in standard physics textbooks. Point out exactly where such a thing becomes logically inconsistent.

What is so "great" about time dilation? It explains a whole bunch of empirical observations. What more can you ask for?

Zz.

 

[JesseM]

I admit that it stumped me for a long time, because of what I see as inconsistent use of primes and for me a much more useful pair of equations would have a more consistent use of primes, similar to the Lorentz transformations.

They are consistent. Unprimed is used in these equations to represent times and lengths in the frame where the ruler/clock is at rest; primed is used to represent the corresponding times and lengths in the frame where the ruler/clock is moving. Can you specify what alternative you're proposing?

 

[Saw]

where in the logical derivation of the consequences did we lose you and they become "inconsistent"? For example, start with a standard derivation of time dilation as presented in standard physics textbooks. Point out exactly where such a thing becomes logically inconsistent.

I would put it the other way round, ZapperZ. I've looked at the transversal light clock example and the longitudinal light clock example by themselves and wondered how one can derive from those examples the consequence that there is RECIPROCAL TD and LC , if you consider those derivations isolatedly.

A different thing is that, if you consider jointly RECIPROCAL TD, LC and RS, the whole system of SR really seems to make sense. Thus the "standard derivation of time dilation as presented in standard physics textbooks" may have a limited use as a way to derive the quantitative aspect of those elements, which however only make sense (as a RECIPROCAL measurement) when they are put together and integrated in a "global system", which by the way does not contemplate any duality of opinions about reality, but, on the contrary, full agreement on what has really happened.

But maybe I am imagining something you are not saying. In your view, the "standard derivation of time dilation as presented in standard physics textbooks" (for instance, the light clock example)..., does it logically prove that there is RECIPROCAL time dilation?

 

[neopolitan]

Would it surprise you to learn that these "time dilation" and "length contraction" are merely consequences of the Lorentz transformation coupled with Einstein's SR postulates?

No, if L=x' and L'=x (or perhaps $$L = \Delta x'$$ and $$L' = \Delta x$$).

Is that what you are saying?

But then where is the consistency of prime notation that JesseM claims in his post?

I do understand that, with the awkward interpretation, time dilation does explain a lot of empirical observations. No problems there. But so would a time equation in the same form as the length equation, and it would not lead to the problem with people thinking "speed of light in another inertial frame is contracted length divided by dilated time ... hang on, that's not invariant!" Then they visit here and have someone metaphorically yelling at them "you are mixing frames!" without actually explaining why you can't use time dilation and length contraction that way.

I know you can't, I just want to know what advantages exist with having time dilation and length contraction expressed the way they are?

Jesse, I can't believe you say the frames are all consistent. We went over it for days, in emails with diagrams. How about I post your very own diagram here to help clarify?

cheers,

neopolitan

 

[ZapperZ]

I would put it the other way round, ZapperZ. I've looked at the transversal light clock example and the longitudinal light clock example by themselves and wondered how one can derive from those examples the consequence that there is RECIPROCAL TD and LC , if you consider those derivations isolatedly.

A different thing is that, if you consider jointly RECIPROCAL TD, LC and RS, the whole system of SR really seems to make sense. Thus the "standard derivation of time dilation as presented in standard physics textbooks" may have a limited use as a way to derive the quantitative aspect of those elements, which however only make sense (as a RECIPROCAL measurement) when they are put together and integrated in a "global system", which by the way does not contemplate any duality of opinions about reality, but, on the contrary, full agreement on what has really happened.

But maybe I am imagining something you are not saying. In your view, the "standard derivation of time dilation as presented in standard physics textbooks" (for instance, the light clock example)..., does it logically prove that there is RECIPROCAL time dilation?

Er.. come again?

There are two aspects to this. One is the logical derivation, i.e. mathematical derivation, based on SR's postulates. It has to based on that because there's nothing mathematically that can derive c being a constant in all frames.

The second is the experimental verification. A logical derivation of anything in physics is no guarantee that it is valid. It is the experimental observation consistent with such result that elevates its validity. It somehow appears as if you want to start from the tail end of it to justify itself, which is fine if you are trying to formulate a new theory. But considering that SR is such a well-established theory with solid foundation, and every one of the consequences can be derived from such foundation, it would be logical to start from there. And that's where I do not get the OP. Is there a problem with the foundation in the first place or is he/she only do not get the consequences? I don't think that is such an unreasonable query.

Zz.

 

[ZapperZ]

No, if L=x' and L'=x (or perhaps $$L = \Delta x'$$ and $$L' = \Delta x$$).

Is that what you are saying?

But then where is the consistency of prime notation that JesseM claims in his post?

I do understand that, with the awkward interpretation, time dilation does explain a lot of empirical observations. No problems there. But so would a time equation in the same form as the length equation, and it would not lead to the problem with people thinking "speed of light in another inertial frame is contracted length divided by dilated time ... hang on, that's not invariant!" Then they visit here and have someone metaphorically yelling at them "you are mixing frames!" without actually explaining why you can't use time dilation and length contraction that way.

I know you can't, I just want to know what advantages exist with having time dilation and length contraction expressed the way they are?

Jesse, I can't believe you say the frames are all consistent. We went over it for days, in emails with diagrams. How about I post your very own diagram here to help clarify?

cheers,

neopolitan

I don't see it.

Forget SR/Lorentz transformation. Start with Galilean transformation. Do you have a problem with that as well? Write down the velocity and displacement of a moving object in two different inertial frames via Galilean transformation. My guess is that you have a problem with that as well, because fundamentally, none of what you wrote above really has anything to do with SR.

At what point do you acknowledge about the empirical verification of these things?

Zz.

 

[Mentz114]

(And by the way, introducing arguments that t in time dilation is the period between tick and tock doesn't really help, because this is more indicative of the confusion since we use clocks everyday to measure the time between events in terms of the number of ticks and tocks rather than in terms of the duration of pause between each tick and tock. Reinterpreting how we use time to make the equation work is not indicative of any greater utility.)

Hard to believe. Can't you see that this is an irrelevance. How else could you express time dilation except as an increase the gap between ticks ? Counting ticks still means you have to multiply the number of ticks by the time between ticks.

When you talk about primes, are you making a point about notation ? You could use some other way to distiguish two frames, but it wouldn't make any difference !

 

[JesseM]

Jesse, I can't believe you say the frames are all consistent. We went over it for days, in emails with diagrams. How about I post your very own diagram here to help clarify?

Yes, and never did you offer any convincing argument that there was something "inconsistent" about the standard definitions. My diagram shows that they are consistent, given the usual definition of "length" and "time interval"--the "spatial analogue of time dilation" and the "temporal analogue of of length contraction" in the diagram don't refer to any commonly-used or intuitive physical quantities (the 'spatial analogue of time dilation' refers to the distance in the primed frame between two events that are simultaneous in the unprimed frame; the 'temporal analogue of length contraction' is even weirder, it refers to the time in the primed frame between two surfaces of simultaneity that cross through the events in the unprimed frame).

 

[neopolitan]

Hard to believe. Can't you see that this is an irrelevance. How else could you express time dilation except as an increase the gap between ticks ? Counting ticks still means you have to multiply the number of ticks by the time between ticks.

When you talk about primes, are you making a point about notation ? You could use some other way to distiguish two frames, but it wouldn't make any difference !

Yes, I know that is what time dilation is (and must be to keep everything right).

What is difficult for people new to SR, the very ones who are being taught time dilation, is that the equation involving t, ostensibly the same t which they may well be used to in Gallilean boosts (x'=x-vt) and the kinetics equations (v = (s_i-s_o)/t and so on), is using a quite different definition of t.

Jesse mentions "time interval", which is fine, I just wonder why we don't use a different symbol for time dilation ($$\tau$$ perhaps) to highlight the difference between "time interval" - time between ticks - and "(measured) time elapsed" - number of ticks.

But none of this answers the original question, what are the benefits of having time dilation and length contraction rather than a pair of equations which would not lead to the continual confusion I alluded to in an earlier post?

(It seems the only responses so far are "you can't do it any other way" and "you are confused". Is there really no other way?)

cheers,

neopolitan

 

[ZapperZ]

Yes, I know that is what time dilation is (and must be to keep everything right).

What is difficult for people new to SR, the very ones who are being taught time dilation, is that the equation involving t, ostensibly the same t which they may well be used to in Gallilean boosts (x'=x-vt) and the kinetics equations (v = (s_i-s_o)/t and so on), is using a quite different definition of t.

Jesse mentions "time interval", which is fine, I just wonder why we don't use a different symbol for time dilation ($$\tau$$ perhaps) to highlight the difference between "time interval" - time between ticks - and "(measured) time elapsed" - number of ticks.

But none of this answers the original question, what are the benefits of having time dilation and length contraction rather than a pair of equations which would not lead to the continual confusion I alluded to in an earlier post?

(It seems the only responses so far are "you can't do it any other way" and "you are confused". Is there really no other way?)

cheers,

neopolitan

This complain has nothing to do with SR. Look at your kinematics problem. You use the SAME thing there! This simply re-enforces my earlier assertion that this isn't about "time dilation" at all. You are simply confused on what we call 'time' in any dynamical system.

Zz.

 

[Mentz114]

Neopolitan,

But none of this answers the original question, what are the benefits of having time dilation and length contraction rather than a pair of equations which would not lead to the continual confusion I alluded to in an earlier post?

There are four equations, one for time each dimension. It's hard to see what you mean.
I don't think most people are 'continually confused'.

 

[neopolitan]

This complain has nothing to do with SR. Look at your kinematics problem. You use the SAME thing there! This simply re-enforces my earlier assertion that this isn't about "time dilation" at all. You are simply confused on what we call 'time' in any dynamical system.

Zz.

Ok, I accept that I may be confused. Let's look at kinematics first.

Say I do a simple experiment with a small car designed to move at a set speed (I don't what that speed is). I run it past two posts (s_o and s_i) and measure the time.

To work out the speed, I use the equation I mentioned before.

How do I work out the time t?

This is my suggestion. I have a stop watch, I start it when the car passes the first post and I stop it when it passes the second post. The value on the watch is then t, which shows me the result in "ticks" each of which will probably be 1ms or 10ms long.

Therefore, t= (number of ticks on my stopwatch).

Not (time between each tick on my stopwatch).

Then I notice there was a separation between me and post s_i and cleverly work out that that means that I don't get to see the car pass the post instantaneously, that there are now simultaneity issues, and therefore I shouldn't really be using gallilean equations, but rather lorentz based ones.

How do I work out the speed now?

I can use the value on my stopwatch, plus the knowledge that the information about the car passing s_i traveled to me at c (approximately, because I am not in a vaccuum).

Still, my value of t is in terms of ticks of my stopwatch, t = (number of ticks of my stopwatch before I receive information about the car passing s_i minus the number of ticks of my stopwatch which elapsed while that information was in transit).

Then, I decide to get more tricky. Because I have been told that for things in motion relative to me time dilates, I want to see some empirical evidence for it.

I put a video camera on post s_i and a stopwatch on the car along with a mechanism which starts both my stopwatch and the stopwatch on the car as the car passes post s_o. I call the car "Prime". I call myself "Unprime". I call the result I calculate from my stopwatch t and the result captured on the video camera as the car passes t'.

I think that the result of my empirical experiment will be that time does not dilate, but rather contracts, since the car's t' will be less than my t.

I can get around that by changing the definition of time in my dynamical system. I don't think that is such a fabulous idea, but I could do it.

Am I simply confused here?

cheers,

neopolitan

 

[JesseM]

But none of this answers the original question, what are the benefits of having time dilation and length contraction rather than a pair of equations which would not lead to the continual confusion I alluded to in an earlier post?

What would be the pair of equations you're proposing, that wouldn't involve quantities which are far more unintuitive to students than "length" and "time interval", and which wouldn't be much more difficult to actually apply to the types of introductory problems found in textbooks? Please write them down.

 

[JesseM]

Ok, I accept that I may be confused. Let's look at kinematics first.

Say I do a simple experiment with a small car designed to move at a set speed (I don't what that speed is). I run it past two posts (s_o and s_i) and measure the time.

To work out the speed, I use the equation I mentioned before.

How do I work out the time t?

This is my suggestion. I have a stop watch, I start it when the car passes the first post and I stop it when it passes the second post. The value on the watch is then t, which shows me the result in "ticks" each of which will probably be 1ms or 10ms long.

Is the watch riding in the car or at rest relative to the posts on the ground? If at rest on the ground, then it must be far from at least one of the posts when the car passes it...how do you ensure that the watch is stopped "when" it passes the post it's not next to? Let's say the watch is next to the first post, so you start it when the car passes...do you stop it when you see the light from the car passing the second post, or do you prearrange things so that it will stop simultaneously with the event of the car passing the second post, relative to the rest frame of the watch and posts? If the latter, then this isn't a novel suggestion, it's the time interval that's always used when calculating speed=distance/time.

Therefore, t= (number of ticks on my stopwatch).

Not (time between each tick on my stopwatch).

The time in speed=distance/time is always the number of ticks on your stopwatch, not "time between each tick on your stopwatch". We'd only be interested in the "time between each tick on your stopwatch" if we knew the number of ticks in the watch's frame and then wanted to figure out the time for the watch to elapse this number of ticks in a different frame where the watch was in motion.

Then I notice there was a separation between me and post s_i and cleverly work out that that means that I don't get to see the car pass the post instantaneously, that there are now simultaneity issues, and therefore I shouldn't really be using gallilean equations, but rather lorentz based ones.

How do I work out the speed now?

I can use the value on my stopwatch, plus the knowledge that the information about the car passing s_i traveled to me at c (approximately, because I am not in a vaccuum).

Still, my value of t is in terms of ticks of my stopwatch, t = (number of ticks of my stopwatch before I receive information about the car passing s_i minus the number of ticks of my stopwatch which elapsed while that information was in transit).

Yes, then in this case you are really calculating the time on your watch between the car passing the first post and the time on your watch that is simultaneous with the car passing the second post in the watch's frame...as I said this is the time interval you'd always want to use when calculating speed=distance/time for the car in the ground frame.

Then, I decide to get more tricky. Because I have been told that for things in motion relative to me time dilates, I want to see some empirical evidence for it.

I put a video camera on post s_i and a stopwatch on the car along with a mechanism which starts both my stopwatch and the stopwatch on the car as the car passes post s_o. I call the car "Prime". I call myself "Unprime". I call the result I calculate from my stopwatch t and the result captured on the video camera as the car passes t'.

I think that the result of my empirical experiment will be that time does not dilate, but rather contracts, since the car's t' will be less than my t.

But that's just because you have "primed" and "unprimed" backwards from the normal convention. The usual convention is that the unprimed t represents the time interval between two events on the worldline of a clock as measured in the clock's rest frame (so both events happen at the same spatial location in the unprimed frame, and t will be equal to the time interval as measured by the clock itself), whereas the primed t' represents the time interval between the same two events in a frame where the clock is moving (so the events happen at different locations in the primed frame). This is consistent with the convention for length contraction, where the unprimed L represents the distance between two ends of an object in the object's own rest frame, and the primed L' represents the distance between the ends of the same object in the frame where the object is in moving.

 

[neopolitan]

Neopolitan,

There are four equations, one for time each dimension. It's hard to see what you mean.
I don't think most people are 'continually confused'.

If you are talking about:

$$t'=\gamma t$$

$$L_{x}'=L_{x} / \gamma$$

$$L_{y}'=L_{y} / \gamma$$

$$L_{z}'=L_{z} / \gamma$$

Then you are confused. I don't think you mean that though.

If you are talking about:

$$t'=\gamma t$$

$$L'=L / \gamma$$

$$t'=\gamma (t-x.\frac{v}{c^{2}}$$

$$x'=\gamma (x-vt)$$

then fair enough.

The pair I am talking about is:

$$t'=\gamma t$$

$$L'=L / \gamma$$

And the (to me more intuitive) pair is:

$$\Delta t'=\Delta t / \gamma$$

$$\Delta x'=\Delta x / \gamma$$

With the explanation that lengths contract in a frame in motion (no different to existing situation) and that clocks slow down in a frame in motion. If you like, you could then talk about how this is equivalent to time intervals dilating, with each tick taking longer in a frame in motion.

Then, when people like chrisc come along (and myself, many moons ago), they won't get into trouble for doing the obvious calculation $$x' / t'$$ and ending up with $$v / \gamma^{2}$$ or worse ... $$c / \gamma^{2}$$.

Hopefully this also answers Jesse's question.

 

[neopolitan]

But that's just because you have "primed" and "unprimed" backwards from the normal convention. The usual convention is that the unprimed t represents the time interval between two events on the worldline of a clock as measured in the clock's rest frame (so both events happen at the same spatial location in the unprimed frame, and t will be equal to the time interval as measured by the clock itself), whereas the primed t' represents the time interval between the same two events in a frame where the clock is moving (so the events happen at different locations in the primed frame). This is consistent with the convention for length contraction, where the unprimed L represents the distance between two ends of an object in the object's own rest frame, and the primed L' represents the distance between the ends of the same object in the frame where the object is in moving.

I know this is the convention. What is the benefit of that convention?

Note that I clearly specified thing so that I would have one primed frame (that of the car) and one unprimed frame (mine) and values from the primed frame would be primed and values from the unprimed frame would be unprimed. Is that not consistent?

The convention is to not really talk about a single frame, but to have a length for which both ends are simultaneous and clock for which ticks are colocated.

The thing that got me thinking about this most recently is chrisc's concern which I think was about light clocks (whether his issue was or was not about light clocks is immaterial either way).

Consider a single light clock. This has both a length (between a tick mirror and a tock mirror) $$\Delta x$$ and consecutive ticks $$\Delta t$$. Lie it down on a carriage of some sort so that the vector $$\Delta x$$ is parallel with the direction of the carriage's potential motion.

The only speed the carriage can have for which the conditions behind time dilation and length contraction can be simultaneously applied to the dimensions of the light clock (I'm only talking little 's' version of simultaneous here, not the strict definition) is zero. Only when the carriage is at rest (relative to are observer) are the ends of the light clock simultaneous (relative to the observer) and consecutive ticks are colocal (relative to the observer).

You can only apply time dilation and length contraction to the light clock in a non-trivial way by mixing frames. And if you try to use those, to work out the speed of the photon in the light clock, you end up with a non-sensical result - because of the frame mixing.

I can see that it is useful to be able to work out the amount by which you would need to slow down a clock which is at rest relative to you to match a clock which is in motion relative to you. But I still don't see the huge benefit associated with the pairing of time dilation and length contraction.

Is it mere orthodoxy? Is it historical? Or is there a real concrete advantage?

cheers,

neopolitan

 

[JesseM]

I know this is the convention. What is the benefit of that convention?

That it would be awfully confusing if primed referred to the rest frame of the clock while unprimed referred to the frame where the ruler was moving, or vice versa.

Note that I clearly specified thing so that I would have one primed frame (that of the car) and one unprimed frame (mine) and values from the primed frame would be primed and values from the unprimed frame would be unprimed. Is that not consistent?

Consistent with what? I thought we were talking about consistency between notation in the length contraction and time dilation equations, I don't know what it would even mean to ask if the time dilation equation alone is "consistent". You're certainly free to refer to the frame of the clock as the primed frame if you like, but then to be consistent you should also use primed to refer to the rest frame of the object whose length you're talking about in the length contraction equation.

The thing that got me thinking about this most recently is chrisc's concern which I think was about light clocks (whether his issue was or was not about light clocks is immaterial either way).

Consider a single light clock. This has both a length (between a tick mirror and a tock mirror) $$\Delta x$$ and consecutive ticks $$\Delta t$$. Lie it down on a carriage of some sort so that the vector $$\Delta x$$ is parallel with the direction of the carriage's potential motion.

The only speed the carriage can have for which the conditions behind time dilation and length contraction can be simultaneously applied to the dimensions of the light clock (I'm only talking little 's' version of simultaneous here, not the strict definition) is zero. Only when the carriage is at rest (relative to are observer) are the ends of the light clock simultaneous (relative to the observer) and consecutive ticks are colocal (relative to the observer).

You can only apply time dilation and length contraction to the light clock in a non-trivial way by mixing frames. And if you try to use those, to work out the speed of the photon in the light clock, you end up with a non-sensical result - because of the frame mixing.

I don't know what you mean by "frame mixing"--don't the time dilation and length contraction equations by definition involve two different frames, one labeled primed and one labeled unprimed? But it's still consistent in the sense that if you use unprimed to refer to the distance between the two mirrors in the clock's rest frame, then unprimed also refers to the time interval between the light going from one mirror to another in the clock's rest frame, and then you can use the time dilation and length contraction equations to get the distance between mirrors and the time between ticks in the frame where the light clock is moving.

I can see that it is useful to be able to work out the amount by which you would need to slow down a clock which is at rest relative to you to match a clock which is in motion relative to you.

I don't understand, the time dilation equation doesn't say anything about slowing down a clock at rest relative to you, it tells you how much time t' will have elapsed in your frame (i.e. how much time elapses on normal unslowed clocks at rest relative to you) when a moving clock ticks forward by some amount t.

But I still don't see the huge benefit associated with the pairing of time dilation and length contraction.

Length contraction and time dilation are both just useful for solving basic problems without using the full Lorentz transformation (for example, if a ship is traveling to a location 12 light year away in the Earth's frame at 0.6c relative to the Earth and you want to know what the ship's clock will read when it gets there, you can figure it out either using the time t' it takes in the Earth's frame and then applying the time dilation equation, or using the distance L' between Earth and the destination in the ship's frame, and then use time = distance/speed in that frame). And if you want to write these equations next to each other, it would be confusing if you didn't use the same convention for which notation you use for the rest frame of the clock/ruler you're talking about.

 

[neopolitan]

I don't know what you mean by "frame mixing"--don't the time dilation and length contraction equations by definition involve two different frames, one labeled primed and one labeled unprimed? But it's still consistent in the sense that if you use unprimed to refer to the distance between the two mirrors in the clock's rest frame, then unprimed also refers to the time interval between the light going from one mirror to another in the clock's rest frame, and then you can use the time dilation and length contraction equations to get the distance between mirrors and the time between ticks in the frame where the light clock is moving.

So, cutting and pasting your words to avoid mistakes:

using "unprimed to refer to the distance between the two mirrors in the clock's rest frame" (L) and noting that "unprimed also refers to the time interval between the light going from one mirror to another in the clock's rest frame" (t) and keeping in mind that this is a light clock where we are using a photon, then c = L / t.

Using your logic, you use the length contraction equation to get "the distance between mirrors" and time dilation to get "the time between ticks in the frame". How far does the photon get in how much time? That would be the speed of light: c = L' / t' = c/$$\gamma^2$$ ?

cheers,

neopolitan

 

[JesseM]

using "unprimed to refer to the distance between the two mirrors in the clock's rest frame" (L) and noting that "unprimed also refers to the time interval between the light going from one mirror to another in the clock's rest frame" (t) and keeping in mind that this is a light clock where we are using a photon, then c = L / t.

Using your logic, you use the length contraction equation to get "the distance between mirrors" and time dilation to get "the time between ticks in the frame". How far does the photon get in how much time? That would be the speed of light: c = L' / t' = c/$$\gamma^2$$ ?

No, because in the primed frame where the light clock is moving, the distance the light travels to get from the left mirror to the right mirror is not equal to the distance between the left and right mirror at a single instant, since both mirrors are moving in this frame. If the whole structure is going from left to right at speed v, and the light is moving at speed c in both directions, then as the light goes from left to right, the distance between the light pulse and the right mirror is shrinking at a "closing speed" of (c - v), while as the light goes from right to left, the distance between the light pulse and the left mirror is shrinking at a closing speed of (c + v). So, the time in this frame for the light to go from left mirror to right and back to left is L'/(c-v) + L'/(c+v) = 2cL'/(c^2 - v^2). So if t' is the time for the light to go from left to right and back in the frame where the light clock is moving, then t' = 2cL'/(c^2 - v^2). So, plugging in $$L' = L*\sqrt{1 - v^2/c^2}$$ and $$t' = t/\sqrt{1 - v^2/c^2}$$ gives:

$$t/\sqrt{1 - v^2/c^2} = 2cL*\sqrt{1 - v^2/c^2}/(c^2 - v^2)$$
multiplying both sides by $$\sqrt{1 - v^2/c^2}$$ gives:
t = 2cL*(1 - v^2/c^2)/(c^2 - v^2)
And since (1 - v^2/c^2) = (1/c^2)*(c^2 - v^2) this simplifies to:
t = 2cL/c^2 = 2L/c, which is exactly what we'd expect to be true in the unprimed frame where the light clock is at rest and the distance between the mirrors is L. Of course you could also reverse this algebra to show that, since the two-way time in the light-clock rest frame is t=2L/c, the two-way time in the frame where the light clock is moving must be t'=2cL'/(c^2 - v^2).

 

[neopolitan]

Yes, Jesse, I know that.

You are obliged to have two way travel to make sense of it.

And you still have the issue with redefining what t means.

This was your benefit of time dilation:

That it would be awfully confusing if primed referred to the rest frame of the clock while unprimed referred to the frame where the ruler was moving, or vice versa.

One primed frame (either moving or not moving, I don't care which). The first authoritive document on SR I read talked about K and K'. One primed frame, one unprimed frame.

Values in the primed frame are primed. Values in the unprimed frame are unprimed.

Why is this so difficult?

Time dilation applies in a clock's rest frame where consecutive ticks are colocal.

Length contraction applies in a length's rest frame where the ends of the length are simultaneous (along with the bits in the middle).

This is on http://en.wikipedia.org/wiki/Special_relativity#Time_dilation_and_length_contraction", and it has been there for a long time:

Writing the Lorentz transformation and its inverse in terms of coordinate differences we get

$$\Delta t' = \gamma . (\Delta t - \frac{v.\Delta x}{c^{2}})$$

$$\Delta x' = \gamma . (\Delta x - v \Delta t)$$

and

$$\Delta t = \gamma . (\Delta t' - \frac{v.\Delta x'}{c^{2}})$$

$$\Delta x = \gamma . (\Delta x' - v \Delta t')$$

Suppose we have a clock at rest in the unprimed system S. Two consecutive ticks of this clock are then characterized by Δx = 0. If we want to know the relation between the times between these ticks as measured in both systems, we can use the first equation and find:

$$\Delta t' = \gamma . \Delta t$$ for events satisfying $$\Delta x = 0$$

This shows that the time Δt' between the two ticks as seen in the 'moving' frame S' is larger than the time Δt between these ticks as measured in the rest frame of the clock. This phenomenon is called time dilation.

Similarly, suppose we have a measuring rod at rest in the unprimed system. In this system, the length of this rod is written as Δx. If we want to find the length of this rod as measured in the 'moving' system S', we must make sure to measure the distances x' to the end points of the rod simultaneously in the primed frame S'. In other words, the measurement is characterized by Δt' = 0, which we can combine with the fourth equation to find the relation between the lengths Δx and Δx':

$$\Delta x' = \frac{\Delta x}{\gamma}$$ for events satisfying $$\Delta t = 0$$

This shows that the length Δx' of the rod as measured in the 'moving' frame S' is shorter than the length Δx in its own rest frame. This phenomenon is called length contraction or Lorentz contraction.

In that text I see the words, I can understand the words, but I see that different versions of the Lorentz transformation are used. It is equivalent to either mixing frames or redefining what t means.

I am not saying it is wrong. It is clearly the preferred way of doing things. I am just not convinced that there is any real benefit in doing things that way.

cheers,

neopolitan

 

[JesseM]

Yes, Jesse, I know that.

Then what were you asking? Why did you say "using your logic ... the speed of light: c = L' / t' " if you knew perfectly well "my logic" would not give me that equation?

You are obliged to have two way travel to make sense of it.

You can make sense of one-way travel too if you keep in mind that stopwatches situated at the position of each mirror which are synchronized in the mirror's frame will be out-of-sync by vL/c^2 in the frame where the mirror is moving at speed v. In this case, in the frame where the mirror is moving, it takes a time of t' = L'/(c-v) for the light to get from the back mirror to the front mirror. So, in this time, the stopwatch at the front mirror has ticked forward from its starting time by t = L*(1 - v^2/c^2)/(c-v). But in the primed frame the front stopwatch started vL/c^2 behind the back watch, so the time on the front watch when the light reaches it is only ahead of the time on the back watch when the light left it by [L*(1 - v^2/c^2)/(c-v)] - [vL/c^2], which can be rewritten as [Lc^2*(1 - v^2/c^2) - vL*(c-v)]/[c^2*(c-v)] = [L*(c^2 - v^2) - vLc + Lv^2]/[c^2*(c - v)] = [Lc*(c-v)]/[c^2*(c-v)] = Lc/c^2 = L/c. And of course, if we consider things in the light clock's rest frame, it makes perfect sense that if we have synchronized stopwatches next to each mirror, the time on the stopwatch next to the front mirror when the light reaches it will be greater than the time on the stopwatch next to the back mirror when the light reaches it by L/c.

And you still have the issue with redefining what t means.

Where did I redefine it?

This was your benefit of time dilation:

That it would be awfully confusing if primed referred to the rest frame of the clock while unprimed referred to the frame where the ruler was moving, or vice versa.

One primed frame (either moving or not moving, I don't care which). The first authoritive document on SR I read talked about K and K'. One primed frame, one unprimed frame.

Values in the primed frame are primed. Values in the unprimed frame are unprimed.

Why is this so difficult?

Why is what so difficult? I have never said anything that contradicted that, have I?

Time dilation applies in a clock's rest frame where consecutive ticks are colocal.

It's not clear what you mean by "applies in". Obviously in the clock's own rest frame, its time is not dilated! It's in the frame where the clock is moving that its ticks are dilated relative to ticks of coordinate time in that frame.

Length contraction applies in a length's rest frame where the ends of the length are simultaneous (along with the bits in the middle).

Same confusion about your use of "applies in".

This is on http://en.wikipedia.org/wiki/Special_relativity#Time_dilation_and_length_contraction", and it has been there for a long time:

In that text I see the words, I can understand the words, but I see that different versions of the Lorentz transformation are used. It is equivalent to either mixing frames or redefining what t means.

What different versions? Can you please be specific about what particular equations on that wikipedia page you think are using contradictory definitions or are mixing frames? Everything looks consistent to me, they are using $$\Delta t$$ to refer to a time interval on a clock at rest in the unprimed frame, and $$\Delta x$$ to refer to the distance between ends of an object at rest in the unprimed frame. And of course, x and t refer to the coordinates of particular events in the unprimed frame (which is a slightly different convention than I was using, but theirs is probably more clear as I didn't use a notation for time intervals that was clearly distinct from notation for time coordinates, although I did use 'L' rather than 'x' for spatial intervals).

I am not saying it is wrong. It is clearly the preferred way of doing things. I am just not convinced that there is any real benefit in doing things that way.

Doing things that way as opposed to what other way? You really need to be way more specific about what exactly you're objecting to and what alternative you think would make more sense, I can't make heads or tails of what you're complaining about.

 

[neopolitan]

It's not clear what you mean by "applies in". Obviously in the clock's own rest frame, its time is not dilated! It's in the frame where the clock is moving that its ticks are dilated relative to ticks of coordinate time in that frame.

Time dilation $$t' = \gamma t$$ applies in the clock's own rest frame where a rest frame is unprimed. The prime is an indication that something different is happening. Same with length contraction, it applies in the frame in the length's own rest frame where the rest frame is unprimed.

What I am saying is that the frames are not the same. You can see that in the wikipedia article quite clearly.

What different versions? Can you please be specific about what particular equations on that wikipedia page you think are using contradictory definitions or are mixing frames? Everything looks consistent to me, they are using $$\Delta t$$ to refer to a time interval on a clock at rest in the unprimed frame, and $$\Delta x$$ to refer to the distance between ends of an object at rest in the unprimed frame. And of course, x and t refer to the coordinates of particular events in the unprimed frame (which is a slightly different convention than I was using, but theirs is probably more clear as I didn't use a notation for time intervals that was clearly distinct from notation for time coordinates, although I did use 'L' rather than 'x' for spatial intervals).

The second pair of equations in the article are the "inverse in terms of coordinate differences".
The first pair of equations give an observer in the unprimed frame the distance and time difference between two events from the perspective of an observer in the primed frame, using the measurements that the observer in the primed frame would use (contracted distances, longer periods between ticks in the primed frame).

The second pair of equations give an observer in the primed frame the distance and time difference between two events from the perspective of an observer in the unprimed frame, using the measurements that the observer in the unprimed frame would use (contracted distances, longer periods between ticks in the unprimed frame).

Because the Lorentz boosts like the Gallilean boosts use a version t such that x/t makes sense, where you could stand there counting ticks and the number of ticks is your value of t, the authors of the wikipedia article had to change frames between the derivations of length contraction and time dilation.

The authors mix frames to get around the redefinition of t problem.

Either t is redefined (from number of ticks to time between ticks) while still using the same sort of notation, or you've got frame mixing.

That's the source of a lot of confusion. And I still can't see what the benefits justifying this confusion are.

cheers,

neopolitan

BTW I usually don't mean "you" personally, but when I said you were obliged to use two way travel to make sense, I meant you personally. You can make sense of it with a one way trip (impersonal "you"). I am assuming that you know that, but also realize that one may have to complicate the issue further by bringing in simultaneity type issues. Anyway, the distance traveled between leaving one mirror and hitting the other mirror divided by the time taken to move between those mirrors will give you the speed of light in all frames. I do hope we don't disagree on that?

When I wrote L'/t' I was meaning (in context) the primed observer's length divided by the primed observer's time. Your equations didn't use the same definitions. Your quoting of me left out, quite inconveniently, the words "the time between ticks in the frame" where it was clear that "the frame" was the frame where the light clock is moving since I was responding to you where you said "you can use the time dilation and length contraction equations to get the distance between mirrors and the time between ticks in the frame where the light clock is moving".

Then you went and used "the time between ticks in the resting frame". So yes, I know what you did. You changed definitions midstream again.

It reminds me of the scene in A Fish Called Wanda, where Kevin Kline is told to raise his other hand so he does, while putting down the first hand. "Don't change definitions." Okay, I'll mix frames then! Tada, same answer! So I have to say, don't mix frames and don't change definitions and then there is no confusion. Or, alternatively, do either but make clear that you are changing definitions or mixing frames and explain why it helps to do so.

It's that last little bit I am after.

 

[JesseM]

Time dilation $$t' = \gamma t$$ applies in the clock's own rest frame where a rest frame is unprimed.

You are just repeating yourself without giving any answer to my question of what "applies in" means. Do you agree that in the clock's rest frame, its ticks are not dilated relative to coordinate time in that frame? Do you agree that the clock's ticks are not "dilated" in any objective frame-independent sense? If so, what do you mean by "time dilation applies in the clock's own rest frame"?

The prime is an indication that something different is happening.

Again, a totally vague and incomprehensible statement if you don't give more specifics. What is different from what, exactly? Primed and unprimed are just arbitrary conventions for identifying two different frames, but each frame's view of the other is totally symmetrical--a clock at rest in the primed frame would be dilated in the unprimed frame just as a clock at rest in the unprimed frame is dilated in the primed frame.

What I am saying is that the frames are not the same.

The laws of physics are the same in each frame, even if the behavior of specific objects is different in different frames.

You can see that in the wikipedia article quite clearly.

Where?

The second pair of equations in the article are the "inverse in terms of coordinate differences".

And when you say they are "not the same", are you referring to the fact that the first pair of equations have minus signs while the second pair have plus signs? If so, this is only because we have defined "v" in a specific way--it is the velocity of the origin of the primed frame along the x-axis of the unprimed frame (so a positive value for v means the origin is moving in the +x direction, a negative value means it's moving in the -x direction). If you instead define v' to be the velocity of the origin of the unprimed frame along the x' axis of the primed frame, then the second pair of equations would have a minus sign instead:
$$\Delta t = \gamma(\Delta t' - v' \Delta x' /c^2)$$
$$\Delta x = \gamma(\Delta x' - v' \Delta t')$$

The first pair of equations give an observer in the unprimed frame the distance and time difference between two events from the perspective of an observer in the primed frame, using the measurements that the observer in the primed frame would use (contracted distances, longer periods between ticks in the primed frame).

The notion of what is determined by an "observer" in a given frame usually is just a cute way of talking about measurements in that frame and that frame alone, so it's totally confusing to talk about an observer in one frame learning the values of measurements made in a different frame. Let's just drop the talk of "observers", OK? The first pair of equations tells us the distance and time intervals between two events in the primed frame if we already know the distance and time intervals between the same two events in the unprimed frame.

The second pair of equations give an observer in the primed frame the distance and time difference between two events from the perspective of an observer in the unprimed frame, using the measurements that the observer in the unprimed frame would use (contracted distances, longer periods between ticks in the unprimed frame).

The second pair of equations tells us the distance and time intervals between two events in the unprimed frame if we already know the distance and time intervals between the same two events in the primed frame. Agreed?

Because the Lorentz boosts like the Gallilean boosts use a version t such that x/t makes sense, where you could stand there counting ticks and the number of ticks is your value of t, the authors of the wikipedia article had to change frames between the derivations of length contraction and time dilation.

Where? I asked you to refer to the specific steps and equations you have a problem with when making complaints, you aren't doing so--if you refuse to do so I will have to bow out of this conversation. I see no way in which your claim makes sense, since in the time dilation and length contraction equations they consistently use $$\Delta t$$ to refer to the time between events on a clock at rest in the unprimed frame, and $$\Delta x$$ to refer to the distance between events on either end of an object which is at rest in the unprimed frame.

Either t is redefined (from number of ticks to time between ticks) while still using the same sort of notation, or you've got frame mixing.

$$\Delta t$$ always refers to the number of ticks of coordinate time in the unprimed frame between two events (and for events which occur at the same position in the unprimed frame, this is equal to the number of ticks between the events on a clock at rest at that position), if you think it's ever used differently you're confused about something.

 

[neopolitan]

And when you say they are "not the same", are you referring to the fact that the first pair of equations have minus signs while the second pair have plus signs? If so, this is only because we have defined "v" in a specific way--it is the velocity of the origin of the primed frame along the x-axis of the unprimed frame (so a positive value for v means the origin is moving in the +x direction, a negative value means it's moving in the -x direction). If you instead define v' to be the velocity of the origin of the unprimed frame along the x' axis of the primed frame, then the second pair of equations would have a minus sign instead:
$$\Delta t = \gamma(\Delta t' - v' \Delta x' /c^2)$$
$$\Delta x = \gamma(\Delta x' - v' \Delta t')$$

Actually no, I was referring to the primes. Good try though.

The notion of what is determined by an "observer" in a given frame usually is just a cute way of talking about measurements in that frame and that frame alone, so it's totally confusing to talk about an observer in one frame learning the values of measurements made in a different frame. Let's just drop the talk of "observers", OK? The first pair of equations tells us the distance and time intervals between two events in the primed frame if we already know the distance and time intervals between the same two events in the unprimed frame.

The second pair of equations tells us the distance and time intervals between two events in the unprimed frame if we already know the distance and time intervals between the same two events in the primed frame. Agreed?

Agreed

$$\Delta t$$ always refers to the number of ticks of coordinate time in the unprimed frame between two events (and for events which occur at the same position in the unprimed frame, this is equal to the number of ticks between the events on a clock at rest at that position), if you think it's ever used differently you're confused about something.

Ok, we are getting somewhere. According to me, how many ticks will I calculate to occur in the primed frame between two events if:
1. $$\Delta t$$ is the number of ticks in the unprimed frame between those two events,
2. a clock in the primed frame is has a speed of v relative to me,
3. I am at rest with the clock in the unprimed frame, and
4. I remember to take into account the inertial motion of the other frame (thus adding or deleting some ticks as required)?

More ticks or less ticks than the unprimed frame?

cheers,

neopolitan

 

[neopolitan]

You are just repeating yourself without giving any answer to my question of what "applies in" means. Do you agree that in the clock's rest frame, its ticks are not dilated relative to coordinate time in that frame? Do you agree that the clock's ticks are not "dilated" in any objective frame-independent sense? If so, what do you mean by "time dilation applies in the clock's own rest frame"?

I mean, that for time dilation the primed frame is the non-rest frame. It is the intent, is it not?

The same with length contraction. The primed frame is the non-rest frame. Perhaps you thought I was saying something more complicated. I wasn't.

cheers,

neopolitan

 

[Ich]

I'd guess you're lost in definitions:
Time dilation: t'=t*gamma - valid for objects at rest in S
Length contraction: L'=L/gamma - length of an object at rest in S (the ends measured simultaneously in S')

What you want to have:
c = l/T with
T = time between emission and absorption of a photon
l = distance between emission and absorption of a photon

Neither T nor l are defined like t or L. The simpler formulas do not apply here.
Use the doppler formula instead:
$$T' = T \sqrt{\frac{1-v}{1+v}}$$
$$l' = l \sqrt{\frac{1-v}{1+v}}$$

to see why you have to use a different formula, draw spacetime diagrams and apply the Lorentz trasformation, from which all those special cases are derived.
Space and time are not separate in SR, so you can't simply define one length and one time interval and apply both to situations that do not match the definition.

 

[JesseM]

Ok, we are getting somewhere. According to me, how many ticks will I calculate to occur in the primed frame between two events if:
1. $$\Delta t$$ is the number of ticks in the unprimed frame between those two events,
2. a clock in the primed frame is has a speed of v relative to me,
3. I am at rest with the clock in the unprimed frame, and
4. I remember to take into account the inertial motion of the other frame (thus adding or deleting some ticks as required)?

More ticks or less ticks than the unprimed frame?

Why are you specifying which frame "you" are at rest in here? That would seem totally irrelevant to what you're asking, nothing would change if you just said "how many ticks will occur in the primed frame between two events if:

1. $$\Delta t$$ is the number of ticks in the unprimed frame between those two events,
2. a clock at rest in the primed frame has a speed of v relative to the unprimed frame"

Anyway, the answer to whether it's more ticks or less ticks than the unprimed frame depends on what events you choose. If you choose two events on the worldline of a clock at rest in the unprimed frame, then it'll be more ticks in the primed frame than the unprimed frame because the clock is slowed down in the primed frame, so its ticks are extended by the amount given by the time dilation equation. If you choose two events on the worldline of a clock at rest in the primed frame, then it'll be less ticks in the primed frame because now the clock is slowed down in the unprimed frame. And if you pick a pair of events that don't occur at the same spatial location in either frame, you have to use the more general equation:

$$\Delta t' = \gamma (\Delta t - v \Delta x/c^2)$$

I mean, that for time dilation the primed frame is the non-rest frame. It is the intent, is it not?

The same with length contraction. The primed frame is the non-rest frame. Perhaps you thought I was saying something more complicated. I wasn't.

I've discussed this with you in the past, but when you talk about "the" rest frame in a given problem, you aren't really making much sense; you seem to have gotten in your head that when analyzing a particular problem we are supposed to pick one frame to label as "the rest frame" (or 'the observer's frame', which may be related to your insertion of an irrelevant observer above), but no such convention exists. The phrase "rest frame" is generally used only in the context of talking about some specific object; for example, if clock A is at rest in the unprimed frame and clock B is at rest in the primed frame, then the unprimed frame is "the rest frame of clock A" and the primed frame is "the rest frame of clock B".

 

[Bob S]

There have been more than a few threads where there clearly is confusion about the use of time dilation and length contraction.

People initially think that:

1. in an frame which is in motion relative to themselves, time dilates and lengths contract; and
2. velocities in a frame which is in motion relative to themselves are contracted lengths divided by dilated time.
neopolitan

A few years ago, experimenters at Brookhaven National Laboratory put a beam of muons (mass = 105.658 MeV) into a magnetic ring and stored them at a gamma of 29.3. The measured lifetime in the ring was about 64.4 microseconds, a factor of 29.3 longer than their measured lifetime at rest (2.2 microseconds). The dilated lifetime gave the experimenters more time to make accurate QED measurements on the muons. During the 64.4 microseconds, the muons traveled about beta*gamma*c*tau meters, where c and tau are the speed of light, and tau is the lifetime at rest. So time dilation works.

 

[neopolitan]

A few years ago, experimenters at Brookhaven National Laboratory put a beam of muons (mass = 105.658 MeV) into a magnetic ring and stored them at a gamma of 29.3. The measured lifetime in the ring was about 64.4 microseconds, a factor of 29.3 longer than their measured lifetime at rest (2.2 microseconds). The dilated lifetime gave the experimenters more time to make accurate QED measurements on the muons. During the 64.4 microseconds, the muons traveled about beta*gamma*c*tau meters, where c and tau are the speed of light, and tau is the lifetime at rest. So time dilation works.

I'm not denying that it works. I don't have a problem with the principle of time dilation. It's pretty much a notation issue.

However, you do give an example of what use time dilation has for which I thank you.

The lifetime at rest is 2.2 microseconds. The "contracted time" for the muons in magnetic ring was ... 2.2 microseconds, yes? If there was a clock stored in that magnetic ring at a gamma of 29.3 it would tick off 2.2 microseconds while clocks not stored in that magnetic ring would tick off 64.4 microseconds. If that clock had a rest length of L in the direction of motion (all sorts of problems here since the clock would have to rotate to keep that length in the direction of motion which is circular, but it's hypothetical), then that length in motion would be L/gamma.

$$L_{in. the. ring}=L_{at. rest. in. the. laboratory} / \gamma$$

$$\Delta t_{in. the. ring}=t_{at. rest. in. the. laboratory} / \gamma$$

I see no problem with using the inverse of "contracted time", or time dilation, to work out that if muons are stored at a gamma of 29.3 relative to the laboratory, then those muons will last 29.3 times longer. But that equation is:

$$\Delta t_{at. rest. in. the. laboratory}=\gamma . t_{in. the. ring}$$

It seems a little odd, under these circumstances, to call $$t_{in. the. ring}$$ a "rest frame" ( or the rest frame of one clock).

cheers,

neopolitan

 

[neopolitan]

Jesse,

Let me repeat the question. I will highlight something for you, hopefully it will answer the question you had before:

According to me, how many ticks will I calculate to occur in the primed frame between two events if:
1. $$\Delta t$$ is the number of ticks in the unprimed frame between those two events,
2. a clock in the primed frame is has a speed of v relative to me,
3. I am at rest with the clock in the unprimed frame, and
4. I remember to take into account the inertial motion of the other frame (thus adding or deleting some ticks as required)?

More ticks or less ticks than the unprimed frame?

cheers,

neopolitan

 

[JesseM]

Jesse,

Let me repeat the question. I will highlight something for you, hopefully it will answer the question you had before:

No, it doesn't. You are asking how many ticks will occur between two specific events "in the primed frame" (your words), so all that matters is the difference in time-coordinate between these events in the primed frame, the fact that you have defined "yourself" to be at rest in the unprimed frame is irrelevant to the problem as you've stated it. If you instead wanted to know how many ticks occur between these events in your rest frame, then the issue of which frame you were at rest in would be relevant, but that isn't what you asked.

 

[Mentz114]

neopolitan:

It seems a little odd, under these circumstances, to call $$t_{in. the. ring}$$ a "rest frame" ( or the rest frame of one clock).

Why ? The muon is at rest in it's own frame.

 

[neopolitan]

I've discussed this with you in the past, but when you talk about "the" rest frame in a given problem, you aren't really making much sense; you seem to have gotten in your head that when analyzing a particular problem we are supposed to pick one frame to label as "the rest frame" (or 'the observer's frame', which may be related to your insertion of an irrelevant observer above), but no such convention exists. The phrase "rest frame" is generally used only in the context of talking about some specific object; for example, if clock A is at rest in the unprimed frame and clock B is at rest in the primed frame, then the unprimed frame is "the rest frame of clock A" and the primed frame is "the rest frame of clock B".

Jesse,

You have an equation, right, $$t' = \gamma t$$. One t is primed, one t is not primed. The equation is discussing the effects of motion on time intervals with the underlying assumption that one value applies to a clock in one frame and one value applies another clock in another frame and so long as $$\gamma$$ is not equal to 1, the clock, and therefore the frames, are not at rest relative to each other. Look at the equation. It is taking the perspective of one clock in it's rest frame. I know you can swap the perspective over, from clock A's perspective to clock B's perspective, if you like - but still t will be the time interval for the clock whose perspective we are examining, in other words the frame in which the clock whose perspective we are examining is at rest. In terms of the equation, we could call the unprimed frame "the rest frame". But it is context.

Take it out of context (ie don't have an equation, just compare the frames) and yes, you can't justify referring to either frame as "the rest frame". It is writing the equation that tags one of the frames as "the rest frame" (in context). I've not talked about the frames at all without referring to a specific equation.

cheers,

neopolitan

 

[JesseM]

However, you do give an example of what use time dilation has for which I thank you.

The lifetime at rest is 2.2 microseconds. The "contracted time" for the muons in magnetic ring was ... 2.2 microseconds, yes? If there was a clock stored in that magnetic ring at a gamma of 29.3 it would tick off 2.2 microseconds while clocks not stored in that magnetic ring would tick off 64.4 microseconds.

Yes, if by "stored in that magnetic ring" you mean "traveling along with a muon in the ring which has been accelerated to a relativistic velocity relative to the lab"

$$\Delta t_{at. rest. in. the. laboratory}=\gamma . t_{in. the. ring}$$

It seems a little odd, under these circumstances, to call $$t_{in. the. ring}$$ a "rest frame" ( or the rest frame of one clock).

Why do you find it odd? Doesn't $$t_{in. the. ring}$$ refer to the time interval in the coordinate system where the clock's position-coordinate is constant over time? That is all that "the rest frame of an object" means", the frame in which it has constant position coordinate (and is therefore at rest in that frame). You seem to be confused about the meaning of the term "rest frame", although I don't really understand what you think it means.