Benefits of time dilation / length contraction pairing?

[JesseM]

That definition is correct, although I would imagine a new student would need to be eased into it.

I can see why you can't make sense of L/t = c = L'/t'.

You specifically want to measure a time interval between two events in the primed frame and then compare that to a time inverval in the unprimed frame.

I wasn't doing that. I was saying that any time inverval in the primed frame between two events which are colocal in the primed frame, will be shorter in the unprimed frame than two analogous (but not the same) events in the unprimed frame. The half-life of one muon in the primed frame (viewed from the primed frame) will be the same as the half-life of a totally different muon in the unprimed frame (viewed from the unprimed frame. (Yes, I know half-lives are statistical, but using a gross misrepresentation here might still be instructive.)

What I am saying is that the half-life of the muon in the primed frame (viewed from the primed frame) will be less than the half-life of the muon in the primed frame (viewed from the unprimed frame).

OK, agree with you so far.

In the example BobS raised earlier in the thread, a muon at a gamma of 29.3 had a measured life time of 64.4ms as opposed to a normal (gamma of 1) life time of 2.2ms.

In the experiment he refers to, I would call the measured lifetime t

OK, as long as you are aware that here you are using the reverse of the "normal" convention, which is to use the unprimed frame for the rest frame of the "clock" (in this case the natural clock provided by the muon's decay) and the primed frame the frame where we are measuring the time interval between events on the worldline of a moving clock. If you want to reverse this and call the muon's rest frame the primed frame, then the "normal" time dilation equation would be written as \Delta t = \Delta t' * \gamma, the "reversed time dilation equation" would be written as \Delta t' = \Delta t / \gamma, and the TAFLC would be written as \Delta t = \Delta t' / \gamma.

and I could use the gamma to calculate what the life time in muon's "rest frame" was (quotation marks because "rest frame" is a bit of a misnomer under the circumstances). I'd prime the rest frame of the muon and leave the laboratory rest frame unprimed. That would give me:

t' = t/gamma = 64.4ms / 29.3 = 2.2ms

Yes. But just to be clear about terminology, do you agree that this is not the TAFLC, but just the reversed version of the regular time dilation equation?

If I had a different experiement, using light clocks, this is how I would be doing it.

At rest in the laboratory, my light clock has a tick time of 2.2ms. That makes the distance between mirrors ct/2 = 330km (giving a L = 660km, the total distance a photon travels between ticks).

Conceptually, put the light clock at a gamma factor of 29.3 (in reality, this would prove difficult).

I will measure, in the laboratory, that the time between ticks of the light clock is now 64.4ms.

This 64.6ms is the t which is equivalent to the t from the muon example. It is not equivalent to the t which I used in ct/2 = 330km (that t was 2.2ms).

What I do know is that, in the laboratory's frame, the photon in the light clock has not traveled 330km in 64.4ms. As you showed before (using time dilation) the photon has to travel much further from one mirror to the other mirror in one direction and a bit less in the other direction.

So the distance traveled between ticks (in the laboratory) is not the same L as before but rather ct where t = 64.4ms ... eg, 19320km.

This L, divided by this t = 19320km/64.4ms = 300000 km/s

Yes.

The distance traveled in the rest frame of the light clock is the old L (330km) and the time a photon takes to travel between them and back again is the old t (2.2ms).

This L, divived by this t = 660km/2.2ms = 300000 km/s

For clarity we can call this distance in the light clock rest frame L' = 660 km and this time t' = 2.2 ms so it maps to your L/t = c = L'/t', correct? In this case, do you agree that t and t' are related not by the TAFLC but by the standard time dilation equation (written with your unusual convention of labeling the clock rest frame as the primed frame) t = t' * gamma? And do you also agree that L and L' are related not by the length contraction equation but by an equation which looks like the "spatial analogue of time dilation" (although I'm not sure L and L' can be assigned the same physical meaning) L = L' * gamma?

As long as you agree with this stuff I have no problem with the L/t = c = L'/t' argument, but I thought you had been saying that the TAFLC was the equation that was useful in understanding the invariance of c, not the time dilation equation. I guess if you want to say that the equation L = L' * gamma is useful for understanding the invariance of the speed of light that would have some truth, although I think this only works when you're talking about the two-way speed away from some fixed point in the clock's frame and back, and as I said I don't know if the physical meaning of L and L' here can be mapped to the "spatial analogue of time dilation" equation even though it looks the same.

Finally, you said earlier: "You specifically want to measure a time interval between two events in the primed frame and then compare that to a time inverval in the unprimed frame. I wasn't doing that." It seems to me you are doing that, with the two events being 1) the event of the photon leaving the bottom mirror of the light clock which is moving in the lab frame, and 2) the event of the photon returning to the bottom mirror of that same clock. The time between these events is t' = 2.2 ms in the clock rest frame and t = 64.4 ms in the lab frame. The part I had not understood was that you were not using L and L' to represent the distance between these events in the two frames, but rather the total distance covered by the photon in each frame between these two events; this would be identical to the distance between the events if the events were on a single straight photon worldline, but since you are talking about the two-way speed of light rather than the one-way speed of light, the photons are reflected so their worldlines aren't straight.

If you want to use the clock in the laboratory you as your reference point, you have to do this:

While a photon in the laboratory moves between mirrors, traveling 660km in 2.2ms - what happens to a photon which is at gamma of 29.3?

If 2.2ms has elapsed in the laboratory, then a period of 2.2ms/29.3 = 75μs will have elapsed in the rest frame of the test clock (the one accelerated to gamma of 29.3).

No, 75 microseconds would represent how much time has elapsed on the test clock (if the test clock had closer mirrors so it could show time-intervals that small) in 2.2 ms of time in the lab frame. In the test clock's own frame, it's the lab clock that's running slow relative to the test clock, so when the lab clock has ticked forward 2.2 ms, the test clock has ticked forward 64.4 ms.

 

[neopolitan]

There still seems to be some confusion.

We talk about L as it has to be a ruler or a rod or a length. They are convenient devices, but L could be a distance between two randomly selected points in a rest frame (I want to say my rest frame, but it can be any rest frame).

We talk about t as if it has to be attached to events, like ticking of a clock, or formally defined events. But t could be the time interval between two randomly selected times.

We can imagine putting two pins on a map and measuring the distance. We have difficulty putting two pins in time and measuring the temporal distance. But I take TAFLC as being for measuring between these two pins in time, in the same was a LC is for measuring between two pins on the map. We take a different perspective on them by putting us and pins into different inertial frames.

If I get myself an inertial frame where two time pins are in the same position, then they will be as far apart in time as they can be. If I get myself an inertial frame where the two length pins are simultaneous, then they will be as far apart in length as they can be.

But, assuming all the pins are in the same frame (ie they share a frame in which the time pins have zero length separation and the length pins have zero time separation), then from any other frame: t' = t/gamma and L'=L/gamma where t and L are the maximum time and length separations for the respective pins.

I'm deliberately using a different approach.

cheers,

neopolitan

 

[neopolitan]

No, 75 microseconds would represent how much time has elapsed on the test clock (if the test clock had closer mirrors so it could show time-intervals that small) in 2.2 ms of time in the lab frame. In the test clock's own frame, it's the lab clock that's running slow relative to the test clock, so when the lab clock has ticked forward 2.2 ms, the test clock has ticked forward 64.4 ms.

Simultaneity issues here. I was only taking the lab's perspective, one perspective at a time. But yes, you can reverse it around, and due to relativity of simultaneity, the muon will decay in its own frame while the lab clock reads 75 microsecond (but since we can't be muons anymore than we can be photons, it makes sense to use the lab perspective. I don't think that particle physicists would report that the muon decays after 75 microseconds on the lab clock when read from the muon's own frame in which a clock, if it could be accelerated to a gamma of 29.3, would read 2.2 ms).

 

[Mentz114]

(but since we can't be muons anymore than we can be photons, it makes sense to use the lab perspective.

Muons have mass, so we can 'be muons'. I thought you might like to know this.

 

[neopolitan]

Muons have mass, so we can 'be muons'. I thought you might like to know this.

Hm, if you were a muon, you wouldn't be one for long.

However, I don't think there is much difference between "I can't be a northern polka dotted, orange bellied, bearded unicorn" and "I can't be a ballet lady". I think there are a few good reasons why I can't be a muon (even though a muon has mass). Equally, I don't think that not having mass is the only thing preventing me from being a photon.

Perhaps I missed a key lecture at uni.

(There's another feeble attempt at sarcasm )

PS Have you got a thing about muons? It's just that you have only popped your head into make comment about them. If they are off limits or something, just let me know and I will use another example.

 

[JesseM]

Damn, I edited this post when I meant to reply to it to add a small comment, hence erasing everything else but the small new comment...I'll have to try to reconstruct it.

 

[Rasalhague]

If I get myself an inertial frame where two time pins are in the same position, then they will be as far apart in time as they can be. If I get myself an inertial frame where the two length pins are simultaneous, then they will be as far apart in length as they can be.

But, assuming all the pins are in the same frame (ie they share a frame in which the time pins have zero length separation and the length pins have zero time separation), then from any other frame: t' = t/gamma and L'=L/gamma where t and L are the maximum time and length separations for the respective pins.

That's a very neat summary. It brings out very clearly where the symmetry lies (between time and space), and where the difference lies (between (1) frames in which a timelike separation has no space component, or frames in which a spacelike separation has no time component, and (2) other frames in which the separation, timelike or spacelike, has a mixture of time and space coordinates). Finger's crossed I've got the terminology corrent there...

 

[Rasalhague]

That's a very neat summary. It brings out very clearly where the symmetry lies (between time and space), and where the difference lies (between (1) frames in which a timelike separation has no space component, or frames in which a spacelike separation has no time component, and (2) other frames in which the separation, timelike or spacelike, has a mixture of time and space coordinates). Finger's crossed I've got the terminology corrent there...

Ah, I just read Jesse's reply after I posted this. I see the point about it being the minimum separation. Taking that into account, it does still seem a satisfying way of looking at it.

 

[Rasalhague]

"given that two events on the worldline of a clock at rest in the primed frame are separated by a coordinate time of \Delta t in the unprimed frame, how much clock time (or coordinate time in the primed frame where the clock is at rest) passes between those two events?" Can you think of a more intuitive way to express the physical significance?

Is this equivalent to saying: "Two events are separated by a timelike interval \Delta \tau. In frame S, this separation has a time component \Delta t > \Delta \tau. Given the value of \Delta t, how can we calculate \Delta \tau? Answer: \Delta \tau = \Delta t / \gamma. The inverse question being: "Given the value of \Delta \tau, how can we calculate \Delta t? Answer: \Delta t' = \Delta \tau * \gamma.<br /> <br /> Alternatively:<br /> <br /> Given two events E_{a1} and E_{a2}, colocal in some frame S, with (time) interval \Delta t, what is the (time) interval \Delta t' in some other frame, moving at constant velocity u relative to S, between two events E_{b1} and E_{b2}, colocal in S', if t_{a1} = t_{b1}, and t_{a2} = t_{b2}?<br /> <br /> \Delta t' = \Delta t / \gamma.<br /> <br /> As opposed to time dilation:<br /> <br /> Given two events E_{a1} and E_{a2}, colocal in some frame S, with (time) interval \Delta t, what is the (time) interval \Delta t' in some other frame S', moving at constant velocity u relative to S, between two events E_{b1} and E_{b2}, colocal in S', if t'_{a1} = t'_{b1}, and t'_{a2} = t'_{b2}?<br /> <br /> \Delta t' = \Delta t * \gamma.<br /> <br /> So would it be fair to say that there really is no fundamental or physical difference between "reverse time dilation" and "temporal analogue of length contraction" ("time contraction")? They ask the same question, only with different names given to the frames. If the problem you're working on only involves one question, or if it only involves asking one type of question of one frame, and the other type of question of the other frame, then you can avoid ever having to use the form \Delta t' = \Delta t / \gamma, and instead always use \Delta t = \Delta t' / \gamma. But if you want to ask both types of question in both directions, then you'd have to use \Delta t' = \Delta t / \gamma, wouldn't you? Or else swap over the labels you've given to the frames as the occasion demands.

 

[Rasalhague]

The garbled text in my previous post should have read:

Is this equivalent to saying: "Two events are separated by a timelike interval \Delta \tau. In frame S, this separation has a time component \Delta t > \Delta \tau. Given the value of \Delta t, how can we calculate \Delta \tau? Answer: \Delta \tau = \Delta t / \gamma. The inverse question being: "Given the value of \Delta \tau, how can we calculate \Delta t? Answer: \Delta t' = \Delta \tau * \gamma.

 

[Rasalhague]

Actually, writing it out in these terms and then thinking about how I'd write out the TAFLC equation in words makes me realize that the question of whether there's really any difference between the TAFLC equation and the reversed time dilation equation is actually rather subtle. If you look at my diagram, you see that the TAFLC isn't really giving you the time-interval between any pair of pink events at all, since none of the events are at the position of the top of the double-headed arrow that I use to represent the delta-t' of the TAFLC equation; it's only if you were to draw a new pair of events that are colocated in the primed frame, at the top and bottom of that double-headed arrow on that diagram, that the TAFLC would tell you the same think about the time between those new events in both frames that the reversed time dilation equation tells you about the time between the colocated events in the primed frame. So I guess what that would mean conceptually is that if you choose your pair of events at the start, then the time dilation equation + reversed time dilation equation tell you everything you need about the relation between the time intervals connecting those specific events in your two frames (one of which must be the frame where they're colocated). In this context the TAFLC equation is actually not telling you about the time-interval between those specific events in either frame, although you could of course draw in some new events such that the times delta-t and delta-t' in the TAFLC equation had the same meaning for that new pair of events that the times delta-t and delta-t' in the time dilation (and reversed time dilation) equation have for the original pair of events. But then if you want to talk about the time between the new pair, why not just start over and have them be the starting events? I guess conceptually what I would say is that to use any of these time equations you should always be clear on what two events you're interested in at the start, and once you've picked them then it's the time dilation and reversed time dilation equation that tell you the relation between the time-intervals in both frames, while the TAFLC is telling you something more abstract about the time in the non-colocated frame between planes of simultaneity from the colocated frame that pass through both events.

But couldn't you look at the conventional time dilation equation in a similar way? In each case you want to know something about the timing of two events. You specify something about the events which you want information about (which other events they have to be simultaneous with, and according to whose definition of simultaneity), in both cases without knowing exactly which events you're looking for, and the equations tell you. It could well be that I'm missing the subtlety though. I need to read these posts more carefully and think this over.

 

[JesseM]

OK, here's the recreation of the last post I accidentally edited away:

There still seems to be some confusion.

We talk about L as it has to be a ruler or a rod or a length. They are convenient devices, but L could be a distance between two randomly selected points in a rest frame (I want to say my rest frame, but it can be any rest frame).

We talk about t as if it has to be attached to events, like ticking of a clock, or formally defined events. But t could be the time interval between two randomly selected times.

We can imagine putting two pins on a map and measuring the distance. We have difficulty putting two pins in time and measuring the temporal distance. But I take TAFLC as being for measuring between these two pins in time, in the same was a LC is for measuring between two pins on the map. We take a different perspective on them by putting us and pins into different inertial frames.

But look at my diagram again. If the pink dots are the pins, with two colocated in the unprimed frame and two simultaneous in the unprimed frame, then it is actually the time dilation equation that compares the time in the two frames between the events that are colocated in the unprimed frame, and the "spatial analogue for time dilation" (SAFTD) equation that compares the distances in the two frames between the events that are simultaneous in the unprimed frame. The TAFLC equation doesn't tell you the time between any pair of pink events in the diagram, although you could invent a new pair of events such that it would--these new events would have to be colocated in the primed frame.

If I get myself an inertial frame where two time pins are in the same position, then they will be as far apart in time as they can be.

Actually that's backwards, the time between events is minimized in the frame where they're at the same position. Suppose I have been moving inertially my whole life, and one event is the event of my birth while the other is the event of my turning 30. The time between these events is 30 years in the frame where I am at rest and they occur at the same location, but in a frame where I am moving there is a greater time between the events because I am aging more slowly.

If I get myself an inertial frame where the two length pins are simultaneous, then they will be as far apart in length as they can be.

That's not quite correct either. If you want to analyze length contraction in terms of just two events rather than three (in the case of three, #1 would be an event on the worldline of the object's left end, #2 would be an event on the worldline of the object's right end that's simultaneous with #1 in the object's rest frame, and #3 would be an event on the worldline of the right end that's simultaneous with #1 in the frame where the object is moving), then you have to pick two events on the worldline of either end of the object that are simultaneous in the frame where the object is moving, but non-simultaneous in the object's rest frame (since both ends of the object have a constant position in the object's rest frame, events on either end will still be separated by the rest length L even if they aren't simultaneous). The distance between these events will be greater in the object's rest frame where they're non-simultaneous (because rest length is greater than moving length), so they aren't at a maximal separation in the frame where the events are simultaneous. In fact it turns out that events will actually have a minimal spatial distance in the frame where they are simultaneous, you can see this by considering the more general equation for the separation between events in two arbitrary frames:

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

If you choose the unprimed frame to be the one where they're simultaneous, then \Delta t = 0 so you're left with \Delta x' = \gamma * \Delta x, which shows that the distance is always greater in the non-simultaneous frame.

Aside from these caveats, I agree with the idea that you can define the meaning of the two frames in equations like time dilation by first picking two events and then making clear which is supposed to be the frame where they are colocated (if they are timelike-separated) or which is supposed to be the frame where they are simultaneous (if spacelike-separated). Writing it out in words, the standard time dilation equation would be:

(time between events in frame where they are not colocated) = (time between events in frame where they are colocal) * gamma

Likewise, the reversed time dilation equation would be:

(time between events in frame where they are colocal) = (time between events in frame where they are not colocated) / gamma

Thinking about writing it in words, it may seem a bit subtle to say what the difference is between the TAFLC equation and the reversed time dilation equation. As I said, if you look at my diagram you see that the double-headed arrow representing the dt' in the TAFLC does not have any of the three pink events at the top end of it; you would have to invent a new pair of events at either end of this double-headed arrow in order to phrase the TAFLC in terms of time intervals between events, and in that case you would write it exactly like the reversed time dilation equation above, except with the understanding that you were now referring to that new pair of events. So the way I would conceptualize this situation is to say that in order to talk about any of these equations, you first have to specify a single pair of events you want to talk about, and then in terms of those specific events the time dilation and reversed time dilation equations tell you everything you want to know about the time interval between the events in two frames (one of which is the one where they're colocated), whereas in terms of those events the TAFLC is telling you something more abstract about the time-interval (in the frame where the events are not colocated) between surfaces of simultaneity from the the frame where the events are colocated. Of course you could start with a new pair of events so that the time interval given by the TAFLC applied to the previous events is just the time interval between the new events in the frame where they're colocated, but then you're really talking about the reversed time dilation equation for these new pair of events, not the TAFLC for them.

But, assuming all the pins are in the same frame (ie they share a frame in which the time pins have zero length separation and the length pins have zero time separation), then from any other frame: t' = t/gamma and L'=L/gamma where t and L are the maximum time and length separations for the respective pins.

As I said above, t and L should be the minimum time and distance separation for the pins, there is no upper limit on their separations (there is an upper limit on the length of a physical object when viewed in different frames, but the concept of the length of an object in different frames is quite different from the concept of the spatial distance between a pair of events in different frames). And if the unprimed frame is the one where the time pins are colocated and the space pins are simultaneous, then the equations above are incorrect, they should be t' = t*gamma and L' = L*gamma, representing the standard time dilation equation along with the SAFTD equation. Do you disagree?

 

[JesseM]

Ah, I just read Jesse's reply after I posted this. I see the point about it being the minimum separation. Taking that into account, it does still seem a satisfying way of looking at it.

Also see the points I made in the re-created version of that post (the original of which I accidentally deleted) about the differences between the concept of the length of a physical object in different frames vs. the concept of the distance between a pair of events in different frames. Even though the length of an object is maximized in its rest frame, the distance between a pair of events is minimized in the frame where they are simultaneous.

 

[Mentz114]

neopolitan;
no, I don't have a thing about muons. I did not introduce the subject so your comment makes no sense. You give the impression that muons can't have a frame of reference, in which you are wrong. I'm trying to shine some light here into your fog of misunderstanding, and you respond with insults and sarcasm.

I enjoyed your little biog about talking to people ( Professors even ) about your doubts and problems with relativity. I hope you get cured soon because it's costing some people an awful lot of effort.

M

 

[JesseM]

Is this equivalent to saying: "Two events are separated by a timelike interval \Delta \tau.

OK, that would be equivalent to the proper time along the worldline of an inertial object that goes from one event to the other, which of course is the same as the coordinate time between the events in that object's rest frame, where the events occur at the same coordinate position.

In frame S, this separation has a time component \Delta t > \Delta \tau. Given the value of \Delta t, how can we calculate \Delta \tau? Answer: \Delta \tau = \Delta t / \gamma. The inverse question being: "Given the value of \Delta \tau, how can we calculate \Delta t? Answer: \Delta t = \Delta \tau * \gamma.

Yes, although your "inverse question" corresponds to the normal time dilation equation (with the most common notation being to use a primed t' where you've used an unprimed t, and an unprimed t where you've used \tau), whereas your first question corresponds to what I've called the "reversed time dilation equation" (where you just divide both sides of the normal time dilation equation by gamma).

Alternatively:

Given two events E_{a1} and E_{a2}, colocal in some frame S, with (time) interval \Delta t, what is the (time) interval \Delta t' in some other frame, moving at constant velocity u relative to S, between two events E_{b1} and E_{b2}, colocal in S', if t_{a1} = t_{b1}, and t_{a2} = t_{b2}?

\Delta t' = \Delta t / \gamma.

Since you wrote t_{a1} = t_{b1} rather than t'_{a1} = t'_{b1}, I take it you want these events to be simultaneous in the unprimed frame rather than the primed frame? If so, then if we want to conceptualize this in terms of the coordinate time in two frames between a single pair of events as in neopolitan's formulation, then we're really talking about the second pair of events E_{b1} and E_{b2} here; we know the time between them in the unprimed frame, and want to know the time between them in the primed frame where they are colocated. So, this would indeed be the "reversed time dilation equation" you have above, but it would be the opposite of the usual convention about primed and unprimed (the usual convention being that the frame in which the two events are colocated would be the unprimed one).

As opposed to time dilation:

Given two events E_{a1} and E_{a2}, colocal in some frame S, with (time) interval \Delta t, what is the (time) interval \Delta t' in some other frame S', moving at constant velocity u relative to S, between two events E_{b1} and E_{b2}, colocal in S', if t'_{a1} = t'_{b1}, and t'_{a2} = t'_{b2}?

\Delta t' = \Delta t * \gamma.

Yes, although if we think in terms of a single pair of events as before, here you've reversed the convention about which frame is the one where they're colocated.

So would it be fair to say that there really is no fundamental or physical difference between "reverse time dilation" and "temporal analogue of length contraction" ("time contraction")? They ask the same question, only with different names given to the frames.

I don't think so--as I said in my post to neopolitan, if you think in terms of starting with a pair of events and then asking various questions about time-intervals involving those specific events, then the TAFLC equation is really asking something more like "in the frame where the events are not colocated, what is the temporal separation between two surfaces of simultaneity from the frame where they are colocated, given that each surface passes through one of the two events?" But this point about starting with a single pair of events brings me to your next post where you were responding to a similar comment from the post I accidentally deleted:

But couldn't you look at the conventional time dilation equation in a similar way? In each case you want to know something about the timing of two events. You specify something about the events which you want information about (which other events they have to be simultaneous with, and according to whose definition of simultaneity), in both cases without knowing exactly which events you're looking for, and the equations tell you. It could well be that I'm missing the subtlety though. I need to read these posts more carefully and think this over.

I think you always have to know what the events are physically, like particular readings on a physical clock, or any other observed events you like, and are then interested in saying various things relating to how different coordinate systems view them, like the difference in coordinate time between the events or which readings on a different physical clock are simultaneous with these events in a particular frame (and what the difference is between the two readings on that clock). I suppose you can ask questions in such a way that you don't know both events in advance, like "which reading on this clock occurs at a time interval of \Delta t after the clock reading 0 in my frame", but for the question to be well-defined it must uniquely determine the events in question even if you don't know them until you do some calculations.

If the problem you're working on only involves one question, or if it only involves asking one type of question of one frame, and the other type of question of the other frame, then you can avoid ever having to use the form \Delta t' = \Delta t / \gamma, and instead always use \Delta t = \Delta t' / \gamma. But if you want to ask both types of question in both directions, then you'd have to use \Delta t' = \Delta t / \gamma, wouldn't you? Or else swap over the labels you've given to the frames as the occasion demands.

But what do you mean by "both types of questions"? What events are you asking questions about? If you're asking about more than a single pair of events then in that case I'd agree you might use both of those equations to talk about time intervals between events, but since you're no longer talking about a single pair of events you'd have to have some different notation to distinguish between verbal formulations like "time-interval in the unprimed frame between events A and B" and "time-interval in the unprimed frame between events C and D"--perhaps you could use \Delta t_{AB} and \Delta t_{AC} in this case. Then if A and B are colocated in the primed frame while C and D are colocated in the unprimed frame, you might write \Delta t'_{AB} = \Delta t_{AB} / \gamma along with \Delta t_{CD} = \Delta t'_{CD} / \gamma, but I would refer to the first as "the reversed time dilation equation for events A and B" and the second as "the reversed time dilation equation for events C and D", in words they would both come out to:

(time between specified events in frame where they are colocated) = (time between specified events in frame where they are not colocated) / gamma

 

[neopolitan]

neopolitan;
no, I don't have a thing about muons. I did not introduce the subject so your comment makes no sense. You give the impression that muons can't have a frame of reference, in which you are wrong. I'm trying to shine some light here into your fog of misunderstanding, and you respond with insults and sarcasm.

I enjoyed your little biog about talking to people ( Professors even ) about your doubts and problems with relativity. I hope you get cured soon because it's costing some people an awful lot of effort.

M

Actually, if you read the text around the comment I made about not being able to be muon, you will see that it was made in the context of a decision about which frame to use. Most readers would be able to interpret from that that I did realize that the muon had a frame of reference. The laboratory frame is a sensible frame. It's not the only frame.

I accept that I may have misunderstandings, but shining light on the blindingly obvious it not helping anyone.

You clearly don't understand the message behind my story about speaking to various people about some "doubts and problems".

As to being cured of my curiosity, did you never have it, or were you cured? ()

Mentz, I know you are curious, I know you think I am obsessing on an unimportant detail. But equally I think you were obsessing on an unimportant detail regarding the muons. It wasn't even me who introduced them. It was BobS. I just thought he raised an interesting and useful real world example.

cheers,

neopolitan

 

[neopolitan]

If you want to use the clock in the laboratory you as your reference point, you have to do this:

While a photon in the laboratory moves between mirrors, traveling 660km in 2.2ms - what happens to a photon which is at gamma of 29.3?

If 2.2ms has elapsed in the laboratory, then a period of 2.2ms/29.3 = 75μs will have elapsed in the rest frame of the test clock (the one accelerated to gamma of 29.3).

No, 75 microseconds would represent how much time has elapsed on the test clock (if the test clock had closer mirrors so it could show time-intervals that small) in 2.2 ms of time in the lab frame. In the test clock's own frame, it's the lab clock that's running slow relative to the test clock, so when the lab clock has ticked forward 2.2 ms, the test clock has ticked forward 64.4 ms.

My fault. I was not clear about photons. It took a moment to see where you didn't agree since you seemed to be saying exactly the same as I said in my quote.

While a photon in the laboratory (in the laboratory light clock) moves between mirrors, traveling 660km in 2.2ms - what happens to a photon in the test frame light clock which is at gamma of 29.3?

The light clock in the test frame is at gamma of 29.3 (it makes no sense to talk about a photon at gamma of 29.3).

Thinking about the light clock in the test frame, while 2.2ms has elapsed in the laboratory (one full in-laboratory tick-tick), the photon has traveled 1/29.3 of the distance it needs to travel for the clock to go through a full tick to tick sequence, which, according the laboratory, is 660km*29.3. According to the laboratory, the photon in the test frame's clock has traveled 660km in 2.2ms. According to the laboratory, the photon in the laboratory frame's clock has traveled 660km in 2.2ms. According to the laboratory, both photons have traveled 600km in 2.2ms.

According to the test frame, what the laboratory frame "thinks" is 660km is actually 660km/29.3 and what the laboratory frame "thinks" is 2.2ms is actually 2.2ms/29.3.

(Aside: You can go through the last two paragraphs and swap the words "test" and "laboratory". The arguments would be the same. To reconcile the different views, you have to use relativity of simultaneity concepts. You shouldn't necessarily forget this next step, but at the moment, it is not necessary.)

If you want to call L'/t' LAFTD/time dilation that is fine. I do see here that that makes sense. But I also see that L'/t' length contraction/TAFLC makes equal sense. (Note that above I have not defined any primed frame or any unprimed frame.)

(660km * 29.3) / (2.2ms * 29.3) = (660km) / (2.2ms) = (660km / 29.3) / (2.2ms / 29.3) = 300000 km/s

So long as no matter what frame you view it from, the photon travels a distance of ct in t and a distance of ct' in t', I am happy - irrespective of how you want to link t and t'.

I prefer keeping in mind that lengths which are not at rest with respect to my rest frame are contracted. So I do prefer "length contraction/TAFCL" (or if you must, you can call it "length contraction/inverse time dilation" but I don't like it, because I interpret time dilation as talking about what happens between two full ticks, not about measured time, eg numbers of ticks or number of graduations between ticks).

You might prefer to think about the fact that compared to your clock, the period between ticks of a clock in motion with respect to you is longer. (Or whatever physical definition you ascribe to time dilation, the point is that you may prefer to keep the time dilation equation whereas I prefer to keep the length contraction equation.)

There is subtle difference in approaches which might be illustrative to highlight. You are focussed very much on the relativity (which is the bit I coloured silver above, so you have to select it to read it). I am focussed very much on the effects of on something which is in motion relative to me or some impartial observer.

Relativity says two things:

Something that is in motion relative to me will be length contracted and experience less time than me, relative to me.

and

The reverse is true, relative to that something.

I am really only looking at the first part, because I know the second part is true, but not terribly useful for working out the extent of that contraction and reduction of time experienced.

You seem to be unable to put that second part aside for a moment, perhaps because you think I think it isn't true. I do think it is true, just not currently helpful (as was the fact that muons have mass as Mentz will have us know, true but not actually helpful).

Again, I hope this helps.

cheers,

neopolitan

 

[bernhard.rothenstein]

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

Is
(T₀/T)(L/L₀=1 an important consequence?
I think that all we discuss there is a conswequence of the standard clock synchronization and of the measurement procedures. In general we ca have length contraction, length dilation and no disrtion at all.

 

[Rasalhague]

Yes, although your "inverse question" corresponds to the normal time dilation equation (with the most common notation being to use a primed t' where you've used an unprimed t, and an unprimed t where you've used ), whereas your first question corresponds to what I've called the "reversed time dilation equation" (where you just divide both sides of the normal time dilation equation by gamma).

Given that this was my attempt at paraphrasing your definition of TAFLC and its inverse (hence the choice of primed and unprimed), I guess that shows I’m still having trouble separating these concepts of TAFLC and inverse time dilation.

(time between events in frame where they are not colocated) = (time between events in frame where they are colocal) * gamma

And (space between events in a frame where they are not simultaneous) = (space between events in a frame where they are simultaneous) * gamma. What does this tell us about the length of an object: if I measure a moving object, this is how long it would be if measured in its rest frame?

(time between events in frame where they are colocal) = (time between events in frame where they are not colocated) / gamma

And (space between events in a frame where they are simultaneous) = (space between events in a frame where they are not simultaneous) / gamma. This being length contraction.

...the concept of the length of an object in different frames is quite different from the concept of the spatial distance between a pair of events in different frames).

I wonder if this is the crucial factor in how the apparent asymmetry comes about between time dilation and length contraction? When the concepts are introduced, in a way that makes one seem somehow parallel to the other, it’s so easy to jump to that conclusion. So would it be correct to say that the ends of an object aren’t events, but that each end of an object occupying some specific location at some specific time does comprise an event (a different event in the case of each end)?

So the way I would conceptualize this situation is to say that in order to talk about any of these equations, you first have to specify a single pair of events you want to talk about, and then in terms of those specific events the time dilation and reversed time dilation equations tell you everything you want to know about the time interval between the events in two frames (one of which is the one where they're colocated), whereas in terms of those events the TAFLC is telling you something more abstract about the time-interval (in the frame where the events are not colocated) between surfaces of simultaneity from the frame where the events are colocated. Of course you could start with a new pair of events so that the time interval given by the TAFLC applied to the previous events is just the time interval between the new events in the frame where they're colocated, but then you're really talking about the reversed time dilation equation for these new pair of events, not the TAFLC for them.

But if we think of, say, the time dilation equation as a function f(t) = t * \gamma which takes as its input some time, and gives as its output some other time, this function has an inverse f^{-1}(t) = t / \gamma, the inverse being also a function over t, the real valued set of all possible time intervals, we can conceptualise both functions as abstract entities, without specifying any particular events until we actually want to calculate something about particular events. In the abstract, they’re functions that tell you something about *any* pair of events. As such, until the events are specified one way or the other--aside from matters of frame-labelling convention--aren’t TAFLC and reverse TD equivalent? And when we do want to specify a pair of events, what’s the difference between performing the same mathematical operation on the same values whether you call it “start[ing] with a new pair of events” or letting the equation tell you about a new pair of events, since, in the latter way of conceptualising it, the events would still be specified uniquely by the question, wouldn't they? (Namely the equation chosen and the value plugged into it.)

But what do you mean by "both types of questions"? What events are you asking questions about?

I meant questions of the type answered by the traditional time dilation equation (or equivalently, I assumed, reverse TAFLC) versus questions of the type answered by reverse time dilation (or equivalenty, I assumed, TAFLC), regardless of how the frames are labelled. Of course, I could well be mistaken to assume that equivalence.

(1) “In Alice’s rest frame, what time on Bob’s watch is simultaneous with Alice’s 4?” Answer: t_{B} = t_{A} / \gamma = 3.2. What do we call this: time contraction, temporal analogue of length contraction, reverse time dilation?

(2) “In Bob’s rest frame, what time on Bob’s watch is simultaneous with Alice’s 4?” Answer t_{B} * \gamma = 5. Time dilation, right? Or is it reversed TAFLC?

If you're asking about more than a single pair of events then in that case I'd agree you might use both of those equations to talk about time intervals between events...

Yes, I can see that if you input the same (nonzero) value into these two equations, you’d be talking about more than a single pair of events.

...but since you're no longer talking about a single pair of events you'd have to have some different notation to distinguish between verbal formulations like "time-interval in the unprimed frame between events A and B" and "time-interval in the unprimed frame between events C and D"--perhaps you could use \Delta t_{AB} and \Delta t_{AC} in this case. Then if A and B are colocated in the primed frame while C and D are colocated in the unprimed frame, you might write \Delta t’_{AB} = \Delta t_{AB} / \gamma along with \Delta t_{CD} = \Delta t’_{CD} / \gamma, but I would refer to the first as "the reversed time dilation equation for events A and B" and the second as "the reversed time dilation equation for events C and D", in words they would both come out to:

(time between specified events in frame where they are colocated) = (time between specified events in frame where they are not colocated) / gamma

So what, if anything, in this situation would you describe as TAFLC? Thanks for your patience, by the way, and sorry if I'm repeating myself or demanding answers to questions you've already answered in detail. Perhaps it'll become clearer to me once I've solved some more problems and got a bit more experience of the sort of questions these concepts are used to deal with, and when I've looked more at time dilation and length contraction in the wider context of the Lorentz transformation and spacetime geometry.

 

[neopolitan]

Is
(T₀/T)(L/L₀=1 an important consequence?
I think that all we discuss there is a conswequence of the standard clock synchronization and of the measurement procedures. In general we ca have length contraction, length dilation and no disrtion at all.

It depends a little on what L_o and T_o are.

I am tempted to think (using standard pairing, time dilation and length contraction):

T = T_o * gamma
L = L_o / gamma

so:

T_o / T = 1 / gamma
L / L_o = 1 / gamma

so:

(T_o / T)(L / L_o) = 1 /(gamma)²

Which is partly why I question it.

Rearranging (T_o / T)(L / L_o) = 1 gives you:

(T_o / L_o)(L / T) = 1

or

L / T = L_o / T_o

Which I think is an important consequence. In much later posts we are nearing a resolution ... maybe :)

For me that discussion could revolve, conceptually, around what a photon does traveling along between two events (but I stress that it doesn't have to). In one frame, it could be said that that photon travels L in time T (so L/T=c). In another frame, it could be said that that same photon travels L_o or L' in time T_o or T' (so that L_o/T_o=c or L'/T'=c).

cheers,

neopolitan

 

[bernhard.rothenstein]

It depends a little on what L_o and T_o are.

I am tempted to think (using standard pairing, time dilation and length contraction):

T = T_o * gamma
L = L_o / gamma

so:

T_o / T = 1 / gamma
L / L_o = 1 / gamma

so:

(T_o / T)(L / L_o) = 1 /(gamma)²

Which is partly why I question it.

Rearranging (T_o / T)(L / L_o) = 1 gives you:

(T_o / L_o)(L / T) = 1

or

L / T = L_o / T_o

Which I think is an important consequence. In much later posts we are nearing a resolution ... maybe :)

For me that discussion could revolve, conceptually, around what a photon does traveling along between two events (but I stress that it doesn't have to). In one frame, it could be said that that photon travels L in time T (so L/T=c). In another frame, it could be said that that same photon travels L_o or L' in time T_o or T' (so that L_o/T_o=c or L'/T'=c).

cheers,

neopolitan

Thank you for your answer. The last case you mention is very interesting, because length and time intervals are related by the Doppler factor in an electromagnetic wave. The light signal generates in I the event (x;ct) whereas in I' the event (x';ct'). The cortresponding Lorentz transformations lead to
x'=g(x-Vt)=gx(1-V/c)
t'=g(t-Vx/cc)=gt(1-V/c)
g standing for the Lorentz factor.
Kind regards

 

[Mentz114]

\mathbf{t'}=\mathbf{t}\cosh(\beta)+\mathbf{x}\sinh(\beta)
\mathbf{x'}=\mathbf{x}\cosh(\beta)+\mathbf{t}\sinh(\beta)

What more needs to be said ?

 

[neopolitan]

\mathbf{t'}=\mathbf{t}\cosh(\beta)+\mathbf{x}\sinh(\beta)
\mathbf{x'}=\mathbf{x}\cosh(\beta)+\mathbf{t}\sinh(\beta)

What more needs to be said ?

Mentz old boy,

You are clearly extremely intelligent, very highly educated and totally untroubled by curiosity not to mention modest. Most of the rest of us would need more than those equations during our years of education even you are able to deduce all that needs to known from them.

Would you replace time dilation and length contraction with those equations? Do you suggest that presenting the new student with those equations would inform them or are you just planning to bludgeon them into conformity?

Since you seem to have said all that needs to be said, I do hope you don't plan to say any more. I am happy for you to leave to my rhetorical questions unaddressed.

cheers,

neopolitan

 

[bernhard.rothenstein]

\mathbf{t'}=\mathbf{t}\cosh(\beta)+\mathbf{x}\sinh(\beta)
\mathbf{x'}=\mathbf{x}\cosh(\beta)+\mathbf{t}\sinh(\beta)

What more needs to be said ?

I think that in order to help the learner there are a lot of thinks which should be mentioned.
1. Length contraction is obtained from the Lorentz transformations if in one of the involved inertial frames a simultaneous detection of the moving rod is performed. Recent papers have shown that the same result could be obtained without imposing the mentioned condition.
2. Time dilation is obtained if in one of the involved inertial frame a proper time interval is measured.
3. Time dilation and length contraction could be derived from thought experiments and that makes the beauty of teching relativity to beginners.
4. If the clocks of the involved inertial frames are standard synchronized there is no time dilation without length contraction.
Kind regards

 

[Mentz114]

Mentz old boy,

You are clearly extremely intelligent, very highly educated and totally untroubled by curiosity not to mention modest. Most of the rest of us would need more than those equations during our years of education even you are able to deduce all that needs to known from them.

Would you replace time dilation and length contraction with those equations? Do you suggest that presenting the new student with those equations would inform them or are you just planning to bludgeon them into conformity?

Since you seem to have said all that needs to be said, I do hope you don't plan to say any more. I am happy for you to leave to my rhetorical questions unaddressed.

cheers,

neopolitan

Thanks.

Would you replace time dilation and length contraction with those equations?

Those equations are length contraction and time dilation.

Do you suggest that presenting the new student with those equations would inform them or are you just planning to bludgeon them into conformity?

This remark first presupposes something then makes a damning inference. Ungentlemanly and very rude.

I am happy for you to leave to my rhetorical questions unaddressed.

Please look up the meaning of 'rhetorical'. Surely you wanted someone to respond.

Please, cut out the personal stuff, ironic or not.

 

[Mentz114]

I think that in order to help the learner there are a lot of things which should be mentioned.
1. Length contraction is obtained from the Lorentz transformations if in one of the involved inertial frames a simultaneous detection of the moving rod is performed. Recent papers have shown that the same result could be obtained without imposing the mentioned condition.
2. Time dilation is obtained if in one of the involved inertial frame a proper time interval is measured.
3. Time dilation and length contraction could be derived from thought experiments and that makes the beauty of teching relativity to beginners.
4. If the clocks of the involved inertial frames are standard synchronized there is no time dilation without length contraction.
Kind regards

Bernhard,
I'm sure you're a dedicated and earnest teacher of the subject, but do beginners have to go into SR as deeply as you enjoy going ?

M

 

[neopolitan]

Mentz,

The equations you provided would not help the new student to SR to understand the physical significance of the standard time dilation and length contraction equations that they are normally presented with.

I am pretty sure that they would confuse. It seems to have confused either you or the author of this site on http://hubpages.com/hub/Hyperbolic-Functions" .

On his site, time dilation is given by cosh u (probably cosh \beta of your equation set, but since you did not define \beta, I don't know).

In the same vein, length contraction (he calls it spatial contraction) is given by sech u (again probably sech \beta).

He shows you graphically what u is in his equations (the area between the asymptote and the x axis). He also clarifies that sech u is the reciprocal of cosh u.

That is slightly more helpful.

I expect that the equation pair you gave really represents the Lorentz Transformations, but in a format which is far less intuitively comprehensible to the new student. I suspect that the equation pair requires you to make reference to the function under which the area \beta is found, namely S² = x² - (ct)² and that where you have written t, you should have written (ct).

But all of this is extraneous to what we were discussing.

cheers,

neopolitan

 

[bernhard.rothenstein]

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

What is so great with time dilation?
1. In teaching it can be derived from the two postulates and from Pythagoras' throrem.
2. It leads directly to length contraction.
3. Length contraction leads directly to the Lorentz transformations.
4. Lorentz transformation lead directly to the formulas that account for all the formulas we encounter in special relativity theory.
Is there to say when it is about its benefits?

 

[neopolitan]

What is so great with time dilation?
1. In teaching it can be derived from the two postulates and from Pythagoras' throrem.
2. It leads directly to length contraction.
3. Length contraction leads directly to the Lorentz transformations.
4. Lorentz transformation lead directly to the formulas that account for all the formulas we encounter in special relativity theory.
Is there to say when it is about its benefits?

I think you are talking about the light clock? or something similar? Someone else stated recently on a thread hereabouts that the light clock derivation has weaknesses. I would think anything similar has similar weaknesses.

The confusion I see comes after getting the students to get shown how time dilation is derived but no clarification is given along the lines that you can't take a contracted length and a dilated time to get a speed which the postulates you started with said was invariant.

Most students won't think more deeply than is required to pass the test and so will learn very little.

Others will instinctively grasp what has not been clarified.

Some, perhaps only a few, will be left with a vague unease because if L and t are such that L/t=c then L'/t' is not c.

I do think that we can derive the Lorentz transformations without even stopping at length contraction and time dilation. Lorentz seemed to and you can go directly from Galilean boosts to Lorentz transformations without having previously derived length contraction or time dilation, you just remove the assumption of instantaneous information transfer and use the first postulate. The second postulate falls out as a consequence.

I'd be happy to dispense with time dilation and length contraction altogether, and just go with Lorentz transformations, as Mentz possibly meant in an earlier post. But this is not the standard approach. Additionally, I would clarify just what it is that the Lorentz transformations can tell you, because if you just plug in t=0 into the spatial transformation, you end up with "length dilation" and that hardly matches with the contraction we expect.

cheers,

neopolitan

 

[bernhard.rothenstein]

I think you are talking about the light clock? or something similar? Someone else stated recently on a thread hereabouts that the light clock derivation has weaknesses. I would think anything similar has similar weaknesses.

The confusion I see comes after getting the students to get shown how time dilation is derived but no clarification is given along the lines that you can't take a contracted length and a dilated time to get a speed which the postulates you started with said was invariant.

Most students won't think more deeply than is required to pass the test and so will learn very little.

Others will instinctively grasp what has not been clarified.

Some, perhaps only a few, will be left with a vague unease because if L and t are such that L/t=c then L'/t' is not c.

I do think that we can derive the Lorentz transformations without even stopping at length contraction and time dilation. Lorentz seemed to and you can go directly from Galilean boosts to Lorentz transformations without having previously derived length contraction or time dilation, you just remove the assumption of instantaneous information transfer and use the first postulate. The second postulate falls out as a consequence.

I'd be happy to dispense with time dilation and length contraction altogether, and just go with Lorentz transformations, as Mentz possibly meant in an earlier post. But this is not the standard approach. Additionally, I would clarify just what it is that the Lorentz transformations can tell you, because if you just plug in t=0 into the spatial transformation, you end up with "length dilation" and that hardly matches with the contraction we expect.

cheers,

neopolitan

Thanks for your answer. As an old teacher of physics I have studied the different ways in which the Lorentz transformations could be derived.
1. I learned a lot from Paul Kard [1] who derives first the formula that accounts for the length contraction, which leads him to the formula that accounts for the Doppler shift which leads to the addition law of relativistic velocities and derives the formula that accounts for the time dilation from the Doppler shift formula. I knew all that from Kard's original papers in Russian.
[1] Leo Karlov, "Paul Kard and the Lorentz-free special relativity," Phys.Educ. 24, 165 (1989)
2. Kalotas and Lee [2] convinced me that the Doppler shift formula could be derived from the formula that accounts for the "Police Radar" an experiment performed in a single inertial reference frame, involving a single clock and so no clock synchronization. The formula that accounts for the Doppler shift is derived by simple injection of the first postulate. He also shows that the Lorentz transformations could be derived from the Doppler shift formula.
[2] T.M. Kalotas and A.R. Lee, "A "two line" derivation of the relativistic longitudinal Doppler formula," Am.J.Phys. 58, 187 (1990)
3. Asher Peres [3] taught me that the basic formulas of relativistic kinematics could be derived from Einstein's postulate: "All the physical laws are the same for all inertial observers,in particular the speed of light is the same" in the following order: radar echo, time dilation, additions of velocities, the Doppler Effect and optical aberration. He does not derive the Lorentz transformations even if starting with one of the basic formulas mentioned above could lead to them.
[3] Asher Peres, "Relativistic telemetry," Am.J.Phys. 55, 516 (1987)

When I started learning English from BBC, Professor Grammar told me that English is a very flexible language. I would say that Special Relativity is a very flexible chapter of physics. We can start with Einstein's postulte, derive the equation that accounts for one of the effects mentioned above and it leads us to the Lorentz transformations.
I would highly appreaciate the criticism of the approaches presented above. My students enjoyed them.
Kind regards and thanks for giving me the opprtunity to discuss about the teaching of special relativity.