Benefits of time dilation / length contraction pairing?by neopolitan Tags: contraction, dilation, length, pairing, time 

#73
Mar1709, 11:38 PM

P: 645

Remember a while back I talked about an apparatus I had. I have it and it is at rest relative to me.
Associated with this apparatus are a length and a time measurement. I called these L and t. I give these to my buddy, and he sets off on a carriage with a speed of v (in a direction that is convenient so that the length I measured as L is parallel to the direction of motion). My buddy will, if he checks, find a length and time measurement of L and t. But while my buddy in motion measures L and t, I will work out that, because he is motion, the length is contracted. I call that L', because I already have an L (unprimed is my frame so I making primed my buddy's frame). I will also work out that, because he is in motion, my buddy's clock will have slowed down. What reads on his clock will less than what I read on mine. If confuses you, and god knows it confuses me, because you have to step back a bit from the intial t I had. So, let's do it another way. Say I have two sets of the apparatus. I keep one, and give the other to my buddy. I know they are identical. I ask him to measure it lengthwise, he gets L and I get L. But if I compare my length to his length (and I can do this with lasers and time measurements in my frame), I will find that he is "confused". His length is actually [tex]L'=L / \gamma[/tex]. (And yes, I know if he does the same thing, he will find that I am "confused".) Time is a little more complex to describe, but equivalent to using lasers and time measurements in my frame. Using a very high quality telescope, I keep track of my buddy's apparatus, most specifically the clock. I note down two times on his clock, [tex]t'_{o}[/tex] and [tex]t'_{i}[/tex] along with the times that I make them (my times, my frame, unprimed). I have to take into account how long it took each of those displayed times on his clock to get to me. I will find that [tex]\Delta t' = \Delta t / \gamma[/tex] (< this is my equation, this is not time dilation!) Now I know that when [tex]\Delta t[/tex] has elapsed in my frame, [tex]\Delta t'[/tex] elapses in his frame. It is not just ticks on clocks, or the time interval between two events  his time dimension is affected. And it is affected in the same way as his spatial dimension is affected. So any speed in his frame will be calculated using contracted length divided by shortened time which will give you the same result as using unaffected length divided by unaffected time. Picking appropriate values of L and [tex]\Delta t[/tex]: [tex]L / \Delta t = c = L' / \Delta t'[/tex] Does that help? cheers, neopolitan 



#74
Mar1809, 12:02 AM

P: 1,400

Wow, thanks for your answer Jesse. That's given me lots to think about! In everyday life we're more used to regarding future time as being what's unpredictable, so it's curious to think of "the state of a surface of constant x" as being more fundamentally impossible to deduce "if all you know is the state of some other surface of constant x". But suppose that, in a deterministic universe, you knew everything about a surface of constant x to some arbitrary degree of precision. You'd know the positions of particles in that surface and be able to say something about other surfaces of constant x by examining the forces operating on the particles in your surface. Admittedly there'd be multiple possible states of the rest of space that could be responsible for the state of your surface of constant x, but then you could likewise have different histories that lead to the same state for some surface of constant t. So is it something about the future specifically, and its predictability, that makes all the difference? If that's even a meaningful question...
Meanwhile, less philosophically, just to check I've understood: is the case where a worldline has multiple pointlike intersections with a surface of constant x only possible for a particle for which there's no inertial frame in which the particle can be said to be at rest (i.e. its worldline isn't a straight line)? (The other two cases you mentionno intersections, one pointlike intersectionbeing possible for a particle which can be described as being at rest in some inertial frame.) 



#75
Mar1809, 01:23 AM

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Perhaps this confusion about what quantities should be primed and what quantities should be unprimed is related to your (so far unexplained) belief that there is something "inconsistent" about the way the standard time dilation and length contraction equations are written? And even if I changed your statement above to "I will work out that, because he is motion, the length is contracted. I call that [tex]L_{cbb}[/tex], because I already have an L", your statement would still be too vague, for exactly the same reason as the statement in your last post was too vague (I offered several possible clarifications so you could pick which one you meant, or offer a different clarification). If [tex]L_{cbb}[/tex] refers to the length of the apparatus in your frame when it's being carried by your buddy, and [tex]L'_{cbb}[/tex] refers to the length your buddy measures the apparatus to be using his own ruler (which is equal to L, the length you measured the apparatus to be using your ruler before you gave it to your buddy, when it was still at rest relative to you), then these will be related by the equation [tex]L_{cbb} = L'_{cbb} / \gamma[/tex], which is just the length contraction equation with slightly different notation. If on the other hand what your buddy "measures" is a distance of L' between two events using his apparatus, then the distance you measure between the same two events will not necessarily be L'/gamma, in fact it could even end up being larger than L'. So you really need to be specific about precisely what is being measured like I keep asking. If that is what you mean by "take into account"and please actually tell me yes or no if it's what you meantthen note that this is exactly the same as asking what times on your clock were simultaneous with his clock reading [tex]t'_{o}[/tex] and [tex]t'_{i}[/tex], using your own frame's definition of simultaneity. So note that although you didn't really respond to my list of possible clarifications, it appears that your meaning is exactly identical to the first one I offered, which I'll put in bold (in the original comment I was using unprimed to refer to the buddy's frame and primed to refer your frame, but since you appear to want to reverse that convention by making times on your buddy's clock primed, I'll change the quote to reflect the idea that times in your frame are unprimed and times in your buddy's are primed): Do you understand and agree with all this? Please tell me yes or no. 



#76
Mar1809, 01:37 AM

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#77
Mar1809, 05:04 AM

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"Picking appropriate values of [tex]L[/tex] and [tex]\Delta t[/tex]" was too vague. The rest of what you were saying was akin to "You can't park four tanks on the rubber dingy you're designing".
Here's what I mean about picking appropriate values, pick any value of [tex]L[/tex], any value you like  in the real world you probably want a really big value, but this is hypothetical world, so it is not so important. Then pick the value of [tex]\Delta t[/tex] so that L/[tex]\Delta t = c[/tex]. If you haven't picked a really big value of [tex]L[/tex], then [tex]/Delta t[/tex] will be pretty damn small so that it will be challenging to take two readings [tex]t_{o}[/tex] and [tex]t_{i}[/tex] where [tex]t_{i}  t_{o} = \Delta t[/tex]  but we are in hypothetical world. We have no argument about length contraction. But do you deny that when I use my readings from my buddy's clock, and take into account the motion that I know he has, that I will get a [tex]/Delta t'[/tex] which is shorter than mine? Do you deny that the extent to which it is shorter is the same as the extent to which L' is shorter than L (where these are given by standard length contraction)? I've described the events, they aren't simultaneous (and in fact, I don't care about simultaneity, I know the time readings on my buddy's clock are not simultaneous with the time readings on mine, the only thing I bother with, or need to bother with, is the extra time the second reading takes to get to me because he has moved during the time). Any discussion of simultaneity in this scenario is a distraction. If you must have some simultaneity, then try thinking that my seeing the time on my buddy's clock is simultaneous with my reading of the time on my clock, but even that is not necessary since I could use a splitframe camera and look at the results afterwards. The bottom line, from you Jesse, is "there is no other way to do it" when the question is "what is the benefit with time dilation". It seems you truly think there is no other option. You have a very long winded way to say it, but I don't think there is any other way to interpret your approach to the original question. And yes, I haven't forgotten the original question. cheers, neopolitan 



#78
Mar1809, 05:42 AM

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#79
Mar1809, 06:08 AM

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JesseM,
You fragment too much. It leads, inexorably, to loss of context. That's why I am not responding to your fragmenting. Look back in previous posts and I explained what I meant about taking readings on my buddy's clock. I made mention of a telescope. But you must have overlooked it in your apparent excitement to demolish any discussion (and I mean that, "discussion", not argument because to demolish an argument you have to make an effort to understand). You make an effort to understand, pose an unfragmented question which indicates that you have made the slightest effort to understand, rather than attack, and I will answer it. cheers, neopolitan 



#80
Mar1809, 09:29 AM

P: 1,400

Now suppose that Alice wonders, "What time does Bob's watch say at the moment which in my rest frame is simultaneous with me asking this question?" The answer to this is given by [tex]t_{B} = \frac{t_{A}}{\gamma}[/tex]. If Alice's watch says 5, Bob's will say 4. "Fancy," thinks Alice. "From my perspective, Bob's watch, which is moving relative to me, is running slow." And, of course, Bob can ask the equivalent questions about the time on Alice's watch with identical results by virtue of the fact that the two frames don't agree on which events are simultaneous (except for those that happen in the same place, such as their synchronisation). But isn't Alice's second question none other than this exotic "temporal analogue for the length contraction equation"? She wants to know "the timeinterval in the primed frame" (the time shown by Bob's watch, indicating a time interval along Bob's worldline) "between two surfaces of constant t in the unprimed frame" (one being the one which Alice and Bob's worldlines intersected when they synchronised watches, the other being Alice's present when she looks at her watch) "which have a temporal distance of [tex]\Delta t[/tex] in the unprimed frame" (the time shown by Alice's watch when she looks at it and makes her query). Is Alice's second question in any way less natural than the first, or a less useful thing to ask of time than of space? I'm puzzled as to how it can be, if it is, as Jesse said, "just a trivial reshuffling of the usual time dilation equation"? 



#81
Mar1809, 01:12 PM

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#82
Mar1809, 04:23 PM

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But you do raise a great point which I hadn't thought of before, which is that the "temporal analogue of the length contraction equation" can always be used to calculate things that you'd normally use the time dilation equation to calculate, provided you reverse the meaning of which frame is primed and which frame is unprimed. Let me give a numerical example similar to yours. Suppose Bob is moving away from Alice at 0.6c and that both their clocks read 0 when they crossed paths as you suggested. But instead of starting Bob's time interval when his clock reads 0 as in your example, suppose we were interested in the time interval on Bob's clock that started with the event of his clock reading [tex]t_{B1}[/tex] = 8 seconds, and ended with his clock reading [tex]t_{B2}[/tex] = 12 seconds, so the length of the interval in Bob's frame is [tex]\Delta t_B[/tex] = ([tex]t_{B2}[/tex]  [tex]t_{B1}[/tex]) = 4 seconds. If Alice wanted to know the time interval [tex]\Delta t_A[/tex] between these same two events in her frame, which is equivalent to wanting to know the time interval between the event [tex]t_{A1}[/tex] on her clock which is simultaneous in her frame with [tex]t_{B1}[/tex] (in this case [tex]t_{A1}[/tex] = 10 seconds) and the event [tex]t_{A2}[/tex] on her clock which is simultaneous in her frame with [tex]t_{B2}[/tex] (in this case [tex]t_{A2}[/tex] = 15 seconds), then she would plug these two different time intervals into the time dilation equation [tex]\Delta t' = \Delta t * \gamma[/tex], treating Bob's frame as unprimed and her frame as primed, which gives [tex]\Delta t_A = \Delta t_B * \gamma[/tex]. If she wanted to reverse this and figure out the time interval [tex]\Delta t_B[/tex] on Bob's clock between two events on [tex]t_{B1}[/tex] and [tex]t_{B2}[/tex] on his clock's worldline that are simultaneous in her frame with two events on her clock's worldline [tex]t_{A1}[/tex] and [tex]t_{A2}[/tex] that are the beginning and end of a time interval [tex]t_A[/tex] (in the example above she would start with times 10 seconds and 15 seconds on her clock and then try to figure out how much time had elapsed on Bob's clock between these moments in her frame), she'd just divide the time dilation equation by gamma so it gives [tex]\Delta t_B[/tex] as a function of [tex]\Delta t_A[/tex], i.e. [tex]\Delta t_B = \frac{\Delta t_A}{\gamma}[/tex]. On the other hand, the "temporal analogue of length contraction" [tex]\Delta t' = \Delta t / \gamma[/tex] would be telling her something conceptually different, assuming she continues to treat Bob's frame as unprimed and her frame as primed. Basically, it would be saying "if you use [tex]t_{A1}[/tex] to label the time on Alice's clock that's simultaneous in Bob's frame with Bob's clock reading [tex]t_{B1}[/tex], and you use [tex]t_{A2}[/tex] to label the time on Alice's clock that's simultaneous in Bob's frame with Bob's clock reading [tex]t_{B2}[/tex], then the time interval on Alice's clock ([tex]t_{A2}  t_{A1}[/tex]) is related to the time interval on Bob's clock ([tex]t_{B2}  t_{B1}[/tex]) by the formula [tex](t_{A2}  t_{A1}) = (t_{B2}  t_{B1}) / \gamma[/tex]. If we use the same numbers for [tex]t_{B1}[/tex] and [tex]t_{B2}[/tex] on Bob's clock as before, namely [tex]t_{B1}[/tex] = 8 seconds and [tex]t_{B2}[/tex] = 12 seconds, then in this case we'd have [tex]t_{A1}[/tex] = 8*0.8 = 6.4 seconds (I just multiplied 8 by 0.8 because I know both clocks read 0 when they were next to each other and Alice's clock is moving at 0.6c in Bob's frame, so the standard time dilation equation tells me her clock is slowed by a factor of 0.8 in his frame) and [tex]t_{A2}[/tex] = 12*0.8 = 9.6 seconds. So the equation [tex](t_{A2}  t_{A1}) = (t_{B2}  t_{B1}) / \gamma[/tex] does work here, since [tex](t_{A2}  t_{A1}[/tex] = 9.6  6.4 = 3.2, [tex]t_{B2}  t_{B1}[/tex] is still 4 seconds, and gamma is still 0.8. But you can see that the time interval in Alice's frame we're talking about now (3.2 seconds) is different than the time interval in Alice's frame we were talking about when we were using the usual time dilation equation (5 seconds). But, that's only because we were treating Alice's frame as the primed frame in both equations! If we reverse the labels and treat Bob's frame as primed and Alice's frame as unprimed, then the standard time dilation equation [tex]\Delta t' = \Delta t * \gamma[/tex] does tell you that when 3.2 seconds have elapsed on Alice's clock, 4 seconds of time have passed in Bob's frame (or equivalently, if you look at the readings on Bob's clock that are simultaneous in Bob's frame with the two readings on Alice's clock, the difference between these two readings on Bob's clock is 4 seconds). So I guess if you take the time dilation equation and divide both sides by gamma to solve for the interval in the primed frame, this is really just equivalent to taking the "temporal equivalent of length contraction" equation and reversing which frame we call primed and which we call unprimed. To me there's still a little bit of a conceptual difference though, in the sense that normally I think of these equations as relating a clock timeinterval to a coordinate timeinterval, with unprimed normally being the clock timeinterval. For instance, when I read the time dilation equation [tex]\Delta t' = \Delta t * \gamma[/tex], I find it most natural to think that [tex]\Delta t[/tex] represents the difference between two clockreadings on a clock at rest in the unprimed frame, and then [tex]\Delta t'[/tex] represents the difference between the coordinate times of these two readings in the primed frame. Of course, because a clock at rest in the primed frame will keep time with coordinate time in that frame, this is equivalent to imagining there's also a clock at rest in the primed frame, and saying [tex]\Delta t'[/tex] represents the difference between two readings on the primed clock that are simultaneous in the primed frame with the two readings on the unprimed clock that were mentioned earlier. The first way of stating it just makes the usefulness of the time dilation equation more intuitive to me; as I said before, physics is all about setting up a spacetime coordinate system and then using equations to figure out how the state of objects in space changes as the timecoordinate increases. 



#83
Mar1809, 06:58 PM

P: 645

JesseM responding to Rasalhague:
There is another way of approaching the relativistic effects other than time dilationlength contraction. That is to use "temporal analogue of the length contraction equation"length contraction. However, using time dilation is more intuitive to you  and possibly also for the majority of people. That said, there is nothing inherently wrong with using a "temporal analogue of the length contraction equation" (although one must note that a different prime convention is required). Rasalhague has shown me that instead of: cheers, neopolitan 



#84
Mar1809, 07:46 PM

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#85
Mar1809, 10:08 PM

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We were probably arguing at cross purposes, frustratingly enough for both of us. I thought you had taken that "temporal equivalent of length contraction" thing onboard a long time ago (it is in your diagram after all). So I was totally confused as to where you were coming from.
Since I thought you had understood the point and were still arguing it, it felt as if you were just trying to play games. That may have been a form of "tranference" (psychological term, relating to ascribing apparent motives of one person to another), since in real life I had a rather difficult person at work doing what I thought you were doing  playing dominance games through irrational argument. I take your point about specifics. You may see that I have tried to be specific with figures in another thread. May I ask why you had not come to the understanding that you just came to, when it seems that both Rasalhague and I did? This is not a hidden "you must be stupid" insult. I find you annoying, as you surely find me, but I don't find you stupid. What I am trying to do is see if you can identify, from the vantage point of someone who has only just came to this understanding, what prevents people from coming to this understanding naturally. Is there a block of some kind? If so, is it pedagogical or psychological? (Clarification follows: I am distinguishing here between pedagogical and psychological, with a definition of "pedagogical" relating to how subjects are taught and "psychological" relating to the different ways in which people think and learn. Specific examples: "whole language" is a pedagogical method for teaching kids to read, moving away from phonics and instead recognition of whole words. As for "psychological", I am a visual, pattern identifying person which means that having a graph in front of me is more useful than a page of numbers. My visual, pattern identifying nature may lead me to link together all things that look the same (like all things with primes against them get grouped).) This is the sort of discussion I really wanted back when I started the thread. Perhaps you might understand why I found the 80 or so posts in between frustrating, even if they were my own fault. cheers, neopolitan 



#86
Mar1909, 03:12 AM

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What I had failed to realize was that if we imagine a physical clock at rest in the primed frame, then the "temporal distance between surfaces of constant t from the unprimed frame" just represents the difference [tex]\Delta t'[/tex] between the clock's readings at the two points where its worldline intersects these surfaces of constant t from the unprimed frame, and that if we then shift our perspective back to the unprimed frame, [tex]\Delta t[/tex] is now just the coordinate time between two readings on the primed clock, so now this is exactly like how I'd conceptualize the physical meaning of the terms in the reversed time dilation equation except with the roles of primed and unprimed reversed. So, this is one or two mental steps from what the TAFLC seemed to mean based on my diagram, and I didn't see the connection until I started working through a numerical example in response to Rasalhague's question. Also, it didn't help that I was used to conceptualizing the standard and reversed time dilation equations as relating a clock timeinterval on a clock at rest in the unprimed frame with a coordinate timeinterval in the primed frame, rather than normally thinking in terms of a clock at rest in the primed frame too. I was aware intellectually of the fact that the coordinate time in the primed frame between two events A and B could be rephrased in terms of readings on a physical clock at rest in the primed frame, specifically the difference between the reading that was simultaneous with A and the reading that was simultaneous with B according to the prime frame's definition of simultaneity. But that seemed like a more complicated way of thinking about the physical meaning of [tex]\Delta t'[/tex] (you can see it took me longer to write it out) so I usually just thought of it in terms of coordinate time. 



#87
Mar1909, 10:04 AM

P: 645

There is something which I find curious. It is a criticism of the pedagogy not of you nor of what time dilation is actually representing. Note that you are "used to conceptualizing the standard and reversed time dilation equations as relating a clock timeinterval on a clock at rest in the unprimed frame with a coordinate timeinterval in the primed frame". It's quite a complex thing to internalise. When being taught, or trying to teach oneself, it is going to be a real uphill struggle to grasp that particular nature of the standard time dilation equation. I certainly struggled with it and it was not helped that I have "back to fundamentals" sort of approach to mathematics which I applied to SR by reading a translation of Einstein's 1905 paper (I use the one at fourmilab.ch). I noted, and agonised over the fact that one of the standard equations is shown in mathematical form (length contraction) but the other is only given in words (on page 10) and this directly follows what was to me the far more intuitive equation  a form of "TAFLC" with [tex]\tau[/tex] instead of t'. I do think there is a source of confusion there. I'd be willing to accept that it just me, but it seems there are many people with some problem or another with SR, which seems really odd. Why SR? Some are kooks for sure, but there are many people who seem to be otherwise able to maintain perfectly normal lives apart from an intuitive feeling that something is just not quite right about SR. I won't say his name, and sadly he has probably passed away with cancer by now, but a professor in a city not far from where I lived up until recently was the lead lecturer for relativity at his university. He generously gave me many hours of his time to discuss my concerns and proposed solutions, and admitted that really, he didn't fully understand it. He did not stop me or explain that my concerns about time dilation were invalid, because he had never intuitively grasped the principles the standard way either. There is a fellow in southern Europe, another physics professor, albeit in a different field who expressed stronger views than I during our discussions that there was something amiss with SR. (I don't think SR is wrong but I do think it could be taught better.) A quantum physics professor in southern England also felt I was onto something with my arguments. If professors of physics don't grasp SR properly, what chance do the average visitors to these forums have? To make sure I am not presenting a biased account, I should clarify that at least four professors I corresponded with gave clear indications that they grasped SR well enough as taught (at least enough as to not be intrigued by my concerns), but sadly they had no time to go into it in depth with me. I had learned SR at university, read up on it, even going back to the original documents (Einstein and Feynman, Feynman because the light clock is sort of his). I had my uneasy feeling despite all this, and being told to go learn SR (again!) didn't really help. Anyways, I've cast away a lot of my original stuff because I can now see that I was looking at the same thing as standard SR from a different perspective (I've not cast away everything, but I may in time cast away even the little that remains) and my deepseated concerns that time dilation could actually be wrong were not justified. However, if this feeling of there being something not quite right (which in my case were, as I said, deepseated and may be equally concerning to others) is due to something as harmless as a pedagogical/psychological issue where some people intuitively think the way you do and others intuitively think another, yet both ways of thinking are completely valid, being just slightly different perspectives on the same thing, then it seems that there is some scope for improvement on how SR is taught. I really do think that your suggestion a long time ago, when we had the discussion in which the diagram I posted here was central, was a good one. You said that you would show your new students a similar diagram and explain that time dilation is not a TAFLC, and is not meant to be. I think you could go a little further and explain the physical significance of the actual TAFLC, and how it relates to length contraction so that c is invariant. That way, you would catch the people like me who feel that TAFLC is a useful equation and gently guide them towards a proper understanding of time dilation. At the same time, you would catch people like you, who go many years without grasping that there is any significance to a TAFLC. Does that sound unreasonable? cheers, neopolitan (I'm trying to get a lot into as few words as possible, it is late and it has been a long day. Sorry if there is anything which is hard to follow.) 



#88
Mar1909, 10:03 PM

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Anyway, if you can see my point that the time dilation and length contraction equation seem fairly "natural" in a universe with absolute space and time, then maybe you can see why, once a physics student has gotten used to the idea of picking a coordinate system and then taking that system's space and time coordinates for granted for the purposes of actual calculating the dynamical behavior of physical systems, then it might seem equally natural to ask how much coordinate time goes by when a certain amount of ticks go by on a moving clock (moving relative to that coordinate system), or how much coordinate space is taken up by a moving ruler. That's the best way I can think of to explain why the equations make intuitive sense to me, but obviously it's subjective so not everyone would have the same intuitions. By the way, I'm about to go on a trip for a few days, so I probably won't be able to continue the discussion until next week sometime. 



#89
Mar1909, 11:44 PM

P: 645

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 halflife of one muon in the primed frame (viewed from the primed frame) will be the same as the halflife of a totally different muon in the unprimed frame (viewed from the unprimed frame. (Yes, I know halflives are statistical, but using a gross misrepresentation here might still be instructive.) What I am saying is that the halflife of the muon in the primed frame (viewed from the primed frame) will be less than the halflife of the muon in the primed frame (viewed from the unprimed frame). 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 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 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 travelled 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 travelled 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 The distance travelled 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 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, travelling 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). The test clock will not have ticked. In the rest frame of the test clock, the test clock's photon will only have travelled a distance of 22.5 km. This time and this distance are t' and L'. L'/t' = 22.5km/75μs = 300000 km/s I hope this helps. cheers, neopolitan 



#90
Mar2009, 01:51 AM

P: 1,400

I suppose "time dilation" and "length contraction" being just a shorthand for the full Lorentz transformation, of use in a special cases, the thing to be learnt is what those special cases are, and (on a more philosophical or abstract level) why a different special case is thus highlighted for time from the special case thus highlighted for space. Regarding which, I've found this a fascinating discussion. 


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