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Time Dilation & Length Contraction, Further Thoughts 
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#1
Dec1812, 09:56 AM

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This is a very common topic especially regarding the apparent slowing down of time, the twins paradox etc.
I have come across the figures for a muon and the relativistic effects that allow it to reach the earths surface despite its very short life. The relevant transforms show length multiplied by the equation and time divided by the same (sqrt(1v2/c2) ). Looking at this I think time expansion is probably a more accurate discription rather than dilation or the more comon use of slowing down. Back to the muon, the figures I saw were (approx) time dilation factor of 22 and length contraction of 10000mts down to 450mts also approx factor of 22. As I see these figures I get 1 muon second = 22 of ours and 1 muon meter = 45mm, it therefore has 22times longer to move 1/22nd distance relative to us in other words it moves very fast and is able to reach the earths surface in its short lifetime. Also time and length contraction are both dependant on velocity and must go hand in hand a fact that does not seem to be taken into account when talking about time slowing for things, normally people moving at near speed of light velocities, there seems to be an obsession with slowing the ageing process, so if time dilates by say a factor of 50 then distance must also contract by the same so you would get 50 times longer to do a journey that would seem 50 times shorter so the total journey would appear to be shrunk by a factor of 2500 (50x50) So if the muon was a person what would we see, after 22 of our seconds their watch would have moved only 1 second but would they actually be 21 seconds younger than the person standing on the earths surface waiting for them, they would certainly have appeared to be moving very fast though. 


#2
Dec1812, 10:37 AM

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You are aware that dilation means expansion, aren't you?
It is common for people to refer to a moving clock running slower as time dilation, and it is because the seconds on the moving clock take longer and are stretched out. However, it's not a squared function. In the earth's rest frame, the muon's time is running slower so it can reach the earth before its half life destroys it but it still has the long distance to go. In the muon's rest frame, its time is normal but the distance to the earth is contracted so its half life is normal but it doesn't have as far to go, or rather, the earth doesn't have as far to go to get to the muon, because remember, we are considering a frame in which the muon is at rest. So if the muon was a person with a clock running 22 times slower than our own according to our rest frame and it/he was coming directly towards us at 99.9% of the speed of light, we would not see its/his clock running slower by a factor of 22. Instead we would see the clock running 44 times faster than our own clocks. Now we're talking about Relativistic Doppler shift, not Time Dilation. We cannot observe Time Dilation, it's a calculation based on a chosen frame of reference and is different with each frame of reference but the Relativistic Doppler shift (what each observer actually sees of the others moving clock compared to their own) is the same no matter what frame of reference is assumed. 


#3
Dec1912, 10:28 AM

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Are you saying that even though time dilation and length contraction are both dependant on velocity you can only have one or the other not both together? You say one is in the earths frame of reference and one in the muons frame why not both in one frame i.e the muons with the earths frame remaining undilated and uncontracted as it is the muon that is moving.



#4
Dec1912, 10:58 AM

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Time Dilation & Length Contraction, Further Thoughts
They are both in both frames but it's only one of them that is required to explain why the muons reach the ground. But remember, in each object's rest frame, it is only the other moving object(s) that are time dilated or length contracted.
So in the earth's rest frame, the muons are both length contracted along the direction of motion and time dilated. However, we don't care about the fact that the muons are compressed instead of symmetrically round (or whatever shape they are in in their own rest frame). We only care that time for them takes longer. And in the muons' rest frame, the earth and the distance to the earth is compressed while the clocks on earth are running slow but the muons don't care about the clocks, they only care that they can survive long enough for the earth to fly up to them because it is much closer than we say it is. 


#5
Dec2012, 11:09 AM

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If the muons clock ticks once for every 22 times of a clock on earth then surely earths clocks should seem to be running faster from the muons point of view. Taking the speed of light as a constant referance frame then from earth to muon we are moving towards it, but from muon to earth we are moving away from it so the transforms should be reversed giving earth clocks moving faster from the muons frame of reference.
Why is length contraction deemed unimportant. If the muon itself is length contracted rather than the distance it travels how does this affect its structure. That is if it contracts/shrinks then its structure must be compressed much like atomic structure is compressed under gravity when a star collapses, looking at the formula it suggests that length contraction at the speed of light should compress any matter into a black hole. So is the moving object contracted, the distance it goes or the space it moves through or all of them. I do understand that it is only in the direction of movement. 


#6
Dec2012, 11:56 AM

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My advice is to draw a spacetime diagram of the muon scenario, first in the Earth's frame, then in the muon's frame. The diagrams will make it clear how length contraction, time dilation, and relativity of simultaneity all fit together into a single coherent picture. Trying to consider any two of them in isolation, without the third, won't work. 


#7
Dec2012, 02:56 PM

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So in your sentence, are you talking about the direction the light is moving when you say "earth to muon"? Then what do you mean by "we are moving towards it"? What is the "it"? And then when you say "muon to earth", is that the direction of the light? So what is "it" that we are moving away from? In the muon's IRF, the earth clocks are time dilated and ticking at 1/22 the rate of the coordinate time. Since the muon is at rest, its clock ticks at the same rate of the coordinate time. I hope you understand that everything is contained in both IRF's and they both explain identically what each observer sees. 


#8
Dec2112, 11:19 AM

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I am getting confused, the muons journey to earth is both time dilated and length contracted due to the fact it is moving faster than the earth. I cannot see how the earth can also be time dilated and length contracted from the muons point of view as it is moving a lot slower, or where relativity of simultaneity comes into it as there is only 1 event and that is the muons trip to earth.
The speed of light as a reference point, the earth is moving slower than the muon so the muons speed is closer to the speed of light so the transforms work as they use increasing velocities i.e. going towards c, but the earth relative to the muon is further away from c or negative, as you cannot have negative velocity then surely the transforms should be reversed to allow for slower velocity, that is time should be multiplied by transform and length divided by as you are moving away from c rather than towards it. If that makes sense. Re rel of sim I have been looking at the train and platform senario, would the person on the platform not see the light waves red/blue shifted and the person on the train see normal wavelength? Thus explaining why the 2 observers see different events 


#9
Dec2112, 12:04 PM

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[Note: I have corrected the definition of event B below; the original definition was wrong.]
To see how relativity of simultaneity comes into it, call the event of the muon being created up in the atmosphere event C, and the event of the muon reaching the Earth's surface event E. Let the unprimed frame be the Earth's frame and the primed frame be the muon's frame. Then we have the following: (1) In the muon's frame (the primed frame), events C and E occur at the same place, and the time interval between them, which is just the proper time elapsed for the muon in flight, is t'. (2) In the Earth's frame (the unprimed frame), events C and E occur at different places, so we have both a time interval t between them (at least, that's the natural way to state it, though as we'll see below, I'm misstating this), and a spatial distance d between them. By the time dilation formula, we find [itex]t = \gamma t'[/itex], where [itex]\gamma[/itex] is the relativistic gamma factor for the muon's velocity relative to the Earth. Since [itex]\gamma > 1[/itex], we have [itex]t > t'[/itex], as expected. (3) To evaluate length contraction, however, we can't use events C and E, because the distance between them is zero in the muon's frame, so that can't possibly be the right distance to compare with d, the distance between C and E in the Earth frame. To resolve this issue, first note that I actually misstated the definition of the distance d above. It isn't the distance between events C and E, because those events are timelike separated, and "distance" only applies between spacelike separated events. The distance d is actually the distance between events D and E, where event D is the event that is at the same spatial location as event C, in the Earth frame (i.e., it is "where the muon was created", in the Earth frame), but happens at the same time, in the Earth frame, as E does. That is how we define the distance the muon has to travel, in the Earth frame. (4) Now, to evaluate length contraction, consider: how do we define the distance in the muon's frame that corresponds to d? That distance would be the distance the Earth has to travel, in the muon's frame; so it will be the distance d' between events B and E, where event B is the event that is at the spatial location of the Earth, in the *muon* frame, at the same time, in the muon frame, as event C. (Note that this is *not* the same as the spatial location of event E in the muon frame; in the muon frame, the Earth moves, so it's at a different spatial location at event B than it is at event E.) If we then apply the length contraction formula to the distances d and d', as defined above, we will find that [itex]d = \gamma d'[/itex], and therefore [itex]d > d'[/itex], as expected; the distance the Earth travels, in the muon's frame, is length contracted compared to the distance the muon travels, in the Earth frame. (5) But note that, in order to get the result we just got for length contraction, we had to use *two different pairs of events*. We used events D and E to get the distance d in the Earth frame, but we used events B and E to get the distance d' in the muon's frame. The reason we had to do that was relativity of simultaneity: distance in the muon's frame is defined using a different simultaneity convention than distance in the Earth frame. (6) And now that we've seen that relativity of simultaneity comes into play in evaluating length contraction, we can see that it also comes into play in evaluating time dilation. Go back and look at item #2 above, and note that I pulled a fast one there: I said that in the Earth frame, events C and E occur at different places, but then I just wrote down a "time interval" between events C and E in the Earth frame, with no further clarification. What I should have done is similar to what I did above for length contraction: the time interval t, in the Earth frame, is not between events C and E, but between events A and E, where A is the event in the Earth frame that is at the Earth's spatial location, but happens at the same time as event C (the muon's creation) does. So again, the time intervals t and t' are between two *different* pairs of events, because time intervals, like distances, are defined using a different simultaneity convention in different frames. So in summary, we have: In the Earth frame: distance d between events D and E; time t between events A and E. In the muon frame: distance d' between events C and B; time t' between events C and E. You can't fit everything together without all three elements: time dilation, length contraction, *and* relativity of simultaneity. 


#10
Dec2112, 12:06 PM

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#11
Dec2212, 04:54 AM

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So the muon's trip to earth can be described as a series of events, starting with its creation miles above the surface of the earth, and as time progresses for the next couple of microseconds, its altitude decreases until it reaches zero altitude. We could have a list of events saying, for example at time zero, it was at an altitude of ten miles, one microsecond later it was at five miles and at a time of two microseconds it was at sea level. Of course, there are an infinite number of events all along the way, but we may choose to list a reasonable subset. One other thing you should be aware of is that if you consider the speed of the muon with respect to the earth's IRF (in which the earth is at rest), then it's exactly the same speed of the earth in the muon's IRF except for the direction, of course. So in the earth's IRF, the speed of the muon is 0.999c because it is going downwards and in the muon's IRF, the speed of the earth is +0.999c because it is going upwards. Now when it comes to time dilation and length contraction, an object in its own IRF will not be subject to these phenomenon because they are functions of speed but the other one will. It's exactly reciprocal because the speed is reciprocal. The muon in the earth's IRF has exactly the same time dilation that the earth has in the muon's IRF. Same with length contraction. 


#12
Dec2212, 04:54 AM

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First, let's look at a spacetime diagram of the earth's IRF showing the motion of the muon. (The letters correspond to the approximate locations of the events that PeterDonis explained in his post #9read it carefully.) I have assumed that the muon is traveling at 0.999c which makes gamma equal to 22.366 and the Relativistic Dopper factor equal to 45.71. I have considered the muon to take two microseconds to reach the earth's surface so that it starts at an altitude of about 44.7 lightmicroseconds. I start the scenario at the point in time when the earth sends a signal to the muon signifying the time on its clock (zero). It then continues to send a new signal every microsecond.
The earth's path (worldline) in the spacetime diagram is shown as a blue line going straight up in time (because it's not moving) along the left side of the diagram at a spatial location of zero. (Since we are assuming that the muon is coming straight down in space, we can ignore the other components of space and focus only on the altitude, shown along the horizontal axis.) The 1microsecond signals are shown as yellow lines propagating diagonally upwards and to the right. There are a total of 90 of them sent during the scenario and they are all received by the muon during its 2microsecond lifetime. This illustrates the Relativistic Doppler which indicates that at 0.999c, each observer will see the other ones clock running 45.71 times its own while they are approaching each other. Now what about Time Dilation? We use the speed of each observer/clock to determine its Time Dilation (gamma) according to the IRF. Since the earth is at zero speed, its Time Dilation is 1, meaning that the progression of its clock (its Proper Time) is in step with the Coordinate Time. This is indicated in the diagram as a series of blue dots, coincident with the progression of the Coordinate Time. Things are different for the muon. It is represented by black in the diagram. Once it starts its trip about half way up the scenario, its path in the spacetime diagram is upwards and to the left. This indicates a decreasing altitude, ending at zero, the location of the surface of the earth, taking 2 microseconds. Because of its speed of 0.999c in the IRF, gamma for it is 22.366 meaning that each microsecond of Proper Time according to the muon's clock will take 22.366 microseconds of Coordinate Time. You can see that the muon begins its existence at just over 44 microseconds of Coordinate time and its first microsecond tick of Proper Time (the black dot in the middle of the black path) occurs at 67 microseconds of Coordinate Time for a delta of just under 23 microseconds. Now the muon is also sending out light signals to the earth, one each microsecond, including one at the beginning of its existence and they travel at c upwards and to the left. Unfortunately, These three light signals propagate so closely to the muon itself that they cannot be distinguished from the muon in a diagram of this scale. So to overcome this problem, I have zoomed in on the detail of first diagram surrounding the events of the muon's arrival on earth: Now we can see the three individual light signals sent out from the muon at 1microsecond intervals and we can see that they are received on earth at the same fast rate as what the muon receives the signals from earth. (You really have to copy the image and print it out so you can draw in extra signals to see that 23 of them would fit between the 89microsecond tick of earth and the 90microsecond tick.) So the bottom line for the earth's IRF is that time is dilated for the muon so it can survive the 45 microseconds it takes to get to the earth, even though its halflife is less than 2 microseconds. 


#13
Dec2212, 04:56 AM

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Now we want to look at the muon's IRF. If we simply transform all the events, using the Lorentz Transformation, in the first IRF by a speed of 0.999c, we get the following spacetime diagram:
This clearly shows us that the earth is moving at an ever increasing distance to the muon which is stationary at about 2000 lightmicroseconds, taking about 2000 microseconds of Coordinate Time at 0.999c to get there. Notice how the Proper Time of earth's clock is Time Dilated by the factor of 22.366 as evidenced, for example by the Coordinate Time of 2000 divided by the final Proper Time of 88.5 yielding 22.6, close enough for eyeballed calculations. But to see the interesting detail, we need to zoom in on the final moments of the earth's path: I have made this diagram to show the last two ticks of the earth's clock to get a perspective of how the timing works. Here we can see that the muon, indicated by the vertical black line is stationary at a location of 1999 lightmicroseconds and lasting for 2 microseconds of Coordinate Time. Now we can see the three individual light signals (shown in black) sent out by the muon at 1 microsecond intervals. We can easily see that they are received at a rate of 1/45th the rate of the Proper Time ticks of the blue earth. We can also see the flood of yellow light signals sent out by the earth, but long ago and far away in this IRF. We zoom in some more to see the details even more clearly: Now we can see the individual yellow light signals coming up from the earth (up from below and to the left of the diagram) and there are the correct number of them. Copy and print out the image if you want to count them out precisely. Here is where we can easily see the Length Contraction of the distance to the earth, as explained by PeterDonis. In the muon's IRF, the distance the earth is away from the muon at its creation, event C, is indicated by the event B, giving a distance of about 2 lightmicroseconds. So that pretty much covers everything. Any questions? 


#14
Dec2212, 04:18 PM

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ghwellsjr, great diagrams! Just one comment to add: in your first diagram, showing events A, C, D, and E in the Earth frame, note that event B would be on the Earth's worldline just a bit below event E. This gives another point of correspondence between that diagram and the diagrams drawn in the muon's frame.



#15
Dec2412, 05:33 AM

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Many thanks for the time you have obviously spent on the above posts, I will need to try and work out where I am, things look more and more complicated and I think its simpler than it may seem.
The effect of the equation squrt 1v2/c2 is fairly self explanitory, it is the application I am having problems following. How can the distance between the muons creation in the upper atmosphere and the earths surface be 0 in the muons frame. Now when it comes to time dilation and length contraction, an object in its own IRF will not be subject to these phenomenon because they are functions of speed but the other one will. It's exactly reciprocal because the speed is reciprocal. The muon in the earth's IRF has exactly the same time dilation that the earth has in the muon's IRF. Same with length contraction. Surely an object in its own IRF would not be aware of any time/length change rather than subject to. Anything that is moving must be subject to time dilation/ length contraction. I can see how velocity affects the object moving but not how the velocity of object A can affect the time dilation/length contraction of object B, while relative velocity may be the same actual velocity clearly is not, I can see how the total can be relative though. 


#16
Dec2412, 11:09 AM

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If you look at either of the last two diagrams, you will see that the muon's creation in the upper atmosphere occurs at event E. The coordinates of that event in this particular IRF that I have defined in which the muon is at rest are (approximately) t=1999 usec and x=1999 lightusecs. At that time, the earth's surface is at (approximately) x=1997 lightusecs. This gives a distance of approximately 2 lightusecs. Note that I said in my earlier posts that these numbers are approximate because we are "eyeballing" them from the diagram. In post #9, section (4), PeterDonis defined an event which he called event B, just for the purpose of determining how far away the earth was at the time of the muon's creation, in the muon's IRF. Nothing is actually happening at this event, that is, even though the event is on the earth, it has no significance or any awareness for any one on the earth. This might seem artificial but it is no different than what he did in the earth's frame when he defined event D in section (3). There again, nothing is happening at that event which has any significance or awareness to anyone in the scenario. Yet, somehow the distance we determine between the earth's surface and the creation of the muon in the earth's IRF seems obvious and natural but doing the same exact thing in the muon's IRF seems artificial and arbitrary. I would encourage you to go back and study PeterDonis's post #9 until this all makes perfect sense to you. It really helps (me, at least), to be able to look at some diagrams for a particular example while studying Peter's explanation. You should keep in mind, as I pointed out earlier, that the particular set of coordinates that I came up with for the earth's IRF and the transformed muon's IRF are just my arbitrary decision. I'm sure Peter (or anyone else) would have come up with a totally different set of coordinates (even for the same example) if he had chosen to do so. But this raises a very important aspect that Peter failed to point out which I would like him to address and that is, how do we determine the exact coordinates for event B? When you draw a diagram for a specific example as I did, we can just follow the horizontal grid line from event E to where it crosses the path of the moving earth and that shows us where event B goes. But it's only approximate because we're eyeballing off a diagram. This is not an issue for determining the exact coordinates for event D in the earth's IRF because, I defined those coordinates when I set up the example. I said the muon was created 2 microseconds (in its IRF) before it reached earth and it traveled at 0.999c straight down. I used the calculation of gamma to determine how long in the earth's IRF it would last and then I used its speed (which is the same as the earth's speed in the muon's IRF) to determine its altitude when it was created (event A). You can carry out this calculation with as much precision as you want. So whatever arbitrary coordinates you want to use for event E, since we know the spatial distance between events A and C, the same spatial distance applies between events E and D, and those two events have the same time coordinate so we're done. However, we cannot do something so easily for determining the coordinates of event B. We have to rely on transformed coordinates, at least, that's how I know to do it. The time coordinate is no problem, it's the same as the time coordinate that was part of the description of the scenario (2 usecs prior to earth impact). But since nothing happened at event B in the earth's IRF, how do we determine its coordinates so that we can transform it to the muon's IRF? I'd like to hear Peter's explanation. Of course, we can always cheat. Since we know the earth's speed in the muon's IRF and we know how much time transpires from its creation to earth impact (2 usecs), we can trivially calculate how far the earth had to travel (d = vt = 0.999c * 2 usecs = 1.998 lightusecs) and this answer must agree exactly with doing it the "hard" way. But cheating doesn't illustrate the length contraction of the earth and the distance to the earth in the muon's IRF which is what we are interested in so I'd very much like to hear how Peter will describe the process. Maybe he will have a better way than the way I have come up with. For convenience, we talk about A's IRF and B's IRF but we simply mean that if we pick an IRF in which A is at rest, then B is moving and if we pick an IRF in which B is at rest, then A is moving. Instead, we could have picked an IRF in which both A and B are moving at the same speed in opposite directions and they would both be subject to the same time dilation and length contraction but we would never say that these different dilations and contractions are due to the motions of one object on the other object, would we? In fact, we can just eliminate one of the objects and see how the other object has different time dilations and length contractions in different IRF's moving at different speeds when it is the only object under consideration. Would that help you to understand the concept? 


#17
Dec2412, 11:30 AM

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#18
Dec2412, 12:15 PM

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