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Quick calculation check please

  1. Feb 18, 2015 #1
    Could someone just check these calculation results for me please. I know this is really basic, and I'm assuming they aren't right but I couldn't see where they were wrong, so any help would be appreciated. (I know I've been given loads on this forum already, and I am grateful).
    Code (Text):

    v    = 29,979,245.8
    gam  = 1.005

    t        0             1              2
    x        0             0              0
    t'       0             1.005          2.010
    x'       0             -30,130,275.7  -60,260,551.4

    t        0             1              2
    x        0             29,979,245.8   59,958,491.6
    t'       0             0.9950         1.990
    x'       0             0              0
     
     
    Last edited: Feb 18, 2015
  2. jcsd
  3. Feb 18, 2015 #2

    jedishrfu

    Staff: Mentor

    I edited your post and used code tags which give you a mono spaced font.

    You may want to clean it up a little more though.
     
  4. Feb 18, 2015 #3
    Thank you :)
     
  5. Feb 18, 2015 #4

    DrGreg

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    Why are you assuming that?

    Note that you have rounded [itex]\gamma[/itex] to 4 significant figures, so don't expect to get more than 4-sig-fig precision from your calculations.
     
  6. Feb 18, 2015 #5
    The problem with it being correct seems to me to be that I could imagine a conveyor belt going at 0.1c relative to the floor. And I could imagine observers along the floor where conveyor belt would have moved to after each second (from the floors perspective). The second table seems to be showing me that for any given observer on the conveyor belt they could consider t'=0 x'=0 to be at the point of passing any observer on the floor, and they will all agree that their clocks are only doing roughly 0.9950 ticks for each tick the floor observer's clocks, and this can be checked each time they subsequently pass an observer (so in table 2 it shows that after it sets t'=0 x'=0 as it passes a floor observer, when it passes the next that next ones clock has gone on a second while its has only progressed 0.995, and when it passes the next, it has also gone on a second while its has only progressed 0.995 since the last time. But it also seems to be saying (in the first table), that if they kept their eye on the clock they were passing when they decided t'=0 x'=0 they would be thinking that for each tick it made their clock had made 1.005 ticks, even though everyone on the conveyor belt can agree that their clocks aren't going faster but slower compared with the ones they were passing, and that can be checked each time they pass a floor observer.

    I think I can see it now from the conveyor belt's perspective the t' isn't for the one that set t'=0 x'=0 it is for the one on the conveyor belt at the x' coordinate given (the one that set t'=0 x'=0 remains at x' =0 from its perspective), but am still not clear on the different ratio.
     
    Last edited: Feb 18, 2015
  7. Feb 19, 2015 #6

    Ibix

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    First a tip to make arithmetic easier: measure distance in light seconds (ls) and time in seconds. Then c=1 ls/s and v=0.1 ls/s.

    Is it the lack of symmetry that's bothering you? By that, I mean were you expecting the second table to look a bit more like the first?

    What's going on here is that you've set up your experiment in a way designed to show you what's going on in the floor frame. In the first table you've got one clock on the floor at x=0 and three clocks on the belt at x'=0, -0.1005, -0.2010. In the second table you've got one clock on the belt at x'=0 and three clocks on the floor at x=0, 0.1000, 0.2000. Basically, you have pre-positioned clocks on the belt so that they pass clocks on the floor at a convenient time for clocks on the floor in both tables. This means that your experimental setup is privileging the floor frame; that's why it kind of looks like there's something special about the floor frame.

    If you use (x',t')=(0,0), (0,1), (0,2) and (x',t')=(0,0), (0.1,1), (0.2,2) you'll have privileged the belt frame. If you then calculate the (x,t) coordinates you'll reproduce your tables upside down, except for a sign.

    Also, draw a space-time diagram. They make visualisation a lot easier.

    Hope that's helpful.
     
  8. Feb 19, 2015 #7
    I think I get it now. If an observer on the floor was to set its clock to 0 as the observer that is passing it on the conveyor belt sets its clock to 0, and the next observer on the floor also sets its clock to zero, then when the observer on the conveyor belt that had set its clock to 0 reaches it, they can compare and see that while its has progressed a second the one on the conveyor belts has only progressed 0.9950.
    But if it were done the other way around and the observers on the floor thought they were stationary and it was the observers on the floor that were moving, and the conveyor belt observers could set their clocks to zero as their clocks showed a certain time, but the observers on the floor might object and say that the clocks on the conveyor belt weren't showing the same time at the same time and if they waited until that to set the clocks to zero they'd be out of synch. Instead the observers on the floor might suggest, that the conveyor belt observers set their clocks to zero as they pass the next floor observer, and then from the conveyor belt perspective each person on the conveyor belt has set its clock to 0 when a one of the travelling floor observers was passing it, and the next was a fixed distance away, and for each of them by the time the next reaches it the passing observer's clocks have ticked 1 while theirs have only ticked 0.995. And that is for all the observers on the conveyor belt from the conveyor belts perspective and they all agree that the distance the next moving floor observer was the same for each of them and that the floor observers were moving at a fixed velocity. From that I would have thought they could have concluded that it takes a fixed amount of time for the floor observers moving at a fixed velocity to go between one conveyor belt observer and another a fixed distance away, and that this could be checked on each passing. But the distance isn't fixed if you use the speed of light as being invariant to measure things. When doing so whether the conveyor belt observers are both equal distance apart or not depending on where you measure it from.
     
  9. Feb 21, 2015 #8

    Ibix

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    Sorry for the slow reply - I typed up a response on my phone yesterday, but the browser crashed just before I sent it and I haven't had a chance to respond since.
    It's very important when talking about "at the same time" to specify who is defining "the same time", since observers in different frames won't necessarily agree. I think you mean "at the same time in the floor frame" here, in which case, I agree.
    I think you've got this, although your description is rather confusing. What I think you are realising is that the floor-based observers see the belt-based clocks ticking slowly, so they are happy that the belt-based clocks only show 0.995s passing in the 1s it takes them to move from one floor-based observer to the next. Meanwhile, the belt-based observers see the distance between adjacent floor-based observers length contracted to 0.0995ls, so are happy that only 0.995s elapses between successive floor-based observers passing them.

    This isn't quite right. Distances are different when measured in different frames whatever method you use to measure them. The frame invariance of the speed of light is one possible starting point to deduce this, but it is not the only one.
     
  10. Feb 23, 2015 #9
    So if I call the observers on the floor Team A, and the observers on the conveyor belt that is going at 0.1c Team B, and let's still imagine that in the Team A reference frame the observers are spaced out 0.1c distance from each other in their frame of reference, and they measure it in feet for example so roughly 98,357,105.643 ft apart, and let's say they it measure out in rulers from their rest frame. Let's also imagine that they have some rulers from Team B's rest frame, and they measure the distance out with however many of them it takes. Team A can agree that the distance of a certain amount of Team B rulers is equal to a certain amount of their rulers, and then it would be a fixed equivalent distance whichever rulers you used wouldn't it, or are you suggesting that there would be a disagreement about the amount of rulers (of either type) laid out on the floor between the Team A observers?
     
  11. Feb 24, 2015 #10

    Ibix

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    Seriously, use a sensible unit for the problem. Using feet to measure experiments on the scale of light seconds is asking for transcription errors and, as Dr Greg noted, pointless unless you retain 11 significant figures on everything.

    You have a conceptual problem here. Rulers are not in one frame or the other, they are in both. That means that "rulers from Team B's rest frame" doesn't make sense. Frames are just choices of coordinates. Imagine that we're discussing a trip. We both have maps on our phones, but yours has flipped into landscape mode. Roads that are going up my screen are going across yours. In other words, my phone is using a reference frame in which north is up, but yours is using one where west is up. It doesn't make sense to talk about "bringing a road from your map to mine". The same is true of relativistic frames.

    Team A could lay out rulers on the floor (which would be at rest in Team A's frame), and would measure 0.1ls between observers. Team B could lay out rulers on the belt (which would be at rest in Team B's frame), and would measure 0.0995ls.

    The teams could then swap rulers, rewind the belt, and repeat if they wish; they'll get the same result.
     
  12. Feb 24, 2015 #11
    In this example though there is no assumption that the speed of light was invariant whatever the rest frame. So it doesn't matter that the light travels the rulers faster in one rest frame than the other. The reason I chose feet instead of meters is that meters are measured in light with the principle that the speed of light is invariant whatever the rest frame. Remember, I had earlier wrote:

    "But the distance isn't fixed if you use the speed of light as being invariant to measure things"

    And you replied:

    "This isn't quite right. Distances are different when measured in different frames whatever method you use to measure them. The frame invariance of the speed of light is one possible starting point to deduce this, but it is not the only one."

    So I've given an example where distance is measured by rulers (in feet), and you haven't explained why the distance would be considered different if the speed of light wasn't considered invariant in each rest frame.
     
  13. Feb 24, 2015 #12

    PAllen

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    Because that's the way the universe works? All physical phenomena transform between frames using the Lorentz transform, irrespective of whether they are based in any way on light. You should be aware that from the indistinguishability of inertial frames, isotropy, and homogeneity, you can derive that either the Galilean transform or the Lorentz transform (with some finite invariant speed), are the only possibilities. Then, all physics will be consistent with one or the other, based on experiment. In our universe, it is the Lorentz transform.
     
  14. Feb 24, 2015 #13

    Ibix

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    Take a 1m ruler. Mark it "1m" on one side and "39.37in" on the other. Do you really think one side will length contract and not the other? If you want to, mount a mirror and clocks on each end and start light pulses bouncing between them, so you're measuring the distance in terms of light speed. How will distance measured one way change without the other changing?

    The meter isn't defined in terms of light speed because of relativity. It's defined that way because we're really good at measuring time, so it makes sense to define distance units in terms of a speed and a time.

    I may have misled you with my point about the frame invariance of "[t]he speed of light [being] one possible starting point". For the record, the speed of light is invariant in all inertial frames. This was one of Einstein's postulates, and if it is not true in any given scenario then we are not discussing relativity.
     
  15. Feb 24, 2015 #14
    I never said it wasn't. All the calculations used the Lorentz transformation.
     
  16. Feb 24, 2015 #15
    I never said that the ruler would shrink on one side and not the other. The point was that what is a meter is determined by light. So with the supposition that the speed of light isn't invariant, a meter ruler would be a different length according to Team A members than Team B members, and while the scenario did show how it could handle them being different lengths (they might not be the same number of feet), I thought it might get confusing to use meters as a measurement since it would sound as though it implied a ruler of the same length (same name) but actually it would be a different length ruler (if the speed of light wasn't considered to be invariant).

    It seems to me that using a meter as a measurement of distance relies on the assumption that clocks don't slow with motion. Whereas using a ruler only relies on the assumption that the distance is a single distance as opposed to multiple distances and which one it is for an observer depending upon the relative motion of their rest frame compared to the rest frame containing the distance. So given that your response was your reply to:

    "So I've given an example where distance is measured by rulers (in feet), and you haven't explained why the distance would be considered different if the speed of light wasn't considered invariant in each rest frame."

    You replied, but I didn't notice your answer. Because remember I had written:

    "But the distance isn't fixed if you use the speed of light as being invariant to measure things"

    So it was clear that I understood that if the speed of light was considered as invariant (as with relativity) that the distance wasn't considered fixed. And yet you replied:

    "This isn't quite right. Distances are different when measured in different frames whatever method you use to measure them. The frame invariance of the speed of light is one possible starting point to deduce this, but it is not the only one."

    So if you just want to say "well in relativity the speed of light is considered to be invariant", then that is fine, but it wouldn't be right to suggest what I had stated wasn't quite right, and state that "distances are different when measured in different frames whatever method you use to measure them" ** if ** one could consider the speed of light to not be invariant and measure distances with rulers, and the distances remain the same. And it is your lack of an answer to why the distances wouldn't be considered to be the same under such circumstances. that led me to write "you replied, but I didn't notice your answer".

    I was thinking that if the team A observers were separated by a distance of 0.1c (in their rest frame) marked out in 1m rulers from Team A's rest frame and 1m rulers from Team B's rest frame, so that if there was any difference in length it would be reflected in the amount of rulers of that type compared to the amount of rulers of the other type, and if the all observers (both team A and team B) were to set their clocks to zero when they next passed an observer (so x = x'=0, t = t' = 0) then they could all agree that when they next passed an observer, the floor observers body clocks were showing t = 1 while the conveyor belt observers body clocks were showing t = 0.9950. What I'm not clear on is why if that is so, it couldn't be concluded that if light wasn't considered to be invariant the distance between the team A members could be considered to be fixed (a certain amount of rulers of whatever type) and that Team B's clocks could be considered to be going slower than Team A's. So is it so, and is it ok to make that conclusion?
     
  17. Feb 24, 2015 #16

    PeterDonis

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    You have to be careful in specifying exactly what is meant by "the speed of light isn't invariant". What appears as "the speed of light" in the table of SI units is just a defined number, 299,792,458; but if you go out and measure some light and how it is moving, you will be using a ruler that is some number of atoms long (and to be perfectly precise, you would have to specify exactly what kind of atoms), and measuring with a clock whose "ticks", if you are using the SI definition of the second, are determined by the frequency of a particular energy level transition in a particular kind of atom. So if Team A and Team B want to compare their measurements, they have to compare how many atoms long their rulers are (and make sure both rulers are made of the same kind of atoms), and exactly what kind of atom and which energy level transition they are using to define their clock ticks.

    What "the speed of light isn't invariant" would mean is that, after both teams have verified that their rulers are exactly the same number of the same kind of atoms long, and that their clocks are using exactly the same energy level transition in the same kind of atom for their tick rate, they get different results for the ratio of (ruler lengths traveled by light) / (clock ticks). The reason SI units now define the speed of light as a particular number is that, whenever such experiments have actually been done, Teams A, B, C, D, .... W, X, Y, Z have all gotten the same number, 299,792,458, for the ratio of (ruler lengths traveled by light) / (clock ticks).

    All of the SR stuff about length contraction, time dilation, etc., applies to rulers and clocks defined in this way: i.e., if we have a ruler that is ##L## atoms long, moving relative to us, then a ruler at rest relative to us will have the same length as the moving ruler (i.e., the ends will line up at a particular instant of our time) if it is only ##L / \gamma## atoms long. And if we have a clock moving relative to us, it will tick ##T## ticks (defined by the appropriate atomic energy level transition) in the same time as an identical clock at rest relative to us will tick ##T \gamma## ticks (defined by the same energy level transition).

    No, it only relies on the assumption that the clock and the ruler are at rest relative to each other and to the experimenter. That is the condition that ensures that you get the ratio 299,792,458 for the ratio of (ruler lengths traveled by light) / (clock ticks) (where the ruler is a particular number of atoms long and the clock ticks are determined by a particular atomic energy level transition). The fact that identical rulers and clocks in motion relative to the experimenter (and to his rulers and clocks) appear shorter and tick slower is not an assumption; it's an experimental fact.
     
  18. Feb 24, 2015 #17
    I don't see that we need to go into the technical details of making rulers of near equal length. You could even use light to make a really long ruler in your rest frame.

    I had said:
    "It seems to me that using a meter as a measurement of distance relies on the assumption that clocks don't slow with motion"
    and you replied:
    It wasn't the shorter rulers and ticks were an assumption, but I was assuming that if light wasn't invariant, and clocks slowed with motion, then a 1m ruler in one rest frame could be a different length from a 1m ruler in another rest frame. And could be measured to be so side by side. The Team B people on the conveyor belt could use their Team B 1m rulers to measure out how far apart the Team A members were, and the Team A members used their Team A 1m rulers to measure how far they are apart, and they could stop the conveyor belt and check. Whether it would be found that the rulers were different lengths when in the same rest frame or whether they were the same length in the same rest frame, and that the team B members were spaced out further than the team A members wouldn't change the thought experiment fact that while in motion the Team B 1m rulers on the conveyor belt measured the distance between the Team A members differently from the Team A 1m rulers. Has there been an experiment to tell btw?

    The team A members have their clocks and rulers at rest relative to each other (and they are the experimenters), and the team B members have their clocks and rulers at rest relative to each other, but they both come up with different answers when it comes to how many meters it is. I'm not suggesting that the distance they are giving is relative, it is just that it seems to me you'd need to get into their rest frame and do an experiment with light and a clock to find the distance (from there to there) that was being talked about, or use the transformation to compare it to a 1m distance in another rest frame. So when I meant a meter as a measurement of distance, I meant from a "there to there" perspective, and if the universe had been different, and the clocks had slowed in a different way, it could have been really impractical to assume that light would always go the same distance each tick of the clock.
     
  19. Feb 24, 2015 #18

    PeterDonis

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    But that would be measuring distance by light travel time, which makes the speed of light invariant by definition; the question "what would it be like if the speed of light was not invariant" would be meaningless. In order to even ask that question, you need to have a way of measuring distance (like "number of atoms long") that isn't dependent on light travel time.

    But what defines a "1m ruler"? Is it "a certain number of atoms long"? If so, what does the speed of light have to do with it? Each Team makes sure their rulers are the same number of atoms long. Then you bring both rulers to rest and compare them.

    This assumes that the process of stopping the conveyor belt does not affect the physical structure of the rulers. Experimentally, this appears to be a good assumption, at least for reasonable accelerations; but it's worth pointing it out since we're trying to be explicit about assumptions.

    Also, experimentally, we find that whenever rulers that are the same number of atoms long are brought to rest and compared, they have the same length (their ends match up), regardless of their past history of relative motion (subject to the above assumption being valid). But, again, that by itself doesn't prove anything about whether the speed of light is invariant; it just proves that "number of atoms long" is a good criterion for how to define the "length" of a ruler. Then you can use these rulers, along with clocks whose tick rates are set by atomic energy level transitions, to measure the speed of light. So whether or not the speed of light is invariant is independent of whether or not rulers of identical construction (same number of atoms) have identical lengths when they are at rest relative to each other.
     
  20. Feb 24, 2015 #19
    I had said:

    "You could even use light to make a really long ruler in your rest frame."

    To which you replied:

    I'm not sure why that would make the speed of light invariant if the rulers could be different lengths. For example, if team A considered there to be 0.1c Team A 1m rulers between the A team members, but the B team considered there to be more Team B 1m rulers, and when they stop the conveyor belt they find they are right. A 1m ruler would the distance light travelled in 1 second in the rest frame divided by 299 792 458, and just considering the distance between the team A members, the ruler length seems to me to be different in Team B's rest frame than in Team A's rest frame.
     
  21. Feb 24, 2015 #20

    PeterDonis

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    I don't understand. You're proposing defining the "length" of the ruler by how long it takes light to go from one end to the other. (There's no other way to make a "ruler" from light.) That means the "speed of light" is "one ruler length per ruler length" by definition; it can't change.
     
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