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Do we have time because we are moving?

  1. Sep 27, 2013 #1
    I searched for an answer already but I didn't find a satisfying answer or even exactly the same question (i believe...if I just didn't get it, please let me know)

    OK, now to my question:

    As it happens, time moves slower with higher speed and/or higher gravity.
    I propose to think of two reference-frames.

    1. on earth, moving with it around the sun and with it around the center of our Galaxy which is moving away from the point of big bang...
    a) earth around sun: approximately 100000 kph
    b) sun around galaxy: approximately 828000 kph
    c) milky way through space: approximately 590 kps

    2. as the second reference frame I chose the center of the kosmos and assume, there won't be any gravitation left because it all flew out into space (which is still accellerating it's expansion afaik)

    If we would take two perfectly synchronised clocks measuring time from 0 to infinity and put one on earch and one at the center of our universe assuming there it has the absolute speed and gravitational pull of zero each. How much would the measured time differ between these within an hour/day/week/month/year/ (earth time)...

    PS: I searched for those numbers, please feel free to replace them with more likely values if I didn't chose well. personally i guess we can't really say how fast our galaxy is moving because we probably can't even see any fixed point in space...But the proposition of space-time really affecting different reference frames to measure the same amount of time differently got me thinking about what might be a "universal standard time".
    Also as some say time only exists because of gravity and in conjunction with space (if i remember correctly). Wouldn't that mean that at my proposed center of the galaxy without any gravity or speed, there wouldn't be any time at all?
    But how does that work?
    If mass and speed slow it down, there should be a maximum "speed of time" unless there is negative mass and speed.

    I hope at least I made some sense here...
    As much as I love thinking about it, I'm really not much of a physicist :( and have too little time to go into detailed studies regarding physics.

    PPS: as english isn't my first language please be patient with me if i made mistakes ;)
     
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  3. Sep 27, 2013 #2

    phinds

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    Your premises are so flawed that the rest of your post is moot.

    (1) All motion is relative, not absolute as you seem to think it is.
    (2) There is no "center of the kosmos/universe" as you seem to think there is
    (3) Time does NOT slow down as you travel faster, it just looks that way to someone in a different inertial frame of reference. You, right this minute, are traveling at .999999c from some frame of reference. From that frame of reference, your time is severely slowed down. From some other frame of reference you are traveling at .9c and your time is only mildly slowed down. From YOUR frame of reference, you time is not slowed down at all. Are any of those three frames of reference more correct than the others? No.
     
  4. Sep 27, 2013 #3

    PeterDonis

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    Taragond, welcome to PF!

    This is true only in a relative sense, and it works differently in the two cases you refer to. In other words:

    * If you are moving relative to me, then your clocks will seem to me to be running slow, and my clocks will seem to you to be running slow. But your clocks will seem to you to be running normally, and my clocks will seem to me to be running normally.

    * If you are closer to a large gravitating mass than I am, and we are not moving relative to each other, then your clocks will seem to me to be running slow, and my clocks will seem to you to be running fast. But your clocks will seem to you to be running normally, and my clocks will seem to me to be running normally.

    All of these speeds are relative; there is no such thing as absolute speed. The speed in a) is the Earth's speed relative to the Sun; the speed in b) is the Sun's speed relative to the center of the galaxy; the speed in c) is the galaxy's speed relative to the cluster of galaxies that it is moving towards (I'm not sure which galaxies exactly, I would have to look it up--I also haven't checked your numbers, but the exact numbers don't really matter for this discussion).

    There is no such thing; the universe (which is what I think you mean by "kosmos") doesn't have a center.

    The fact that the universe's expansion is accelerating does not mean there is no gravitation left. In fact, the acceleration of the universe's expansion is a form of gravitation (though it's a form that doesn't work the same as the ordinary form of gravitation that we are used to).

    It's impossible to answer this question because, as I said above, there is no center of the universe. That means there is also no such thing as absolute time; there is no "true" time that all other times can be measured relative to.

    Can you give a specific reference for this? "As some say" doesn't help very much, since the statement as you give it is vague and could mean a number of different things.

    Speed is relative, as I said above. Mass is not; if there were only a single isolated mass in the entire universe, there would indeed be a "maximum speed of time", which would be approached as a limit as you got farther and farther away from the mass. However, our universe does not have a single isolated mass; on average, over large distances, it has a relatively constant density of matter everywhere. In such a universe there is no "maximum speed of time"; it's all relative.

    It's fine to ask questions about things, but you should realize that the concepts you are asking questions about are hard to understand without at least some detailed study. So you may find that you can't really get a satisfying answer to some of your questions without spending a fair bit of time.
     
  5. Sep 27, 2013 #4

    ghwellsjr

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    Do we have time because we are moving?

    Is this another one of those questions inspired by Brian Greene?
     
  6. Sep 27, 2013 #5
    Hi, thanks for your fast reply but i hoped for a few more details on how my thinking is wrong (anf of course WHY)

    (1) That sounds as if we can be absolutely sure about that. Of course I know all motion is relative to another. But can we be sure there is not even a theoretical standing still in our universe at all?

    (2) Isn't there or are we simply not able to detect one? By which theory do you conclude there isn't any center? If there was a big bang, wouldn't the center of that "explosion" also be a kind of center of the universe? Am I still hanging on an old flawed theory or did I misunderstand it?

    (3) Isn't it true, that if you put perfectly synchronised watches one on earth and one on a satellite and bring them back together later, they aren't in sync any more?
    Aren't the clocks of our gps-satellites going faster to compensate for the speed they have orbiting us?
    Also I didn't ask for "the correct" time but for the difference between them. And that's why i specified frames of reference.
    If that's correct so far, am I thinking so flawed that it's not possible to deduct a difference between two hypothetical reference-frames?

    And isn't it possible to calculate how different those clocks would be?

    Ay, have to read some posts to get the hang of it ;)
     
    Last edited: Sep 27, 2013
  7. Sep 27, 2013 #6
    Thank you PeterDonis,
    that's what I hoped for.

    That's actually pretty much the way I understood it, with the distinction of the actual difference in measured time if you bring the clocks from different reference frames together afterwards...see my last post about gps and co.

    ok, just wanted to give numbers as a basis for calculations if possible but you are right, for the discussen itself they don't matter at all

    Also ok by me, should have called it a hypothetical center but if that's not possible either, I'm ok with that as well...
    Regarding this maybe I should ask if "we" are still considering a big bang or if we moved past that already...

    uh, i should have recognised that by myself...accelleration means working forces...of course (facepalm)

    sorry about that. I saw a few dokumentations recently and one said that time probably began it's own existence with the beginning of the universe, didn't exist before and therefore...there is no "before" the big bang...

    I was aware of that, but from my point of view it's really hard to find the right point to start. As my time is limited I just can't afford to try to learn everything from scratch as much as i want to... :(
     
  8. Sep 27, 2013 #7
    Hubble proved that either the universe doesn't have a centre, or that everywhere is the centre. I too used to think the big bang was an explosion such that space and time would be projected in the shape of a sphere, thus producing a single initial point of creation, when observed, would be the centre of the universe.

    This simply isn't the case, as far as we can tell the observable universe is flat...
     
    Last edited: Sep 27, 2013
  9. Sep 27, 2013 #8

    bapowell

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    The universe having no center has nothing to do with the universe being flat. The balloon analogy, which imperfectly but sufficiently models an expanding spherical universe, illustrates this point.

    Taragond: The big bang did not occur at a single point in a pre-existing space -- it occurred everywhere -- including where you happen to be right now! The observable universe is a uniformly expanding, homogeneous space. The big bang is just the name cosmologists assign to this model of a universe, once hot and dense, that has been expanding and cooling throughout its history.
     
    Last edited: Sep 27, 2013
  10. Sep 27, 2013 #9

    PeterDonis

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    The Big Bang model is still the accepted model of how the universe began. There are various details still being worked out, but none of them change the overall conclusion that the Big Bang is how the universe began (though some of them redefine exactly what our "universe" that began with the Big Bang is within a larger scheme--see below).

    This is true of what might be called the "standard" model of the Big Bang, but there are other models in which it is not true, such as the "eternal inflation" model in which what we call the "universe" is just one "bubble" within a larger universe in which these inflating bubbles are constantly forming. (String theory's idea of the "landscape" of all possible universes consistent with string theory is also somewhat similar to this.) These are among those details that I referred to above, that are still being worked out.
     
  11. Sep 27, 2013 #10
    I never said it did, I was explaining to the OP how it didn't explode in a spherical shape like he thought.
     
  12. Sep 27, 2013 #11
    No, because the experimentally verified, theory of relativity tells us that all motion is relative to something.

    Observations show that all galaxies (and galaxy clusters) are moving away from each other at the same rate relative to their distance apart. There is no observable central point from which you can look out to see more distance galaxies moving away at a different rate. In other words since that observation will be the same from any point in the Universe, how do you determine and define a center.

    Another issue is if the Universe is finite or infinite in size, there is some disagreement; however NASA WMAP data indicates the geometry of the universe is flat and the models that most support that geometry would indicate the Universe is infinite in size. How do you define a center in something infinite in size?


    Yes, yes, and yes as long as you remember it is a matter of perspective. Keep in mind the realtive aspect of time and motion, from the GPS satellite perspective their clock is not fast the Earth clocks are slow. I think the poster was just trying to remind you time only functions at different rate outside your frame of reference and it never change's in it.
     
  13. Sep 27, 2013 #12

    bapowell

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    The surface of a sphere is finite, and yet lacks a center. Also, there are flat universes which are compact, like a torus. So the question of whether the universe is infinite or finite doesn't really help here.
     
  14. Sep 27, 2013 #13
    Well ok. As we have multiple not yet proven models, I'm not entirely sure what prove there is for simply stating there is no center. But as we surely can't find one I will gladly forget about it.

    Am I correct assuming that if we were to reach lightspeed (yep, I know, infinite mass...) time would be standing still if observed from our perspective?

    Also what I just don't get into my head, If I were travelling at .5c (for the sake of it not being possible to travel at full light speed). And would turn on head-lights the resulting light would travel away from me with perceived c but also an outside observer standing still would see the same light moving with lightspeed as well an not at 1.5*c? (the last part doesn't need further explaining, what baffles me is why it wouldn't seem to me that it travels at .5c?)

    Now to my other issue with the universe...just got it thanks to you guys ;)
    What now? We know how that it didn't expand spherical?
    And how did it happen everywhere? I understood that once it was compressed in a massive singularity.
    Am I seeing space wrong? I assumed, space is just there...no matter no energy, just space. where we are matter and energy fill space...while before the big bang there would be space without anything else in it.
    So with your balloon analogy. The balloon would represent the universe. And when it is expanding, it is expanding where? Outside of that ballon would still be space, just with nothing at all in it, or not?

    To get somewhat back to my opening premises and maybe better understand time:
    a) traveling with .5c
    b) traveling with .5c in the opposite direction
    c) standing still
    Questions:
    1. would a) and b) see each other as traveling with c?
    2. how would the clocks of a) and b) desynchronize from c) if brought together into the same reference frame after say a hundred years from c) point of view?
    3. If it's all relative to the point of view, wouldn't that result in a time difference between a) and b) as well? even a higher one compared to the differences between a):c) and b):c) which should be the same?

    PS: I needed some time to formulate this, partly because i was in a late conversation about a software-project for two hours, so between starting and finishing it there were more posts.
    Therefore, thank you very much Maxila for answering question 3 of post 5, I already assumed it before my new premises but now i could just start without it :)

    In addition I would like to ask about how we can be sure theory of relativity is the end of it? If we were looking from outside the universe (assuming there is an outside :-p) we could probably see a center which from the inside isn't possible to determine. and If we saw that center from the inside it would of course move in relation to us and we wouldn't know that it's standing still (if of course it can be said that the universe itself isn't moving, but thats much further than i wanted to go in the first place where I just tried to understand time and what effects it how)
     
  15. Sep 27, 2013 #14

    bapowell

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    OK, but it really isn't just an assumption. Sure, there are models that are non-isotropic, but they fail to match observations. We conclude that there is no center because as far back as we can see -- even out to the CMB itself -- the universe appears homogeneous.

    No, since the equations of special relativity break down at v=c. They simply don't work anymore.

    NO! This is the main result of special relativity -- the speed of light in vacuum is equal to c for *all* inertial observers!

    There was no singularity. The singularity that everyone associates with the big bang does not refer to a physical reality since it's impossible. Instead, the big bang singularity in cosmology (like the black hole singularity) is signalling that the theory is breaking down at these points. The theory is not applicable, and the physics it predicts is wrong. We simply don't know what happened at t=0.

    No, this is the wrong picture. The big bang was not an explosion occurring in a pre-existing space. The big bang is the initial expansion of the space itself.

    The universe is the surface of the balloon, so it's necessarily a lower-dimensional analogy. Replace the 2-D surface of the balloon with a 3-sphere and you'll get the model appropriate to our universe. The downside to the balloon analogy is that it implies -- wrongly -- that there is a space inside and outside the balloon. Interestingly, this is not necessary for the universe (for the balloon either, mathematically). It is possible to describe the expanding surface of an n-sphere in n-dimensional space -- no higher dimensional ambient space required!

    No. Look up the relativistic addition of velocities.

    How were they all originally synchronized? Bringing them all together into the same reference frame likely requires someone to change direction, which requires acceleration. This complicates the problem somewhat. See the Twin Paradox.

    I don't know why you think one would see a center. Why do you suppose this is somehow more natural than a manifold without a center? Where's the center of the surface of the Earth?
     
  16. Sep 27, 2013 #15
    so...do most assume there is no "outside"?

    oops, ok that is something i will have to wrap my head around before i can assume anything else about it...it just feels easier to imagine an infinite universe with no ends than one where you can walk around in an endless "straight circle"...

    After reading on wikipedia (i hope http://en.wikipedia.org/wiki/Velocity-addition_formula is what you meant) my thinking would go as follows:
    the light leaving observer a's reference frame becomes its own reference frame?
    and because of the time dilution every reference frame would measuere the speed of said light at the same velocity?
    also, even looking it up i sadly can't make anything of the formulas at what speed would i measure someone coming straight at me with .5c while i'm heading straight towards him at .5c?

    ok, they were synchronized at oberserver c) then a) and b) accelerate to .5c in opposite direction where each of them stop when about a lightyear away. then they turn and accelerate the same way to get back to c) where they compare their clocks ( I would just say they accelerate instantly just to simplify the calculation if that is mathematically possible, i know it isn't physically)

    well, probably because it's more natural to me, but you are right of course, i don't have any knowledge about how the universe looks like as a whole...if it's infinite it wouldn't even have a shape at all but would you deem it more likely it is just infinite in any direction or do you suppose if you would fly in one direction you eventually would get back at where you started?
     
  17. Sep 27, 2013 #16

    PeterDonis

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    No, because "time" from the perspective of an object moving at lightspeed is a meaningless concept. For example, there is no such thing as "time from a photon's perspective".

    What this really means is that objects that move at lightspeed are fundamentally different, physically, from objects that move slower than light. "Time", or more precisely "proper time", which is the technical term for "time from the object's perspective", is a concept that only applies to the latter kind of object.
     
  18. Sep 28, 2013 #17

    Nugatory

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    His speed is .5c relative to what? And your speed is .5c relative to what? Any time you're thinking about speeds without paying attention to what they're relative to, you'll find yourself in trouble with relativity.

    So let's try phrasing your question more precisely: there is an observer, who is (of course) at rest relative to himself. You are approaching this observer from his left at .5c relative to him. Someone else is approaching him from his right at .5c relative to him. What do you see?

    Well, you are rest relative to you. You see the observer rushing towards you at .5c relative to you. And beyond the observer you see the other guy, moving even faster relative to you. How fast? That's what the velocity addition formula is for:
    [tex]
    \frac{u+v}{1+\frac{uv}{c^2}} \Rightarrow \frac{.5c+.5c}{1+\frac{(.5c)(.5c)}{c^2}} = .8c
    [/tex]
     
  19. Sep 28, 2013 #18

    ghwellsjr

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    I think it might help you to understand with the aid of a series of spacetime diagrams.

    First, I'll draw one that describes your scenario. There is an outside observer standing still at the coordinate position of zero depicted in blue. The dots along his worldline indicate one-nanosecond increments of time. You are depicted in red and your dots are Time Dilated to 15% longer distances due to your speed of 0.5c. (The calculation is for the Lorentz factor, in case you're interested.) At the origin of the diagram (when all the coordinates and clocks equal zero and where the observer is), you turn on your head-lights we'll say for a brief moment just to make an identifiable burst or flash of light which is shown as the thin red line extending up and to the right along a 45-degree diagonal. Light travels at one foot per nanosecond. Here's the first spacetime diagram:

    attachment.php?attachmentid=62251&stc=1&d=1380352805.png

    Have you ever considered what it takes to "perceive" or "see" (your words) a flash of light? As a matter of fact, once it leaves you, there is no way for you to watch it as it travels away from you, is there? Instead, what you can do is place an object along the light's path and then you will be able to see the reflected flash when it finally gets back to you. Then, you can measure how far away the object was from you with your ruler or yardstick and you can measure how long the total round trip time was for the flash to leave your vicinity, travel to the object and reflect off of it and travel back to you. If you make the assumption (Einstein's convention) that the light took the same amount of time to travel away from you as it did to travel back to you, then you can easily determine the speed of light from your measurements.

    Let's see how this works for the outside observer as he measures the flash from one of your head-lights. Let's assume that he has placed a reflector one yard (three feet) away from himself. Then the light will reflect back to him as shown in this diagram (we'll assume the flash from the other head-light continues on):

    attachment.php?attachmentid=62252&stc=1&d=1380352805.png

    The green worldline is his reflector. Notice it is also stationary in his rest frame and it remains 3 feet away from him. The light takes 6 nsecs to make the round trip from the origin to the green reflector and back to him. He assumes that the light spent half its time getting to the reflector and half it time getting back and concludes that the light traveled 3ft/3nsec for a speed of 1 foot per nanosecond, the correct answer.

    Now let's see how you would make a similar measurement. We'll start over again and focus on just what you would do. First off, we need to transform the scenario to your rest frame using the Lorentz Transformation process (which is just a set of equations that we apply to all the events (dots) in the diagram. Notice how your time is no longer dilated but lines up with the Coordinate time while the observer's time is dilated. You will have stuck a reflector (shown in black) on a structure that extends out one yard in front of your car and you exactly perform the same measurements that the original observer did and arrive at the same conclusion that light travels at one foot per nanosecond. Does this make sense to you?

    attachment.php?attachmentid=62253&stc=1&d=1380352805.png

    Now we go back to the original rest frame to see what your measurement looks like in the starting frame:

    attachment.php?attachmentid=62254&stc=1&d=1380353651.png

    You can see that in this frame, your reflector has moved up and to the right but closer to you, allowing light to travel at "c" and still arrive at the same 6-nsec final time.

    Now for some real fun. I'm going to show the frame of the first diagram with all the obects, observers, and light paths and show how in this one frame, both the observer and you determine the speed of light to be c:

    attachment.php?attachmentid=62255&stc=1&d=1380353651.png

    You should be able to identify all the details of the two sets of measurements by the observer and you and the calculations leading to the speed of light. If so, you are ready to see the last diagram, that of your rest frame with both sets of measurement going on at the same time and both leading to the same conclusion for the speed of light, c:

    attachment.php?attachmentid=62256&stc=1&d=1380353651.png

    Does this help clear up your confusion?
     

    Attached Files:

  20. Sep 28, 2013 #19
    Both speeds are relative to c) which i assumed to be standing still

    wow, that adds up to .8c?
    What if:

    a) and b) are both half a lightyear away from c) while a whole lightyear away from each other.
    Then both are instantly accelerated to .5c towards c)
    Both should arrive at c) one year later?

    So if c) wasn't there, they would arrive at each other exactly one year later? If they were able to measure their exact distance before accelerating they would conclude that they approached each other with c, wouldn't they?

    @ghwellsjr: wow, that was really helpful... while I was aware of the difficulty to "see" light in a vacuum I didn't think it would matter for theorization.

    But it raised one question.
    I assumed before that the light would seem to be moving away from me with .5c
    They way some of those diagrams look could suggest (as I suspected before) that the light travels a longer way to the mirror I have on my "car" as it travels back to my sensor (but only to the observer outside because in this construct, my mirror doesnt measure the time when it reflects the light)
    Is it just very probable or can we be absolutely sure that if:
    I were to travel at .5c with my long straight train.
    Two extremely prezise synchronized clocks, both with a flashlight and a sensor are placed a fair distance away on that train so there can't be any difference between the way towards a mirror and back to a sensor, that they would measure the same amount of time for the light of each others flashlight reaching the other sensor?

    PS: Sorry if I'm annoying ;)
     
  21. Sep 28, 2013 #20

    Nugatory

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    Sure does... But don't take my word for it, try it yourself.

    Also, it's interesting to see what happens if either of the speeds are ##c##; that covers the situation in which both you and an observer moving relative to you get the same speed ##c## for a light signal.
     
  22. Sep 28, 2013 #21

    ghwellsjr

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    It's not just difficult, it's impossible, not only to see the propagation of light but also to know where it is at any given time unless we do as Einstein said to do, we define the time that light arrives at a distant locatoin.

    You can make that assumption, but if you do, you will be lead to the conclusion that it travels instantly on its way back to you, based on your measurement of the roundtrip time, correct? For example, if you measured 6 nsecs for it to make the roundtrip between your car and your mirror three feet away, and you assume that it traveled at 0.5c away from you, then you are saying that it took 6 nsec to get there but since you see it get back to you in 6 nsec, the trip back has to be instantaneous.

    That is correct, your mirror doesn't measure the time when the light is reflected. And remember, the outside observer has the same problem you have, he can't see the propagation of light either and he has to make an assumption just like you do.

    As I said before, "there is no way for you to watch it as it travels away from you" and that includes there is no way for you to know how light moves away from you.

    You suggest two extremely precise synchronized clocks a fair distance away from each other but you haven't described how you can get them to be synchronized so that you can make the measurement and that is the issue. Einstein's solution is for you to make the measurement and if it doesn't come out so that the times are equal, you adjust one of the clocks so that it does come out equal the next time you make the measurement. In other words, making the measurement repeatedly is how you get the clocks to be synchronized. Does that make sense to you?

    As long as you're learning, you're not annoying.
     
    Last edited: Sep 28, 2013
  23. Sep 28, 2013 #22

    ghwellsjr

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    In my previous post, I made some diagrams to show how you and the outside observer both determine the speed of light by making measurements, making an assumption and doing a calculation.

    It's different when observers are trying to determine the speeds of other observers. They can't just make observations, they have to actively generate some signals that we call radar signals and measure the timings of the reflections and then make some more assumptions about the radar signals and do some calculations. This is known as the radar method and I used it to answer the same question as yours that someone else asked a while back. I also made several spacetime diagrams which I hope made it all clear. Have a look at this thread:

    Frames of reference for speed?

    Let me know if this answers your question.

    I also answered similar questions to yours in this thread:

    Triplet Paradox

    That thread has a lot of side issues so I would suggest that you just zip past them and get to where I show my diagrams on page 6 and 8. (That thread was what motivated me to write my software to do these diagrams and you can see how my diagrams have evolved.)

    Let me know if you have any more questions.
     
  24. Sep 28, 2013 #23
    If c) was there, its perfectly alright that he measure both a) and b) as moving at .5c. It is also true that if a) and b) were "still," (meaning sharing c)'s reference frame) they could measure the distance between them as 1LY. However once moving, from the perspective of a) and b), the original distance between them would no longer have been 1LY, and they would not be traveling at 1c towards each other, as they changed their frame of reference (even though that is exactly what c) measures)
     
  25. Sep 28, 2013 #24

    Nugso

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    Hello. I have a question I'd like to ask if I may, so basically if you and I have watches and you were to move relative to me with a velocity of 0.9c, I'd see your watch run slower than it would run when you don't move relative to me, and you'd still see your own watch run normal? Did I get it right? If yes, how would I see it run slow? Doesn't something like a battery of the watch have to be low so that I can see your watch run slower?
     
  26. Oct 5, 2013 #25

    ghwellsjr

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    Yes, according to the frame in which all three were originally at rest and according to c)'s clock (since he remained at rest), but not according to the clocks that a) and b) carry with them. According to their clocks, they arrive in 10.4 months. This can be determined by dividing 12 months by the Lorentz Factor at 0.5c (12/1.15471=10.4).

    Of course if we removed c) and his clock from the scenario, then there won't be any clock that reads one year when a) and b) unite but it's still the case that in the original IRF the time is one year when they arrive together and it's still the case that their own clocks read 10.4 months.

    They can measure their exact distance between each other all along the way but they won't measure that they approached at c. In fact, the maximum speed that they each see the other one approaching is 0.8c (the relativistic sum of 0.5c and 0.5c) but they also measure two other speeds. They can also measure their distance to c) and his approach to them (maximum of 0.5c). Here, I'll show you how this works.

    First we start with a spacetime diagram depicting your scenario in the original Inertial Reference Frame (IRF). The thick blue line represents a), the thick black line represents c) and the thick green line represents b). The dots show one-month intervals of the passage of time for each observer and his clock. I have also drawn in some radar signals to show how a) measures how far away c) is as a function of his own time:
    attachment.php?attachmentid=62502&stc=1&d=1381005301.png

    It's important to realize that a) has been sending and receiving radar signals continually during the entire scenario and he would typically process them all but that would make for a very cluttered diagram so I only show enough to get the relevant point across. Here is a list of the measurements a) makes along with his calculations:

    Code (Text):

    Radar   Radar   Calculated   Calculated   Black's
    Sent    Rcvd      Time        Distance     Time
    -24     -12       -18            6         -18
    -12       0        -6            6          -6
      0       6.9       3.45         3.45        6
     10.4    10.4      10.4          0          12
     
    Just by way of reminder, the Calculated Time is the average of the Radar Sent and Radar Rcvd times and the Calculated Distance is the difference between those two times divided by two. Black's Time is simply what a) observes at the Radar Rcvd time.

    Now all the time that a) is making the radar measurements to c), he is doing the same thing with b). Here's a diagram showing these measurements:

    attachment.php?attachmentid=62529&stc=1&d=1381037404.png

    And here is his list of measurements and calculations:

    Code (Text):

    Radar   Radar   Calculated   Calculated   Green's
    Sent    Rcvd      Time        Distance     Time
    -24      0        -12           12         -12
    -12      6.9       -2.55         9.45        0
      0      9.2        4.6          4.6         6.9
     10.4   10.4       10.4          0          10.4
     
    Now before I show you a diagram depicting the results from the two tables, I want to show you another IRF for the final rest frame of a):

    attachment.php?attachmentid=62530&stc=1&d=1381037404.png

    Note how all the measurements from the previous IRF continue to be the same in this IRF and particularly note the measurements made by a) while he was at rest at the top end of the diagram.

    Now here is the diagram showing what a) determines from his measurements, the assumption that a radar signal always takes the same amount of time to reach its target as it does to return the echo, the observation of the target's clock and the calculations defined as the radar method:
    attachment.php?attachmentid=62531&stc=1&d=1381037404.png

    Note how a) is at rest in this non-inertial frame and how c) and b) are moving towards him. Even though this frame is non-inertial, it still adheres to the defined rule that light (and radar signals) always travel at c. However, even though we see the dots indicating deviations from coordinate time, they aren't always dilated, sometimes they are constricted. This is because this frame was not derived from an application of the Lorentz Transformation.

    Now I want to show you that the radar signals are depicted faithfully in this non-inertial frame, just like they are in any IRF. Here's the first diagram repeating measurements of c):

    attachment.php?attachmentid=62532&stc=1&d=1381037404.png

    And finally a diagram repeating the measurements of b):

    attachment.php?attachmentid=62533&stc=1&d=1381037404.png

    In particular, note that the top end of this diagram exactly matches the top end of the third diagram in this post (not the coordinates, just the relative shapes and positions of all the features) and the bottom end of this diagram exactly matches the bottom end of the second diagram.

    If you want, you can determine the intermediate speeds of b) and c) as they approach a). And I'm sure you're aware that these diagrams are mirror images of what we would draw for b) measuring the positions and speeds of a) and c).

    Any questions?
     

    Attached Files:

    Last edited: Oct 6, 2013
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