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An illustration of relativity with rulers and clocks

  1. Jan 8, 2005 #1


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    This is an explanation and illustration of how different reference frames assign coordinates to points in space and time, which I wrote up a while ago on a different forum. I figure it might be interesting to people here, especially in terms of showing how two different observers can both see the other observer's clocks slowing down and rulers shrinking in relation to their own, without leading to any inconsistencies (the key is to understand how different reference frames also define simultaneity differently). Here it is:

    A reference frame is really just a coordinate system for marking the location and time of different events. You can think of the physical basis of these coordinates in terms of each observer having a network of rulers and clocks spread throughout space, so if you imagine taking a picture of an event as it happens, its coordinates are defined by the markings on the ruler and the time on the clock right next to the event in the photo. Each observer uses a network of rulers and clocks which are at rest relative to himself. Clocks at different locations must all be synchronized, and in classical physics this would be pretty simple--you could just bring two clocks together to check that they're in sync, then move them to different positions. But time dilation due to movement makes sychronization a little trickier. Einstein's idea was that you could have each observer synchronize their own set of clocks using light signals--pick a point halfway between two clocks, send a flash of light from that point in both directions, and if both clocks read the same time when the light reaches them, they are considered synchronized. But unlike the Galilean frames used in classical physics, this means that the different Lorentzian frames used in relativity will not agree about simultaneity of pairs of events. For example, if an observer on a spaceship sees a flash of light go off in the middle of the ship, then he should synchronize clocks at the front and back of the ship by making sure they both read the same time when the light hits them. But for another observer who sees the ship moving, the back of the ship is moving towards the point where the flash happened and the front is moving away from that point, so his own network of clocks will be synchronized in such a way that the clock next to the event "light hits the back of the ship" will read an earlier time than the clock next to the event "light hits the front of the ship".

    Here are some diagrams which may help explain how different reference frames could each see the other's rulers contracted and the other's times dilated without leading to any contradictions. In this example, we have two rulers with clocks mounted on them moving alongside each other, and in order to make the math work out neatly, the relative velocity of the two rulers is (square root of 3)/2 * light speed, or about 259.628 meters per microsecond. This means that each ruler will observe the other one’s clocks tick exactly half as fast as their own, and will see the other ruler's distance-markings to be squashed by a factor of two. Also, I have drawn the markings on the rulers at intervals of 173.085 meters apart—the reason for this is again just to make things work out neatly, it will mean that observers on each ruler will see the other ruler moving at 1.5 markings/microsecond relative to themselves, and that an observer on one ruler will see clocks on the other ruler that are this distance apart (as measured by his own ruler) to be out-of-sync by exactly 1 microsecond, some more nice round numbers.

    Given all this, here is how the situation would look at 0 microseconds, 1 microsecond, and 2 microseconds, in the frame of ruler A:


    And here’s how the situation would look at 0 microseconds, 1 microsecond, and 2 microseconds, in the frame of ruler B:


    Some things to notice in these diagrams:

    1. in each ruler's frame, it is at rest while the other ruler is moving sideways at 259.6 meters/microsecond (ruler A sees ruler B moving to the right, while ruler B sees ruler A moving to the left).

    2. In each ruler's frame, its own clocks are all synchronized, but the other ruler's clocks are all out-of-sync.

    3. In each ruler's frame, each individual clock on the other ruler ticks at half the normal rate. For example, in the diagram of ruler A’s frame, look at the clock with the green hand on the -519.3 meter mark on ruler B--this clock first reads 1.5 microseconds, then 2 microseconds, then 2.5 microseconds. Likewise, in the diagram of ruler B’s frame, look at the clock with the green hand on the 519.3 meter mark on ruler A—this clock also goes from 1.5 microseconds to 2 microseconds to 2.5 microseconds.

    4. Despite these differences, they always agree on which events on their own ruler coincide in time and location with which events on the other. If you have a particular clock at a particular location on one ruler showing a particular time, then if you look at the clock right next to it on the other ruler at that moment, you will get the same answer to what that other clock reads and what marking it’s on regardless of which frame you’re using. Here’s one example:


    You can look at the diagrams to see other examples, or extrapolate them further in a given direction then I actually drew them, and to later times, to find even more. In every case, two events which coincide in one reference frame also coincide in the other.
  2. jcsd
  3. Feb 24, 2008 #2
    I like it ^^, but I dont quite understand the clocks on the bottom bit, they seem to start with a skewing factor of 2,1.5 at t=0?
  4. Mar 4, 2008 #3
    Very educational.. thanks a million.

    But the numbers seem a little contrived because at no point do the two observers at their respective 0.0m mark agree on times or distances except at time t=0. Consider A to be stationary. Then when A believes time=1 microsecond in A's context B believes that only 0.5 microsecs have past, and while A believes that B has travelled 259.6m in that 1 second B believes they have travelled 519m.

    Does not the argument come down to the fact that the changes are continuous, and if the two observers find themselves at odds as to time and distance travelled there must exist some point at where were they but at that point in time-space they would both agree on the precise distance travelled and the precise time taken. There being one point in time-space for any set time/distance is not that remarkable. This seems to be related to the jordan curve theory in mathematics which roughly says that if you have a continuous function f with f(x) < A < f(y) for some x and some y, there must exist a value z such that f(z) = A. Your diagrams seem to simply highlight the appropriate value for z in the circumstance considered.

    Secondly a personal beef of mine. Why must we talk about your velocity in terms of my
    stationary time frame, and my stationary distances, when I'm not the one going places. Observer B travels in my frame 259.6m/microsec and it takes him 0.5microsecs to do this. Thus his velocity should be considered 259.6*2m/microsec. If I wanted to get to alpha-centuri the two parameters of interest to me are how far a-priori (standing still) I have to go, and having set off at 259m/microsec - how long will it take me to get there. Or equally well having arrived there and being asked what my velocity was why should I reply 259.6m/microsec. Isn't it a lot more logical to say, again standing still, well Earth is s km away and it took me t hours to get here so my "effective" velocity was s/t km/h -- ie distance travelled/time taken.

    I don't know how many times I've heard it wrongly said that the speed of light is a limiting factor on how far one could travel in a year or a lifetime. People would not fall into this fallacy if they were instead told that there was no limit to how fast one could go, or how quickly one could get between point A and point B, and in particular that they could certainly travel more than one light year, in a year of their time given only propulsion sufficient to arrive at a point where time dilation more than compensated for the desired distance to be travelled in a given positive time. Could we not perhaps as a compromise talk about "perceived" velocity and "effective" velocity.

    Last edited: Mar 4, 2008
  5. Mar 4, 2008 #4


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    You can do this as long as you have a preferred reference frame, in which you CAN define what is the space-distance between far-away objects. In your example, you tacitly assume that your end point, and your start point, are stationary (or almost so) WRT a preferred frame. In that case, you can define their "absolute distance". You then divide their "absolute distance" by your eigentime, and that gives your "effective velocity". But if your starting point (earth, say), and your end point (say, Captain Kirk's spaceship) are not stationary in the same frame, how are you going to define the earth-spaceship distance ? In what frame ? Earth's ? The spaceship's ?
  6. Mar 4, 2008 #5


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    The method you suggest for measuring motion can be used and it is called "proper velocity" or "celerity" by some authors. As you correctly point out, the celerity of light is infinite, so there is no limit on a particle's celerity. Also the momentum of a particle is equal to the product of its "rest" mass and its celerity. It should be realised that celerity is a hybrid concept: the distance is measured by observer A but the time is measured by observer B. (The distance measured by observer B is zero.) As vanesch has pointed out, you need to choose an external reference A in order to assign a distance; observer B cannot determine a celerity without reference to observer A.

    Although you rarely see "celerity" or "proper velocity" mentioned by name, it is in fact used, in a slightly hidden way, in 4-dimensional formulations of relativity. Celerity is just "the space component of 4-velocity" (relative to a chosen inertial frame), for anyone who understands 4-vector jargon.
  7. Mar 4, 2008 #6
    True enough. At the theoretical level I agree, but any practical application of the theory has to take the distance to be travelled and the time dilation into consideration, once relative velocities make the time dilation significant. Purely mundane issues such as lifetime of componentry, instructions as to the time to fire rockets to turn round and start slowing down, telemetry alterations in on-board computers due to altered cpu-clock rates, and altered distances, all suggest that the observer should put themselves in the context of the package making the trip, rather than focus on the way the package appears to them as it wizzes by, if they actually want to get a package from point A to point B. We already have to do all this for such things as GPS satellite technology. At a minimum all should maintain two separate notions of time and travel time, so that the masses do not as otherwise often tends to happen imagine the two times to be one and the same thing.

    The language should make it easier for all to appreciate that there is no fundamental contradiction in scripts such as "the vaccine has a shelf life of a month, fortunately we can get it to you before its expiry date has passed, but regretably you'll all have died of the disease 10,000 years before it arrives, because you are 10 light years away".

    The general lay view that the speed of light limits how far you can go in a given time, has it completely backwards, because what it really limits is how soon from the point of view of delivery the package sent will arrive. And since there is no one (that we know of) out their waiting for the package that is not in and of itself an absolute impediment to space travel. For those who desperately wanted to make it to the other end of the universe the lifetime of the universe might become a serious concern, but for those who wanted to emigrate to nearby stars, it is not theory that says it can't be done, but merely lack of technology capable of providing acceleration and deceleration of 1g or so for a month or so.

    Thanks for confirming that Star Treks warp factor 9 (ie velocities of 9*c are not strictly needed to span stellar distances in reasonable amounts of time).
  8. Mar 4, 2008 #7
    The bottom clock is skewed with respect to the fixed clock just as are the distances. The clocks that change are essentially the computed clock times for that position in space-time given that the ruler is moving at the indicated velocity with respect to the stationary observer.

    It might help to look at the two red marks, on the two rulers where observers A and B are permanently located. At time t=0 in A's frame the entire universe is currently existing at that given time. He is imagining that the same clock time currently is "happening" on alpha-centuri as at his location behind the tracks (presuming alpha-centuri is stationary with respect to him). At A's time t=1 B's clock has advanced half as fast so B's clock is now at 0.5, where B's red line is on B's ruler. That is B occupies a position in his frame with coordinates in his frame of t=0.5 and x=0.

    Within B's frame the clock times would be the same everywhere, but in A's frame, the minute that A grasps that B's clock must be running slow, it follows that B's means of measuring distance must also be skewed, since an obvious/certain method of getting the accurate distance measurements is to translate from the time it takes light to travel that distance, back into meters or whatever. If I say something is one light second away it is approximately 186,000 miles away. Now obviously if my means of measuring a second is skewed so that my second is twice as long as yours, I won't measure distances the same way as you, which explains why you will think I'm getting all my distances wrong.

    The last part of the puzzle (which I don't fully understand) is that there is a metric which both A and B will agree on examining either ruler. That is that x^2+y^2+z^2 - t^2 will be the same. So having established that distances will appear different to B as determined by A, A can then plot the values by which time will be non-sychronous for B with respect to A.

    So for example the speed of light is ~ 299.8 m/microsecond. At time 0 for A and B the lower clock reads 2 because:

    (-346.2/299.8)^2 - 0^2 == (-692.3/299.8)^2 - t'^2

    1.333 == 5.332 - t^2

    So t^2 == 4 so t = +/- 2.

    It is not clear to me why the sign is chosen as +ve, but then it is not clear to me how we assign time a sign in the first place.

    As a second example at time 2 in Rulers A frame at the bottom of the page where B's time is given as 5 we have:

    (-173.1/299.8)^2 - 2^2 == (-1,384.7/299.8)^2 - t'^2

    0.333 - 4 = 21.332 - t'^2

    t'^2 = 25 so t = +/-5.

    It is the final bit of the argument that I don't understand. I don't see what is the significance of there being a given fixed point where the times and distances agree for any time and any distance. If the times agree so then have to the absolute distances for in one dimension we know that x^2-t^2 = x'^2 - t'^2. And there has to be a point between A's current time and B's current time if they differ where they are the same, if the change in time (over time) is continuous. The only somewhat suprising part is that these actual points are at nice convenient integers, but that may be merely a consequence of the velocity chosen, which is indeed something which invites the appearance of such convenient integers.
  9. Mar 4, 2008 #8


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    [STAR_TREK_NERD_FACTOR=10]This is a critical and important error that cannot be allowed to stand on the Physics Forums! http://home.att.net/~srschmitt/script_warpcalc.html" [Broken]. The warp factors are cubed to get the multiple of c. The scientific accuracy of the warp factors is indisputable and above reproach as it has been verified on a weekly basis by at least two generations of intrepid spacefarers and eager fans![/STAR_TREK_NERD_FACTOR]
    Last edited by a moderator: May 3, 2017
  10. Mar 4, 2008 #9
    I date reality from 1963 when at age 10 when I saw the very first episode of Dr Who. My sister who was only 7 at the time meanwhile remembers the same date somewhat less enthusiastically as the date when her desire to watch Emerald Soup were thwarted. We both have long memories.

    http://www.teletronic.co.uk/tvt63ab8.htm [Broken]

    Get to my age and the past can really bring a lump to ones throat. So real yet so very unreachable. Warp speeds were never that important when one had a time machine stuck in the appearance of what was intended to be suitable camaflage for London 1963.. namely a London police box. It sure would look out of place were it to materialise in London 2008. Feel a bit out of place myself frankly. Back then travelling this far into the future was beyond my ability to contemplate. 1984 was a long way past the other end of the beyond. The only imaginable future was 1970, which for me was the date the US had promised to have put a man on the moon by, and that was a date I waited years to arrive. Rather suprised at how far one travels in a single lifetime. Time fascinates me.
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