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Distance in accelerated frames

  1. Dec 20, 2006 #1


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    OK, this is a spin off from another thread, but it's an interesting topic in its own right.

    What I'm going to try to do is to give a rather physical interpretation of accelerated frames, specifically the Rindler metric, and a physical (rather than mathematical) interpretation of the Rindler coordinates and the associated Rindler distance.

    We start out with an observer with a constant proper acceleration through a flat, Minkowski space-time. It can be shown that the equations of motion for such an observer are:

    x^2 - c^2 t^2 = c^4 / a^2

    see http://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity)
    (assuming nobody vandalizes it - I'll have to learn the trick of linking to specific versions sometime) and


    which gives the same equation in alternate form, solving for x(t) as d(t).

    We will call this observer our "reference observer". We can imagine this observer as being the "bottom" of the elevator.

    Next, we need to construct a set of observers who are "stationary" with respect to this accelerating observer. There are several ways that we could accomplish this, but one reasonable way of doing this is to say that an observer is "stationary" relative to our accelerating observer if there is a constant two-way time delay for light - i.e. if the "radar distance" from our reference observer is constant. This is the definition we will use for the remainder of the post.

    Note that the radar distance will turn out to be different from our Rindler distance, but they will share the feature that if the Rindler distance is constant, the radar distance is also constant.

    One can work out the paths through space-time given this approach. One will get the same answer as the Wikipedia gets


    that the "stationary" observers all follow hyperbolic trajectories, with the same asymptotes as shown in the diagram. I.e in the above diagram, x=1 is the path followed by our reference observer, and x=2 is the path of an observer at some constant distance "above" our reference observer. We can call observer 2 the "top of the elevator".

    I think this has been done in some old posts (with a certain amount of discussion going on before it was realized that two different seeming results were actually identical.

    A couple of points need to be made:

    1) It's a bit confusing to have the curve labeled as x=1 our "reference observer". This is related to the fact that our equations of motion do not have a solution for (x=0, t=0). One can re-write the equations to get around this confusion, but it's more trouble than its worth. I hope people can just deal with the fact that the curve for x=1 is our "reference observer" or "the bottom of the elevator", and that the curve for x=2 is "the top of the elevator".

    2) The observers who maintain the same distance will not have the same accelerations! The observer at x=2 in our diagram, for instance, will be accelerating at half the rate of observer 1. This is necessary to maintain a constant distance. Why do we maintain a constant distance? Because we want to. The point of being in an elevator is that the roof is stationary (in some sense) with respect to the floor.

    3) Clocks for the different observers will not run at the same rate. While the radar distance from our the bottom to the top of the elevator is a constant (and so is the Rindler distance), signals sent from the bottom to the top of the elevator will always be redshifted, and signals sent from the top of the elevator back to the bottom will be blue-shifted. These red and blueshifts are an indication that the clocks do not tick at the same rates (I hope this is clear as written and doesn't need more explanation. But I'm afraid that it might not be. But I'm not going to go into more detail unless asked.)

    Now that we've set up the notion of a 'stationary observer', let's talk about how we measure distances.

    The standard unit of distance is one wavelength of some particular cesium source. We can imagine a sequence of cesium sources, all stationary (per our "radar distance" notion of stationary) with respect to our reference observer. These observers will also be radar stationary with respect to each other. These observers form the basis of our Rindler coordinate system - we will give each of them a constant "distance coordinate" via a procedure outlined below.

    The way we get the distance from the top to the bottom of the elevator is conceptually simple. Each of these stationary observers measures the distance to the next nearest observer in the chain, arranged as a straight line. (It's easiest if we make the problem only two dimensional, with one space dimension and one time dimension, then all our observers will automatically be in a straight line.) (expand: Basically, the defintion of distance in terms of wavelength means that our observers use radar methods to calculate distance. The distance will be equal to the total round trip time of the radar signal (as measured by a local cesium clock!), divided by 2, and multiplied by the speed of light.

    Note that because of "gravitational time dilation", or the red/blue shifts that I mentioned, each of these observers thinks that the other observers clocks are off. That's why we have to go into such detail of specifying which clock we use. We do not use a single clock - we divide the distance up into a large number of segments so that our clocks are all very close together, and we use a clock that's co-located at the position where we measure distance.

    Note also that nothing substantial would change if we replaced our radar and cesium clock setup with a series of physical platinum iridium bars. Atoms are atoms, and the atoms in the platinum-iridium bar would experience the same relativistic effects that the cesium atoms do.
    The main difference would be that our radar-bar would be much more rigid than any physical bar could realistically be. This is because the speed of propagation in a physical bar is limited to the speed of sound in a material, and this is much much slower than the speed of light. The signals in our radar setup are the maximum physically possible - so our radar setup acts a like an idealized physical bar would be if the speed of sound in the bar were equal to the speed of light.

    The fact that all our clocks tick at different rates leads to the differences between the "radar distance" and the "Rindler distance". For short distances, it's important to note that the two methods agree - it's only for long distances that there is a difference. If we restrict ourselves to elevators such that acceleration*height << c^2, we will not see any significant difference between Rindler distance and radar distance.

    And that's there is all to it. The distance (measured in this elaborate way, with an infinite sequence of observers) from our reference observer to any stationary observer is the Rindler coordinate of that observer. Having a notion of distance, we can also note that the speed of light is still isotropic in the elevator (sometimes people get confused about that, if so I may have to write more, but I won't unless asked), and that we can synchronize nearby clocks by a light signal emitted by a source at the midpoint. To synchronize clocks that are far apart, we need to construct a chain of intermediaries, as we did to measure distance.

    Actually there could be more to say :-) but this post is already long enough, and we've hopefully met the main goal about setting up a physical interpretation of the Rindler coordinate system.
    Last edited: Dec 21, 2006
  2. jcsd
  3. Dec 20, 2006 #2

    Chris Hillman

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    Quick HOWTO: Citing a specific version of a Wikipedia article

    I agree and am glad to see you are interested!

    I'll have to return when I can find the time to give your post the attention it deserves (due to instabilities this might take a few days), but let me quickly point out that this is easy and convenient:

    1. websurfers who use Firefox can search in pane at Wikipedia at upper right for "hyperbolic motion", which takes one right to http://en.wikipedia.org/wiki/Hyperbolic_motion, which is always the URL of "the current version" (which might be vandalized or otherwise compromised); websurfers who use another browser may need to first go to en.wikipedia.org,

    2. in the left sidebar of this (or any) Wikipedia page, press the "permanent link" button; at the moment I wrote this post, this action took one to http://en.wikipedia.org/w/index.php?title=Hyperbolic_motion&oldid=90597850 which URL cites a particular version (ID 90597850),

    2'. even better, in this (or any) Wikipedia page, press the history button at top center; at the time of writing I see that User:Michael Hardy, who is well known as a trustworthy and expert math guru at WP, fairly recently edited the page, so if I don't have time to study recent versions in detail, I might chose to cite http://en.wikipedia.org/w/index.php?title=Hyperbolic_motion&oldid=29542474; if I had a few more seconds, I'd use the buttons to compute the diff with http://en.wikipedia.org/w/index.php?title=Hyperbolic_motion&oldid=65630277, i.e. to see the effect of the changes made by User:Rgdboer (not currently known to me; regretably, the default should be to trust only editors whose work you know you can trust) right after MH's last edit.

    OK, I'll come back and comment on something substantial as soon as I can!
  4. Dec 21, 2006 #3
    I'm trying to get through your explanation. If you write x=1,
    it means X=c^2/a =1 which would mean that a =9*10^16 m/s^2 which would correspond to a tremendous acceleration (right ?), while further on you mention that the accelerations are small enough such that the iridium bar does not deform. Is this not a contradiction ?

    I can also not see in the diagram that the distance between the two curves remains the same (as time progresses).

    I'm probably making some wrong interpretation. Could you clarify this ?

    Last edited: Dec 21, 2006
  5. Dec 21, 2006 #4


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    It's a question of scale

    Use units of light years for distance, years for times, and light years/year^2 for acceleration.

    Then an acceleration of 1 (1 light year/year^2) is conveniently approximately 1 gravity.

    (Type 1 light year / year^2 = into Google to use it's handy unit coversion feature and find that it's more exactly equal to 9.5 m/s^2)

    As far as working out the path of the radar-stationary observer, it's discussed on this thread as an incidental problem to a different discussion


    As you see, after some discussion George Jones, Hukyl, and I came to a common answer. (It took me a while to realize that our answers were the same). A formal proof that our different-looking answers are the same is given in post #14.

    There is quite a bit of math to the thread. Hurkyl's observations in this thread in post #11


    are a much quicker way to get to the right answer than the approach I took, if one is familiar with the concept of "lines of simultaneity". But you can do it the hard way too, the answer comes out the same.

    If you read the thread, you'll also see George's usenet post where he worked out the problem (and some errata, corrections George found to his post).

    Yet another source that comes to the same conclusion is the oft-mentioned

    Last edited: Dec 21, 2006
  6. Dec 21, 2006 #5

    Chris Hillman

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    This was easy!

    Hi again, pervect,

    OK, I just read this post and I agree with everything you say, except that for some reason I seem to have broken my ability to see images in the WP (probably I misconfigured my browser or some firewall to reject images from wikipedia.org; now I'm too lazy to figure out what I might have done and to undo it).

    I'd just add that the world lines of the Rindler observers form a rigid congruence (vanishing expansion tensor), while the Bell observers form a nonrigid congruence (nonvanishing expansion tensor). I guess you could say the strictly speaking, specifying a congruence in a Lorentzian manifold and computing its expansion tensor are mathematical operations with geometric interpretations independent of any physical interpretation, but what I said about about "rigidity" acquires a reasonable physical interpretation in the context of str. However, the observation about Bell observers has been called "paradoxical" since it surprises many when they first hear this. Oh, one more comment: after setting c=1, the quantity 1/a in the first equation would be the radius of curvature{/I] in the sense of the theory of curves in (semi)-Riemannian geometry, so a is the path curvature. This is purely mathematical, if you like, but in str, the path curvature has an important physical interpretation; it equals the magnitude of acceleration.
  7. Dec 26, 2006 #6
    Well since you write:
    I am a bit confused.

    Clearly from a physical perspective the speed of light is not isotropic for a proper constant accelerating observer. Afteral there is a Rindler horizon behind him and not in front of him!
    Last edited: Dec 26, 2006
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