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Length contraction of falling things

  1. Jun 17, 2012 #1
    1: Does an observer, standing on the moon, see a brick, that is falling straight down, to contract?

    2: Does an observer, standing on the moon, see a light pulse, that is falling straight down, to contract?
     
  2. jcsd
  3. Jun 17, 2012 #2
    Yes.

    The answer to this question is much more complicated, the detailed mathematical explanation can be found here, I just posted it a few days ago.
     
  4. Jun 17, 2012 #3
    Oh yes, the contraction is proportional to blue shift. And the simplest blue shift case is very simple.

    Here's something people here may disagree with:

    The contraction happens because the font of the light pulse moves slower than the rear of the light pulse. And that happens because light slows down in a gravity field.
     
  5. Jun 17, 2012 #4
    I don't know where you get this but it is wrong.

    You are making up stuff.
     
  6. Jun 17, 2012 #5
  7. Jun 17, 2012 #6
  8. Jun 18, 2012 #7

    pervect

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    What's your motivation for asking the question?

    If you're trying to understand SR better, dragging GR into it isn't really a good way to proceed. You're much better off handling the simpler case in flat space-time where there's no gravity to confuse the issue.

    If you're trying to understand GR better, it would be a good idea to sketch exactly how you propose to measure the length of the object remotely. I can warn you in advance that the notion of "distance" in general relativity is ambiguous, mainly due to differing notions of simultaneity. Some people understand the warning, some don't, and I get a bit frustrated trying to explain the problem to those who don't. This is really a SR issue, by the way, but it's an important one to understand before you try to tackle GR.

    I can tell you how in general I would go about "setting up" the problem. The first step is to decide what notion of simultaneity you want to use, and why. So I would start by assuming that the moon is the most significant gravitating body, and that we can ignore the Earth. Right away, I can see a possible conflict with your question, perhaps you didn't intend the moon to be the most significant gravitating body? In that case, the issue of how to determine the notion of simultaneity would require some thought. If the moon is the most significant source of gravity, however, there is an obvious way to proceed, that is to use a clock synchronization scheme compatible with "static observers". It's less obvious how to proceed if the moon is not the most significant source of gravity.

    Once you have decided on the notion of simultaneity you're going to use, the simplest procedure is to construct a spatial geodesic curve. (Not a space-time geodesic, but a spatial geodesic!). Constructing this geodesic will require you to use the metric induced on your spatial hypersurface by your space-time geometry. Constructing the geodesic will require some knowledge of the geodesic equations, and how to solve them. Solving them directly is usually difficult, and taking advantage of conserved quantities via means of Killing Vectors or some equivalent procedure, is highly recommended.

    Next you need to confirm that the problem that you've set up is really one-dimensional. If it is, then you mark on this spatial geodesic curve the position of the front end of the rod, and another mark for the rear end of the rod, "at the same time", according to your notion of simultaneity. If both ends of the object aren't on a single curve, then your problem isn't one dimensional. But let's assume that it is. Then the length along the curve (the spatial geodesic) between the marks on the front and rear will be the "length of the rod".

    I'm afraid I can't give an exact formula for the result you'd get if you followed this procedure, I'd actually have to work it. My own view is that the exercise of thinking about what you want to measure is considerably more helpful than going through the detailed calculations of calculating the number. Furthermore, if you don't understand what all the steps I've outlined mean (and some of them might require a certain amount of expertise), just giving you the number won't really accomplish much.
     
    Last edited: Jun 18, 2012
  9. Jun 18, 2012 #8

    I could use some new facts in my thought experiment building hobby.

    Let's say the light pulse is catched into a box which is just large enough. This is done at many different altitudes, with many light pulses.

    Then all boxes are moved into same place and compared.

    I would guess the boxes from lower altitudes are smaller, and light in these boxes has more energy.
     
  10. Jun 19, 2012 #9

    pervect

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    If you raise or lower a meter stick, oriented normally to the gravitational force so the change in stress doesn't cause the length to vary, I can't see any reason for it to change it's length.

    So I would say that the same applies to a box, say a microwave cavity filled with one wavelength of microwave radiation. The local physics will be the same, the cavity/box won't change length, and a local frequency counter would measure the same frequency, the constant length of the cavity divided by c, no matter what the height of the box was.

    The energy-at-infinity would change as you raised the box, but I think this could eventually be traced back to the work done in raising the radiation in the box. It will take more work to raise a box full of microwave radiation than one that's empty. I'm not aware of ay formal p roof of the matter, but I don't see how it could be otherwise.
     
  11. Jun 19, 2012 #10
    People can disagree with anything - and often it very much depends on definitions of words. :tongue2:
    But it's a simple fact that if according to a distant observer a stationary meter rod is length contracted and a clock at one end is ticking slower, that then with that reference system the return speed of light can only be measured as c if the light according to the distant observer is slowed down with gamma squared (using a standard and consistent meaning of words).

    PS: As Shapiro's phrased it, "the speed of a light wave depends on the strength of the gravitational potential along its path" - http://prl.aps.org/pdf/PRL/v13/i26/p789_1 [Broken] .
    See also discussions on "Shapiro time delay" and note that according to GR the rod will not be contracted if held parallel to the surface.
     
    Last edited by a moderator: May 6, 2017
  12. Jun 19, 2012 #11

    Dale

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    Similarly to your misconception about clocks slowing down, in GR light (in vacuum) also does not ever slow down in a coordinate independent sense. It always travels along null geodesics.

    Even in flat spacetime, it is possible to write down coordinates in which the coordinate speed of light is faster or slower than c, but these are obviously coordinate-dependent statements. Anywhere in any spacetime that you find a set of coordinates where light slows down there are also other sets of coordinates where it does not.
     
  13. Jun 19, 2012 #12
    Agree entirely with that observation. And it agrees I believe with what #10 was intending to convey. One just needs to be careful in defining precisely the relative nature of 'length change' in such situations.
     
  14. Jun 19, 2012 #13
    In SC's transverse length in coordinate measure expressly is invariant wrt potential, but not if that rod is radially oriented (and that's with 'stress' subtracted out). But 'radially shortened' is imo only of value in the sense that integrating over an extended radial path r2 to r1 (coordinate measured), there is more proper distance covered in the interval r2-r1 than if gravity were switched off. We all agree that locally no length or time distortions can logically be evident.
    Which is distinctly different to OP's scenario - free-falling light pulse measured at different heights by a 'hovering ruler' (the box in effect).
    Sure, and that gets back to arguments I was making elsewhere re 'charged BH' and elsewhere - EM field energy associated with charges/currents of a non-free-fall system, is depressed in a gravitational potential by redshift factor. Not so for geodesic propagating light beam.
     
    Last edited: Jun 19, 2012
  15. Jun 19, 2012 #14
    Size yes, but energy I don't know - that's a tricky topic. My guess would be that the total energy doesn't change.
     
  16. Jun 19, 2012 #15

    HallsofIvy

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    Even in "thought" experiments, you cannot just ignore basic facts of physics. You cannot "catch" light in a box and move it.
     
  17. Jun 19, 2012 #16
  18. Jun 19, 2012 #17

    Well, then I guess I must be saying things in a coordinate dependent sense. Is that a bad thing?
     
  19. Jun 19, 2012 #18
    Well of cource it changes. Light that falls, and bounces back, does not change. Light that falls, and is lifted back, does change.
     
  20. Jun 19, 2012 #19

    Dale

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    No, it isn't a bad thing as long as you understand what you are doing and are clear about it. Otherwise it leads to confusion, which I believe is reflected in your writing.
     
    Last edited: Jun 19, 2012
  21. Jun 19, 2012 #20

    Dale

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    My guess would be the same as harrylin's. I don't think that the "of course" is warranted, unless you have a completely foolproof derivation in three lines or less.
     
    Last edited: Jun 19, 2012
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