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Confused in basic SR

  1. May 16, 2007 #1
    I'm trying to understand SR better however I think I've got myself confused in some ways.

    Firstly, I can't think of exactly why a mechanical wave like sound should follow gallilean relativity (besides that F = ma is used on the "mechanical pieces" of the wave and we know F = ma obeys gallilean relativity), yet maxwell's equations don't. What's different regarding these two waves?

    Secondly is a bit about the twin paradox (Sorry ;]): We assume the standard "paradox" where one twin goes out (straight line, short/quick accelerations) and comes back. One twin will feel accelerations but as I take it, it's the velocity that will "age" the twin, but what is difference between the velocity of the true departing twin perceived by the "stationary twin" (say, on earth) and the velocity of the "stationary twin" as perceived by the twin who feels accelerations?
     
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  3. May 16, 2007 #2
    What do you exactly mean with euclidian relativity? Do you mean Galilean mechanic? And what is your trouble exactly?
     
  4. May 16, 2007 #3

    Doc Al

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    The difference is speed. Sound waves, and other things going slowly compared to light speed, appear to obey galilean addition of velocity. But that's only approximately true due to the low speeds involved--everything really obeys relativistic addition of velocities.

    I'm sure our experts will give you a more complete answer, but I'll just note that the observations made by the "stationary" twin are all made from a single inertial frame, while those made by the "moving" twin are not.
     
  5. May 16, 2007 #4
    The situation with the twins is not symmetric.

    From the very begining you have two possible frames - the one in which the Earth twin and the distant Star are not moving and the one in which the 'traveling' twin is not moving. They can't be both inertial since two inertial frames must have a constant relative velocity. Their relative velocity changes in direction when the traveling twin reaches the star and goes back. In order to use the SR formulas you have to decide from the onset which frame is the inertial one and we pick up the frame of Earth and the Star since they both don't have engines and are in a free-fall motion.

    On the other hand had we decided from the onset that the traveling twin frame was the inertial one because the Earth and Star have some engines and propel back and forth creating the impression that the 'traveling' twin goes to the Star and back, we would have gotten exactly the opposite result, that the Earth twin is younger than the traveling.

    So the asymmetry of the result is a consequence of our asymmetric choice which frame is inertial and writing the SR formulas in that frame or other inertial frames that move with constant velocities with respect to it.
     
    Last edited: May 16, 2007
  6. May 17, 2007 #5

    JesseM

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    It's not so much the waves that are different, it's just that the laws of physics have the property of "Lorentz symmetry" which means they obey the same equations in different inertial coordinate systems related by the Lorentz transformation:

    [tex]x' = \gamma (x - vt)[/tex]
    [tex]t' = \gamma (t - vx/c^2)[/tex]
    where [tex]\gamma = 1/\sqrt{1 - v^2/c^2}[/tex]

    On the other hand, the laws of physics would not obey the same equations if you created a similar coordinate system based on some other speed s, like the speed of sound:

    [tex]x' = \gamma (x - vt)[/tex]
    [tex]t' = \gamma (t - vx/s^2)[/tex]
    where [tex]\gamma = 1/\sqrt{1 - v^2/s^2}[/tex]

    That's just because the laws of physics don't have the property of being symmetric relative to this coordinate transformation, this is not one of the symmetries of nature.
    Each twin's velocity will be different in different inertial coordinate systems, and you can analyze how much each twin ages using whichever inertial coordinate system you want. But as long as you're using inertial coordinate systems, the twin who accelerates will always change velocities when he does so, while the twin who moves inertially will have a constant velocity. For example, you could pick an inertial coordinate system where the twin on the ship is actually at rest during the outbound leg of the trip, and the Earth is moving away from it at relativistic speed--in this frame, the twin on the ship will actually age faster than the Earth twin during the outbound leg, not slower. But when the twin on the ship turns around and returns to Earth for the inbound leg, in this frame the twin on the ship will be moving even faster than the Earth, and thus aging slower, and if you actually calculate how much each twin ages during the inbound leg and add that to how much they age during the outbound leg, you'll still find that the twin on the ship has aged less in total, by exactly the same amount that you'd have predicted if you used the same procedure in the Earth's inertial frame.
     
  7. May 17, 2007 #6
    "Secondly is a bit about the twin paradox (Sorry ;]): We assume the standard "paradox" where one twin goes out (straight line, short/quick accelerations) and comes back. One twin will feel accelerations but as I take it, it's the velocity that will "age" the twin, but what is difference between the velocity of the true departing twin perceived by the "stationary twin" (say, on earth) and the velocity of the "stationary twin" as perceived by the twin who feels accelerations?"

    That is actually an unanswered question - just what role acceleration plays in time dilation. Einstein says in his 1905 paper that two separated clocks at rest in the same frame will read differently when one is moved toward the other until they meet - this involves a start-up acceleration but the amount of time dilation is given as a function of the velocity (assuming the acceleration is short). On the other hand, as Jesse points out, either twin could initially considered himself to be at rest and calculate how much time the other twin loses during the first part of the trip. In such an experiment, however, the initial conditions do not specify any acceleration - rather, the problem starts with the two twins in relative motion. Some well noted writers insist that a consideration of which twin initally accelerated is critical to avoid the paradox - that is, at the end of the first phase (reaching the turnaround point) two clocks cannot each be slower than the other. Where the times are compared in the same frame (e.g., the twin that accelerated stops and checks his time with a clock in the earth frame before starting his return trip) at the end of the first leg, any difference must be attributed to asymmetry caused by the initial acceleration. Real age differences always involve an acceleration somewhere - whereas measurements made between two relatively moving inertial frames always result in an apparent slowing of the other guys clock.
     
    Last edited: May 17, 2007
  8. May 17, 2007 #7

    JesseM

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    This may be a philosophical question you have, but physicists do not consider it an "unanswered question."
    Which writers say this, specifically? They must be writers who disagree with the relativity of simultaneity, and thus disagree with the theory of relativity itself. If no frame's definition of simultaneity is to be preferred over any others, then there will be frames where the Earth-twin is older than the travelling twin "at the same moment" that the travelling twin turns around, and other frames where the Earth-twin is younger "at the same moment".
    If one accepts the relativity of simultaneity, one can only talk about a "real" (i.e. frame-independent) age difference if the twins reunite at a single position in space, not when they are far apart.
     
  9. May 17, 2007 #8
    It is fairly straight-forward:

    The factor between the future accumulation of proper time for a clock that performs an instant acceleration and the future accumulation of proper time for a clock in the prior (perhaps even instantly comoving) frame of the first clock changes. The factor decreases when the direction of the acceleration is away from this other clock it increases when the direction is towards this other clock. The factor is never larger than one.
    Conform the equivalence principle we can see the same effect near a gravitational field. Here the factor decreases as well for a clock that is closer to the center of a gravitational field.
     
    Last edited: May 17, 2007
  10. May 17, 2007 #9
    Medium.

    The difference is that classical or mechanical waves (eg. sound) move at a constant speed with respect to a physical medium (eg. air), and you can devise experiments to measure the velocity of the medium itself. But with light, it travels at a constant and invariant speed with respect to every observer, which doesn't even make sense according to Galilean relativity.
     
  11. May 17, 2007 #10
    The special relativity applies to both.
    Galilean relativity applies approximatively for low-speed objects like sound waves.
    Galilean relativity does not apply for high-speed object because it becomes a bad approximation for high speeds.
    Galilean relativity is totally wrong very close to the speed of light.
     
  12. May 17, 2007 #11
    One important aspect of any answer is that light waves don't need a medium to travel through while a mechanical wave does. In the absense of a medium there is no preferred frame while in the presence of a medium there is a prefered frame, i.e. the rest frame of the medium.
    Its not the acceleration per se that causes the time dilation effect by the non-symmetry of the motion of the traveling twin. The traveling twin has a bent worldline whereas the stay at home twin travels on a straight world line.

    This would actually make a great web page. I've been putting it off for some time but perhaps now is the time to start. Thanks.

    Pete
     
  13. May 21, 2007 #12
     
    Last edited: May 21, 2007
  14. May 21, 2007 #13

    JesseM

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    I agree acceleration is necessary to explain the age difference, but I was asking about some more specific statements of yours: "Some well noted writers insist that a consideration of which twin initally accelerated is critical to avoid the paradox - that is, at the end of the first phase (reaching the turnaround point) two clocks cannot each be slower than the other." Which writers say that which twin initially accelerated (as opposed to which twin accelerated to turn around once they had moved a large distance apart) is critical to avoid paradox? And which writers say that at the turnaround point "two clocks cannot each be slower than the other"? That latter statement was what I was referring to as a disagreement with the relativity of simultaneity, since the relativity of simultaneity means that different frames disagree about which twin has aged more at the time of the turnaround.
    It can't be compared in any absolute way, according to relativity. You can compare the time on the traveling clock with the time on the earth clock in the earth's rest frame, or you can compare it in the rest frame of the ship, and you'll get different answers.
    The Altair clock only reads the same as the earth clock in the earth/Altair rest frame. There is nothing that makes this frame physically preferred over any other inertial frame, according to relativity.
    I entirely agree with Rindler here, but this quote doesn't support the statements I was asking you about above.
     
  15. May 21, 2007 #14
    In SR, the role acceleration plays in determining real age difference is not agreed upon - many writers insist that SR alone is sufficient to un-paradox the twin paradox. Others, which I have quoted above, insist that acceleration is necessary. Both get the same answer but by entirely different process - can you say there is no difference between 1) observing a spaceship passing earth at 0.1c and 2) starting with the rocket clock at rest on earth and initially synchronized with earth clocks, then quickly accelerating to 0.1c - we assume here that both rockets travel to a distant point P and stop and we compare the accumulated time on the rocket clock with the earth clock (e.g., by sending signals that indicated how many hours have been logged on the rocket clock). In the first case there is no bases for assuming any age difference inasmuch as we do not have a way to distinguish between the inertial frame of the rocket and the earth-P frame. In the second case there is an asymmetry introduced by the initial acceleration.
     
  16. May 21, 2007 #15
    "It can't be compared in any absolute way, according to relativity. You can compare the time on the traveling clock with the time on the earth clock in the earth's rest frame, or you can compare it in the rest frame of the ship, and you'll get different answers"

    Yes - I am using the earth frame as the convenient frame in which to make comparisons - we start with all clocks at rest in the earth frame and we end up with all clocks in the earth frame at rest - but not necessarily close together

    My point in raising the issue is directed at the questions posed by cscott - the literature is divided between those that resolutely insist SR is sufficient to explain the clock paradox and those that assert it is really a problem that must be treated by GR.
     
  17. May 21, 2007 #16
    Jesse: "Which writers say that which twin initially accelerated (as opposed to which twin accelerated to turn around once they had moved a large distance apart) is critical to avoid paradox? And which writers say that at the turnaround point "two clocks cannot each be slower than the other"? That latter statement was what I was referring to as a disagreement with the relativity of simultaneity, since the relativity of simultaneity means that different frames disagree about which twin has aged more at the time of the turnaround. "

    I am drawing a distinction between measurements made while the clocks are in relative motion (since each will see the other running slow, and two clocks cannot each be running slower than the other) and the age difference that will be measured when the traveling clock reaches the end of the outward journey, and all clocks are temporarily at rest in the earth frame. In the latter case, the traveling clock will have accumulated less time assuming initial synchronization of all clocks in the earth rest frame. I see no ambiguity so long as the comparisons are made in the same frame in which all clocks were initially synchronized. The amount of time lost on the one way trip is 1/2 that for the round trip assuming the other factors are the same (speed, path length etc)
     
    Last edited: May 21, 2007
  18. May 21, 2007 #17

    robphy

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    I believe that part of this apparent division relies on the what is meant by "SR" and by "GR".

    Does "SR" mean "the geometry of zero-curvature spacetime with R4 topology" [what I will call the modern geometric interpretation] or does "SR" mean that only algebraic,trigonometric, and geometric methods of [a single] Minkowski vector space (akin to methods of Euclidean geometry) [what I will call the older Einsteinian interpretation or SR-methods]?

    Certainly one can use GR-methods (with, say, non-Cartesian coordinates and related Christoffel symbols) with the flat SR-spacetime [as one can use Riemannian methods with flat Euclidean space].

    The classical twin paradox is a problem on a flat SR-spacetime with R4 topology. While algebraic,trigonometric, and geometric methods of [a single] Minkowski vector space are quite capable to explain it, one can (if desired) go further with GR-methods [on this flat SR-spacetime] to discuss additional details not easily done with those SR-methods (for example, how to generalize to non-SR spacetimes).
     
  19. May 21, 2007 #18

    JesseM

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    Well, anyone who accepts relativity would disagree that "two clocks cannot each be running slower than the other" (in different inertial frames), I don't have a problem with this any more than I have a problem with the idea that in each clock's frame the other clock's velocity is higher, and there is no absolute truth about which clock "really" has a higher velocity.

    Again, this is the thing I was asking you to back up when you said that "Some well noted writers insist that a consideration of which twin initally accelerated is critical to avoid the paradox - that is, at the end of the first phase (reaching the turnaround point) two clocks cannot each be slower than the other." Here it makes it sound as though these "well noted writers" agree with your claim that "the paradox" which must be avoided is the idea that two clocks are each running slower than the other. Did you not mean to imply that, and were you just adding the part about there being a problem with the rate of the clocks as your own opinion, not one shared by any of these "well noted writers"? If it's not just your personal opinion, than please show me a quote where any "well noted writer" suggests there must be an absolute truth about which of two separated clocks is running slower.
    But it's totally arbitrary to choose the Earth/Altair frame and say that this frame's view of which clock accumulated less time is the "true" one. There is no physical motivation for it, it's just a matter of your own prejudices. As I recall, you also admitted that you had no general way to pick which frame's view is the "true" one on post #187 of this thread. As an example of the problem here, imagine that Earth and Altair pass by two other star systems, Murth and Faltair, which are moving at a high velocity relative to them...if Earth first passes next to Murth and the time is noted on Murth's clock, then Earth passes next to Faltair and the time is noted on Faltair's clock, then according to the synchronized clocks in the Murth/Faltair frame, less time has passed on Earth than passed in their frame...but during exactly the same flyby, if Murth first passes by Altair and then passes by Earth, then according to synchronized clocks in the Earth/Altair frame, less time passed by on Murth's clock than passed in the Earth/Altair frame. If you think there must be a truth about whether Murth's clock or Earth's clock is running slower, how would you decide it?
     
  20. May 21, 2007 #19
    Sorry but I cannot possibly agree with that.

    I think you are mixing up the light signals from the clocks with the clocks themselves. When two clocks are moving with respect to each other the light signals make it appear that each clock is running slower.

    It is a simple fact that due to the relativity principle we cannot detect which clock is actually running slower, if any. If one of them has accelerated away from the other in the past then that one is certainly accumulating less time compared to the one that did not accelerate.
     
  21. May 21, 2007 #20

    JesseM

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    Agree with what? I said 'anyone who accepts relativity would disagree that "two clocks cannot each be running slower than the other" (in different inertial frames)'. So are you saying that you agree with yogi it's impossible for two clocks to each be running slower than the other, even if we analyze the situation from two different inertial frames? Are you agreeing with him that there must be an absolute truth about which clock is 'really' running slower?
    But the rate of ticking of a clock in a given inertial frame does not depend on what you see with light signals, it depends on measurements made on a network of clocks at rest in that frame which have been "synchronized" in that frame using the Einstein synchronization convention. As I'm sure you know, the relativistic Doppler effect indicates that a clock moving towards you will actually appear to be ticking faster than your own clock if you just look at it with your eyes (or a telescope), but in your rest frame it is ticking slower than your own clock, and measurements on a network of clocks at rest and synchronized in your frame would confirm this.
    Yes, I agree. But that's what yogi was disagreeing with.
    You're not saying it's "accumulating less time" in any absolute sense, I hope? If a clock accelerated away from the Earth and then moved away at constant velocity, in the frame where the clock was at rest after its acceleration, the Earth's clock would be accumulating less time than the traveling clock, and this frame is just as good as any other inertial frame in SR.
     
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