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I'm currently learning special relativity in high school and we only primarily deal with what happens when an object is moving at constant relativistic speeds. But what if the object slowed back down to a stop?

I observe length contraction for an object at relativistic speeds, when it returns to be stationary, does it go back to its original length?

Qmechanic
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Aean
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4 Answers4

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Calculating the effect of acceleration in special relativity is straightforward, but I suspect the algebra is a bit much at high school level. See John Baez's article on the Relativistic Rocket for a summary, or see Chapter 6 of Gravitation by Misner, Thorne and Wheeler for a more detailed analysis.

When you're first introduced to SR you tend to be told about time dilation and length contraction and given formulae to calculate them. However this is at best an oversimplification and at worst actively misleading. When you're looking at some object moving relative to you you do indeed measure the object's length to be contracted, but what actually happens is that the two end points in the object's rest frame transform into points at slightly different times in your rest frame. You measure the object to be contracted because you're measuring the end points at slightly different times. There is no sense in which the object is squeezed by it's high velocity.

Any object has a proper length, which is equal to its length in its rest frame. The proper length is an invariant and all observers will measure the same proper length regardless of their relative velocity. If you consider proper length then the object is not contracted.

Anyhow, the answer to your question is that when the object comes to a stop relative to you its length has not changed. This is because it never did change - the change you measured was due to the coordinates you were using not matching the coordinates the object was using. When the object comes to rest in your frame you and the object are using the same coordinates (at worst differing in the position of the origin) so both of you measure the length to be the proper length.

John Rennie
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    +1 for actively misleading. MTW's Gravitation may be a bit much for high school, but there is an excellent introductory book by Taylor and Wheeler, "Spacetime Physics", which explains the physics of what actually happens, complete with diagrams, problems etc. It should be quite accessible for a high school student. – Anton Tykhyy Mar 28 '14 at 10:54
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    I also strongly recommend Taylor and Wheeler's book Exploring Black Holes. This is a bit harder, but anyone who did maths to an advanced level at high school (i.e. in the UK did A level maths) should be able to get somethong from it. – John Rennie Mar 28 '14 at 12:11
  • "Anyhow, the answer to your question is that when the object comes to a stop relative to you its length has not changed. This is because it never did change" I think this is likely to mislead the OP. What happens to the length of the object (according to you) when it comes to a stop (relative to you) depends entirely on the details of the decelration and can be anything at all. I think you are presuming that the OP intends for the entire object to decelerate "all at once", but of course "all at once" is itself a relative concept.... – WillO Sep 29 '19 at 04:34
  • Not wanting to belabor this too much, but I think it's particularly beneficial for beginners to be reminded that although relativity sometimes contradicts common sense, one ought not abandon common sense altogether. Common sense tells you correctly that if the back of the object deceleartes before the front, the object gets longer, and if both ends decelerate simulataneously, the length can't change. It's good for beginners to be reminded that although relativiity is counterintuitive sometimes, it still has to make sense, and sometimes that requirement is enough to solve a problem. – WillO Sep 29 '19 at 04:38
  • Wait, so breakthrough moment. You're just measuring the front and back at different times!? Now length contraction makes perfect sense. Thank you so much. – Curious Layman Aug 13 '21 at 06:32
  • @JimmyG. for the details see this answer. – John Rennie Aug 13 '21 at 06:38
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Everything remains relativistic. However, the variation between the Newtonian and Einstein relativity is of the order of $v^2/c^2$. So when this number drops below observable error, you can't tell the difference.

wendy.krieger
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If you are the observer and you are observing a car move in front of you, then the clock in the car will appear to move slower than the clock in your reference frame and the length of the car will appear to be contracted to you in the direction of motion. All these effects get magnified as the velocity of the car approaches that of light. In the same way, all these effects get less and less magnified if the velocity decreases and completely stops when there is no relative velocity between you and the car.

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Take the extreme case where a pole moving rightward (relative to some fixed frame) stops abruptly. If the left end stops before the right end, then (obviously) the pole becomes longer. If the two ends stop at the same moment, then (obviously) its length can't change. If the right end stops before the left end, then (obviously) the pole gets shorter.

Different observers will disagree about when each end stops and hence can disagree about whether stopping causes the pole to shrink, grow, or remain the same length as it was.

WillO
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