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A question about the equivalence principle. 
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#1
May2412, 04:54 AM

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I had a physics test at school recently. One of the questions was based on the equivalence principle, going something like this: Two clocks in a space ship that is accelerating. One at the bottom and one at the top of the space ship. Now think that the space ship is so far away from any object in space, that it is not affected by any gravitational force.
It is my understanding that according to the equivalence principle, one can not be able to do any test that suggests that you are no longer in a gravitational field, but in an accelerated system. And according to the theory of relativity, the clock furthest down in the gravitational field will go slower than any clock higher up. Yet it makes no sense to me that the same rules would apply in an accelerated system. The correct answer supposedly is that the clock at the bottom of the ship will slow down. What am I missing? Would the clock at the bottom of the ship really slow down? Making the rules of time dilation apply in an accelerated system, the same way it does in an gravitational field? 


#2
May2412, 05:10 AM

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#3
May2412, 05:31 AM

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The elegant way to do it is to determine the transform between the inertial and accelerating systems and then evaluate the spacetime interval for a stationary object. The brute force way to do it is to calculate the Doppler shift between the upper and lower clocks. 


#4
May2412, 06:00 AM

P: 3,967

A question about the equivalence principle.
Consider the point of view of an inertial (non accelerating) observer who watches the rocket accelerating. If the length of the rocket appears constant to the accelerating observers on board the rocket, then the rocket appears to be length contracting according to the inertial observer. This means the back end of the rocket is accelerating faster than the front end and the back end of the rocket is moving faster than the front end at any given time according to that inertial observer, so the back end experiences more time dilation relative to the front end.



#5
May2412, 06:47 AM

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#6
May2412, 06:55 AM

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We can compute the discrepancy between the clock rates at the front and the rear of the rocket using SR time dilation and length contraction. But the Doppler shift derivation works whether or not the rocket undergoes lengthcontraction: Imagine, instead of a single rocket with two clocks, we have two identical rockets, one above the other. They launch identically, and the distance between the rockets remains constant (as measured in the launch frame). Then even though the distance between the rockets remains constant, the Doppler shift explanation still holds: if the rear rocket sends signals at rate one per second up toward the front, the front will receive them at a rate slower than one per second. I'm just leaving this as a puzzleI'm not saying the Doppler shift explanation is wrong, but it seems weird that it works whether or not there is length contraction, while the other explanation relies only on length contraction (and differential time dilation). 


#7
May2412, 07:16 AM

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#8
May2412, 07:25 AM

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#9
May2412, 09:48 AM

P: 484

I have always thought that when somebody carries the clock from the back end of the rocket to the front end of the rocket, that is when the clock experiences velocity time dilation. But now I see that if I use the noncontracted lenght, then the time dilation is too large. If I use the contracted length, then the time dilation is too small. 


#10
May2412, 12:19 PM

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#11
May2412, 01:25 PM

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#12
May2412, 01:52 PM

P: 3,967

However, in the particular example we gave where the rocket is not length contracting in the inertial launch reference frame, the rocket will actually be length expanding in the rocket frame, but that is probably an unnecessary distraction and there will still be time dilation red shift over and above the classical Doppler shift. P.S. I think you are also right to have concerns about the length contraction explanation (when no length contraction is measured in the rocket frame undergoing Born rigid acceleration) so that makes the explanation a bit weak. It does explain how the inertial observer in the launch frame concludes that the clocks at the back and front are running at different rates, and if we allow for the change in simultaneity when transforming to the rocket reference frame the conclusion still holds, but the explanation becomes more convoluted then while the Doppler explanation seems more direct. However, we still have to refer to the inertial reference frame to observe the different velocities of the back and front ends of the rocket, even for the Doppler explanation. 


#13
May2412, 03:16 PM

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#14
May2412, 10:09 PM

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#15
May2412, 11:39 PM

P: 1,162

It would seem to also apply in the other direction with the exception that in that case the difference in time dilation would cancel out part of the red shift due to the increased velocity It also seems like there would be more red shift due to velocity measured in the front ship than blue shift due purely to velocity measured in the rear ship. With the dilation factor working oppositely I guess it would take calculations for specific conditions to determine what the relationship would be. Perhaps 


#16
May2512, 12:09 PM

P: 3,967

The final attached chart is the length contraction case which is associated with the Rindler Metric and Born Rigid acceleration. In this case the distance between rockets and the blue or red shift from neighbouring rockets, remains constant over time, in an accelerating rocket reference frame. The green curves are the "lines of equal proper time" for the rocket and plotted at 1/2 unit time intervals. For those that might be interested, these curves are plotted using these parametric equations: [tex]x = r * \cosh (T/r) [/tex] [tex]t = r * sinh(T/r) [/tex] where r is the parametric variable and T is a constant (The proper time). r is also the nominal radius and is inversely proportional to the constant proper acceleration of a given rocket. 


#17
May2512, 10:06 PM

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#18
May2512, 10:38 PM

P: 1,162

Having thought a little more it appears that in any case the overall observed Doppler shift would not be a meaningful indication of the relative dilation, as a large part of the effect is purely classical doppler and even if you went through the business of separating out this component, you would be left with a pure dilation evaluation that was comparing the clock rates at two different intervals on the respective world lines and still have the problem of interpreting that into relative rates at the "same" time in the accelerating frame. Regarding the Rindler clocks and the constant relative Doppler: I find this curious and would like to know more but maybe not in this thread. ;) 


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