A question about the equivalence principle.

I have had a closer look and it looks like I was a bit hasty and you (and stevendaryl) are right. The rear observer does observe blue shift from signals coming from the leading rocket, even if there is no length contraction. This can be seen from examining the attached graph "f2b" which plots the paths (blue) of two rockets with equal and constant proper acceleration as seen in an inertial reference frame. The horizontal green lines are lines of equal proper time on the accelerating rockets at one unit time intervals. On this graph the diagonal black lines are signals sent from the front rocket to the rear rocket at regular unit time intervals. One interesting aspect is the the blue shift seen by the rear rocket reduces asymptotically towards unity as time progresses and the proper separation of the rockets increases.
This is correct too. The leading rocket does observe red shift of signals coming from the rear rocket with or without length contraction. The attached chart "b2f" charts the no length contraction case where the rear rocket sends signals at regular unit time intervals to the front rocket. Again there is an interesting aspect, in that the red shift increases dramatically over time, which is the exact opposite of how the blue shift seen by the rear rocket evolves.

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.

Let's say we have a non-length-contracting accelerating rocket.

If a clock is thown from the rear of the rocket to the front of the rocket, it will be time dilated during the flight an amount that makes the reading of the clock to be the same as if the clock would have been in a gravity well.

If this same clock is thrown back to the rear from the front, it will be time dilated less than everything else in the rocket during the flight, and this effect is larger than the effect of the first throw, and the reading of the clock will be the same as if the clock had visited upper regions of a gravity well.

Hi Take a look at yuiop's diagrams a couple of posts back. Compare the relative path lengths between forward and backward travel and see if you still think the front to back travel would have greater effect on elapsed time than the travel from back to front.

Hi WHy do you thnk there would be a difference between a single rocket and two rockets if the distance remained constant in the reference frame in both cases ?

In the two rocket case, the distance between the two rockets remains constant (in the launch frame), while in the one rocket case, the distance from the bottom of the rocket to the top decreases with time (in the launch frame). But in the latter case, the rate of decrease starts off zero, so it makes no difference to the Doppler shift initially.

Yes. In fact, the fact that the clock "lower" in the accelerating system slows down can be derived purely using SR without any reference to the equivalence principle or the Einstein field equations.

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.

Okay, I did some calculations and I think I have a good grasp now on the clock discrepancies in the accelerated rocket. There are two effects going on:

1. Because of relativity of simultaneity, if two clocks are synchronized in the "launch" frame, then the forward clock will be farther ahead, as measured in the instantaneous rest frame.
2. The front clock is traveling slightly slower than the rear clock (as measured in any inertial frame), and so experiences slightly less time dilation.

Here's what's interesting: Early on, effect number 1 is the most important, and effect number 2 is negligible. Much later, when the rocket is traveling very close to the speed of light, effect number 2 is most important, and effect number 1 is negligible.

If by 1) you mean relative to a MCIRF I think your right but that doesn't really address the question of the mechanism of why the clocks would have different rates, which relative velocity does.

The ship starts out with a certain synch. If there is no resynchronization it will maintain that relationship throughout the course , right?
So initially there will be small discrepancy between it and succeeding MCIRF's But as velocity increases that discrepancy will also increase because the relationship of the ship to the MCIRF is the same as the relationship of that MCIRF to the initial frame. So adding in the time dilation acts to increase that discrepancy , the front clock keeps running even further ahead of the rear and the front of the MCIRF
So to figure out dilation you would have to eliminate the ever increasing effect of desynchronization if you are going to use a MCIRF as a reference. Or so it seems to me.

Well, consider the events
e_1: the rear clock shows time 12:00
e_2: the front clock shows time 12:00

If there is no length contraction, then e_1 and e_2 are simultaneous in the launch frame. But in the frame in which the rocket is momentarily at rest, e_2 happens before e_1. So in that frame, the front clock runs ahead of the rear clock.

If there is length contraction, then it means that even in the launch frame, event e_2 happens before e_1.

I'm not exactly sure what you are saying. But if I did the calculation correctly, the exact result for the time τ on a clock aboard a Born-rigid rocket of constant proper acceleration is this:

τ = R/c arcsinh(ct/R)

where t is the time in the launch frame, and where R is a distance computed by:

R = c²/a + x

where x = the height of the clock above the rear of the rocket, and a = the acceleration felt by the rear of the rocket.

Hi Take a look at yuiop's diagrams a couple of posts back. Compare the relative path lengths between forward and backward travel and see if you still think the front to back travel would have greater effect on elapsed time than the travel from back to front.

Okay, I did some calculations and I think I have a good grasp now on the clock discrepancies in the accelerated rocket. There are two effects going on:

1. Because of relativity of simultaneity, if two clocks are synchronized in the "launch" frame, then the forward clock will be farther ahead, as measured in the instantaneous rest frame.
2. The front clock is traveling slightly slower than the rear clock (as measured in any inertial frame), and so experiences slightly less time dilation.

Here's what's interesting: Early on, effect number 1 is the most important, and effect number 2 is negligible. Much later, when the rocket is traveling very close to the speed of light, effect number 2 is most important, and effect number 1 is negligible.

I thought we were no longer talking about the case of non contraction.

And I agreed with your thought regarding the desynchronization relative to the momentarily comoving frame. I disagreed with your assumption that that effect would dimish over time.
IMO it would increase throughout the complete course of acceleration. Why do you think it would diminish?

Do you know of any treatment done through Doppler that has been worked out for review??

The brute force approach as you called it. It certainly seems messy in principle and hard to interpret but I would like to know the method of approach.