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First, the challenge again:
The setup: somewhere in gravity-free space, fit the following sensors into the nose cone of a long rocket: atomic clock as master time base, RF Doppler speedometer and an accelerometer, all interfaced to a computer with a telemetry channel back to base station. Do the same in the tail, obviously out of harms way from the exhaust. Synchronize the atomic clock in the tail to the master clock (i.e., set it to the same reading) and then let it run on it's own.
Assumptions: it's a "long playing" rocket that is programmed to profile the thrust for a low, constant acceleration of (say) 1g for a long time, despite becoming less massive as fuel burns off. The rocket is not appreciably compressed in the longitudinal direction by the acceleration and the lengthwise transient oscillation and vibrations are negligible, i.e., it's a "near rigid body" problem.
The experiment: the rocket is ignited and both computers (nose and tail) start to record the time, speed and acceleration in their respective locations. After many moons, the rocket stops accelerating and at some short (identical) time after detecting the end of acceleration, each computer radios its recorded data back to base station for analysis.
The challenge: Qualitatively predict:
a) how the recorded speed profiles (against recorded time) will differ, if at all;
b) how the recorded acceleration profiles will differ, if at all;
c) how the atomic clocks will differ in time recording for the events of "acceleration starts" and "acceleration stops", if at all.
The short solution:
a) The recorded speed profiles of the nose and tail will be the same, barring a short delay (tens of milliseconds) while the thrust applied to or removed from the tail propagated to the nose at the speed of sound in the rocket body.
b) The acceleration will in principle be different, but with about one part in 10^14 and utterly immeasurable with accelerometers.
c) If the acceleration continued for 3 years, the nose clock will record the duration of the acceleration as one microsecond longer than the tail clock. This is so because the nose clock will gain time on the tail clock by about one part in 10^14.
The long solution:
Figure 1 below shows the (highly exaggerated) situation graphically, taken from a presentation by John Mallinckrodt, Professor of Physics, California State Polytechnic University, Pomona.
Figure 1: 
The "flashing rod" (bold red lines) in this diagram represents our rocket. The thin red lines are lines of simultaneity between the front and the rear at successive time intervals.
While the acceleration lasts, the world lines of the front and the rear of the rocket are hyperbolas that share the same vertex (or focus), with the equation:
t^2 = x^2 - σ^2,
where σ is the distance from the vertex at time zero (i.e. rocket launch) and units are chosen so that c=1, e.g., time in seconds and distance in light seconds. When the acceleration stops, the world lines obviously become straight lines.
The salient points of this diagram are:
· Each red line of simultaneity cuts the world lines of the two ends at the same angle - meaning that the two ends measure the same speed at moments that they perceive as simultaneous. The base station will not agree that they measured those speeds simultaneously.
· The proper (felt) acceleration is proportional to 1/σ, so it is always smaller for the front end than for the rear end. So the two accelerometers measure different accelerations.
· The time of the front end gets ahead of the rear end; in other words, the originally synchronized clocks do not remain synchronized and the front clock measures a longer acceleration time than the rear clock. This is consistent with the fact that the front of the rocket measures a lower acceleration over a longer time.
Conclusion
All of the above are solid mainstream relativity, although it may be relatively difficult to accept at face value. To make it more palatable, remember that the equivalence between acceleration and gravity also dictates this behavior. Over short distances, a normal, non-uniform gravitational field causes the same result for accelerometers and clocks as what happens in lengthwise linear acceleration. This is perhaps the most important insight to be taken from this rather 'esoteric through experiment'.
Linear acceleration is discussed more fully on the website Relativity 4 Engineers.
The Blog is open for questions (or overripe tomatoes…)
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Re: Rocket Challenge Solution
