Re: speed of length contraction - explanation/contradiction
Earlier you suggested accelerating all parts of the rod simultaneously. After the rod has achieved the new velocity along its entire length, then it will be contracted by the factor (1 - (v/c)^2)^.5.
This is basically correct, but it has an semantic ambiguity. The word 'simultaneous' has an objective meaning only for events that occur at the same point of space. The word simultaneous is subjective for all events occurring in different spatial locations. If two events are simultaneous to all observers, then they have to occur at the same point in space. If events occur very far apart in space, then the events occur at the same time in only one inertial frame. For the infinite number of other inertial frames, the two events occur at different times.
This is a very, very long rod made up an an infinite number of infinitesimal segments. Suppose we have two segments that are very far apart. There is only one inertial frame where the two segments start accelerating at the same time. The two segments start accelerating at different times in every other inertial frames. This is why simultaneous has a subjective meaning in this case.
A segment starts to accelerate when a mechanical force starts to act on that segment. If the instruments of one inertial frame applies force at the same time, then all segments can start accelerating at the same time according to that inertial frame. However, the forces are independent by definition. There is no causal connection between the start of acceleration. In order to synchronize the motion of segments, a force has to propagate from one segment to the other. The mechanical forces in this case are determined by the elasticity and density of the segments. The propagation of the mechanical force in this case is the speed of sound.
However, this probably isn't what the OP had in mind. Let us consider a case more similar to what the OP has in mind.
Consider a long rod in free fall in outer space. Choose an inertial frame where the observer is at the center of mass of the rod. Let us say that every non-overlapping segment of the rod has a node of the inertial frame nearby. Each node of the inertial frame has an identical laser, very small, that pushes the rod segment with light pressure. The lasers independently start pushing the segments at the same time as observed in this one inertial frame.
Note that in this case there doesn't have to be a sound wave carrying mechanical force from one segments to another. The lasers turn on 'at the same time' by an amazing coincidence. It is an amazing coincidnece only in the inertial frame where the center of mass of the rod is at rest. There is no causal connection between the mechanical forces that start pushing each segment.
Now, I add some instruments that measure mechanical stress in the rod. I hypothesize that the rod is made of transparent material that is isotropic. There is no optical birefringence in the rod unless there is stress. Stress induces optical birefringence in the rod segment. Stress induced birefringence is known in every transparent material on earth.
If a rod segment is in equilibrium, then it has no birefringence.
The segments of the rod show birefringence when stressed. I assume that each segment of rod is sandwiched by infinitesimal crossed polarizers. An LED and photodiode near each segment measrues the birefringence of each segmetn. So now the observer in the reference frame can reconstruct a stress profile for the rod. Two cases
1) Administrator case where external force is applied to one end of the rod.
Consider what happens when one laser beam is applied to one end of the rod. The entire rod doesn't accelerate at the same time. A sound wave has to pass through the rod to make the whole rod accelerate. So the birefringence in the rod is no longer uniform. It shows sinusoidal banding because of the sound wave.
If a sound wave passed down the rod, then the shape of the sound wave could be determined from the stress profile as a function of time. Thus, the start of acceleration of each rod segment would be indicated by the start of stress as measured by laser transmission. The wave front would be seen to move down the rod at the speed of sound. To measure the new length of the rod, one would have to wait until the energy of the sound wave dissipated. The rod would have to lose heat energy to reach the same temperature as before a force was applied.
2) OP case where each segment of the rod is accelerated at the same time as seen in one reference frame.
Let us look at one inertial frame where all segments of the rod are accelerated at the same time. This is the a causal case, since there is no one force that actually moves the rod.
The rod is one length while it is stationary. It has a certain geometric length before the lasers turn on. The geometric length is its equilibrium length before the lasers turn on. Furthermore, its geometric length remains the same while the lasers alone are doing the acceleration. However, the geometric length is no longer the equilibrium length. The rod segmetns show birefringence. So the geometric length is longer than the equilibrium length.
The rod is accelerated to the same velocity as in case (1). Unlike in case (1), the rod now shows a uniform birefringence. There is extra elastic energy in the rod because of the stress.
Now, turn off the lasers off. The rod is now allowed to shrink or expand, as determined by elasticity. It vibrates for a while. The birefringence shows the vibration. However, the vibrations eventually have to damp. Energy is radiated into space. Eventually, the rod is in equilibrium at the same temperature as before. The state of equilibrium is indicated by the absence of stress.
The geometric length of the rod is once again the same as the equilibrium length of the rod. The equilibrium length of the rod after acceleration is shorter than the equilibrium length of the rod before acceleration. The rod has shrunk as determined by the Lorentz length contraction formula.
What I always emphasize is that in dynamics, geometry is never independent of the forces that hold bodies together. So to precisely model the dynamics behavior of the rod, one needs to monitor the forces that hold the body together. This is as true in Newtonian mechanics as it is in relativity. Someone who has never analyzed stress in Newtonian physics is likely to be surprised when it reappears in relativistic physics.
Mechanical forces appear in the last quarter of Einstein's 1905 article on relativity. The first half of the paper discusses 'geometry' alone. It is tempting to drop the last half of the paper, looking only at the first half. However, it was not placed near the end because it is unimportant. The laws of mechanical force close the equations of motion. If a reader ignores mechanical force entirely, then the first half of the article is under determined. If you take into account both parts of the paper, relativity becomes exactly determined. The paradoxes of relativity arise because the geometry of relativity is under-determined.
Einstein never said that the length of a body is determined by only time and space. The shape of a body is determined by the directions of the internal forces that hold it together. The direction of the internal forces are determined by time and space. Space and time tell you how, not why. So when looking for 'why' the shape of a body happens, one has to look at the mechanical forces.
I am sure that you understand