Born Rigid Motion - observations
pervect said:
Does anyone have any good English language references (textbook references or journal articles) for linear Born rigid motion, preferably at an elementary level (i.e. just SR, no tensors).
No, but I found an interesting application of Born rigid motion/Rindler frames in a paper titled "
Electrodynamics of hyperbolically accelerated charges V. The field of a charge in the Rindler space and the Milne space" by E. Eriksen and Ø. Grøn in Annals of Physics 313 (2004) 147–196, available here: http://http://www.hep.princeton.edu/~mcdonald/examples/EM/eriksen_ap_313_147_04.pdf" .
This prompted me to play around with that old 'enigma', Bell's spaceship 'paradox'. I've done a spacetime diagram (attached) and would appreciate comments on some of my observations. I've calculated the case of two identically accelerating spaceships in line astern formation (constant proper acceleration), with a string attached to the leading ship only. I assumed that the string is accelerating Born-rigidly by some unspecified means, surely not by merely being pulled.
After two time units (years in this case), the accelerations of the two spaceship stop simultaneously in the reference frame. I've used the 'easy' proper acceleration of a=1 ly/y^2 ~ 1g for both ships, with an enormous string of 0.5 ly long, which is also the initial separation of the two spaceships. The loose end of the string is hence accelerating at a constant a'=2g, from the hyperbolic motion relationship x^2 - t^2 =1/a^2.
Observations:
- The two spaceships stay at a constant coordinate length apart, while the proper length between them increases during the acceleration.
- The loose end of the string reaches the final velocity of the two ships after 1 coordinate year and then moves at that constant velocity.
- The string's Lorentz contraction observed by the coordinate frame continues until the lead ship's acceleration stops, which is when the lead ship and the string front end have caught up with the velocity of the string tail end.
- The string is then observed by the reference frame as normally Lorentz contracted, as per normal relative inertial movement.
What is interesting about these (rather 'old hat') observations is that if you lose sight of the fact that we are dealing with Born rigid acceleration and start thinking in terms of a 'rigid string' being pulled, there is an apparent causality violation - in the reference frame, the tail end of the string stops accelerating a full year before the cause of the acceleration stops, the cause being the lead ship pulling the string.
However, Born-rigid acceleration means that each part of the string are being accelerated by an appropriate force, in order to keep the string completely stress free while it Lorentz contracts in the reference frame. Those forces are right there, on the spot, so no causality problems. As a matter of fact, a real string that is being pulled can never be observed to act in this way, due to speed of sound in the string that is much smaller than c and the effect of pulling (or not) will take many years to propagate down a string of 0.5 ly long!
I would appreciate feedback on anything that I've got wrong in this analysis.
-J
PS: apologies for the 'wrong-way-around' spacetime diagram, with the time axis horizontal; it was just easier to plot it this way out of a spreadsheet, because x is defined for all values of t, but the reverse is not true.
