I How specifically does an accelerated uniform rod Lorentz contract?

[Orodruin]

In a particular MCIF, yes; but you are really just using that frame's time to define "at the same time", instead of using the proper times along each worldline.

Yes, it is just a particular simultaneity convention set up in such a way that "simultaneous" events on the rod have the same velocity.

 

[vanhees71]

The Born rigid body is fictitious and not so easy to treat as it might seem. It's a fascinating mathematical subject and a now classical result of mathematical physics to establish which motions such a fictitious body can make at all (Herglotz, Fritz Noether ~1910), but physicswise it's ficititious. I found the above cited treatment of an elastic body which comes closest to a kind of rigidity in assuming an ideal fluid with the speed of sound equal the speed of light. It's of course also fictitious but makes the issues much clearer since it's closer to a real "quasirigid" body.

 

[Orodruin]

The Born rigid body is fictitious

I think this is true for most (if not all) models of physical phenomena. It is then a question only if the model is close enough to reality to actually make useful predictions.

 

[DrGreg]

The Born rigid body is fictitious

I think this is true for most (if not all) models of physical phenomena. It is then a question only if the model is close enough to reality to actually make useful predictions.

I suppose this depends on how you apply the term "Born rigidity". If you're thinking of it as an intrinsic material property of an object, then, yes it is fictitious. But if you think of it as a type of motion, then there are circumstances in which an object can achieve Born rigid motion, if not exactly but to a very good approximation.

If you apply a force to an object that previously was subject to no force, then initially the object will deform and pressure waves will travel through the object. But if you continue to apply a constant proper force and wait for a long enough time, eventually the object will reach an equilibrium state (to a good approximation) and pressure waves will have reduced to a negligible level. The object will now differ from a truly Born rigid object by a negligible amount.

 

[vanhees71]

I think this is true for most (if not all) models of physical phenomena. It is then a question only if the model is close enough to reality to actually make useful predictions.

Well, as I said the model of an ideal fluid with $c$ as speed of sound is also fictitious, but it's not as fictitious as a Born rigid body, contradicting the fact how spinning tops behave, which are well described in non-relativistic mechanics within the model of a rigid body, but within relativity you have to use elastic bodies.

 

[vanhees71]

If you apply a force to an object that previously was subject to no force, then initially the object will deform and pressure waves will travel through the object. But if you continue to apply a constant proper force and wait for a long enough time, eventually the object will reach an equilibrium state (to a good approximation) and pressure waves will have reduced to a negligible level. The object will now differ from a truly Born rigid object by a negligible amount.

Sure, that's one of the motions that are allowed for a Born rigid body. Nevertheless to set it in motion from rest you usually must assume some elasticity in the beginning.