From: draper@sscvx1.ssc.gov
Subject: Re: Davis Mechanics (was Re: John Woods Campbell)
Organization: Superconducting Super Collider Laboratory
Date: Thu, 7 Jan 1993 15:33:07 GMT

In article <1993Jan6.214249.14902@blaze.cs.jhu.edu>, arromdee@jyusenkyou.cs.jhu.edu (Ken Arromdee) writes:
>>This is one I can't figure out.  Consider a steel bar, in free space, a
>>long way from anywhere, one light-year long.  place a large rocket
>>against one end, and turn it on.  Let it run for 364 days.  Turn it off.
>>What has happened?  Try it again with an incompressible rod - if the rod
>>moves, the message reached the far end faster than light.  If the rod
>>doesn't move, where'd the momentum go?
> 
> You can't make an incompressible rod, precisely for this reason.
> 
> Thought experiments allow us to assume things that get around real-world
> engineering limitations, but not to make assumptions which are equivalent to
> violations of physics; in this case, an incompressible rod is one.
> 
> So the rocket-powered end is moving and the other isn't.  The local motion of
> the matter that is at any moment located in the propagating wave is associated
> with momentum, and this momentum remains the same as the wave progresses.
> --

Stiffness is one of those things one has to watch VERY CAREFULLY when
relativistic effects are to be taken into account. A board sliding along a
table with a hole in it will, in one frame, appear to remain stiff and fall
into the hole. In another frame, the board will appear to sag and dip into the
hole -- resolving the so-called "skateboard and manhole" paradox. The effect is
due to relativity of simultaneity, but it tosses a caution onto the definition
of stiffness and compressibility at relativistic speeds.

An interesting ramification of an accelerating rod is that there is a limit on
how long a rod one can accelerate. The reference is E. Taylor and A. French,
Am.J.Phys. 51 (1983), 889. The paradox comes from the question of how it is a
long rod of length L contracts as it accelerates from rest in time T to a
velocity where it has contracted to L/2. If the front of the rod moves a
distance X in time T, then the rear of the rod will have moved a distance X+L/2
in time T. If L is arbitrarily large, then the rear of the rod can in principle
move faster than the speed of light ==> paradox! 

I will try to explain this in as clear English as possible, considering the
cross-posts. The way the contraction happens is as follows. According to
someone riding on the rod, we can accelerate the rod and get it to hold
together by giving the front and back ends equal velocity kicks at the same
time. However, for someone who remains at rest, however, the relativity of
simultaneity predicts that the kick at the back end will happen earlier than
the kick at the front end, the net result being that, as the rod continues to
accelerate in these little steps, the back end "catches up" with the front-end
and the rod contracts. How much earlier the back kick happens compared to the
front kick depends on the length of the rod. This is the fun part. If the rod
is long enough, for given velocity kicks at set intervals at the front end, the
kicks at the back end will pile up on top of each other, so that the limit is
that the back end accelerates to its final velocity !instantaneously! in this
frame of reference. Technically, the limit to the proper acceleration is
a*<=c^2/L* where L* is the proper length of the rod. That is, a rod of length
L* cannot possibly accelerate faster than this without falling apart or
imposing faster-than-light communication. This fact is probably the sternest
filter of unrealistic rocket drives I know.

Paul Draper
University of Texas at Arlington
Mythink, not UTAthink, not SSCthink, not GOVthink.
Newsgroups: rec.arts.sf.written,rec.arts.sf.science,sci.physics,sci.skeptic