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On 06/15/2010 10:50 AM, William Maddox wrote:
To go along with the astronaut/twin "paradox" there is a less well known
paradox involving length contraction. It is known as the rocket-rope
paradox and as Bell's spaceship paradox. If interested see this website:
http://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox
That wikipedia article says the paradox ...
"was first designed by E. Dewan and M. Beran in 1959 as an
argument for the physical reality of length contraction."
I suppose y'all can predict where I come down on this.
I don't think length contraction has any physical reality.
In my experience, all the arguments for its physical reality
are grossly flawed. It's just a question of how many
femtoseconds of thought are required to find the flaw.
The fact is, if you rotate a ruler, "the" length of the
ruler does not change. The projection of the ruler onto
this-or-that coordinate system might change, but the actual
length -- the proper length -- does not change.
Similarly if you boost a ruler, "the" length of the ruler
does not change. The projection onto this-or-that coordinate
system might change, but the actual length -- the proper
length -- does not change.
It has been known for more than 2300 years that the shadow
of a thing is not the same as the real thing. It has been
known for more than 100 years that in spacetime, physics
is much more closely connected to the proper length and
proper time than it is to the projections onto whatever
coordinate frame (if any!) is being used at the moment.
Relativity was not invented in 1905. In fact it has been
clearly understood since 1632 that the laws of physics are
invariant with respect to a change in velocity.
However, there is another invariance, arguably more important,
and even older. The laws of physics are invariant with respect
to a change in /position/. Even in 1632 this had been a well-
established idea for thousands of years.
Minkowski (1908) said quite clearly that the right way to think
about relativity -- including position, time, and velocity --
was in terms of invariances.
http://de.wikisource.org/wiki/Raum_und_Zeit_(Minkowski)
I mention this because it makes it super-easy to analyze the
spaceship/rope riddle. One spaceship accelerates for one day
of its proper time. The other spaceship accelerates for one
day of /its/ proper time. The two motions are congruent,
differing only in a change of position. We can represent this
using a super-simple spacetime diagram:
A' B'
/ /
/ /
| |
| |
A B
The initial length of the rope is the proper distance between
A and B. The final length of the rope is the proper distance
between A' and B'. We can easily evaluate both of these lengths
in the lab frame. But proper length is a Lorentz scalar, so
it is the same in /any/ frame. So the rope does not stretch.
Not even a little bit.
This completes the analysis. The whole analysis takes only
a couple of paragraphs and involves little more than a few
fundamental principles, such as invariance of the laws with
respect to change of position, and Lorentz invariance of the
proper length.
As usual, a correct and complete analysis does not require
any Lorentz transformations.
As usual, a spacetime diagram makes it much easier to see
what is going on.
Meanwhile, here are some tangentially-unrelated remarks:
The analysis in the wikipedia article is bogus because
it fails to account for the acceleration, and therefore
makes incorrect assumptions about simultaneity. We
know from the notorious "traveling twins" scenario that
acceleration must redshift one clock relative to another. This can be seen as a particularly simple example of a
gravitational redshift, since general relativity tells
us that a gravitational field is locally indistinguishable
from an accelerated reference frame. (If the last sentence
isn't helpful to you, feel free to ignore it.)
The size of the shift depends on the acceleration vector
*and* the relative position vector. A' thinks B' is blue
shifted relative to A', while B' thinks A' is red shifted
relative to B'. (In the language of gravitational red
shifts, B is higher in the gravitational potential.)
In any case, this whole discussion of relative time
shifts is irrelevant. It is an unnecessary complication.
The only thing that really matters is each ship's notion
of its own proper time, which is not complicated at all.
Each ship accelerates for one day of its own proper time,
and that's all that need be said.
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