Re: Simple length problem - for Todd, Harald, thinkers
- From: David <dseppala@xxxxxxxxxxxxx>
- Date: Sat, 24 Jun 2006 15:02:40 GMT
On Fri, 23 Jun 2006 14:49:59 -0500, "Todd" <nospam@xxxxxxxxxx> wrote:
"David" <dseppala@xxxxxxxxxxxxx> wrote in message
news:2gln92hbuma63r6k999aj30hhrn6ilf4p7@xxxxxxxxxx
On Thu, 22 Jun 2006 10:13:54 -0500, "Todd" <nospam@xxxxxxxxxx> wrote:
[snip due to length]
Yes, that question is essentially different. In your example when the
strings are moved from frame to frame the change in length is simply
the change expected using the Lorentz transform. Nothing unusual
there. But let me explain the problem I have with the cylinder
example by doing a dialog between me, the experimenter, explaining my
experimental test results to my manager. Then you can see what is
bothering me about this example. Both me and my manager are in the
moving frame and our comments are from the moving frame point of view.
M. In your experiments you had a rotating cylinder aligned along the
x-axis, and you simultaneously, as measured in our frame, attached all
points along the bottom of a rigid rod to the surface of the rotating
cylinder. The rod iss the same length as the cylinder, and extends
from one end of the cylinder to the other. The rod and cylinder have
zero relative velocity along the x-axis. You attached the rod
parallel to the x-axis on to the surface of the cylinder. The
cylinder and rod have a velocity V along the x-axis relative to our
frame. How did the test results go?
D. I did the experiment with rigid rods made out of various
materials. I discovered that when I tried attaching some of the rods
as you described, some of the rods broke and bent. When we brought
these back to our lab, we determined that the rods were broken and
bent due to stretching along the length of the rod and twisting.
M. Wait, you were supposed to attach all points simutaneously so that
the straight rod remained straight after the attachment.
D. Yes, that's what we did, but some rods broke due to stretching and
twisting. So we did some experiments with the rods that didn't break.
And we got some unusual results.
M. What do you mean?
D. Well we had a rod that was well within its elastic limits. When
we attached that rod, the length before attachment was L, the same
length as the cylinder. After the attachment this rod was a straight
line extending from one end of the cylinder to the other end. We
removed the rod by simultaneously unattaching all points. We waited
for a long while and then measured that the rod was the same length
before and after attachment.
M. So what's so unusual about that?
D. Nothing there, but one of the rods we attached was at its elastic
limit. If it got stretched or twisted any amount the rod would not
return to its previous shape. The chemical bonds get out of alignment
so it won't return to its original shape if it is stretched.
M. Well, in your experiment the rod is a straight line before and
after attachment, and its the same length as the cylinder before and
after attachment, so what's the problem.
D. When we simultaneously attached all points of this rod, everything
was as before in the previous experiment, but when we removed the rod,
it was not a straight line and it was longer than the length of the
cylinder.
M. Not a straight line? What shape was it?
D. Well, surprisingly, it's shape was a spiral the same diameter as
the cylinder. It was as if this rod was wrapped around the cylinder
instead of being placed in a straight line on the cylinder. And its
length was stretched just as if it was a spiral wrapped around the
cylinder from end to end.
M. That is strange.
D. Well we did further testing. One test we did was cut the attached
rod into many very short segments. We then removed the force keeping
these segments attached to the rotating cylinder. When we did this
the segments flew off the cylinder in the direction of their
instantaneous velocity.
M. That's the way its supposed to work, so what.
D. Well we did the same thing with a very long, very rigid rod
instead of a segmented rod. When we removed the force keeping the
long rod attached to the surface of the cylinder, this straight rod
did not go flying off. Instead it continued to rotate about the
cylinder as if it was still attached?
M. How can that be?
D. Then we did another experiment with even more bizarre results. We
had a long straight electro-magnet the length of the cylinder. When
all points of the rotating straight rod were in close proximity to the
magnet, we thought the magnet had enough strength to pull the rod off
the surface of the cylinder, but it didn't.
M. The rod was probably moving too fast past the magnet.
D. That's what we thought at first. However, in every experiment the
straight rod acted like it was a spiral and not a straight rod. So
we asked how does a spiral rod behave. So instead of attaching all
points simultaneously to the surface, we attached one end first and
let the rod wrap itself around the cylinder. And we discovered that
depending on the length of the cylinder and rod, if we had a certain
delay in attaching one end before the other, the attached rod had
some very unusual properties.
M. Like what.
D. Remember all those materials that broke and bent when we kept the
shape the same by attaching all points simultaneously?
M. Yeah, they must have really been mangled when you wrapped them
around the cylinder in a spiral stretching and changing their shape.
D. Actually, not a single one broke. And we could not detect any
stretching.
M. Wait. Before they were attached, they were the same length as the
cylinder, and they were straight. And to spiral around the cylinder,
the length increases. But they weren't stretched and they weren't
bent?
D. That's right. And now we repeated the magnet experiment. Remember
in the first experiment when all points of the straight rod were in
close proximity to the magnet, the magnet did not have enough strength
to remove the straight rod from the surface of the cylinder?
M. Yeah. Let me guess what happens with a spiral rod and the magnet.
Well this spiral rod has the same angular velocity as the straight
rod, and hardly any points of this spiral rod simultaneously are in
close proximity to the magnet, so the magnet probably had hardly any
effect whatsoever.
D. Well surprisingly, the magnet now pulled the spiral rod off the
surface of the cylinder eventhough both the spiral rod and straight
rod were identical in mass and material composition. And in doing so,
the magnet straightened the spiral rod into a straight line just as
had been before it was attached to the cylinder. And what's more
remarkable, we had rods of various materials with a metal strip on
them. And each of these were unbent by the magnet into a straight
line independent of the material composition! And the magnet didn't
require any more force to unbend a diamond spiral or a steel spiral or
a wood spiral.
M. That doesn't make sense.
D. And when we repeated the experiment of removing the force
attaching the rod to the surface of the cylinder, we expected the
spiral rod to keep rotating around the cylinder. But instead it
unbent itself and flew off as a straight rod! And that happened for
every different material we used.
M. Now let me get this straight. In all the experiments in which you
attached the rod so that it remained straight after attachment, the
rod behaved as if it was a spiral rod, and in all the experiments you
did in which the rod was a spiral after attachment, the rod behaved as
if it were a straight rod. And when the length remained the same
before and after attachment, the rod was stretched and twisted. And
when the length after attachment was longer in the spiral case, the
rod did not stretch.
D. That's right.
M. That sounds totally backwards. Were all your measurements
repeatable?
D. Yes.
M. Exactly how did you measure that the rod was a straight line after
simultaneously attaching all points to the surface of the cylinder?
D. Well we actually didn't measure that. We used a theoretical
computation of simultaneous events in different inertial frames.
See Todd, that is what I don't grasp in these cylinder problems. Dirk,
Eric, Hogbin, et al. think that all of these posts have been the same
problem and that they've been brilliantly explained. But I still
haven't understood why others don't agree with the manager's comment
that this sounds totally backwards.
David
Well, there is a lot going on here - you are dealing with a situation that
in my opinion is more complicated than what you seem to imply. However, I
think that at least part of the explanation of why the moving frame
observers see the experiments as yielding results that you feel are
'backwards' can be given without getting into rotational motion of the
cylinders. The wrapping of the wires around the cylinder adds an additional
twist, so to speak, that brings in additional complications. All I'll try
to do here is to look at a simpler example that has some of the
characteristics of your more complicated example. Even this simpler example
is a bit tedious to go through.
In this experiment we have two long rails sitting at rest in the rest frame.
The rails are oriented in the y-direction and the rails are separated in the
x-direction by a distance L according to the rest frame. Each rail contains
a little cart that can move along its rail in the y-direction. Initially
the two carts are at rest at y = 0 (i.e., the carts are both sitting on the
x-axis a distance L apart. A lightweight spring connects the two carts.
The spring is oriented along the x-axis and is neither stretched nor
compressed (no tension). From the moving frame viewpoint the carts, the
spring, and the rails are all moving parallel to the x-axis in the positive
x-direction (I think of the moving frame as moving in the negative
x-direction relative to the rest frame.) The rails are separated in this
frame by L/g where g is the gamma factor relating the two frames. Observers
in the moving frame agree that the spring has no tension even though they
see the spring as Lorentz contracted.
The carts are equipped with motors that allow them to accelerate in the
y-direction along the rails. We now consider two experiments.
EXPERIMENT 1. Simultaneously according to the *rest* frame, the carts start
accelerating with identical y-components of acceleration until each cart has
acquired some specified value of the y-component of velocity and then the
carts continue to move with constant speed along the rails. At any given
time in the rest frame, the two carts have the same y-coordinate. So the
spring remains oriented parallel to the x-axis and no tension develops in
the spring.
According to the moving frame the cart on the left starts accelerating first
and gets ahead of the other cart in the y-direction. The x-separation of
the carts of course does not change as the carts remain on the rails. So,
according to the moving frame the distance between the carts increases.
Hence the spring gets longer as the carts accelerate according to the moving
frame and the spring ends up with a slanted orientation across the rails.
Nevertheless, no tension develops in the spring despite the fact that it
''stretches'' according to the moving frame.
EXPERIMENT 2. Start over exactly as before except the carts now begin to
accelerate simultaneously according to the moving frame. In the moving
frame both carts undergo identical accelerations for the same time
interval - starting and ending their accelerations simultaneously. Hence,
in the moving frame, neither cart gets ahead of the other in the y-direction
and the x-separation remains constant. So, in the moving frame the spring
does not change its length. Nevertheless, the spring develops a real
tension - enough to break the spring if the material of the spring is too
weak.
According to the rest frame, the cart on the right starts accelerating first
and gets ahead of the other cart in the y-direction. The x-separation doesn't
change. So, according to the rest frame, the spring gets longer. Observers
in the rest frame say its natural for tension to develop in the spring.
Summary:
In the rest frame nothing 'strange' appears to happen. In experiment 1,
according to the rest frame, the spring didn't stretch and no tension
developed in the spring. In experiment 2, according to the rest frame, the
spring stretched and the spring developed tension.
In the moving frame things appear to be 'backwards'. In experiment 1
according to the moving frame the spring 'stretched' (i.e., 'got longer ')
yet it did not develop any tension. In Case 2 according to the moving frame
the spring did not change its length yet it developed tension.
Of course, we know that these 'backwards' results of the moving frame are
not really paradoxical. They are natural consequences of the relativity of
simultaneity.
Todd, here's what I don't understand.
If for example, the moving frame observers say the rod is a straight
line, and the moving frame experimenters bring back a broken curved
rod to their lab, why is that logical? Why don't you consider that a
negative consequence of relativity instead of something that supports
the theory?
If for example, the moving frame observers say the rod never gets
picked up by the magnet when all points of the rod are in close
proximity to the magnet. The magnet only will pick up the rod if the
majority of the mass is far removed from the magnet and only a small
portion of the mass is near the magent. Why do you consider that
logical and something that supports the theory instead of a negative
consequence of relativitity?
That is what I don't understand.
David
.
When you attach your rod to the rotating cylinder (either simultaneously
according to the rest frame or simultaneously according to the moving frame)
then it is very similar to accelerating the pieces of the rod in the
y-direction like the carts on the rails. The spring simulates the elastic
forces between elements of the rod. But there are additional complications
in your case due to the wrapping of the rod around the cylinder.
Todd
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