Energy stored in a length contracted pole
Jartza
If you are not familiar with "ladder paradox" or "barn - pole
paradox", then it is a good idea to check out this:
http://math.ucr.edu/home/baez/physics/Relativity/SR/barn_pole.html
We cool a 10 m long pole almost to absolute zero temperature.
Then we accelerate the pole to very large speed.
(0.99 c is fast enough)
Then the pole goes into a 5 m long tightly fitting tube.
(5 m is the rest length of the tube, which is at rest)
Then the kinetic energy of every molecule of the pole is extracted, this
is done simultaneously for every molecule, the simultaneity
here is the simultaneity in the frame of the tube, which is at rest.
Then a lid is put on the tube, so it's a closed tube.
Now there is huge pressure in the tube.
(because a 10 m long pole is in a 5 m long tube)
The energy to generate the pressure came ... from where?
Eric Gisse
Except the pole isn't 10m in your reference frame. It is 5m.
There is no pressure.
Tom Roberts
> We cool a 10 m long pole almost to absolute zero temperature.
> Then we accelerate the pole to very large speed.
> (0.99 c is fast enough)
> Then the pole goes into a 5 m long tightly fitting tube.
> (5 m is the rest length of the tube, which is at rest)
> Then the kinetic energy of every molecule of the pole is extracted, this
> is done simultaneously for every molecule, the simultaneity
> here is the simultaneity in the frame of the tube, which is at rest.
> Then a lid is put on the tube, so it's a closed tube.
> Now there is huge pressure in the tube.
> (because a 10 m long pole is in a 5 m long tube)
> The energy to generate the pressure came ... from where?
From the way you stopped the pole inside the tube.
[I assume a speed of 0.866 c, so gamma = 2; not your 0.99 c.
This makes the description easier, as the pole just fits.]
Let me designate the initial inertial frame of the pole as "the pole frame";
this is an inertial frame that does not slow down as the pole is stopped.
You start decelerating the pole simultaneously in the tube frame, all along its
length, so let me imagine this starts when it is 1 mm from fitting inside the
tube (this takes enormous forces, but no matter). In the pole frame, however,
the force on the front end starts before any other force on the pole, and the
initiation of force on the rest of the pole successively moves along the pole to
the back end; for each point along the the pole, initiation of force occurs when
the corresponding point of the tube is 0.5 mm away. In the pole frame it's
obvious that the front of the pole is pushed back towards the back end by those
enormous forces, thus compressing the pole. In the pole frame, the tube appears
to be 2.5 meters long, so the initiation of force on the front end occurs when
the back end of the tube is 7.5005 meters from the back end of the pole;
initiation of force on the back end of the pole occurs when the back end of the
tube is 0.5 mm from the back end of the pole.
Remember that what happens to an object is what is described in its own rest
frame -- descriptions from other frames are irrelevant to the object. Note the
pole is NOT at rest in the pole frame during its deceleration, so one really
needs to consider each small region along the length of the pole; each region is
changing rest frames as it decelerates, and they each do so at different times
in their own instantaneous rest frames. But each such region will see force
applied to its front, and only later to its rear, so each region gets
compressed. If you think about this carefully, you will realize that it works
out correctly, and each small region of the rod is compressed the same as all
the others, starting when it is 0.5 mm from its final position in the tube and
ending when it reaches that position.
Exercise for the reader: explain why I did not need to mention
the compressibility of the pole, or the strength of its inter-
atomic bonds, or the strength of the force required to stop it.
Hint: consider the trajectory of each individual point of the
pole; given the description, how does mass affect this path?
Tom Roberts
Jartza
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On 04/16/2011 08:54 AM, Eric Gisse wrote:
> Except the pole isn't 10m in your reference frame. It is 5m.
>
> There is no pressure.
Let me rephrase the story of the pole:
A 100 m long pole is accelerated to speed 0.866 c, which makes
the pole a 50 m long pole.
Now the pole is stopped in a very very very _VERY_ short time.
Obviously immediately after the stopping the length of the pole
is 50 m, and the pole is exploding.
From where comes the energy of the explosion?
[[Mod. note --
1. When moving at 0.866 c, the pole is (still) 100 m long in its own
rest frame.
2. After it's stopped, the pole is (still) 100 m long in its own
rest frame (which is a different inertial frame from the inertial
frame of #1).
3. So, there's no explosion.
4. If you actually want to *understand* special relativity, your
best bet is tutoring by a knowledgable teacher, or failing that,
*carefully* reading a *good* book or two. My favorite introductory
special relativity books for this purpose are Taylor & Wheeler
"Spacetime Physics" 2nd edition and N. David Mermin, "Space and
Time in Special Relativity".
-- jt]]
Daryl McCullough
>Let me rephrase the story of the pole:
>
>A 100 m long pole is accelerated to speed 0.866 c, which makes
>the pole a 50 m long pole.
>
>Now the pole is stopped in a very very very _VERY_ short time.
>Obviously immediately after the stopping the length of the pole
>is 50 m, and the pole is exploding.
>
> From where comes the energy of the explosion?
>
>[[Mod. note --
>1. When moving at 0.866 c, the pole is (still) 100 m long in its own
> rest frame.
>2. After it's stopped, the pole is (still) 100 m long in its own
> rest frame (which is a different inertial frame from the inertial
> frame of #1).
>3. So, there's no explosion.
I don't understand the moderator's note, at all. As measured
in the initial frame, there is a time that the pole is traveling at
0.866c, and its length is 50 meters. There is a later time
in which the pole is at rest, and its length is 100 meters.
In this frame, the pole has expanded from 50 meters to 100
meters. That's an explosion. If someone tried to enclose
the pole in a strong box 50 meters in length, then either
the box will break, or the pole will, or both. That's not
some kind of illusion, it's a real catastrophe.
But there is no real question about "where the energy
comes from" for this catastrophe--from the point of
view of the initial frame, the moving pole has
enormous kinetic energy, and if you attempt to stop
it, that energy will go into smashing up the pole and
whatever tried to stop it.
--
Daryl McCullough
Ithaca, NY
Jartza
> Exercise for the reader: explain why I did not need to mention
> the compressibility of the pole, or the strength of its inter-
> atomic bonds, or the strength of the force required to stop it.
> Hint: consider the trajectory of each individual point of the
> pole; given the description, how does mass affect this path?
>
>
> Tom Roberts
>
In the tube frame:
Compressibility has no effect on pressure energy, because pressure
energy is the difference of acceleration energy and braking energy,
which both depend on mass and velocity only.
In the tube frame:
The force fields responsible of incompressibility of the pole
have inertia, and the rear part of a force field keeps on
moving, when the front part has already stopped. So these kind
of force fields can't resist the compression of the pole in the
case of the extremely fast braking.
In the pole frame:
Well, I don't know. You tell me.
Tom Roberts
> Jartza says...
>
>> Let me rephrase the story of the pole:
>>
>> A 100 m long pole is accelerated to speed 0.866 c, which makes
>> the pole a 50 m long pole.
>>
>> Now the pole is stopped in a very very very _VERY_ short time.
>> Obviously immediately after the stopping the length of the pole
>> is 50 m, and the pole is exploding.
>>
>> From where comes the energy of the explosion?
>>
>> [[Mod. note --
>> 1. When moving at 0.866 c, the pole is (still) 100 m long in its own
>> rest frame.
>> 2. After it's stopped, the pole is (still) 100 m long in its own
>> rest frame (which is a different inertial frame from the inertial
>> frame of #1).
>> 3. So, there's no explosion.
>
> I don't understand the moderator's note, at all.
The moderator clearly only discussed the situation before stopping, and the
situation well after the pole is stopped, with an implicit assumption that the
pole remained a pole and achieved its normal (equilibrium) length. This ignores
what happens during the stopping, which is indeed an explosion (but there's no
explosion after it is stopped and equilibrated, assuming it does equilibrate).
As my earlier post discussed, given the way Jartza stopped the pole, there are
ENORMOUS internal strains induced in the pole, and no actual pole could sustain
them (nor could any conceivable real apparatus do what was described).
> As measured
> in the initial frame, there is a time that the pole is traveling at
> 0.866c, and its length is 50 meters. There is a later time
> in which the pole is at rest, and its length is 100 meters.
> In this frame, the pole has expanded from 50 meters to 100
> meters. That's an explosion.
Sure.
> If someone tried to enclose
> the pole in a strong box 50 meters in length, then either
> the box will break, or the pole will, or both. That's not
> some kind of illusion, it's a real catastrophe.
Yes. The challenge is getting the pole inside and then stopping it.
> But there is no real question about "where the energy
> comes from" for this catastrophe--from the point of
> view of the initial frame, the moving pole has
> enormous kinetic energy, and if you attempt to stop
> it, that energy will go into smashing up the pole and
> whatever tried to stop it.
From the point of view of the pole, it never has any kinetic energy. What
causes it to "explode" is the enormous forces impressed on it to make it stop
within the tube. Because after it is stopped those forces are removed, and the
internal strains in the pole will certainly result in a spectacular explosion.
Remembering that this is all a gedanken, of course.
As I said before, in describing what happens to the pole, the only relevant
frame is the pole's rest frame, which of course changes while it is being
stopped, and differs for each small region of the pole. But for each small
region of the pole, the retarding force is applied to its front before its back,
and is removed right when the region has been compressed to half its original
(equilibrium) length -- that's an enormous strain all along the pole.
Tom Roberts
Tom Roberts
> On 04/17/2011 10:48 AM, Tom Roberts wrote:
>> Exercise for the reader: explain why I did not need to mention
>> the compressibility of the pole, or the strength of its inter-
>> atomic bonds, or the strength of the force required to stop it.
>> Hint: consider the trajectory of each individual point of the
>> pole; given the description, how does mass affect this path?
>
No. Just go back and look at the description of how you stopped the pole (plus
my additions to make it complete) -- it essentially describes the trajectories
of the individual small regions of the pole, without mentioning the magnitude of
the forces involved. In essence, their accelerations were specified, so the
forces on the individual regions of the pole scale with the mass of that region.
The inter-regional forces holding the pole together are VASTLY smaller than the
forces required to stop the pole, so treating the regions as independent is
justified.
For "region" read "molecule" if that helps, but I wanted to keep
QM out of the discussion.
Note also that physical phenomena are completely independent of the frame used
to describe them. So your attempt to give two different explanations in the two
frames cannot possibly be correct.
One can obtain different values for measurements of physical
quantities when using instruments at rest in different frames, but
that is a real (physical) difference in the relationship between
phenomenon and instrument.
Tom Roberts
ben6993
>
> - Show quoted text -
In the observer's frame the pole is 50m long at 0.886c. In the pole's
frame the pole is 100m long, at rest.
Then, forces are applied to slow the pole in the observer's frame and
those forces compress the pole to 50m in the pole's frame by the time
it stops moving in the observer's frame. Say, in this unrealistic
exercise, that it is assumed that there is no elastic rebound in
length back to greater than 50m in the pole's frame, because the
compressive forces are so strong and damaging to the pole's structure.
While the pole shrinks from 100m to 50m in the pole's frame, the pole
starts at length 50m and ends at length 50m in the observer's frame.
At some time inbetween the pole is say 80m long in the pole's frame
and it appears less than 80m long in the observer's frame due to
relativistic contraction. But it won't be as small as 40m as the speed
had reduced to below .886c. Say it is still 50m.
So it could be possible to control the application of the forces so
that the pole's length appears to be 50m long to the observer at all
times? In that case the pole would neatly fit into a 50m long
container (as measured by the observer) at any time in its travel.
Tom Roberts
> [... see previous posts for context]
> In the observer's frame the pole is 50m long at 0.886c. In the pole's
> frame the pole is 100m long, at rest.
>
> Then, forces are applied to slow the pole in the observer's frame and
> those forces compress the pole to 50m in the pole's frame by the time
> it stops moving in the observer's frame. Say, in this unrealistic
> exercise, that it is assumed that there is no elastic rebound in
> length back to greater than 50m in the pole's frame, because the
> compressive forces are so strong and damaging to the pole's structure.
>
> While the pole shrinks from 100m to 50m in the pole's frame, the pole
> starts at length 50m and ends at length 50m in the observer's frame.
>
> At some time inbetween the pole is say 80m long in the pole's frame
> and it appears less than 80m long in the observer's frame due to
> relativistic contraction. But it won't be as small as 40m as the speed
> had reduced to below .886c. Say it is still 50m.
>
> So it could be possible to control the application of the forces so
> that the pole's length appears to be 50m long to the observer at all
> times? In that case the pole would neatly fit into a 50m long
> container (as measured by the observer) at any time in its travel.
Given no rebound (e.g. the pole is made up of non-interacting dust particles), I
suspect that in the original description, the pole remains 50 m long in the
observer's frame (that of the tube). As the front of the pole is forced to
compress in its own instantaneous rest frame, its frame is also slowing down
relative to the observer's frame, and I suspect the change in its proper length
is exactly compensated by the change in gamma. This occurs all along the pole,
of course, but at different times and places for each region of the pole (as
seen by each individual region of the pole).
I have not attempted to do the math to prove that these effects cancel. But in
any case, if the pole goes from full speed to rest in 1 mm of the tube/observer
frame, it's clear that its length cannot change by more than 1 mm in that frame,
and is 50 m both before and afterwards.
Note also that there is no INERTIAL FRAME in which the pole is ever both at rest
and 80 m long -- during its compression/stopping the different regions of the
pole are at rest in different instantaneously co-moving frames, so there is no
"pole frame" during the compression/stopping.
Tom Roberts
ben6993
>
> - Show quoted text -
Thank you. I see what you mean about the pole not being at rest and
at the same time being 80m long in any inertial frame.
This is the same issue as the so called bee and train paradox. A bee
hits a train head on in a collision and is knocked backwards.
Therefore the train is momentarily stopped by the bee. A lot of
people cannot agree with that on first thoughts, and the danger is of
labelling things and quantising the object by thinking of it as
immutable as the label. A train is a very hard, heavy, solid object
and 'cannot' be stopped by a mere bee. Except it is only a little bit
of the train that is stopped in the leading atoms, or maybe even the
outer electrons in those leading atoms, which act rather like a
trampoline to bounce the bee backwards.
You are correct. Even while thinking of the pole as being compressed
from 100m to 50m, I was still probably thinking of it as a single
label 'pole' and therefore somehow still at rest in its own frame.
Either that or else I just neglected to think it through at all. It
is better to think in terms of a collection of dust particles, as you
said, than a pole to avoid quantising the 'pole' in one's thoughts.