Re: Einstein's math and physical objects

From: Todd (nope_at_nospam.com)
Date: 01/17/05 

Date: Mon, 17 Jan 2005 16:42:50 GMT

<dseppala@austin.rr.com> wrote in message
news:41ebc446.313067671@news-server.austin.rr.com...
> On Mon, 17 Jan 2005 12:50:10 +0100, "Harry" <harald.vanlintel@epfl.ch>
> wrote:
>
>>
>>"Todd" <nope@nospam.com> wrote in message
>>news:05wGd.9492$eT5.4591@attbi_s51...
>>>
>>> "harry" <harald.vanlintel@epfl.ch> wrote in message
>>> news:1105888400.022235.231360@c13g2000cwb.googlegroups.com...
>>> > [Tod]:
>>> > "Let me avoid the acceleration by bringing about the final state in a
>>> > somewhat different way. Imagine that the disks are at rest in frame B
>>> > and
>>> > the disks are not rotating. Diametrically opposite wires connect the
>>> > disks
>>> > as before. Frame B is moving in the positive x-direction relative to
>>> > frame
>>> > A. Thus, A sees the disks sliding in the positive x-direction, not
>>> > rotating, and the wires parallel to the x-axis."
>>> >
>>> > Oh oh, I'm afraid you now messed up - for without rotation the wires
>>> > are simply crossed for all observers.
>>> > This isn't going to help...
>>> >
>>> > Harald
>>>
>>>
>>> I still think I'm right! But maybe you can explain where I'm going
>>> wrong.
>>
>>I now see that I misread diametrically for diagonally! Sorry.
>>
>>> I'm not sure what you're referring to when you say 'without rotation'.
>>> There is no rotation initially, but there is rotation in the final
>>> state.
>>>
>>> Again, the disks are initially sliding along the x-axis in frame A with
>>> no
>>> rotation. The wires are strung between them parallel to the x-axis.
>>Frame
>>> B sees the disks and wires all at rest - no sliding and no rotation.
>>
>>OK, thus wires parallel along x and x'.
>>
>>> Now, we introduce the rotation. More precisely, we imagine that torques
>>are
>>> applied to the two disks _simultaneously in frame A_ so that the disks
>>> obtain identical rotations simultaneously in frame A. No external
>>> torques
>>> or forces are applied to the wires. The wires feel only their internal
>>> stresses and the forces of attachment to the disks.
>>>
>>> Imagine what happens in frame B. Due to relativity of simultaneity, the
>>> disk that has the greater x-coordinate starts rotating first. In this
>>frame
>>> the disks are not sliding. So, it's just like you where holding the
>>> disks
>>> in front of you and turning one of the disks without turning the other.
>>So,
>>> in frame B the wires will assume a simple crossed configuration.
>>
>>Right.
>>
>>> Once the
>>> other disk also begins to rotate in frame B, no further twisting of the
>>> wires will occur and the wires and disks simply maintain their crossed
>>> configuration while the whole thing rotates (from the point of view of
>>> B).
>>
>>Exactly. That is non-relativistic mechanics.
>>
>>> In frame A, the final configuration of the wires must be the Lorentz
>>> transform of the configuration in B. If I'm not mistaken, this will be
>>the
>>> conical helix shape. Any points where the wires touch in B, they will
>>also
>>> touch in A.
>>>
>>> Todd
>>
>>OK I'm again with you! Indeed this is the same end condition as the
>>original
>>paradox.
>>As mentioned before, the problem of the paradox is to identify the
>>centripetal forces on the wires in frame A.
>>
>>Harald
>>
>>
> Harald, can you please clarify the following. Let's assume the wires
> are very long so we can examine things at non-relativistic speeds.
> And lets assume the wires as measured in the final reference frame,
> after the acceleration has stopped and the wries have reached a
> steady-state condition, and the two disks have a relative rotation
> angle of 180 degrees, and that the wires take the shortest path and
> indeed cross. If you are in the first reference frame, and are at the
> points where the wires cross, do you see the wires cross or merely
> touch? By cross, let's say the final reference frame observer sees
> the wires in the x-y plane over some small distance and he sees one
> wire go from a positive y to a negative y coordinate over some small
> distance, and he see's the other wire go from a negative y to a
> positive y coordinate over the same span.
> If you are in the original reference frame at that same position
> and at that same time, what do you see? I don't see how you can
> possibly see the wires merely touch, instead of cross. If they do
> cross (one wire going from positive y to negative y while the other
> wire goes from negative y to postive y), that means there is another
> location where they must also cross as observed by this oberver (since
> the disks have zero relative rotation angle for observers in the
> original frame). We can do the observation of this point
> experimentally. We simply assume the wires segment we are looking at
> are 1 meter long, and the relative velocity between the two frames is
> 3 meters per second, and the rotation speed is say 120 revolutions per
> second. I don't see how we would see anything substantially different.
> Can you clarify?
> Thanks,
> David
>

I'll try to describe it the way I see it.

In frame B (the 'final' frame) the wires between the disks are straight
lines that touch halfway between the disks. So they form an 'X' in frame B.
As the system rotates, B-observers see the X rotate such that the wires
generate a full cone with vertex halfway between the disks.

In frame A (the 'original' frame) the configuration of the wires may be
obtained in the following way. Imagine slicing the cone in frame B into
many thin disks perpendicular to the x-axis. In frame A this cone will
still look like a cone except shortened along the x-axis by Lorentz
contraction. But due to relativity of simultaneity, consecutive
cross-sectional disks in A will be rotated relative to one another compared
to B. The wires in A will lie on the cone, but they will 'helically wind'
around the cone.

At the point where the wires touch, the wires will still appear to form an X
in frame A (just as in B) as long as you are looking at small enough
segments of the wires so that these segments appear straight.

It is similar for the case where the relative rotation between the physical
disks is greater than 180 degrees. Thus suppose the relative rotation is
720 degrees. In frame B, the wires are straight lines except near the
middle where they wrap around each other 'twice' over a very short distance.
In frame A the wires also wrap around each other 'twice' near the middle.
There would be very little difference between frame A and frame B in the
appearance of the wires where they are wrapping around each other. If the
wires wrap twice around each other in a right-hand sense in frame B they
also wrap essentially twice around each other in a right-hand sense in frame
A. However, as you move away from the middle in frame A, the wires would
'unwind' helically in a left-hand sense. Thus, the wires would make one
complete left-hand helical wind around the cone as you go from the first
disk to the middle where they touch, the wires then wind around each other
twice in a right-hand sense over a short distance, and finally the wires
would make another complete left-hand helical winding on the cone as you go
from the middle to the last disk.

Todd

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