Re: Einstein's math and physical objects

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

Date: Wed, 05 Jan 2005 18:09:19 GMT

<dseppala@austin.rr.com> wrote in message
news:41daa7b2.1483740@news-server.austin.rr.com...

> Let there be two circular disks in the y-z plane, with their centers
> on the x-axis. Let one disk be at x=0 and let the other disk be at
> x=L. I'll call the disk at x=0, disk A, and I'll call the other disk,
> disk B. Now stretch a wire from the top of disk A to the top of disk
> B. And stretch a wire from the bottom of disk A to the bottom of disk
> B. These two wires are attached to points 180 degrees apart on each
> disk. We note the following physical property. If one disk is
> rotated 180 degrees or more with respect to the other disk, the two
> wires will cross each other (in the steady-state condition). At time
> t0, we start both disks rotating at identical rates. Let's make the
> rotation speed N revolutions per minute. When a steady-state
> condition is achieved, the two wires do not cross each other.
>
> Let the mass of each disk be much greater than the mass of the
> attached wires. Now at some time tA we accelerate both disks and the
> attached wires to some velocity V along the x-axis. Both disks are
> accelerated in an identical fashion. We note that as viewed in the
> original reference frame the distance between the disks remains L (the
> wires, according to Einstein stretch), and we note that in terms of
> the rotation rates, the two disks remain in phase. That is, the
> relative rotation angle between the two disks remains constant (zero),
> as measured in this frame. When the end of one wire is at the top of
> disk A, the other end of that same wire is at the top of disk B (as
> measured in this frame). And when one wire is at the top of disk A,
> the other wire is at the bottom of disk A (and the other end of the
> wire is at the bottom of disk B).
>
> Now let's say the velocity V and distance L are such that simultaneous
> events measured in the original reference frame, are measured to be
> one second apart in a frame that is moving with V relative to the
> x-axis. This implies that the rotation angle of the two disks is not
> at zero degrees as measured in this frame. Depending on the rotation
> rate N we pick, we can make this angle greater than 180 degrees (or
> substantially larger than 180 degrees). In this frame, when the end
> of one wire is at the top of disk A, the other end of that wire is not
> at the top of disk B (unless the rotation angle of B and A differ by
> exactly N revolutions). As stated previously, if the two disks are
> 180 degrees or more out of phase, the two wires must cross in the
> steady-state condition. If the wires are attached 180 degrees apart
> on each disk, there is no physical way to keep the stretched wires
> from crossing each other in the steady-state condition if the disks
> get out of phase by 180 or more degrees. But they never cross as
> viewed in the original reference frame. They only cross if the two
> disks have a relative angle greater than 180 degrees between them,
> which they don't. Its non-sensical to have both physical conditions
> simultaneously true. Please explain how this is resovled using
> Einstein's notions of space and time.

It might help to imagine replacing the disks by a long circular cylinder.
The cylinder is initially at rest in the earth frame and oriented with its
axis of symmetry along the x-axis. You paint two red stripes along the
surface of the cylinder parallel to the x-axis such that the stripes are
diametrically opposite.

Now imagine that somehow the cylinder is accelerated along the x-axis such
that each point of the cylinder has identical acceleration relative to the
earth frame for the same amount of time (relative to the earth frame). This
will cause tensile stress to build up in the cylinder, but we can ignore it
since this stress has nothing to do with your argument. We may assume that
the material of the cylinder has enough elasticity that it can adjust to the
stress. The cylinder will maintain its length relative to the earth frame
but it will stretch to a longer length in the final rest frame of the
cylinder.

In the earth frame the two red stripes will still be parallel to the x-axis
and diametrically opposite. In the final rest frame of the cylinder, the
stripes will spiral around the axis of the cylinder in a double-helix
pattern (as Paul and Tom have already pointed out). The stripes will still
be on the surface of the cylinder and they will not touch each other
anywhere. The two wires in your example will do the same thing.

Todd

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