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Re: non-relative angular velocity



David ...

Thanks you for the detailed explanation. Since arguments such as those
that you propose in your reply are not usually available in our
textbooks, perhaps you would consider submitting an article on
this topic to the American Journal of Physics.

Herb

On Thu, 03 Feb 2000 12:23:51 -0500 David Bowman
<David_Bowman@GEORGETOWNCOLLEGE.EDU> writes:
Regarding Ludwik's question:

According to David the singularity is a relativistic dilemma.
Foucault was not aware of relativity. Are you addressing
the same topic?

John Denker wrote:
...

If I may be so bold as to attempt to speak for John, I suspect that
he
was addressing a related--but not identical--topic. I think John's
comment was meant to be an argument showing that there are actual
physical effects associated with different states of absolute
rotation
(as manifested by Foucault's pendulum) due to the form of the laws
of
nature being dependent on the state of rotation of the coordinate
system
used (and these effects, or their absence, can be used to define an
absolute state of non-rotation about which all other rotating
coordinate
system's can have their absolute rate of rotation measured).

In my point I was (implicitly) granting Herb the right to use a
rotating
coordinate system if he wanted to (as long as he used the proper
form
of the laws of nature that appear in such a frame), but was pointing
out
that the use of such a rotating frame is not necessarily globally
extendable to all of spacetime and, as such, is not equivalent,
*after*
the idiosyncratic frame-dependent fictitious forces are properly
taken
into account, to a frame which is not rotating. A rotating frame is
not useful for describing regions of space which are sufficiently
far
from the rotation axis, because in such regions *all four* spacetime
coordinates (the three 'spatial coordinates *and* the 'temporal'
coordinate) are separately *spacelike* when they are varied (when
holding
all the other coordinates fixed during the variation) and this gives
the metric all negative diagonal coefficients for the square
increment of
proper time.

It should be noted that very large distances from the rotation axis
can
be described using such a rigidly rotating coordinate system. It's
just
that such a description will involve a supposedly 'time' coordinate
which
is really spacelike, and all actual timelike-separated events out
there
*must* have a description which includes a subtle and judicious
increment
of *both* the 'temporal' *and* the 'spatial' coordinates. What
causes
this is that the transformation to the rotating spatial coordinates
is
such that the new coordinates are not orthogonal between the
temporal and
spatial directions in spacetime. This makes certain cross-terms
between
the temporal and spatial coordinates appear in the metric form. It
ends
up that there is no transformation of the temporal coordinate that
can be made which will make it orthogonal to the new rotating
spatial
coordinates. For distances equal to [omega]/c from the rotation
axis
this supposedly temporal coordinate becomes null, and for greater
distances it becomes actually spacelike. Essentially, at such
distances
the nonorthogonal 'time' parameter becomes so tilted w.r.t. the
spatial
directions that it tilts by more than 45 degrees w.r.t space and
ends up
being a space-like parameter. Thus any description using a rigidly
rotating coordinate system for sufficiently distant phenomena will
necessarily be one where all 4 spacetime coordinates are spacelike
in nature, and such a description has no 'time' parameter which we
can
conveniently understand as describing the 'flow of time'. I would
not
consider such a situation equivalent (in any sense) to a nonrotating
frame--even after we grant corrections due to such mundane effects
as the
usual fictitious forces that arise in such coordinate systems.

David Bowman
David_Bowman@georgetowncollege.edu