Re: non-relative angular velocity

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

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