Re: relativistic centrifuge

From: Old Man (nomail_at_nomail.net)
Date: 10/11/04 

Date: Sun, 10 Oct 2004 21:02:18 -0500

"Edward Green" <spamspamspam3@netzero.com> wrote in message
news:eca320d0.0410101021.2ee424e@posting.google.com...
> "Old Man" <nomail@nomail.net> wrote in message
news:<a9ydnZyRCdFxSvXcRVn-iQ@prairiewave.com>...
>
> > Old Man wonders why, for a centrifuge of radius, R, and
> > acceleration, a, the ratio of proper time, t_p, to that at the
> > center of the centrifuge, t_0, isn't given by
> >
> > t_p / t_0 = sqrt{[ 1 - ( 2 a R / c^2 ) ] [ 1 - ( a R / c^2 ) ]}*
>
> I don't know: just how far is Old Man prepared to discuss the basic
> comceptual problems of GR and SR with me? I ask this not from the
> stand point of a know-it-all, but as someone who both is conscious of
> periodic confusions in his own mind, and does not feel confident all
> such features have been purged from the minds of those not considered
> unqualified to comment on these things -- aside from obvious
> crackpots.
>
> In http://scienceworld.wolfram.com/physics/GeneralRelativity.html, we
> meet the following claim:
>
> "The fundamental principle of general relativity asserts that
> accelerated reference frames and reference frames in gravitation
> fields are equivalent. General relativity states that clocks run
> slower in strong gravitational fields (or highly accelerated
> frames)..."
>
> I was on the point of criticizing this as follows: in the absence of
> gravitational fields GR reduces to SR. And SR includes no component
> which predicts time dilation based on acceleration per se. Therefore
> neither does GR, and the claim is false.

For global observations, like time dilation, SR and GR
aren't required to agree. The EP of GR concerns local
observations and has naught to with global observations.

Other than that, Old Man is even less certain than Ed.
Here's why:

WRT a centrifuge:

When applying a = v^2 / R, for gravitational time dilation,
we have a factor of 2 in the term

GR: (2 G M) / (R c^2) => (2 a R) / (c^2),

whereas for relativistic time dilation, we have,

SR: (v / c)^2 => (a R) / (c^2)

wherein there is no factor of 2. Therefore, GR and SR
don't predict the same global result, but locally, there's no
difficulty. Still, Old Man wonders if we haven't used an
erroneous radius, R, because, for centrifugal force ( |v| is
constant ),

F = dp / dt = m (dv / dt) (gamma)

thus, for a = dv / dt = v^2 / R_p, we get ( R_p is the
proper radius and, R_0, the coordinate radius)

R_0 / R_p = (gamma)

This is correct in the case of a uniform magnetic field,
whereof F = q (v x B). See "Gravity" by J. B. Hartle,
ISBN 0-8053-8662-9

So, In the substations,

(2 G M / R) => (2 a R) and (v^2) => (a R),

is R to be taken as the coordinate radius in both cases ?
The same question applies to the acceleration, a.

However, this correction isn't enough to make-up that
factor of 2. After all this rigmarole, no matter what, it
seems that the global predictions of GR and SR, WRT
time dilation for a centrifuge, remain divergent. That's
OK with Old Man.

> However, I see a small fly in my ointment. In arguments supporting
> the equivalence principle, it is common to imagine an accelerating
> spaceship, and
> calculate the rate of receipt at the bow of pulses periodically
> emitted at the stern, and conclude that the rate of receipt is smaller
> than that of emission as controlled by a clock at the stern. So here
> is an effect "in an accelerated frame" predicted purely from SR.
> However, it would be misleading to offer this effect as, by
> implication, something peculiar to GR.
>
> I _think_ it is true that the rate of an accelerating clock is handled
> correctly in SR, and that there are no factors beyond (1) the relative
> velocity of the clock, and (2) any direct effects of acceleration on
> the mechanism of the clock.
>
> This latter takes us perilously close to the Scylla of so-called
> Lorentz Aether Theory -- which is a point on which I appear somewhat
> cranky, since I insist it is an approach not wrong in spirit nor
> alternative to the meat of SR, but an alternative and equivalent way
> of understanding what is going on: an antithesis which a full
> synthesis must include. If we tell the tale of the accelerating
> spaceship within a single atom, we may well have a kind of Lorentzian
> Aether Theory for the case of acceleration, which may leave us with a
> semantic issue whether this is a fundamental effect of acceleration
> which must be added by hand into SR -- a harbinger of GR -- or a
> simple consequence of the physics of an accelerated body, analyzed in
> SR.
>
> Anyway, moving on, consider this alternative account:
>
> http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html
>
> "It is a common misconception that Special Relativity cannot handle
> accelerating objects or accelerating reference frames. It is claimed
> that general relativity is required because special relativity only
> applies to inertial frames. This is not true. Special relativity
> treats accelerating frames differently from inertial frames but can
> still deal with them. Accelerating objects can be dealt with without
> even calling upon accelerating frames."
>
> Ok. So far, so good.
>
> "The difference between general and special relativity is that in the
> general theory all frames of reference including spinning and
> accelerating frames are treated on an equal footing. In special
> relativity accelerating frames are different from inertial frames.
> Velocities are relative but acceleration is treated as absolute. In
> general relativity all motion is relative."
>
> Whoa-oh. Not so good!
>
> The alleged distinction refers merely to an accidental formal
> difference between the presentations, and not to new physics. GR is
> expressed in a way which handles all frames equivalently, but the same
> mechanism could be applied to a purely flat world, ruled by SR.
> Furthermore acceleration is absolute even in GR: physics continues to
> be measureably different on a bucket rotating relative to a first
> bucket, whereas physics is not different on a bucket translating
> relative to a first bucket. Local experiments can in general
> distinguish the first two cases but not the second -- our now unified
> formalism notwithstanding.
>
> * (You mean of course, aside from a symbolic factor of "^-1").

That acceleration and gravitation are guaranteed to be locally
indistinguishable, doesn't imply that the globally observed time
dilation for acceleration is equal to that of gravitation.

[Old Man]

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