[ExI] Relativity in Linearly Moving and Rotating Frames

[Lee Corbin]

Damien B. writes


> At 08:00 PM 9/28/2008 -0700, Lee wrote:
> 
>>The same holds true for whether it is the Earth that goes around
>>the sun, or vice-versa. To me, it is *not* a matter of whose views
>>become rarer. If the Church had prevailed in the 1600s and 
>>successfully suppressed all over the world the view that the Earth 
>>goes around the sun, then the Church would still be wrong.
> 
> Are you quite sure that's unequivocally true?

Yes, for all unaccelerated frames (though perhaps, to be on the safe
side, I should say "nearby" unaccelerated frames, e.g., within a 
lightyear of Sol).

> What does a Special Relativist say about that?

He or she can't say much; it's as though they write off accelerating
frames in SR.  It's GR (which E managed to build on top of SR,
it seems to me, that can deal with rotating frames of reference---
rotation being another form of acceleration).

> (And for extra marks--I don't have the 
> answer to this on the tip of my tongue, due to my problems with 
> visualization--once you grasp that the Earth is rotating once a day, 
> how obvious is it which body is orbiting the other?

One is supposed to be able to shut oneself up in a laboratory
and confirm whether or not one is in an accelerated frame.
(Foucault managed to do this using the whole Earth, or at 
least the latitude of Europe, this way.)  If there are mysterious
forces that cause objects to descend towards one wall, i.e.
"fall", then by the Equivalence Principle of GR one is either
in an accelerating spaceship or elevator, or one is having
a force from the outside being applied to one's laboratory
(e.g., the ground exerting pressure on the bottom of the
laboratory, which prevents one from achieving free fall
towards the nearby gravitating source). 

Or the same thing said with spiffier language using the curvature
of space. Wheeler's immortal words: matter tells space how to
curve and space tells matter how to move, or something like that.

In GR (as opposed to SR), it is perfectly valid, however, to 
go about using accelerated frames of reference, (though because
of SR usage, I might be misspeaking slightly). Einstein was a
follower of Mach, who claimed that if you were on a merry-go-round
and felt a centrifugal force, theoretically it might be that you were
standing still and the entire universe was rotating about you.

The Lens-Thirring effect is believed in by practically everyone, even
though I don't think it has been measured. (The newer satellites are
trying to do just that, I think.)  The simplest idea is this: suppose that
you had a Foucault pendulum going at the exact center of the North Pole.
The pendulum, being the law-obeying prudent type, says to itself "well,
I see a conflict. I'm getting information from a lot of stars that is saying
that the Earth is rotating and they're not, but there is a huge, huge object
right next to me that says it's standing still and all those stars are 
rotating around it. Who to believe?"  

It's predicted that the pendulum is going to be affected by the Earth
in this mysterious manner, and with respect to the fixed stars, it would
very, very slowly rotate as though the Earth were "dragging" it along
ever so slightly. Still too small an effect to measure today, though.

> In the deep 
> future when Earth is tidally locked, how self-evident will any of this appear?)

A Foucault pendulum would no longer rotate slowly depending on
latitude (at present, it doesn't rotate at the equator and *does*
rotate once every 24 hours at the north pole). Yet if the Earth
were tidally locked to the sun, then the Foucault pendulum would
still have far more "faith" in the fixed stars than it does in the sun,
and would keep aligned with the former (except for a very, very,
very tiny "frame-dragging" effect from Lens-Thirring). 

Exactly what messages the stars are sending is to me very exciting
stuff. Unlike ordinary gravitation, it falls off at 1/r instead of 1/r^2.
(At 1/r^2, I understand, the effect of all those stars would be too
weak, and the pendulum would "believe" the Earth.) Ciufolini and
Wheeler talk a lot about this, in damn near the most exciting
science prose I've ever read in their "Gravitation and Inertia"
http://press.princeton.edu/titles/5635.html , which you might want
to check out of a library some time. (Unfortunately, about 90%
of the math is over my head, but 20 or so introductory pages
are great.)

Lee