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Re: [Phys-L] Sun going around the Earth?



Perhaps what your interlocutor meant is that it's always possible to
choose a coordinate system for the spacetime metric such that (a) it
looks like the special-relativistic Minkowski ("flat") metric at a
point P, and (b) even better, the first derivatives of the metric
vanish in this coordinate system, so that the metric looks Minkowski in
a neighborhood of P. This choice of coordinate system isn't even
unique.

One consequence of your coordinate system transformation is that the
geodesic equation becomes trivial at P, because the Christoffel symbols
that appear in it (and which depend on first derivatives of the metric)
vanish. So an object (the Earth) that is initially motionless in this
coordinate system will remain at rest; in fact, you've defined a
locally inertial reference frame.

Now, you haven't made the metric *actually* flat at this point. The
Riemannian curvature depends on second derivatives of the metric, which
you can't entirely get rid of through coordinate transformations. So
tidal effects still exist on your motionless Earth, for example.

The consequence of making your metric look simple at point P is that
away from P it looks messier. (On extragalactic scales, does it look
Messier?) This leads to terms in the geodesic equation for the Sun
that would produce its "orbit" around the Earth. In Newtonian physics,
the equivalent is the appearance of the "fictitious" (ugh, bad
term) forces in accelerated reference frames.


On Wed, 25 Mar 2015 08:59:48 -0400
Brian Blais <bblais@bryant.edu> wrote:

So, I was having an argument with someone about establishing that the
Earth goes around the Sun, and he claimed that he could write down a
perfectly consistent general relativistic framework where the Sun
would be going around the Earth. Not having the time, I didn't get a
chance to see this done, but I was wondering whether it *could* be
done - even in principle. If so, then is the geocentric model "just
as good" as the heliocentric model - in the sense of "just as
consistent with reality" - as opposed to the "just as convenient"
sense?

The discussion was part of a more general discussion of epistemology,
which is perhaps the only time that anyone argues such things. :)

Anyone ever seen a treatment of this?

thanks,

Brian Blais