Re: Misinterpretation of the radial parameter in the Schwarzschild solution?

Ilja Schmelzer wrote:
"Tom Roberts" <tjroberts137@xxxxxxxxxxxxx> schrieb
Indeed, charts are not geometrical objects, and are merely labels for
the points; geodesics are geometric objects -- defining the manifold in
terms of geodesics is MUCH better,

brrr.

Yeah. I spoke too loosely.


As I have already explained, the definition of a manifold via charts is a
correct way to do it.

In math, perhaps, though I have my doubts. In physics that is like saying the real world is defined by the coordinates -- utter nonsense.

The manifold exists completely independent of ANY coordinates, though you can use any coordinates you like, and define the manifold by them if you wish. But you MUST be sure that your coordinates span the entire manifold, and the Schw. coordinates don't -- mathematicians can study all sorts of manifolds with holes, cuts, and omitted regions, but as a model of the universe such arbitrary omissions are unphysical. It simply does not make sense for a geodesic path to leave and/or enter the manifold, if the manifold is supposed to be a model of the universe (unless it intersects a singularity, where the theory breaks down).


But defining manifolds in terms of geodesics is certainly false.

I spoke overly loosely. But any geodesic in the manifold should remain in the manifold until it hits a singularity -- any other manner of leaving the manifold is unphysical[#]; but if one claims the Schw. coordinates define the entire manifold then that happens at r=2M (and worse, the geodesic immediately "re-enters" the manifold after leaving).

[#] Actually, singularities are unphysical, too. We usually
consider them to indicate that the theory is incomplete and
does not apply there....


Because
a manifold does not have a metric, and, therefore, no geodesics. The metric
is an additional structure on the manifold.

Yes. In GR we always discuss manifolds with metric, even though we rarely mention it -- that is implicit in the theoretical context.


Tom Roberts
.

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