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

JanPB says...

I.Vecchi wrote:

The Schwarzschild chart is smooth over the Schwarzchild surface

What do you mean by "Schwarzschild chart" exactly? The one he has in
his original paper? It's not smooth at the horizon either...

I think the problem here is that the manifold has not been explicitly
defined. I.Vecchi is *identifying* the manifold with the set of 4-tuples
<t,r,theta,phi> with -infinity < t < +infinity, 0 < r < +infinity,
0 <= theta <= pi, 0 <= phi < 2pi. (With the usual smooth structure
on subsets of R^4).

In other words, he's saying that the manifold is just R^4,
minus the one-dimensional locus corresponding to the singularity
at r=0.

Kruskal spacetime can be identified with the set of points
(T,R,theta,phi) with -infinity < R < +infinity,
- square-root(1+R^2) < T < square-root(1+R^2),
0 <= theta <= pi, 0 <= phi < 2pi.

That's a different manifold, and there is no smooth map
between them.

--
Daryl McCullough
Ithaca, NY

.

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