Re: Misinterpretation of the radial parameter in the Schwarzschild solution?
- From: "I.Vecchi" <tttito@xxxxxxxxx>
- Date: 15 Aug 2006 09:02:42 -0700
Tom Roberts ha scritto:
xxein@xxxxxxxxxxxxx wrote:
TR said, in essence, that there was a disconnect between r<2m and r>2m.
NOT AT ALL!!! You need to learn how to read more accurately.
There _is_ no discontinuity, "disconnect", or any other defect in the
manifold at the boundary between r<2M and r>2M. But there is no simple
coordinate system that covers both regions or includes that boundary; it
requires more complicated coordinates, such as Eddelston-Finkelstein or
Kruskal-Szerkes coordinates.
You are talking about a different manifold then.
A differentiable manifold is defined by a set of charts, which provide
local coordinate systems. You may obtain equivalent atlases (i.e. sets
of charts corresponding to the same manifold) by diffeomorphisms on the
charts (i.e. smooth change of coordinates) as long as they fit where
the charts overlap. Diffeomorphisms preserve both the differentiable
structure and the metric of a pseudo-Riemannian manifold. However the
Kruskal map, being singular at the Schwarzschild surface, is not a
diffeomorphism and it yields a manifold that is different from the
original one.
Two atlases that are not equivalent define different manifolds. OK?
IV
.
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