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

LEJ Brouwer wrote:
JanPB wrote:
LEJ Brouwer wrote:
Are the details of human record-keeping purely 'bureaucratic', or is it
the underlying nature of physical spacetime that ensures that
coordinates are naturally measurable by us in this way?

Well, the key phrase is "naturally" :-) What happens here is that what
appears "natural" do a distant observer turns out to be simply
underestimating the amount and the type of curvature that Einstein's
equation forces on spacetime, so the "natural" method of keeping track
of "space" turns out to run out of steam past certain values of the
radial coordinate. It just happens that strong gravity is less
intuitive than initially anticipated so the observer then presumably
goes back to his filing system and relabels events in a more sensible
way, e.g. by relabeling the spheres of symmetry or by dropping the
insistence on orthogonality of time and space cordinates (which were
introduced purely as mathematical "bureaucratic" simplifications of the
derivation).

But it is always possible to make a choice of coordinates locally such
that the geometry is locally Minkowski - yet here you seem to be saying
that it is not?

An observer can choose approximately Minkowskian coordinates in his
proximity. So a "Schwarzschild observer" (i.e., very distant) cannot
extend his local Minkowskian system past the horizon which is too far
away.

The time and space coordinates are what we physically
observe - they are not just mathematical simplifications. R and T (or U
and V) of the Kruskal spacetime are an example of mathematical
abstractions which do not have a direct connection with physical
observations (unless we map back to Schwarzschild coordinates, or
similar).

I think their physicality is the same, they are just different.

And note that the mapping from Schwarzschild to Kruskal
spacetime (and vice versa) is ill-defined at the event horizon (as it
has to be, as the Schwarzschild coordinates break down there). I should
have made more of this anomaly when the matter was discussed earlier.

This is just a mapping going bad, like 1/x going bad at zero. It
doesn't make the origin in the real line to disappear. The reason it
seems so jarring I think is that it starts out what seems like a very
reasonable chart (Minkowski at infinity, etc.) but then the curvature
really plays tricks with our intuition. It's *just* that, there is
nothing singular there otherwise. You pick wrong coordinates you pays
the consequences. The horizon _is_ physically distinguished but is not
singular.

--
Jan Bielawski

.

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