Re: error in mathematics behind black holes

Juan R. wrote:
Tom Roberts ha escrito:
The terminology "coordinate singularity" is both necessary and accurate
-- the singularity at r=2M is in the COORDINATES, not in the manifold
itself.

My point was not that. My point is that by "coordinate singularity"
many people think that the problem is only on the coordinates and think
one can study the region r =< 2M with a change of coordinates, which is
_not_ true.

It _is_ true. In the sense that an analyst external to the manifold can indeed study that region. This is, after all, how physics is done for situations like this -- this manifold is a highly idealized case that is incompatible with the word we inhabit.


A far away Schw observer cannot detect signals being sent
from the region close to r = 2M and this is independent if (s)he is
using Schw or Kruskal or some other set of coordinates.

Sure. But that does not mean the region of the manifold r<2M "does not exist", it merely means it is unobservable to this particular observer. To other observers, such as infalling ones, that region _is_ observable, and that's how we know it is part of the manifold, and therefore exists.


Some people think that a Schwarzschild observer could study the region
r =< 2M by just a change of coordinates. This is reinforced by some
relativists saying: "look scalars as R_abcdR^abcd are finite for r = 2M
therefore the Schwarzschild solution can be extended using different
'labels' with a better behavior".

Yes. Those "relativists" are of course external to the manifold, and are able to observe any part of it at leisure. As, for instance, one can imagine examining Euclidean 3-space in its entirety. This is what abstract mathematics is, and in this case physics is not far removed from that.


My point was that a Schwarzschild observer _cannot_ study the region r
=< 2M by a change of coordinates because the singularity on the
original Schwarzschild metric will be translated to the transformation
equations, breaking just at r = 2M.

But an observer in the manifold can indeed study that region, as long as she is willing to fall into the black hole.

You are attempting to enshrine an observer far away with some sort of mystical special properties. That's silly.


Now a crucial question, is the singularity at r = 0 real for _any_
observer? You appear to claim "yes!" but it may be noticed that a
Schwarzschild observer cannot detect stuff on/beyond the Schwarzschild
horizon.

Certainly any observer sufficiently near r=0 will be able to "observe" the singularity there. Of course such an observer will be annihilated quite soon.... Just because some particular distant observer cannot observe something does not mean it is not "real" or does not exist....


If the singularity at r = 0 cannot be physically detected by a
Schwarszchild observer -and this does not change with a change to
Kruskal coordinates- then the singularity does not exist for that
observer

You use "exist" in a very strange way. The rest of us would use "observe" instead, and that makes it a completely different situation. Yes, a distant observer cannot _observe_ the interior of the horizon, but that has no bearing on the question of whether or not that region _exists_. Existence is quite independent of observer.


However, the singularity may be observed -if GR is right-
by a Kruskal observer beyond the horizon.

No. Any observer external to the horizon cannot observe anything inside the horizon. That is, there is no future-pointing timelike or null path from inside the horizon to outside the horizon. This is completely independent of coordinates (as should be obvious from the way I stated it).


This is independent of coordinates -- the horizon is a geometrical
aspect of the manifold. As I said above, you are confusing two different
aspects of this manifold.

As said both are closely related, the horizon on Kruskal coordinates is
on r = 2M because the singularity of the Schwazschild exterior metric
is on r = 2M.

No -- your "because" is completely wrong, and is essentially backwards. The horizon is independent of coordinates and is a geometrical property of the manifold, the coordinate singularity at r=2M is not.


[*] So far as i know there exists not universally accepted definition
of singularity on GR, is there?

Yes. A a sufficient condition: if any curvature invariant diverges as a given locus is approached then the manifold is singular at that locus.


Moreover, the Ivannenko
textbook on gravitation i already cited in this newsgroup state the
same that i am saying: singularity at r = 2M is observer dependant.

There is no question but that that particular coordinate singularity is only present for coordinate systems related to the Schw. coordinates. That's why we call it a "coordinate singularity". <shrug>

Your notion of "observer dependence" is not very well defined,
and you have made some clear mistakes attempting to apply it;
coordinate dependence, on the other hand, is well defined.


Tom Roberts
.

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