Re: error in mathematics behind black holes

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. The singularity at r=0 is in the manifold[#]. We need
terminology to distinguish these two important cases, and "coordinate
singularity" is what we use.

[#] well, not really: one must omit this locus from the
manifold because the locus is singular. Note this is not
a single point, nor is it a timelike line; this singularity
is a 3d spacelike surface (look at a Kruskal diagram).

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. 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.

This interpretation is reinforced when it is commonly said that scalars
as R_abcdR^abcd remain finite when r = 2M and therefore the problem
"was" on the coordinate. However the transformation from singular
spacetimes to regular spacetimes is done via singular transformation
equations.

You are confused. It is not possible to "transform" the horizon r=2M to
or from Schw. coordinates, because those coordinates are not valid
there. There are two disjoint regions in which the Schw. coordinates are
valid: r>2M and 0<r<2M; they are not valid at r=2M.

You completely misinterpreted I said. I already said in previous posts
that Schwarzschild (exterior) solution is defined for r > 2M several
times. Above I was citing a common -but unaccurate- claim ***on
both relativistic and popular literature***.

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".

This point of view is also reinforced with the popular comparison with
polar coordinates and Earth maps being done by many relativists -see
comments below-. Some people think that they could study the region r
= 2M as Earth poles can be studied when changing coordinates. I read
often this kind of stuff in this newgroup too.

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. I state this many times and do not
understand why some of you try to find some error here.

The correct interpretation is that the concept of singularity is
observer dependant.

No. That is blatantly wrong. COORDINATE singularities are "observer
dependent", but real singularities are not. In Schw. spacetime, the
coordinate singularity at r=2M does not appear in Kruskal coordinates.
But the curvature singularity at r=0 does.

You are misreading again even if one asummes a universal concept of
singularity was accepted by all physicists and mathematicians [*].

If your 'coordinate' singularity r = 2M was detected by observers A, B
but was not detected by P, Q then the singularity is dependant of the
observer. I can state that in mathematical form like

singularity = f(observer)

with f some function. If the singularity at r = 0 were detected by all
observers, this does NOT invalidate above premise because then, f may
be some kind of _constant_ function; thus, changing the variable
'observer' does not change the result of the function: 'singularity'.
Everything of this iff asuming you completely right about
singularities.

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. This is closely related to Penrose's CCC for BH.

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. However, the singularity may be observed -if GR is right-
by a Kruskal observer beyond the horizon. One conclude that singularity
at r = 0 is also detected by some observers but is not for others.
Again there is not absolute singularities just relative ones.

It is interesting that Synge also had a similar thought about
singularities being observer dependant instead absolute [**].

For an observer in a Kruskal frame there is not
singularity at U = V = 0, but for a observer at exterior Schwarzschild
frame the singularity continues at r = 2M.

Usually when one says "singularity" without modification, one means a
real singularity, not a purely coordinate one like this. You have
confused yourself by using sloppy terminology.

First a remark. since I did explicit that I was writing about the
singularity at r = 2M, I do not see how any confusion with the
singularity at r = 0 could arise; what is more, I did explicit that I
was writing about the *exterior* Schwarzschild solution.

It may funny to believe that my terminology is "sloppy". The syntax is
formally correct

<singularity> ==> <'coordinate' singularity> | <'real' singularity>

Your convention is, hovewer, sintatically ambiguous

<singularity> ==> <coordinate singularity> | <singularity>

The problem is not just
mathematical, the problem is associated with the physical nature of the
r = 2M surface, light cannot cross the surface to the observer and
cannot send causal information to the observer.

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.

An observer on (r,t) 'sees' the singularity. It is not an artifact for
her/him. It is real and with physical interpretation.

It is only as real as is the coordinate singularity in polar coordinates
at r=0, or the coordinate singularity for longitude and latitude at the
north and south poles -- i.e. not "real" at all.

This very typical relativists' argument misses at least a point; one
physical. I can receive physical signals from anyone on a Earth pole, a
Schwarschild observer _cannot_ receive information from an observer on
r = 2M; that is independent of the system of coordinates being used by
the Schwarschild observer: Schwarzschild coordinates, Kruskal
coordinates, or any other coordinates.


-----

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

[**] The same Synge that advanced the basis for posterior work of
Kruskal and Szekeres on the extended Schw. 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.

.

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