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Misinterpretation of the radial parameter in the Schwarzschild solution?
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Phineas T Puddleduck
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PostPosted: Thu Jul 06, 2006 11:11 am    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

On 6/7/06 11:58, in article
1152183483.961504.63390@j8g2000cwa.googlegroups.com, "LEJ Brouwer"
<intuitionist1@yahoo.com> wrote:

Quote:
The moderators of s.p.research have decided that Steve Carlip's
response has settled this issue, and refuse to allow further
discussion. I disagree, so I am continuing the thread here, and would
like to invite Steve to respond if he wishes - Sabbir.
=============================================

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koobee.wublee@gmail.com
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PostPosted: Thu Jul 06, 2006 5:02 pm    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

"LEJ Brouwer" <intuitionist1@yahoo.com> wrote in message
news:1152183483.961504.63390@j8g2000cwa.googlegroups.com...

Quote:
The moderators of s.p.research have decided that Steve Carlip's
response has settled this issue, and refuse to allow further
discussion. I disagree, so I am continuing the thread here, and would
like to invite Steve to respond if he wishes - Sabbir.
=============================================

The following papers claim that
there is an error in the interpretation of the radial coordinate 'r' in
the standard Schwarzschild metric:

L. S. Abrams, "Black holes: The legacy of Hilbert's error", Can. J.
Phys. 23 (1923) 43, http://arxiv.org/abs/gr-qc/0102055

S. Antoci, "David Hilbert and the origin of the 'Schwarzschild
solution'", http://arxiv.org/abs/physics/0310104

S. J. Crothers, "On the general solution to Einstein's vacuum field and
its implications for relativistic degeneracy", Prog. Phys. 1 (2006)
68-73.

Einstein Field Equations are

R_ij - R g_ij / 2 = constant Lm g _ij

Where

** i, j have variations from 0 to 3 each.
** R_ij = Ricci tensor
** R = g^uv R_uv
** g_ij = metric = curvature correction factors
** Lm = Lagrangian of mass energy

Lm = constant rho

Where

** rho = mass density if inside an object
** rho = (- 1 / r^3) if in free space near an object

To make things much simpler, we allow only one spatial and one temporal
dimensions and also only diagonal metric. The field equations above
simplify to the following after many steps of still very tedious
caculations. Concentrating with solutions in free space and matching
with the Newtonian result, we have

- @^2g_00/@r^2 / (2 g_00 g_11) - @g_00/@r @(1/(g_00 g_11))/@r / 4 = 2 G
M / r^3 / c^2

Where

** @/@r = partial derivative operator with respect to r

Set the following.

g_00 g_11 = -1

Then, the equation becomes

@^2g_00/@r^2 = - 4 G M / r^3 / c^2

Solving that very simple differential equation above, we get

g_00 = k1 + k2 r - 2 G M / r / c^2

Where

** k1, k2 = integration constants

Applying the boundary condition at (g_00 = 1) where (r = infinity), you
get

g_00 = 1 - 2 G M / r / c^2

This means Hilbert is correct. However, if you decide that the
Newtonian gravitational law is only an approximation which it mostly
likely is anyway, the Poisson equation in free space has the following
form.

@^2g_00/@r^2 = - 4 G M / r^3 / c^2 + (smaller components)

Where

** 4 G M / r^3 / c^2 >> (smaller components)

This will open up an infinite number of solutions in which
Schwarzschild's original solution is just one of them. The
Schwarzschild metric today is valid only if we take the Newtonian
result exactly as it is.

An interesting scenario is when setting (g_11 = -1) or flat space, the
field equation becomes

@^2g_00/@r^2 + (@g_00/@r)^2 / (4 g_00) = - 2 G M g_00 / r^3 / c^2

To solve the differential equation above, it appears to be quite
challenging. I have a feeling that it would not even lead to anything
remotely logical in nature. I will attempt to solve it later on when I
go out jogging with my dog using Taylor series approximations. If you
are interested, I can post my finding.
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shuba
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PostPosted: Thu Jul 06, 2006 10:09 pm    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

Sabbir Rahman wrote:

Quote:
The moderators of s.p.research have decided that Steve Carlip's
response has settled this issue, and refuse to allow further
discussion. I disagree, so I am continuing the thread here, and would
like to invite Steve to respond if he wishes - Sabbir.

Ah, this is the same stunt Sabbir pulled in 2001, back before
he decided to use the handle of a dead mathematician. Usenet
theatre at its finest!

http://groups.google.com/group/sci.physics/msg/79a4b9362363fd93

Quote:
L. S. Abrams, "Black holes: The legacy of Hilbert's error", Can. J.
Phys. 23 (1923) 43, http://arxiv.org/abs/gr-qc/0102055

Judging from the citebase citations to that paper, this appears
to (once again) have everything to do with the discredited ideas
of a certain Abhas Mitra, just as in 2001. It's good this was posted,
as otherwise some people might have been led to believe that
Sabbir's purpose in bringing up the articles on s.p.research was
to solicit informed opinion.

Mitra has been ripped apart by Carlip, Hillman, Baez, and many
others. The monomaniacal posturing of Sabbir Rahman just
enhances the effect. In the words of Baez, in a recent (2004)
foray into this newsgroup (responding to Sabbir and Mitra),

"Dignity, eh? If I had any "dignity" I wouldn't be posting here."
http://groups.google.com/group/sci.physics.relativity/msg/ca9aef3604acf620


---Tim Shuba---
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Tom Roberts
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Posts: 1399

PostPosted: Fri Jul 07, 2006 3:21 am    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

LEJ Brouwer wrote:
Quote:
Have we really all been making this silly mathematical error, and is
our present understanding of the simplest classical black hole way off
the mark?

No. If what you (those papers) claim were true, than any pair of
timelike objects approaching r=2M from different directions would
approach arbitrarily close to each other. This does not happen, and the
distance between such infalling geodesics does not go to zero for all
possible pairs of geodesics.

More remarkably, for the approach to r=0 such objects can have an
arbitrarily large distance between them.

The whole confusion is about coordinates that behave badly at r=2M. The
Schwarzschild coordinates are INVALID there, and that destroys the whole
"argument". Use coordinate-independent notions (such as distance between
geodesics, curvature tensors, etc.) and the truth comes out.

Steve Carlip and the moderators got it right.


Tom Roberts
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LEJ Brouwer
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PostPosted: Fri Jul 07, 2006 4:35 am    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

shuba wrote:
Quote:
Sabbir Rahman wrote:

The moderators of s.p.research have decided that Steve Carlip's
response has settled this issue, and refuse to allow further
discussion. I disagree, so I am continuing the thread here, and would
like to invite Steve to respond if he wishes - Sabbir.

Ah, this is the same stunt Sabbir pulled in 2001, back before
he decided to use the handle of a dead mathematician. Usenet
theatre at its finest!

Indeed, a 'stunt' which made pretty clear to all the double standards
of the moderators of that group.

Quote:
http://groups.google.com/group/sci.physics/msg/79a4b9362363fd93

L. S. Abrams, "Black holes: The legacy of Hilbert's error", Can. J.
Phys. 23 (1923) 43, http://arxiv.org/abs/gr-qc/0102055

Judging from the citebase citations to that paper, this appears
to (once again) have everything to do with the discredited ideas
of a certain Abhas Mitra, just as in 2001. It's good this was posted,
as otherwise some people might have been led to believe that
Sabbir's purpose in bringing up the articles on s.p.research was
to solicit informed opinion.

Mitra has been ripped apart by Carlip, Hillman, Baez, and many
others. The monomaniacal posturing of Sabbir Rahman just
enhances the effect. In the words of Baez, in a recent (2004)
foray into this newsgroup (responding to Sabbir and Mitra),

"Dignity, eh? If I had any "dignity" I wouldn't be posting here."
http://groups.google.com/group/sci.physics.relativity/msg/ca9aef3604acf620


---Tim Shuba---

Tim,

It is true that Abhas was subjected to an unreasonable amount of abuse
from a couple of the individuals you mention (Steve Carlip was not one
of them), but he certainly did not lose the technical battle, and
responded satisfactorily to all objections made. Maybe you did not have
the patience to read the entire thread?

As for John Baez's lame effort, his "proof" of Mitra's error amounted
to the dubious claim that lim f(x)/g(x) -> Infty if g(x) -> 0,
carelessly overlooking the fact that f(x) -> 0 in the same limit. This
was also the reason for the subsequent thread entitled "John Baez
teaches calculus".

As for Abhas's supposedly "discredited ideas", his work has being
published in several established journals, and has most recently been
accepted by Phys Rev D and MNRAS:

http://arxiv.org/find/grp_q-bio,grp_cs,grp_physics,grp_math,grp_nlin/1/all:+AND+mitra+abhas/0/1/0/all/0/1

He is currently visiting the Max Planck Institute in Heidelberg.

I find it curious that you feel justified in judging the technical
merits of a paper on the arxiv by the number of citations alone. This
is remarkably similar to the method of Hillman and Baez, who also have
the curious habit of reading a paper only *after* they have ripped its
contents apart. So might I suggest that you also try reading Abrams
before passing judgment?

Note that Abrams original paper actually predates Mitra's work by many
years (the correct journal reference is actually Can. J. Phys. 67
(1989) 919 - apologies for the incorrect reference given above).

- Sabbir.
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LEJ Brouwer
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PostPosted: Fri Jul 07, 2006 4:49 am    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

Tom Roberts wrote:
Quote:
LEJ Brouwer wrote:
Have we really all been making this silly mathematical error, and is
our present understanding of the simplest classical black hole way off
the mark?

No. If what you (those papers) claim were true, than any pair of
timelike objects approaching r=2M from different directions would
approach arbitrarily close to each other. This does not happen, and the
distance between such infalling geodesics does not go to zero for all
possible pairs of geodesics.

No. I think we both agreed that this was an unusual situation where
apparently the event horizon has finite area (hence the separated
geodesics) but zero radial distance from the pointlike mass. Note that
we are dealing with a non-Euclidean metric here, so the radial distance
from the origin and the radius associated with the square root of the
area need not be the same. Note also that the metric becomes
ill-defined in this limit (the EH forms at t=Infty according to an
external observer), and that is why I proposed the existence of a phase
transition occuring at the formation of the event horizon, so that this
singular limit is not precisely realised.

Quote:

More remarkably, for the approach to r=0 such objects can have an
arbitrarily large distance between them.

It is not remarkable because there is no interior solution for r<2m.

Quote:
The whole confusion is about coordinates that behave badly at r=2M. The
Schwarzschild coordinates are INVALID there, and that destroys the whole
"argument". Use coordinate-independent notions (such as distance between
geodesics, curvature tensors, etc.) and the truth comes out.

No it's not. The confusion is due to the use of radial parameters
outside of their physically valid range.

Quote:
Steve Carlip and the moderators got it right.

No they didn't.

Quote:
Tom Roberts

- Sabbir.
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Ben Rudiak-Gould
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Posts: 382

PostPosted: Fri Jul 07, 2006 9:18 pm    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

LEJ Brouwer wrote:
Quote:
I think we both agreed that this was an unusual situation where
apparently the event horizon has finite area (hence the separated
geodesics) but zero radial distance from the pointlike mass.

The radial "distance" from the center is timelike, which is not the same as
being zero.

Do you agree that the event horizon is at r = 2m (not r = 0) in
Schwarzschild coordinates? I.e. is this whole debate over how one ought to
define "radius"?

Quote:
Note also that the metric becomes ill-defined in this limit (the EH forms
at t=Infty according to an external observer),

Event horizons don't form. They aren't dynamical objects. They can't be
defined in terms of any local property of the geometry or the metric.

-- Ben
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LEJ Brouwer
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PostPosted: Sat Jul 08, 2006 11:24 am    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

Ben Rudiak-Gould wrote:
Quote:
LEJ Brouwer wrote:
I think we both agreed that this was an unusual situation where
apparently the event horizon has finite area (hence the separated
geodesics) but zero radial distance from the pointlike mass.

The radial "distance" from the center is timelike, which is not the same as
being zero.

Do you agree that the event horizon is at r = 2m (not r = 0) in
Schwarzschild coordinates? I.e. is this whole debate over how one ought to
define "radius"?

Yes, the event horizon is at r=2m in Schwarzschild coordinates, and
this coincides with the position of the mass, i.e. r=2m _is_ the
origin/centre, not r=0. Values of r<2m are physically meaningless, and
there is no 'interior' solution at all.

Quote:

Note also that the metric becomes ill-defined in this limit (the EH forms
at t=Infty according to an external observer),

Event horizons don't form. They aren't dynamical objects. They can't be
defined in terms of any local property of the geometry or the metric.

Not so - the norm of the 4-acceleration on a test particle is a local,
invariant, intrinsic quantity that diverges on the event horizon. See
sections 3 and 4 of,

Antoci and Liebscher, "Reinstating Schwarzschild's original manifold
and its singularity", http://arxiv.org/abs/gr-qc/0406090

Quote:
-- Ben

- Sabbir
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Tom Roberts
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PostPosted: Sat Jul 08, 2006 7:01 pm    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

LEJ Brouwer wrote:
Quote:
Yes, the event horizon is at r=2m in Schwarzschild coordinates, and
this coincides with the position of the mass, i.e. r=2m _is_ the
origin/centre, not r=0. Values of r<2m are physically meaningless, and
there is no 'interior' solution at all.

You are confusing your coordinate chart with the manifold itself. Yes,
indeed, the exterior Schw. coordinates are invalid for r<2M.

But as is well known, coordinates are human artifacts and do not
determine the geometry of the manifold (or any other properties).
Transform to other coordinates, such as Kruskal coordinates, and it
becomes QUITE CLEAR that the manifold does not "end" at r=2M as you
claim. In particular, there are infalling timelike geodesic paths that
go right through the surface r=2M and continue all the way to a limit
point at r=0. Moreover, Ricci=0 at every point of every such geodesic
(i.e. there is no energy density anywhere for r>0).

Note that the locus r=0 must be deleted from the manifold. Note I did
not say "point", as this is most definitely not a point. While one might
naively expect it to be a 1-d timelike locus (all possible values of t),
in fact it is a 2-d spacelike locus and that disconnect is at base why
it must be deleted from the manifold (please do remember that for r<2M
the Schw. coordinate t is spacelike).


Quote:
Not so - the norm of the 4-acceleration on a test particle is a local,
invariant, intrinsic quantity that diverges on the event horizon.

That is plain and simply not true. An infalling timelike geodesic path
can go right through the horizon, and it has zero 4-acceleration
everywhere (because it is a timelike geodesic).

The abstract of gr-qc/0406090 discusses a test particle at rest relative
to the black hole, and points out that such worldlines have divergent
4-accelerations for positions approaching the horizon. I suspect you
think that is universal. Not so.

[That paper considers this a "defect", but in fact it is
quite reasonable and well known.]


You, and the papers you quote, have very serious misunderstandings about
the basics of GR.


Quote:
See
sections 3 and 4 of,
Antoci and Liebscher, "Reinstating Schwarzschild's original manifold
and its singularity", http://arxiv.org/abs/gr-qc/0406090

I'll look at it when I get a chance....


Tom Roberts
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LEJ Brouwer
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PostPosted: Sat Jul 08, 2006 8:50 pm    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

Tom Roberts wrote:
Quote:
LEJ Brouwer wrote:
Yes, the event horizon is at r=2m in Schwarzschild coordinates, and
this coincides with the position of the mass, i.e. r=2m _is_ the
origin/centre, not r=0. Values of r<2m are physically meaningless, and
there is no 'interior' solution at all.

You are confusing your coordinate chart with the manifold itself. Yes,
indeed, the exterior Schw. coordinates are invalid for r<2M.

But as is well known, coordinates are human artifacts and do not
determine the geometry of the manifold (or any other properties).
Transform to other coordinates, such as Kruskal coordinates, and it
becomes QUITE CLEAR that the manifold does not "end" at r=2M as you
claim. In particular, there are infalling timelike geodesic paths that
go right through the surface r=2M and continue all the way to a limit
point at r=0. Moreover, Ricci=0 at every point of every such geodesic
(i.e. there is no energy density anywhere for r>0).

Tom, I don't claim that the manifold ends at r=2M. Rather, I agree with
you totally and accept the Kruskal extension and that the exterior
solution is double-sheeted. I actually DISAGREE with Abrams et alii on
this point. I think they made a mistake. However, they are _correct_
that the the coordinates r<2M are invalid (note that no point on the
exterior Kruskal-extended solution has r<2M). It is a shame that there
are mistakes in the works I refer to (it seems that no work on this
surprisingly tricky subject has been perfect), just as there have been
clear mistakes in Hilbert's work and those who followed him. Antoci's
paper on Hilbert's derivation makes this point very clearly.

[Note that the very fact that the Schwarzschild solution had to be
extended by Kruskal and Szekeres means that the Schwarzschild/Hilbert
solution is *not* the most general spherically symmetric solution to
the problem - the reason that _this_ mistake was made was because of
the prior assumption, which is subtly hidden in the way that the
problem is usually formulated, and which again depends upon the
misinterpretation of r as the radial coordinate as opposed to an
apriori unknown parametristation of the radius, that there can only be
single-sheeted solution. It was not until Synge's derivation in 1974,
that the most general solution (i.e. the Kruskal-extened one) was
derived from first principles.]

As in all matters, we should be objective about this, and take what is
right and leave what is wrong. Until we are willing to do this and
accept that some of the things we have 'learnt' in the past may in fact
be mistaken, we are going to have great difficulty making progress in
our understanding.

Quote:

Note that the locus r=0 must be deleted from the manifold. Note I did
not say "point", as this is most definitely not a point. While one might
naively expect it to be a 1-d timelike locus (all possible values of t),
in fact it is a 2-d spacelike locus and that disconnect is at base why
it must be deleted from the manifold (please do remember that for r<2M
the Schw. coordinate t is spacelike).

Yes, r=0 is a singularity and is physically unreasonable. Indeed
pointlike masses themselves are physically unreasonable. In a more
realistic scenario of gravitational collapse we will have, say, a
spherical distribution of matter collapsing until an event horizon is
formed at time t=Infty. To an external observer (though I guess there
won't be any around to watch at t=Infty), all of the mass contained
within the event horizon will appear to be contained at a point,
admittedly with apparently finite area. My proposal as to what must
happen to explain the lack of metric/curvature singularities at the
event horizon is that a topological phase transition occurs as a result
of which the mass contained within the event horizon becomes physically
separated from the exterior solution (including its Kruskal extension).

The only consistent picture that comes to mind is that any particles
falling in radially towards the event horizon will not experience any
singularity, but will reappear apparently travelling radially outwards
and backwards in time on the other sheet to an external observer. An
infalling particle will actually look like a particle-antiparticle
annihilation event, though the infalling particle itself will be
totally oblivious of how its trajectory appears to external observers.
As for the mass in the interior of the event horizon, it will not
actually be concentrated at a point but will occupy a finite volume
within which it will continue to gravitate normally, as an isolated
system from the exterior manifold. [This interior manifold should also
be time non-orientable like the exterior manifold.]

Quote:
Not so - the norm of the 4-acceleration on a test particle is a local,
invariant, intrinsic quantity that diverges on the event horizon.

That is plain and simply not true. An infalling timelike geodesic path
can go right through the horizon, and it has zero 4-acceleration
everywhere (because it is a timelike geodesic).

How can spacelike and timelike directions suddenly swap places if there
is nothing unusual going on at the event horizon? This is completely
unphysically, yet is accepted in a completely matter-of-fact way in all
standard textbooks. Anyone considering this objectively should
immediately smell a rat here.

Quote:
The abstract of gr-qc/0406090 discusses a test particle at rest relative
to the black hole, and points out that such worldlines have divergent
4-accelerations for positions approaching the horizon. I suspect you
think that is universal. Not so.

[That paper considers this a "defect", but in fact it is
quite reasonable and well known.]


You, and the papers you quote, have very serious misunderstandings about
the basics of GR.

I will need to take a look at the paper again as it is a while since I
last read it, but even if I were to concede this last point, I will not
concede the main point I am making which is the fact that r<2M has no
meaning for the Schwarzschild coordinates, so that the usual interior
solutions are invalid.

I am not certainly not God (wa aoudhu bika min dhalik), and make no
personal claims to infallibility, but the mistake that Hilbert made,
which everyone after him followed, seems *very* clear, and so one could
say that Hilbert and all of his followers also have 'serious
misunderstandings about the basics of GR'. But I am not going to say
that because 'to err is human', and sometimes simple mistakes do
propagate for a long time without being noticed. There is a problem
here which needs to be addressed - petty name-calling and pretending
that nothing is wrong is not going to change that.

Quote:
See
sections 3 and 4 of,
Antoci and Liebscher, "Reinstating Schwarzschild's original manifold
and its singularity", http://arxiv.org/abs/gr-qc/0406090

I'll look at it when I get a chance....

Okay, I appreciate it.

Quote:
Tom Roberts

Best wishes,

Sabbir.
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Jan Bielawski
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PostPosted: Sun Jul 09, 2006 4:25 am    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

LEJ Brouwer wrote:
Quote:

[Note that the very fact that the Schwarzschild solution had to be
extended by Kruskal and Szekeres means that the Schwarzschild/Hilbert
solution is *not* the most general spherically symmetric solution to
the problem - the reason that _this_ mistake was made

It's not a mistake, Schwarzschild's coordinate system is simply a chart
covering a subset covered by Kruskal and Szekeres. One counterintuitive
aspect of the Schwarzschild is that what appears as a "line" r=2M is
really a "point" in 1+1 (really a 2-sphere, or the Einstein-Rosen
"neck") in a manner similar to the Mercator representation of earth's
north pole (say) as a line. On top of that, Schwarzschild omits the
most important portion of the horizon through which matter and light
signals fall(!) So these are very counterintuitive coordinates.

Quote:
was because of
the prior assumption, which is subtly hidden in the way that the
problem is usually formulated, and which again depends upon the
misinterpretation of r as the radial coordinate as opposed to an
apriori unknown parametristation of the radius,

I'm not sure why you keep repeating that - the usual derivation only
assumes the radial coordinate is related to the sphere surface area via
the usual 4 pi r^2. It's just a matheatical assumption, nobody ever
claimed this coordinate was supposed to have a meaning besides that.

--
Jan Bielawski
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LEJ Brouwer
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PostPosted: Sun Jul 09, 2006 5:38 am    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

JanPB wrote:
Quote:
LEJ Brouwer wrote:

[Note that the very fact that the Schwarzschild solution had to be
extended by Kruskal and Szekeres means that the Schwarzschild/Hilbert
solution is *not* the most general spherically symmetric solution to
the problem - the reason that _this_ mistake was made

It's not a mistake, Schwarzschild's coordinate system is simply a chart
covering a subset covered by Kruskal and Szekeres. One counterintuitive
aspect of the Schwarzschild is that what appears as a "line" r=2M is
really a "point" in 1+1 (really a 2-sphere, or the Einstein-Rosen
"neck") in a manner similar to the Mercator representation of earth's
north pole (say) as a line. On top of that, Schwarzschild omits the
most important portion of the horizon through which matter and light
signals fall(!) So these are very counterintuitive coordinates.

One of the points that I have been trying to make is that matter and
light do not fall through the event horizon, because there is (a) no
interior solution since r<2M is not allowed, and (b) timelike and
spacelike directions cannot just swap places, which is what would have
to happen if matter were to fall past the horizon. In any case, it is
invalid to apply a singular transformation to a singular coordinate
choice to produce a 'nonsingular' coordinate system. As I have already
said, that is like dividing infinity by infinity and believing the
answer.

Quote:

was because of
the prior assumption, which is subtly hidden in the way that the
problem is usually formulated, and which again depends upon the
misinterpretation of r as the radial coordinate as opposed to an
apriori unknown parametristation of the radius,

I'm not sure why you keep repeating that - the usual derivation only
assumes the radial coordinate is related to the sphere surface area via
the usual 4 pi r^2. It's just a matheatical assumption, nobody ever
claimed this coordinate was supposed to have a meaning besides that.

The use of the metric in the form ds^2 = A(r) dt^2 - B(r) dr^2 - r^2
dA^2 contains the (incorrect) hidden assumption that 0<r<Infty, and
that there is a one-to-one mapping between values of r and 2-spheres.
This immediately disallows solutions like that of Kruskal where the
solution manifold 'folds back' on itself, so that there is more than
one 2-sphere for a given r. And that is why the standard solution (i.e.
the Schwarzschild solution) needs to be analytically extended (though
the process by which this is done, as I have hinted at is rather
dubious). A correct definition of a spherically symmetric metric was
given by Synge in 1974, and as a result he was able to derive the
Kruskal-extended solution directly (though he still misinterprets the
radial coordinates):

J. L. Synge, "Model universes with spherical symmetry", Ann. di Mat.
Pura ed App., 98 (1974) 239-255.

Quote:
--
Jan Bielawski

- Sabbir.
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Jan Bielawski
science forum Guru


Joined: 08 May 2005
Posts: 388

PostPosted: Sun Jul 09, 2006 6:24 pm    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

LEJ Brouwer wrote:
Quote:
JanPB wrote:

It's not a mistake, Schwarzschild's coordinate system is simply a chart
covering a subset covered by Kruskal and Szekeres. One counterintuitive
aspect of the Schwarzschild is that what appears as a "line" r=2M is
really a "point" in 1+1 (really a 2-sphere, or the Einstein-Rosen
"neck") in a manner similar to the Mercator representation of earth's
north pole (say) as a line. On top of that, Schwarzschild omits the
most important portion of the horizon through which matter and light
signals fall(!) So these are very counterintuitive coordinates.

One of the points that I have been trying to make is that matter and
light do not fall through the event horizon, because there is (a) no
interior solution since r<2M is not allowed,

What do you mean "not allowed"? True, the metric for r<2M does not
satisfy the constraint you set up for yourself (namely, it's not
static) and it may contain other hidden assumptions (like r<infty) but
so what? It's still a solution to Einstein's equation and the
requirement for being static, etc., was just a simplifying (i.e., ad
hoc, random, man-made) assumption which in the end just turned out to
be an overkill. So you ended up with a more general solution covering a
larger region (the extra region - the interior - is disconnected, in
the topological sense, from the static exterior).

Quote:
and (b) timelike and
spacelike directions cannot just swap places, which is what would have
to happen if matter were to fall past the horizon.

This is a mathematical feature of the Schwarzschild representation of
the metric. Nothing really swaps places, it's just that the
mathematical process of solving the relevant differential equations
mirrors one another over the two components of the disconnected domain
so the *letters* r and t end up denoting different things for r<2M
and r>2M due to what is basically an abuse of notation. One reason for
the confusion is that this sort of "switch" in the signature is rarely
seen elsewhere so people tend to see something physical going on here.
The domain of this chart is topologically disconnected and the equation
describing a particular representation of the metric over such domain
is free to switch the letters denoting the coordinates over each
connected component as it pleases, without violating continuity. If it
bothers you, just switch the letters r and t in the interior
Schwarzschild formula. (It would be nice if GR textbooks did it, it
also would be nice if pop-sci books stopped incessantly harping on this
nonexistent "switch" - it's about as bad as the dreadful rubber sheet
so-called model.)

Quote:
In any case, it is
invalid to apply a singular transformation to a singular coordinate
choice to produce a 'nonsingular' coordinate system.

Why? Imagine your manifold is the real line minus the origin (zero).
You have 1/x there as a coordinate chart. It's singular at 0. Someone
then comes in and says: "OK, here I have the entire real line
(including zero) with the identity chart. Now if I temporarily ignore
the zero in my manifold, then it will be diffeomorphic to yours - in
fact 1/x is the diffeomorphism. But my manifold has another chart - the
identity - which covers the whole thing! So here I applied a singular
transformation to a singular coordinate system to produce a nonsingular
one."
_This is differential geometry 101_

Quote:
As I have already
said, that is like dividing infinity by infinity and believing the
answer.

No, these guys who wrote those papers need to study the basics more.
Spivak is back in print in a nicely TeX-reset 3rd edition so there is
no excuse not to read it (first two volumes)! This is all really basic
and must be clearly understood before climbing that GR mountain.

Quote:
was because of
the prior assumption, which is subtly hidden in the way that the
problem is usually formulated, and which again depends upon the
misinterpretation of r as the radial coordinate as opposed to an
apriori unknown parametristation of the radius,

I'm not sure why you keep repeating that - the usual derivation only
assumes the radial coordinate is related to the sphere surface area via
the usual 4 pi r^2. It's just a matheatical assumption, nobody ever
claimed this coordinate was supposed to have a meaning besides that.

The use of the metric in the form ds^2 = A(r) dt^2 - B(r) dr^2 - r^2
dA^2 contains the (incorrect) hidden assumption that 0<r<Infty, and
that there is a one-to-one mapping between values of r and 2-spheres.
This immediately disallows solutions like that of Kruskal where the
solution manifold 'folds back' on itself, so that there is more than
one 2-sphere for a given r.

Yes, so what? You've constrained yourself to seek only solutions of
certain type, over certain domain, that's what you got (more or less,
you actually got a bit more: the non-static interior). This is not
"incorrect", it's a man-made self-limitation and it resulted in a
coordinate chart valid over a domain which - it turns out - is a subset
of some other domain over which another coordinate representation *of
the same metric* is valid.
This is again the basics - see Spivak volume 1.

Quote:
And that is why the standard solution (i.e.
the Schwarzschild solution) needs to be analytically extended (though
the process by which this is done, as I have hinted at is rather
dubious).

No, it's logically and mathematically very plain and boring. Even to
call analytical extension a "process" is a bit of a stretch because it
suggests some definite procedure is applied whose validity might then
be questioned. In fact one merely gets hold of *another* solution
(another coordinate representation of the metric) and then notices that
its domain is larger than the original domain. Such new solution is
then referred to as an "extension", as if some sort of action of
"extending" has taken place. This is just reification, scientists like
to talk like that.

Quote:
A correct definition of a spherically symmetric metric was
given by Synge in 1974, and as a result he was able to derive the
Kruskal-extended solution directly (though he still misinterprets the
radial coordinates):

J. L. Synge, "Model universes with spherical symmetry", Ann. di Mat.
Pura ed App., 98 (1974) 239-255.

That's all fine. It just doesn't mean what you seem to think it means.
It's all far less sinister :-)

--
Jan Bielawski
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LEJ Brouwer
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Joined: 07 May 2005
Posts: 120

PostPosted: Mon Jul 10, 2006 1:14 am    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

The following is a response to tessel's posting to the thread now
called "Flogging the Xprint" on sci.physics.research:

tessel@um.bot wrote:
Quote:
Sabbir Rahman ("LEJ Brouwer") asked about a group of arXiv eprints which
(as I think he knows) have previously been castigated in this newsgroup
for committing elementary student errors.

I referenced preprints by Abrams, Antoci and Crothers. I did a search
in sci.physics.research for each of these authors, and none of their
papers have been discussed at all, let alone been 'castigated'. Given
that your first statement is blatantly untrue, this does not leave much
hope for the objectivity of the rest of your post.

Quote:
Steve Carlip briefly explained the most fundamental of these errors, but
unfortunately the comment by Mark Hopkins muddies the waters by going off
in a different direction (and by stating as established "truth" a
controversial assertion).

I have responded to these assertions on sci.physics.relativity. The
moderators can explain why I am not being allowed to reply on
s.p.research.

Quote:
I just want to make sure that everyone understands the only really
important point here: the papers Rahman mentioned belong to a group of
papers which are founded upon serious but elementary misconceptions. I
call these "Xprints" in homage to a Monty Python skit.

Yes, "bicycle repair man" was one of my favourites.

Quote:
No bits were actually harmed in producing this post :-/

=== The papers ===

It turns out there is a sizable group of arXiv eprints

http://arxiv.org/find/gr-qc/1/au:+Antoci_S/0/1/0/all/0/1
http://www.arxiv.org/find/gr-qc/1/au:+Liebscher_D/0/1/0/all/0/1
http://arxiv.org/find/gr-qc/1/au:+Mihich_L/0/1/0/all/0/1
http://arxiv.org/find/grp_physics/1/au:+Loinger_A/0/1/0/all/0/1
http://arxiv.org/find/grp_physics/1/au:+Abbassi_A_H/0/1/0/all/0/1

http://arxiv.org/find/grp_physics/1/au:+Zakir_Zahid/0/1/0/all/0/1
(I now see that his 1999 eprints have been withdrawn)

http://arxiv.org/find/grp_physics/1/au:+Abrams_L/0/1/0/all/0/1
http://arxiv.org/find/grp_physics/1/au:+Mitra_Abhas/0/1/0/all/0/1

and even published papers, including several by Leonard S. Abrams and
Stephen J. Crothers, which

Well, I am referring to specific papers by Abrams, Antoci and Crothers,
so I am not sure why you are dragging a whole bunch of other authors
into the discussion.

Quote:
1. claim that all the standard gtr textbooks misrepresent and misinterpret
the Schwarzschild vacuum solution, following a "mistake" they generally
attribute to Hilbert 1917,

2. repeat a number of -elementary- mistakes, all involving fundamental
misconceptions concerning the roles played by atlases, pullbacks, and
diffeomorphisms in manifold theory, and consequently,

3. confuse the concept of coordinate singularity with the concept of a
singularity in some physical field (or in a geometric quantity such as the
metric or curvature tensor).

These papers often use absurdly overcomplicated notation which can obscure
the elementary nature of their most fundamental errors. (Some of these
papers also commit more sophisticated errors--- but here, I just want to
make sure everyone understands the -simple- stuff!)

So you have just made a long list of claims about these papers, without
proving any of them claims. Shall we just assume that your claims
represent incontrovertible facts, as if handed down from God, or are
you actually going to justify your claims? Also, I don't know about the
other authors you reference, but the papers I referred to seem pretty
clear in terms of notation used.

Quote:
One of the above cited authors, Salvatore Antoci, has also translated some
relevant historical papers into English and posted them to the arXiv
(unfortunately adding misleading "editorial comments"):

Unfortunately you omit to mention which 'editorial comments' you find
to be misleading or why.

Quote:
Karl Schwarzschild,
On the gravitational field of a mass point according to Einstein's theory
http://www.arxiv.org/abs/physics/9905030

Karl Schwarzschild,
On the gravitational field of a sphere of incompressible fluid according
to Einstein's theory
http://www.arxiv.org/abs/physics/9912033

Marcel Brillouin,
The singular points of Einstein's Universe,
http://www.arxiv.org/abs/physics/0002009

=== The problems ===

Recall that in the now standard "Scharzschild exterior chart" for the
Schwarzschild vacuum, the line element expressing the Schwarzschild vacuum
solution takes the form

ds^2 = -(1-2m/r) dt^2 + 1/(1-2m/r) dr^2 + r^2 dOmega^2,

2m < r < infty, 0 < theta < pi, -pi < phi < pi

I have often argued that physicists must learn the habit of -always-
stating coordinate ranges, since -failing- to do so can easily mislead
students, while -following- this precept can prevent embarrassing errors.

Well, essentially that's what I have been arguing all along.

Quote:
For example, in (2) of gr-qc/0310104, the stated range 0 < r < infty is
-incorrect- because of the coordinate singularity at r=2m. In fact, the
range 0 < r < 2m gives Schwarschild coordinates for a perfectly valid
-interior- patch, but this patch is -disjoint- from the -exterior- patch
above!

I see, in one breath you mention that the solution is only valid for 2m
< r < infty, and in the next breath you say that, 'in fact' it is valid
for 0 < r < 2m as well. Talk about inconsistent logic. Using the words
'in fact' prior to making a claim does not constitute a proof of that
claim.

Quote:
This is not just a quibble because this oversight is in fact an
integral part of a mistake made in this paper, as we shall see.

The only oversight here is your own.

In fact, the range 0 < r < 2m gives coordinates for a patch that does
not physically exist. Since you haven't bothered reading the papers I
mentioned to find out why, and are intent on misleading everyone else
who hasn't, I shall explain once again why 0 < r < 2m does not
correspond to any physical region of spacetime.

Starting with a general spherically symmetric metric:

ds^2 = A(r)dt^2 - B(r)dr^2 - C(r) dOmega^2, (eqn 3 of Crothers)

where A,B,C>0, and applying the same transformation r* = sqrt{C} used
by Hilbert (we write r* instead of r to avoid confusion with r above),
the general solution is,

ds^2 = (1 - 2m/sqrt{C}) dt^2 - (sqrt{C} / (sqrt{C} - 2m)) (C'^2/4C)
dr^2 - C dOmega^2 (eqn 7 of Crothers).

Now, if the coordinate associated with the point mass is r=sqrt{C}=r0
(which may be chosen such that r0=0), then we can define the 'proper
radius' from the mass at r0 to any r>r0 to be the integral from ro to r
of sqrt{-grr}:,

R = int sqrt{-grr} dr from r0 to r (eqn 11 of Crothers)

Given the boundary condition that R->0 as r->r0, this implies the
condition C(r0)=4m^2, and that R is given by:

R(r) = sqrt{sqrt{C}*(sqrt{C} - 2m)} + 2m ln
|(sqrt{sqrt{C}}+sqrt{(sqrt{C} - 2m)} / sqrt{2m}| (eqn 14 of Crothers)

where we must have C(r0)=4m^2, i.e. r*(r0) = 2m, so that the radial
position of the point mass in the Schwarzschild solution must be at r*
= 2m (which also coincides with the event horizon). Hilbert's error was
to confuse r* with r.

One cannot consider radii with r* < 2m, because these radii make no
physical sense, as they would have to have a smaller radius than radius
corresponding to the position of the point mass which defines the
origin of our space.

It's as simple as that. So there is NO solution for r* < 2m, because r*
< 2m DOES NOT PHYSICALLY EXIST.

If you do not understand this proof, then I suggest you read through
Crother's paper again, bearing in mind Antoci's clear explanation of
the incorrect assumption Hilbert made on setting r = sqrt{C}.

Quote:
Recall also that the "Schwarzschild radial coordinate" r has a simple
geometric interpretation, which was apparently first pointed out by
Hilbert 1917, namely: the locus t=t0, r=r0 is a geometric sphere with line
element

dsigma^2 = r0^2 (dtheta^2 + sin(theta)^2 dphi^2)

and with surface area 4 pi r0^2.

That's fine, but it does not affect the 'proper radius' argument
outlined above.

Quote:
Now, in his 1923 paper (posted to the arXiv by Antoci), Brillouin makes
the simple substitution

R = r-2m,

2m < r < infty

This defines a diffeomorphism (2m,infty) -> (0,infty), i.e. a coordinate
transformation. This substitution brings the line element into the form

ds^2 = -1/(1+2m/R) dt^2 + (1+2m/R) dR^2 + (R+2m)^2 dOmega^2,

0 < R < infty, 0 < theta < pi, -pi < phi < phi

(In modern language, we have pulled back the metric tensor under our
diffeomorphism.) This is a new coordinate chart covering precisely the
same region as the old chart. Geoemetrically speaking, this expression
gives the -same- metric tensor (placed on some underlying smooth manifold,
if you like) as before--- it has just been expressed in a new chart.
Actually, this new chart doesn't really improve anything, because it also
has a coordinate singularity at the event horizon; it simply relabels the
locus r=2m as the locus R=0. Unfortunately, Brillouin failed to recognize
this.

As I already mentioned in response to Steve Carlip, it does improve
things in the sense that we are less likely to assume that -2m < r < 0
is physical, when it is not. Hilbert and indeed pretty much everyone
else made the incorrect association of r with the proper radius, and
this choice of coordinates makes such an error less easy to make as we
are unlikely to associate -2m < r < 0 with the proper radius.

Quote:
In addition, Brillouin incorrectly used the range 0 < r < infty in the
original chart,

And so did Hilbert.

Quote:
and he then became confused by the fact that the
coordinate vector field @/@t is timelike "outside" and spacelike "inside"
the horizon, while @/@r is spacelike "outside" but timelike "inside".

He was absolutely correct to be confused. The confusion disappears when
we realise that 0 < r < 2m is physically invalid.

Quote:
This
observation, which unfortunately is often repeated verbatim by modern
physicists--- who ought to know better--- doesn't make sense as stated
since these are in fact -two disjoint charts- neither of which can be
extended as they stand through the event horizon (to do that you have to
adopt a more suitable chart, such as the charts introduced by Painleve
1921, Eddington 1922, or LeMaitre 1933, all of which are well defined at
r=2m, and in fact -overlap- both the interior and exterior regions, which
is: they can be used to -extend- from the exterior to the interior, or
vice versa).

Of course this is all rubbish as we have just shown that there is no
'interior region'

Quote:
Brillouin felt that this means that the event horizon is some kind of
"impassable physical barrier". But of course, according to gtr this is
wrong, as all the modern textbooks explain.

You mean, as all the modern textbooks 'fudge'. Brillouin was absolutely
right for the reasons mentioned above.

Quote:
(The notion of frame fields, a way of "decorating" the Lorentzian manifold
by a structure which can be drawn, and which also has an immediate
physical meaning in terms of the physical experience of some class of
observers, renders such issues transparent. Due to lack of time, this
important notion is rarely taught in introductory gtr courses, but see the
new book by Eric Poisson.)

Even worse, Brillouin apparently thought that the locus t=t0, R=0 is a
geometric -point-. In fact, it is a -sphere- with surface area A = 8m^2,
as is carefully explained in standard textbooks like Misner, Thorne, &
Wheeler, Gravitation, 1973. Several contemporary authors have repeated
this second error of Brillouin, some even arguing that since "the point"
R=0 "has no interior" (sic), the interior region must not exist!

Adding an exclamation mark does not make the statement any less true.
Brillouin was correct because the locus is at proper radius (as defined
above, or according to any other sensible measure of radial distance
from the origin) is zero. It is also true that the locus has finite
area, and this apparent inconsistency is a reflection of the
singularity in the metric at that point.

Quote:
The papers by Antoci et al., Loinger, and Abrams all use a much more
confusing notation but make essentially the same basic errors, although
they are a bit harder to spot.

That's complete rubbish.

Quote:
These authors make much of the fact that Schwarzschild's original paper
used a more general coordinate chart than is found in modern textbooks.
The implicit argument seems to be that since these physics textbooks
oversimplify the historical details, they must be "lying" about the
physics too! ;-/

So you say that the papers accuse the textbooks of 'lying'? Where? In
any case, Crothers lists a set of solutions which is even more general
than Schwarzschild's - see eqn 17 of his paper, and following that
summarises very neatly how earlier descriptions correspond to varous
parameter choices. I think that the fact that he was able to do this is
highly praiseworthy.

Quote:
To obtain this more general chart from the now standard Schwarzschild
exterior chart, put

rho = (a^3 + r^3)^(1/3)


I think you mean rho = (a^3 + r^3)^(2/3) here?

Quote:
where a is a second parameter (the first being the mass parameter m).

a=2m by definition. a is not a 'second parameter'. Unless you mean a is
the same as my r0 above - in which case the above equation is wrong.
Reference please?

Quote:
It
should be immmediately apparent that this second parameter merely -adjusts
the radial coordinate-, but does not affect any physics. (To be truly
fussy I should write "rho_a" since changing "a" gives a -different- radial
coordinate--- but not, of course, a different manifold!)

I have lost you now - what work are you referring to, or are you just
making this up as you go along?

Quote:
But in gr-qc/0102084, Antoci et al. incorrectly state (I have slightly
changed the notation to agree with that used here): "for different values
of a... the solutions are geometrically and physically different". This
claim exhibits the same fundamental misconception about diffeomorphisms
and coordinate charts in manifold theory as Brillouin 1923.

Although I can't find the statement in the paper you refer to, the
statement is correct as 'a' is proportional to the mass 'm' as
mentioned above. Maybe you are confusing notation again?

Quote:
If we carry out the coordinate transformation, the line element now takes
the form

ds^2 = -(1-2m/(rho^3+a^3)^(1/3)) dt^2

+ rho^4/(rho^3+a^3)/((rho^3+a^3)^(1/3) -2m) drho^2

+ (rho^3+a^3)^(2/3) dOmega^2,

(8m^3-a^3)^(1/3) < rho < infty

As before, this is the -same- metric tensor, namely the metric tensor
defining the Schwarzschild vacuum solution; it has simply been
re-expressed in new coordinates. Unfortunately, many physicists
carelessly speak of "the new metric" in cases like this, which can easily
confuse students--- or authors like Antoci!

Hilbert noticed that if we set a = 0, the somewhat complicated line
element we just obtained simplifies considerably, giving the now standard
"Schwarzschild exterior chart". Even better, with a = 0, rho = r now
acquires the memorable geometric interpretation mentioned above.

OTH, if we set a = 2m, we obtain a chart valid on the range 0 < rho
infty. Here, the locus rho = 0 corresponds to the event horizon. This
chart is the one favored by Antoci et al., but to paraphrase Carlip's
comment, using this chart is a -really bad idea-, because it -greatly-
increases the complexity of the components of the metric and other
quantities, while yielding no compensatory advantage whatever!

Again, I disagree - see my comments above.

Quote:
For example, the magnitude of the acceleration of static test particles is

m/(rho^3 + 8 m^3)^(1/2)/(rho^3 + 8 m^3)^(1/3)-2m))^(1/2)

(Expanding this in powers of 1/rho confirms that the parameter m has the
same interpretation in terms of the mass of the central object as it has
in the standard Schwarzschild exterior chart.) And the tidal tensor as
measured by static observers is

m/(rho^3+a^3) diag(-2,1,1)

Here, the meaning of rho is that the surface area of the sphere t=t0,
rho=rho0 is

A = 4 pi (rho0^3 + 8 m^3)^(2/3)

Compare these expressions with (respectively):

m/r/sqrt(1-2m/r)

m/r^3 diag(-2,1,1)

A = 4 pi r0^2

This shows why the mainstream has been -wise- to adopt the coordinate
normalization suggested by Hilbert!

(I say "normalization" because the only remaining coordinate freedom in
the standard chart corresponds to the time translation symmetry and the
spherical symmetry which were assumed in deriving the solution.)

But authors like Antoci and Abrams insist that Hilbert made some kind of
mistake and that all the textbooks are wrong :-/

Several of these authors also claim that there is something wrong with
Eddington's 1922 extension of the exterior chart past the event horizon.
Recall the Carter-Penrose conformal diagram:

future
singularity

888888888888 i^+ ("future timelike infinity")
/\ /\
/ \ future / \
/ \ int. / \
/ \ / \ scri^+ ("future null infinity")
/ second \ / first \
/ exterior \/ exterior \
\ region /\ region / i^0 ("spatial infinity", r = infty)
\ / \ /
\ / \ /
\ / past \ / scri^- ("past null infinity")
\ / int. \ /
\/ \/
888888888888 i^- ("past timelike infinity")

past
singularity

Eddington's extension covers the region

888888888888
/\**********/\
/ \********/**\
/ \******/****\
/ \****/******\
/ \**/********\
/ \/**********\
\ /\**********/
\ / \********/
\ / \******/
\ / \****/
\ / \**/
\/ \/
888888888888

Some of these careless/confused authors note that the coordinate
transformation (a diffeomorphism defined on the exterior region) between
the Schwarzschild and Eddington charts is -not defined- on the event
horizon. They claim that this implies that the extension is spurious!

But this is analogous to considering a locally flat chart covering the
left half plane

ds^2 = dxi^2/xi^2 + dy^2,

0 < xi < infty, -infty < y < infty

and claiming that this cannot be extended to the usual Euclidean plane
because the diffeomorphism x = 1/xi is well-defined only on the right half
plane. Such a claim would of course be nonsense, and would exhibit a
fundamental misconception concerning the role of diffeomorphisms in
manifold theory. In this example, note also that 0 < xi < infty and
-infty < xi < 0 give -two- nonoverlapping charts; it would be quite wrong
to give the range as -infty < xi < infty! But this is precisely analogous
to one of the elementary errors which is committed by Antoci et al., as
noted above!

So the proper response to these authors is that -of course- the coordinate
transformations in question are not defined on the horizon! That is why
we say this locus represents a "coordinate singularity"! :-/

Abhas Mitra makes an odd computational error in the Kruskal-Szekeres
chart. His error is esssentially a goof using elementary calculus to
compute a limit of an implicitly defined function. Since we extensively
discussed this error in this very newgroup, many years hence, I will just
say that it can be helpful to use the Lambert W function to express the
Kruskal-Szekeres line element -explicitly-, rather than use an implicit
formulation as in Misner, Thorne, & Wheeler and other textbooks.

The Lambert W function is the special function given by solving w = z
exp(z) for z. This is a multivalued function with a branch point at -1/e,
but the principal branch is both real valued and single valued on
(-1/e,infty), which is just the range we need to write down the
Schwarzschild metric tensor on the maximal analytic extension in terms of
the Kruskal-Szekeres chart.

(BTW, the KS chart is not quite a -global chart-, since it still has
coordinate singularites at the "international date line" of each of our
nested spheres, but it comes close.)

Mitra claims that the tangent vector to the world line of a freely and
radially infalling test particle becomes null on the horizon. Here too,
frame fields are useful in verifying that this is incorrect, because we
can -draw- the frame fields and then we "see" the result: a (correct)
computation of the Kruskal-Szekeres components of the timelike unit vector
field in question shows that these infalling timelike geodesics are
perfectly well behaved at the horizon.

None of the above is related to the point I am trying to make. For one
thing, I do not reject the Kruskal extension to the exterior solution
(and as I said, I DISAGREE with Abrams et alii on this point) - I
reject the physical existence of an interior solution, and secondly, I
am not defending Abhas Mitra here - he is quite capable of defending
himself, and if you would like to point out his mistakes to him, I am
sure he will be glad to respond to you in person.

Quote:

=== The Lessons ===

What can we learn from reviewing this protracted parade of goofs?

That the physics establishment is more than capable of screwing up en
masse.

- Sabbir.
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LEJ Brouwer
science forum Guru Wannabe


Joined: 07 May 2005
Posts: 120

PostPosted: Mon Jul 10, 2006 1:23 am    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

JanPB wrote:
Quote:
LEJ Brouwer wrote:
JanPB wrote:

What do you mean "not allowed"? True, the metric for r<2M does not
satisfy the constraint you set up for yourself (namely, it's not
static) and it may contain other hidden assumptions (like r<infty) but
so what? It's still a solution to Einstein's equation and the
requirement for being static, etc., was just a simplifying (i.e., ad
hoc, random, man-made) assumption which in the end just turned out to
be an overkill. So you ended up with a more general solution covering a
larger region (the extra region - the interior - is disconnected, in
the topological sense, from the static exterior).

I refer you to the reply I gave to T.Essel's post. It is clearly not a
solution to the Einstein equation if the interior region yuo refer to
does not even physically exist.

Quote:
and (b) timelike and
spacelike directions cannot just swap places, which is what would have
to happen if matter were to fall past the horizon.

This is a mathematical feature of the Schwarzschild representation of
the metric. Nothing really swaps places, it's just that the
mathematical process of solving the relevant differential equations
mirrors one another over the two components of the disconnected domain
so the *letters* r and t end up denoting different things for r<2M
and r>2M due to what is basically an abuse of notation.

I see. And I take it that you will also remember to swap these letters
in the Einstein field equations? Why not go ahead and also swap M and
theta? In fact, why not change 'ds^2' to 'sd^2' too, just for the hell
of it? They are just letters after all.

- Sabbir.
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