"Hilbert's error" a bogus problem in "pathological science"?

On Aug 12, 2006, at 10:50 AM, Paul Zielinski wrote:

Jack Sarfatti wrote:
Carlip's analysis is obviously correct.

The problem is that Carlip's posted "analysis" doesn't address Abrams'
actual argument.

Sure it does. This is a trite bogus example of pathological science as
explained by Sharon Weinberger in "Imaginary Weapons."

Jack, you can only get so far by citing authorities. At some point you
have to study the actual problem as it appears in the published
literature. Carlip is just jerking his knee here, in a newsgroup posting.

Carlip raised all the points I did earlier.

The whole point is that you cannot get Hilbert's SSS solution from
Schwarzschild's simply by performing a coordinate transformation. So you
and Carlip have unwittingly scored an own goal.

A Red Herring.


1. You write Einstein's vacuum equation.

Ruv = 0

2. You write the trial solution


ds^2 = (1 - 2m/r)dt^2 - (1 - 2m/r)^-1dr^2 + r^2[d(polar)^2 +
sin^2(polar)d(azimuthal)^2]

polar = latitude
azimuthal = longitude

3. As Matt Visser does in "Lorentzian Wormholes" & in other text books
you compute the connection field, the 4th rank Riemann tensor, contract
the latter and voila

Ruv = 0

Therefore, formally it is a SSS solution - at least for 2m/r 1.

Now what does it mean physically?

4. You go to Wheeler's high school book with Taylor "Exploring
Blackholes" and to Kip Thorne's "Black Holes and Time Warps".



There you find out that for r 2m you imagine a Bucky Fuller geodesic
dome "shell frame" concentric to but outside of the lightlike event
horizon at r = 2m, the one-way membrane trapped surface for lightlike
and timelike worldlines.

?
http://www.yanku.com/images/bucky-dome.jpg

Such a stationary mechanical structure would collapse outside near an
actual black black hole and it cannot exist inside the event horizon.
You can emulate it with hovering LNIF rockets firing their engines
toward the center.

r = book keeper radial coordinate = Schwarzschild radial coordinate.

Circumference C of Bucky dome "shell frame" about a great circle using a
tape measure is

C = 2pir and area of the dome is A = 4pir^2.

The Bucky dome global shell frame is approximated by a network of
hovering LNIFs on non-geodesic world lines r = constant 2m.

For a concentric Bucky dome at r - dr, drop a plumbob down from r to r -
dr, you use a length of string

dr(shell) = (1 - 2m/r)^-1/2dr

r/2m 1

Note for a LIF dropped from rest from 2m/r 1, i.e. "rain LIF"

dr = dr(rain LIF)

for a direct physical meaning of 'dr".

Most formal GCTs are completely USELESS physically, operationally.
In particular

r'^3 = r^3 + (2m)^3

cited in pathological papers on the bogus "Hilbert's error" brouhaha
have no significant physical meaning.

Physics is an empirical science that must be rooted firmly in sound
doable in principle operational procedures in PW Bridgman's sense -
otherwise it is mere formal math games of no real interest to physical
science.


?http://nrumiano.free.fr/Images/lightcones_E.gif

Note that in the above diagram the light cones tip over more and more
closer and closer to event horizon approaching from the outside. The
forward cone is also narrowing in this representation not to be confused
with the Penrose diagram below. The "Time-Space" orientations in the
picture are only meaningful for hovering distant asymptotically flat
outside the event horizon using "book keeper global coordinates"
(Wheeler), r, latitude, longitude, t. Note that the forward light cones
touching the vertical r = 0 singularity line are tipped over a full 90
degrees compared to what they are far outside the event horizon in
asymptotic global flat spacetime. Therefore, r = 0 is at right angles to
the center of symmetry of the local light cone touching it - exactly
like t = constant outside the event horizon. That is the r & t book
keeper coordinates interchange roles inside the event horizon compared
to outside and this interchange time -- space is maximal at the
singularity r = 0 that is now parallel to a spacelike worldline outside
the local light cone not a timelike world line as in flat spacetime
where r = constant is a timelike worldline inside the local light cone.

See "Inside a Black Hole" http://nrumiano.free.fr/Estars/int_bh.html

Abandon all hope, ye who enter here. Synchronized clocks are at the
nodes of the dome. Wheeler tells exactly how to synchronize them with
light pulses from a reference clock.
In fact

?http://nrumiano.free.fr/Images/penrose_stat_E.gif

Light rays are at +-45 deg everywhere in the above Penrose CONFORMAL
diagram. The event horizon r = 2m is lightlike. Outside the event
horizon r = constant 2m is timelike, i.e. local angle is less than 45
deg inside local light cone. Inside event horizon r = constant 2m is
spacelike, i.e. local angle is greater than 45 deg outside local light
cone. Outside the event horizon t = constant is spacelike outside local
light cone. Inside the event horizon t = const is timelike inside local
light cone.

Note when a coordinate world line x = constant is timelike, its dual 3D
hypersurface is spacelike. Similarly, when world line x = constant is
spacelike, its dual 3D hypersurface is timelike. For now forget the 2
polar angles. Simplify to 1 + 1 spacetime of a string. Therefore, inside
the event horizon the worldlines r = constant are like the world lines t
= constant outside the event horizon. The curvature singularity at r = 0
is inevitable like the flow of time outside the event horizon. All
RETARDED timelike and lightlike worldlines must hit it. Spacelike world
lines need not hit it.




And you want to talk about "pathological science"?! How about "pretzel
logic"?

Dishonest argument. You hand wave with polemics and never actually show
anything with the math.

If Carlip is claiming that Abrams was wrong that Hilbert's manifold is
diffeomorphically
inequivalent to Schwarzschild's, then he doesn't know what he's talking
about. However,
contrary to the impression you're trying to create here, I don't think
Carlip has actually said that.

Z.

This "Hilbert's error" issue is bogus and is an example of "pathological
science."

On Aug 12, 2006, at 1:20 AM, Paul Zielinski wrote:


That a coordinate transformation alone cannot convert Schwarzschild's
manifold into
Hilbert's inequivalent manifold? But I basically agree with this,
although I understand that there are some subtle aspects of such
transformation where there are singularities on the manifold. Fiziev
discusses this issue in his arXiv papers. It is not as simple and
straightforward as you seem to think.

Too bad Hilbert mishandled his r - r* "coordinate transformation"


Complete nonsense. (written by Jack)

Paul continues

and consequently messed up Schwarzschild's 1916 solution in his infamous
1917 paper, leading to a spacetime with a completely different topology.

I think like Steve Carlip you are missing the point -- because you
haven't studied the
problem.


It's not *my* position. Here you are being almost as simplistic as Hudson!

This was *Leonard Abrams'* position. I didn't invent this thesis.
Abrams has been published in leading physics journals, and quite a few
highly qualified working mathematical physicists currently agree with
his position on Hilbert vs. Schwarzschild.

The torch as now passed to people like Antoci, Loinger, and Fiziev.
These people are widely published. They have academic positions. Don't
try and tell me these people are all "crackpots".

If they're right about Hilbert's error, then the only thing that is
truly "crackpot" here is the orthodox SSS model, accepted uncritically
as a "prediction" of Einstein's theory. The fact of the matter, which
everyone invested in black holes would like to go away, is that there
are much simpler and chronogeometrically better-behaved SSS solutions
to Einstein's field equations. The canonical solution is not unique, and
it has some very undesirable properties.

Wheeler gives the correct chronogeometrics as I indicated above. Abrams
paper et-al are "pathological science" IMO.


First of all a timelike naked singularity is a breakdown of classical
general relativity as Hawking stated in that quote I cited. (Jack earlierz)


This is elementary high school math.


If it threw Hilbert, than it couldn't have been that obvious. And even
if Hilbert was right, then that would make Schwarzschild wrong. So in
that case it stumped Schwarzschild.

Jack, I'm sorry, but I don't think you've understood the problem.

In any case, if this issue of diffeomorphic inequivalence of the two
SSS manifolds is high school math as you claim, then why is it taking so
long for you people to figure it out? Why did it take so long for
gravitational physicists and mathematicians to distinguish between, for
example, curvature singularities from coordinate singularities? Why is
the mathematical definition of a curvature singularity so tricky? Why
doesn't anyone even seem to know that what is routinely called the
"Schwarzschild solution" in textbooks is not Schwarzschild's actual
solution at all?

I'd love to see a debate on this between Fiziev and Carlip.

Z.

On Aug 11, 2006, at 7:24 PM, Paul Zielinski wrote:

I'm already quite familar with this post from Carlip. His argument is
no good.

Abrams' position is precisely that it was Hilbert's mishandling of the
coordinate transformation r - r* that created a different topology --
not the actual transformation itself.

So again, you've misunderstood the problem. I need to talk to Waldyr
about this directly. I think he, like Carlip, has missed the point of
Abrams' argument.

Z.

Jack Sarfatti wrote:



Begin forwarded message:

From: "Waldyr A. Rodrigues Jr."
Date: August 11, 2006 5:40:45 PM PDT
To: "'Jack Sarfatti'"
Subject: RES: Hilbert's error... I agree with S. Carlip - no such error

Thanks, but, of course, I already read all that...

-----Mensagem original-----
De: Jack Sarfatti ]
Enviada em: sexta-feira, 11 de agosto de 2006 20:00
Para:
Assunto: Hilbert's error... I agree with S. Carlip - no such error

"Hilbert's error" is what Sharon Weinberger calls "pathological
physics" in "Imaginary Weapons". Other examples are IMHO "Hafnium
bomb", the Puthoff "PV approach to metric engineering" and the Haisch
approach to "orgin of inertia" from random transverse EM virtual
photon "friction" ignoring the standard model's explanation from the
macro-quantum Vacuum ODLRO electroweak Higgs field. The origin of
invariant rest mass is coherent not random incoherent ZPF (a small
effect compared to the main coherent one).

S. Carlip basically repeats what I have said, but better



Misinterpretation of the radial parameter in the Schwarzshild
solution?


Subject: Misinterpretation of the radial parameter in the
Schwarzshild solution?
From:
Date: Tue, 4 Jul 2006 03:57:29 +0000 (UTC)
Approved: (sci.physics.research)
Message-ID:
Newsgroups: sci.physics.research
Organization: University of California, Davis
References:
.com115081755 1. 10 5








00cwm.googlegroups.com
Sender:
User-Agent: tin/1.4.5-20010409 ("One More Nightmare") (UNIX) (Linux/
2.4.21-37.0.1.EL (i686))


LEJ Brouwer wrote:
[...]

The reason I am interested is because 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.




In particular they show, in a rather simple fashion, that the event
horizon is at radius zero, coinciding with the position the point



mass

itself, and actually appears pointlike to an external observer.




These papers are complete nonsense. In particular, the authors seem
not to understand the basic fact that physics does not depend on what
coordinates one chooses to use.

It is trivially true that if one changes coordinates in the standard
Schwarzschild solution from r to r-2m, then the horizon is at r=0.
It is also trivially true that this does not change the spacetime
geometry -- the horizon is still a lightlike surface, with an area at
fixed time of 4m^2. Choosing a coordinate that makes the horizon
look like a point doesn't make it a point -- it just means that you've
made a dumb coordinate choice.




Exactly. I said this several times to no avail.


They claim that the reason that the original misinterpretation



occurred

is because Hilbert incorrectly assumed a priori that the 'r' which
appears in the metric must be the radial coordinate (in fact, it



need

only parametrise the radii to ensure a spherically symmetric



solution).

It is radial in the sense that the set of points at constant r and t
is a two-sphere of area 4pi r^2. It is not "radial distance," but
no one has claimed that it is.




Exactly. I said this several times to no avail.
If the authors would simply read Wheeler's high school text
"Exploring Black Holes" they would understand the meaning of
"r" as a convenient "global book keeping radial coordinate"
with a clear operational meaning. 2pir is the measured circumference
measured in the non-inertial "shell" frame. For example put a tape
measure
around the equator of the Earth's surface.

dr(shell LNIF) = (1 - 2m/r)^-1/2dr

dr(shell) is actual small radial distance measured by a plumb bob
lowered to a nearby concentric shell (like Bucky Fuller geodesic
domes for example with clocks).

Note also the geodesic LIF "rain frames" i.e. test particle dropped
from rest at infinity, then

dr(rain LIF) = dr(Book keeper) = dr

r = Schwarzschild radial coordinate i.e. C = 2pir (A = 4pir^2) in
HOVERING rest "shell" LNIF frames.

Had the authors thought physically they would not have made such a
silly suggestion.



The careful analysis of Abrams et al shows that the point mass



actually

resides at r=2m, which therefore corresponds to the true origin, so
that there is in fact no 'interior' solution.




This analysis is not "careful" -- it's mathematically awful. How can
a "point mass" reside at a two-sphere of finite area?




Exactly. I said this several times to no avail.

What sense does
it make to say that a mass resides at a position at which the Ricci
tensor is zero?

Abrams makes an elementary mistake. He concludes that r=2m (in
standard
Schwarzschild coordinates) is singular because the "radius" of a
circle
around this "point" goes to zero as r-2m while its "circumference"
does not. But this is not a singularity -- it's just a reflection
of the fact that r=2m is a two-sphere, not a point.




Exactly. I said this several times to no avail.


If the event horizon is at the origin, and there is no interior
solution, then this tends to raise the question, "well, where does a
radially infalling particle actually go?". Does it just bounce



off the

'brick wall' (or rather, 'brick point')?




To answer this, you just compute the motion. You find that it falls
right past the "origin," with nothing peculiar happening there. (Of
course, you can insist on using bad coordinates, but that's your own
fault...).

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.

Steve Carlip

Science cannot solve the ultimate mystery of nature.

And that is because, in the last analysis, we ourselves are part of the
mystery that we are trying to solve.

-- Max Planck

--
Ahmed Ouahi, Architect
Best Regards!


"Jack Sarfatti" wrote in message
t...

On Aug 12, 2006, at 10:50 AM, Paul Zielinski wrote:

Jack Sarfatti wrote:
Carlip's analysis is obviously correct.

The problem is that Carlip's posted "analysis" doesn't address Abrams'
actual argument.

Sure it does. This is a trite bogus example of pathological science as
explained by Sharon Weinberger in "Imaginary Weapons."

Jack, you can only get so far by citing authorities. At some point you
have to study the actual problem as it appears in the published
literature. Carlip is just jerking his knee here, in a newsgroup posting.

Carlip raised all the points I did earlier.

The whole point is that you cannot get Hilbert's SSS solution from
Schwarzschild's simply by performing a coordinate transformation. So you
and Carlip have unwittingly scored an own goal.

A Red Herring.


1. You write Einstein's vacuum equation.

Ruv = 0

2. You write the trial solution


ds^2 = (1 - 2m/r)dt^2 - (1 - 2m/r)^-1dr^2 + r^2[d(polar)^2 +
sin^2(polar)d(azimuthal)^2]

polar = latitude
azimuthal = longitude

3. As Matt Visser does in "Lorentzian Wormholes" & in other text books
you compute the connection field, the 4th rank Riemann tensor, contract
the latter and voila

Ruv = 0

Therefore, formally it is a SSS solution - at least for 2m/r 1.

Now what does it mean physically?

4. You go to Wheeler's high school book with Taylor "Exploring
Blackholes" and to Kip Thorne's "Black Holes and Time Warps".



There you find out that for r 2m you imagine a Bucky Fuller geodesic
dome "shell frame" concentric to but outside of the lightlike event
horizon at r = 2m, the one-way membrane trapped surface for lightlike
and timelike worldlines.

?
http://www.yanku.com/images/bucky-dome.jpg

Such a stationary mechanical structure would collapse outside near an
actual black black hole and it cannot exist inside the event horizon.
You can emulate it with hovering LNIF rockets firing their engines
toward the center.

r = book keeper radial coordinate = Schwarzschild radial coordinate.

Circumference C of Bucky dome "shell frame" about a great circle using a
tape measure is

C = 2pir and area of the dome is A = 4pir^2.

The Bucky dome global shell frame is approximated by a network of
hovering LNIFs on non-geodesic world lines r = constant 2m.

For a concentric Bucky dome at r - dr, drop a plumbob down from r to r -
dr, you use a length of string

dr(shell) = (1 - 2m/r)^-1/2dr

r/2m 1

Note for a LIF dropped from rest from 2m/r 1, i.e. "rain LIF"

dr = dr(rain LIF)

for a direct physical meaning of 'dr".

Most formal GCTs are completely USELESS physically, operationally.
In particular

r'^3 = r^3 + (2m)^3

cited in pathological papers on the bogus "Hilbert's error" brouhaha
have no significant physical meaning.

Physics is an empirical science that must be rooted firmly in sound
doable in principle operational procedures in PW Bridgman's sense -
otherwise it is mere formal math games of no real interest to physical
science.


?http://nrumiano.free.fr/Images/lightcones_E.gif

Note that in the above diagram the light cones tip over more and more
closer and closer to event horizon approaching from the outside. The
forward cone is also narrowing in this representation not to be confused
with the Penrose diagram below. The "Time-Space" orientations in the
picture are only meaningful for hovering distant asymptotically flat
outside the event horizon using "book keeper global coordinates"
(Wheeler), r, latitude, longitude, t. Note that the forward light cones
touching the vertical r = 0 singularity line are tipped over a full 90
degrees compared to what they are far outside the event horizon in
asymptotic global flat spacetime. Therefore, r = 0 is at right angles to
the center of symmetry of the local light cone touching it - exactly
like t = constant outside the event horizon. That is the r & t book
keeper coordinates interchange roles inside the event horizon compared
to outside and this interchange time -- space is maximal at the
singularity r = 0 that is now parallel to a spacelike worldline outside
the local light cone not a timelike world line as in flat spacetime
where r = constant is a timelike worldline inside the local light cone.

See "Inside a Black Hole" http://nrumiano.free.fr/Estars/int_bh.html

Abandon all hope, ye who enter here. Synchronized clocks are at the
nodes of the dome. Wheeler tells exactly how to synchronize them with
light pulses from a reference clock.
In fact

?http://nrumiano.free.fr/Images/penrose_stat_E.gif

Light rays are at +-45 deg everywhere in the above Penrose CONFORMAL
diagram. The event horizon r = 2m is lightlike. Outside the event
horizon r = constant 2m is timelike, i.e. local angle is less than 45
deg inside local light cone. Inside event horizon r = constant 2m is
spacelike, i.e. local angle is greater than 45 deg outside local light
cone. Outside the event horizon t = constant is spacelike outside local
light cone. Inside the event horizon t = const is timelike inside local
light cone.

Note when a coordinate world line x = constant is timelike, its dual 3D
hypersurface is spacelike. Similarly, when world line x = constant is
spacelike, its dual 3D hypersurface is timelike. For now forget the 2
polar angles. Simplify to 1 + 1 spacetime of a string. Therefore, inside
the event horizon the worldlines r = constant are like the world lines t
= constant outside the event horizon. The curvature singularity at r = 0
is inevitable like the flow of time outside the event horizon. All
RETARDED timelike and lightlike worldlines must hit it. Spacelike world
lines need not hit it.




And you want to talk about "pathological science"?! How about "pretzel
logic"?

Dishonest argument. You hand wave with polemics and never actually show
anything with the math.

If Carlip is claiming that Abrams was wrong that Hilbert's manifold is
diffeomorphically
inequivalent to Schwarzschild's, then he doesn't know what he's talking
about. However,
contrary to the impression you're trying to create here, I don't think
Carlip has actually said that.

Z.

This "Hilbert's error" issue is bogus and is an example of "pathological
science."

On Aug 12, 2006, at 1:20 AM, Paul Zielinski wrote:


That a coordinate transformation alone cannot convert Schwarzschild's
manifold into
Hilbert's inequivalent manifold? But I basically agree with this,
although I understand that there are some subtle aspects of such
transformation where there are singularities on the manifold. Fiziev
discusses this issue in his arXiv papers. It is not as simple and
straightforward as you seem to think.

Too bad Hilbert mishandled his r - r* "coordinate transformation"


Complete nonsense. (written by Jack)

Paul continues

and consequently messed up Schwarzschild's 1916 solution in his infamous
1917 paper, leading to a spacetime with a completely different topology.

I think like Steve Carlip you are missing the point -- because you
haven't studied the
problem.


It's not *my* position. Here you are being almost as simplistic as Hudson!

This was *Leonard Abrams'* position. I didn't invent this thesis.
Abrams has been published in leading physics journals, and quite a few
highly qualified working mathematical physicists currently agree with
his position on Hilbert vs. Schwarzschild.

The torch as now passed to people like Antoci, Loinger, and Fiziev.
These people are widely published. They have academic positions. Don't
try and tell me these people are all "crackpots".

If they're right about Hilbert's error, then the only thing that is
truly "crackpot" here is the orthodox SSS model, accepted uncritically
as a "prediction" of Einstein's theory. The fact of the matter, which
everyone invested in black holes would like to go away, is that there
are much simpler and chronogeometrically better-behaved SSS solutions
to Einstein's field equations. The canonical solution is not unique, and
it has some very undesirable properties.

Wheeler gives the correct chronogeometrics as I indicated above. Abrams
paper et-al are "pathological science" IMO.


First of all a timelike naked singularity is a breakdown of classical
general relativity as Hawking stated in that quote I cited. (Jack
earlierz)


This is elementary high school math.


If it threw Hilbert, than it couldn't have been that obvious. And even
if Hilbert was right, then that would make Schwarzschild wrong. So in
that case it stumped Schwarzschild.

Jack, I'm sorry, but I don't think you've understood the problem.

In any case, if this issue of diffeomorphic inequivalence of the two
SSS manifolds is high school math as you claim, then why is it taking so
long for you people to figure it out? Why did it take so long for
gravitational physicists and mathematicians to distinguish between, for
example, curvature singularities from coordinate singularities? Why is
the mathematical definition of a curvature singularity so tricky? Why
doesn't anyone even seem to know that what is routinely called the
"Schwarzschild solution" in textbooks is not Schwarzschild's actual
solution at all?

I'd love to see a debate on this between Fiziev and Carlip.

Z.

On Aug 11, 2006, at 7:24 PM, Paul Zielinski wrote:

I'm already quite familar with this post from Carlip. His argument is
no good.

Abrams' position is precisely that it was Hilbert's mishandling of the
coordinate transformation r - r* that created a different topology --
not the actual transformation itself.

So again, you've misunderstood the problem. I need to talk to Waldyr
about this directly. I think he, like Carlip, has missed the point of
Abrams' argument.

Z.

Jack Sarfatti wrote:



Begin forwarded message:

From: "Waldyr A. Rodrigues Jr."
Date: August 11, 2006 5:40:45 PM PDT
To: "'Jack Sarfatti'"
Subject: RES: Hilbert's error... I agree with S. Carlip - no such error

Thanks, but, of course, I already read all that...

-----Mensagem original-----
De: Jack Sarfatti ]
Enviada em: sexta-feira, 11 de agosto de 2006 20:00
Para:
Assunto: Hilbert's error... I agree with S. Carlip - no such error

"Hilbert's error" is what Sharon Weinberger calls "pathological
physics" in "Imaginary Weapons". Other examples are IMHO "Hafnium
bomb", the Puthoff "PV approach to metric engineering" and the Haisch
approach to "orgin of inertia" from random transverse EM virtual
photon "friction" ignoring the standard model's explanation from the
macro-quantum Vacuum ODLRO electroweak Higgs field. The origin of
invariant rest mass is coherent not random incoherent ZPF (a small
effect compared to the main coherent one).

S. Carlip basically repeats what I have said, but better



Misinterpretation of the radial parameter in the Schwarzshild
solution?


Subject: Misinterpretation of the radial parameter in the
Schwarzshild solution?
From:
Date: Tue, 4 Jul 2006 03:57:29 +0000 (UTC)
Approved: (sci.physics.research)
Message-ID:
Newsgroups: sci.physics.research
Organization: University of California, Davis
References:
.com115081755 1. 10 5








00cwm.googlegroups.com
Sender:
User-Agent: tin/1.4.5-20010409 ("One More Nightmare") (UNIX) (Linux/
2.4.21-37.0.1.EL (i686))


LEJ Brouwer wrote:
[...]

The reason I am interested is because 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.




In particular they show, in a rather simple fashion, that the event
horizon is at radius zero, coinciding with the position the point



mass

itself, and actually appears pointlike to an external observer.




These papers are complete nonsense. In particular, the authors seem
not to understand the basic fact that physics does not depend on what
coordinates one chooses to use.

It is trivially true that if one changes coordinates in the standard
Schwarzschild solution from r to r-2m, then the horizon is at r=0.
It is also trivially true that this does not change the spacetime
geometry -- the horizon is still a lightlike surface, with an area at
fixed time of 4m^2. Choosing a coordinate that makes the horizon
look like a point doesn't make it a point -- it just means that you've
made a dumb coordinate choice.




Exactly. I said this several times to no avail.


They claim that the reason that the original misinterpretation



occurred

is because Hilbert incorrectly assumed a priori that the 'r' which
appears in the metric must be the radial coordinate (in fact, it



need

only parametrise the radii to ensure a spherically symmetric



solution).

It is radial in the sense that the set of points at constant r and t
is a two-sphere of area 4pi r^2. It is not "radial distance," but
no one has claimed that it is.




Exactly. I said this several times to no avail.
If the authors would simply read Wheeler's high school text
"Exploring Black Holes" they would understand the meaning of
"r" as a convenient "global book keeping radial coordinate"
with a clear operational meaning. 2pir is the measured circumference
measured in the non-inertial "shell" frame. For example put a tape
measure
around the equator of the Earth's surface.

dr(shell LNIF) = (1 - 2m/r)^-1/2dr

dr(shell) is actual small radial distance measured by a plumb bob
lowered to a nearby concentric shell (like Bucky Fuller geodesic
domes for example with clocks).

Note also the geodesic LIF "rain frames" i.e. test particle dropped
from rest at infinity, then

dr(rain LIF) = dr(Book keeper) = dr

r = Schwarzschild radial coordinate i.e. C = 2pir (A = 4pir^2) in
HOVERING rest "shell" LNIF frames.

Had the authors thought physically they would not have made such a
silly suggestion.



The careful analysis of Abrams et al shows that the point mass



actually

resides at r=2m, which therefore corresponds to the true origin, so
that there is in fact no 'interior' solution.




This analysis is not "careful" -- it's mathematically awful. How can
a "point mass" reside at a two-sphere of finite area?




Exactly. I said this several times to no avail.

What sense does
it make to say that a mass resides at a position at which the Ricci
tensor is zero?

Abrams makes an elementary mistake. He concludes that r=2m (in
standard
Schwarzschild coordinates) is singular because the "radius" of a
circle
around this "point" goes to zero as r-2m while its "circumference"
does not. But this is not a singularity -- it's just a reflection
of the fact that r=2m is a two-sphere, not a point.




Exactly. I said this several times to no avail.


If the event horizon is at the origin, and there is no interior
solution, then this tends to raise the question, "well, where does a
radially infalling particle actually go?". Does it just bounce



off the

'brick wall' (or rather, 'brick point')?




To answer this, you just compute the motion. You find that it falls
right past the "origin," with nothing peculiar happening there. (Of
course, you can insist on using bad coordinates, but that's your own
fault...).

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.

Steve Carlip