A 'brick wall' horizon?

LEJ Brouwer

In Susskind & Lindesay's "Black Holes, Information Theory and the
String Theory Revolution" they write in section 9.1 regarding the
interpretation of the event horizon of a black hole,

"There are two more possibilities worth pointing out. One is that the
horizon is not penetrable. In other words, from the viewpoints of an
in-falling system, the horizon bounces everything out. A freely falling
observer would encounter a "brick wall" just above the horizon."

They go on to say that the idea was never seriously entertained, but
provide no references as to why the possibility was brought up in the
first place or by whom.

Does anyone know where this idea of an impenetrable horizon came from,
and/or have any references to where it was discussed?

boson boss

LEJ Brouwer wrote:
> In Susskind & Lindesay's "Black Holes, Information Theory and the
> String Theory Revolution" they write in section 9.1 regarding the
> interpretation of the event horizon of a black hole,

In Seer & Prophet's "Three Things No One's Ever Seen" they write in
section Messiah 9 verse 1 regarding the interpretation of the event
horizon of a black hole,

"Though shalt know the law from above, There are two more
possibilities worth pointing out. One is as God said
My horizon is not penetrable. In other words, from the
viewpoints of an in-falling system, the horizon bounces
everything out. A freely falling observer would encounter a
"brick wall" just above the horizon."

> They go on to say that the idea was never seriously entertained, but
> provide no references as to why the possibility was brought up in the
> first place or by whom.

So you do question The Wall or is it your head too soft? :-))

> Does anyone know where this idea of an impenetrable horizon came from,
> and/or have any references to where it was discussed?

http://arxiv.org/abs/hep-th/9604130
http://www2.udec.cl/~mariaant/0011024.pdf

LEJ Brouwer

boson boss wrote:
> So you do question The Wall or is it your head too soft? :-))

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.

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).
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. In particular, they
mention that Schwarzschild's original paper never allowed for an
interior solution either.

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')? (I do not agree with the above
papers that the Kruskal extension is invalid - it is absolutely
necessary to have a consistent well-defined timelike direction).

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?

Cheers,

Sabbir.

carlip-nospam@physics.ucdavis.edu

LEJ Brouwer <intuitionist1@yahoo.com> 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.

> 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.

> 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? 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.

> 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

markwh04@yahoo.com

carlip-nospam@physics.ucdavis.edu wrote:
> > 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.

The radial coordinate in Schwarzschild coordinates is simply the square
root of the quotient of the area of the sphere of symmetry by 4 pi.

> 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.

One can always define the *topology* such that the sphere of symmetry
at the event horizon is a single point. More generally, a solution only
defines a spacetime up to homeomorphism. Like it or not, that's a major
gap in the underlying theory, since there are no principles, in
general, to determine which topology should apply to a given spacetime:
the universal covering topology or one of its homeomorphic images.

The solution with the event horizon identified as a single point is
simply a different solution than that with the topology normally
ascribed to a black hole. There is no principle to say whether it, or
the other solution, or any of the infinite number of alternatives
arrived at by various quotienting operations, should be the "real"
solution.

In the longer run, the question is moot anyhow. None of them are the
"real" solution. The "real" solution is a Hawking-evaporating black
hole that evaporates away in finite time and has no trapped region at
all.

carlip-nospam@physics.ucdavis.edu

markwh04@yahoo.com wrote:
> carlip-nospam@physics.ucdavis.edu wrote:

[...]
>> 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.

> One can always define the *topology* such that the sphere of symmetry
> at the event horizon is a single point. More generally, a solution only
> defines a spacetime up to homeomorphism. Like it or not, that's a major
> gap in the underlying theory, since there are no principles, in
> general, to determine which topology should apply to a given spacetime:
> the universal covering topology or one of its homeomorphic images.

No, you can't do this in GR -- the point set topology is not uniquely
determined by the metric, but it is also not independent of the metric.

In this case, if you identify the two-sphere at r=2m, t=const. with a
point, the Schwarzschild metric is singular at that point. This means
that the point is not part of the manifold described by the solution of
the field equations, and must be removed. Once the point is removed,
there are standard techniques to analyze the nature of the resulting
boundary (e.g., by treating it as a b-boundary). These show that the
boundary is not a point, and that the geometry can be extended across
the boundary. Do this, and you get the standard Kruskal-Szekeres
solution.

There are certainly situations like the one you describe, in which the
field equations do not tell you whether to use a non-simply connected
manifold or one of its covers. But that's only relevant when the metric
is nonsingular on all of the choices. When the metric is singular, the
theory *does* tell you what to do -- you must remove the points at which
the singularity occurs, and then determine whether you can continue the
solution past the resulting boundary in a nonsingular way.

Steve Carlip

tessel@um.bot

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.

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 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.

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:+An.../0/1/0/all/0/1
http://www.arxiv.org/find/gr-qc/1/au.../0/1/0/all/0/1
http://arxiv.org/find/gr-qc/1/au:+Mi.../0/1/0/all/0/1
http://arxiv.org/find/grp_physics/1/.../0/1/0/all/0/1
http://arxiv.org/find/grp_physics/1/.../0/1/0/all/0/1

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

http://arxiv.org/find/grp_physics/1/.../0/1/0/all/0/1
http://arxiv.org/find/grp_physics/1/.../0/1/0/all/0/1

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

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!)

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"):

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.

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! 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.

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.

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.

In addition, Brillouin incorrectly used the range 0 < r < infty in the
original chart, 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". 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).

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.

(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!

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.

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! ;-/

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

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

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

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.

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!

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.

=== The Lessons ===

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

The fact that such obviously wrong papers continue to be produced,
published, and cited is dismaying, because one major goal of providing
electronic archives is to make it easier to find/obtain/study relevant
previous work, yet this kind of rampant repetition of old errors suggests
that some "researchers" have forgotten that -reading- is the most
important part of library research!

This raises a disturbing question: by drastically lowering the threshold
of pain involved in simply -finding- and -obtaining- relevant prior work,
while leaving unaltered the threshold of pain involved in -reading- what
one has obtained, has the advent of the arXiv had the unexpected and
paradoxical effect of -decreasing- knowledge of the research literature
among researchers? If we make "the easy part" of library research -too
easy-, will the next generation fail to take the trouble to read the
contents of our libraries (that's "the hard part" of library research), on
the grounds that actually -reading- the literature would constitute an
unacceptable burden on the time and energy of busy scholars?

The fact that few of the papers/eprints listed above have been voluntarily
withdrawn further suggests that the arXiv may benefit from some more
coercive method of pruning.

"T. Essel"

tessel@um.bot

A new paper by Malcolm MacCallum (a coauthor of the second edition of "the
exact solutions book" and well known as a veteran researcher in classical
gravitation) was posted to the arXiv earlier this week:

http://www.arxiv.org/abs/gr-qc/0608033

This paper is a critique of some of the papers by Antoci et al.; the bulk
is devoted to less elementary errors but an appendix discusses some of the
more elementary points which I noted in my previous post.

In a less salutory development, some fairly new Mitra-related Wikipedia
articles exhibit a cranky slant:

[[Abhas Mitra]]

[[Magnetospheric eternally collapsing object]]

A word of warning: random sampling of English language papers published in
the Indian subcontinent suggests the existence of an entire genre
amounting to a "war of words" between Indian and Pakistani "patriots" in
their respective newspapers. The newspaper article cited in the first
article seems to belong to this genre, which can be recognized by an
overheated "cheerleading" rhetoric suggestive of boozy football fans
(except that the opposing "teams" are brandishing nuclear weapons). I
could be wrong, but it might be worth looking into some of the trouble
caused at en.wikipedia.org by past India-Pakistan flame wars, before
trying to decide whether it is really worthwhile to even try to ameliorate
the partisan slant of the current version of the first article.

"T. Essel"