# A 'brick wall' horizon?

by LEJ Brouwer
Tags: brick, horizon, wall
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 P: n/a 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?
 P: n/a 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
 P: n/a 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.
 P: n/a A 'brick wall' horizon? 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. > 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
 P: n/a 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.
 P: n/a markwh04@yahoo.com 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
 P: n/a 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"
 P: n/a 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"

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