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