I draw the forum’s attention to the fact that the so-called Schwarzschild solution is not Schwarzschild’s solution. This can be easily verified by reading Schwarzschild’s original paper. Furthermore, the all too frequent claim that Schwarzschild obtained the black hole from his solution is patently false, since Schwarzschild did not breathe a single word about such an object, because one cannot get a black hole from Schwarzschild’s original solution. Schwarzschild’s original solution is regular everywhere in the range 0 < r < infinity. The standard line-element of the standard metric erroneously named for Schwarzschild, does occur in Schwarzschild’s original paper, in terms of what he called his auxiliary parameter R, defined on alpha < R < infinity, where his alpha = 2m. The foregoing facts have been ignored by the relativists. That’s not scientific method, is it? The relativists have confounded Schwarzschild’s R with his r and thereby bungled their analysis. The so-called Schwarzschild metric is actually a corruption of Schwarzschild’s 1915 solution, and of the solution obtained independently by Johannes Droste in May 1916. Droste’s solution is Schwarzschild’s auxiliary parameter solution. The so-called Schwarzschild solution was obtained by David Hilbert in December 1916, but he botched the interpretation of the variable r therein, and most of the relativists have done the same since. It is from Hilbert’s solution, on 0 < r < infinity, that the black hole was ‘’derived’’. It is a fact, very simply demonstrated, that Hilbert’s solution is incompatible with Schwarzschild’s true solution since the one cannot be obtained from the other by an admissible transformation of coordinates. Droste’s solution is, of course, compatible with Schwarzschild. One cannot get a black hole from Droste’s solution either, since it is Schwarzschild’s solution in the latter’s auxiliary parameter, and Schwarzschild’s solution has no black hole. It is fair and accurate to say that the black hole has become a scientific fraud that makes Piltdown Man look like a pimple on an elephant’s rump. You can verify my claims by reading Schwarzschild’s original paper, which you can get at www.geocities.com/theometria/schwarzschild.pdf and Droste’s paper at www.geocities.com/theometria/Droste.pdf A documented and therefore irrefutable history of the genesis of the “black hole” fraud can be obtained at www.geocities.com/theometria/holes.pdf and a correct description of the fundamental geometry of Einstein’s gravitational field can be had at www.geocities.com/theometria/inRussia.pdf It is evident that the relativists cannot be trusted to do science or to tell the truth. Here is another recent example of claims that are patently false. See this article: http://www.abc.net.au/science/news/stories/s1619786.htm and note the claims about Einstein. However, Einstein never claimed that General Relativity admitted the possibility of the black hole, but, on the contrary, claimed that it excludes the black hole. I refer you to his paper "On a stationary system with spherical symmetry consisting of many gravitating masses", Annals of Mathematics, Vol 40, No. 4, October, 1939. In the penultimate paragraph of that paper Einstein says, "The essential result of this investigation is a clear understanding as to why the 'Schwarzschild singularities' do not exist in physical reality." In other words, according to Einstein, black holes do not exist. I also refer you to the paper by Einstein and Rosen, "The particle problem in the general theory of relativity", Physical Review, Vol. 48, July 1, 1935, wherein they state, "For these reasons writers have occasionally noted the possibility that material particles might be considered as singularities of the field. This point of view, however, we cannot accept at all. For a singularity brings so much arbitrariness into the theory that it actually nullifies its laws." And further, that "Every field theory, in our opinion, must therefore adhere to the fundamental principle that singularities of the field are to be excluded." Nonetheless, the article cited above would have us believe that Einstein not only accepted the idea of the black hole, but worked on its physics! Is this honest? Is it competent? And here is another gem. I remark that Einstein's pseudo-tensor is meaningless, because it requires, by application of Euler's theorem, the existence of a 1st order intrinsic differential invariant, depending only upon the components of the metric tensor and their 1st derivatives; but, as has been proved long ago (in 1900) by the pure mathematicians (Ricci and Levi-Civita), such an invariant does not exist! Therefore, Einstein’s pseudo-tensor cannot be used to substantiate anything at all; let alone the localisation of gravitational energy, but curiously, that has not stopped the physicists from using it to do so. Is that competent? Is it honest? It is about time the relativists came back to reality and the art of rational conjecture, instead of peddling falsehoods to all and sundry for vainglory, self-aggrandizement and money. Stephen J. Crothers thenarmis@yahoo.com 15th June 2006
I don't know enough about the history or the details of the Schwarzschild solution to address the rest of your post (except to note that this section of an online book on relativity says 'Interestingly, the solution in Schwarzschild's 1916 paper was not presented in terms of what we today call Schwarzschild coordinates. Those were introduced a year later by Droste.', so apparently this is not controversial or hidden, although it sounds like it's just a difference in the coordinates used to describe the same spacetime), but I wanted to comment on this: It's true that Einstein did not believe in black holes, but I don't see where the article claims he did. Are you referring to this quote? The context is talking about a simulation of the gravitational waves that arise from colliding black holes, and the preceding paragraph sets up the quote by saying they've found "a basic gravity wave signature" of such a collision, so I think "Einstein's prediction" refers to the idea of gravitational waves, not to black holes. Einstein certainly did argue that gravitational waves were a feature of general relativity--he included them in his theory in 1916, although there was a brief period in 1936 where he thought he had found a proof they couldn't exist (see here for the story).
The history of the 'Schwarzchild solution' is irrelevant to the question ot its validity. I don't know the history, but I do know that what we call the Schwarzchild solution is a valid solution to the Einstein equation, it is not overly complicated this to show mathematically. Do you have any problems with the mathematical details or just the history?
Okay I read some of your papers, in particular the 'inRussia' paper which seems to get the the heart of your objection. I think you analysis is off. You state that starting from a Minkowski metric you can say that 0 < r < inf and then if you transform your radial co-ordinate you can find that the new 'radial co-ordinate' has some odd range of validity, causing it to lose meaning beyond a real valued parameter. The problem is that you could equally go back the other way, i.e. start with the transformed metric and find that the Minkowski line element was not valid. You are therefore claiming implicitally that there is somthing special about the Minkowski metric, that it is the 'real' metric and all others must be consistant with it under you non dodgy range of validity rule. However there is nothing special about the Minkowski metric in GR. It is merely a co-ordinate system that is as valid as any other. In order to get from a mathematical co-ordinate system to a physical prediction (velocity, force etc experienced by or on an observer) you need to use the orthonormal basis for that co-ordinate system. This accounts for the different way in which coordinate systems slice up space time and result in the same answer regardless of the coordinates used. There is no one 'right' way to define distance in GR. All coordinate systems will have different definitions, but providing they use the correct orthornormal basis will give identical physical predictions. As I said in my above post, what we call the Schwarschild solution can be shown to come directly from the Einstein equations as the unique solution for a spherically symetric static spacet-time. There are many co-ordinate transformations, such as the Eddington-Finklestein or Kruskal co-ordinates, which are more convenient for certain situations, but they all give the same physical predictions, even though the way they slice up space-time appears different. If you attempt to apply naive generlisations about the meaning of metric co-efficients you will find apparent conflict, but used as intended there is not underlying conflict between different ways of slicing up space-time with different co-ordinates. A case in point is the comparison between the Minkowski metric and the FRW metric for an empty universe. On the one hand we have a simple SR looking metric, on the other hand we have a metric with expanding space and all the rest of it. Surely they cannot be physically equivalent since they measure time and distance is different ways! But of course they are indentical and used properly give exactly the same physical predictions. There is not one 'right' way to define distance in GR!
On black hole binaries and other fancies The scientists now routinely make the claims that black hole binary systems, black hole collisions and black hole mergers not only occur in Nature, but will also provide sources of gravitational radiation, detectable by the LIGO and LISA projects. See for example the relevant LISA sites: http://sci.esa.int/science-e/www/area/index.cfm?fareaid=27 http://lisa.jpl.nasa.gov/ http://www.lisa-science.org/newsletter There is, I submit, a fundamental conceptual error with the now commonplace claims for black hole binaries, black hole collisions, and black hole mergers. By what rigorous means do the scientists maintain that the supposed problems are well posed? In other words, by what rigorous arguments do they hold that black hole binaries or black hole collisions are well-defined scenarios? I remark that the black hole is alleged to be a consequence of Einstein’s General Relativity. Assuming for the sake of argument that this is correct, it is evident that the black hole is the result of a solution to Einstein’s field equations for the configuration of a single spherically symmetric gravitating body interacting with a test particle in vacuum. It is not a solution for the interaction of two comparable masses, such as two black holes. Before one can talk of black hole binaries or black hole collisions it must first be demonstrated that Einstein’s field equations admit of solutions for multi-body configurations of gravitationally coupled spherically symmetric comparable masses. This can be done in two possible ways, in principle: 1) by deducing a particular solution, or 2) proving an existence theorem. There are however, no known solutions to the field equations for the interaction of two or more spherically symmetric comparable masses, so option 1) has never been met. In fact, it is not even known if Einstein’s field equations admit of such solutions as no existence theorem has ever been adduced, so option 2) has never been met either. Furthermore, one cannot simply assert by analogy with Newton’s gravitation that a black hole can be a component of a binary system or that black holes can collide or merge. Consequently, all talk of black hole binaries, black hole mergers, black hole collisions, etc. does not deal with well-defined problems at all. The now commonplace claims and arguments by a great many investigators are meaningless. They are without any scientific justification. One cannot test General Relativity by means of a chimera. Stephen J. Crothers. thenarmis@yahoo.com 15th June 2006
Your objections are incorrect. Satisfaction of the field equations is a necessary but insufficient condition for a proposed solution for Einstein's gravitational field. There exists an infinite number of solutions to the field equations that are not solutions for the gravitational field. The structure of the lin-element determines the entire geometry, not any particular coordinate system. Hence, Ricci flatness and satisfaction of the field equations is intrinsic to the structure of the line-element. The intrinsic geometry of the line-element for Einstein's gravitational field fixes two radii, a radius of curvature and a proper radius. These are never the same, except in the infinitely far field where the gravitational field is asymptotically Minkowski (i.e. Euclidean). You have not understood the geometrical structure of type 1 spherically symmetric Einstein spaces, and hence your erroneous remarks about distance in Einstein's gravitational field. I repeat that Schwarzschild's solution admits no black hole and Hilbert's solution is incompatible with Schwarzschild as the one cannot be obtainecd from the other by an admissible transformation of coordinates. You have not understood the relevant papers. The claims in the cited article are that the projects will verify Einstein's work in relation to black hole physics. That is patently false, as stated by Einstein himself.
The Schwarzschild solution is a perfectly valid solution of Einstein's field equations. It is, as has been remarked, the unique spherically symmetric vacuum solution to Einstein's equations. The remarks that are basically incoherrent. It is particularly unclear what Crother thinks he means when he talks about "the" gravitational field.
That is your interpretation of the quote. A more plausible interpretation is that they are simply talking about verifying the idea of gravitational waves, which Einstein did predict. Of course, Einstein was not aware of later results which strengthened the idea of black holes as a realistic physical solution. The wikipedia page on the history of general relativity says: Of course wikipedia is user-edited and not 100% trustworthy, so you should check these yourself. But I do remember some other results that strengthened the conclusion that black holes were realistic--for example, I believe it was Penrose who showed, using topological methods, that the formation of a central singularity is inevitable once a collapsing star becomes compressed beyond a certain point, regardless of whether the collapse is spherically symmetrical or more irregular (a google search shows this is called the 'singularity theorem' and was proved by Penrose in 1964, using global methods). And the no-hair theorem showed that whenever a black hole singularity does form, the black hole's properties do not depend on any characteristics of the original collapsing star aside from mass, charge, and angular momentum. Are you aware of these various post-Einstein results? Do you dispute them?
Yep. The Schwarzschild solution is a perfectly valid solution to Einstein's field equations, but it does assume perfect spherical symmetry. The actual solution for a physical collapse is unlikely to be spherically symmetrical inside the event horizon - small departures from spherical symmetry will cause the interior solution of a black hole to become asymmetrical. Current models favor the BKL singularity as the likely interior solution for a black hole. http://scienceworld.wolfram.com/physics/BKLSingularity.html The exterior solution for a black hole will be the usual Schwarzschild solution - in fact, the "no-hair" theorem that you mention proves that in the exterior region of the black hole, any small departures from symmetry will be self-correcting. There's been a lot of work done on the subject of black holes since Einstein's day. Mr. Crother's appears to be stuck in the past, as well as being a "conspiracy theory buff. Why do I call him a "conspiracy theory buff? This is why: Frankly, I'm a bit surprised our moderators have not seen fit to take any action against this crank.
A similar thought passed through my mind. Crothers, regardless of who actually derived it, the Schwarzchild solution is inevitable (given the symmetries) thanks to Birkhoff's Theorem (Wiki page).