The Gravitational Static Two Body Problem

0:00 I will speak, in essence, on the purely gravitational, static two-body. And probably most of you will realize that there is no such thing. I mean, this is really what the topic is all about. I mean, that both in Newtonian theory and in Einstein's theory, it is impossible to have a purely gravitational case of two bodies, except for the minor exception, which I will come back to later. And therefore, why should one be concerned? and what happened when people didn't realize that this was an impossible situation. Now, in honor of our country, which we are my first and my last transparency in being French, for scientific reasons, not just that. So the first question is which relativist realized first that this static two-body problem does not exist. Any suggestions from the audience? What? Close. Lagrange. Lagrange. So, I will have at various points have to come back to the general problem of motion in addition to the two-body problem. and of that of course I talked about at the first conference and I will therefore minimize my comments only to put certain things into perspective the two body problem was really attacked very early in very general terms

2:30 by Levi Civita and by Weil who dealt with the general problem of axially symmetric static solutions and they realized very soon that you could relate this to two-dimensional solutions of Laplace's equation and you could provide the exact solutions in the static case with two apologies to Professor Singh from his book this is Peyton what I am going to talk about then is the situation illustrated on the right where you have two bodies separated this way and so what is essential for our discussion is that in what I'm going to say situation in which you can separate these two bodies here by a plane. This is not necessarily the case. You could, for example, two constants be experimented shells. So that's a different question which I will come back to very briefly. Now, Levy-Civita gave general solutions, and so did Weil in 1917-18. and at one point Lady Civita criticized Weill's work and saying that he had not taken account of certain things and Weill obviously was annoyed and wrote a response. I want to interpose on this that there was a war going on. Weill was in Switzerland, which was Newfoundland, Lady Civita was in Italy, which was on one side And Weill's paper, given the amount of physics, which was on the other side. But apparently there was no difficulty of communication. I mean, Levy-Civita knew about Weill's work. Now, Weill got annoyed and tried to clarify the situation as follows. That while Mr. Levy-Civita looks for solutions of the homogeneous equations of gravitation, which are valid where the energy momentum tensor vanishes, in short, in the empty space equations, I aim at determining the field of given

5:00 axially symmetric masses and charges. When the mass distribution in the image space of the canonical coordinates is given, there exists one and only one system of stresses satisfying the condition 2, this condition 2 meaning that the diagonal terms was equal to zero, which balances the gravitational forces of the masses. So it was perfectly clear to him that he could not have a static solution unless you had stresses. Your abscess or his? His abscess. So, now, this was 1919. 19. I should add that, as I had discussed in some detail two years ago, Eddington had realized that the law of motion in some sense is contained in the Einstein's field equations as early as 1918. Weil had realized it more or less at the same time and in his round-side he had discussed it at length in 1919 and then in subsequent editions even more so so Weil knew that not just this much about the problem of motion but a lot more but since this was a question of discussing static solutions this is his emphasis this is late there is a slight comment in a side to modern times I wish to point out Weill had said that there is a unique system of stresses. Now, this is not quite so, and I'm just, again, this view ecologist at Penrose. This is taken from the Weill centenary volume, and he says that for two mass points, you can interpret the solution two different ways, either by having a rod, or what is more commonly referred to as a strap, between the two, or instead one could reinterpret and have the two held like he calls it fishing line so that's it in short it is not quite a unique situation but this comes about

7:30 this is 1985 and this connects somewhere by Zipoy etc in the 80s which I have no possibility of discussing so the question is First of all, I want to give a very brief summary of who worked on this, just in case most of you associate the two-body problem with the paper by Bach and Weil, and then also with the solution given by Curzon. Now, actually, I've already quoted Weil 1919, so Weil knew in 1919, even though usually it too. But this is a list of people who worked on two-body stress in the purely gravitational problem, and I separated them into people who did realize that you need stresses in people who didn't. Now, the first one who worked was Frost. In 1915, he published an approximate solution of the two-body problem. What he was interested in, he said, I want I need to so that I can discuss the motion of a test particle in the field of two centers. So this is an approximate solution. He at that time did not realize that in these tests. Weil, I mentioned, Bach and Weil gave an exact solution, and they did realize it, what was going on. I also want to mention that I have been unable to get any information about who Bach was. I assumed he was a student of Weill's I could not find any biographical data on him afterwards it is also very peculiar this Bach and Weill is not the usual paper with a joint authorship but it is a paper by Bach with a nachdrag by Weill and most peculiarly in Weill's collected works it were a paper by write. So I don't know what went on there and to what extent the author should really have been clarified. Bach gave a solution for two spheres, or kugel-ähnlich, I mean, the case where it's similar to spheres, whatever that means, and

10:00 in his discussion he makes clear that he knows you need stresses for this. And then while apparently peaked because of Lady Civita, then elaborates in this Nachtrag on, yes, you need stresses, etc., etc. A supposedly exact solution was given by Dreftz in 1922, the German mathematician. It's wrong, so let's forget about it. He did not realize anything about the need for stresses. the next thing is a French thesis by Levinson and some of you with good memories you may remember that I talked about Levinson in the connection with the EIH equations that's Horace Levinson not Norman Levinson he wrote the test Université which French people sneer upon it's a short thesis I don't know why he worked in France at the time it is not clear from the thesis whether he worked with anybody it was sponsored by Borrell but I don't think that that means anything so it is an approximate solution and he has no idea of the need for stress Palatini gave an exact solution in 1923 he also didn't realize it Chazee, without initially knowing about Paratini's work, gave one in 1923, a very general solution. When I say a very general solution, I have to say how the solutions of Laplace's two-dimensional equations are connected with the actual physical problem. then you want to, most of you will remember that when you want to describe two mass points what you do is you take two rods solve the process for two rods and then do the appropriate thing to come back to the original problem now what Palatini did is he solved these not for the usual cylindrical coordinates but he used ellipsoidal coordinates and so he had a number of parameters Chalzi did the same more generally, and then at one point he realized that Palatini had

12:30 worked on this and that Palatini was a special case. It's clear that if you do one rod to get a strong solution, is it obvious then that if you do two rods that that physically corresponds to two mass? It's not obvious to me. Should I return in 10 minutes and say yes, it is obvious? so I think it was obvious to those people no questions asked also true sorry for a typo here this man is Stranio, not Stranio he's an Italian physicist who also worked on this and who also realized that you will need stresses. I will have another transparency on that in a minute. I just want to finish this. Curzon, who was characterized by Sanchez-Lon, is an obscure English mathematician, did the solution, which by now is well known. He did not realize the need for stresses. he gave zero references not even to where I will again come back to his paper later Silberstein about whom most of my talk will be wrote to Einstein in 1933 giving essentially the same solution as Curzon and published in 1936 the story which has to be discussed in detail Now, Stan Ayer, about whom, again, I know very little, was very clear on the problem. He said, he has repeatedly stressed the great difficulty of principle, which renders almost neither the possibility of applying the theory of very rigorously to concrete problems. It says it gives an infinity of formal solutions, one, this is only under the condition that such masses can remain constantly in equilibrium by virtue of their mutual interactions alone.

15:00 That's his emphasis. But such mass distributions are absolutely incompatible with Einstein's theory of gravitation. Let us consider two free mass points for which Einstein's equations have zero tensor. the energy momentum the two points attract each other and move toward each other the field cannot be static to adopt equation two which is an equation containing t i k one could adopt the hypothesis of fixing the two masses for example by means of inserting a strut which is supposed to have no mass but such a strut would be subject to a compressive force so i mean he's even more clear than right on this which is again contained in Chazee's solution by the way Darmoire was mentioned repeatedly in this conference Darmoire gave a series of in which among other things he discussed this problem and he mentions of course Chazy and he mentions Palatini and Sternio he does not mention Curzon so he did not know about it now the question is as an aside is what solutions can you get without stress static solutions requiring no stresses. Now the mass point gross there and he did the calculations in 1915, Schwarzschild in 16 you can have charged bodies in which case of course the electric force can keep equilibrium with the gravitational force this was done by Curzon as I said Curzon did not discuss the question of stresses at all but his solution for the electric charged bodies is correct does not contain any stress And then Majumda and Papapetou in 1947 working independently gave fairly general solutions for the charged bodies. Then you can have masses of both sides. Of course, it's unphysical, but I mean, since we are discussing just on what's... When you don't need stresses. This was done by Hoffman in 1959.

17:30 He presented it at the Royal Monk Conference. Singh discusses it in his book in 1960 and there's a very detailed paper by Israel and come much later and then the question is masses which cannot be separated by plane which I mentioned there you can have a Newtonian gravitational equilibrium that was discussed by Marder in 1959 as I noted all these solutions correspond to Newtonian equilibrium solutions so we have a complete analogy Now, I just want to show the first page of Curzon's paper, as I said, zero references. the thing which Levi-Ciliter and while labored on is contained in here there's no comment, he just says this can be done with Laplace's equation, then he goes on gives a solution which I will not discuss because we will have to discuss it later in connection with Lieberstein, anyhow it's only a four page paper it does contain the problem of electric the telecharge because I said it's correct let me digress on the problem of motion in general as it was understood or should have been understood in the 20s had mentioned, Eddington and Weill both understood it perfectly well, so did a number of other authors. This contained in a number of textbooks gave derivations of either the geodesic law or even an approximation of Newton's law in the early 20s. The paper usually referred to is Einstein and Germer, who came in 1927, and who essentially taught as if there had been no previous work.

20:00 And I want to read some of that because it is of importance for when we get back to the Einstein-Zimperstein concept so the Einstein-Glamour paper says the following it had a very clear statement of the problem of motion at first it restates that both in Newton's theory of gravitation contained in the field theory and in the Maxwell-Lorent theory motion are independent turning to general relativity it distinguishes three ways of approaching the question whether there are true existed dualism fields or law of motion the first approach patterned after newtonian theory takes the empty space field equations and the law of geodesics as a material of point as independent the second takes the basic field equation as an energy-momentum tensor and requires that t nu nu be singularity-free and expressible through continuous field quantities, which in turn are determined by some partial differential equations. Einstein and Germer then state, without any calculation, if one assumes that matter is arranged around narrow world tubes, one obtains from this by an elementary consideration the theorem that the axes of these world tubes are geodesic lines in the absence of a electromagnetic field. This means the law of motion is a consequence of the field law. Now, as I said before already, this is a rather astonishing statement considering that Einstein had given no indication in the decades since he had postulated the geodesic law that this was to earth but in any case it's immediately rejected by Einstein who said all attempts of the last few years to explain the elementary particles of matter by continuous fields have failed the suspicion that this may not be at all the right way for understanding the material particles has become very strong in us we are thus that the third approach any field variables apart from the gravitational and the electromagnetic field,

22:30 except possibly the cosmological term, but assumes singular world lines. And then, with their emphasis, it has turned out to be probable that the law of motion of the singularities is completely determined by the field equations and the character of the singularities without the necessity of additional assumptions. I have to read a little more of this to understand Einstein's later point of view Einstein had misunderstood the implications of the linear approximation of general relativity and so he wrote we had thought much earlier of the possibility that the law of motion of the singularities may be contained in the field equation of gravitation seem to speak against it and scared us off. The field law of gravitation can be approximated for the actually existing cases very closely by a linear law. The linear field law, however, like the electrodynamic one, allows arbitrarily moving singularity. It seems obvious that one can proceed from such an approximate solution by successive approximations to an exact one differing from it only slightly. If this were the case, then it would be possible to have that a field corresponds to the exact sort of equation that is arbitrarily prescribed motion of the singularities, and thus the law of motion of the singularities would not be contained in the field equation. But that this cannot be solved follows from investigations of x to symmetric static gravitational fields for which we can thank Weil, Levi, Celita, and Bach. They then refer to these papers and give a short proof that the singular point can be in equilibrium only if the external field strength vanishes at that point. hit the ceiling when he saw this paper and he dashed off a letter. Dear colleague, Herrglatz gave me the gallus of your note on the law of motion and general relativity. I thank you very much for this and also for the support you give it by my old idea about matter. However, I must confess that I did not understand what in it goes beyond my earlier developments.

25:00 And then he says, in my addendum to the paper by Bach, you quoted the integral over d-gamma is interpreted as a force on the body embedded in the gravitational field. Brief review on this in the realm of Zeitmateria, and then he goes on knowing that Einstein wouldn't look at the book he then outlines the whole argument so this is where we were in 27 now Mr. Silberstein who has been to my surprise mentioned in various different contexts in this conference because usually one doesn't talk about the game at all Frankenstein was a Polish physicist, he was seven years older than Einstein, one of the very few people I know who are older than Einstein who worked in general relativity. He had various minor positions in Poland, in Italy, and in England, and then in the early 20s he went to Rochester to work at Kodak and stayed there until the end and working mainly in optics. Whether this was by choice or because he couldn't get an academic position I don't know. I was unable to get any information either either from Kodak or from his family. By the time I got to these things, only his grandchildren were alive, and I got no response from them. He died seven years before Einstein, so exactly at the same age as Einstein, but shifted seven years. He wrote a book on special relativity very early, He wrote a book on general relativity, which was published in 1922. I want to show you the part of the introduction, just to show you that there was no animosity in that book at all. He apologizes for some of my readers, we miss perhaps in this volume the enthusiastic tone which usually permeates the books and pamphlets

27:30 project, with the notable exception of Einstein's own writing. Yet the author is the last man to be blind to the admirable boldness and the severe architectonic beauty of Einstein's theory. But it has seemed that beauties of such a kind are rather enhanced and obscured by the adoption of a sober tone in an apparently cold form of presentation. it's a book it presents a series of lectures it's a short book what is notably missing is the problem of motion on the other hand at least I have not seen anywhere else such an early mention of the problem of elementary flatness which we get put later on He's, I don't know if, he may have been the first one to use the term, I don't know. He has had correspondence with Einstein as early as 1918. I don't know whether they ever met. they certainly did not meet in the course of the controversy but he when Einstein came to the United States he wrote in December 33 now I want to tell you that except for the paragraph I will give you in translation which was in German, Silberstein always wrote in English, Einstein always wrote in German, which is precisely why I want to add a light touch in this paragraph, because there Silberstein writes, after greeting him, Einstein, when he's arrived to the United States, he says, I have gradually lost the fluency in the German language, I take the liberty of writing in English, the more so since you yourself probably are making more and more use of the slang. So he proceeds from then on in English, and Einstein responds in German.

30:00 So I always have to give you a translation. and it's a long letter in which he gives a two body solution but he gives it in the form of a question can you read it right there what he says is he asks about the question about the physical admissibility of solutions of the empty space equation because he says he had a two body solution as nonsense because you can't have a steady solution so what is Einstein's criterion for admissibility of solutions so I just want to go and point out something which is very hard to me but it is important namely yes yes because because it's a question of notation. Notice, we said e to the minus two nu, e to the minus two nu. Remember that. All right. So he asked about the admissibility. besides what he said, it would therefore seem necessary to set up some general criterion of admissibility of a solution of the field equation, always supposing that the theory of relativity is to be a self-contained doctrine, not borrowing special information from other sources. Einstein, I want to, before giving you some translation, I want to just show you this answer where you see the difficulties of discussing such things in these Gilbert notes. So, yeah, but I mean, that's what I have, but, yeah, there's no copy of the letter actually said. Well, it's always without formulas. I mean, you know the type thing, so it's without formulas. so I have to show you the actual so there he says as I said in German at first I was taken aback

32:30 by your static example since I believed you that the space outside the mass points is regular I was even more astonished since I myself had shown earlier that singularities will appear already in calculating the second approximation actually however the solution given by you is singular I'm sorry, I'm not using typos on this, as shown by the following considerations. So I'm back to the draft. What he says, take a small circle, and you should have a dimensary flatness, and you... I was just pointing of what's in the left. Anyhow, so he says you take a small circle here around the axis, and if you calculate this, you don't get 2 pi, therefore it's not that no elementary flatness and the solution is wrong. However. However. Dear Professor Einstein. I'm sorry to say you are quite wrong. you have inadvertently misplaced the two exponents nu and nu and that's true Einstein misunderstood and so he goes through this and says if you do it correct you get 2 pi there's nothing wrong please keep in mind the date, this is December Einstein scribbles on this letter Stimmt. Dear Mr. Zimmerstein, this is, by the way, the 24th of December, Christmas Eve. I beg you to excuse my mistake. So it is true that the existent static solution is only two-point-like singularity.

35:00 What does this signify for the general theory? that the general basis of the theory implies the correct law of motion. He sticks to that. A rigorous theory would doubtless proceed as follows. In a pure gravitational field, there are no masses. Singularities must be excluded in principle in the theory, because if the kind of singularity is not separately stipulated, which in a general theory would be arbitrary and ugly, the singularity would signify a place of lawlessness of an arbitrary boundary condition on the surface of a world to turning a singularity. In any case, your investigation shows clearly how carefully one has to handle singularities, and how empty is a feed theory which allows singularities without precisely stipulating their character. When he throws up his hands, you have a correct solution, so we shouldn't use singularities. And then he elaborates on that, and Gibberstein responds that Dear Professor Einstein, many thanks for your excellent letter of the 24th. I fully agree with you. It seems that for the present, the best plan is to make the complete field equations, i.e. this TIK unequal 0, the master equations of the theory, and if somebody finds solutions of RIK in 0 with singularities, he has to test them by considering these singularities as small regions, slender world tubes, seats of TIK, and therefore to show, to verify that these very regions are thin geodesic tubes of the field represented by that solution. In fina, the search for solutions of RIT may still be interesting and useful, but it must submit its results to this test. This is December 3rd of 1933, and there the matter rests for two years. Einstein Einstein having given up on singularities now works with Rosen and publishes the famous Rosen the bridge paper which was submitted

37:30 received in May 8th 1935 here's a paper it's in July 1, 1935 and then he discusses for these reasons writers have occasion and noted the possibility that material particles might be considered as singularities of the field this point of view however we cannot accept it all for singularity brings so much arbitrariness into the theory that it actually nullifies its loss. A pretty confirmation of this was imparted in a letter to one of the authors by L. Silberstein. Then he discusses that the David Civita by the Pepper and Milneau solution, which I think is perfectly acceptable and doesn't make any sense, so therefore this paper. Now, I should, just as an aside, I skipped it because it's not our main problem, but in the same letter of December 30, 1933, Silberstein proudly announces, I have proved that the most general spherical-semitic solution of euphetic relations is that. Einstein does not comment. He's the one, obviously, has heard of Bergkamp-Tieren. This is 1933. and later, Silberstein publishes this result in the Philosophical Magazine in 1936, and obviously the referee either, so alright, now there is some correspondence in between which has nothing to do with this problem, what is relevant, there are two things which are relevant, one is in the meantime, of course things go on in Germany and Lieberstein asks Einstein for helping for doing something for refugees and then he tries to get together with him and says

40:00 he can't come so they do not meet nothing happens and in September 35 Silverstein announces that he wants to publish his solution and on November 25th 1935 he submits his paper to the physical review takes another look because he writes to him that Mr. Silberstein, your paper has a bedenklichen mark. And he repeats his original argument with new and new interchange. So Silberstein And it's a ceiling. In short, he says, have you forgotten? Have you really forgotten? He clarified this three years ago. Well, Einstein had forgotten, but he does take a further look, and now he gets a turn. And in the meantime, various things happen. concerning the solutions corresponding to two mass points I'm greatly puzzled by your statement in the last letter namely the perimeter over diameter is uninclused pi is it possible that you have quite forgotten that you have made the very same objections in December 1933 so that was exactly three years ago and that I have then shown that you have just made a character error misquoting my formula may that you have then

42:30 january 34 written me along that apologizing hastily for your mistake and so on so on the 30th of december which is essentially immediately an Einstein response. Dear Mr. Silberstein, now I remember that you gave me your argument And then he says, I have to show the German one because that's where the diagrams are. if you take these are the positions of the two mass points if you consider this alpha then you get something or otherwise on the axis which is plus or minus cosine alpha and so one has to take the same for all of space you must get the same sine if you don't want to have a discontinuity in the first differential portion. But however one chooses the sign, one cannot obtain that the thing vanishes always on the axis. The choice of sign carries parts of the strata on the fishing lines? No, I don't think so. How do the two-star have a two-star come up here? Well, there's a square root. Yeah, oh

45:00 Dear Professor Einstein, many thanks for your prompt answer to my letter. I am sorry to say that you are again wrong. in fact et cetera what he says one should say take absolute value of course I'll show it here in English I am not yet giving up the hope of convincing you of your error, etc. I mean, the details are not important, guys. The important thing is the last sentence. If you still don't admit your error, I will no longer write about it. I only beg you not to conclude my agreement from such files. This must end is not happening. first of all he argues that this is wrong so he says what's wrong with the dick's continuity why don't you consider this as an admissible gravitational field surrounding two mass centers after all Then something happens. some reporters show up in Princeton and they question Einstein and he said something about Silberstein

47:30 Silberstein writes on February 10th, 1936 you have told some reporters in Princeton February 9th that my conclusion was based on an error, etc. in the non-euclideanism of these turklets correct did not i tell you that it can be abolished your behavior now in relation to my last letter and fear princeton report the strikesmiths quite Then, in the next letter he writes, I feel justified in assuming that you don't desire to continue any direct correspondence with me and that you prefer to drop your previous principle of fair scientific discussion and embark upon a non-geodesical in plain English crooked with your previous friend So, on March 7th, there's an item in the Toronto paper, just to give you an idea, a fatal blow to relativity issue here. Silverstein now asserts that relativity obviously belongs to that its solutions show, and so on. In the meantime, his paper has appeared, and Einstein and Rosen have answered in another publication in the Physician Review. i only show it because i don't think i'll ever have occasion to be able to show a paper which in full goes on a single time this is the einstein-rosen letter to silberstein

50:00 about silberstein i'm sorry to say repeating the argument this cosine alpha minus one and there the matter rests nothing further is said except that Einstein defends why he talked to reporters He says, I have alerted you in two letters in detail to your mistake and advised you to withdraw a publication. In addition, the newspapers contained the idiotic claim, Blötsig, that I had revised the general theory of relativity because of an earlier letter by you. By this you made it necessary for me to correct your errors publicly. Pauli, for example, told me that I should absolutely do this since the error was not so obvious that it could be noticed immediately by any knowledge of a reader. Whether I will answer later publications by you on this subject will depend on whether I consider it necessary. There were no further publications on the subject, as I said, nothing further was saved by Einstein, Nothing happens until 1941 when Silberstein takes up some objection to some minor objection to the EIH, a letter to Messers, Einstein, Unfeld, and Hoffman, which had nothing anymore to do with our problem. the long and then after that they have a friendly correspondence after 1941 nothing acrimonious anymore ok so this is a story now as I said I should also go back to Curzon Curzon essentially is not quoted until the 60s or 70s.

52:30 Nobody at that time realized that there was a Curzon solution. Nobody objected to a Curzon solution. There was no realization that Dilberstein and Curzon had anything to do with each other. But of course, now people, when they always think that it was a Dilberstein discussion discussion which clarified the matter and all i can say is that really everything which had to be said scientifically on the matter was said in the early 20s and this was really an afterthought by two people who had both stopped reading the literature and then as he went back and forth until finally Einstein sat down and looked at the problem in peace two years too late and after he had detoured on the Einstein-Rosen paper. Since I mentioned Rosen I also want to mention something find peculiar in that obviously I never worked with Einstein but apparently whenever he had somebody he worked with he talked with him only on some very narrow subjects they were working with because I mentioned that the Einstein grammar paper gave a proof that a single particle in the gravitational field cannot be addressed Rosen, who had worked with Einstein in the 30s in 1949, publishes a paper in honor of Einstein in which he proves that a single part of human gravitation cannot be addressed. I wrote about that in my previous talk, and I sent it to Rosen, and I asked him whether he had any objections to that statement, and he didn't. So apparently he, the problem of, this aspect of the problem of motion was not discussed with Rose when he was there. He said forgot, but usually the one thing he did remember was the Einstein-Germer paper, but apparently he did not tell Rosen about the Einstein-Germer paper.

55:00 uh all right so this is the end oh no it's not the end of the story final transparency the conclusions on the problem Damois in 1927 in that book wrote it is possible in the theory of general relativity to balance the forces of gravitation and to maintain two axially symmetric bodies at rest by a system of stresses whose global effect on either body by Newton's law. Nevertheless, this solution, as elegant and satisfactory as it is, can only be considered as an encouraging contribution to the real two-body problem. The real problem of the free motion of two masses requiring a d square with two mass cubes is not at all involved. Even about the problem of two equal masses rotating around the mass in the circle, one knows that d is nothing. Conclusion 1988. I just wondered, do you get the impression from the later correspondents that during the period between 36 and 41, Zilberstein realized and accepted his mistake? No. Or he just didn't feel it was worthwhile discussing that? No, there's no indication. He never withdrew in print his claims. No. And then, I think I mentioned, I wrote to Kodak a couple of years ago, and asked him, They must have liked us, and Sieberstein worked there for maybe two decades, and of the consultant they are finding young people's best, and they have any material, and they said they have nothing, so.

57:30 Just at this point, you tried the University of Toronto, and you had a young people in life, for sure. No, I have not tried the University of Toronto, but of course by that time, he was no wrong But still, I mean, you might find narrow... Yeah, I read that, but I know of nothing. By the way, this textbook is excellent. Yeah, the textbook is good. And as I said, I mean, it's peculiar that it does mention elementary flatness, and then the argument... I mean, at the first, the argument all essentially revolves around the mental flatness, and that's what it is to me. Is there anything else? The next session, which will begin, I believe, around 11.30, is one which will take place this afternoon. But this afternoon begins at 11.30. Thank you. Thank you. Thank you.