H.A. Lorentz's Attempt at a Coordinate-Free Formulation of GR

0:00 In 1960, the state of affairs in mathematics, more specifically a differential geometry, was hardly such that one could expect such an ambitious program to succeed. One of the key concepts in any coordinate-free formulation of general relativity, namely the notion of parallel displacement, was not yet available. It was not introduced until one year later, in 1970, by Lady Chiquita. Not surprisingly, Lawrence, at several points of his article, runs into serious difficulties because he does not know about parallel displacement, and so eventually he fails to reach his goal. Cornel's almost inevitably major appearance. In this talk, I want to discuss some important features of Lawrence's approach towards general relativity, and I will concentrate on his geometrical interpretation of the curvature scalar. It is in this area that his problems with parallel displacement most clearly manifest themselves. But before I discuss the contents of Lawrence's paper, I want to address another issue, namely the apparent discrepancy between Lawrence's involvement with general relativity and his attitude towards special relativity. It is well known that Lawrence, to the end of his life, persisted in his belief in the existence of an ether. based theory all kinds of kinematical effects from special relativity are explained either as dynamical effects or as artifacts of measurement. One might even doubt whether Lawrence ever came to understand that in special relativity things like length contraction or time dilutation can simply be explained as geometrical features of Minkowski in space-time. Given this attitude towards special relativity it is somewhat puzzling that Lawrence of all people should foundation of general relativity. In order to understand Lawrence's concern, we should take a closer look at his motivation in casting general relativity and coordinate-free geometrical form. Nowadays, the purpose of eliminating all reference to coordinates from a theory is to bring out more clearly the geometrical content of the theory. Lawrence, on the other hand, is not interested, at least is not primarily interested, in this intrinsic value of the For Lorentz, the geometrical approach is just a way to guarantee general covariance right from the start. Lorentz writes, and I quote,

2:30 In such a geometric treatment, the introduction of coordinates will be of secondary importance, dot, dot, dot. So we are sure beforehand that the general covariance of the equations, unquote. But then again we can ask, why did Lorentz feel so strongly about general covariance? From his point of view, there seems to be absolutely nothing wrong with privileged frames of reference. And as a matter of fact, in their extensive correspondence on general relativity, Lawrence more than once pointed out to Einstein that in almost every physical situation, there are privileged frames of reference. But, as Lawrence realized, reference frames can be physically distinct while being empirically equivalent. And at least for Lawrence, it is the empirical equivalent which provides the rationale for the monogeneral covariance. As Lawrence explicitly states in his article, it was a remark made by Einstein in their correspondence has convinced him of the need for general covariance. Einstein had noticed that physical theories essentially only do with point coincidences such as the intersection of the world lines of material particles. Einstein's remark made in a letter to Eryphes, which Lawrence also read, has to do with the resolution of the famous Hall argument, the argument that generally covariance field equations do not uniquely determine the gravitational field as Einstein at that point felt they should. This argument seems to have reconciled Einstein, at least for a while, to the restricted covariance of the field equations of the Einstein-Roseman theory. But since the field equations of the final theory are generally covariant, what Einstein in 1915 clearly no longer believed that the whole argument forms a valid objection against general covariance. As he explained to Ehrenfest, uniquely determined. The only things which have to be uniquely determined are point coincidences. Now, remarkably, Lawrence, in 1915, thought of virtually the same objection against generally covariant field equations as Einstein's some two years earlier. When he read Einstein's remarks about the whole argument in the letter to Irrenfest, he immediately grasped Einstein's point and was very excited about it. As we know from a letter he wrote to Irrenfest that very same day, Lawrence had immediately written to Einstein to congratulate But not surprisingly enough, in arguing for general covariance in his own articles, Lawrence stresses the fundamental importance of point coincidences. In summary, Lawrence's involvement with general relativity did not force him to give up the

5:00 cherished notion of an ether, or more generally, the notion of permanent strains of reference. Einstein's remark about the importance of point coincidences, however, had convinced of the fact that at an empirical level all reference frames are equivalent. Therefore, all physical laws should be generally covariant. And in order to guarantee that the equations really are independent of an arbitrary choice of coordinates, Lorenz set out to give a coordinate free geographical formulation of Einstein's theory. So much for his motivation. I now turn to the contents of Lorenz's paper. In fact, I will only discuss the sections on the curvature scale in detail, but to give you an impression, though, range of topics Lawrence addresses I will start with the global survey of Lawrence's article. The title simply is on Einstein's theory of gravitation. It was submitted in four consecutive communications to the Dutch Academy of Sciences between February and October 1960. I've already discussed the first three sections of the first part where Lawrence argues for the or covariance. In sections 4 and 5, Lawrence has introduced the important concept of indicatrices. They are to take over the role of the metric tensor in Lawrence's formalism. The idea is very simple. At every point P of space-time, Lawrence considers the set of points lying at some fixed infinitesimally small distance, plus or minus epsilon, from P. In local Lawrence frames at P, these hyper-surfaces have the shape of mass shells in energy-momentum space, The part in the time-like region is called the endocatrix, and the parts in the space-like region is called the conjugate endocatrix. The purpose of endocatrixes and conjugate endocatrixes is to provide natural units, as Lawrence calls it, for measuring length, area, and volumes. For instance, consider the line segment PR, it is measured in units PQ, and now ascribing the length epsilon to PQ, it is easily seen that PR obtains the length 3 epsilon, et cetera, et cetera. Using length, areas, and volumes expressed in natural units, Lawrence goes on to define various Lagrangians in the geometry. The Lagrangian for the set of point particles,

7:30 for the gravitational field, and for the electromagnetic field. After having defined various Lagrangians, Lorentz derived the field equations from a variational principle. And then the remainder of his article is devoted to some inconclusive considerations on the energy momentum of the gravitational field. I will now turn to Lawrence's geometrical interpretation of the curvature scale. At the beginning of section 7 of his article, Lawrence tells us that he knows from a paper by Hilbert, as well as from his correspondence with Einstein, that one should take the curvature scale as the Lagrangian density for the gravitational field. Since both Einstein and Hilbert had already communicated the analytic expression for the curvature scale, Lawrence is only concerned with his geometrical interpretation. Einstein himself, in his papers from 1913 to 1915, does not pay any attention to the geometrical meaning of the quantities he borrows from differential geometry. Whenever he wants to use some quantities, he only asks whether it has the desired transformation properties, and whether it fits in with the natural generalization with a natural generalization of Newton's gravitational theory. This is only natural when we take into account that Eisen and Grossman took the approach to differential geometry from a paper by Ritchie and Levy-Chivita in 1901. Of course, Professor Norton told me this is only part of the story, but I didn't know that when I was preparing my talk. So, in Richard and Levy-Chivita's papers, as is well known, the foundations are laid for modern tensor calculs. In their approach, traditional geometrical methods largely recede into the background and give way to purely analytical ones. To give an example, to show that some quantity has geometrical substance, all one has to do is to invoke its covariance properties. to show that the quantity represents some geometrical entity which can somehow be visualized. So much for Einstein's geometrical background. Lorenz on the other hand, as we know from his notebooks, deposited at the State Archive in the Hague, learned differential geometry from the older group by Bianchi.

10:00 The original Italian edition dates from 1886 and only deals with the theory of surfaces. In 1899, a German translation appeared to which two chapters on n-dimensional manifold were added. But even in this updated German edition, the nature of the work is still much more geometrical than in Ritchie and Levi-Civita's paper. Now, using Bianchi's book, we can reconstruct Lawrence's interpretation of the curvature scale. Lawrence's own article is rather cryptic at this point. Furthermore, he doesn't cite any literature on differential geometry. Lawrence apparently assumed that these ideas were well known among his colleagues. To the modern reader, however, Lawrence's reasoning does not look familiar at all. This is probably because of the fact that in modern textbooks, the geometrical interpretation of the curvature scalar is seldom found. It has become common practice to turn to the free unprotected curvature tensor instead, which can be given an elegant geometrical interpretation with the help of the notion of parallel displacement. And shortly after the appearance of Le Cifita's a pop-making paper, this interpretation really seems to have eclipsed the original interpretation of Hermannian curvature, which draws heavily on the theory of curved surfaces. For instance, Hermann Weil, in the first edition of his influential book Round Side Materia, in 1980, already introduces Riemannian curvature in the context of parallel displacement. But in 1960, however, just before Leif-Chief does work, Lawrence could safely assume that his readers would know what he was talking about. Most of the ideas he uses go back directly to Gauss and Riemann, the founding fathers of differential geometry. And I have outlined Lawrence's interpretation of the curvature scalar on the transparency. The basic idea is, and I'll read it slowly to you while you study this drawing here. So the basic idea is that the curvature scalar at an arbitrary point P in four-dimensional spacetime equals the double sum of the Gaussian curvature of six hypersurfaces made up of geodesics in the direction spanned by a pair of vectors chosen from a set of four independent I hope you can follow that. It's all in the figure here.

12:30 This idea goes back to Riemann. Schematically, the chain by which the Gaussian curvature of these hypersurfaces linked to the Riemannian curvature of space-time as a whole is given in this line. Here, R1212 is the curvature tensor of the surface, and R0sigma minu is the fully co-variant form of the curvature tensor of space-time as a whole. The actual derivation goes as follows. The curvature scalar is obtained by contracting the curvature tensor. Using a local Lorentz frame, or a fear bias, we would nowadays say, we can write the components of the metric tensor in this form. Inserting this expression into the equation here, we obtain this form. Now, when a square of the norm of the factor psi a is denoted by delta a, I've written that here, this equation can be rewritten in this form. Because of the symmetries of the curvature tensor, this sum effectively contains only six terms, each of which should be taken twice. Now listen to them here. AB is 0, 1, 0, 2, 0, 3, 1, 2, 1, 2, 3, and 2, 3. I will call all the terms in this sum KAB. What we will have to do now is to prove that KAB is the Gaussian curvature of this geographic hyperservice. This proof can be split into three parts. I ain't going to say very much about it, just to give you the general idea. First it can be shown that the Gaussian curvature, which as usual is defined as the product of the so-called two-principle curvature, is equal to the ratio of the determinants of the second and first fundamental forms. That's the first line here. Then, one can prove that the determinant of the second fundamental form is just the curvature tensor of the surface, independent component R1212. Now finally it can be shown that there is a simple relation between the curvature tensor of these geodetic hyperservices and the curvature tensor of space-time as a whole. That's 0.3 here. Combining these three results we obtain this expression here for the Gaussian curvature. You see that it is almost identical to the expression occurring in the Sun here. The

15:00 Here we have the determinant of the components of the metric. And here we have these quantities, delta A, delta B. Now, up to now, I have said nothing about the coordinates I'm using in this calculation. And it's actually so that in this coordinates, the metric at the point P is diagonal. So, it's just a product of plus or minus one. You see that it corresponds with the delta A, delta B we have here. But these expressions really are equivalent, and this concludes the proof of Lorentz's claim that you can build up the Riemannian curvature of space-time as a whole from Gaussian curvatures of these geographic hyperservices. Lorentz now wants to give an operational meaning to this still highly abstract concept of Gaussian curvature. In fact, he wants to generalize the famous theorem by Gauss in the theory of curved surfaces on the relation between Gaussian curvature and the so-called angular excess. Consider some geoletic triangle PQR. A geoletic triangle, of course, is a triangle which has three geoletics as its size. In curved spacetime, the sum of the angles of the triangle deviates from the Euclidean value pi. The amount by which it differs from pi says, and it's denoted by E, for this quantity, Gauss derives the relation E is kappa times delta, where kappa is the Gaussian curvature of the surface at this point. I should say that this geometric triangle is infinitesimally small. And delta A is the infinitesimal area of this surface. Lawrence now wants to use this equation to replace the abstract concept curvature by the more concrete concept of angular excess, which probably can give an operational meaning. Now, this last step in Lawrence's interpretation of the curvature scale is very interesting, albeit in a somewhat negative way. When I saw this last step, I

17:30 felt the suspicion that in the end, Lawrence's whole interpretation of the curvature scale can simply be subsumed under the modern interpretation of Hermannian curvature in terms of parallel I must confess that I can only present a kind of a hand-waving argument to substantiate this claim. My argument will be suggested, though, at all. Consider a factor A at a point B. We are going to transport it around a closed circuit, P, Q, R, S, P. And when we get back to P, the factor A has changed to the factor A bar. For the change, delta A we can write, suppressing all indices. This, of course, is the shaky part of the argument. Delta A is A bar minus A is the curvature tensor times the factor A times these displacements DX1 and DX2. Now insert that delta A can also be written as A times E. E is here, here is the angular excess, the deviation from the Euclidean value 2 times pi, we would expect in this case. When we insert delta A is... When we insert this equation into this one, the fact that A drops out at the left and the right hand side, and what is left is this equation. e is r times dx1 dx2 which of course is just the area of this square pqrs now we see that this equation has exactly the same form as the equation from which Gauss has derived for geodetic triangles so it seems that Lawrence's interpretation of Hermannian curvature can indeed be assumed under the modern interpretation of Hermannian curvature interpretation has some obvious advantages over Lorentz's. We can deal with space-time as a whole, rather than have to cut it up first in two-dimensional sub-manifolds. And furthermore, we don't have to restrict ourselves to geodetic triangles. Any closed circle actually will do. What this admittedly tentative argument shows is that one is seriously handicapped when trying to give a geometrical coordinate-free formulation

20:00 of general relativity, while lacking the concept of parallel displacement. Now let me finish my talk by quoting a passage from Lawrence's article which clearly shows that he was actually aware of these problems. I'll leave it to you. It must be remarked that if an equation like 10, in which we are concerned with the composition of factors at different points, a definite meaning we must know which components are to be considered as having the same direction this has been determined by the introduction of coordinates so lawrence's allegedly coordinate free formulation of general relativity is not coordinate free after all whether or not two factors at different points are considered as being parallel to one another depends on an arbitrary choice of a coordinate system from a modern point of view this of course is a fatal of a flaw. Lacking the concept of parallel displacement, however, one can probably do no better. So to sum up, working just prior to Leif Chivita's findings, Lawrence's pro-get was doomed at the outset. Thank you very much. Thank you very much for your question. Are there any questions? Just a question. Is Lawrence Is it hyper-English? It is. All articles in the Sidningsverslagen are appearing in a simultaneous translation. I have a copy with it if you're interested. I'd be very interested. I'm not sure if you have understood your technical argument about the Gaussian curvature and the principal radius of colors. because in Gauss it works in two-dimensional surfaces and you multiply the two radius of curvature and then what does Lorentz do? Does he multiply the successive principal curvatures defined along the various hypersurfaces? he deals with one service at a time to build up the Ramanian curvature scale you need six hyper-services of the kind I told you about and then what you have to do is to embed such a hyper-services in Euclidean space and there you get the whole machinery of the theory of curve-services and you define the principle of curve-services

22:30 and you multiply them, etc, etc no, no, no after you've obtained the Gaussian curvature for one hyper-surfaces, you do it for the other five ones, and then you add them together. I can show you the transparency. So the idea is that this expression should somehow be proven that this is equal to the Gaussian curvature of such a It's such a geodentic hyperservice. And this is done in the . I can show you the detailed calculations here. May I tell you, attention to the fact that the way our witnesses of a new step in a, so to say, in a boom, in the re-appreciation of Lawrence, especially in the field I think that this renaissance of appreciation of Lorentz was commenced by, as I think, one of us as young, who was a Fulbright fellow in the Netherlands, and now she works in Princeton University. and she is the editor of the Lorenz Selected Papers, and it was followed by Professor Koch's paper on Lorenz, General Relativistic Achievement, and also the next paper, which was submitted only two months after Koch's paper, is mine and also in the archive, this on the press, also on the connection between Einstein and Lorenz, which overlaps in some historical parts with the lecture, but it's not so deep in its technical detail. And I Something begins in the appreciation of Lorentz, and so we may end the traditional concept of Lorentz as a classical physicist, a traditional physicist as the opponent of everything which is not just relativistic.

25:00 So I'm very happy and enthusiastic to hear this fine lecture. Maybe I may add something to that. First I think that we are determined to pursue this line of research together and hopefully also maybe in collaboration with you because we could get together on that. And also, I think the forthcoming publication of the correspondence of Lawrence which I'm working on will, I think, and I hope, contribute also to this re-evaluation, especially with Lawrence's work in general relativity, because both the correspondence between Lawrence and Einstein and the correspondence between Lawrence and Ehrenfest I think we should move on to the next speaker. The next speaker is Professor Kichinashimi of the group and parents who will speak on the other place in the first place. Thank you. More than 60 years of the draft formulated special dimitivistic equations for Wohlenbeck and Goodsmith's thinning electrons. No consensus exists as to what should be the coupling of electron and gravitation. Essentially, there exist two descriptions, one in the space-time V4 of general relativity

27:30 and the other in an Einstein-Kartan space-time U4 with torsion, such that when the gravitational field is switched off, the first goes over to the special relativistic description and later not. Indeed, measurement of the precession of the spinning particle or a Stern-Gerlach type experiment may, in principle, settle between these two descriptions but remain We note that general relativity has the advantage of presently being the only relativistic theory of gravitation, accounting for all known observational tests, while Einstein-Carton theory is as attractive as it is apparently in the trend of gauge theories. Now, the Iraq equations are four linear partial differential equations for what was in 1927 a new object, the four spinner, although this bird was coined only in 1929 by Van der Waarden who also developed a spinner calculus. Spinners were in fact discovered in 1930 by L.D. Carton in his classification of simple representations of simple Lie algebras. Their nature was investigated by many others and clearly elucidated by Brouwer and White in 1935. Spinners are the geometrical objects defining the irreducible representation space of the Clifford group in the end, the universal covering group of the orthogonal group S over N. Moreover, in the case of N equal to 4, in generally even, the 4 spinner may be constructed out of the reduced spinners

30:00 or half spinners, to use the terminology of Cheval. Thus, when as early as in 1928, the extension of Dirac theory to general relativity was considered even before its standard interpretation were established, one had to face two chief problems. Definition of spinner fields on a manifold and derivation of the field equations they obeyed. The evolution of these problems owes much to Erman Weil, and we now sketch it here. To define the spinner-spinaries in a manifold, two different approaches were used. orthonormal tetrad approach respecting the group properties of spinners and the general covariant approach using the classical Freemanian techniques. In the orthonormal tetrad approach, first used by Fork in 1928, the transformation law of a spinner under a rotation derives from that of the vector corresponding to the spinner and its conjugate thus introducing an indetermination leading in the differentiation of spinners to four new quantities which he hopes to relate to the electromagnetic potential. However, Hermann Weil in 1929 connects the space transformation of the spinner C with the gauge transformation and shows the necessity of replacing in the Dirac-Libb-Ladon-Gen of an electron in an electromagnetic field the derivatives dkc by dk plus e akc, where ak denaps the electromagnetic potential. He remarks also that ak plays the same role as the connection in general relativity

32:30 and thus appears to be the first promoter of gate fields. The general co-ordination approach was initiated by Tetrod in 1928 and developed by Schrodinger so as to reconcile with Fogs' attempt. However, Infeld and Wanderwarden in 1933 took a more drastic view. They consider spinners and tensors as autonomous objects with respect to transformational laws, but connected via some mixed quantities in the basic two-spinner space, thus allowing a rich structure on the manifold. This view was nevertheless amended by Bergman, Bede, and Yale, who introduced some out-of-conditions which made these spinners equivalent in B4 to those obtained by the ONT approach. Now, Elie Carthon in 1938, like Hermann Weil, insists on the determination of spinners with respect to the orthogonal group and not to the general linear group, as it follows from the general covariant approach. Indeed, the last theorem of Cappan's book asserts the impossibility for spinners to define the linear representation space of the general linear group. He again stresses these difficulties in the postumous edition of 1974, not in the edition of 1931. Cela tient à ce que les spinners fournies rotation ecclidienne, mais ne peut fournir aucune représentation du groupe AFI. On peut, du reste, justifier cette impossibilité par des raisons orthologiques. I translate this stems from the fact that spinners provide with a linear representation of the water-warner group, but not with a linear representation of the general linear group. Moreover, one Chayam justified this impossibility with purely topological reasons.

35:00 Seemingly, these topological reasons are those given by Afliger in 66 and Milner in 63, that is the varnishing of the two first two stifle-wisne characteristics. A systematic introduction of spinners according to Elie Capitan, is given in Lichner Rubik's in 1964. However, Marcel Gris finds Elie Capitan too severe towards the physicist who sticks to general covariance as a guiding principle, and probably for analogous reason, Brill and Wheeler and R. Penrose adopt the Amander general covariant approach. Penrose and Greenler in 1984 find even more useful to consider two spinners as primitive elements. They give also an axiomatic approach to the covariant differentiation of spinners and show that uniqueness implies the the conservation of the inner matrix and the varnishing of the torsion. Now to the field equations. If the first atoms obtained formally general adivistic description by replacing ordinary derivatives gk by covariant derivatives nablaki in special relativity description relativistic description that of wild of 1929 give general relativistic description and energy momentum contribution of Dirac fields through a variational principle of Hilbert type that is using a Lagrangian formalism with tetrault fields Ea and matter fields C as independent variables. The connection gamma having been expressed in terms of Ea and its derivatives. Otherwise, if we accept studies of Dirac equation and Einstein-Meyer theories or in Kapton's book There was a pause until 1948, when Weill returns to the problem and displays a different set of field equations, invalidating the Riemannian structure by using the mixed variation.

37:30 That is a Lagrangian formalism where we vary independently the tetraphy fields, the matter fields, and the connection. General relativity can then only be recovered by adding some terms to the Lagrangian. It is interesting to note that Weil has republished the same paper in the Physical Review in 1950 as many misprints in the preceding one appeared in the Mexican Review obscured its content. Unnoticed by Brill and Wheeler in the study of the Netrino, Weill's paper was compiled with Tetrault's idea of the spin giving rise to a non-symmetric energy tensor by G. V. Tiama in 1962 who re-derived Weill's equation relating spin and torsion and introduced a force coupling curvature and torsion. This theory is independently obtained by Keeble in 61 by extending the Young-Nilz procedure to Lorentz-Krupp. The new equations known as Einstein-Kartan-Skyama-Keeble equations, LCS-K, presuppose an Einstein-Kartan space-time with a metric-preserving non-symmetric connection of which the anti-symmetric part is nevertheless arbitrary. This should be noted that when we write B, K, J, E, C equals 0, the equation leaves totally arbitrary the torsion. One may remark that when no matter is present, general relativistic description obtains. Einstein, Carton, Schiava, Kiebel,

40:00 Lagrangian may be reduced to general relativistic Lagrangian plus quadratic terms in torsion, so that subtracting them one gets general relativistic description. These ideas are more or less reformulated without almost no real progress until early 70s, as if the community was as perplex as Hermann Goyle in 1950. General relativistic description continues to be an object of study, as in Lysnerodricks 1964. The first interesting step in the understanding of his theory was made in 1975 by Korsinsky, who studies through the influence of different constraints, euclidicity of the connection, varnishing of torsion, etc., in a nine-strand cartoon space-time, although euclidicity constraint was passingly considered in 1971 by Lunoire. The technique of introducing constraints is exploited in a number of papers by Ale and his co-workers from 1971 onwards. To establish, Yale and other co-workers try to establish the rational of the relation spin coercion, while Nestor in 1977 displays an effective equivalence between Einstein-Carton's gamma-article and general relativistic description. So that, Lorentzian and Einstein-Carton's space-time may equally serve for the description of spin particle. In fact, as they have shown elsewhere in 1986, the systematic use of Lagrangian multipliers in space-time with a metric preserved

42:30 in connection leads, if you use the Lagrangian gravitation matter under constraint, leads to general relativity when the matter part of the Lagrangian does not depend on the connection gamma. To Einstein-Tatton's gamma-tubule theory, when these parts depend on gamma as a consequence of minimal coupling. To general relativistic description, when matter part depends on gamma and the torsion is vanishing, then this last constraint leads to the Rosenfeld the well-informed asymmetrization of the energy tensor. when gravitational field is switched off, general relativistic description goes over to special relativistic description, whereas Einstein-Carton's scam article gives a teleparallelism theorem, that is, the curvature null and Tarton null punishing, which has, in contrast to special relativistic description, no experimental basis. Torson introduced the unwanted Eisenberg-Pauli type terms in hierarchy cases. However, some kind of local equivalence principle may be preserved by requiring that Einstein-Kartanskyama-Tibul Lagrangian reduced to special relativistic Lagrangian when connection is varnishing, as assumed by von der Heide in 1975. On the other hand, in order to settle between these descriptions, the motion of spin particles has been studied through different methods, conservation laws, Nicholson-Power-Betro approach, post-Newtonian methods, classical limit of quantum equations motion through W, K, B or Corbens algebraic method. The others who study these equations are indicated there. Aldama Sotman, Coel, Yaskin, Steger, Liebscher, Nisch, Rump and Wodrecht. on his last paper or published in a conference by Professor Bergman and Sabata in 1980.

45:00 Therefore, generally, it appears that Stern-Gerlach type experiment or measurement of spin precession of a Dirac particle may give the right answer. Although, the relation of the torsion responsible for this effect to the source is far from being clear. Therefore, general relativistic description and the whole weight of general relativity in its favor, while Einstein-Carton schema achievable, seems attractive as one hopes that Einstein-Carton space-time may be the basis for Pointer-Aid-Gaith theory. However, as was shown by Martin Schwetzer in 1980, the existing so-called Poincaré-Gerge theory just combines local Lorentz invariance with general covariance to get local Poincaré invariance. Moreover, the arbitrariness of the torsion invariance is a very important part of the equation. How could spin be directly related to the gravitational potential or come to some new quantity as the torsion? This debate will not be further conducted here. I would rather conclude by observing that was indeed an amazement for me that very wise contributions to this field are now being ignored by the present protagonists of Einstein I should like to point out that at some point, I think that it's about topological considerations, there was some confusion which I should know, but I would like to comment on the following

47:30 practice in recent years, that she's beginning in 1977. always likes to object to the statement that there are no spinners for the general linear component to value representation. To a linear group, and you point out that this is more accurate, that this is only to of the linear group, and therefore you can then, without the introduction of the pure He has a few of these arguments, first, only for the group GL2R, and succeeded now to go to GL3R and GL4R, so we are in four dimensions now in this case. but that's an interesting historical point the historical point is that Neumann says this restriction to the finite number of components in Carton's book is not very clear I mean, where the theorem appears first it's not mentioned just because the whole book is only a finite number of components it's self-evident Later on, the illustration of the final number of components appears explicitly, and so there is no doubt about this. So, nevertheless, this way of lettering leaves some possibility open for discussion. However, I found a paper of Carton in 1936. It's an address to the Con-Mathematical Congress in Boston. And there he, again, gives the argument, and he explicitly says, a finite number of conforms. He gives the argument to complex implications. And when I read further a little bit, then I found a remark concerning the general linear group and its covering groups. And again, it is expressed in a way one has to guess.

50:00 I see from the clear terminology which we have nowadays. So maybe historians can see from this paper what was going on at this time. I can give only not a good comment because I had the opportunity of hearing Neymar When speaking of these things, I don't say I was really convinced, but maybe I didn't understand that. I never had paper in my hands. So, and he was, no, I'm not sure he is right, but I post on my mind. In this question, the situation is relatively clear because in Dimension 2, Dimension 2 is exceptional in general for the Spinner's situation. Dimension 2 is exceptional. And the situation that may not describe is not possible except in the Dimension 2. It is not very interesting. You change the definition of the spinors, of course. The definition of the spinors by the spinors is changed. But if you change to an infinite linear optimization, at least dimensional linear optimization, we don't have time, of the good time, the dimension of the two. I just want to make two comments. first is that in the direct equation coupling to torsion is only to one irrepressible part of the whole torsion no the axial part so you could have torsion even in the easy sk theory without noticing it because if the axial part is vanishing it doesn't occur in direct equation no you know that there is a paper by Ray and Snelly probably in 84 or 83 or something that very clearly shows

52:30 that if spin as to be related to torsion it must be the totally anti-symmetric part of the torsion right only that part only that part of torsion if the geometry is a geometry with torsion and the totally unsymmetric part is not there and of course we can't measure it by an electron if the other two parts are there, the tensor part and the vector part okay, yes, we agree and the second comment is that the status of the Poincare-Lieck H-theory is far from clear I mean, first, fall to the dynamics, and second, you want to describe gravitation as a case theory, and it's not clear if you should take a Poincaré group or a Poincaré group broken down with a large group or another group, conform group, a general linear group as well. I mean, that's part of the idea. Now, the only Poincade-Gaith theory which is available in the literature is that of A, and Schweitzer has proved that it is not a real Poincade-Gaith theory. That's what I was telling. I don't completely agree because the theory discussed in the literature is a nine parameter, an equation is nine free parameters in addition to Are you speaking of four categories theory in general or just in the case of ECSK? In general, not ECSK, in general. That's a nine parameter nine free parameter theory But the problem is this, to know how to gauge, how you gauge the translation. The way was given by Hale by taking some equations. And Schwarzscher, in the same context as the book by Bergman and Sabata, spin, torsion, and supergravity. In this book, Martin Schwarzscher shows that it is only local Poincarei invariance which is obtained and not a global Poincarei invariance. This is exactly the trend of the argument of Martin Schwarzscher in this paper.

55:00 The next speaker, Professor Shretniaga of the William University of Krakow will speak on the Meyer of Hutchins work on the general motion in the general motion. Thank you. Thank you. Thank you. Thank you.