Brief Report on the First International Scientific Workshop “Geometry of Finsler Spaces with the Berwald-Moor’s Metric”
The first international scientific workshop took place on October 15-22, 2005 in Cairo, Egypt. It was organized by the “Finsler Prize” noncommercial foundation for the development of the Finsler geometry research. It was also supported by Moscow Bauman State University (rector I.B. Fedorov, head of the Physics department A.N. Morozov, fellows T.M. Gladysheva, V.O. Gladyshev, D.G. Pavlov). This workshop was a logical continuation of the work of the corresponding Section of the International Conference “Physical Interpretations of the Relativity Theory” (Moscow, 2005). The participants of the workshop were scientists from Russia, Romania, China and Great Britain.
In the introductory talk D. Pavlov discussed the space with Berwald-Moor’s metric from the point of view of the observer living in it. It was shown that in the broad range of the parameters it is impossible to distinguish this space from the space-time of special relativity. The differences appear only for some specific conditions. This means that the geometry of the real space-time may be regarded on the four levels: first is the classical Galileo-Newton geometry on the base of the linear metric form, second is the Minkowski-Einstein geometry on the base of the second order form, third could appear to be connected to the Chernov’s cubic metric, and finally, the fourth is the geometry with Berwald-Moor’s metric which has the fourth order metric form. The fourth level is beautiful and simple and simultaneously the most substantial level. This is due to the fact that the geometry of such space is closely connected to the commutative and associative algebra of the hyper complex numbers (that are not Hamilton’s quaternions), the so called quadrahyperbolic algebra. The simplicity of the quadra numbers does not mean that the space connected with them is trivial. Besides the usual isometric and conformal transformations, the preferential role is played by several classes of reflections that are absent in the quadratic geometries.
The last problem was discussed in detail by G. Garas’ko who established the correlation between the generalized conformal transformations in the space of quadranumbers and generalized analytical functions of these numbers.
G. Asanov in his talk gave the explicit formulas for the transformations of the coordinates in the Berwald-Moor space which are analogous to the Lorentz transformations in the Minkowski space. The analogous relations in another form were given earlier by G. Bogoslovski and G. Garas’ko, the fact that lead to the priority argument.
G. Bogoslovski reported on the changes that take place with the equations connecting the 4-momentum of a particle with its velocity in the space with Berwald-Moor’s metric. This talk demonstrated the perspectives of the use of the Berwald-Moor’s metric in the investigations based on the Hamilton’s formalism.
As it was shown by V. Chernov, the geometry of spaces with Berwald-Moor’s metric has a close and natural connection not only with the Galileo’s and Minkowski’s geometries already used in Physics, but also with the geometry whose metric is based on the third power of the components of the vector. Such metric function produces a space which is an intermediate case between the space-time of the special relativity and the space with Berwald-Moor’s metric.
In his talk S. Lebedev introduced the many index generalizations not only for the metric tensor, but for the connection coefficients too. This develops the qualitative approach suggested by D. Pavlov. The application of such methods for the calculation of the curvature tensor could lead to the non-zero constant value for the scalar curvature of the indicatrices in the Berwald-Moor spaces. When the traditional methods based on the two-index metric tensor and three-index connection coefficients are used, this value appears to be zero.
The final stroke on behalf of the Russian theorists was performed by S. Siparov who had drawn the attention of the participants to the fundamental problems which appear in Physics when the Finsler geometry is used. Simultaneously, he discussed several situations that could be effectively investigated with regard to the possible anisotropy of the space-time. A simple approach to the construction of the canonical equations for the Berwald-Moor spaces was presented.
The only experimental research presented in the workshop was performed by V. Pancheliuga and S. Shnol’. They investigated the time dependence of the alpha decay and found out that the fine structure of the partition function of the decay rate is very sensitive to the space orientation of the special experimental set up. The authors concluded that a certain global anisotropy affects all the processes, and it could possibly be linked to a Finsler type of the geometry of the real space-time.
Other experimental data that point at the possible anisotropy were obtained by J.-P. Luminet (France) and D. MacMillan (USA). These researchers had been invited to take part in the workshop but they could not come because the invitations came too late. Hopefully, they would come next year when the next workshop would take place in Cairo.
The Romanian scientists and their leader G. Atanasiu (head of the Geometry Department of Brasov State University) presented a series of talks dedicated to the development of the notion of scalar polyproduct introduced by D. Pavlov. This notion generalizes the regular scalar product of the Riemannian space for the more complicated case of Finsler space with the non-quadratic metric.
In the report by V. Balan and N. Brinzei the vector bundles endowed with (h,v)-metrics were presented. The vertical part was provided with the flag-Finsler Berwald-Moore metric, while the horizontal part was Synge-relativistic optics metric. The last one was presented as a polyproduct reduced to a more common form. For this model the extended Einstein equation and Maxwell equation were constructed.
G. Atanasiu, V. Balan and M. Neagu introduced the 4-polyforms for momentum that were analogous to the corresponding coordinate polyproducts and investigated their applications for the generalized Hamilton spaces. This talk had much in common with that presented by S. Lebedev and this points at the consistency of the approach used.
In the talk given by M. Paun the invariant Finsler reference frames for Berwald-Moor metric were presented. In the review talks by G. Atanasiu and N. Brinzei and G. Atanasiu and V. Balan the Berwald-Moor metric for the tangent and cotangent bundles of the second order were discussed.
The discussion of the results obtained by the Romanian colleagues lead to the agreement to sign a three year contract between the “Finsler Prize” foundation and Brasov University (Romania) concerning the research in the field of spaces with Berwald-Moor’s and Chernov’s metrics.
In the talk given by Chinese scientist K. Mo it was shown that all the bi-invariant Finsler metrics on the compact connected Lee group belong to the Berwald type metrics. This talk has a general mathematical character.
English scientist M. Wright who is a well known organizer of the international seminars and workshops dedicated to the discussion of the most actual problems in modern Physics and Mathematics took part in the workshop as an observer. In the end he expressed a wish to take part in the further developments in this field and to become an official representative of the “Finsler Prize” foundation in Great Britain and France. The similar wish was expressed by D.Maslennikov who lives now in Egypt.
The participants visited the main pyramids of Egypt. They were accompanied by the specialists in Egyptian culture T. Sherkova and O. Krugliakov. The participants were also shown the documental movie “Forbidden issues of History” (with English translation). This movie was shot last year during the expedition to Egypt. A. Skliarov who is the author of several books on the alternative history of Egypt and the producer of the movie (and by the way the graduate of Moscow Physical-Technical Institute) commented upon the problems of the origin of the pyramids. These commentaries produced a vivid discussion.
The talks and excursions were shot by the professional cameraman S. Bondarenko. The Organizing Committee plans to present the information in the Internet and produce a movie about the most interesting ideas mentioned during the workshop. The special volume of the “Hyper complex numbers in Geometry and Physics” journal will be dedicated to the results of the workshop.
Full Seminar Program