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- MUALLIM | УЧИТЕЛЬ | TEACHER №4 | 2021
| Summary and conclusions
As it is well known, the G2-symmetry-reduced vacuum Einstein equations or electrovacuum Einstein-Maxwell equations are integrable and admit various solution generating procedures, which allow to construct large families of exact solutions starting from arbitrarily chosen “beginning” (of “seed”, or “background”) solution. Each of such solution generating procedures can be considered as transformations of the corresponding solution spaces described in terms of transformations of “coordinates” characterizing every local solution. In the entire solution spaces of the integrable reductions of Einstein’s field equations, the monodromy data for the fundamental solution of associated “spectral problems” can be used as the “coordinates” in the infinite-dimensional solution spaces. In this paper, we considered more simple construction of such “coordinates” which exist in the (infinite-dimensional) subspaces of electrovacuum solutions for which the gravitational and electromagnetic fields possess a regular behaviour near degenerate orbits of the space-time isometry group G215. In the corresponding two-dimensional orbit space, the degenerate orbits constitute the lines which can be considered as the boundaries of this orbit space. In the infinite-dimensional spaces of solutions of these types, the values of the Ernest potentials on such boundaries in the orbit spaces can serve as “coordinates” of each solution. The solution generating transformations considered MUALLIM | УЧИТЕЛЬ | TEACHER №4 | 2021 35 above was described in terms of these “coordinates” by simple expressions in which a particular choice of the beginning (initial, or background) solution is not assumed. The beginning solution in these expressions is represented also by its boundary values of the Ernst potentials which can be chosen as arbitrary functions of the parameter along the boundary. It is clear that many physical parameters of generating solutions can be calculated directly from these boundary values of the Ernst potentials and the detail knowledge of the components of the solution on the whole orbit space is not necessary for these. Besides that, the explicit form of each of these solution generating procedures in terms of these “coordinates” allows to compare different suggested solution generating procedures, to find the relations between numerous constant parameters introduced by different methods as well as to determine various physical and geometrical properties of generating solutions (such as e.g., cylindrical wave profiles on the axis of symmetry, multipole moments of asymptotically flat fields, appearance of horizons in stationary axisymmetric fields and others) even before a detail calculation of all components of generated solution. It is necessary to note that in this paper we have not discussed the known methods of the other type (also based on the integrability of the equations under consideration) which allow a direct construction of multiparametric families of solutions, but which do not admit arbitrary choice of some beginning (background) solution. Among these, we can mention the methods for construction of solutions for boundary value problems. Some examples of such subspaces of solutions are cylindrical waves, stationary axisymmetric fields created by compact sources and considered near some intervals on the axis between or outside the sources, some cosmological-like solutions, plane waves near the “focusing singularities”, solutions with Killing horizons and some others. Besides that, there exist the integral equation methods for a direct construction of solutions. Among these, there is a scalar linear integral equation method which was suggested by N.Sibgatullin [53] for construction of stationary axisymmetric electrovacuum fields. Actually, Sibgatullin started from the solution generating method suggested earlier by Hauser and Ernst for effecting Kinnersley-Chitre transformations and reduced considerably their matrix linear singular integral equation to a scalar one, using the choice of Minkowski spacetime as the initial solution. During the last decades the Sibgatullin’s integral equation was used frequently enough in the literature by some authors for construction of particular stationary axisymmetric asymptotically flat solutions with various rational structures of the Ernst potentials on the axis. Another method of direct construction of solutions of integral reductions of Einstein’s field equations (not only for the case of electrovacuum) which we do not discuss here, is the monodromy transform approach. This approach is also based on a reformulation of integral reductions of Einstein’s field equations in terms of a linear singular integral equations, but in contrast to the Hauser and Ernst approach, the construction of these integral equations does not assume any restrictions on the entire space of local solutions of these field equations. One of general applications of this approach, which can be found in [54], is a construction in a unified (determinant) form of a huge class of electrovacuum solutions with arbitrary rational structure of the Ernst potentials on degenerate orbits of space-times isometry group G2. This class includes hierarchies of soliton and non-soliton solutions, stationary axisymmetric solutions (which are not necessary asymptotically flat), as well as various types of waves and cosmological solutions which admit G2-symmetry. Using this method, more singular types of solutions for interacting waves and inhomogeneous cosmologies were found. жүктеу/скачать 3,08 Mb. 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