Axially symmetric gravitational fields. II. Stationary solutions of the vacuum Einstein equations

Stationary axially-symmetric space-times possess two commuting Killing vectors which make it possible to introduce two coordinates t (time) and varphi (azimuthal angle) in such a way that the vectors are exactly partial_t and partial_varphi. Static axially-symmetric solutions of the Einstein equation provide a special class of manifolds in which the two vectors are orthogonal to each other and, hence, the coordinate system which includes t and varphi can be chosen to be orthogonal. The Einstein equation in this case reduces to the Laplace equation, therefore the whole space of solutions, called the Weyl class, is known. The Weyl class of solutions was described in the previous part of the work [Part I, Gravit. Cosmol. {bf 8} (2002), no.~4, 249--260; [msn] MR1982698 (2004e:83013) [/msn]], and the part under consideration describes approaches to the vacuum Einstein equation in the case of non-static space-time. par In this area particular families of solutions have been obtained by different methods. In the non-static case the original Einstein equation reduces to a system of two partial differential equations for two unknown functions of two coordinates (other than t and varphi). Further development of this approach consists in transforming this system into various forms and attempts to solve it. In this part of the work a detailed description of achievements of this development is given. Contrary to the authors' statement that the work "is an attempt at systematization of known special solutionsdots", it is in fact an attempt to represent all known solutions as if they have been obtained along the lines of straightforward reduction of the original Einstein equation to the system of two partial differential equations or its equivalents like the Ernst equation. Though it is known that, on the one hand, very few of the known solutions possess a genuine physical meaning, and that all physically meaningful solutions were obtained before the Ernst equation was published, the authors represent all of them as if they are obtained by solving the Ernst equation. par {Reviewer's remarks: Systematization of something would be based on a certain idea of how to systematize it. The idea could be physical, mathematical or chronological; correspondingly, known solutions could be systematized by physical properties, by methods used or by the date of obtaining them. None of these possibilities has been used in this work. Physical properties of space-time models provided by known solutions are not considered, and the fact that very few of them (Taub-NUT, Kerr and Kerr-NUT solutions) possess clear physical meaning, have not been mentioned. Only one method of straightforward reduction is considered while it is known that physically meaningful solutions have been obtained before the Ernst equation was published in 1968 [F. J. Ernst, Phys. Rev. {bf 167} (1968), no. 5, 1175--1178; Phys. Rev. {bf 168} (1968), no. 5, 1415--1417]. Thereby, chronology is also broken and thus it is not clear what kind of a systematization is presented. Though the methods by which physically meaningful solutions have been obtained are not generally known, it is seen that these methods apparently differ from the method of straightforward reduction, for example, from the work [A. H. Taub, Ann. of Math. (2) {bf 53} (1951), 472--490; [msn] MR0041565 (12,865b) [/msn]] to which the authors, where mainly space-time symmetries were studied. Nor are any signs of straightforward reduction seen in the works of R. P. Kerr [Phys. Rev. Lett. {bf 11} (1963), 237--238; [msn] MR0156674 (27 #6594) [/msn]] and M. Demianski and E. T. Newman [Bull. Acad. Pol. Sci. Sér. Sci. Math. Astron. Phys. {bf 14} (1966), 653--657; Zbl 0184.55102] in which the Kerr-NUT solution was presented for the first time. Thus, this work is rather a good text on the straightforward reduction approach than a review and systematization of known solutions.}

Gutsunaev Ts.I. , Hassan N.Sh.
Turakulov Zafar Ya.
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