Antisymmetric tensor generalizations of affine vector fields (Q2795558)

Tensor generalizations of affine vector fields called symmetric and antisymmetric affine tensor fields are discussed as symmetries of spacetimes. The authors review the properties of the symmetric ones, which have been studied in earlier works, and investigate the properties of the antisymmetric ones, which are the main subject in this paper. It is shown that antisymmetric affine tensor fields are closely related to one-lower-rank antisymmetric tensor fields which are parallely transported along geodesics. It is also shown that the number of linearly independent rank-\(p\) antisymmetric affine tensor fields in \(n\) dimensions is bounded by \((n + 1)!/p!(n - p)!\). The authors derive the integrability conditions for antisymmetric affine tensor fields. Using the integrability conditions, they discuss the existence of antisymmetric affine tensor fields on various spacetimes.NEWLINENEWLINENEWLINEThe paper present interest not only for its possible connection with spacetime symmetries, but also from the bibliographical point of view.NEWLINENEWLINENEWLINEThe interest for Killing and Killing-Yano tensor fields of 2-nd rank was very great during the 1968--1978 years, when the separability of the Hamilton-Jacobi equation for a test particle in a Kerr geometry was discovered by Carter (1968). Subsequently a lot of work was given by the mathematical physicists from UK, such as Penrose (1970), Penrose and Floyd (1972), Walker and Penrose (1970, 1972). Later the same mathematical apparatus was applied to separability of the Dirac equation in the Kerr spacetime and its type-D generalizations. Parallely, with less physical significance, the Japanese mathematicians Tachibana-Shun ichi (1968) and Kashiwada Toyoko (1968) discussed and developed the well-known research by Kentaro Yano (1952, 1956).NEWLINENEWLINENEWLINEIn some other Countries, such as USSR, Germany, Italy, France, Romania, Poland the interest for Killing and Killing-Yano tensors was high during the 80-th. For a review, see [the reviewer, Квантовые частицы в полях Еинштеина-Максвелла (Russian). Kishinev: ``Shtiintsa'' (1989; Zbl 1102.81313)], as well as few later papers by M. Visinescu and coauthors (1993--2000) and D. Baleanu and coauthors (1994, 1998, 1999) with interest to some particular spacetimes, admitting Killing and Killing-Yano tensors. One has to mention that the group of V. G. Bagrov in a number of publications studied the inverse problem: Suppose a particular Killing or Killing-Yano tensor is given. Find all the vacuum (resp. electro-vacuum) spacetimes (resp. electromagnetic fields) allowing the given symmetry.NEWLINENEWLINENEWLINEThe value of the present paper consists in a) a general approach to the problem as well as some historical and bibliographic findings. Namely, the authors found a few very early papers by P. Stachel (1895) and S. Bochner (1948), which can not be found in known databases with mathematical literature, including Zentralblatt für Mathematik and Jahrbuch fur Mathematik. These papers are earlier than the well-known work by Kentaro Yano (1952), treating Killing and Killing-Yano tensors and the respective symmetries of spacetimes.NEWLINENEWLINENEWLINEI would like to mention that the number of publications on this topic is sometimes increasing drastically, such that there is need to introduce an elementary and simple presentation of the Killing and Killing-Yano type symmetries into textbooks on General Relativity, such as the well-known textbooks by Landau and Lifshits (1974 and before), S. Weinberg (1974) and R. Adler, M. Bazin and M. Schiffer (1975) at least as a problem in the chapter where the separation of variables in the equations of motion for particles and light rays at least in type-D spacetimes (Kerr spacetime) is treated. In this order of ideas the reviewer has written in 1981 a letter in Prof. L. P. Pitaevskij in Moscow (Institute of Theoretical Physics), which was engaged in the updating of the books by L. D. Landau and I. M. Lifshits, but no answer was given.