Finding collineations of Kimura metrics
After a brief review of the various types of geometrical symmetries, the authors concentrate on a class of static spherically symmetric spacetime metrics due to Kimura and given by \[ ds^2= {r^2\over b}dt^2-{1\over b^2r^2} dr^2-r^2 (d\theta^2+ \sin^2 \theta d\varphi^2) \tag{1} \] where \(b\) is a nonzero constant. Then, by using the computer package Dimsym, they study the existence of Killing, conformal, affine and projective vector fields and also conformal collineations and Killing tensors for this metric. They find: (i) a 4-dimensional Lie algebra of Killing vector fields (but this is obvious since (1) is static and spherically symmetric and any further Killing symmetry would, using elementary Killing theory, produce a contradiction); (ii) that there are no proper affine vector fields (but again this is obvious since the metric (1) is not metrically decomposable [\textit{G. S. Hall} and \textit{J. da Costa}, J. Math. Phys. 29, 2465-2472 (1988; Zbl 0661.53017); \textit{G. S. Hall}, \textit{D. J. Low} and \textit{J. R. Pulham}, J. Math. Phys. 35, 5930-5944 (1994; Zbl 0818.53023)]); (iii) that each conformal collineation is a conformal vector field (again well-known since in all spacetimes except those of Petrov type \(N\) or 0 -- which (1) is not -- the Weyl tensor component \(C^a_{bcd}\) determines the metric up to a conformal factor [\textit{G. S. Hall}, in: Classical general relativity, Proc. Conf., London 1983, 103-120 (1984; Zbl 0577.53021)]); (iv) results regarding conformal and projective symmetry which are apparently new; and (v) results regarding the existence of Killing tensors. The paper concludes with a discussion of other Kimura type metrics and a discussion of the symmetries they admit. That the third and fourth of these (in the authors' numbering) admit a proper affine vector field is again well-known since each of these metrics is metrically decomposable.
scientific article; zbMATH DE number 1206889
scientific article