\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
The nonlinear superposition of the -metric and the Kerr metric results in a -Kerr metric that represents a deformed Kerr black hole with , where qââ>ââ0 is proportional to the nonrelativistic quadrupole moment of the collapsed configuration. We study this spacetime and determine q + such that for q, 0ââ<ââqââ<ââq +â, the outer spacetime singularity remains a null hypersurface. In this case, -Kerr spacetime represents a generalized black hole, namely, an asymptotically flat, stationary and axisymmetric vacuum solution of general relativity for which the outer singularity is a closed null hypersurface. For an approximate variant of -Kerr spacetime characterized by mass M, quadrupole parameter q and angular momentum parameter a, where the latter two parameters are treated to first and second orders of approximation, respectively, we analytically determine the quasinormal mode (QNM) frequencies in the ray approach using the light-ring method as well as in the complementary wave approach for massless scalar field perturbations in the aââ=ââ0 limit. The QNM frequencies of -Kerr spacetime turn out to be nearly the same as those of the rotating HartleâThorne spacetime
\end{document}