Accurate Quasinormal Modes of the Five-Dimensional Schwarzschild-Tangherlini Black Holes

The objective of this paper is to construct the accurate (say, to 11 decimal places) frequencies of the quasinormal modes of the 5-dimensional Schwarzschild-Tangherlini black hole using three major techniques: the Hill determinant method, the continued fractions method and the WKB-Pad\'e method and to discuss the limitations of each. It is shown that for the massless scalar, gravitational tensor, gravitational vector and electromagnetic vector perturbations considered in this paper, the Hill determinant method and the method of continued fractions (both with the convergence acceleration) always give identical results, whereas the WKB-Pad\'e method gives the results that are amazingly accurate in most cases. Notable exception are the gravitational vector perturbations ($j =2$ and $\ell = 2 $), for which the WKB-Pad\'e approach apparently does not work. Here we have interesting situation in which the WKB-based methods (WKB-Pad\'e and WKB-Borel-Le Roy) give the complex frequency that differs from the from the result obtained within the framework of the continued fraction method and the Hill determinant method. For the fundamental mode, deviation of the real part of frequency from the exact value is $0.5\%$ whereas the deviation of the imaginary part is $2.7\%.$ For $\ell \geq 3$ the accuracy of the WKB results is similar again to the accuracy obtained for other perturbations. The case of the gravitational scalar perturbations is briefly discussed.