Which coordinate representations can give correct Hawking temperature of Kerr black hole by tunneling approach

Some authors found that, in different coordinates, the tunneling approach gives different Hawking temperature for the Schwarzschild black hole recently. In this paper, by studying the Hawking radiation of the Kerr black hole arising from the scalar and Dirac particles, we find that, to obtain the Hawking temperature by using tunneling effect, the coordinate representations for the stationary Kerr black hole should satisfy two conditions: (a) to keep the Killing vectors x_(t)^m{{\xi_{(t)}^\mu}} and x_(j)^m{{\xi_{(\varphi)}^\mu}} invariant; and (b) the radial coordinate transformation is a regular and non-zero function.