The stability of the Schwarzschild black hole is restudied in the Painlevé coordinates. Using the Painlevé time coordinate to define the initial time, we reconsider the odd perturbation and find that the Schwarzschild black hole in the Painlevé coordinates is unstable. The Painlevé metric in this paper corresponds to the white-hole-connected region of the Schwarzschild black hole （r 〉 2m） and the odd perturbation may be regarded as the angular perturbation. Therefore, the white-hole-connected region of the Schwarzschild black hole is unstable with respect to the rotating perturbation.
The stability problem of the Rindler spacetime is carefully studies by using the scalar wave perturbation. Using two different coordinate systems, the scalar wave equation is investigated. The results are different in the two cases. They are analysed and compared with each other in detail. The following conclusions are obtained： （a） the Rindler spacetime as a whole is not stable; （b） the Rindler spacetime can exist stably only as part of the Minkowski spacetime, and the Minkowski spacetime can be a real entity independently; （c） there are some defects for the scalar wave equation written by the Rindler coordinates, and it is unsuitable for the investigation of the stability properties of the Rindler spacetime. All these results may shed some light on the stability properties of the Schwarzschild black hole. It is natural and reasonable for one to infer that： （a） perhaps the Regge-Wheeler equation is not sufficient to determine the stable properties; （b） the Schwarzschild black hole as a whole might be really unstable; （c） the Kruskal spacetime is stable and can exist as a real physical entity; whereas the Schwarzschild black hole can occur only as part of the Kruskal spacetime.
