D ec 2 00 5 The stable problem of the black - hole connected region in the Schwarzschild black hole ∗

The stability of the Schwarzschild black hole is studied. Using the Painlevé coordinate, our region can be defined as the black-hole-connected region (r > 2m, see text) of the Schwarzschild black hole or the white-hole-connected region (r > 2m, see text) of the Schwarzschild black hole. We study the stable problems of the black-holeconnected region . The conclusions are: (1) in the black-hole-connected region , the initially regular perturbation fields must have real frequency or complex frequency whose imaginary must not be greater than − 1 4m , so the black-hole-connected region is stable in physicist’ viewpoint; (2) On the contrary, in the mathematicians’ viewpoint, the existence of the real frequencies means that the stable problem is unsolved by the linear perturbation method in the black-hole-connected region . PACC:0420-q Studies on the stability of the Schwarzschild black hole are of great importance both in theoretical and cosmological back-grounds. The Schwarzschild black hole is the only candidate for the spherically static vacuum space-time. It is generally believed that it is the ultimate fate of massive star after getting off its angular momentum. Also, Many theoretical results heavily rely on their applications to the the Schwarzschild black hole. Regge and Wheeler first studied the problem, and divided the perturbations into odd and even ones [1]. The odd one is really the angular perturbation to the metric, while even one corresponds to the radial perturbation to the metric [2]. The odd perturbation equation is the well-known Regge-Wheeler equation. Vishveshwara made the study further by transforming the perturbation quantities to the Kruskal reference frame, and tried to find the real divergence at r = 2m from the spurious one caused by the improper choice of coordinate due to the Schwarzschild metric’s ill-defined-ness at r = 2m [3]. Later, Price also studied the problem carefully [4] and Wald studied from the mathematical background [5]. In this paper, we give full consideration on the perturbation fields with complex frequency. Using the Painlevé coordinate, our region can be defined as the black-hole-connected region (r > 2m, see text) of the Schwarzschild black hole or the white-hole-connected region (r > 2m, see text) of the Schwarzschild black hole. We study the stable problems of the two kinds of region. In reference [6], Stewart applied the Liapounoff theorem to define dynamical stability of a black-hole. First, according Stewart, the normal mode of the perturbation fields to the Schwarzschild black-hole is the perturbation fields Ψ with time-dependence of e−ikt which are bounded at the boundaries of the event horizon r = 2m and the infinity r → ∞. The range of permitted frequency is defined as the spectrum S of the Schwarzschild black-hole. Then, for the Schwarzschild black-hole, it could be obtained by the Liapounoff theorem that[6]: ∗E-mail of Tian: [email protected], [email protected]