2 00 5 Is the Schwarzschild black hole really stable ? ∗

The stability of the Schwarzschild black hole is studied. Regge and Wheeler treated the problem first at 1957 and obtained the dynamical equations for the small perturbation. There are two kinds of perturbations: odd one and even one. Using the Painlevé coordinate, we reconsider the odd perturbation and find that: the white-hole-connected universe (r > 2m, see text) is unstable. Because the odd perturbation may be regarded as the angular perturbation, therefore, the physical mean to it may be that the whitehole-connected universe is unstable with respect to the rotating perturbation. PACC:0420-q Studies concerning the stability of the Schwarzschild black hole may step back to the work of the Regge and Wheeler, who first divided the perturbation into odd and even ones [1]. Later, it is found that odd one is really the angular perturbation to the metric, while even one corresponds to the radial perturbation to the metric [2]. 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 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]: (1)if ∃k ∈ S with Ik > 0, the Schwarzschild black-hole is dynamically unstable, (2)if Ik < 0 for ∀k ∈ S , and the normal modes are complete, then, the Schwarzschild black-hole is dynamically stable, (3)if Ik ≤ 0 for ∀k ∈ S , and there is at least one real frequency k ∈ S, the linearized approach could not decide the stability of the Schwarzschild black-hole. Vishveshwara proved the normal mode of the Schwarzschild black-hole could be real. Therefore, strictly speaking, the stability of the Schwarzschild black-hole is unsolved according to this criterion of Stewart’s definition. Here, we reconsider the stability problem of the Schwarzschild black hole using the Painlevé coordinate metric(see following). The conclusions are: for thewhite-hole-connected universe of the Schwarzschild space-time(see following), there exists frequencies with Ik > 0. So, the white-hole-connected universe of the Schwarzschild space-time is unstable. ∗E-mail of Tian: [email protected], [email protected], [email protected] 1 Recently, in studying the Hawking radiation as tunnelling [7], the Schwarzschild coordinates is replaced by the Painlevé coordinates, which were discovered independent by Painlevé in 1921[8] and Kraus, Wilczck in 1994 [9], [10]. The Painlevé metric is stationary and regular at the horizon [9], [10]. This good quality makes it more suitable for studying the stability of the Schwarzschild black hole. In the following, we first introduce the Painlevé coordinate metric for the Schwarzschild black hole, then using the Vishveshwara’s result, we transform the odd perturbation quantities to the Painlevé coordinate system, and study the problem. The Schwarzschild metric is ds = −(1− 2m r )dts + (1− 2m r )−1dr2 + rdΩ. (1) By tp = ts − [ 2 √ 2mr + 2m ln √ r − √ 2m √ r + √ 2m ] , (2) we obtain the Painlevé metric of the black hole, ds = − ( 1− 2m r )