ar X iv : g r - qc / 0 51 11 21 v 1 2 2 N ov 2 00 5 Does the Schwarzschild black hole really exist ?

we use the Kruskal time coordinate T to define the initial time. By this way, it naturally divides the stable study into one connected with the two regions: the whitehole-connected region and the black-hole-connected region. The union of the two regions covers the Schwarzschild space-time (r ≥ 2m). We also obtain the very reasonable conclusion: the white-hole-connected region is instable; whereas the black-hole-connected region is stable. If we take the instability with caution and seriousness, it might be not unreasonable to regard that the Schwarzschild black hole might be instable too. PACC:0420-q In general relativity, black holes could be produced by the complete gravitational collapse of sufficiently massive stars. Whether or not the black holes exist depends on their stability against small perturbation. The Schwarzschild black hole is the only vacuum spherically symmetric solution of the Einstein’s equation. It is the final fate of a spherical star after its complete gravitational collapse. Of course, the perturbation study to the Schwarzschild blackhole is of critical importance due to its very special status in general relativity. It is treated and studied by many people since Regge-Wheeler’s original paper in 1957[1]. Generally, the problem of the stability of the Schwarzschild black hole is believed to be settled down[2]-[4]: it is absolutely stable against small perturbation. Here we study the problem again. Due to its metric component g00 = 0 at the horizon r = 2m, the Schwarzschild time coordinate t lose its meaning at the horizon. This is the drawback of the Schwarzschild time coordinate t. Actually, this drawback is closely connected with the boundary problem at the horizon r = 2m in the stability study. People have not consider its effect in all previous stability study[1]-[4]: t = 0 is taken for granted as the initial time at the horizon. It is natural for one to suspect its rationality. One might guess its influence on the stability study. It would best for one to use some ”good” time to reconsider the stability problem and compare the two results. It is the main work in this paper. By using the more reasonable and suitable time coordinate, the Kruskal time coordinate T , to define the initial time, we study the stable problem and obtain the conclusion completely opposite to the common belief: the Schwarzschild black hole might be instable from some viewpoint (see the text). This unexpected conclusion would urge people to study the perturbation of the Kerr black hole, which is the second simplest black hole and topologically different from the Schwarzschild black hole. More importantly, the Kerr black hole has intrinsic angular momentum produced more possible from the spinning star’s complete gravitational collapse. In Regge and Wheeler’s original paper, they deduced perturbation into two kinds[1]: the odd one and the even one, corresponding respectively to the angular and radial metric perturbation[3]. In the paper, we study the odd perturbation. For the Schwarzschild black hole, the perturbation equations were obtained in the Schwarzschild coordinates: ds = −(1− 2m r )dt + (1− 2m r )dr + rdΩ, (1) and the odd perturbation equation is the well-known Regge-Wheeler equation[1]: dQ dr + [