Klein - Gordon equation and the stable problem in the Rindler space - time ∗

The Klein-Gordon equation in the Rindler space-time is studied carefully. It is shown that the stable properties depend on using what time coordinate to define the initial time. If we use the Rindler time, the scalar field is stable. Alternatively, if we use the Minkowski time, the scalar field may be regarded unstable to some extent. Furthermore, the complete extension of the Rindler space time is the Minkowski space time, we could also study the stable problem of the Rindler space time by the Klein-Gordon equation completely in the Minkowski coordinates system. The results support that the Rindler space time is really unstable. This in turn might cast some lights on the stable problem of the Schwarzschild black-hole, which not only in many aspects shares the similar geometrical properties with the Rindler space time but also has the very same situation in stable study as that in Rindler space time. So, it is not unreasonable to infer that the Schwarzschild black hole might really be unstable in comparison with the case in Rindler space time. Of course, one must go further to get the conclusion definitely. PACS: 0420-q, 04.07.Bw, 97.60.-s We have studied the stable problem in the Schwarzschild black hole recently[1]-[4]. Usually, researchers gets the conclusion that it is stable by using the Schwarzschild time t = 0 to define the initial time[6],[8]. It is recently noticed that the Schwarzschild time t = 0 could not be used to study the stable problem due to the fact that g00|r=2m = 0 makes the Schwarzschild time t loses its meaning at the horizon r = 2m. By using the Kruskal time coordinate T = C = const, the stable properties depend on the sign of the initial time T = C = const: the Schwarzschild black hole is stable when T = C ≥ 0, whereas the Schwarzschild black hole is unstable when T = C ≤ 0[1]-[4]. These unexpected results are in contrast with the conclusion taken as granted that the Schwarzschild black hole is stable. Which one is really correct? There still is not a convincing and satisfying answer to it due to the fact it is almost impossible to restudy the problem completely in the Kruskal coordinates whose background metric is varying with time[6]. The Rindler space time is one part of the Minkowski space time whose constant spatial coordinates describe accelerated observers with constant accelerations. The interest in Rindler space time lies in its similar geometrical structure with the Schwarzschild black-hole. Actually, the Hawking radiation in the Schwarzschild black-hole is closely connected with the Unruh effect in the Rindler space time[5],[7]. Apart from the similar geometrical properties with the Schwarzschild black-hole, the Rindler space time is mathematically simple to study the physical problem and has closed form for them. Here we study the Klein-Gordon equation and the stable problem of the Rindler space time, and try to find what effect the horizon of the space time may have on its stable study. Subsequently we want to obtain some clues to the stable study of the Schwarzschild black hole which is similar very much with that of the Rindler space time[1]-[4]. ∗E-mail of Tian: [email protected], [email protected]