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

TL;DR

This paper investigates the stability of scalar fields in Rindler space-time using the Klein-Gordon equation, revealing that stability depends on the choice of time coordinate and suggesting possible instability of Schwarzschild black holes.

Contribution

It demonstrates how the stability of scalar fields in Rindler space depends on the time coordinate and extends the analysis to Minkowski coordinates, providing insights into black hole stability.

Findings

Scalar field stability depends on the time coordinate used.

Rindler space is potentially unstable when analyzed in Minkowski coordinates.

Results imply Schwarzschild black holes might also be unstable.

Abstract

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.