Stability of the Schwarzschild Metric

Abstract

The stability of the Schwarzschild exterior metric against small perturbations is investigated. The perturbations superposed on the Schwarzschild background metric are the same as those given by Regge and Wheeler, consisting of odd- and even-parity classes, and with the time dependence $\mathrm{exp}(\ensuremath{-}ikt)$, where $k$ is the frequency. An analysis of the Einstein field equations computed to first order in the perturbations away from the Schwarzschild background metric shows that when the frequency is made purely imaginary, the solutions that vanish at large values of $r$, conforming to the requirement of asymptotic flatness, will diverge near the Schwarzschild surface in the Kruskal coordinates. Since the background metric itself is finite at this surface, the above behavior of the perturbation clearly contradicts the basic assumption that the perturbations are small compared to the background metric. Thus perturbations with imaginary frequencies that grow exponentially with time are physically unacceptable, and hence the metric is stable. Perturbations with real values of $k$ representing gravitational waves are also examined. It is shown that the only nontrivial stationary perturbation that can exist is one that is due to the rotation of the source, which is given by the odd perturbation with the angular momentum $l=1$. The significance of solutions with complex frequencies is pointed out, as is the lack of a theorem (completeness of the eigenfunction) for the even-parity case to parallel the Sturm-Liouville theory, which is applicable to the odd-parity case. Such a theorem would be required to convert the computations indicating stability as given here into a fully rigorous stability theorem.