The Stability of the Schwarzschild Metric

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1968

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Abstract

The stability of the Schwarzschild exterior metric against small perturbations is investigated. The exterior extending from the Schwarzschild radius r =2m to spatial infinity is visualized as having been produced by a spherically symmetric mass distribution that collapsed into the Schwarzschild horizon in the remote past. As a preamble to the stability analysis, the phenomenon of spherically symmetric gravitational collapse is discussed under the conditions of zero pressure, absence of rotation and adiabatic flow. This is followed by a brief study of the Kruskal coordinates in which the apparent singularity at r = 2m is no longer present; the process of spherical collapse and the consequent production of the Schwarzschild empty space geometry down to the Schwarzschild horizon are depicted on the Kruskal diagram. 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 exp(-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 even at the initial instant t = 0. Since the background metric itself is finite at this surface, the above behaviour 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. In the case of the odd perturbations, the above proof of stability is made rigorous by showing that the radial functions for real values of k form a complete set, by superposition of which any well behaved initial perturbation can be represented so that the time development of such a perturbation is non-divergent, since each of the component modes is purely oscillatory in time. A similar rigorous extension of the proof of stability has not been possible in the case of the even perturbations because the frequency (or k2) does not appear linearly in the differential equation. A study of stationary perturbations (k = 0) shows that the only nontrivial stationary perturbation that can exist is that due to the rotation of the source which is given by the odd perturbation with the angular momentum £ = 1. Finally, complex frequencies are introduced under the boundary conditions of only outgoing waves at infinity and purely incoming waves at the Schwarzschild surface. The physical significance of this situation is discussed and its connection with phenomena such as radiation damping and resonance scattering, and with the idea of causality is pointed out.

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