Stability of self-gravitating magnetic monopoles

The stability of a spherically symmetric self-gravitating magnetic monopole is examined in the thin wall approximation: modeling the interior false vacuum as a region of de Sitter space; the exterior as an asymptotically flat region of the Reissner-Nordström geometry; and the boundary separating the two as a charged domain wall. There remains only to determine how the wall gets embedded in these two geometries. In this approximation, the ratio k of the false vacuum to surface energy densities is a measure of the symmetry breaking scale $\eta .$ Solutions are characterized by this ratio, the charge on the wall Q, and the value of the conserved total energy M. We find that for each fixed k and Q up to some critical value, there exists a unique globally static solution, with $M\simeq Q^{3/2};$ any stable radial excitation has M bounded above by Q, the value assumed in an extremal Reissner-Nordström geometry, and these are the only solutions with $M<Q.\mathrm{}$ As M is raised above Q a black hole forms in the exterior: (i) for low Q or k, the wall is crushed; (ii) for higher values, it oscillates inside the black hole. If the mass is not too high the former solutions coexist with an inflating bounce; (iii) for k, Q or M outside the above regimes, there is a unique inflating solution. In case (i) the course of the bounce lies within a single asymptotically flat region (AFR) and it resembles closely the bounce exhibited by a false vacuum bubble (with $Q=0).\mathrm{}$ In case (iii) the course of the bounce spans two consecutive AFRs. However, for an asymptotic observer it resembles a monotonic false vacuum bubble.