A note on the stability of generic spherically symmetric thin-shell wormhole supported by a false vacuum

In the stability analysis of the Schwarzschild thin-shell wormhole (TSW) performed by Poisson and Visser in Poisson and Visser (1995) the equation of state (EoS) of the surface matter on the throat satisfies beta(2)(sigma)=(alpha p)/(alpha sigma) in which sigma and p are the surface energy density and the surface transverse pressure. In this Letter, we show that beta(2)(sigma)=-1 has to be excluded as it results in the radius of the equilibrium a(0)=3M a radius which was problematic and later on was treated by Varela in Varela (2015). Furthermore, we present a formalism on the mechanical stability of generic spherically symmetric TSWs supported by a surface energy-momentum tensor of the form sigma=-p corresponding to beta(2)(sigma)=-1. Such an EoS with sigma>0 describes the so-called vacuum matter or dark energy, however, here it is called false vacuum because sigma<0. We obtain the general conditions upon which such a TSW remains stable against a mechanical linear perturbation. In particular, we apply our formalism to first the TSW constructed in the Reissner-Nordstr & ouml;m (RN) spacetime and then in the Bronnikov-Zaslavskii (BZ) closed black hole. We show that in both cases stable TSW exists, however, for the former TSW, it corresponds to the non-black hole RN solution.