Stability of Thin Shell and Wormhole Configurations: Schwarzschild, Schwarzschild -- (Anti-) de Sitter, and FLRW Spacetimes

TL;DR

The paper develops a comprehensive stability analysis for thin-shell wormholes and black holes formed by joining Schwarzschild, Schwarzschild--(A)dS, and FLRW spacetimes using the Darmois--Israel formalism. Stability is analyzed via an effective potential $V(R)$, shell mass functions $M$, and conservation identities, yielding a taxonomy of junctions and mapping stability regions in parameter space. Across all wormhole configurations, no stable regions lie within the causal (sound-speed) domain, while black-hole configurations can exhibit stability outside this region or around asymptotes; FLRW junctions behave similarly to Schwarzschild cases with expansion effects modifying the effective mass. The results imply that, under linearized, spherically symmetric perturbations, thin-shell wormholes remain unstable in the physical causal regime for the configurations studied, and provide detailed criteria and plots to delineate stability landscapes across diverse spacetimes.

Abstract

The stability of thin shell wormholes and black holes to linearized spherically symmetric perturbations about a static equilibrium is analyzed. Thin shell formalism is explored and junctions formed from combinations of Schwarzschild, Schwarzschild - de Sitter, and Schwarzschild - anti-de Sitter, as well as Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetimes are considered. The regions of stability for these different combinations are thoroughly described and plotted as a function of mass ratios of the Schwarzschild masses and radii of the wormhole throats. A taxonomy of the qualitative features of the various configurations and parameter spaces is developed, illustrating the stability regions when present. The considered wormholes are all found to be unstable in the causal region.

Figures (18)

• Figure 1: Schwarzschild -- Schwarzschild Junction: The areas shaded in green and blue correspond to the stability regions for black hole and wormhole, respectively. The shaded region in gray represents the causal region in both cases. This color scheme is used consistently throughout this paper. For the black hole, $M\neq0$ anywhere and there is no stability flip. Stability partially intersects the Causal Region. For the wormhole there is an asymptote present and stability is separated into two disjoint regions, neither of which intersect the Causal Region. For 3-D plot, see figure (\ref{['fig:SS_3d']}).
• Figure 2: Schwarzschild - de Sitter -- Schwarzschild Junction: For the black hole, there are points where $M=0$ and a stability flip is present without an asymptote. This occurs at the vertical line. A Schwarzschild - de Sitter -- Schwarzschild junction implies $[C] =-0.001$ which does not fulfill the asymptote condition for any beta. For both plots the effects of the de Sitter Horizon can be seen by the divergence around $\alpha \approx 54$. For 3-D plot, see figure (\ref{['fig:SdsS_3d']})
• Figure 3: Schwarzschild Anti-de Sitter -- Schwarzschild Junction: Black hole has an asymptote as $[C] =0.001$ which does fulfill the asymptote condition. A de Sitter horizon is not present as $\Lambda^\pm\leq0$. For 3-D plot, see figure (\ref{['fig:SadsS_3d']}).
• Figure 4: Schwarzschild -- Schwarzschild - de Sitter Junction: Black hole is similar to Schwarzschild - anti-de Sitter -- Schwarzschild (figure \ref{['fig:SadsS']}) with a similar asymptote, though in this case the de Sitter Horizon is also present. Again $[C] =0.001$ in this case, fulfilling the asymptote condition. De Sitter Horizon is present.
• Figure 5: Schwarzschild -- Schwarzschild - anti-de Sitter Junction: Black hole is similar to Schwarzschild - de Sitter -- Schwarzschild (figure \ref{['fig:SdsS']}) but does not have a de Sitter Horizon. $[C]=-0.001$ and there is no asymptote.