Static Dark Fluid Thin Shells in Schwarzschild-de Sitter Spacetimes: Stability and Black Hole Shadows

Abstract

We study the existence and radial stability of static, spherically symmetric thin shells joining two Schwarzschild--de~Sitter (SdS) spacetimes $(m_\pm,Λ_\pm)$. Using the Israel junction formalism, we map the stable equilibria ($V_{\mathrm{eff}}&#39;&#39;>0$) of the effective potential. Near the equilibrium radius $R_0$ the shell&#39;s surface density $σ$ and pressure $p$ obey the linearized barotropic law $p=p_0+c_s^2(σ-σ_0)$, with sound speed $c_s^2=λc^2$. Since $c_s^2$ is independent of the equilibrium ratio $w_0\equiv p_0/(σ_0 c^2)$, tension shells ($w_0<0$) stay radially stable with real $c_s$. Fixing $Λ_+$ so that its vacuum energy density equals the critical density (Planck~2018), and taking $m_-$ representative of astrophysical black holes, we systematically map the stable equilibria $(R_0,σ_0)$ over $(m_\pm,Λ_\pm,λ,w_0)$ and find that stable shells with $σ_0>0$ and $0<λ\le1$ exist only for $m_+/m_->1$, at three scales -- the photon sphere, the SdS static radius, and the cosmological horizon. At $λ=1$ the numerical windows, checked against the analytic test-shell bounds, are $(1-\sqrt{13})/6\lesssim w_0\lesssim 1/2$ ($Λ_+=Λ_-$), $-2/3\lesssim w_0\lesssim 1/2$ ($Λ_+>Λ_-$), and $0\lesssim w_0$ ($Λ_+<Λ_-$). Positive-pressure shells ($0\lesssim w_0\lesssim 1/2$) sit near the photon sphere and those with $w_0\gtrsim1/2$ near the static radius scale, while tension shells reach the cosmological horizon scale for $Λ_+=Λ_-$, only the static radius scale for $Λ_+>Λ_-$, and are absent for $Λ_+<Λ_-$. Finally, we compute the dark fluid shell&#39;s imprint on the SdS black-hole shadow seen by a static observer at varying radial distance.