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 $(m_\pm,\Lambda_\pm)$. Using the Israel junction formalism, we map the stable equilibria ($V_{\mathrm{eff}}''>0$) of the effective potential. Near the equilibrium radius $R_0$ the shell's surface density $\sigma$ and pressure $p$ obey the linearized barotropic law $p=p_0+c_s^2(\sigma-\sigma_0)$, with sound speed $c_s^2=\lambda c^2$. Since $c_s^2$ is independent of the equilibrium ratio $w_0\equiv p_0/(\sigma_0 c^2)$, tension shells ($w_0<0$) stay radially stable with real $c_s$. Fixing $\Lambda_+$ 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 $(R_0,\sigma_0)$ over $(m_\pm,\Lambda_\pm,\lambda,w_0)$ and find that stable shells with $\sigma_0>0$ and $0<\lambda\le1$ exist only for $m_+/m_->1$, at three scales -- the photon sphere, the SdS static radius, and the cosmological horizon. At $\lambda=1$ the numerical windows, checked against the analytic test-shell bounds, are $(1-\sqrt{13})/6\lesssim w_0\lesssim 1/2$ ($\Lambda_+=\Lambda_-$), $-2/3\lesssim w_0\lesssim 1/2$ ($\Lambda_+>\Lambda_-$), and $0\lesssim w_0$ ($\Lambda_+<\Lambda_-$). 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 $\Lambda_+=\Lambda_-$, only the static radius scale for $\Lambda_+>\Lambda_-$, and are absent for $\Lambda_+<\Lambda_-$. Finally, we compute the dark fluid shell's imprint on the SdS black-hole shadow seen by a static observer at varying radial distance.