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 $(m_{\pm },\hat{I}{›}_{\pm })$. Using the Israel junction formalism, we map the stable equilibria ($V_{\mathrm{eff}}^{\prime \prime }>0$) of the effective potential. Near the equilibrium radius $R_0$ the shell's surface density $\ddot{I}ƒ$ and pressure $p$ obey the linearized barotropic law $p=p_0+c_s^2(\ddot{I}ƒ-\ddot{I}{ƒ}_0)$, with sound speed $c_s^2=\hat{I}»c^2$. Since $c_s^2$ is independent of the equilibrium ratio $w_0\equiv p_0/(\ddot{I}{ƒ}_0{c}^2)$, tension shells ($w_0<0$) stay radially stable with real $c_s$. Fixing $\hat{I}{›}_{+}$ 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 $(R_0,\ddot{I}{ƒ}_0)$ over $(m_{\pm },\hat{I}{›}_{\pm },\hat{I}»,w_0)$ and find that stable shells with $\ddot{I}{ƒ}_0>0$ and $0<\hat{I}»\le 1$ exist only for $m_{+}/m_{-}>1$, at three scales -- the photon sphere, the SdS static radius, and the cosmological horizon. At $\hat{I}»=1$ the numerical windows, checked against the analytic test-shell bounds, are $(1-\sqrt{13})/6\lesssim w_0\lesssim 1/2$ ($\hat{I}{›}_{+}=\hat{I}{›}_{-}$), $-2/3\lesssim w_0\lesssim 1/2$ ($\hat{I}{›}_{+}>\hat{I}{›}_{-}$), and $0\lesssim w_0$ ($\hat{I}{›}_{+}<\hat{I}{›}_{-}$). Positive-pressure shells ($0\lesssim w_0\lesssim 1/2$) sit near the photon sphere and those with $w_0\gtrsim 1/2$ near the static radius scale, while tension shells reach the cosmological horizon scale for $\hat{I}{›}_{+}=\hat{I}{›}_{-}$, only the static radius scale for $\hat{I}{›}_{+}>\hat{I}{›}_{-}$, and are absent for $\hat{I}{›}_{+}<\hat{I}{›}_{-}$. Finally, we compute the dark fluid shell's imprint on the SdS black-hole shadow seen by a static observer at varying radial distance.

The manuscript consists of 14 pages and 5 figures. The numerical analysis file used for the construction of the figures may be found at https://github.com/Dimitrios1993/Static-Thin-Shells-in-SdS-Spacetimes