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 separating two Schwarzschild--de Sitter spacetimes with parameters $(m_\pm,Λ_\pm)$. Using the Israel junction formalism and a linear barotropic equation of state $p = λ(σ- σ_1)c^2$, we decouple the sound speed $c_s^2 = λc^2$ from the equilibrium equation-of-state parameter $w_0 \equiv p_0 / (σ_0 c^2)$ and derive the effective potential governing radial dynamics. For observationally motivated parameters, stable configurations with $σ_0>0$ and $0<λ\leq1$ exist only when $m_+/m_- > 1$. Three distinct stability windows emerge, when $λ=1$: $-3/7 \lesssim w_0 \lesssim 1/2$ for $Λ_+ = Λ_-$, $-2/3 \lesssim w_0 \lesssim 1/2$ for $Λ_+ > Λ_-$ and $0 \lesssim w_0 \lesssim 1$ for $Λ_+ < Λ_-$. Positive-pressure shells ($w_0>0$) reside near the photon sphere, whereas negative-pressure shells ($w_0<0$) extend outward, reaching either the cosmological horizon or the static radius. Stability relies on the variation of $w(σ)$ with the surface energy density. Negative pressure (tension) stabilizes the system because the tension increases during expansion. Conversely, positive pressure stabilizes the system because the pressure increases during contraction. Finally, a static, stable, dark, fluid thin shell acts as a gravitational refractive layer that enlarges the black hole's shadow for a distant, static observer outside the shell. The effect depends on the shell radius $R_0$, the background parameters $(m_{\pm}, Λ_{\pm})$, and the equation-of-state. Dark fluid shells can be considered as theoretical toy models that illustrate qualitative effects. Future high-resolution black hole shadow observations could, in principle, use such models to explore how different equations of state might influence observable signatures.

Figures (7)

• Figure 1: We set $m_- = 10^{10}\,M_\odot$ and consider a cosmological-constant-dominated universe with $\rho_{\Lambda_+} = \rho_{\rm crit} = 3H_0^2/(8\pi G)$Planck2018. We numerically solve for $(R_0,\sigma_0)$ the system of Eqs. (\ref{['systemA']}) and (\ref{['systemB']}) while requiring that Eq. (\ref{['Veffpp0']}) be positive and that condition Eq. (\ref{['sigma0cond']}) be satisfied and we plot the stability regions in the $(w_0,\lambda)$ parameter space (left column), together with the corresponding equilibrium radius $R_0$ distribution over the stable region (middle column), and the surface energy density $\sigma_0$ distribution (right column) for each $( w_0,\lambda)$ in the stable domain. The rows correspond to the cases: (a) $m_+/m_- = \Lambda_+/\Lambda_- = 1.25$, (b) $m_+/m_- = 1.25, \Lambda_+/\Lambda_- = 0.75$, (c) $m_+/m_- = 1.25, \Lambda_{\pm}=0$, and (d) $m_+/m_- = 1.25, \Lambda_+/\Lambda_- = 1.0$. The colorbars represent the equilibrium radius in units of the Schwarzschild radius ($R_s$) and the surface density in units of the characteristic scale $m_-/R_s^2$. The reddish or white areas in the distribution plots indicate regions where either (i) no static equilibrium with $\sigma_0 > 0$ exists, or (ii) a static equilibrium exists but is unstable ($V_{\mathrm{eff}}"(R_0) < 0$).
• Figure 2: We set $m_- = 10^{10}\,M_\odot$ and consider a cosmological-constant-dominated universe with $\rho_{\Lambda_+} = \rho_{\rm crit} = 3H_0^2/(8\pi G)$Planck2018. By imposing $w_0 = \lambda$, i.e., assuming Eq. (\ref{['sigmaperivpaper']}), we numerically solve the system of Eqs. (\ref{['systemA']}) and (\ref{['systemB']}) for $(R_0, \sigma_0)$. In doing so, we require that Eq. (\ref{['Veffpp0']}) be positive and that the condition in Eq. (\ref{['sigma0cond']}) be satisfied. We then plot the stability regions in the $(w_0, \Lambda_+/\Lambda_-)$ parameter space (left column), together with the corresponding equilibrium radius $R_0$ distribution over the stable region (middle column), and the surface energy density $\sigma_0$ distribution (right column) for each $(w_0, \Lambda_+/\Lambda_-)$ within the stable domain. The top row corresponds to the mass ratio $m_+/m_- = 1.25$, while the bottom row corresponds to $m_+/m_- = 0.75$. Left column: Stability regions in the parameter space spanned by the equation-of-state parameter $w_0$ and the cosmological constant ratio $\Lambda_+/\Lambda_-$. Green areas represent stable configurations ($V_{\mathrm{eff}}" > 0$), whereas red areas indicate instability. Middle and right columns: Distributions of the normalized equilibrium radius $R_0$ (measured in units of the Schwarzschild radius $R_s$) and the surface energy density $\sigma_0$, respectively. The heatmaps are shown only for configurations that satisfy the stability conditions.
• Figure 3: We set $m_- = 10^{10}\,M_\odot$ in a cosmological-constant-dominated universe with $\rho_{\Lambda_+} = \rho_{\rm crit} = 3H_0^2/(8\pi G)$Planck2018, and consider parameters satisfying $m_+/m_- = \Lambda_+/\Lambda_- = 1.001$. The static radius in the corresponding SdS spacetimes is $R_{\rm st} \simeq 6.52\times10^{21}\,\mathrm{m}$, defined as $R_{\rm st} \equiv \left[\frac{3 G m_-}{\Lambda c^2}\right]^{1/3}$ with $\Lambda = (\Lambda_+ + \Lambda_-)/2$. For $-2/3 < w_0 < 0$, stable static shells exist near $R_{\rm st}$, with $w_0 = -2/3$ marking the stability boundary when $\lambda = 1$. In the narrower range $0.01 \lesssim w_0 < 0.5$, the stable radii cluster around $R \sim 10^{13-14}\,\rm m$, approaching the photon sphere $3G m_-/c^2 \simeq 4.43 \times 10^{13}\,\rm m$ at $w_0 = 0.5$ (but always exceeding itBrady:1991np). This illustrates how the stability region depends on $m_+/m_- > 1$ and $\Lambda_+/\Lambda_- > 1$.
• Figure 4: Normalized stable shell radius $R_0$ as a function of the equation of state parameter $w_0$ for a fixed sound speed $\lambda = 1$. The figure displays three distinct mass-vacuum energy configurations. Top Row (Case 1): A slight mass and vacuum energy excess ($m_+/m_- = 1.001, \Lambda_+/\Lambda_- = 1.001$). The left panel shows the negative pressure branch ($w_0 < 0$) normalized to the static radius $R_{st}$, while the right panel shows the positive pressure branch ($w_0 > 0$) normalized to the photon ring radius $3Gm_-/c^2$. Middle Row (Case 2): A significant mass excess with equal vacuum energies ($m_+/m_- = 1.001, \Lambda_+/\Lambda_- = 1.0$). The left panel displays the negative pressure branch normalized to the cosmic horizon $R_{ch}$, and the right panel shows the positive pressure branch normalized to the photon ring. Bottom Row (Case 3): A mass excess coupled with a vacuum energy deficit ($m_+/m_- = 1.001, \Lambda_+/\Lambda_- = 0.999$). The left panel shows the small astrophysical branch stable for $0 < w_0 < 0.5$, while the right panel shows the large cosmological branch stable for $0.5 < w_0 < 1$.
• Figure 5: Stability analysis of the shell for the case where $m_+/m_-=1.001$ and $\Lambda_+/\Lambda_-=1.001$. (a) The effective potential shifted by the perturbation energy, $(V_{\text{eff}}(R) - \epsilon^2)/c^2$. The shell is confined between the turning points $d_1$ and $d_2$ where the potential becomes negative. (b) Solutions to the exact equation of motion $\ddot{R} = -V'_{\text{eff}}(R)$. For initial velocities $\dot{R} < \sqrt{2}\epsilon_{\text{crit}}$ (blue, $\epsilon = 100$ m/s), the motion remains bounded (stable). Conversely, if $\dot{R} \ge \sqrt{2}\epsilon_{\text{crit}}$ (red, $\epsilon = 2.92715\times 10^5$ m/s), the shell overcomes the potential barrier and escapes to cosmic horizon (unbounded instability).