Static Charged Polytropic Spheres with a Cosmological Constant: Physical Acceptability and Trapped Orbits

TL;DR

This work extends static, spherically symmetric fluid models in general relativity by combining a polytropic EOS $p\propto\rho^{\Gamma}$ with a power-law charge distribution $q\propto r^{n}$ in the presence of a cosmological constant $\Lambda$. By reformulating the generalized TOV equation into a master equation for the mass profile $m(r)$ and solving it numerically, the authors characterize physically acceptable charged polytropic configurations and compute their density, pressure, and metric functions, ensuring subluminal sound speeds and satisfaction of energy conditions. They then study internal trapping of circular geodesics by constructing an effective potential for neutral and charged, massive and massless test particles, revealing trapping regions across broad $(n,\Gamma)$ ranges and showing that increasing $Q$ generally suppresses trapping while increasing the boundary radius $r_b$ enhances it. The results indicate that trapped orbits can occur in many physically viable models, with the cosmological constant playing a minor role for realistic $\Lambda$ values, and highlight future work on stability analyses and anisotropic generalizations. These insights enhance understanding of ultra-compact objects and potential neutrino or wave trapping in charged polytropic spacetimes with $\Lambda$.

Abstract

We consider static charged fluid spheres with a cosmological constant. We assume a polytropic equation of state, $p \propto ρ^Γ$, and a power law charge distribution, $q\propto r^n$. Using this, we convert the generalised Tolman-Oppenheimer-Volkoff equation into a differential equation for the mass profile. By solving this equation numerically, we analyse both physical and geometric properties of these charged fluid spheres for different values of $n$ and $Γ$. By imposing subluminal sound speeds and energy conditions, we restrict ourselves to configurations that are physically acceptable. Then, within these physical models, we study internal trapping of circular geodesics and find the trapping regions in the $n$-$Γ$ parameter space. Going beyond the traditionally studied case of null geodesics, we consider orbits of charged and/or massive particles. We show that for neutral null particles (and only for them), their trapping depends purely on the properties of the space-time. In the other three cases, properties such as the particle's own charge and/or energy also play a role. In general, we find that trapping of all types of particles is allowed for a broad range of $n$ and $Γ$.

Figures (21)

• Figure 1: The mass profile, $m(r)$, of various charged polytropic configurations (different values of $n$ and $\Gamma$) obtained by solving equation \ref{['masseq']}. The central density, $\rho_0$, and total charge, $Q$, correspond to their typical values for ultra compact objects mathews1997ray2003ray2004. The boundary radius is, $r_b = 10^4$ m and $\Lambda = 10^{-52}$ m$^{-2}$.
• Figure 2: The density profile, $\rho(r)$, for various polytropic configurations. We present the cases where the energy density decreases monotonically throughout the extent of the fluid sphere.
• Figure 3: The radial profile for pressure, $p(r)$, for various polytropic configurations. We present the cases where the pressure decreases monotonically throughout the extent of the fluid sphere.
• Figure 4: The speed of sound squared, $c_s^2(r)$, as a function of the radial coordinate, $r$. For all the presented cases, we find subluminal sound speeds ($c_s^2 < 1$), i.e., causality is not violated.
• Figure 5: The bulk modulus, $B(r)$, as a function of $r$ for various charged polytropic configurations.