A Precise Ultra - Dense Star Model on Spheroidal Space-Time

TL;DR

The paper presents a precise closed-form interior solution for a relativistic superdense star whose interior geometry is a three-spheroid in spheroidal space-time, governed by the Einstein field equations. By adopting a $t_{1}$-constant three-spheroid framework and solving the resulting ODE for $\psi = e^{\nu/2}$, the authors obtain an explicit metric and a physically admissible density profile parameterized by $\lambda = \rho_a/\rho_0$, linking central and boundary densities. The solution satisfies standard energy conditions and hydrostatic equilibrium, with matching to the exterior Schwarzschild solution and a mass expression $m = \frac{35}{2R_1^2}\frac{a^3}{[1 + 34(a^2/R_1^2)]}$. A stability analysis based on Chandrasekhar’s radial pulsations indicates dynamical stability for $\lambda = 0.4$ (and $k_1 = -34$), supporting the model as a physically viable analytic description of ultra-dense stellar interiors and yielding neutron-star-like mass-radius behavior in the appropriate parameter regime.

Abstract

This study presents a static, spherically symmetric configuration in which the interior geometry of a relativistic superdense star is modeled as a three-spheroid with constant $t_1$. The model is constructed using an analytical closed-form solution to Einstein's field equations. Assuming a characteristic density of $ρ_{a}= 2\times10^{14} \mathrm{g\,cm^{-3}}$, we compute the total mass and radius of the star based on a prescribed set of structural parameters that influence the density profile. The resulting stellar configurations exhibit boundary radii and total masses comparable to those of neutron stars with vanishing charge density. New exact solutions are obtained by solving the relevant second-order ordinary differential equations. We demonstrate that these solutions satisfy the standard energy conditions and maintain hydrostatic equilibrium throughout the stellar interior. All physical requirements remain valid at every point within the configuration. In this framework, the parameter $λ=\frac{ρ_{a}}{ρ_{0}}$ serves as a key determinant of the mass-radius relationship. We further assess the suitability of the model for representing a relativistic superdense star and analyze its stability under radial perturbations. The investigation indicates that, for the configuration to remain dynamically stable, the density ratio between the outer and inner regions must take the value $λ= 0.4$.

Figures (2)

• Figure 1: (a) Graph of $\hat{\rho }$=$\frac{8G\pi}{c^{2}}\rho_{1}\times10^{4}$ against radius $\hat{p}=\frac{8G\pi}{c^{4}}p _{1}\times10^{4}$
• Figure 2: $\lambda$ =0.4 , $k$=-34 The functions $\hat{\rho}$, $\hat{p}$, and $\hat{\rho} - 3\hat{p}$ are plotted as functions of the radial coordinate $r$.