The representation and computational efficiency of the Tolman-Oppenheimer-Volkoff equations in isotropic coordinates

TL;DR

This paper introduces an analytical approach for solving the Tolman-Oppenheimer-Volkoff equations in isotropic coordinates to model relativistic stars, analyzing its computational efficiency and accuracy compared to existing methods.

Contribution

It presents a novel direct solution method in isotropic coordinates and evaluates its computational performance and accuracy relative to traditional curvature coordinate approaches.

Findings

Computational time increases with matter stiffness.

Mass difference initially grows with central energy density, then stabilizes.

Proposed method's accuracy is comparable to existing packages.

Abstract

This study aims to provide an analytical scheme for computing equilibrium configurations of relativistic stars by solving the Tolman-Oppenheimer-Volkoff equations directly in isotropic polar coordinates, as opposed to the commonly applied methods of rescaling the radial profile of corresponding solutions obtained in curvature coordinates. This study also provides evidence that the differential equation for gravitational mass may be replaced by an algebraic expression relating the metric potential to the energy density in the form of the quartic equation. Nevertheless, the greater computational expense of evaluating the algebraic equation renders its application less efficient. A further objective of this study was to evaluate the performance of the present computational scheme in the computational time and numerical accuracy. Our results indicate that the computational time increases with the stiffness of the constituent matter inside the star. Conversely, the absolute difference between the gravitational mass obtained by the proposed method and that computed via the use of LORENE packages initially increases rapidly with the central energy density, but the rate of growth subsequently declines as the maximum mass configuration is approached.