Spinning solutions in general relativity with infinite central density

TL;DR

This paper develops a numerical method using MSQI coordinates with a logarithmic radial scale to simulate rotating polytropes in general relativity, enabling analysis of solutions with extremely high central densities and confirming stability criteria.

Contribution

The paper introduces a novel numerical approach with logarithmic radial coordinates for simulating highly dense rotating relativistic stars, extending the range of resolvable solutions and validating stability analysis.

Findings

Successfully resolved near-singular mass distributions at the center.

Found good agreement between analytical small-radius behavior and numerical results.

Confirmed stability limit at the first maximum in mass versus central density.

Abstract

This paper presents general relativistic numerical simulations of uniformly rotating polytropes. Equations are developed using MSQI coordinates, but taking a logarithm of the radial coordinate. The result is relatively simple elliptical differential equations. Due to the logarithmic scale, we can resolve solutions with near-singular mass distributions near their center, while the solution domain extends many orders of magnitude larger than the radius of the distribution (to connect with flat space-time). Rotating solutions are found with very high central energy densities for a range of adiabatic exponents. Analytically, assuming the pressure is proportional to the energy density (which is true for polytropes in the limit of large energy density), we determine the small radius behavior of the metric potentials and energy density. This small radius behavior agrees well with the small radius behavior of large central density numerical results, lending confidence to our numerical approach. We compare results with rotating solutions available in the literature, which show good agreement. We study the stability of spherical solutions: instability sets in at the first maximum in mass versus central energy density; this is also consistent with results in the literature, and further lends confidence to the numerical approach.