Slowly rotating covariant anisotropic objects

TL;DR

The paper extends the Hartle slow-rotation formalism to anisotropic relativistic stars using a covariant $\mathcal C$-star equation of state, combining a covariant anisotropic EOS with a polytropic radial pressure to compute rotational perturbations up to second order. By numerically integrating the extended structure equations, it determines surface and integral properties such as the moment of inertia, mass change, and the mass quadrupole moment, and analyzes the monopole and quadrupole perturbations. A key result is that highly anisotropic $\mathcal C$-stars can become prolate at sufficient compactness, with the surface ellipticity turning negative, while the Kerr factor $QM/J^2$ can approach the Kerr black hole value for strong anisotropy. These findings connect rotating ultracompact configurations to black-hole-like behavior, though the stable branch remains bounded well below the Buchdahl/black-hole limit due to stability constraints. The work provides detailed insights into how covariant anisotropy shapes frame-dragging, deformation, and gravitational multipole moments in slowly rotating relativistic stars.

Abstract

The equilibrium configurations of slowly rotating anisotropic self-gravitating fluids are computed using the extended Hartle structure equations, including anisotropic effects, derived in our previous paper. We focus on the so-called $\mathcal{C}$-star, whose anisotropic pressure follows a fully covariant equation of state (EoS), while a standard polytrope describes the radial pressure. We determine surface and integral properties, such as the moment of inertia, mass change, mass quadrupole moment, and ellipticity. Notably, for certain values of the compactness parameter, highly anisotropic $\mathcal{C}$-stars exhibit a prolate shape rather than the typical oblate form, an intriguing behavior also observed in other anisotropic systems like Bowers-Liang spheres and stars governed by a quasi-local EoS. Although the $\mathcal{C}$-stars considered in this study are limited by stability criteria and cannot sustain compactness beyond $M/R\approx0.38$, we found indications that certain rotational perturbations exhibit similarities to those observed in other ultracompact systems approaching the black hole limit.

Figures (19)

• Figure 1: Mass-radius diagram for $\mathcal{C}$-stars, for different values of the anisotropy parameter $\mathcal{C}$. We consider a polytrope with $\kappa=100$ and $n=1$. The maximum of the curves indicates the onset of instability. The color filled horizontal bands indicate the observational mass constraints from the pulsars PSR J0952-0607 Romani:2022jhd and PSR J0348-0432 Demorest:2010bx.
• Figure 2: Compactness $GM/c^2 R$, as a function of the central density, for the same values of the anisotropic parameter $\mathcal{C}$ as in Fig. \ref{['fig:mass-rad']}. Note that, for a given value of the central density, as the anisotropy increases, the compactness also increases.
• Figure 3: Radial profiles of the $\varpi$ function, in the unit $\Omega$, for different compactness, for various values of the anisotropy parameter $\mathcal{C}$. Observe that as the compactness increases, $\varpi$ at the origin decreases. Notice that the highest compactness configurations we consider here are determined by the $M(R)$ stability criteria, so we do not have configurations especially close to the limiting compactness when the central pressure diverges for that particular EOS, nor to the BH limit.
• Figure 4: Left panel: surface value of the $\widetilde{\varpi}$ function, as a function of compactness, for various values of the anisotropy parameter $\mathcal{C}$. Right panel: function $\varpi$ evaluated at the origin, as a function of the compactness, for the same values of $\mathcal{C}$ as in the left panel. We measure $\varpi(0)$ in the unit $\Omega$. Observe that at any given anisotropy, $\varpi(0)$ decreases with compactness, and at any given compactness $\varpi(0)$ increases with anisotropy.
• Figure 5: Normalized moment of inertia $I/MR^2$ of $\mathcal{C}$-stars, as a function of the compactness, for various values of the anisotropy parameter $\mathcal{C}$.