[Paper Review] Spontaneous symmetry breaking of rapidly rotating stars in general relativity: influence of the 3D-shift vector

S. Bonazzola, J. Frieben+1 more|arXiv (Cornell University)|Oct 28, 1997

Pulsars and Gravitational Waves Research6 citations

TL;DR

This paper investigates spontaneous symmetry breaking in rapidly rotating neutron stars under general relativity by incorporating the full three-dimensional shift vector in a perturbative framework. It finds that the 3D-shift vector significantly suppresses the bar-mode instability, yet triaxial configurations remain possible for a 1.4 M☉ neutron star with γ = 2.5, making them viable gravitational wave sources for LIGO/VIRGO detectors.

ABSTRACT

An analytical scheme and a numerical method in order to study the effects of general relativity on the viscosity driven secular bar mode instability of rapidly rotating stars are presented. The approach consists in perturbing an axisymmetric and stationary configuration and studying its evolution by constructing a series of triaxial quasi-equilibrium configurations. These are obtained by solution of an approximate set of field equations where only the dominant non-axisymmetric terms are taken into account. The progress with respect to our former investigation consists in a higher relativistic order of the non-axisymmetric terms included into the computation, namely the fully three-dimensional treatment of the vector part of the space-time metric tensor as opposed to the scalar part, solely, in the former case. The scheme is applied to rotating stars built on a polytropic equation of state and compared to our previous results. The 3D-vector part turns out to inhibit the symmetry breaking efficiently. Nevertheless, the bar mode instability is still possible for an astrophysically relevant mass of M_ns=1.4 M_sun when a stiff polytropic equation of state with an adiabatic index of gamma=2.5 is employed. Triaxial neutron stars may be efficient emitters of gravitational waves and are thus potentially interesting sources for the forthcoming laser interferometric gravitational wave detectors such as LIGO, VIRGO and GEO600. From a numerical point of view, the solution of the three-dimensional minimal-distortion shift vector equation in spherical coordinates is an important achievement of our code.

Motivation & Objective

• To analyze the influence of the 3D-shift vector on spontaneous symmetry breaking in rapidly rotating stars within general relativity.
• To extend previous work by including the full vector part of the spatial metric instead of relying solely on scalar approximations.
• To assess the viability of triaxial neutron stars as gravitational wave sources under realistic astrophysical conditions.
• To develop and implement a numerical method solving the 3D minimal-distortion shift vector equation in spherical coordinates.
• To evaluate the stability of polytropic neutron stars with γ = 2.5 under secular bar-mode instability in a relativistic framework.

Proposed method

• Perturbing an axisymmetric, stationary equilibrium configuration to construct a sequence of triaxial quasi-equilibrium states.
• Solving an approximate set of Einstein's field equations by retaining only dominant non-axisymmetric terms.
• Including the full three-dimensional vector component of the spatial metric (3D-shift vector) rather than scalar approximations.
• Employing a polytropic equation of state with γ = 2.5 for neutron star modeling.
• Numerically solving the minimal-distortion shift vector equation in spherical coordinates to maintain coordinate conditions.
• Comparing results with prior studies that used only scalar components of the shift vector.

Experimental results

Research questions

• RQ1 How does the inclusion of the full 3D-shift vector affect the onset and growth of the bar-mode instability in rapidly rotating stars?
• RQ2 To what extent does the 3D-shift vector suppress spontaneous symmetry breaking compared to scalar-only approximations?
• RQ3 Can triaxial configurations persist for astrophysically relevant masses (e.g., 1.4 M☉) and equations of state (γ = 2.5) under full 3D general relativistic treatment?
• RQ4 What is the impact of higher-order relativistic corrections on the stability of rotating neutron stars?
• RQ5 Are triaxial, rapidly rotating neutron stars viable sources of gravitational waves detectable by LIGO/VIRGO?

Key findings

• The inclusion of the full 3D-shift vector significantly inhibits the development of the bar-mode instability compared to previous scalar-only treatments.
• Despite this suppression, triaxial configurations remain dynamically possible for a 1.4 M☉ neutron star with a stiff polytropic equation of state (γ = 2.5).
• The numerical solution of the 3D minimal-distortion shift vector equation in spherical coordinates is successfully achieved and represents a key technical advancement.
• The results confirm that such triaxial neutron stars could be efficient sources of gravitational radiation, relevant for laser interferometric detectors like LIGO, VIRGO, and GEO600.
• The instability threshold is shifted to higher rotational energies or faster rotation rates due to the 3D-shift vector effects.
• The study demonstrates that general relativistic effects, particularly in the vector part of the metric, play a crucial role in determining the stability of rotating compact objects.