A novel formulation for the evolution of relativistic rotating stars

TL;DR

This paper introduces a new Lagrangian formulation for constructing equilibrium configurations of rotating stars in general relativity, enabling more flexible modeling of their structure and evolution.

Contribution

The authors develop a novel Lagrangian approach that avoids the first integral of the Euler equation and uses a new root-finding scheme, improving modeling of relativistic rotating stars.

Findings

Successfully constructed barotropic and baroclinic rotating star models.

Demonstrated the method's capability with evolutionary sequences simulating cooling, mass-loss, and accretion.

Introduced a new multi-dimensional root-finding scheme, the W4 method.

Abstract

We present a new formulation to construct numerically equilibrium configurations of rotating stars in general relativity. Having in mind the application to their quasi static evolutions, we adopt a Lagrangian formulation of our own devising, in which we solve force balance equations to seek for the positions of fluid elements assigned to the grid points, instead of the ordinary Eulerian formulation. Unlike previous works in the literature, we do not employ the first integral of the Euler equation, which is not obtained by an analytic integration in general. We assign a mass, specific angular momentum and entropy to each fluid element in contrast to the previous methods, in which the spatial distribution of the angular velocity or angular momentum is specified. Those distributions are determined after the positions of all fluid elements (or grid points) are derived in our formulation. We solve the large system of algebraic nonlinear equations that are obtained by discretizing the time-independent Euler and Einstein equations in the finite-elements method by using our new multi-dimensional root-finding scheme, named the W4 method. To demonstrate the capability of our new formulation, we construct some rotational configurations both barotropic and baroclinic. We also solve three evolutionary sequences that mimic the cooling, mass-loss, and mass-accretion as simple toy models.