A novel Lagrangian formulation to construct relativistic rotating stars: towards its application to their evolution calculations

Abstract

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 on a secular time-scale, we adopt a Lagrangian formulation of our own devising, in which we solve force-balance equations to seek for the positions of fluid elements corresponding 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 analytically in general. We assign a mass, specific angular momentum and entropy to each fluid element in contrast to the previous Eulerian methods, in which the spatial distribution of the angular velocity or angular momentum is specified. These 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 non-linear equations that are obtained by discretizing the time-independent Euler and Einstein equations in the finite-element method by using our new multidimensional root-finding scheme, named the W4 method. To demonstrate the capability of our new formulation, we construct some rotational configurations, both barotropic and baroclinic. As toy models, we also solve three evolutionary sequences that mimic the cooling, mass-loss, and mass-accretion.