Formulation for nonaxisymmetric uniformly rotating equilibrium configurations in the second post-Newtonian approximation of general relativity

TL;DR

This paper develops a formalism to compute equilibrium configurations of uniformly rotating fluids in the second post-Newtonian approximation of general relativity, enabling accurate modeling of complex astrophysical objects.

Contribution

It introduces a method to solve 29 Poisson equations for nonaxisymmetric rotating bodies in relativistic gravity, applicable to binary neutron stars and similar systems.

Findings

Formalism accurately solves Poisson equations with decreasing source terms

Applicable to nonaxisymmetric configurations like binary neutron stars

Facilitates modeling of relativistic rotating equilibrium states

Abstract

We present a formalism to obtain equilibrium configurations of uniformly rotating fluid in the second post-Newtonian approximation of general relativity. In our formalism, we need to solve 29 Poisson equations, but their source terms decrease rapidly enough at the external region of the matter(i.e., at worst $O(r^{-4})$). Hence these Poisson equations can be solved accurately as the boundary value problem using standard numerical methods.This formalism will be useful to obtain nonaxisymmetric uniformly rotating equilibrium configurations such as synchronized binary neutron stars just before merging and the Jacobi ellipsoid.