Hamilton-Jacobi hydrodynamics of pulsating relativistic stars

TL;DR

This paper introduces a new well-posed hydrodynamic scheme for simulating pulsating relativistic stars, addressing the fluid-vacuum boundary problem in numerical relativity with improved stability and mode resolution.

Contribution

It formulates a canonical hydrodynamic scheme based on variational principles and Hamilton-Jacobi equations, enhancing simulation stability at the vacuum boundary.

Findings

The scheme is well-posed and suitable for neutron-star simulations.

It preserves irrotational flows using variational principles.

It resolves more radial oscillation modes than standard methods.

Abstract

The dynamics of self-gravitating fluid bodies is described by the Euler-Einstein system of partial differential equations. The break-down of well-posedness on the fluid-vacuum interface remains a challenging open problem, which is manifested in simulations of oscillating or inspiraling binary neutron-stars. We formulate and implement a well-posed canonical hydrodynamic scheme, suitable for neutron-star simulations in numerical general relativity. The scheme uses a variational principle by Carter-Lichnerowicz stating that barotropic fluid motions are conformally geodesic and Helmholtz's third theorem stating that initially irrotational flows remain irrotational. We apply this scheme in 3+1 numerical general relativity to evolve the canonical momentum of a fluid element via the Hamilton-Jacobi equation. We explore a regularization scheme for the Euler equations, that uses a fiducial atmosphere in hydrostatic equilibrium and allows the pressure to vanish, while preserving strong hyperbolicity on the vacuum boundary. The new regularization scheme resolves a larger number of radial oscillation modes compared to standard, non-equilibrium atmosphere treatments.