Discontinuous Galerkin methods for general-relativistic hydrodynamics: formulation and application to spherically symmetric spacetimes

We have developed the formalism necessary to employ the discontinuous-Galerkin approach in general-relativistic hydrodynamics. The formalism is firstly presented in a general 4-dimensional setting and then specialized to the case of spherical symmetry within a 3+1 splitting of spacetime. As a direct application, we have constructed a one-dimensional code, EDGES, which has been used to asses the viability of these methods via a series of tests involving highly relativistic flows in strong gravity. Our results show that discontinuous Galerkin methods are able not only to handle strong relativistic shock waves but, at the same time, to attain very high orders of accuracy and exponential convergence rates in smooth regions of the flow. Given these promising prospects and their affinity with a pseudospectral solution of the Einstein equations, discontinuous Galerkin methods could represent a new paradigm for the accurate numerical modelling in relativistic astrophysics.Comment: 24 pages, 19 figures. Small changes; matches version to appear in PR

Twisted-torus configurations with large toroidal magnetic fields in relativistic stars

Understanding the properties of the internal magnetic field of neutron stars remains a theoretical challenge. Over the last years, twisted-torus geometries have been considered both in Newtonian and general-relativistic equilibrium models, as they represent a potentially good description of neutron star interiors. All of these works have found an apparent intrinsic limitation to geometries that are poloidal-field-dominated, with a toroidal-to-poloidal energy ratio inside the star that are <10%, unless surface currents are included and magnetic fields are allowed to be discontinuous. This limitation is in stark contrast with the general expectation that much higher toroidal fields should be present in the stellar interior and casts doubt about the stability and hence realism of these configurations. We here discuss how to overcome this limitation by adopting a new prescription for the azimuthal currents that leads to magnetized equilibria where the toroidal-to-total magnetic-field energy ratio can be as high as 90%, thus including geometries that are toroidal-field-dominated. Moreover, our results show that for a fixed exterior magnetic-field strength, a higher toroidal-field energy implies a much higher total magnetic energy stored in the star, with a potentially strong impact on the expected electromagnetic and gravitational-wave emission from highly magnetized neutron stars.Comment: 5 pages, 3 figures, 1 tabl

Distortion of neutron stars with a toroidal magnetic field

Models of rotating relativistic stars with a toroidal magnetic field have been computed for a sample of eight equations of state of cold dense matter. Non-rotating models admit important levels of magnetization and quadrupole distortion accompanied by a seemingly unlimited growth in size. Rotating models reach the mass-shedding limit at smaller angular velocities than in the non-magnetized case according to the larger circumferential equatorial radius induced by the magnetic field. Moreover, they can be classified as prolate-prolate, oblate-prolate, or oblate-oblate with respect to surface deformation and quadrupole distortion. Simple expressions for surface and quadrupole deformation are provided that are valid up to magnetar field strengths and rapid rotation.Comment: 3 pages, 2 figures, to appear in the Proceedings of the 13th Marcel Grossmann Meeting held in Stockholm 1-7 July, 2012, based on arXiv:1207.4035. Updated to final published versio

Equilibrium models of relativistic stars with a toroidal magnetic field

We have computed models of rotating relativistic stars with a toroidal magnetic field and investigated the combined effects of magnetic field and rotation on the apparent shape (i.e. the surface deformation), which could be relevant for the electromagnetic emission, and on the internal matter distribution (i.e. the quadrupole distortion), which could be relevant for the emission of gravitational waves. Using a sample of eight different cold nuclear physics equations of state, we have computed models of maximum field strength, as well as the distortion coefficients for the surface and the quadrupolar deformations. Surprisingly, we find that non-rotating models admit arbitrary levels of magnetization, accompanied by a growth of size and quadrupole distortion to which we could not find a limit. Rotating models, on the other hand, are subject to a mass-shedding limit at frequencies well below the corresponding ones for unmagnetized stars. Overall, the space of solutions can be split into three distinct classes for which the surface deformation and the quadrupole distortion are either prolate and prolate, oblate and prolate, or oblate and oblate, respectively. We also derive a simple formula expressing the relativistic distortion coefficients, which allows one to compute the surface deformation and the quadrupole distortion up to significant levels of rotation and magnetization, essentially covering all known magnetars. Such a formula replaces Newtonian equivalent expressions that overestimate the magnetic quadrupole distortion by about a factor of 6 and are inadequate for strongly relativistic objects like neutron stars.Comment: Published version with additional correction of remaining typo