0000000000560235
AUTHOR
Eric Gourgoulhon
Improved constrained scheme for the Einstein equations: An approach to the uniqueness issue
Uniqueness problems in the elliptic sector of constrained formulations of Einstein equations have a dramatic effect on the physical validity of some numerical solutions, for instance when calculating the spacetime of very compact stars or nascent black holes. The fully constrained formulation (FCF) proposed by Bonazzola, Gourgoulhon, Grandcl\'ement, and Novak is one of these formulations. It contains, as a particular case, the approximation of the conformal flatness condition (CFC) which, in the last ten years, has been used in many astrophysical applications. The elliptic part of the FCF basically shares the same differential operators as the elliptic equations in CFC scheme. We present he…
Mathematical Issues in a Fully-Constrained Formulation of Einstein Equations
Bonazzola, Gourgoulhon, Grandcl\'ement, and Novak [Phys. Rev. D {\bf 70}, 104007 (2004)] proposed a new formulation for 3+1 numerical relativity. Einstein equations result, according to that formalism, in a coupled elliptic-hyperbolic system. We have carried out a preliminary analysis of the mathematical structure of that system, in particular focusing on the equations governing the evolution for the deviation of a conformal metric from a flat fiducial one. The choice of a Dirac's gauge for the spatial coordinates guarantees the mathematical characterization of that system as a (strongly) hyperbolic system of conservation laws. In the presence of boundaries, this characterization also depen…
Analysis of the Characteristics in the Meudon Constrained Evolution Scheme
A first analysis of the characteristics associated with the evolving modes in the constraint evolution scheme proposed by the Meudon group in 2004 is presented. The system is written in a first-order hyperbolic form and a so-called generalized Dirac gauge is considered. Applications to inner boundary conditions in an excised approach to black hole evolutions are discussed.
Evolutionary sequences of rotating protoneutron stars
We investigate the evolution of rigidly and differentially rotating protoneutron stars (PNSs) during the first twenty seconds of their life. We solve the equations describing stationary axisymmetric configurations in general relativity coupled to a finite temperature, relativistic equation of state, to obtain a sequence of quasi-equilibrium configurations describing the evolution of newly born neutron stars. Our estimates show that the scale of variation of the angular velocity in a PNSs is of the order of 7-10 km. We obtain the maximum rotation frequency that can be reached as the protoneutron stars deleptonizes and cools down, as well as other relevant parameters such as total angular mom…
Trapping Horizons as inner boundary conditions for black hole spacetimes
We present a set of inner boundary conditions for the numerical construction of dynamical black hole space-times, when employing a 3+1 constrained evolution scheme and an excision technique. These inner boundary conditions are heuristically motivated by the dynamical trapping horizon framework and are enforced in an elliptic subsystem of the full Einstein equation. In the stationary limit they reduce to existing isolated horizon boundary conditions. A characteristic analysis completes the discussion of inner boundary conditions for the radiative modes.