An excision scheme for black holes in constrained evolution formulations: spherically symmetric case

Excision techniques are used in order to deal with black holes in numerical simulations of Einstein equations and consist in removing a topological sphere containing the physical singularity from the numerical domain, applying instead appropriate boundary conditions at the excised surface. In this work we present recent developments of this technique in the case of constrained formulations of Einstein equations and for spherically symmetric spacetimes. We present a new set of boundary conditions to apply to the elliptic system in the fully-constrained formalism of Bonazzola et al. (2004), at an arbitrary coordinate sphere inside the apparent horizon. Analytical properties of this system of boundary conditions are studied and, under some assumptions, an exponential convergence toward the stationary solution is exhibited for the vacuum spacetime. This is verified in numerical examples, together with the applicability in the case of the accretion of a scalar field onto a Schwarzschild black hole. We also present the successful use of the excision technique in the collapse of a neutron star to a black hole, when excision is switched on during the simulation, after the formation of the apparent horizon. This allows the accretion of matter remaining outside the excision surface and for the stable long-term evolution of the newly formed black hole.Comment: 14 pages, 9 figures. New section added and changes included according to published articl

Evolution formalisms of Einstein equations: Numerical and Geometrical Issues

The topic treated along this thesis is the theoretical and numerical study of formalisms of Einstein equations, with the final aim of applications to black holes and gravitational waves. The General Relativity theory of Einstein (1915) postulated that light and trajectories of all particles are curved by the geometry of spacetime. Schwarzschild a few months later and Kerr in 1963 found solutions which describe non-rotating and rotating black holes. From an astrophysical point of view, a stellar black hole can be seen as the final result of some kind of collapse of massive stars or merger of compact binaries objects. One of the predicted consequences of General Relativity, not detected yet, is the existence of gravitational waves. This is the only direct method for detecting black holes. These waves can be viewed as ripples in the curvature of spacetime caused by non-spherically symmetric accelerations of matter. The first indirect detection was in 1974 by Hulse and Taylor, and they were awarded the Nobel. Huge experimental, theoretical and numerical efforts have been carried out in the last forty years, from the resonant bars of Weber to the future space-based interferometers as LISA. The General Relativity theory describes scenarios involving strong gravitational fields and velocities close to light velocity. The different formalisms lead to write Einstein equations as a set of partial differential equations. We must recognize the capability of the most used ones, as the so-called BSSN (Baumgarte-Shapiro-Shibata-Nakamura), crucial in the recent simulations of binary black holes. One of the recent formalisms is the FCF (Fully Constrained Formalism), which will be object of study along the thesis. In FCF, Einstein equations are written as a set of elliptic-hyperbolic equations, where the constraints are solved in each time step. It is a natural generalization of the relativistic approximation CFC (Conformally Flat Condition), used in many astrophysical applications. The theoretical work done in the thesis is very important, as the proof of the local existence of maximal slicings in spherically symmetric spacetimes. Moreover, the resulting equations in FCF have been studied mathematically. On one hand, the introduction of a new vector allows rewriting the elliptic equations such that local uniqueness is guaranteed and the equations form a hierarchical system. This is a very important in order to guarantee the well-posedness of the whole system. Numerical problems appear as consequence of the theoretical ones, and it was no possible to compute the migration test of a rotating neutron star and the spherical and rotational collapse to a black hole in the CFC approximation (and, so, in the FCF). The hyperbolic properties of the evolution system have also been studied. The explicit expressions of the eigenvalues are very useful in the study of inner boundary conditions of trapping horizons in which the singularity is removed from the numerical grid. The numerical work done in the thesis has as objective the extension of the CoCoNuT code to the FCF, in order to simulate non-vacuum dynamical spacetimes, including magnetic fields. We have performed the evolution of Teukolsky waves, analytical solution in vacuum and in linear regime, and the evolution of stationary rotating and perturbed rotating neutron stars. The next step will be the extraction of the gravitational signal in astrophysical scenarios and to compare the results with other approximations, as the quadrupole formula.El tema de la tesis es el estudio teórico y numérico de los formalismos de las ecuaciones de Einstein, con aplicaciones a la formación de agujeros negros y generación de ondas gravitatorias. La teoría de la Relatividad General de Einstein (1915) postulaba que la luz y las trayectorias de las partículas eran curvadas por la geometría del espacio tiempo. Schwarzschild (1915) y Kerr (1963) encontraron las soluciones que describen agujeros negros estático y en rotación. Desde un punto de vista astrofísico, un agujero negro estelar es el resultado de algunos tipos de colapso o la fusión de binarias de objetos compactos. Las ondas gravitatorias, predichas por la Relatividad General, aún no detectadas, son el único método directo para detectar agujeros negros. Son arrugas en la curvatura del espacio-tiempo. La primera detección indirecta por Hulse y Taylor (1974) les valió el Nobel. Enormes esfuerzos experimentales se han llevado a cabo en los últimos cuarenta años, desde las barras resonantes de Weber hasta los futuros observatorios espaciales como LISA. La Relatividad General describe escenarios que involucran campos gravitatorios intensos y velocidades próximas a la de la luz. En los diferentes formalismos las ecuaciones de Einstein se escriben como diferentes sistemas de ecuaciones en derivadas parciales. BSSN ha sido crucial en las recientes simulaciones de binarias de agujeros negros. FCF, introducido recientemente, ha sido objeto de estudio en la tesis. Las ligaduras se resuelven en cada paso de tiempo y es una generalización natural de la aproximación relativista CFC. El trabajo teórico realizado es muy importante: la prueba de la existencia local de foliaciones maximales en espacios-tiempo con simetría esférica; la introducción de un campo vectorial en las ecuaciones elípticas de FCF, que permite garantizar la unicidad local; el estudio de la hiperbolicidad de las ecuaciones de evolución en FCF, con aplicación a horizontes atrapados de agujeros negros. El trabajo numérico se centra en la extensión del código numérico CoCoNuT a la formulación FCF, para poder simular espacios-tiempo dinámicos con materia, incluyendo campos magnéticos. Varios tests satisfactorios permiten pensar en la extracción de la radiación gravitatoria en escenarios más complejos

Master Majorana neutrino mass parametrization

After introducing a master formula for the Majorana neutrino mass matrix we present a master parametrization for the Yukawa matrices automatically in agreement with neutrino oscillation data. This parametrization can be used for any model that induces Majorana neutrino masses. The application of the master parametrization is also illustrated in an example model, with special focus on its lepton flavor violating phenomenology.Comment: 5 pages, 3 figures and 2 tables; v2: minor corrections, matches published versio

General parametrization of Majorana neutrino mass models

We discuss a general formula which allows to automatically reproduce experimental data for Majorana neutrino mass models, while keeping the complete set of the remaining model parameters free for general scans, as necessary in order to provide reliable predictions for observables outside the neutrino sector. We provide a proof of this master parametrization and show how to apply it for several well-known neutrino mass models from the literature. We also discuss a list of special cases, in which the Yukawa couplings have to fulfill some particular additional conditions.Comment: 35 pages, 8 figures, 8 tables; v2: minor changes, matches published versio

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 here a reformulation of the elliptic sector of CFC that has the fundamental property of overcoming the local uniqueness problems. The correct behavior of our new formulation is confirmed by means of a battery of numerical simulations. Finally, we extend these ideas to FCF, complementing the mathematical analysis carried out in previous studies.Comment: 17 pages, 5 figures. Minor changes to be consistent with published versio

On the convexity of Relativistic Ideal Magnetohydrodynamics

We analyze the influence of the magnetic field in the convexity properties of the relativistic magnetohydrodynamics system of equations. To this purpose we use the approach of Lax, based on the analysis of the linearly degenerate/genuinely non-linear nature of the characteristic fields. Degenerate and non-degenerate states are discussed separately and the non-relativistic, unmagnetized limits are properly recovered. The characteristic fields corresponding to the material and Alfv\'en waves are linearly degenerate and, then, not affected by the convexity issue. The analysis of the characteristic fields associated with the magnetosonic waves reveals, however, a dependence of the convexity condition on the magnetic field. The result is expressed in the form of a generalized fundamental derivative written as the sum of two terms. The first one is the generalized fundamental derivative in the case of purely hydrodynamical (relativistic) flow. The second one contains the effects of the magnetic field. The analysis of this term shows that it is always positive leading to the remarkable result that the presence of a magnetic field in the fluid reduces the domain of thermodynamical states for which the EOS is non-convex.Comment: 14 pages. Submitted to Classical and Quantum Gravit

On the local existence of maximal slicings in spherically symmetric spacetimes

In this talk we show that any spherically symmetric spacetime admits locally a maximal spacelike slicing. The above condition is reduced to solve a decoupled system of first order quasi-linear partial differential equations. The solution may be accomplished analytical or numerically. We provide a general procedure to construct such maximal slicings.Comment: 4 pages. Accepted for publication in Journal of Physics: Conference Series, Proceedings of the Spanish Relativity Meeting ERE200

Scheduled Relaxation Jacobi method: Improvements and applications

Elliptic partial differential equations (ePDEs) appear in a wide variety of areas of mathematics, physics and engineering. Typically, ePDEs must be solved numerically, which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution. The Scheduled Relaxation Jacobi (SRJ) is a promising class of methods, atypical for combining simplicity and efficiency, that has been recently introduced for solving linear Poisson-like ePDEs. The SRJ methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances. The efficiency in the reduction of the residual increases with the number of levels employed in the algorithm. Applying the original methodology to compute the algorithm parameters with more than 5 levels notably hinders obtaining optimal SRJ schemes, as the mixed (non- linear) algebraic-differential system of equations from which they result becomes notably stiff. Here we present a new methodology for obtaining the parameters of SRJ schemes that overcomes the limitations of the original algorithm and provide parameters for SRJ schemes with up to 15 levels and resolutions of up to 2^15 points per dimension, allowing for acceleration factors larger than several hundreds with respect to the Jacobi method for typical resolutions and, in some high resolution cases, close to 1000. Most of the success in finding SRJ optimal schemes with more than 10 levels is based on an analytic reduction of the complexity of the previously mentioned system of equations. Furthermore, we extend the original algorithm to apply it to certain systems of non-linear ePDEs