Evolution formalisms of Einstein equations: Numerical and Geometrical Issues

Abstract

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.