Simulações numéricas no formalismo BSSN em simetria esférica: colapso de um campo escalar e condições de calibre

Abstract

In this work, we provide a brief introduction to the 3+1 formalism of General Relativity (GR), focusing on the decomposition of Einstein’s equations and gauge conditions. Next, we introduce the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formalism for solving a problem in GR under strong-field conditions. This problem involves a scalar field without self-gravitating mass. To address it, we discuss the theoretical foundations and advantages of the BSSN formalism, including its application in spherical coordinates. We derive the basic equations for the scalar field and concentrate on analyzing its behavior. We also investigate gravitational collapse and the initial conditions leading to the dispersion of the scalar field at infinity. Additionally, we present spectral methods for numerically solving the equations, focusing on the Galerkin-Collocation method. We conduct numerical experiments to validate the codes built based on the BSSN formalism in spherical symmetry, including convergence tests for different gauge conditions. At the end of the work, we present the main conclusions of the study, highlighting satisfactory results and future prospects for simulating other more complex astrophysical systems.