3+1 formalism in General Relativity

Abstract

The study of new solutions in General Relativity motivated investigations of Cauchy problems for alternative gravitational regimes, which led to the need to elaborate techniques to split the spacetime into three plus one dimensions. The 3+1 formalism arises as a mathematical tool for decomposing the components of the metric and the curvature. The central mechanism found in the literature is known as the ADM formalism, developed initially as an attempt to construct a theory of canonical quantum gravity, which later its formulation was applied to evolve the Einstein equation numerically. As a subfield of Gravitation, the Numerical Relativity investigates phenomena in strong field systems and other scenarios which are not possible to solve analytically. In this dissertation, we will present an introduction to the 3+1 formalism, as well as gauge and initial conditions to prepare some gravitational systems. To consolidate this theoretical approach, we will show techniques of Numerical Relativity to evolve the spacetime from of given initial configurations.