Aspectes geomètrics de la teoria de la relativitat general

Abstract

The purpose of my final degree project is to study the necessary mathematical tools to understand the theory of relativity postulated by Albert Einstein. It is divided in two blocks: special relativity and general relativity Special relativity theory was introduced in 1905 to replace the Newtonian theory of relativity. It consists of two principles which can be summarized as follows: - Every physical law can be formulated identically in every non-accelerating reference frame. - The speed of light remains constant in every non-accelerating reference frame. With these two postulates Einstein realized that time and space are not independent. Hermann Minkowski introduced the concept of Minkowski spacetime, an affine space with a nondegenerate metric. Points in spacetime represent events that occur at a certain instant of time and position. As we will see, the necessary mathematical tools to understand this part of the project are linear algebra and affine geometry. Special relativity was valid in reference frames without the action of gravity. Einstein realized that in infinitesimal regions we can neglect the effect of gravity and, therefore, we can apply special relativity. In 1915, he was able to introduce the effect of gravitational fields in what is called general theory of relativity, or simply general relativity. It states that gravity is a manifestation of the curvature of spacetime, which he postulates that must be a Lorentzian manifold. And reciprocally, the effect of gravitational fields in spacetime is the prodution of curvature. Einstein’s equation allows the study of the relationship between mass-energy and curvature. Solutions of this equation, like Schwarzschild and Robertson-Walker spacetimes, permitted the development of cosmology, the science that tries to describe the universe as a whole. To understand the second part of this project is necessary a knowledge of semi-Riemannian geometry. Concepts such as differentiable manifolds, metric tensors, geodesics and curvature are essential to develop this theory.