[en] The main goal of this work is to construct the space-time, a topological space used for physical modelling. In 1905, Albert Einstein published a revolutionary article that changed the perception of space and time. He proposed that there is a maximum speed that anything can travel, the speed of light, and its value is constant for any observer. This caused interest to many scientists to study a new type of model based on Einstein’s ideas. The phenomena happening on a velocity close to the speed of light contradict the theory of Newton’s mechanics, which uses Euclidean spaces for modelling. The first space proposed to model these effects is called Minkowski space-time, which is R 4 with a non Euclidean metric (this will be explained in detail in this work). The theory that studies physical phenomena based on Einstein’s ideas, and modelled in the Minkowski space-time, is called the Theory of special relativity. However, this theory was not enough. The Minkowski space-time can only describe physical phenomena when there is no gravity involved. It took 10 years for Einstein to create the theory of general relativity, which is compatible with Newton’s theory of gravitation. The intuitive idea is that, in presence of masses, the space-time is no longer "flat" (R 4 has null curvature) and it becomes a "curved" topological space. Hence, in order to construct the space-time, we shall introduce the ba- sics of differential geometry, which studies the concepts of differential calculus applied to curved spaces. Such spaces are called differentiable manifolds. Concepts as the curvature and the metric are defined using tensor algebra, hence, we will also study the basics of tensor calculus and algebra.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Carlos Currás Bosch