Uma introdução ao estudo dos tensores: aplicações na teoria da relatividade

Abstract

Tensors are mathematical objects used in physics that generalize scalars, vectors, and matrices. They allow physical laws to remain invariant when analyzed in any coordinate system. This tool is applied in the study of three important tensors in the Theory of Relativity: Incoherent Matter (Dust), Perfect Fluid, and Electromagnetic Field. In this context, the present work aimed to develop tensor algebra and its mentioned applications. In the first part, some equations that are often presented in a compact form in many reference materials were fully developed to enhance understanding, and the concepts involved in the use of these expressions were addressed in simpler language. In the second part, each tensor was explored starting from the quantities that define them, as well as the steps required for their construction, also using clearer language. Based on what was done, after developing tensor algebra, the tensor for Incoherent Matter was structured using a vector field of four-velocity and a scalar field of proper density. Next, the tensor for the Perfect Fluid was constructed by adding a scalar pressure field to the four-velocity and proper density fields. Finally, Maxwell’s equations were converted into the tensor formalism. With these procedures, it was possible to facilitate the understanding of certain concepts and equations involved in tensor analysis and its applications.