A equação de Dirac em referenciais acelerados

Abstract

In this dissertation, we discuss some important concepts of the geometry of curves in Minkowski space, and from these concepts we formulate the Serret-Frenet equations in Minkowski’s four-dimensional space-time. We state some important theorems, among them, the fundamental theorem of the local theory of curves in 𝑀. We also show a general case, where the Serret-Frenet vectors are written in terms of the observer’s eigencoordinates. We solved the Serret-Frenet equations for the particular case of plane curves, and from that case, it was possible to clearly analyze the case in which the observer’s acceleration is inversely proportional to the proper time. From the Serret-Frenet formalism, we present and analyze the Dirac equation in a frame adapted to any accelerated observer. From this equation and an idealized model in which a laboratory on Earth describes a circular and uniform motion in the absence of the gravitational field, we were able to examine the effects of the contribution of proper time for two specific cases, namely, when the observer is on the surface. , and the other when the observer is at the center of the Earth. Based on Dirac’s equations, we describe the accelerated motion of an observer, both for the general case, where there is no restriction on the rotation of the triad, and for the case in which the triad rotates obeying the Serret-Frenet equations. Using data such as Earth’s radius and angular frequency, it was possible to obtain the solution of the approximate version of Dirac’s equation for a rotating observer.