Remarks on the Dirac equation in a class of black holes with a cloud of strings

Abstract

The main goal of this dissertation was to obtain the metric of the spacetime generated by a rotating, charged gravitational body surrounded by a cloud of strings and analyze the Dirac equation and its solutions in the gravitational field generated by this body, which corresponds to a charged rotating black hole with a cloud of strings, i.e., the Kerr-Newman black hole with the addition of a cloud of strings as a source for the Einstein field equations. In order to do that, we firstly reviewed, in the second chapter, some fundamental physical concepts that were important for the further investigations, the physical concept of a black hole and the physical concept of a cloud of strings. We also obtained, for the sake of completeness, the metric for the Schwarzschild black hole surrounded by a cloud of strings (Letelier’s spacetime) as well as the Reissner-Nordstrom’s black hole surrounded by a cloud of strings. In order to illustrate the method developed by Janis and Newman [1], we obtained the Kerr metric and then used this method to find a new solution of the Einstein field equations, generated by a charged rotating gravitational body surrounded by a cloud of strings, which we called Kerr-Newman black hole with a cloud of strings. We aimed to obtain such background for our application of Dirac equation in the fourth chapter. In the third chapter, we reviewed the special relativistic wave equation for spin-1/2 particles, called Dirac equation, since its obtaining, up to its form invariance under Lorentz transformations. In this same third chapter, we discussed the procedure that generalizes this wave equation for curved spacetimes, obtaining the so-called general relativistic Dirac equation, which is covariant under both the group of general coordinate transformations for the entire manifold, and the group of local Lorentz transformations defined at each and every point of the manifold individually. Having obtained this Dirac equation for curved spacetimes, and getting to know how to formulate it once the spacetime metric is given, we could apply this equation, at the fourth chapter, for our desired spacetime configuration, which is the Kerr-Newman spacetime with a cloud of strings as obtained in chapter two. As the result of this application, we finally managed to separate the solution into the angular and radial parts and compare with recent studies performed by Kraniotis [2] for Kerr-Newman black hole and concluded that the equations are formally similar, and correspond to a Generalized Heun Equation. Therefore, the solutions for the Dirac equation in the background spacetime under consideration, are also given in terms of the Generalized Heun Functions as well. We analyzed these solutions and all particular cases, namely, uncharged, non rotating, and uncharged and non rotating black holes with cloud of strings, and pointed out the signature of the strings and consequently, their physical role. These results call attention to the application power of the Generalized Heun Equation, what may permit to obtain other results, such as analytical solutions for the Dirac equation in others spacetime configurations. We also analyzed the particular cases corresponding to Kerr, Reissner-Nordström and Schwarzschild black holes surrounded by cloud of strings, whose results are also given in terms of Generalized Heun Functions. The analytical solutions at the vicinity of some specific reference points, such as the Cauchy horizon, event horizon, and at the infinity were obtained and analyzed and the signature of the presence of the cloud of strings was pointed out.