Universe phenomenology as understood from gravitational theories with non-vanishing torsion: cosmology and black holes

Beckering Vinckers, Ulrich Karoo

Universe phenomenology as understood from gravitational theories with non-vanishing torsion: cosmology and black holes

Master Thesis

2021

Abstract

In this thesis, we study gravitational theories for which the natural choice of an affine connection is metric compatible while not being symmetric. More specifically, we study gravitational theories constructed on the Riemann-Cartan and Weitzenböck space-times. Firstly, we outline the mathematical notions needed to construct a definition of space-time. Following this, we introduce the space-time definitions to be made use of throughout this thesis. We then discuss the notions of extremal and auto-parallel curves on the Riemann-Cartan space-time. It is noted that test particles follow extremal curves which are auto-parallel curves of the LeviCivita connection. Therefore, one must turn to the standard, torsion-free Raychaudhuri equation when studying the focusing conditions that arise in theories constructed on the Riemann-Cartan or Weitzenböck space-times. Once we have introduced the definitions of the relevant space-times, we move on to review some of the gravitational theories that involve non-vanishing torsion. We first review the Einstein-Cartan theory and two of its modifications. We then review the so-called f(T) theories of gravity before discussing the focusing conditions that arise in this context. By making use of the f(T) field equations together with the torsion-free Raychaudhuri equation, we derive for the first time the f(T) focusing conditions for a one-parameter dependent congruence of timelike auto-parallel curves of the LeviCivita connection. We then study these focusing conditions for three bi-parametric cosmological models. Finally, we turn our attention back to the Einstein-Cartan theory and derive the Arnowitt-DeserMisner formulation of this theory. By making use of this formulation, we derive for the first time the Generalised-Baumgarte-Shapiro-Shibata-Nakamura formulation of the Einstein-Cartan theory. We then consider the case of a vacuum in spherical symmetry and construct a 1-dimensional code to evolve the system numerically. We leave the inclusion of torsion into this code as the subject for future work.

Keywords

Mathematics and Applied Mathematics

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