6D Anti-de Sitter Space Solutions to Einstein's Field Equation with a Scalar Field

Abstract

Abstract : The purpose of this Trident Scholar project was to study a scalar field in six dimensional Anti-de Sitter space by extending the Randall-Sundrum model. This model included a single scalar field and two compactible extra dimensions. One of these extra dimensions was defined by periodic boundary conditions. The other extra dimension was compactible and stabilized by a scalar field in the space. The shape of the six-dimensional space was defined by its metric, a mathematical structure that described how the length scale changes as a function of position in space and time. The metric was required to satisfy a differential equation known as the Einstein Field equation. By starting with some known facts about the structure of the metric, the Einstein equation was decomposed into a system of differential equations that were solved to find the final solution for the metric. In addition to the requirement of the Einstein Field equation, once the scalar field was added to the system, it needed to satisfy its own differential equation, the Klein-Gordon equation. Perturbation methods were used to simultaneously solve the Einstein Field equation and the Klein-Gordon equation to find the back reaction of the energy due to the scalar field on the six-dimensional Anti-de Sitter space metric. This process gave a new metric for the space that included the effect of the scalar field. The physical characteristics of the newly calculated space were explored to ensure that it satisfied the hierarchy problem as well as to determine how the laws of physics were affected by the warping of the space.