New Class of Exact Solutions to Einstein-Maxwell-dilaton Theory on Four-dimensional Bianchi Type IX Geometry A Thesis Submitted to the College of Graduate and Postdoctoral Studies in Partial Fulfillment of the Requirements for the degree of Master of Science in the Department of Physics & Engineering Physics University of Saskatchewan Saskatoon By Bardia Hosseinzadeh Fahim c Bardia Hosseinzadeh Fahim, August 2020. All rights reserved. Permission to Use In presenting this thesis in partial fulfilment of the requirements for a Postgraduate degree from the University of Saskatchewan, I agree that the Libraries of this University may make it freely available for inspection. I further agree that permission for copying of this thesis in any manner, in whole or in part, for scholarly purposes may be granted by the professor or professors who supervised my thesis work or, in their absence, by the Head of the Department or the Dean of the College in which my thesis work was done. It is understood that any copying or publication or use of this thesis or parts thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of Saskatchewan in any scholarly use which may be made of any material in my thesis. Requests for permission to copy or to make other use of material in this thesis in whole or part should be addressed to: Head of the Department of Physics & Engineering Physics 163 Physics Building 116 Science Place University of Saskatchewan Saskatoon, Saskatchewan Canada S7N 5E2 Or Dean College of Graduate and Postdoctoral Studies University of Saskatchewan 116 Thorvaldson Building, 110 Science Place Saskatoon, Saskatchewan S7N 5C9 Canada i Abstract We construct new classes of cosmological solution to the five dimensional Einstein-Maxwell-dilaton theory, that are non-stationary and almost conformally regular everywhere. The base geometry for the solutions is the four-dimensional Bianchi type IX geometry. In the theory, the dilaton field is coupled to both the electromagnetic field and the cosmological constant term, with two different coupling constants. We consider all possible solutions with different values of the coupling constants, where the cosmological constant takes any positive, negative or zero values. In the ansatzes for the metric, dilaton and electromagnetic fields, we consider dependence on time and two spatial directions. We also consider a special case of the Bianchi type IX geometry, in which the geometry reduces to that of Eguchi-Hanson type II geometry and find a more general solution to the theory. ii Acknowledgements Foremost, I would like to express my sincere gratitude to my supervisor, Dr. Masoud Ghezelbash, who expertly guided me through my education and research. His support, patience, motivation and enthusiasm helped me in all the time of my research. My appreciation also extends to my committee members, Dr. Rainer Dick, Dr. Rob Pywell, Dr. Artur Sowa, Dr. Mark Boland and Dr. Alexander Koustov for their encouragement and insightful comments. Nobody has been more important to me in the pursuit of this project than the members of my family. I would like to thank my parents and my brother, whose love and guidance are with me in whatever I pursue. I would also like to thank my friend and colleague Ashkan Alibabaei for the discussions and support through my research. iii Contents Permission to Use i Abstract ii Acknowledgements iii Contents iv List of Figures vi List of Abbreviations viii 1 Introduction 1 2 General Relativity 5 2.1 Newtonian Gravity . 5 2.2 From Special to General Relativity . 7 2.3 Four-vectors and the Covariant Derivative . 10 2.4 Geodesic, Parallel Transportation and the Curvature Tensor . 11 2.5 Einstein's Gravitational Field Equations . 14 2.6 Schwarzschild Geometry . 17 2.7 Friedmann-Robertson-Walker Geometry . 21 3 Bianchi Geometry 23 3.1 Classification of the Bianchi Spaces . 23 3.2 Bianchi type IX Geometry . 28 3.3 Atiyah-Hitchin Metric . 30 3.4 Triaxial Bianchi Type IX Geometry . 35 4 Einstein-Maxwell-dilaton Theory 39 4.1 Hamilton's Principle and Classical Field Theory . 39 4.2 Maxwell's Action . 41 4.3 Einstein-Hilbert Action . 42 4.4 Einstein-Maxwell-dilaton Theory . 44 4.5 Exact Solutions to the Einstein-Maxwell-dilaton Theory Based on the Four-Dimensional Eguchi- Hanson Type II Geometry . 45 4.6 Exact Solutions to the Einstein-Maxwell-dilaton Theory Based on the Four-Dimensional Taub- NUT Geometry . 48 5 New Classes of Exact Solutions to the Einstein-Maxwell-dilaton Theory on a Four- Dimensional Bianchi Type IX Geometry 51 5.1 The Exact Solutions to the Einstein-Maxwell-dilaton Theory, Based on the Bianchi Type IX Geometry, with Two Different Coupling Constants a and b .................... 51 5.2 Exact Solutions to the Einstein-Maxwell-dilaton Theory Based on the Bianchi Type IX Ge- ometry, Where the Non-zero Coupling Constants are Equal a = b 6= 0 . 59 5.3 Exact Solutions Where the Two Coupling Constant are Equal to Zero . 63 5.4 More General Solutions to the Einstein-Maxwell-dilaton Theory Based on the Eguchi-Hanson Type II Geometry for Non-equal Coupling Constants . 67 5.5 More General Solutions where the Non-zero Coupling Constants are Equal . 68 5.6 More General Solutions where the Coupling Constants are Zero . 69 iv 5.7 Uplifting to Higher Dimensions . 70 5.8 Janis-Newman Method . 72 6 Conclusions 77 7 Appendix 79 References 80 v List of Figures 2.1 First image of a super massive black hole which was taken by Event Horizon Telescope. 21 3.1 The behaviour of the Jacobi elliptic function sn with respect to φ. 33 3.2 The behaviour of the Jacobi elliptic function cn with respect to φ. 34 3.3 The behaviour of the Jacobi elliptic function dn with respect to φ, where we set the constant k = 0:7. ................................................ 34 3.4 The behaviour of the Jacobi elliptic functions sn, cn and dn (which are shown by blue, green and red in the graph, respectively) with respect to φ, where we set the constant k = 0:7. 35 3.5 The behaviour of the function J(r) with respect to the coordinate r, where we set the constants k = 0:5 and c =1. .......................................... 37 4.1 The behaviour of the function f(r) with respect to the r coordinate, where we set the constant c =1. ................................................. 46 4.2 The metric function H(r; θ) as a function of the coordinates r and θ, where the constants are set g+ = 2, g− = 3, a = 1 and c =1. ................................ 47 5.1 The metric function H(r; θ) as a function of the coordinates r and θ, where we set j+ = 0:5, j− = 15, c = 1 and a =1. ...................................... 55 5.2 The behaviour of the metric function R(t) with respect to the time coordinate t, where we set the constants η = 2, # = 2 and a =1. ............................... 56 5.3 The c-function for the five-dimensional spacetime, where the coupling constants are not equal. 57 5.4 The dilaton field φ(t; r; θ) as a function of the coordinates r and θ for two different time slices, t = 1 and t = 3, which are the lower and the upper surface, respectively. We set the constants j+ = 0:5, j− = 15, a = 1, η = 1 and # =2.............................. 58 5.5 The r-component of the electric field as a function of the coordinates r and θ for t = 1. We set the constants as j+ = 0:5, j− = 15, a = 1, η = 1, # = 2, c = 1. 58 5.6 The θ-component of the electric field as a function of the coordinates r and θ for t = 1. We set the constants as j+ = 0:5, j− = 15, a = 1, η = 1, # = 2, c = 1. 59 5.7 The metric function H(t; r; θ) as a function of the coordinates r and θ, where we set = 1, µ = 2, g+ = 0:5, g− = 15 and a =1. ................................ 60 5.8 The c-function for the five-dimensional spacetime, where the coupling constants are equal. 61 5.9 The dilaton field φ(t; r; θ) for two different time slices t = 1 and t = 2 (upper and lower hypersurfaces respectively), where g+ = 0:5, g− = 15, a = 1, = 1 and µ = 2. 62 5.10 The r-component of the electric field as a function of the coordinates r and θ, where t = 1. We set the constants as g+ = 0:5, g− = 15, a = 1, = 1, µ = 2, c = 1. 62 5.11 The θ-component of the electric field as a function of the coordinates r and θ, where t = 1. We set the constants as g+ = 0:5, g− = 15, a = 1, = 1, µ = 2, c = 1. 63 5.12 The metric function H(t; r; θ) as a function of the coordinates r and θ for two different time slices t = 1 and t = 2 (upper and lower surfaces, respectively), where γ = +1 and the constants are set f+ = 0:5, f− = 15 and Λ = 2. 64 5.13 The metric function H(t; r; θ) as a function of the coordinates r and θ for two different time slices t = 1 and t = 2 (lower and upper surfaces, respectively), where γ = −1 and the constants are set f+ = 0:5, f− = 15 and Λ = 2. 65 5.14 The c-functions with γ = +1 (left) and γ = −1 (right). 65 5.15 The behaviour of the r-component of the electric field with respect to r and θ coordinates for t = 1.