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Numerical Methods for Rotating Compact Objects in Modified Gravity

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Numerical Methods for Rotating Compact Objects in Modified Gravity NUMERICAL METHODS FOR ROTATING COMPACT OBJECTS IN MODIFIED GRAVITY THEORIES by Andrew Patrick Kyung Sullivan A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics MONTANA STATE UNIVERSITY Bozeman, Montana April 2020 c COPYRIGHT by Andrew Patrick Kyung Sullivan 2020 All Rights Reserved ii DEDICATION I dedicate this to my mom, whose trips to the local library to checkout science books as a child were an enormous inspiration. iii ACKNOWLEDGEMENTS I would like to thank my advisor, Dr. Nicol´asYunes whose support and mentorship has been invaluable. Thank you for always pushing me to be the best researcher I can be. I would like to thank the members of my committee, especially my co-chair Dr. Neil Cornish for the constructive comments on my work. I would like to thank Margaret Jarrett of the MSU Physics department for her help and support. I would like to thank my friends who have supported me along the way especially Dimitry Ayzenberg whose conversations and company helped during the early years of my graduate career. Finally I would like to thank my family; my parents, my brother, and sister whose continual support has been immeasurable. iv TABLE OF CONTENTS 1. INTRODUCTION ........................................................................................1 2. SLOWLY-ROTATING NEUTRON STARS IN MASSIVE BIGRAV- ITY ...........................................................................................................12 Contribution of Authors and Co-Authors ..................................................... 12 Manuscript Information Page ...................................................................... 13 Abstract .................................................................................................... 14 Introduction............................................................................................... 14 Massive Bigravity ....................................................................................... 22 Slowly-rotating Neutron stars in Massive Bigravity within the Hartle-Thorne Approximation.......................................................... 27 Metric Tensors for Slowly-Rotating Spacetimes ..................................... 27 Neutron Star Stress-Energy Tensor....................................................... 28 Interaction Tensor ............................................................................... 29 Modified Field Equations..................................................................... 32 Recasting of Modified Field Equations.................................................. 35 Numerical Construction of Neutron Stars in Massive Bigravity...................... 36 Zeroth-Order in Rotation..................................................................... 37 Local analysis at r = 0 and r = ................................................ 37 Numerical Solutions .....................................................................1 40 First-Order in Rotation ....................................................................... 45 Local Analysis at r = 0 and r = ............................................... 45 Numerical Solution.......................................................................1 47 Conclusions and Future Directions............................................................... 49 Acknowledgments ....................................................................................... 52 3. NUMERICAL BLACK HOLE SOLUTIONS IN MODIFIED GRAV- ITY THEORIES: SPHERICAL SYMMETRY CASE...............................................................53 Contribution of Authors and Co-Authors ..................................................... 53 Manuscript Information Page ...................................................................... 54 Abstract .................................................................................................... 55 Introduction............................................................................................... 55 Numerical Methods..................................................................................... 61 Validation .................................................................................................. 62 Toy Problem ....................................................................................... 63 Schwarzschild Black Hole..................................................................... 68 Spherically Symmetric Black Holes in Scalar-Gauss-Bonnet Gravity............... 73 v TABLE OF CONTENTS { CONTINUED Action and Field equations .................................................................. 73 Linear Scalar-Gauss-Bonnet Gravity..................................................... 75 Einstein-dilaton-Gauss-Bonnet Gravity................................................. 82 Properties of Solution ................................................................................. 85 Fitting Function.................................................................................. 85 ISCO.................................................................................................. 87 Light Ring .......................................................................................... 90 Naked Singularity................................................................................ 91 Conclusions................................................................................................ 93 Acknowledgments ....................................................................................... 95 4. NUMERICAL BLACK HOLE SOLUTIONS IN MODIFIED GRAV- ITY THEORIES: AXIAL SYMMETRY CASE........................................................................96 Contribution of Authors and Co-Authors ..................................................... 96 Manuscript Information Page ...................................................................... 97 Abstract .................................................................................................... 98 Introduction............................................................................................... 98 Numerical Methods................................................................................... 105 Validation ................................................................................................ 108 Axially Symmetric Black Holes in Scalar-Gauss-Bonnet Gravity .................. 115 Action and Field equations ................................................................ 115 Linear Scalar-Gauss-Bonnet Gravity................................................... 118 Einstein-dilaton-Gauss-Bonnet Gravity............................................... 126 Properties of Solution ............................................................................... 126 Fitting Function................................................................................ 128 Marginal Stable Circular Orbits ......................................................... 134 Light Ring ........................................................................................ 135 Innermost Stable Circular Orbit......................................................... 136 Conclusions.............................................................................................. 137 Acknowledgments ..................................................................................... 138 5. CONCLUSION......................................................................................... 139 REFERENCES CITED.................................................................................. 141 APPENDICES .............................................................................................. 152 APPENDIX A : Sample Functions ............................................................ 153 vi TABLE OF CONTENTS { CONTINUED APPENDIX B : Numerical Methods.......................................................... 155 Iterative Relaxation Method in the Continuum Limit .......................... 156 Discrete Representation through Newton's Polynomials ....................... 158 Derivatives of Newton Polynomials and Discretization Error ................ 162 Minimization of the Discretization Error............................................. 164 Minimization of the Residual Vector................................................... 167 Linear System Solvers........................................................................ 168 APPENDIX C : Sample Fitting Coefficients for Spherically Symmetric Solutions ........................................................................... 169 APPENDIX D : Sample Fitting Coefficients for Axially Symmetric Solutions ........................................................................... 172 vii LIST OF TABLES Table Page C.1 Fitting Coefficients for ai;j ............................................................. 170 C.2 Fitting Coefficients for bi;j ............................................................. 171 C.3 Fitting Coefficients for ci;j ............................................................. 171 D.1 Fitting Coefficients for ai;j ............................................................. 173 viii LIST OF FIGURES Figure Page 2.1 The mass-radius relation (left) and the moment of inertia-mass relation (right) using three realistic equations of state with varying m and all other coupling parameters fixed. The mass of the star M is obtained from the mass function of the physical met- ric evaluated at
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