An elastic constitutive model of spacetime and its applications By Tichomir G. Tenev A Dissertation Submitted to the Faculty of Mississippi State University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Computational Engineering in the Department of Computational Engineering Mississippi State, Mississippi December 2018 Copyright by Tichomir G. Tenev 2018 An elastic constitutive model of spacetime and its applications By Tichomir G. Tenev Approved: Mark F. Horstemeyer (Major Professor) Anzhong Wang (Committee Member) Tomasz Haupt (Committee Member) Shantia Yarahmadian (Committee Member) Jason M. Keith Dean Bagley College of Engineering Name: Tichomir G. Tenev Date of Degree: December 14, 2018 Institution: Mississippi State University Major Field: Computational Engineering Major Professor: Mark F. Horstemeyer Title of Study: An elastic constitutive model of spacetime and its applications Pages of Study: 208 Candidate for Degree of Doctor of Philosophy We introduce an elastic constitutive model of gravity that enables the interpretation of cosmological observations in terms of established ideas from Solid Mechanics and multi- scale modeling. The behavior of physical space is identified with that of a material-like medium called “cosmic fabric,” which exhibits constitutive behavior. This cosmic fabric is a solid hyperplate that is broad in the three ordinary spatial dimensions and thin in a fourth hyperspatial dimension. Matter in space is treated as fabric inclusions that prescribe in-plane (three-dimensional) strain causing the transverse bending of the fabric into the fourth hyperspatial dimension. The linearized Einstein-Hilbert action, which governs the dynamics of physical space, is derived from postulating Hooke’s Law for the fabric, and the Schwarzschild metric is recovered from investigating matter-fabric interactions. At the continuum length scale, the Principle of Relativity is shown to apply for both moving and stationary observers alike, so that the fabric’s rest reference frame remains observationally indistinguishable at such a length scale. Within the Cosmic Fabric paradigm, the structural properties of space at different hierarchical length scales can be investigated using theoreti- cal notions and computational tools from solid mechanics to address outstanding problems in cosmology and fundamental physics. For example, we propose and offer theoretical sup- port for the “Inherent Structure Hypothesis”, which states that the gravitational anomalies currently attributed to dark matter may in fact be manifestations of the inherent (undefor- med) curvature of space. In addition, we develop a numerical framework wherein one can perform numerical “experiments” to investigate the implications of said hypothesis. Key words: modified gravity, constitutive model, spacetime, cosmic fabric, hierarchical length scales, dark matter, particle-mesh method DEDICATION To the Lord Jesus Christ through Whom all things were made and without Whom nothing was made that was made. In Him we live and move and have our being. ii ACKNOWLEDGEMENTS Before all, I thank my Heavenly Father Who is the source of all good and perfect gifts, and Who provided the time and finances, the ideas, environment, and people to allow me to accomplish this dissertation work. May it be pleasing to Him. Amen. Mark Horstemeyer encouraged me to move to Mississippi and subsequently became my doctoral advisor and collaborator. I thank him for this and for teaching me mechanical engineering and what it means to be a scientist. I am also thankful to the members of my graduate committee for their continual super- vision and encouragement. Anzhong Wang introduced me to the General Relativity com- munity and guided me in that area. Shantia Yarahmadian supervised my study in applied math, and shared his insights with me on the nature of time. Byron Williams introduced me to Mississippi State’s graduate program and became a committee member in addition to becoming a good friend. Tomasz Haupt stepped in during the final months of my program to take Byron’s place on the committee whose job had taken him elsewhere. I would like to acknowledge the wonderful group of colleagues with whom I fellows- hipped at the University’s Center for Advanced Vehicular Systems (CAVS): Denver Seeley, Robert Allen, Heechen Cho, Sungkwan Moon, Nayeon Lee, and others. John Baumgardner, Russ Humphreys, and Andrew McIntosh offered unfiltered feed- back and from the beginning were a source of inspiration and encouragement. iii Also, I would like to acknowledge Tamas Morvai for the helpful discussions on the nature of time, and Youseff Hammi for his gracious help with Abaqus. Finally, I would like to acknowledge my wife, Rong Tenev, without whose support, deep insight, and encouragement, this work would not have been possible. iv TABLE OF CONTENTS DEDICATION . ii ACKNOWLEDGEMENTS . iii LIST OF TABLES . ix LIST OF FIGURES . .x LIST OF SYMBOLS, ABBREVIATIONS, AND NOMENCLATURE . xii CHAPTER I. INTRODUCTION . .1 1.1 Early Material Models of Cosmic Space . .3 1.2 Relativity and the Material View of Space . .5 1.3 Recent Research Relating Mechanics and Relativity . .8 1.4 The Hierarchical Length Scale Structure of Space . 10 1.5 Computational Methods . 14 1.6 Dark Matter . 16 1.7 Assumptions and Limitations . 19 1.8 Contributions and Organization of the Dissertation . 20 II. BACKGROUND . 22 2.1 Differential Geometry . 22 2.1.1 Contravariant and Covariant Representation of Points, Re- ciprocal Basis . 22 2.1.2 Metric and Metric Tensor . 24 2.1.2.1 Relationship to the Dot Product of Vectors . 26 2.1.2.2 Raising and Lowering of Indexes . 27 2.1.2.3 Flat Space with Cartesian Coordinates . 28 2.1.2.4 Non-Cartesian Coordinates . 29 2.1.3 Affine Connection and Covariant Derivative . 30 2.1.4 Intrinsic Curvature . 32 v 2.2 Continuum Mechanics . 33 2.2.1 Deformation, Material and Reference Coordinates . 33 2.2.2 Small Strain . 35 2.2.3 Stress and Elastic Moduli . 36 2.3 General Relativity . 37 III. COSMIC FABRIC MODEL OF GRAVITY . 41 3.1 Formulation of the Cosmic Fabric Model of Gravity . 44 3.2 Coordinate Assignment and Reference Space . 44 3.2.1 Postulates . 46 3.2.1.1 Elastic Thin Hyperplate . 46 3.2.1.2 Inclusions . 47 3.2.1.3 Lapse Rate . 48 3.2.2 Linearized Spacetime Metric . 50 3.2.3 Bending Energy Density . 51 3.2.4 Membrane Energy Density . 55 3.2.5 Lagrangian Density . 57 3.3 Discussion . 58 3.3.1 Fabric Strain and Gravitational Potential . 59 3.3.2 Poisson’s Ratio and the Substructure of Space . 59 3.3.3 Fabric Vibrations and Gravitational Waves . 60 3.3.4 Elastic Modulus and Density of Free Space . 60 3.3.5 Gravitational Waves . 61 3.4 Summary and Conclusion . 64 IV. RECOVERING RELATIVITY FROM THE COSMIC FABRIC MODEL 67 4.1 Invariance of the Speed of Signals . 68 4.1.1 Stationary Observers at Different Locations . 69 4.1.2 Moving Observer . 71 4.2 Lorentz Transformations . 74 4.3 Discussion . 77 4.4 Conclusion . 80 V. SPACETIME METRIC OF A SPHERICALLY SYMMETRIC INCLUSION 81 5.1 Geometry of Spherically Symmetric Deformation . 83 5.1.1 Material (Spatial) Metric Tensor . 85 5.1.2 Intrinsic Curvature . 87 5.2 Spacetime Metric due to Spherically Symmetric Inclusion . 89 5.2.1 Formulation of the Governing Equations . 90 5.2.2 Transverse Displacement . 94 vi 5.2.3 Volumetric Strain . 95 5.2.4 Radial Stretch . 96 5.2.5 Metric . 97 5.3 Membrane Energy and the Discrete Nature of Matter . 100 5.3.1 Total Membrane Energy of a Homogeneous Spherically Sym- metric Body . 101 5.3.2 Discrete Substructure . 108 5.4 Discussion . 110 5.4.1 Comparison with the Schwarzschild Metric . 111 5.4.2 Flamm’s Paraboloid . 113 5.5 Conclusion . 115 VI. DARK MATTER EFFECT . 118 6.1 Introduction . 118 6.2 Spherically symmetric inherent curvature . 123 6.2.1 Derivatives of Radial Functions . 125 6.2.2 Derivatives of the Reference Coordinates . 125 6.2.3 Spatial Metric Tensor . 126 6.2.4 Christoffel Symbols . 127 6.3 Gravity in the context of inherent spherically symmetric curvature 128 6.4 Discussion . 133 6.4.1 Conditions for observational equivalence and falsifiability . 133 6.4.2 Comparison with Modified Newtonian Dynamics (MOND) 136 6.4.3 Implication to cosmological models . 142 6.5 Summary and Conclusion . 143 VII. NUMERICAL FRAMEWORK FOR WEAK GRAVITY DYNAMICS . 145 7.1 Mathematical Model and Discretization . 148 7.1.1 Discretization of First and Second Derivatives . 152 7.1.2 Discretization of the Undeformed Metric and Christoffel Symbols . 154 7.1.3 Discretization of the Geodesic Equation . 155 7.1.4 Discretization of the Poisson Equation . 158 7.1.5 Density Computation . 159 7.2 Numerical Framework . 162 7.2.1 Input Parameters . 162 7.2.2 Preprocessing Stage . 163 7.2.3 Solving Stage . 164 7.2.4 Post Processing Stage . 165 7.2.5 Validation . 166 7.3 Numerical Experiments . 167 vii 7.3.1 Validation of the Framework by Simulating the Sun’s Gravity 167 7.3.2 Modeling the Effect of Inherent Curvature on the Gravity of Messier 33 . 172 7.4 Discussion . 178 7.5 Summary and Conclusion . 180 VIII. CONCLUSIONS . 182 IX. FUTURE RESEARCH . 189 9.1 Further Development of the Cosmic Fabric model . 189 9.2 Further Research into the Inherent Structure Hypothesis . 192 9.3 Additional Categories of Computational Simulations . 192 9.4 Implications to Cosmology and Fundamental Physics . 194 viii LIST OF TABLES 3.1 Comparison between the General Relativity and Solid Mechanics Perspectives 65 5.1 Kinematic variables pertaining to a spherically symmetric deformation of a hypersurface. 84 5.2 Geometric parameters pertaining to spherically symmetric gravitating body of mass M................................... 93 5.3 Length scales and energy scales of common sub-atomic particles . 107 6.1 Apparent masses and sizes of galaxies 25×103 - 13:4×109 light years from Earth .