Wormholes, Warp Drives and Energy Conditions (Fundamental Theories of Physics, 189)

Top researchers in the field of gravitation present the state-of-the-art topics outlined in this book, ranging from the stability of rotating wormholes solutions supported by ghost scalar fields, modified gravity applied to wormholes, the study of novel semi-classical and nonlinear energy conditions, to the applications of quantum effects and the superluminal version of the __warp drive__ in modified spacetime. Based on Einstein's field equations, this cutting-edge research area explores the more far-fetched theoretical outcomes of General Relativity and relates them to quantum field theory. This includes quantum energy inequalities, flux energy conditions, and wormhole curvature, and sheds light on not just the theoretical physics but also on the possible applications to warp drives and time travel. This book extensively explores the physical properties and characteristics of these 'exotic spacetimes,' describing in detail the general relativistic geometries that generate closed timelike curves. Preface 6 Acknowledgements 9 Contents 10 Acronyms 12 1 Introduction 14 1.1 Historical Background 14 1.2 State of the Art: Wormhole Geometries and Warp Drive Spacetimes 16 References 19 Part I Traversable Wormholes 21 2 Wormhole Basics 22 2.1 Static and Spherically Symmetric Traversable Wormholes 22 2.1.1 Spacetime Metric 22 2.1.2 The Mathematics of Embedding 23 2.1.3 Equations of Structure for the Wormhole 25 2.1.4 Stress--Energy Tensor 27 2.1.5 Exotic Matter and Modified Gravity 28 2.1.6 Traversability Conditions 31 2.2 Dynamic Spherically Symmetric Thin-Shell Traversable Wormholes 34 2.2.1 Generic Static Spherically Symmetric Spacetimes 35 2.2.2 Extrinsic Curvature 35 2.2.3 Lanczos Equations: Surface Stress--Energy 37 2.2.4 Conservation Identity 38 2.2.5 Equation of Motion 39 2.2.6 Linearized Equation of Motion 41 2.2.7 The Master Equations 42 2.2.8 Discussion 43 References 44 3 Rotating Wormholes 46 3.1 Introduction 46 3.2 Rotating Wormholes in Four Dimensions 48 3.2.1 Theoretical Setting 48 3.2.2 Symmetric Wormholes 54 3.2.3 Nonsymmetric Wormholes 59 3.3 Rotating Wormholes in Five Dimensions 61 3.3.1 Theoretical Setting 62 3.3.2 Wormholes 65 3.3.3 Stability 67 3.4 Conclusions and Outlook 70 References 71 4 Astrophysical Signatures of Thin Accretion Disks in Wormhole Spacetimes 73 4.1 Introduction 73 4.2 Electromagnetic Radiation Properties of Thin Accretion Disks in General Relativistic Spacetimes 78 4.2.1 Marginally Stable Orbits 78 4.2.2 Physical Properties of Thin Accretion Disks 79 4.3 Electromagnetic Signatures of Accretion Disks in Static Wormhole Geometries 83 4.3.1 Static Spherically Symmetric Wormhole Geometries 83 4.3.2 Electromagnetic Signatures of Static Spherically Symmetric Wormhole Geometries 84 4.4 Accretion Disk Properties in Rotating Wormhole Geometries 88 4.4.1 Stationary Axially Symmetric Wormhole Spacetimes 88 4.4.2 Effective Potential for Rotating Wormhole Geometries 89 4.4.3 Electromagnetic Signatures of Thin Accretion Disks in Rotating Wormhole Geometries 89 4.5 Discussions and Final Remarks 95 References 97 5 Horndeski Wormholes 99 5.1 Introduction 99 5.2 Action and Field Equations 100 5.3 Static Spherically Symmetric Asymptotically Flat Wormholes 101 5.3.1 Wormhole Configuration 102 5.3.2 Initial Condition Analysis 103 5.3.3 Asymptotic Analysis 105 5.3.4 Exact Wormhole Solution with η=0 106 5.3.5 Numerical Analysis 107 5.4 Exact Black Hole and Wormhole Solutions in Horndeski Theory 109 5.4.1 Schwarzschild Coordinates: ρ(r)=r 111 5.4.2 Wormhole Configuration: ρ(r)=sqrtr2+a2 114 5.5 Summary and Discussion 116 References 118 6 Self-Sustained Traversable Wormholes 120 6.1 Introduction 120 6.2 Effective Einstein Equations and the Hamiltonian 122 6.2.1 Einstein Equations and the Hamiltonian 122 6.2.2 The Effective Einstein Equations 123 6.3 Energy Density Calculation in Schrödinger Representation 125 6.4 The Transverse Traceless (TT) Spin 2 Operator for the Traversable Wormhole and the W.K.B. Approximation 128 6.5 One Loop Energy Regularization and Renormalization 130 6.5.1 Specific Case: Ellis Wormhole 132 6.5.2 Specific Case: pr=ωρ 132 6.5.3 Specific Case: pr=ω(r) ρ 133 6.5.4 Specific Case: Wormhole Geometry Induced by Noncommutative Geometry 134 6.6 Distorting Gravity 137 6.6.1 Gravity's Rainbow 138 6.6.2 Noncommutative Geometries 141 6.7 Summary and Conclusions 141 References 143 7 Trapped Ghosts as Sources for Wormholes and Regular Black Holes. The Stability Problem 145 7.1 Introduction 145 7.2 Basic Equations. NEC Violation and the Necessity of a Nonzero Potential 147 7.2.1 General Relations 147 7.2.2 Solutions with a Massless Scalar 149 7.2.3 Equations with an Arbitrary Potential 151 7.3 Models with a Trapped Ghost 152 7.4 Spherically Symmetric Perturbations 156 7.4.1 Perturbation Equations 156 7.4.2 Gauge-Invariant Perturbations 159 7.5 Throats, Trapped Ghosts and Stability 160 7.5.1 Perturbations Near a Throat 160 7.5.2 Perturbations Near the Surface h = 0 162 7.6 Conclusion 164 References 165 8 Geons in Palatini Theories of Gravity 168 8.1 Introduction 168 8.2 Microstructures, Holes, Metric-Affine Geometry, and Wormholes 170 8.3 Analytical Models with Nonmetricity 171 8.3.1 f(R) Theories with Nonlinear Electromagnetic Fields and Anisotropic Fluids 172 8.3.2 Quadratic Gravity with Maxwell and Born--Infeld Electrodynamics 180 8.3.3 Born--Infeld Gravity with Electromagnetic Fields 190 8.3.4 Higher Dimensional Generalizations 191 8.4 Conclusions 193 References 194 Part II Energy Conditions 198 9 Classical and Semi-classical Energy Conditions 199 9.1 Introduction 199 9.2 Standard Energy Conditions 199 9.2.1 The Hawking--Ellis Type I --- Type IV Classification 202 9.2.2 Alternative Canonical Form 206 9.2.3 Limitations of the Standard Energy Conditions 207 9.3 Averaged Energy Conditions 209 9.4 Nonlinear Energy Conditions 210 9.5 Semi-classical Energy Conditions 212 9.6 Discussion 215 References 215 10 Quantum Energy Inequalities 220 10.1 Introduction 220 10.2 Violation of the Energy Conditions in QFT 222 10.3 An Example of a QEI and Its Consequences 226 10.3.1 QEIs Obeyed by a Free Scalar Field in Minkowski Spacetime 226 10.3.2 Properties and Consequences of the QEI 227 10.3.3 Some History and References 231 10.4 Derivation of a QEI 232 10.4.1 Background and Preliminaries on QFT in Curved Spacetimes 232 10.4.2 The QEI Derivation 235 10.4.3 Specific Examples 239 10.5 Further Developments 242 10.5.1 Quantum Energy Inequalities Beyond Free QFT 242 10.5.2 Higher Moments and Probability Distributions 247 10.6 Applications of the QEIs 249 10.6.1 Exotic Spacetimes 250 10.6.2 States of Low Energy 251 10.6.3 Averaged Null Energy Condition 252 10.6.4 Singularity Theorems 252 10.6.5 Applications of the Probability Distribution 253 References 256 Part III Warp Drive 260 11 Warp Drive Basics 261 11.1 Introduction 261 11.2 Warp Drive Spacetime 263 11.2.1 Alcubierre Warp Drive 263 11.2.2 Superluminal Travel in the Warp Drive 264 11.2.3 The Violation of the Energy Conditions 266 11.2.4 The Quantum Inequality Applied to the Warp Drive 268 11.3 Linearized Warp Drive 270 11.4 The Horizon Problem 272 11.5 Superluminal Subway: The Krasnikov Tube 273 11.5.1 The Two-Dimensional Krasnikov Solution 273 11.5.2 Superluminal Travel Within the Krasnikov Tube 275 11.5.3 The Four-Dimensional Generalization 277 11.6 Closed Timelike Curves 278 11.6.1 The Warp Drive 278 11.6.2 The Krasnikov Tube 280 11.7 Summary and Conclusion 281 References 282 12 Probing Faster than Light Travel and Chronology Protection with Superluminal Warp Drives 284 12.1 Introduction 284 12.2 Time Machines: Basic Technical Definitions 285 12.3 Causality Challenging Spacetimes 286 12.3.1 Time Machine Rotating Solutions 286 12.3.2 Faster than Light Travel Spacetimes 287 12.4 Time Machines from Faster than Light Travel 290 12.4.1 The Warp Drive Case 291 12.4.2 The Wormhole Case 292 12.4.3 Some Analogue Gravity Lessons 292 12.5 Time Travel Paradoxes and Possible Solutions 293 12.6 Superluminal Warp Drive Instabilities: A Pre-emptive Chronology Protection at Work? 296 12.6.1 Light Ray Propagation 296 12.6.2 Renormalized Stress--Energy Tensor 297 12.6.3 Physical Interpretation 298 12.7 Warp Drive Instabilities Under Dispersion 300 12.8 Conclusions 301 References 302 Index 304 Front Matter....Pages i-xiv Introduction....Pages 1-7 Front Matter....Pages 9-9 Wormhole Basics....Pages 11-34 Rotating Wormholes....Pages 35-61 Astrophysical Signatures of Thin Accretion Disks in Wormhole Spacetimes....Pages 63-88 Horndeski Wormholes....Pages 89-109 Self-Sustained Traversable Wormholes....Pages 111-135 Trapped Ghosts as Sources for Wormholes and Regular Black Holes. The Stability Problem....Pages 137-159 Geons in Palatini Theories of Gravity....Pages 161-190 Front Matter....Pages 191-191 Classical and Semi-classical Energy Conditions....Pages 193-213 Quantum Energy Inequalities....Pages 215-254 Front Matter....Pages 255-255 Warp Drive Basics....Pages 257-279 Probing Faster than Light Travel and Chronology Protection with Superluminal Warp Drives....Pages 281-300 Back Matter....Pages 301-303