Special Relativity (eBook)

Preface 6
Contents 8
List of Contributors 14
Part I Historical and Philosophical Aspects 18
Isotropy of Inertia: A Sensitive Early Experimental Test 20
1 Introduction 20
2 Early Ideas 21
3 Possibilities for Experiments 21
4 Some Factors Expected to A.ect Sensitivity in a Simple NMR Measurement 22
5 Development of the Experimental Technique 22
6 Initial Observations 24
8 Experimental Procedure 26
9 Discussion of Experimental Results 29
10 Interpretation 29
11 Some Personal Remarks 30
Acknowledgements 30
References 30
The Challenge of Practice: Einstein, Technological Development and Conceptual Innovation 32
1 Knowledge and Power in the Scienti.c Revolution 32
2 Contrasting Intuitions on the Cascade Model 34
3 Poincar ´ e, Einstein, Distant Simultaneity, 37
and the Synchronization of Clocks 37
4 The Emerging Rule of Global Time 41
5 Technology-Based Concepts and the Rise of Operationalism 42
6 Technological Problems, Technological Solutions, and Scientific Progress 45
References 47
Part II Foundation and Formalism 50
Foundations of Special Relativity Theory 52
1 Introduction 52
2 Inertial Frames 53
3 Poincar ´ e Transformations 53
4 Minkowski Spacetime 56
5 Axiomatics 57
6 The Principle of Special Relativity and Its Limits 57
7 Examples 58
8 Accelerated Frames of Reference 58
9 SR Causality 59
References 60
Algebraic and Geometric Structures in Special Relativity 62
1 Introduction 62
2 Some Remarks on Symmetry and Covariance 63
3 The Impact of the Relativity Principle on the Automorphism Group of Spacetime 66
4 Algebraic Structures of Minkowski Space 72
5 Geometric Structures in Minkowski Space 88
A Appendices 115
Acknowledgements 125
References 125
Quantum Theory in Accelerated Frames of Reference 129
1 Introduction 129
2 Hypothesis of Locality 130
3 Acceleration Tensor 132
4 Nonlocality 133
5 Inertial Properties of a Dirac Particle 136
6 Rotation 137
7 Sagnac E.ect 138
8 Spin-Rotation Coupling 139
9 Translational Acceleration 142
10 Discussion 146
References 146
Vacuum Fluctuations, Geometric Modular Action and Relativistic Quantum Information Theory 150
1 Introduction 150
2 From Quantum Mechanics and Special Relativity to Quantum Field Theory 154
3 The Reeh–Schlieder–Theorem and Geometric Modular Action 163
4 Relativistic Quantum Information Theory: Distillability in Quantum Field Theory 171
References 177
Spacetime Metric from Local and Linear Electrodynamics: A New Axiomatic Scheme 180
1 Introduction 180
2 Spacetime 181
3 Matter – Electrically Charged and Neutral 182
4 Electric Charge Conservation 183
5 Charge Active: Excitation 183
6 Charge Passive: Field Strength 184
7 Magnetic Flux Conservation 185
8 Premetric Electrodynamics 185
9 The Excitation is Local and Linear in the Field Strength 187
10 Propagation of Electromagnetic Rays ( Light ) 190
11 No Birefringence in Vacuum and the Light Cone 192
12 Dilaton, Metric, Axion 197
13 Setting the Scale 198
14 Discussion 199
15 Summary 201
Acknowledgments 201
References 201
Part III Violations of Lorentz Invariance? 206
Overview of the Standard Model Extension: Implications and Phenomenology of Lorentz Violation 208
1 Introduction 208
2 Motivations 211
3 Constructing the SME 214
4 Spontaneous Lorentz Violation 220
5 Phenomenology 229
6 Tests in QED 232
7 Conclusions 238
References 239
Anything Beyond Special Relativity? 244
1 Introduction and Summary 244
2 Some Key Aspects of Beyond-Special-Relativity Research 249
3 More on the Quantum-Gravity Intuition 256
4 More on the Quantum-Gravity-Inspired DSR Scenario 261
5 More on the Similarities with Beyond-Standard-Model Research 289
6 Another Century? 291
References 292
Doubly Special Relativity as a Limit of Gravity 296
1 Introduction 296
2 Postulates of Doubly Special Relativity 297
3 Constrained BF Action for Gravity 301
4 DSR from 2+1 Dimensional Gravity 307
5 Conclusions 312
Acknowledgements 313
References 313
Corrections to Flat-Space Particle Dynamics Arising from Space Granularity 316
1 Introduction 316
2 Basic Elements from Loop Quantum Gravity (LQG) 321
3 A Kinematical Estimation of the Semiclassical Limit 329
4 Phenomenological Aspects 335
Acknowledgements 357
References 357
Part IV Experimental Search 364
Test Theories for Lorentz Invariance 366
1 Introduction 366
2 Test Theories 368
3 Model-Independent Descriptions of LI Tests 371
4 The General Frame for Kinematical Test Theories 381
5 The Test Theory of Robertson 384
6 The General Formalism 393
7 The Mansouri-Sexl Test Theory 396
8 Discussion 398
Acknowledgements 400
References 400
Test of Lorentz Invariance Using a Continuously Rotating Optical Resonator 402
1 Introduction 402
2 Setup 404
3 LLI-Violation Signal According to SME 406
4 LLI-Violation Signal According to RMS 411
5 Data Analysis 413
6 Outlook 415
References 417
A Precision Test of the Isotropy of the Speed of Light Using Rotating Cryogenic Optical Cavities 418
1 Introduction 418
2 Experimental Setup 419
3 Characterization of the Setup 424
4 Data Collection and Analysis 427
5 Conclusions 430
Acknowledgments 431
References 431
Rotating Resonator-Oscillator Experiments to Test Lorentz Invariance in Electrodynamics 433
1 Introduction 433
2 Common Test Theories to Characterize Lorentz Invariance 434
3 Applying the SME to Resonator Experiments 441
4 Comparison of Sensitivity of Various Resonator Experiments in the SME 450
5 Applying the RMS to Whispering Gallery Mode Resonator Experiments 454
6 The University of Western Australia Rotating Experiment 456
7 Data Analysis and Interpretation of Results 462
8 Summary 465
Acknowledgments 467
References 467
Recent Experimental Tests of Special Relativity 468
1 Introduction 468
2 Theoretical Frameworks 469
3 Michelson-Morley and Kennedy-Thorndike Tests 476
4 Atomic Clock Test of Lorentz Invariance 485
in the SME Matter Sector 485
5 Conclusion 492
References 494
Experimental Test of Time Dilation by Laser Spectroscopy on Fast Ion Beams 496
1 Introduction 496
2 Principle of the Ives Stilwell Experiment 497
3 Ives-Stilwell Experiment at Storage Rings 498
4 Outlook 507
Acknowledgments 509
References 509
Tests of Lorentz Symmetry in the Spin-Coupling Sector 510
1 Introduction 510
2 129Xe/3He maser (Harvard-Smithsonian Center for Astrophysics) 511
3 Hydrogen Maser 514
4 Spin-Torsion Pendula (University of Washington and Tsing-Hua University) 516
References 521
Do Evanescent Modes Violate Relativistic Causality? 523
1 Introduction 523
2 Wave Propagation 525
3 Photonic Barriers, Examples of Evanescent Modes 527
4 Evanescent Modes Are not Observable 532
5 Velocities, Delay Times, and Signals 533
6 Partial Re.ection: An Experimental Method to Demonstrate Superluminal Signal Velocity of Evanescent Modes 539
7 Evanescent Modes a Near Field Phenomenon 541
8 Superluminal Signals Do not Violate Primitive Causality 544
9 Summary 546
Acknowledgement 547
References 547
Lecture Notes in Physics 549

Overview of the Standard Model Extension: Implications and Phenomenology of Lorentz Violation (p. 191-192)
R. Bluhm

Colby College, Waterville, ME 04901, USA
rtbluhm@colby.edu

Abstract. The Standard Model Extension (SME) provides the most general observerindependent field theoretical framework for investigations of Lorentz violation. The SME lagrangian by definition contains all Lorentz-violating interaction terms that can be written as observer scalars and that involve particle fields in the Standard Model and gravitational fields in a generalized theory of gravity. This includes all possible terms that could arise from a process of spontaneous Lorentz violation in the context of a more fundamental theory, as well as terms that explicitly break Lorentz symmetry. An overview of the SME is presented, including its motivations and construction. Some of the theoretical issues arising in the case of spontaneous Lorentz violation are discussed, including the question of what happens to the Nambu-Goldstone modes when Lorentz symmetry is spontaneously violated and whether a Higgs mechanism can occur. A minimal version of the SME in flat Minkowski spacetime that maintains gauge invariance and power-counting renormalizability is used to search for leading-order signals of Lorentz violation. Recent Lorentz tests in QED systems are examined, including experiments with photons, particle and atomic experiments, proposed experiments in space, and experiments with a spin-polarized torsion pendulum.

1 Introduction

It has been 100 years since Einstein published his first papers on special relativity [1]. This theory is based on the principle of Lorentz invariance, that the laws of physics and the speed of light are the same in all inertial frames. A few years after Einstein’s initial work, Minkowski showed that a new spacetime geometry emerges from special relativity. In this context, Lorentz symmetry is an exact spacetime symmetry that maintains the form of the Minkowski metric in different Cartesian-coordinate frames. In the years 1907–1915, Einstein developed the general theory of relativity as a new theory of gravity. In general relativity, spacetime is described in terms of a metric that is a solution of Einstein’s equations.

The geometry is Riemannian, and the physics is invariant under general coordinate transformations. Lorentz symmetry, on the other hand, becomes a local symmetry. At each point on the spacetime manifold, local coordinate frames can be found in which the metric becomes the Minkowski metric. However, the choice of the local frame is not unique, and local Lorentz transformations provide the link between physically equivalent local frames.

The Standard Model (SM) of particle physics is a fully relativistic theory. The SM in Minkowski spacetime is invariant under global Lorentz transformations, whereas in a Riemannian spacetime the particle interactions must remain invariant under both general coordinate transformations and local Lorentz transformations. Particle fields are also invariant under gauge transformations. Exact symmetry under local gauge transformations leads to the existence of massless gauge fields, such as the photon. However, spontaneous breaking of local gauge symmetry in the electroweak theory involves the Higgs mechanism, in which the gauge fields can acquire a mass.

Classical gravitational interactions can be described in a form analogous to gauge theory by using a vierbein formalism [2]. This also permits a straightforward treatment of fermions in curved spacetimes. Covariant derivatives of tensors in the local Lorentz frame involve introducing the spin connection.