Mass and Motion in General Relativity (eBook)

Mass and Motion in General Relativity 1
Mass and Motion in General Relativity 5
Contents 9
The Higgs Mechanism and the Origin of Mass 19
1 The Standard Model and the Generation of Particle Masses 19
1.1 The Elementary Particles and Their Interactions 20
1.2 The Standard Model of Particle Physics 21
1.3 The Higgs Mechanism for Mass Generation 23
2 The Profile of the Higgs Particle 26
2.1 Characteristics of the Higgs Boson 26
2.2 Constraints on the Higgs Boson Mass 27
2.3 The Higgs Decay Modes and Their Rates 29
3 Higgs Production at the LHC 31
3.1 The Large Hadron Collider 31
3.2 The Production of the Higgs Boson 32
3.3 Detection of the Higgs Boson 33
3.4 Determination of the Higgs Boson Properties 36
4 The Higgs Beyond the Standard Model 38
5 Conclusions 40
References 41
Testing Basic Laws of Gravitation – Are Our Postulates on Dynamics and Gravitation Supported by Experimental Evidence? 42
1 Introduction – Why Gravity Is So Exceptional 42
2 Key Features of Gravity 44
3 Standard Tests of the Foundations of Special and General Relativity 45
3.1 Tests of Special Relativity 45
3.1.1 The Constancy of the Speed of Light 45
3.1.2 The Relativity Principle 46
3.1.3 The Consequence 47
3.2 Tests of the Universality of Free Fall 47
3.3 Tests of the Universality of the Gravitational Redshift 48
3.4 The Consequence 49
4 Tests of Predictions 51
4.1 The Gravitational Redshift 52
4.2 Light Deflection 53
4.3 Perihelion/Periastron Shift 54
4.4 Gravitational Time Delay 55
4.4.1 Direct Measurement 55
4.4.2 Measurement of Frequency Change 55
4.4.3 Remarks 57
4.5 Lense–Thirring Effect 57
4.6 Schiff Effect 58
4.7 The Strong Equivalence Principle 58
5 Why New Tests? 59
5.1 Dark Clouds – Problems with GR 59
5.1.1 Dark Matter 59
5.1.2 Dark Energy 60
5.1.3 Pioneer Anomaly 60
5.1.4 Flyby Anomaly 61
5.1.5 Increase of Astronomical Unit 61
5.1.6 Quadrupole and Octupole Anomaly 62
5.2 The Search for Quantum Gravity 62
5.3 Possible New Effects 62
6 How to Search for ``New Physics'' 63
6.1 Better Accuracy and Sensitivity 63
6.2 Extreme Situations 64
6.2.1 Extreme High Energy 64
6.2.2 Extreme Low Energy 64
6.2.3 Large Distances 65
6.2.4 Small Accelerations 65
6.2.5 Large Accelerations 65
6.2.6 Strong Gravitational Fields 65
6.3 Investigation of ``Exotic'' Issues 66
7 Testing ``Exotic'' but Fundamental Issues 67
7.1 Active and Passive Mass 67
7.2 Active and Passive Charge 68
7.3 Active and Passive Magnetic Moment 69
7.4 Charge Conservation 69
7.5 Small Accelerations 70
7.6 Test of the Inertial Law 71
7.6.1 Higher Order Equation of Motion for Classical Particles 72
7.6.2 Higher Order Equation of Motion for Quantum Particles 73
7.7 Can Gravity Be Transformed Away? 74
7.7.1 Finsler Geometry 74
7.7.2 Testing Finslerian Anisotropy in Tangent Space 75
7.7.3 Finslerian Geodesic Equation 75
8 Summary 77
References 77
Mass Metrology and the International System of Units (SI) 83
1 Introduction 83
2 The SI 84
2.1 Base Units/Base Quantities 84
2.2 Gaussian Units 86
2.3 Planck Units, Natural Units, and Atomic Units 87
3 Practical Reasons for Redefining the Kilogram 87
3.1 Internal Evidence Among 1 kg Artifact Mass Standards 87
3.2 Fundamental Constants 89
3.3 Electrical Metrology 91
3.4 Relative Atomic Masses 93
4 Routes to a New Kilogram 94
5 Realizing a New Kilogram Definition in Practice 95
5.1 Watt Balances 96
5.2 Silicon X-Ray Crystal Density (XRCD) 97
5.3 Experimental Results 98
6 Proposals for a New SI 99
6.1 Consensus Building and Formal Approval 99
6.2 An SI Based on Defined Values of a Set of Constants 100
7 Conclusion 100
References 101
Mass and Angular Momentum in General Relativity 103
1 Issues Around the Notion of Gravitational Energy in General Relativity 104
1.1 Energy–Momentum Density for Matter Fields 104
1.2 Problems when Defining a Gravitational Energy–Momentum 106
1.2.1 Nonlocal Character of the Gravitational Energy 107
1.3 Notation 108
1.3.1 3+1 Decompositions 108
1.3.2 Closed 2-Surfaces 109
2 Spacetimes with Killing Vectors: Komar Quantities 110
2.1 Komar Mass 110
2.2 Komar Angular Momentum 111
3 Total Mass of Isolated Systems in General Relativity 111
3.1 Asymptotic Flatness Characterization of Isolated Systems 111
3.2 Asymptotic Euclidean Slices 112
3.2.1 Asymptotic Symmetries at Spatial Infinity 113
3.3 ADM Quantities 113
3.3.1 ADM Energy 115
3.3.2 ADM 4-Momentum and ADM Mass 118
3.3.3 ADM Angular Momentum 120
3.4 Bondi Energy and Linear Momentum 121
3.4.1 Null Infinity 122
3.4.2 Symmetries at Null Infinity 122
3.4.3 Bondi–Sachs 4-Momentum 123
4 Notions of Mass for Bounded Regions: Quasi-Local Masses 124
4.1 Ingredients in the Quasi-Local Constructions 124
4.2 Some Relevant Quasi-Local Masses 125
4.2.1 Round Spheres: Misner–Sharp Energy 125
4.2.2 Brown–York Energy 125
4.2.3 Hawking, Geroch, and Hayward Energies 127
4.2.4 Bartnik Mass 129
4.3 Some Remarks on Quasi-Local Angular Momentum 130
4.4 A Study Case: Quasi-Local Mass of Black Hole IHs 131
4.4.1 A Brief Review of IHs 131
4.4.2 An Overview of the Hamiltonian Analysis of IHs 132
5 Global and Quasi-Local Quantities in Black Hole Physics 134
5.1 Penrose Inequality: a Claim for an Improved Mass Positivity Result for Black Holes 135
5.2 Black Hole (Thermo-)dynamics 135
5.3 Black Hole Extremality: a Mass–Angular Momentum Inequality 137
6 Conclusions 137
References 139
Post-Newtonian Theory and the Two-Body Problem 141
1 Introduction 141
2 Post-Newtonian Formalism 144
2.1 Einstein Field Equations 144
2.2 Post-Newtonian Iteration in the Near Zone 147
2.3 Post-Newtonian Expansion Calculated by Matching 151
2.4 Multipole Moments of a Post-Newtonian Source 155
2.5 Radiation Field and Polarization Waveforms 159
2.6 Radiative Moments Versus Source Moments 161
3 Inspiralling Compact Binaries 163
3.1 Stress–Energy Tensor of Spinning Particles 163
3.2 Hadamard Regularization 166
3.3 Dimensional Regularization 169
3.4 Energy and Flux of Compact Binaries 172
3.5 Waveform of Compact Binaries 176
3.6 Spin–Orbit Contributions in the Energy and Flux 178
References 180
Post-Newtonian Methods: Analytic Results on the Binary Problem 183
1 Introduction 183
2 Systems in Newtonian Gravity in Canonical Form 185
3 Canonical General Relativity and PN Expansions 187
3.1 Canonical Variables of the Gravitational Field 189
3.2 Brill–Lindquist Initial-Value Solution for Binary Black Holes 191
3.3 Skeleton Hamiltonian 192
3.4 Functional Representation of Compact Objects 195
3.5 PN Expansion of the Routh Functional 201
3.6 Near-Zone Energy Loss Versus Far-Zone Energy Flux 201
4 Binary Point Masses to Higher PN Order 203
4.1 Conservative Hamiltonians 203
4.2 Dynamical Invariants 204
4.3 ISCO and the PN Framework 206
4.4 PN Dissipative Binary Dynamics 208
5 Toward Binary Spinning Black Holes 208
5.1 Approximate Hamiltonians for Spinning Binaries 212
6 Lorentz-Covariant Approach and PN Expansions 216
6.1 PM and PN Expansions 218
6.2 PN Expansion in the Near Zone 219
6.3 PN Expansion in the Far Zone 221
7 Energy Loss and Gravitational Wave Emission 222
7.1 Orbital Decay to 4 PN Order 222
7.2 Gravitational Waveform to 1.5 PN Order 223
References 225
The Effective One-Body Description of the Two-Body Problem 227
1 Introduction 227
2 Motion and Radiation of Binary Black Holes: PN Expanded Results 229
3 Conservative Dynamics of Binary Black Holes: the EOB Approach 231
4 Description of Radiation–Reaction Effects in the EOB Approach 240
4.1 Resummation of Taylor Using a One-Parameter Family of Padé Approximants: Tuning vpole 243
4.2 Parameter-Free Resummation of Waveform and Energy Flux 246
5 EOB Dynamics and Waveforms 254
5.1 Post–Post-Circular Initial Data 254
5.2 EOB Waveforms 255
5.3 EOB Dynamics 257
6 EOB and NR Waveforms 259
7 Conclusions 264
References 265
Introduction to Gravitational Self-Force 269
1 Motion of Bodies in General Relativity 269
2 Point Particles in General Relativity 270
3 Point Particles in Linearized Gravity 271
4 Lorenz Gauge Relaxation 272
5 Hadamard Expansions 272
5.1 Hadamard Expansions for a Point Particle Source 274
6 Equations of Motion Including Self-Force 275
6.1 The MiSaTaQuWa Equations 275
6.2 The Detweiler–Whiting Reformulation 276
7 How Should Gravitational Self-Force Be Derived? 277
References 278
Derivation of Gravitational Self-Force 279
1 Difficulties with Usual Derivations 279
2 Rigorous Derivation Requirements 280
3 Limits of Spacetimes 280
4 Our Basic Assumptions 281
4.1 Additional Uniformity Requirement 281
5 Geodesic Motion 282
6 Corrections to Motion 283
6.1 Calculation of the Perturbed Motion 284
7 Interpretation of Results 285
8 Self-Consistent Equations 285
9 Summary 286
References 286
Elementary Development of the Gravitational Self-Force 287
1 Introduction 287
1.1 Outline 289
1.2 Notation 290
2 Newtonian Examples of Self-Force and Gauge Issues 291
3 Classical Electromagnetic Self-Force 293
4 A Toy Problem with Two Length Scales That Creates a Challenge for Numerical Analysis 294
4.1 An Approach Which Avoids the Small Length Scale 295
4.2 An Alternative That Resolves Boundary Condition Issues 297
5 Perturbation Theory 298
5.1 Standard Perturbation Theory in General Relativity 299
5.2 An Application of Perturbation Theory: Locally Inertial Coordinates 301
5.3 Metric Perturbations in the Neighborhood of a Point Mass 303
5.4 A Small Object Moving Through Spacetime 305
6 Self-Force from Gravitational Perturbation Theory 307
6.1 Dissipative and Conservative Parts 308
6.2 Gravitational Self-Force Implementations 309
6.2.1 Field Regularization Via the Effective Source 309
6.2.2 Mode-Sum Regularization 310
6.2.3 The Gravitational Self-Force Actually Resulting in Acceleration 310
7 Perturbative Gauge Transformations 311
8 Gauge Confusion and the Gravitational Self-Force 313
9 Steps in the Analysis of the Gravitational Self-Force 314
10 Applications 316
10.1 Gravitational Self-Force Effects on Circular Orbits of the Schwarzschild Geometry 316
10.2 Field Regularization Via the Effective Source 317
11 Concluding Remarks 320
References 322
Constructing the Self-Force 324
1 Introduction 324
2 Geometric Elements 326
3 Coordinate Systems 327
4 Field Equation and Particle Motion 331
5 Retarded Green's Function 331
6 Alternate Green's Function 333
7 Fields Near the World Line 334
8 Self-Force 336
9 Axiomatic Approach 337
10 Conclusion 339
References 340
Computational Methods for the Self-Force in Black Hole Spacetimes 341
1 Introduction and Overview 341
1.1 The MiSaTaQuWa Formula 343
1.2 Gauge Dependence 344
1.3 Implementation Strategies 345
1.3.1 Quasi-Local Calculations 346
1.3.2 Weak-Field Analysis 346
1.3.3 Radiation-Gauge Regularization 346
1.3.4 Mode-Sum Method 346
1.3.5 ``Puncture'' Methods 347
2 Mode-Sum Method 349
2.1 An Elementary Example 350
2.2 The Mode-Sum Formula 352
2.3 Derivation of the Regularization Parameters 353
3 Numerical Implementation Strategies 357
3.1 Overcoming the Gauge Problem 358
3.1.1 Self-Force in an ``Hybrid'' Gauge 358
3.1.2 Generalized SF and Gauge Invariants 359
3.1.3 Radiation-Gauge Regularization 360
3.1.4 Direct Lorenz-Gauge Implementation 360
3.2 Numerical Representation of the Point Particle 361
3.2.1 Particle Representation in the Time Domain 361
3.2.2 Particle Representation in the Frequency Domain: the High-Frequency Problem and Its Resolution 364
4 An Example: Gravitational Self-Force in Schwarzschild Via 1+1D Evolution in Lorenz Gauge 366
4.1 Lorenz-Gauge Formulation 366
4.2 Numerical Implementation 368
5 Toward Self-Force Calculations in Kerr: the Puncture Method and m-Mode Regularization 370
5.1 Puncture Method in 2+1D 370
5.2 m-Mode Regularization 372
6 Reflections and Prospects 374
References 378
Radiation Reaction and Energy–Momentum Conservation 381
1 Introduction 381
2 Energy–Momentum Balance Equation 383
2.1 Decomposition of the Stress Tensor 385
2.2 Bound Momentum 388
2.3 The Rest Frame (Nonrelativistic Limit) 391
3 Flat Dimensions Other than Four 392
4 Local Method for Curved Space-Time 393
4.1 Hadamard Expansion in Any Dimensions 394
4.2 Divergences 395
4.3 Four Dimensions 398
4.4 Self and Radiative Forces in Curved Space-Time 400
5 Gravitational Radiation 401
5.1 Bianchi Identity 401
5.2 Vacuum Background 403
5.3 Gravitational Radiation for Non-Geodesic Motion 404
6 Conclusions 405
References 406
The State of Current Self-Force Research 408
1 Introduction 408
2 The Teukolsky Equation 410
2.1 The Inhomogeneous Teukolsky Equation with a Distributional Source 410
2.2 Adiabatic Waveforms 411
2.3 Numerical Solution of the Teukolsky Equation 412
2.4 The Linearized Einstein Equations 413
3 Frequency-Domain Calculations of the Self-Force 414
3.1 Mode-Sum Regularization 414
3.2 The Detweiler–Whiting Regular Part of the Self-Force 415
4 Time-Domain Calculations of the Self-Force 416
4.1 1+1D Numerical Simulations 416
4.2 2+1D Numerical Simulations 417
4.2.1 m-Mode Regularization 418
4.2.2 The Square of the Geodesic Distance 419
4.2.3 The Puncture Function 420
5 Post-adiabatic Self-Force-Driven Orbital Evolution 421
5.1 The Importance of Second-Order Self-Forces 421
5.2 Conservative Self-Force Effects 425
References 426
High-Accuracy Comparison Between the Post-Newtonianand Self-Force Dynamics of Black-Hole Binaries 428
1 Introduction and Motivation 429
2 The Gauge-Invariant Redshift Observable 431
3 Regularization Issues in the SF and PN Formalisms 432
4 Circular Orbits in the Perturbed Schwarzschild Geometry 434
5 Overview of the 3PN Calculation 436
5.1 Iterative PN Computation of the Metric 436
5.2 The Example of the Zeroth-Order Iteration 439
6 Logarithmic Terms at 4PN and 5PN Orders 440
6.1 Physical Origin of Logarithmic Terms 440
6.2 Expression of the Near-Zone Metric 442
7 Post-Newtonian Results for the Redshift Observable 443
8 Numerical Evaluation of Post-Newtonian Coefficients 446
8.1 Overview 447
8.2 Framework for Evaluating PN Coefficients Numerically 448
8.3 Consistency Between Analytically and Numerically Determined PN Coefficients 450
8.4 Determining Higher Order PN Terms Numerically 451
8.5 Summary 452
References 454
LISA and Capture Sources 456
1 LISA – A Mission to Detect and Observe Gravitational Waves 456
1.1 Mission Concept 457
1.2 Sensitivity 458
1.3 Measurement Principle 459
2 Capture Sources 461
3 Science Return 462
4 Detection 464
4.1 Capture Rates 464
4.2 Signal Characteristics 465
4.3 Data Analysis 467
5 Summary and Conclusions 469
References 470
Motion in Alternative Theories of Gravity 473
1 Introduction 473
2 Modifying the Matter Action 474
3 Modified Motion in Metric Theories? 476
4 Scalar-Tensor Theories of Gravity 480
4.1 Weak-Field Predictions 481
4.2 Strong-Field Predictions 483
4.3 Binary-Pulsar Tests 484
4.4 Black Holes in Scalar-Tensor Gravity 488
5 Extended Bodies 489
6 Modified Newtonian Dynamics 491
6.1 Mass-Dependent Models? 492
6.2 Aquadratic Lagrangians or k-Essence 493
6.3 Difficulties 494
6.4 Nonminimal Couplings 497
7 Conclusions 498
References 499
Mass, Inertia, and Gravitation 502
1 Introduction 502
2 Vacuum Fluctuations and Inertia 505
2.1 Linear Response Formalism 505
2.2 Response to Motions 509
2.3 Relativity of Motion 513
2.4 Inertia of Vacuum Fields 515
3 Mass as a Quantum Observable 519
3.1 Quantum Fluctuations of Mass 519
3.2 Mass and Conformal Symmetries 521
4 Metric Extensions of GR 525
4.1 Radiative Corrections 526
4.2 Anomalous Curvatures 529
4.3 Phenomenology in the Solar System 531
5 Conclusion 537
References 538
Motion in Quantum Gravity 542
1 Introduction 542
1.1 The Problem of Defining Motion in Quantum Gravity 542
1.2 Quantum Gravity 543
1.3 Three-Dimensional Quantum Gravity Is a Fruitful Toy Model 545
1.4 Outline of the Article 546
2 Casting an Eye Over Loop Quantum Gravity 547
2.1 The Classical Theory: Main Ingredients 547
2.2 The Route to the Quantization of Gravity 549
2.3 Spin-Networks Are States of Quantum Geometry 550
2.4 The Problem of the Hamiltonian Constraint 552
3 Three-Dimensional Euclidean Quantum Gravity 553
3.1 Construction of the Noncommutative Space 554
3.1.1 Quantum Gravity and Noncommutativity 554
3.1.2 The Quantum Double Plays the Role of the Isometry Algebra 555
3.1.3 The Quantum Geometry Defined by Its Momenta Space 556
3.1.4 The Fuzzy Space Formulation 557
3.1.5 Relation to the Classical Geometry 558
3.2 Constructing the Quantum Dynamics 561
3.2.1 An Integral on the Quantum Space to Define the Action 561
3.2.2 Derivative Operators to Define the Dynamics 561
3.2.3 Free Field: Solutions and Properties 562
3.3 Particles Evolving in the Fuzzy Space 563
3.4 Reduction to One Dimension 564
3.4.1 Dynamics of a Particle: Linear versus Nonlinear 566
3.4.2 Background Independent Motion 568
4 Discussion 569
References 570
Free Fall and Self-Force: an Historical Perspective 571
1 Introduction 572
2 The Historical Heritage 573
3 Uniqueness of Acceleration and the Newtonian Back-Action 575
4 The Controversy on the Repulsion and on the Particle Velocity at the Horizon 579
5 Black Hole Perturbations 584
6 Numerical Solution 588
7 Relativistic Radial Fall Affected by the Falling Mass 592
7.1 The Self-Force 592
7.2 The Pragmatic Approach 597
8 The State of the Art 600
8.1 Trajectory 601
8.2 Regularisation Parameters 602
8.3 Effect of Radiation Reaction on the Waveforms During Plunge 602
9 Beyond the State of the Art: the Self-Consistent Prescription 603
10 Conclusions 605
References 607
Index 614