Metric Theories of Gravity: Perturbations and Conservation Laws
621
Metric Theories of Gravity: Perturbations and Conservation Laws
621eBook
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Overview
By focusing on the mostly used variational methods, this monograph aspires to give a unified description and comparison of various ways of constructing conserved quantities for perturbations and to study symmetries in general relativity and modified theories of gravity. The main emphasis lies on the field-theoretical covariant formulation of perturbations, the canonical Noether approach and the Belinfante procedure of symmetrisation. The general formalism is applied to build the gauge-invariant cosmological perturbation theory, conserved currents and superpotentials to describe physically important solutions of gravity theories. Meticulous attention is given to the construction of conserved quantities in asymptotically-flat spacetimes as well as in asymptotically constant curvature spacetimes such as the Anti-de Sitter space. Significant part of the book can be used in graduate courses on conservation laws in general relativity.
THE SERIES: DE GRUYTER STUDIES IN MATHEMATICAL PHYSICS
The series is devoted to the publication of monographs and high-level texts in mathematical physics. They cover topics and methods in fields of current interest, with an emphasis on didactical presentation. The series will enable readers to understand, apply, and develop further, with sufficient rigor, mathematical methods to given problems in physics. The works in this series are aimed at advanced students and researchers in mathematical and theoretical physics. They can also serve as secondary reading for lectures and seminars at advanced levels.
Product Details
| ISBN-13: | 9783110383409 |
|---|---|
| Publisher: | De Gruyter |
| Publication date: | 04/24/2017 |
| Series: | De Gruyter Studies in Mathematical Physics , #38 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 621 |
| File size: | 30 MB |
| Note: | This product may take a few minutes to download. |
| Age Range: | 18 Years |
About the Author
Alexander N. Petrov, Sergey M. Kopeikin, Robert R. Lompay, Bayram Tekin.
Table of Contents
List of Figures xx
Primary notations xxi
1 Conservation laws in theoretical physics: A brief introduction 1
1.1 Conserved quantities in classical mechanics 1
1.1.1 Some basic notions of non-relativistic classical mechanics 1
1.1.2 The least action principle 2
1.1.3 Noether's theorem in classical mechanics 5
1.1.4 Conserved quantities for a system of non-relativistic particles 12
1.1.5 The Minkowski space and the Poincare group 15
1.1.6 A point-like particle in special relativity 17
1.1.7 Conserved quantities for a system of relativistic particles 20
1.2 Field theory in the Minkowski space 22
1.2.1 The action 22
1.2.2 Variational field equations 26
1.2.3 The Noether theorems 29
1.2.4 Conserved quantities in field theories 38
1.2.5 Examples of field theories in the Minkowski space 48
1.3 General relativity: fundamental mathematical relations 52
1.3.1 Lagrangians for the gravitational sector of general relativity 52
1.3.2 The Einstein equations 55
1.4 Classical conserved quantities in general relativity 60
1.4.1 The third Noether's theorem 60
1.4.2 Pseudotensors and superpotentials 64
1.5 Applications 76
1.5.1 Linearized general relativity 76
1.5.2 Weak gravitational waves in general relativity 80
1.5.3 The energy of an isolated gravitating system in general relativity 84
2 Field-theoretical formulation of general relativity: The theory 89
2.1 Development of the field-theoretical formulation 89
2.1.1 Geometrical formalism and field theories 89
2.1.2 Earlier perturbative formulations of general relativity 89
2.1.3 Deser's field-theoretical model 92
2.1.4 Various methods of the construction 95
2.2 The field-theoretical formulation of general relativity 97
2.2.1 A dynamical Lagrangian 98
2.2.2 The Einstein equations in the field-theoretical formulation 102
2.2.3 Functional expansions 105
2.2.4 Gauge transformations and their properties 107
2.2.5 Differential conservation laws 113
2.2.6 Different variants of the field-theoretical formulation in general relativity 118
2.2.7 The background as an auxiliary structure 121
2.3 Metric perturbations as compensating fields 123
2.3.1 "Localization" of background Killing vectors 123
2.3.2 The total action 128
2.3.3 Discussion of the results 129
2.4 The Babak-Grishchuk gravity with a non-zero graviton mass 132
2.4.1 Second derivatives in the energy-momentum tensor 132
2.4.2 Modified Lagrangian and equations 133
2.4.3 Non-zero masses of gravitons 135
2.4.4 Black holes and cosmology in massive gravity 138
2.4.5 Gauge invariance in the Babak-Grishchuk modifications 141
3 Asymptotically flat spacetime in the field-theoretical formulation 143
3.1 The Arnowitt-Deser-Misner formulation of general relativity 143
3.1.1 The ADM action principle 143
3.1.2 Asymptotically flat spacetime at spatial infinity in general relativity 154
3.1.3 The ADM definition of conserved quantities 156
3.1.4 The Regge-Teitelboim modification 163
3.2 An isolated system in the Lagrangian description 165
3.2.1 Asymptotically flat spacetime as a field configuration 166
3.2.2 Global conserved quantities 169
3.2.3 The parity conditions 171
3.2.4 Gauge invariance of the motion integrals 173
3.2.5 Concluding remarks 177
3.3 An isolated system in the Hamiltonian description 177
3.3.1 The difference between the canonical and symmetric currents 178
3.3.2 Phase variables and their asymptotic behaviour 180
3.3.3 Global conserved integrals 182
3.3.4 Gauge invariance of global integrals 185
4 Exact solutions of general relativity in the field-theoretical formalism 189
4.1 The Schwarzschild solution 189
4.1.1 The total energy 189
4.1.2 The energy distribution for the Schwarzschild black hole 191
4.1.3 The Schwarzschild black hole as a point particle 198
4.1.4 The Schwarzschild solution and the harmonic gauge fixing 203
4.2 Other exact solutions of general relativity 207
4.2.1 The Friedmann solution for a closed universe 207
4.2.2 The Abbott-Deser superpotential and its generalizations 210
4.2.3 The total mass of the Schwarzschild-AdS black hole 214
5 Field-theoretical derivation of cosmological perturbations 217
5.1 Introduction: Post-Newtonian, post-Minkowskian and post-Friedmanninan approximations in cosmology 217
5.2 Lagrangian and field variables 226
5.2.1 Action functional 227
5.2.2 Lagrangian of the ideal fluid 227
5.2.3 Lagrangian of scalar field 230
5.2.4 Lagrangian of a localized astronomical system 231
5.3 Background manifold 232
5.3.1 Hubble flow 232
5.3.2 Friedmann-Lemître-Robertson-Walker metric 233
5.3.3 Christoffel symbols and covariant derivatives 236
5.3.4 Riemann tensor 238
5.3.5 The Friedmann equations 239
5.3.6 Hydrodynamic equations of the ideal fluid 241
5.3.7 Scalar field equations 242
5.3.8 Equations of motion of matter of the localized astronomical system 243
5.4 Lagrangian perturbations of FLRW manifold 244
5.4.1 The concept of perturbations 244
5.4.2 The background field equations 248
5.4.3 The dynamic Lagrangian for perturbations 248
5.4.4 The Lagrangian equations for gravitational field perturbations 250
5.4.5 The Lagrangian equations for dark matter perturbations 254
5.4.6 The Lagrangian equations for dark energy perturbations 255
5.4.7 Linearized post-Newtonian equations for field variables 256
5.5 Gauge-invariant scalars and field equations in 1+3 threading formalism 260
5.5.1 Threading decomposition of the metric perturbations 260
5.5.2 Gauge transformation of the field variables 262
5.5.3 Gauge-invariant scalars 264
5.5.4 Field equations for the gauge-invariant scalar perturbations 267
5.5.5 Field equations for vector perturbations 270
5.5.6 Field equations for tensor perturbations 271
5.5.7 Residual gauge freedom 272
5.6 Post-Newtonian field equations in a spatially-flat universe 273
5.6.1 Cosmological parameters and scalar field potential 273
5.6.2 Conformal cosmological perturbations 276
5.6.3 Post-Newtonian held equations in conformal spacetime 279
5.6.4 Residual gauge freedom in the conformal spacetime 284
5.7 Decoupled system of the post-Newtonian field equations 285
5.7.1 The universe governed by dark matter and cosmological constant 285
5.7.2 The universe governed by dark energy 290
5.7.3 Post-Newtonian potentials in the linearized Hubble approximation 291
6 Currents and superpotentials on arbitrary backgrounds: Three approaches 299
6.1 The Katz, Bicák and Lynden-Bell conservation laws 300
6.1.1 A bi-metric KBL Lagrangian 300
6.1.2 KBL conserved quantities 302
6.2 The Belinfante procedure 306
6.2.1 The Belinfante symmetrization in general relativity 306
6.2.2 The Belinfante method applied to the KBL model 307
6.3 Currents and superpotentials in the field-theoretical formulation 311
6.3.1 Noether's procedure applied to the field-theoretical model 311
6.3.2 A family of conserved quantities and the Boulware-Deser ambiguity 316
6.3.3 Comments on conserved quantities of three types 318
6.4 Criteria for the choice of conserved quantities 321
6.4.1 Tests of consistency 321
6.4.2 The Reissner-Nordström solution 323
6.4.3 The Kerr solution 325
6.4.4 The total KBL energy for the S-AdS solution 327
6.5 The FLRW solution as a perturbation on the de Sitter background 327
6.5.1 Spatially conformal mappings of FLRW spacetime onto de Sitter space 328
6.5.2 Superpotentials and conserved currents 330
6.6 Integral constraints for linear perturbations on FLRW backgrounds 332
6.6.1 A FLRW background and its conformal Killing vectors 334
6.6.2 Integral relations for linear perturbations 337
6.6.3 Possible applications 341
7 Conservation laws in an arbitrary multi-dimensional metric theory 343
7.1 Covariant Noether's procedure in an arbitrary field theory 343
7.1.1 Covariant identities and identically conserved quantities 344
7.1.2 Another variant of covariantization 349
7.1.3 A new family of the identically conserved covariant Nother quantities 352
7.1.4 A Belinfante corrected family of identically conserved quantities 354
7.2 Conservation laws for perturbations: Three approaches 356
7.2.1 An arbitrary metric theory in n dimensions 356
7.2.2 Canonical conserved quantities for perturbations 358
7.2.3 The Belinfante corrected conserved quantities 362
7.2.4 The field-theoretical formulation for perturbations 363
7.2.5 Currents and superpotentials in the field-theoretical formulation 367
8 Conserved quantities in the Einstein-Gauss-Bonnet gravity 371
8.1 Superpotentials and currents in the EGB gravity 371
8.1.1 Action and field equations in the EGB gravity 371
8.1.2 Three types of superpotentials 372
8.1.3 Three types of currents 377
8.2 Conserved charges in the EGB gravity 381
8.2.1 Charges for isolated systems 381
8.2.2 Superpotentials for static spherically symmetric solutions 382
8.2.3 Mass of the Schwarzschitd-AdS black hole 387
8.3 Interpretation of the Maeda-Dadhich exotic solutions 394
8.3.1 Kaluza-Klein type 3D black holes 395
8.3.2 Mass for the static Maeda-Dadhich objects 400
8.3.3 Mass and mass flux for the radiative Maeda-Dadhich objects 404
9 Generic gravity: Particle content, weak field limits, conserved charges 408
9.1 Introduction: Raisons d'être of modified gravity theory 408
9.1.1 Conventions 411
9.2 Particle spectrum and stability of vacuum in quadratic gravity 411
9.2.1 Curvature tensors at second order in perturbation theory 412
9.2.2 Field equations and the vacuum structure 414
9.2.3 Linearization of quadratic gravity 416
9.2.4 Explicit check of linearized Bianchi identity 418
9.2.5 Degrees of freedom of quadratic gravity in AdS 419
9.3 Particle spectrum of f(R μνσρ ) gravity in (A)dS 426
9.3.1 Linearization of the field equations 427
9.3.2 Lovelock gravity 433
9.3.3 Propagator structure of the Lovelock theory 434
9.4 Weak field limits: Potential energy from tree-level gravitons 439
9.4.1 Potential energy from the scattering amplitude 442
9.4.2 Decomposition of the graviton field and tree-level scattering amplitude 444
9.4.3 Van Dam-Veltman-Zakharov discontinuity 447
9.4.4 New massive gravity redux 449
9.4.5 Spin-spin, spin-orbit, orbit-orbit interactions 452
9.4.6 Gravitomagnetic effects in general relativity 454
9.4.7 Gravitomagnetic effects in massive gravity 455
9.4.8 Gravitomagnetic effects in quadratic gravity 456
9.4.9 Photon-photon scattering in massless and massive gravity 457
9.5 Conserved charges in generic gravity 458
9.5.1 Mass and angular momenta of Kerr-AdS black holes in n dimensions 465
9.5.2 Conserved charges in quadratic gravity in AdS 469
9.6 Miscellaneous issues about conserved charges 473
9.6.1 Conserved charges of f(Riemann) theories 474
9.6.2 Conserved charges of topologically massive gravity 477
9.6.3 Conserved charges from the symplectic structure for generic backgrounds 479
9.6.4 Generic scalar-tensor theory in n dimensions 487
10 Conservation laws in covariant field theories with gauge symmetries 489
10.1 Conserved quantities in generally-covariant Yang-Mills theories 489
10.1.1 The Yang-Mills theories 489
10.1.2 Field equations and the Noether current 492
10.1.3 Conserved quantities corresponding to the gauge invariance 495
10.1.4 Conserved quantities corresponding to the diffeomorphism invariance 497
10.1.5 Modified Lie derivative 500
10.1.6 Commutator of the modified variations Δ ξ in d-t basis 503
10.1.7 Commutator of the modified variations Δ ξ in D-t basis 506
10.1.8 The functoriatity condition 508
10.2 Conservation laws in the tetrad formalism of general relativity 510
10.2.1 Tetrads and gravitational field 510
10.2.2 Connections and derivatives 512
10.2.3 Variation of the Hilbert action 514
10.2.4 The Noether current 517
10.2.5 Conserved quantities corresponding to the local Lorentz invariance 519
10.2.6 Conserved quantities corresponding to the diffeomorphism invariance 519
10.2.7 The Kosmann lift and the Komar superpotential 521
10.3 Fiber bundles and the Noether theorem 525
10.3.1 Diffeomorphisms, automorphisms and functorial lift 525
10.3.2 Field theories without intrinsic gauge symmetry as natural field theories 527
10.3.3 Field theories with intrinsic gauge symmetry as gauge-natural field theories 528
10.3.4 Fixing the horizontal lift 531
Appendix A Tensor quantities and tensor operations 533
A.1 Tensors and tensor densities 533
A.2 Derivatives 538
A.2.1 Covariant derivatives and the Christoffel symbols 538
A.2.2 The curvature tensor 540
A.2.3 Lie derivative 540
A.2.4 Variational and Lagrangian derivatives 542
A.3 Introduction to economic tensor operations 546
A.3.1 Economic index notations 546
A.3.2 Algebra of economic index notations 548
A.3.3 Covariant expressions 549
Appendix B Retarded functions 553
B.1 Lorentz invariance of retarded potentials 553
B.2 Retarded solution of the sound-wave equation 556
Appendix C Auxiliary expressions in EGB gravity 561
Bibliography 567
Index 589