Metric Theories of Gravity : Perturbations and Conservation Laws.
- 1st ed.
- 1 online resource (622 pages)
- De Gruyter Studies in Mathematical Physics Series ; v.38 .
- De Gruyter Studies in Mathematical Physics Series .
Intro -- Contents -- List of Figures -- Primary notations -- 1. Conservation laws in theoretical physics: A brief introduction -- 1.1 Conserved quantities in classical mechanics -- 1.1.1 Some basic notions of non-relativistic classical mechanics -- 1.1.2 The least action principle -- 1.1.3 Noether's theorem in classical mechanics -- 1.1.4 Conserved quantities for a system of non-relativistic particles -- 1.1.5 The Minkowski space and the Poincaré group -- 1.1.6 A point-like particle in special relativity -- 1.1.7 Conserved quantities for a system of relativistic particles -- 1.2 Field theory in the Minkowski space -- 1.2.1 The action -- 1.2.2 Variational field equations -- 1.2.3 The Noether theorems -- 1.2.4 Conserved quantities in field theories -- 1.2.5 Examples of field theories in the Minkowski space -- 1.3 General relativity: fundamental mathematical relations -- 1.3.1 Lagrangians for the gravitational sector of general relativity -- 1.3.2 The Einstein equations -- 1.4 Classical conserved quantities in general relativity -- 1.4.1 The third Noether's theorem -- 1.4.2 Pseudotensors and superpotentials -- 1.5 Applications -- 1.5.1 Linearized general relativity -- 1.5.2 Weak gravitational waves in general relativity -- 1.5.3 The energy of an isolated gravitating system in general relativity -- 2. Field-theoretical formulation of general relativity: The theory -- 2.1 Development of the field-theoretical formulation -- 2.1.1 Geometrical formalism and field theories -- 2.1.2 Earlier perturbative formulations of general relativity -- 2.1.3 Deser's field-theoretical model -- 2.1.4 Various methods of the construction -- 2.2 The field-theoretical formulation of general relativity -- 2.2.1 A dynamical Lagrangian -- 2.2.2 The Einstein equations in the field-theoretical formulation -- 2.2.3 Functional expansions. 2.2.4 Gauge transformations and their properties -- 2.2.5 Differential conservation laws -- 2.2.6 Different variants of the field-theoretical formulation in general relativity -- 2.2.7 The background as an auxiliary structure -- 2.3 Metric perturbations as compensating fields -- 2.3.1 "Localization" of background Killing vectors -- 2.3.2 The total action -- 2.3.3 Discussion of the results -- 2.4 The Babak-Grishchuk gravity with a non-zero graviton mass -- 2.4.1 Second derivatives in the energy-momentum tensor -- 2.4.2 Modified Lagrangian and equations -- 2.4.3 Non-zero masses of gravitons -- 2.4.4 Black holes and cosmology in massive gravity -- 2.4.5 Gauge invariance in the Babak-Grishchuk modifications -- 3. Asymptotically flat spacetime in the field-theoretical formulation -- 3.1 The Arnowitt-Deser-Misner formulation of general relativity -- 3.1.1 The ADM action principle -- 3.1.2 Asymptotically flat spacetime at spatial infinity in general relativity -- 3.1.3 The ADM definition of conserved quantities -- 3.1.4 The Regge-Teitelboim modification -- 3.2 An isolated system in the Lagrangian description -- 3.2.1 Asymptotically flat spacetime as a field configuration -- 3.2.2 Global conserved quantities -- 3.2.3 The parity conditions -- 3.2.4 Gauge invariance of the motion integrals -- 3.2.5 Concluding remarks -- 3.3 An isolated system in the Hamiltonian description -- 3.3.1 The difference between the canonical and symmetric currents -- 3.3.2 Phase variables and their asymptotic behaviour -- 3.3.3 Global conserved integrals -- 3.3.4 Gauge invariance of global integrals -- 4. Exact solutions of general relativity in the field-theoretical formalism -- 4.1 The Schwarzschild solution -- 4.1.1 The total energy -- 4.1.2 The energy distribution for the Schwarzschild black hole -- 4.1.3 The Schwarzschild black hole as a point particle. 4.1.4 The Schwarzschild solution and the harmonic gauge fixing -- 4.2 Other exact solutions of general relativity -- 4.2.1 The Friedmann solution for a closed universe -- 4.2.2 The Abbott-Deser superpotential and its generalizations -- 4.2.3 The total mass of the Schwarzschild-AdS black hole -- 5. Field-theoretical derivation of cosmological perturbations -- 5.1 Introduction: Post-Newtonian, post-Minkowskian and post-Friedmanninan approximations in cosmology -- 5.2 Lagrangian and field variables -- 5.2.1 Action functional -- 5.2.2 Lagrangian of the ideal fluid -- 5.2.3 Lagrangian of scalar field -- 5.2.4 Lagrangian of a localized astronomical system -- 5.3 Background manifold -- 5.3.1 Hubble flow -- 5.3.2 Friedmann-Lemître-Robertson-Walker metric -- 5.3.3 Christoffel symbols and covariant derivatives -- 5.3.4 Riemann tensor -- 5.3.5 The Friedmann equations -- 5.3.6 Hydrodynamic equations of the ideal fluid -- 5.3.7 Scalar field equations -- 5.3.8 Equations of motion of matter of the localized astronomical system -- 5.4 Lagrangian perturbations of FLRW manifold -- 5.4.1 The concept of perturbations -- 5.4.2 The background field equations -- 5.4.3 The dynamic Lagrangian for perturbations -- 5.4.4 The Lagrangian equations for gravitational field perturbations -- 5.4.5 The Lagrangian equations for dark matter perturbations -- 5.4.6 The Lagrangian equations for dark energy perturbations -- 5.4.7 Linearized post-Newtonian equations for field variables -- 5.5 Gauge-invariant scalars and field equations in 1+3 threading formalism -- 5.5.1 Threading decomposition of the metric perturbations -- 5.5.2 Gauge transformation of the field variables -- 5.5.3 Gauge-invariant scalars -- 5.5.4 Field equations for the gauge-invariant scalar perturbations -- 5.5.5 Field equations for vector perturbations -- 5.5.6 Field equations for tensor perturbations. 5.5.7 Residual gauge freedom -- 5.6 Post-Newtonian field equations in a spatially-flat universe -- 5.6.1 Cosmological parameters and scalar field potential -- 5.6.2 Conformal cosmological perturbations -- 5.6.3 Post-Newtonian field equations in conformal spacetime -- 5.6.4 Residual gauge freedom in the conformal spacetime -- 5.7 Decoupled system of the post-Newtonian field equations -- 5.7.1 The universe governed by dark matter and cosmological constant -- 5.7.2 The universe governed by dark energy -- 5.7.3 Post-Newtonian potentials in the linearized Hubble approximation -- 6. Currents and superpotentials on arbitrary backgrounds: Three approaches -- 6.1 The Katz, Bi?cák and Lynden-Bell conservation laws -- 6.1.1 A bi-metric KBL Lagrangian -- 6.1.2 KBL conserved quantities -- 6.2 The Belinfante procedure -- 6.2.1 The Belinfante symmetrization in general relativity -- 6.2.2 The Belinfante method applied to the KBL model -- 6.3 Currents and superpotentials in the field-theoretical formulation -- 6.3.1 Noether's procedure applied to the field-theoretical model -- 6.3.2 A family of conserved quantities and the Boulware-Deser ambiguity -- 6.3.3 Comments on conserved quantities of three types -- 6.4 Criteria for the choice of conserved quantities -- 6.4.1 Tests of consistency -- 6.4.2 The Reissner-Nordström solution -- 6.4.3 The Kerr solution -- 6.4.4 The total KBL energy for the S-AdS solution -- 6.5 The FLRW solution as a perturbation on the de Sitter background -- 6.5.1 Spatially conformal mappings of FLRW spacetime onto de Sitter space -- 6.5.2 Superpotentials and conserved currents -- 6.6 Integral constraints for linear perturbations on FLRW backgrounds -- 6.6.1 A FLRW background and its conformal Killing vectors -- 6.6.2 Integral relations for linear perturbations -- 6.6.3 Possible applications. 7. Conservation laws in an arbitrary multi-dimensional metric theory -- 7.1 Covariant Noether's procedure in an arbitrary field theory -- 7.1.1 Covariant identities and identically conserved quantities -- 7.1.2 Another variant of covariantization -- 7.1.3 A new family of the identically conserved covariant Nother quantities -- 7.1.4 A Belinfante corrected family of identically conserved quantities -- 7.2 Conservation laws for perturbations: Three approaches -- 7.2.1 An arbitrary metric theory in n dimensions -- 7.2.2 Canonical conserved quantities for perturbations -- 7.2.3 The Belinfante corrected conserved quantities -- 7.2.4 The field-theoretical formulation for perturbations -- 7.2.5 Currents and superpotentials in the field-theoretical formulation -- 8. Conserved quantities in the Einstein-Gauss-Bonnet gravity -- 8.1 Superpotentials and currents in the EGB gravity -- 8.1.1 Action and field equations in the EGB gravity -- 8.1.2 Three types of superpotentials -- 8.1.3 Three types of currents -- 8.2 Conserved charges in the EGB gravity -- 8.2.1 Charges for isolated systems -- 8.2.2 Superpotentials for static spherically symmetric solutions -- 8.2.3 Mass of the Schwarzschild-AdS black hole -- 8.3 Interpretation of the Maeda-Dadhich exotic solutions -- 8.3.1 Kaluza-Klein type 3D black holes -- 8.3.2 Mass for the static Maeda-Dadhich objects -- 8.3.3 Mass and mass flux for the radiative Maeda-Dadhich objects -- 9. Generic gravity: Particle content, weak field limits, conserved charges -- 9.1 Introduction: Raisons d'être of modified gravity theory -- 9.1.1 Conventions -- 9.2 Particle spectrum and stability of vacuum in quadratic gravity -- 9.2.1 Curvature tensors at second order in perturbation theory -- 9.2.2 Field equations and the vacuum structure -- 9.2.3 Linearization of quadratic gravity -- 9.2.4 Explicit check of linearized Bianchi identity. 9.2.5 Degrees of freedom of quadratic gravity in AdS.
9783110351781
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