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

9.3 Particle spectrum of f (R,-31) gravity in (A)dS

9.3.1 Linearization of the field equations

9.3.2 Lovelock gravity

9.3.3 Propagator structure of the Lovelock theory

9.4 Weak field limits: Potential energy from tree-level gravitons

9.4.1 Potential energy from the scattering amplitude

9.4.2 Decomposition of the graviton field and tree-level scattering amplitude

9.4.3 van Dam-Veltman-Zakharov discontinuity

9.4.4 New massive gravity redux

9.4.5 Spin-spin, spin-orbit, orbit-orbit interactions

9.4.6 Gravitomagnetic effects in general relativity

9.4.7 Gravitomagnetic effects in massive gravity

9.4.8 Gravitomagnetic effects in quadratic gravity

9.4.9 Photon-photon scattering in massless and massive gravity

9.5 Conserved charges in generic gravity

9.5.1 Mass and angular momenta of Kerr-AdS black holes in n dimensions

9.5.2 Conserved charges in quadratic gravity in AdS

9.6 Miscellaneous issues about conserved charges

9.6.1 Conserved charges of f (Riemann) theories

9.6.2 Conserved charges of topologically massive gravity

9.6.3 Conserved charges from the symplectic structure for generic backgrounds

9.6.4 Generic scalar-tensor theory in n dimensions

10. Conservation laws in covariant field theories with gauge symmetries

10.1 Conserved quantities in generally-covariant Yang–Mills theories

10.1.1 The Yang–Mills theories

10.1.2 Field equations and the Noether current

10.1.3 Conserved quantities corresponding to the gauge invariance

10.1.4 Conserved quantities corresponding to the diffeomorphism invariance

10.1.5 Modified Lie derivative

10.1.6 Commutator of the modified variations B. in ??-t basis

10.1.7 Commutator of the modified variations B. in D-t basis

10.1.8 The functoriality condition

10.2 Conservation laws in the tetrad formalism of general relativity

10.2.1 Tetrads and gravitational field

10.2.2 Connections and derivatives

10.2.3 Variation of the Hilbert action

10.2.4 The Noether current

10.2.5 Conserved quantities corresponding to the local Lorentz invariance

10.2.6 Conserved quantities corresponding to the diffeomorphism invariance

10.2.7 The Kosmann lift and the Komar superpotential

10.3 Fiber bundles and the Noether theorem

10.3.1 Diffeomorphisms, automorphisms and functorial lift

10.3.2 Field theories without intrinsic gauge symmetry as natural field theories

10.3.3 Field theories with intrinsic gauge symmetry as gauge-natural field theories

10.3.4 Fixing the horizontal lift

Appendix A: Tensor quantities and tensor operations

A.1 Tensors and tensor densities

A.2 Derivatives

A.2.1 Covariant derivatives and the Christoffel symbols

A.2.2 The curvature tensor

A.2.3 Lie derivative

A.2.4 Variational and Lagrangian derivatives

A.3 Introduction to economic tensor operations

A.3.1 Economic index notations

A.3.2 Algebra of economic index notations

A.3.3 Covariant expressions

Appendix B: Retarded functions

B.1 Lorentz invariance of retarded potentials

B.2 Retarded solution of the sound-wave equation

Appendix C: Auxiliary expressions in EGB gravity

Bibliography

Index