Noether symmetries in theories of gravity : with applications to astrophysics and cosmology / Francesco Bajardi, Salvatore Capozziello.

Format:

Book

Author/Creator:

Bajardi, Francesco, author.

Capozziello, Salvatore, author.

Series:

Cambridge monographs on mathematical physics.

Cambridge monographs on mathematical physics

Language:

English

Subjects (All):

General relativity (Physics).

Noether's theorem.

Astrophysics.

Cosmology.

Physical Description:

1 online resource (xxiii, 426 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge : Cambridge University Press, 2023.

Summary:

This volume summarizes the many alternatives and extensions to Einstein's General Theory of Relativity, and shows how symmetry principles can be applied to identify physically viable models. The first part of the book establishes the foundations of classical field theory, providing an introduction to symmetry groups and the Noether theorems. A quick overview of general relativity is provided, including discussion of its successes and shortcomings, then several theories of gravity are presented and their main features are summarized. In the second part, the 'Noether Symmetry Approach' is applied to theories of gravity to identify those which contain symmetries. In the third part of the book these selected models are tested through comparison with the latest experiments and observations. This constrains the free parameters in the selected models to fit the current data, demonstrating a useful approach that will allow researchers to construct and constrain modified gravity models for further applications.

Contents:

Cover

Half-title page

Series page

Title page

Copyright page

Dedication

Contents

Preface

Acknowledgments

Amalie Emmy Noether: A Life for Mathematics

Notation

Acronyms

Part I Preliminaries

1 The Concept of Symmetry

1.1 Symmetries in Physics

1.1.1 The Unitary Group

1.1.2 The Translation Group

1.1.3 The Rotation Group

1.1.4 The Lorentz Group

1.1.5 The Poincaré Group

2 The Two Noether Theorems

2.1 Noether's First Theorem

2.1.1 First Demonstration

2.1.2 Second Demonstration

2.1.3 Internal Symmetries

2.2 Noether's Second Theorem

2.3 The Lie Derivative: Applications and Beyond

2.4 The Noether-Bessel-Hagen Theorem: Symmetries of Equations of Motion

3 Applications of Noether's First Theorem to Fields and Particles

3.1 The Free Scalar Field: The U(1) Gauge Invariance

3.2 Invariance under Translations and Rotations

3.2.1 Space-Time Translations

3.2.2 Rotations

3.3 The Lorentz Invariance

3.3.1 The Lorentz Invariance for the Electromagnetic Field

3.3.2 The Gauge Invariance for the Electromagnetic Field

3.4 The Spontaneous Symmetry Breaking

3.5 The Noether Theorem for Particles

3.5.1 Free Particle

3.5.2 Harmonic Oscillator

4 Theories of Gravity: An Overview

4.1 An Overview of General Relativity: Successes and Shortcomings

4.2 Different Viewpoints in Theories of Gravity

4.3 Curvature Extensions

4.4 Scalar-Tensor Gravity

4.5 The Palatini Formalism

4.6 Teleparallel Equivalent of General Relativity and the Extension to f(T) Gravity

4.7 Symmetric Teleparallel Equivalent of General Relativity

4.8 The Geometric Trinity of Gravity

5 Toward Quantum Gravity

5.1 Quantum Cosmology: Everything from Nothing

5.2 Gauge Theories of Gravity

Part II The Noether Symmetry Approach.

6 From the Noether Theorem to the Noether Symmetry Approach

6.1 The First Noether Theorem for Canonical Lagrangians

6.1.1 Internal Symmetries

6.2 Particle Lagrangian with Unknown Potential

6.3 Application to the Point like Canonical Lagrangian

6.4 The Noether Symmetry Approach for Theories of Gravity

7 The Extensions of GR, TEGR, and STEGR

7.1 GR Extension: The Case of f(R) Gravity

7.2 Teleparallel Extension: The Case of f(T) Gravity

7.2.1 TEGR Extension with Boundary Term: f(T,B) Gravity

7.2.2 f(R,T) Gravity

7.3 Geometric Extensions of STEGR

8 Higher-Order Extensions with the Gauss-Bonnet Invariant

8.1 f(R, G) Gravity

8.1.1 R+ f(G) Gravity

8.1.2 f(G) Gravity

8.2 Teleparallel Equivalent of Gauss-Bonnet Gravity

9 Extensions with Higher Derivatives of R and T

9.1 f(R,□R) Gravity

9.2 f(T,□T) Gravity

10 Scalar-Tensor Theories of Gravity

10.1 The Curvature Scalar Coupled to a Scalar Field

10.1.1 Generalization to Higher Dimensions

10.1.2 Conformal Transformations

10.2 Hybrid Gravity

10.3 The Torsion Scalar Coupled to a Scalar Field

10.4 The Gauss-Bonnet Invariant Coupled to a Scalar Field

10.5 Equivalence among Scalar-Tensor Theories by Noether Symmetries

10.6 Horndeski Gravity

11 Nonlocal Gravity

11.1 Nonlocality in Physics

11.2 Infinite Derivatives Theories of Gravity

11.3 Integral Kernel Theories of Gravity

11.4 Nonlocality with Curvature

11.5 Nonlocality with Gauss-Bonnet Scalar

11.5.1 f(G,□[sup(-1)] G) Gravity

11.5.2 General Relativity with Nonlocal Gauss-Bonnet Corrections

11.6 Nonlocality with Torsion

12 Noether Symmetries in Bianchi Universes

12.1 The Bianchi Classification of Space-Times

12.2 Noether Symmetries in Bianchi Space-Times

12.3 Results

12.4 Examples of Exact Integration.

13 The Noether Approach in Spherical Symmetry

13.1 Spherical Symmetry in f(R) Gravity

13.1.1 Axial Symmetry from Spherical Symmetry

13.2 Spherical Symmetry in f(T,B) Gravity

13.3 Spherical Symmetry in f(G) Gravity

Part III Applications

14 Applications to Solar System, Stars, and Our Galaxy

14.1 Solar System Tests in Modified Gravity

14.1.1 Solar System Constraints in f(R) Gravity

14.2 Mass-Radius Relation of Neutron Stars in f(R) Gravity

14.3 Gravitational Lensing in f(R) Gravity

14.4 Constraining Nonlocal Gravity by S2 Star Orbit

15 Applications to Galaxies

15.1 The Missing Matter Problem by f(R) Gravity

15.2 The Fundamental Plane by f(R) Gravity

16 Applications to Cosmology

16.1 Inflation and Cosmological Perturbations in Scalar-Tensor Gravity

16.2 Inflation in f(R,G) Gravity

16.2.1 Inflation in f(G) Gravity

16.2.2 Inflation in {R+ f(G)} Gravity

16.2.3 Inflation in f(R) Gravity

16.3 Generalized Energy Conditions and Cosmology

16.3.1 Energy Conditions in f(G) Cosmology

16.3.2 Energy Conditions in {R + f(G)} Cosmology

16.3.3 Energy Conditions in f(R) Cosmology

16.4 Geometric Quintessence

16.4.1 Quintessence in f(R) Gravity

16.4.2 Quintessence in f(R,G) Gravity

16.4.3 Quintessence in Extended Teleparallel Gravity

17 Applications to Quantum Cosmology

17.1 Quantum Cosmology in f(R) Gravity

17.2 Quantum Cosmology in f(T) Gravity

17.3 Quantum Cosmology in f(G) Gravity

17.4 Quantum Cosmology in Higher-Derivatives Gravity

17.4.1 Higher-Order Teleparallel Equivalent

17.5 Quantum Cosmology in Scalar-Tensor Gravity

17.5.1 Curvature Scalar-Tensor Gravity

17.5.2 Torsion Scalar-Tensor Gravity

17.5.3 Equivalence of R, T, and G Scalar-Tensor Gravity via Hamiltonian Dynamics

18 Strings, Swampland, Renormalizability, and Viability.

18.1 String-Dilaton Cosmology and Scale Factor Duality

18.1.1 Noether Symmetry for String-Dilaton Lagrangian

18.1.2 The Wave Function of the Universe for String-Dilaton Cosmology

18.2 String-Dilaton Cosmology and Swampland Conjecture

18.2.1 The Swampland Conjecture in f(R) Gravity

18.3 Renormalizability

18.4 Viability Conditions

18.4.1 Weak Field Limit and Solar System Tests

18.4.2 Cosmological Dynamics

18.4.3 Instabilities and Ghosts

18.4.4 Cosmological Perturbations

18.4.5 The Cauchy Problem

Epilogue

Appendices

A: Variational Principles

B: Differential Forms and Variations of Gauge Actions

C: Noether-Bessel-Hagen Symmetries in Scalar-Tensor Cosmology

References

Index.

Notes:

Title from publisher's bibliographic system (viewed on 10 Nov 2022).

Includes bibliographical references and index.

ISBN:

1-009-20873-X

1-009-20872-1