Noether symmetries in theories of gravity: with applications to astrophysics and cosmology
This volume summarizes the many modified theories of gravity and shows how to select physically viable models using symmetry principles
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge ; New York ; Port Melbourne ; New Delhi ; Singapore
Cambridge University Press
2023
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| Series: | Cambridge monographs on mathematical physics
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | This volume summarizes the many modified theories of gravity and shows how to select physically viable models using symmetry principles |
| Physical Description: | xxiii, 426 Seiten Illustrationen, Diagramme |
| ISBN: | 9781009208741 |
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| adam_text | Contents Preface Acknowledgments Amalie Emmy Noether: A Life for Mathematics Notation Acronyms page xv xviii xix xxii xxiv Part I PRELIMINARIES 1 1 The Concept of Symmetry 3 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 5 11 12 12 13 15 2 The Two Noether Theorems 16 2.1 Noether’s First Theorem 2.1.1 First Demonstration 2.1.2 Second Demonstration 2.1.3 Internal Symmetries Noether’s Second Theorem The Lie Derivative: Applications and Beyond The Noether-Bessel-Hagen Theorem: Symmetries of Equations of Motion 28 17 17 19 21 23 25 2.2 2.3 2.4 3 Applications of Noether’s First Theorem to Fields and Particles 34 3.1 3.2 The Free Scalar Field: The [7(1) Gauge Invariance Invariance under Translations and Rotations 3.2.1 Space-Time Translations 3.2.2 Rotations The Lorentz Invariance 34 36 36 38 40 3.3
x Contents 3.4 3.5 3.3.1 The Lorentz Invariance for the Electromagnetic Field 3.3.2 The Gauge Invariance for the Electromagnetic Field The Spontaneous Symmetry Breaking The Noether Theorem for Particles 3.5.1 Free Particle 3.5.2 Harmonic Oscillator 41 41 43 45 46 46 4 Theories of Gravity: An Overview 48 4.1 4.2 4.3 4.4 4.5 4.6 48 61 66 70 73 4.7 4.8 An Overview of General Relativity: Successesand Shortcomings Different Viewpoints in Theories of Gravity Curvature Extensions Scalar-Tensor Gravity The Palatini Formalism Teleparallel Equivalent of General Relativity and the Extension to ƒ( T) Gravity Symmetric Teleparallel Equivalent of GeneralRelativity The Geometric Trinity of Gravity 5 Toward Quantum Gravity 88 5.1 5.2 Quantum Cosmology: Everything from Nothing Gauge Theories of Gravity 94 99 Part II THE NOETHER SYMMETRY APPROACH 75 81 84 105 6 From the Noether Theorem to the Noether Symmetry Approach 107 6.1 6.2 6.3 6.4 The First Noether Theorem for Canonical Lagrangians 6.1.1 Internal Symmetries Particle Lagrangian with Unknown Potential Application to the Point like Canonical Lagrangian The Noether Symmetry Approach for Theories of Gravity 109 110 112 116 117 7 The Extensions of GR, TEGR, and STEGR 123 7.1 7.2 7.3 GR Extension: The Case of ƒ (R) Gravity Teleparallel Extension: The Case of ƒ ( T) Gravity 7.2.1 TEGR Extension with Boundary Term: f(T, B) Gravity 7.2.2 f(R, T) Gravity Geometric Extensions of STEGR 123 130 132 135 138 8 Higher-Order Extensions with the Gauss—Bonnet Invariant 143 8.1 ДЛ,.?) Gravity 8.1.1 R + ք(Ձ) Gravity 8.1.2 Л50 Gravity Teleparallel
Equivalent of Gauss-Bonnet Gravity 8.2 146 153 154 160
xi Contents 9 9.1 9.2 Extensions with Higher Derivatives of Rand T f{R, ĽLR) Gravity /(Τ,ΠΤ) 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 165 166 170 174 175 180 184 185 187 191 195 198 11.6 Nonlocal Gravity Nonlocality in Physics Infinite Derivatives Theories of Gravity Integral Kernel Theories of Gravity Nonlocality with Curvature Nonlocality with Gauss-Bonnet Scalar 11.5.1 Д^, ü՜1^ Gravity 11.5.2 General Relativity with Nonlocal Gauss-Bonnet Corrections Nonlocality with Torsion 12 12.1 12.2 12.3 12.4 Noether Symmetries in Bianchi Universes The Bianchi Classification of Space-Times Noether Symmetries in Bianchi Space-Times Results Examples of Exact Integration 226 227 230 233 236 13 13.1 The Noether Approach in Spherical Symmetry Spherical Symmetry in f(R) Gravity 13.1.1 Axial Symmetry from Spherical Symmetry Spherical Symmetry in f(T, B) Gravity Spherical Symmetry in /(5) Gravity 238 240 246 248 254 11 11.1 11.2 11.3 11.4 11.5 13.2 13.3 Part HI APPLICATIONS 14 14.1 14.2 Applications to Solar System, Stars, andOur Galaxy Solar System Tests in Modified Gravity 14.1.1 Solar System Constraints in fiR) Gravity Mass-Radius Relation of Neutron Stars in ƒ (R) Gravity 202 202 204 206 207 213 213 219 220 263 265 267 273 277
xii Contents 14.3 14.4 Gravitational Lensing in ƒ (R) Gravity Constraining Nonlocal Gravity by S2 Star Orbit 281 287 15 Applications to Galaxies 294 15.1 15.2 The Missing Matter Problem by f(R) Gravity The Fundamental Plane by ƒ (R) Gravity 295 303 16 Applications to Cosmology 310 Inflation and Cosmological Perturbations in Scalar-Tensor Gravity 16.2 Inflation in f(R, 3) Gravity 16.2.1 Inflation in ƒ(£) Gravity 16.2.2 Inflation in {R + ƒ(5)} Gravity 16.2.3 Inflation in ƒ (R) Gravity 16.3 Generalized Energy Conditions and Cosmology 16.3.1 Energy Conditions in f(S) Cosmology 16.3.2 Energy Conditions in {R + f(3j} 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, S í Gravity 16.4.3 Quintessence in Extended Teleparallel Gravity 16.1 311 316 318 319 320 320 321 326 328 329 330 331 333 17 Applications to Quantum Cosmology 334 17.1 17.2 17.3 17.4 Quantum Cosmology in ƒ(R) Gravity Quantum Cosmology in f(T) Gravity Quantum Cosmology in 1(3) Gravity Quantum Cosmology inHigher-Derivatives Gravity 17.4.1 Higher-Order Teleparallel Equivalent 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 3 Scalar-Tensor Gravity via Hamiltonian Dynamics 348 335 337 339 341 343 345 345 346 18 Strings, Swampland, Renormalizability, and Viability 351 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 String-Dilaton Cosmology and Swampland Conjecture 18.2.1 The Swampland Conjecture in f(R) Gravity Renormalizability 352 356 17.5 18.2 18.3 358 360 361 364
Contents xiii 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 368 369 370 371 372 373 EPILOGUE 376 APPENDICES 378 A: Variational Principles 378 B: Differential Forms and Variations of Gauge Actions 388 C: Noether-Bessel-Hagen Symmetries in Scalar-Tensor Cosmology 394 References Index 398 424
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| adam_txt |
Contents Preface Acknowledgments Amalie Emmy Noether: A Life for Mathematics Notation Acronyms page xv xviii xix xxii xxiv Part I PRELIMINARIES 1 1 The Concept of Symmetry 3 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 5 11 12 12 13 15 2 The Two Noether Theorems 16 2.1 Noether’s First Theorem 2.1.1 First Demonstration 2.1.2 Second Demonstration 2.1.3 Internal Symmetries Noether’s Second Theorem The Lie Derivative: Applications and Beyond The Noether-Bessel-Hagen Theorem: Symmetries of Equations of Motion 28 17 17 19 21 23 25 2.2 2.3 2.4 3 Applications of Noether’s First Theorem to Fields and Particles 34 3.1 3.2 The Free Scalar Field: The [7(1) Gauge Invariance Invariance under Translations and Rotations 3.2.1 Space-Time Translations 3.2.2 Rotations The Lorentz Invariance 34 36 36 38 40 3.3
x Contents 3.4 3.5 3.3.1 The Lorentz Invariance for the Electromagnetic Field 3.3.2 The Gauge Invariance for the Electromagnetic Field The Spontaneous Symmetry Breaking The Noether Theorem for Particles 3.5.1 Free Particle 3.5.2 Harmonic Oscillator 41 41 43 45 46 46 4 Theories of Gravity: An Overview 48 4.1 4.2 4.3 4.4 4.5 4.6 48 61 66 70 73 4.7 4.8 An Overview of General Relativity: Successesand Shortcomings Different Viewpoints in Theories of Gravity Curvature Extensions Scalar-Tensor Gravity The Palatini Formalism Teleparallel Equivalent of General Relativity and the Extension to ƒ( T) Gravity Symmetric Teleparallel Equivalent of GeneralRelativity The Geometric Trinity of Gravity 5 Toward Quantum Gravity 88 5.1 5.2 Quantum Cosmology: Everything from Nothing Gauge Theories of Gravity 94 99 Part II THE NOETHER SYMMETRY APPROACH 75 81 84 105 6 From the Noether Theorem to the Noether Symmetry Approach 107 6.1 6.2 6.3 6.4 The First Noether Theorem for Canonical Lagrangians 6.1.1 Internal Symmetries Particle Lagrangian with Unknown Potential Application to the Point like Canonical Lagrangian The Noether Symmetry Approach for Theories of Gravity 109 110 112 116 117 7 The Extensions of GR, TEGR, and STEGR 123 7.1 7.2 7.3 GR Extension: The Case of ƒ (R) Gravity Teleparallel Extension: The Case of ƒ ( T) Gravity 7.2.1 TEGR Extension with Boundary Term: f(T, B) Gravity 7.2.2 f(R, T) Gravity Geometric Extensions of STEGR 123 130 132 135 138 8 Higher-Order Extensions with the Gauss—Bonnet Invariant 143 8.1 ДЛ,.?) Gravity 8.1.1 R + ք(Ձ) Gravity 8.1.2 Л50 Gravity Teleparallel
Equivalent of Gauss-Bonnet Gravity 8.2 146 153 154 160
xi Contents 9 9.1 9.2 Extensions with Higher Derivatives of Rand T f{R, ĽLR) Gravity /(Τ,ΠΤ) 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 165 166 170 174 175 180 184 185 187 191 195 198 11.6 Nonlocal Gravity Nonlocality in Physics Infinite Derivatives Theories of Gravity Integral Kernel Theories of Gravity Nonlocality with Curvature Nonlocality with Gauss-Bonnet Scalar 11.5.1 Д^, ü՜1^ Gravity 11.5.2 General Relativity with Nonlocal Gauss-Bonnet Corrections Nonlocality with Torsion 12 12.1 12.2 12.3 12.4 Noether Symmetries in Bianchi Universes The Bianchi Classification of Space-Times Noether Symmetries in Bianchi Space-Times Results Examples of Exact Integration 226 227 230 233 236 13 13.1 The Noether Approach in Spherical Symmetry Spherical Symmetry in f(R) Gravity 13.1.1 Axial Symmetry from Spherical Symmetry Spherical Symmetry in f(T, B) Gravity Spherical Symmetry in /(5) Gravity 238 240 246 248 254 11 11.1 11.2 11.3 11.4 11.5 13.2 13.3 Part HI APPLICATIONS 14 14.1 14.2 Applications to Solar System, Stars, andOur Galaxy Solar System Tests in Modified Gravity 14.1.1 Solar System Constraints in fiR) Gravity Mass-Radius Relation of Neutron Stars in ƒ (R) Gravity 202 202 204 206 207 213 213 219 220 263 265 267 273 277
xii Contents 14.3 14.4 Gravitational Lensing in ƒ (R) Gravity Constraining Nonlocal Gravity by S2 Star Orbit 281 287 15 Applications to Galaxies 294 15.1 15.2 The Missing Matter Problem by f(R) Gravity The Fundamental Plane by ƒ (R) Gravity 295 303 16 Applications to Cosmology 310 Inflation and Cosmological Perturbations in Scalar-Tensor Gravity 16.2 Inflation in f(R, 3) Gravity 16.2.1 Inflation in ƒ(£) Gravity 16.2.2 Inflation in {R + ƒ(5)} Gravity 16.2.3 Inflation in ƒ (R) Gravity 16.3 Generalized Energy Conditions and Cosmology 16.3.1 Energy Conditions in f(S) Cosmology 16.3.2 Energy Conditions in {R + f(3j} 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, S'í Gravity 16.4.3 Quintessence in Extended Teleparallel Gravity 16.1 311 316 318 319 320 320 321 326 328 329 330 331 333 17 Applications to Quantum Cosmology 334 17.1 17.2 17.3 17.4 Quantum Cosmology in ƒ(R) Gravity Quantum Cosmology in f(T) Gravity Quantum Cosmology in 1(3) Gravity Quantum Cosmology inHigher-Derivatives Gravity 17.4.1 Higher-Order Teleparallel Equivalent 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 3 Scalar-Tensor Gravity via Hamiltonian Dynamics 348 335 337 339 341 343 345 345 346 18 Strings, Swampland, Renormalizability, and Viability 351 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 String-Dilaton Cosmology and Swampland Conjecture 18.2.1 The Swampland Conjecture in f(R) Gravity Renormalizability 352 356 17.5 18.2 18.3 358 360 361 364
Contents xiii 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 368 369 370 371 372 373 EPILOGUE 376 APPENDICES 378 A: Variational Principles 378 B: Differential Forms and Variations of Gauge Actions 388 C: Noether-Bessel-Hagen Symmetries in Scalar-Tensor Cosmology 394 References Index 398 424 |
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| spelling | Bajardi, Francesco ca. 20./21. Jh. Verfasser (DE-588)1278625879 aut Noether symmetries in theories of gravity with applications to astrophysics and cosmology Francesco Bajardi, Scuola Superiore Meridionale di Napoli, Salvatore Capozziello, Università degli Studi di Napoli "Federico II" Cambridge ; New York ; Port Melbourne ; New Delhi ; Singapore Cambridge University Press 2023 ©2023 xxiii, 426 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Cambridge monographs on mathematical physics This volume summarizes the many modified theories of gravity and shows how to select physically viable models using symmetry principles Kosmologie (DE-588)4114294-9 gnd rswk-swf Gravitationstheorie (DE-588)4158117-9 gnd rswk-swf Symmetrie (DE-588)4058724-1 gnd rswk-swf Gravitationstheorie (DE-588)4158117-9 s Symmetrie (DE-588)4058724-1 s Kosmologie (DE-588)4114294-9 s DE-604 Capozziello, Salvatore ca. 20./21. Jh. Verfasser (DE-588)142835277 aut Erscheint auch als Online-Ausgabe 978-1-00-920873-4 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034254521&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bajardi, Francesco ca. 20./21. Jh Capozziello, Salvatore ca. 20./21. Jh Noether symmetries in theories of gravity with applications to astrophysics and cosmology Kosmologie (DE-588)4114294-9 gnd Gravitationstheorie (DE-588)4158117-9 gnd Symmetrie (DE-588)4058724-1 gnd |
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| title | Noether symmetries in theories of gravity with applications to astrophysics and cosmology |
| title_auth | Noether symmetries in theories of gravity with applications to astrophysics and cosmology |
| title_exact_search | Noether symmetries in theories of gravity with applications to astrophysics and cosmology |
| title_exact_search_txtP | Noether symmetries in theories of gravity with applications to astrophysics and cosmology |
| title_full | Noether symmetries in theories of gravity with applications to astrophysics and cosmology Francesco Bajardi, Scuola Superiore Meridionale di Napoli, Salvatore Capozziello, Università degli Studi di Napoli "Federico II" |
| title_fullStr | Noether symmetries in theories of gravity with applications to astrophysics and cosmology Francesco Bajardi, Scuola Superiore Meridionale di Napoli, Salvatore Capozziello, Università degli Studi di Napoli "Federico II" |
| title_full_unstemmed | Noether symmetries in theories of gravity with applications to astrophysics and cosmology Francesco Bajardi, Scuola Superiore Meridionale di Napoli, Salvatore Capozziello, Università degli Studi di Napoli "Federico II" |
| title_short | Noether symmetries in theories of gravity |
| title_sort | noether symmetries in theories of gravity with applications to astrophysics and cosmology |
| title_sub | with applications to astrophysics and cosmology |
| topic | Kosmologie (DE-588)4114294-9 gnd Gravitationstheorie (DE-588)4158117-9 gnd Symmetrie (DE-588)4058724-1 gnd |
| topic_facet | Kosmologie Gravitationstheorie Symmetrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034254521&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bajardifrancesco noethersymmetriesintheoriesofgravitywithapplicationstoastrophysicsandcosmology AT capozziellosalvatore noethersymmetriesintheoriesofgravitywithapplicationstoastrophysicsandcosmology |