General relativity and its applications: black holes, compact stars and gravitational waves
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Boca Raton ; London ; New York
CRC Press, Taylor & Francis Group
2021
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| Ausgabe: | First edition |
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| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | Literaturverzeichnis Seite 461-467 |
| Beschreibung: | xviii, 426 Seiten Illustrationen, Diagramme Breite 178 mm, Hoehe 254 mm |
| ISBN: | 9780367625320 9781138589773 |
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Datensatz im Suchindex
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| adam_text | Contents Preface xiii Notation and conventions Chapter 1.1 1 ■ Introduction xvii 1 NON-EUCLIDEAN GEOMETRIES 1 1.1.1 1.1.2 1.1.3 4 5 5 The metric tensor in different coordinate frames The Gaussian curvature Pseudo-Euclidean geometries and spacetime 1.2 NEWTONIAN THEORY AND ITS SHORTCOMINGS 6 1.3 THE ROLE OF THE EQUIVALENCE PRINCIPLE 9 1.4 GEODESIC EQUATIONS AS A CONSEQUENCE OF THE EQUIVALENCE PRINCIPLE 11 LOCALLY INERTIAL FRAMES 13 1.5 Chapter 2.1 2.2 2.3 2.4 2.5 2 ■ Elements of differential geometry 17 TOPOLOGICAL SPACES, MAPPING, MANIFOLDS 17 2.1.1 2.1.2 2.1.3 2.1.4 17 18 22 26 Topological spaces Mapping Manifolds and differentiable manifolds Diffeomorphisms VECTORS 27 2.2.1 2.2.2 27 29 The traditional definition of a vector A geometrical definition ONE-FORMS 39 2.3.1 2.3.2 39 43 One-forms as geometrical objects Vector fields and one-form fields TENSORS 46 2.4.1 2.4.2 46 51 Geometrical definition of a tensor Symmetries of a tensor THE METRIC TENSOR AND ITS PROPERTIES 54
vi ■ Contents Chapter 3 ■ Affine connection and parallel transport______________________ 63 3.1 THE COVARIANT DERIVATIVE OF VECTORS 63 3.2 THE COVARIANT DERIVATIVE OF SCALARS AND ONE-FORMS 67 3.3 SYMMETRIES OF CHRISTOFFEĽS SYMBOLS 68 3.4 TRANSFORMATION RULES FOR CHRISTOFFEĽS SYMBOLS 69 3.5 THE COVARIANT DERIVATIVE OF TENSORS 70 3.6 CHRISTOFFEĽS SYMBOLS IN TERMS OF THE METRIC TENSOR 72 3.7 PARALLEL TRANSPORT 76 3.7.1 Parallel transport of a vector along a closed path on a twosphere 78 3.8 GEODESIC EQUATION 81 3.9 FERMI COORDINATES 82 3.10 NON-COORDINATE BASES Chapter 84 4 ■ The curvature tensor______________________________ ________ 87 4.1 PARALLEL TRANSPORT ALONG A LOOP 87 4.2 SYMMETRIES OF THE RIEMANN TENSOR 91 4.3 THE RIEMANN TENSOR GIVES THE COMMUTATOR OF COVARIANT DERIVATIVES 91 4.4 THE BIANCHI IDENTITIES 92 4.5 THE EQUATION OF GEODESIC DEVIATION 92 Chapter 5 ■ The stress-energy tensor ______ 97 5.1 5.2 THE STRESS-ENERGY TENSOR IN FIAT SPACETIME IS Ta@ A TENSOR? 97 100 5.3 DOES 104 5.4 IS Γα% = 0 A CONSERVATION LAW? Chapter Ta? SATISFY A CONSERVATION LAW? 6 ■ The Einstein equations 107 109 6.1 GEODESIC EQUATIONS IN THE WEAK-FIELD, STATIONARY LIMIT 110 6.2 EINSTEIN S FIELD EQUATIONS 112 6.3 GAUGE INVARIANCE OF EINSTEIN S EQUATIONS 117 6.4 THE HARMONIC GAUGE 118 Chapter 7 ■ Einstein s equations and variational principles______ ________ 121 7.1 EULER-LAGRANGE S EQUATIONS IN SPECIAL RELATIVITY 121 7.2 EULER-LAGRANGE S EQUATIONS IN CURVED SPACETIME 122 7.3 EINSTEIN S EQUATIONS IN VACUUM 124 7.4 EINSTEIN S EQUATIONS WITH SOURCES 128 7.4.1 The stress-energy tensor in some
relevant cases 129 EINSTEIN S EQUATIONS IN THE PALATINI FORMALISM 131 7.5
Contents ■ vii Chapter 8 ■ Symmetries__________________________________________________ 135 8.1 KILLING VECTOR FIELDS 135 8.2 KILLING VECTOR FIELDS AND THE CHOICE OF COORDINATE SYS TEMS 139 KILLING VECTOR FIELDS AND CONSERVATION LAWS 142 8.3.1 8.3.2 142 143 8.3 8.4 HYPERSURFACE-ORTHOGONAL VECTOR FIELDS 8.4.1 8.4.2 8.5 Conserved quantities in geodesic motion Conserved quantities from the stress-energy tensor Frobenius’ theorem 145 Hypersurface-orthogonal vector fields and the choice of coor dinate systems 146 DIFFEOMORPHISM INVARIANCE OF GENERAL RELATIVITY Chapter 144 9 ■ The Schwarzschild solution 148 151 9.1 STATIC AND SPHERICALLY SYMMETRIC SPACETIMES 151 9.2 THE SCHWARZSCHILD SOLUTION 154 9.3 SINGULARITIES OF THE SCHWARZSCHILD SOLUTION 159 9.4 SPACELIKE, TIMELIKE, AND NULL HYPERSURFACES 160 9.4.1 162 9.5 SINGULARITIES IN GENERAL RELATIVITY 9.5.1 9.5.2 9.5.3 9.5.4 9.5.5 9.6 Constant radius hypersurfaces in Schwarzschild’s spacetime Geodesic completeness How to remove a coordinate singularity Extension of the Rindler spacetime Extension of the Schwarzschild spacetime Eddington-Finkeistein coordinates THE BIRKHOFF THEOREM Chapter 10· Geodesic 10.1 10.2 motion in Schwarzschild s spacetime A VARIATIONAL PRINCIPLE FOR GEODESIC MOTION EQUATIONS OF MOTION 163 163 164 166 170 176 178 181 181 182 10.3 THE CONSTANTS OF GEODESIC MOTION 185 10.4 ORBITS OF MASSLESS PARTICLES 188 10.5 ORBITS OF MASSIVE PARTICLES 190 10.6 RADIAL CAPTURE OF A MASSIVE PARTICLE 193 Chapter 11· 11.1 Kinematical tests of General Relativity GRAVITATIONAL SHIFT OF SPECTRAL LINES 11.1.1 11.1.2 Redshift of
spectral lines in the weak-field limit Redshift of spectral lines in a strong gravitational field 197 197 201 201 11.2 THE DEFLECTION OF LIGHT 203 11.3 PERIASTRON PRECESSION 207
viii ■ Contents 11.4 THE SHAPIRO TIME DELAY 11.5 THE SHADOW OF A BLACK HOLE 11.5.1 212 The accretion disk of a black hole 214 218 Chapter 12· Gravitational waves_________________________________________ 221 12.1 12.2 PERTURBATIVE APPROACH 221 GRAVITATIONAL WAVES AS PERTURBATIONS OF FLAT SPACETIME 223 12.3 HOW TO CHOOSE THE HARMONIC GAUGE 228 12.4 PLANE GRAVITATIONAL WAVES 229 12.5 THE TT GAUGE 230 12.6 12.7 12.8 HOW DOES A GRAVITATIONAL WAVE AFFECT THE MOTION OF A SINGLE PARTICLE GEODESIC DEVIATION INDUCED BY A GRAVITATIONAL WAVE GRAVITATIONAL WAVES AND MICHELSON INTERFEROMETERS Chapter 13· Gravitational waves 13.1 13.2 232 239 in the quadrupede approximation______ 243 THE WEAK-FIELD, SLOW-MOTION APPROXIMATION THE QUADRUPOLE FORMULA 13.2.1 13.2.2 13.2.3 232 The tensor-viriai theorem The quadrupole moment tensor Absence of monopolar and dipolar gravitational waves 243 245 245 247 247 13.3 HOW TO TRANSFORM TO THE TT GAUGE 248 13.4 GRAVITATIONAL WAVES EMITTED BY A HARMONIC OSCILLATOR 250 13.5 GRAVITATIONAL WAVE EMITTED BY A BINARY SYSTEM IN CIRCULAR ORBIT 252 13.6 ENERGY CARRIED BY A GRAVITATIONAL WAVE 13.6.1 13.6.2 The stress-energy pseudo-tensor of the gravitational field Energy flux of a gravitational wave Chapter 14· Gravitational wave sources______________________________ 257 257 261 267 14.1 EVOLUTION OF A COMPACT BINARY SYSTEM 14.2 GRAVITATIONAL WAVES FROM INSPIRALLING COMPACT OBJECTS 271 14.2.1 14.2.2 14.2.3 14.2.4 14.2.5 September 14th, 2015: the detection of gravitational waves The chirp mass and the luminosity distance A lower bound for the total mass of the system
The final stages of the inspirai Merger and ringdown: identifying the nature of coalescing compact objects 14.2.6 More signals from coalescences 14.3 GRAVITATIONAL WAVES FROM ROTATING COMPACT STARS 14.3.1 14.3.2 Stars rigidly rotating around a principal axis Wobbling stars 267 273 274 277 278 279 280 284 284 289
Contents ■ ix 14.4 COSMOLOGICAL PARAMETERS 14.4.1 14.4.2 14.4.3 14.4.4 The cosmological redshift The Hubble constant Luminosity distance Standard sirens: coalescing binaries as standard candles Chapter 15· Gravitational 292 293 294 295 297 waves from oscillating black holes___________ 299 15.1 A TOY MODEL: SCALAR PERTURBATIONS 299 15.2 PERTURBATIONS OF THE SCHWARZSCHILDSPACETIME 304 15.2.1 15.2.2 15.2.3 15.2.4 15.2.5 15.2.6 15.3 MASTER EQUATIONS FOR AXIAL AND POLARPERTURBATIONS 15.3.1 15.3.2 15.4 Linearized Einstein’s equations in vacuum Harmonic decomposition The Regge-Wheeler gauge Perturbations with l = 0 and 1 = 1 Linearized Einstein’s equations in vacuum Separation of the angular dependence The Regge-Wheeler equation for the axial perturbations The Zerilli equation for the polar perturbations 304 305 308 310 310 311 312 313 313 THE QUASI-NORMAL MODES OF A SCHWARZSCHILD BLACK HOLE 315 Chapter 16· Compact stars 321 16.1 STELLAR EVOLUTION IN A NUTSHELL 321 16.2 WHITE DWARFS 324 16.2.1 16.2.2 16.2.3 16.2.4 16.2.5 16.2.6 16.3 The discovery of white dwarfs Degenerate gas in quantum mechanics A criterion for degeneracy The equation of state of a fully degenerate gas of fermions The structure of a white dwarf The Chandrasekhar limit NEUTRON STARS 16.3.1 16.3.2 16.3.3 16.3.4 16.3.5 16.3.6 16.3.7 16.3.8 16.3.9 The discovery of neutron stars The internal structure of a neutron star Thermodynamics of perfect fluids in General Relativity The stress-energy tensor of a perfect fluid The equations of stellar structure in General Relativity The Schwarzschild solution for a homogeneous
star Relativistic polytropes Buchdahl’s theorem Stability of a compact star 324 325 327 330 334 338 343 343 343 346 352 353 358 360 365 367
x ■ Contents Chapter 17 ■ The far-field 17.1 limit of an isolated, stationary object_________ 371 THE WEAK-FIELD CASE 17.1.1 17.1.2 The multipolar expansion The metric of the far-field limit in polar coordinates 372 373 377 17.2 THE STRONG-FIELD CASE 378 17.3 MASS AND ANGULAR MOMENTUM OF AN ISOLATED OBJECT 382 17.4 PRECESSION OF A GYROSCOPE IN A GRAVITATIONAL FIELD 386 17.4.1 Gyroscopes in the gravitational field of a rotating body: the Lense-Thirring precession 387 17.4.2 Moving gyroscopes: geodesic precession 388 17.4.3 Measurement of geodesic and Lense-Thirring frequencies: Gravity Probe В 392 Chapter 18· The Kerr solution________________________________________ 395 18.1 OBSERVATIONAL EVIDENCE FOR ROTATING BLACK HOLES 395 18.2 THE KERR METRIC IN THE BOYER-LINDQUIST COORDINATES 396 18.3 SYMMETRIES 399 18.4 BLACK HOLE HORIZONS 399 18.4.1 18.4.2 Horizon structure How to remove the singularity at the horizons 400 401 18.5 FRAME DRAGGING 405 18.6 THE ERGOSPHERE 406 18.6.1 18.7 Static and stationary observers 408 THE SINGULARITY OF THE KERR SPACETIME 409 18.7.1 18.7.2 18.8 THE INTERIOR OF AN ETERNAL KERR BLACK HOLE 18.8.1 18.8.2 18.9 The Kerr-Schild coordinates The metric in Kerr-Schild’s coordinates Extensions of the Kerr metric Causality violations GENERAL BLACK HOLE SOLUTIONS Chapter 19· Geodesic motion in Kerr s spacetime________________ 409 411 413 414 416 417 421 19.1 THE EQUATIONS FOR t AND φ 422 19.2 THE EQUATIONS FOR f AND Θ 423 19.3 EQUATORIAL GEODESICS 426 19.3.1 19.3.2 19.3.3 19.3.4 19.3.5 19.3.6 Potentials for equatorial geodesics Null geodesics Energy of a
particle in Kerr’s spacetime Timelike geodesics Emission of light from the ergosphere Generalization of Kepler’s third law in Kerr’s spacetime 429 430 434 436 439 441
Contents ■ xi 19.4 ENERGY EXTRACTION FROM BLACK HOLES 19.4.1 19.4.2 The Penrose process Superradiant scattering 442 442 444 Chapter 20 ■ Black hole thermodynamics___________________________ 447 20.1 IRREDUCIBLE MASS AND BLACK HOLE AREA THEOREM 447 20.2 THE LAWS OF BLACK HOLE THERMODYNAMICS 452 20.3 THE GENERALIZED SECOND LAW OF THERMODYNAMICS 456 20.4 THE HAWKING RADIATION 459 Bibliography 461 Index 469
General Relativity and its Applications Black Holes, Compact Stars and Gravitational Waves Containing the latest, groundbreaking discoveries in the field, this text outlines the basics of Einstein’s theory of gravity with a focus on its most important astrophysical consequences, including stellar structure, black holes and gravitational waves. Blending advanced topics - usually not found in introductory textbooks - with examples, pedagogical boxes, mathematical tools and practical applications of the theory, this textbook maximises learning opportunities and is ideal for master and graduate students in Physics and Astronomy. Key features: • Provides a self-contained and consistent treatment of the subject that does not require previous knowledge of the fie!d. • Explores the subject with a new focus on gravitational waves and astrophysical relativity, unlike current introductory textbooks. • Fully up-to-date, containing the latest developments and discoveries in the field. Valeria Ferrari is Professor of Theoretical Physics at Sapienza, University of Rome (Italy). She performed research in various fields of General Relativity, with special focus on black hole and stellar perturbations, and on gravitational wave sources. She has published over 125 papers in international journals, given over 100 invited talks, in addition to coordinating international research projects, the Virgo-Ego Scientific forum, and serving as part of several journal Editorial Boards. She co-authored textbooks in Physics for undergraduate students. She is a founding member of the Amaldi Research Center
for Gravitational Physics at Sapienza and co-Chair of the European Research Network COST Action GWverse. Leonardo Gualtieri is Associate Professor at Sapienza, University of Rome (Italy). His main research interests are gravitational theory and gravitational waves, with special focus on gravitational wave sources as probes of fundamental physics. He has published over 100 papers in international journals and given over 30 invited talks, in addition to coordinating TEONGRAV, the INFN network of Italian groups studying gravitational wave sources. Paolo Pani is Associate Professor of Theoretical Physics at Sapienza, University of Rome (Italy) and Junior Fellow at Sapienza’s School for Advanced Studies. He coordinates the DarkGRA project “Unveiling the dark universe with gravitational waves” funded by the European Research Council (ERC). He received the SIGRAV Prize and the Outstanding Referee award from the American Physics Society. He is co-author of the book “Superradiance” and of over 100 scientific publications on black-hole physics and gravitational-wave phenomenology, and their connections to fundamental physics. Physics CRC Press Taylor և Francis Group an informa business WWW. routledge.com CRC Press titles are available as eBook editions in a range of digital formats 9780367625320
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Contents Preface xiii Notation and conventions Chapter 1.1 1 ■ Introduction xvii 1 NON-EUCLIDEAN GEOMETRIES 1 1.1.1 1.1.2 1.1.3 4 5 5 The metric tensor in different coordinate frames The Gaussian curvature Pseudo-Euclidean geometries and spacetime 1.2 NEWTONIAN THEORY AND ITS SHORTCOMINGS 6 1.3 THE ROLE OF THE EQUIVALENCE PRINCIPLE 9 1.4 GEODESIC EQUATIONS AS A CONSEQUENCE OF THE EQUIVALENCE PRINCIPLE 11 LOCALLY INERTIAL FRAMES 13 1.5 Chapter 2.1 2.2 2.3 2.4 2.5 2 ■ Elements of differential geometry 17 TOPOLOGICAL SPACES, MAPPING, MANIFOLDS 17 2.1.1 2.1.2 2.1.3 2.1.4 17 18 22 26 Topological spaces Mapping Manifolds and differentiable manifolds Diffeomorphisms VECTORS 27 2.2.1 2.2.2 27 29 The traditional definition of a vector A geometrical definition ONE-FORMS 39 2.3.1 2.3.2 39 43 One-forms as geometrical objects Vector fields and one-form fields TENSORS 46 2.4.1 2.4.2 46 51 Geometrical definition of a tensor Symmetries of a tensor THE METRIC TENSOR AND ITS PROPERTIES 54
vi ■ Contents Chapter 3 ■ Affine connection and parallel transport_ 63 3.1 THE COVARIANT DERIVATIVE OF VECTORS 63 3.2 THE COVARIANT DERIVATIVE OF SCALARS AND ONE-FORMS 67 3.3 SYMMETRIES OF CHRISTOFFEĽS SYMBOLS 68 3.4 TRANSFORMATION RULES FOR CHRISTOFFEĽS SYMBOLS 69 3.5 THE COVARIANT DERIVATIVE OF TENSORS 70 3.6 CHRISTOFFEĽS SYMBOLS IN TERMS OF THE METRIC TENSOR 72 3.7 PARALLEL TRANSPORT 76 3.7.1 Parallel transport of a vector along a closed path on a twosphere 78 3.8 GEODESIC EQUATION 81 3.9 FERMI COORDINATES 82 3.10 NON-COORDINATE BASES Chapter 84 4 ■ The curvature tensor_ _ 87 4.1 PARALLEL TRANSPORT ALONG A LOOP 87 4.2 SYMMETRIES OF THE RIEMANN TENSOR 91 4.3 THE RIEMANN TENSOR GIVES THE COMMUTATOR OF COVARIANT DERIVATIVES 91 4.4 THE BIANCHI IDENTITIES 92 4.5 THE EQUATION OF GEODESIC DEVIATION 92 Chapter 5 ■ The stress-energy tensor _ 97 5.1 5.2 THE STRESS-ENERGY TENSOR IN FIAT SPACETIME IS Ta@ A TENSOR? 97 100 5.3 DOES 104 5.4 IS Γα% = 0 A CONSERVATION LAW? Chapter Ta? SATISFY A CONSERVATION LAW? 6 ■ The Einstein equations 107 109 6.1 GEODESIC EQUATIONS IN THE WEAK-FIELD, STATIONARY LIMIT 110 6.2 EINSTEIN'S FIELD EQUATIONS 112 6.3 GAUGE INVARIANCE OF EINSTEIN'S EQUATIONS 117 6.4 THE HARMONIC GAUGE 118 Chapter 7 ■ Einstein's equations and variational principles_ _ 121 7.1 EULER-LAGRANGE'S EQUATIONS IN SPECIAL RELATIVITY 121 7.2 EULER-LAGRANGE'S EQUATIONS IN CURVED SPACETIME 122 7.3 EINSTEIN'S EQUATIONS IN VACUUM 124 7.4 EINSTEIN'S EQUATIONS WITH SOURCES 128 7.4.1 The stress-energy tensor in some
relevant cases 129 EINSTEIN'S EQUATIONS IN THE PALATINI FORMALISM 131 7.5
Contents ■ vii Chapter 8 ■ Symmetries_ 135 8.1 KILLING VECTOR FIELDS 135 8.2 KILLING VECTOR FIELDS AND THE CHOICE OF COORDINATE SYS TEMS 139 KILLING VECTOR FIELDS AND CONSERVATION LAWS 142 8.3.1 8.3.2 142 143 8.3 8.4 HYPERSURFACE-ORTHOGONAL VECTOR FIELDS 8.4.1 8.4.2 8.5 Conserved quantities in geodesic motion Conserved quantities from the stress-energy tensor Frobenius’ theorem 145 Hypersurface-orthogonal vector fields and the choice of coor dinate systems 146 DIFFEOMORPHISM INVARIANCE OF GENERAL RELATIVITY Chapter 144 9 ■ The Schwarzschild solution 148 151 9.1 STATIC AND SPHERICALLY SYMMETRIC SPACETIMES 151 9.2 THE SCHWARZSCHILD SOLUTION 154 9.3 SINGULARITIES OF THE SCHWARZSCHILD SOLUTION 159 9.4 SPACELIKE, TIMELIKE, AND NULL HYPERSURFACES 160 9.4.1 162 9.5 SINGULARITIES IN GENERAL RELATIVITY 9.5.1 9.5.2 9.5.3 9.5.4 9.5.5 9.6 Constant radius hypersurfaces in Schwarzschild’s spacetime Geodesic completeness How to remove a coordinate singularity Extension of the Rindler spacetime Extension of the Schwarzschild spacetime Eddington-Finkeistein coordinates THE BIRKHOFF THEOREM Chapter 10· Geodesic 10.1 10.2 motion in Schwarzschild's spacetime A VARIATIONAL PRINCIPLE FOR GEODESIC MOTION EQUATIONS OF MOTION 163 163 164 166 170 176 178 181 181 182 10.3 THE CONSTANTS OF GEODESIC MOTION 185 10.4 ORBITS OF MASSLESS PARTICLES 188 10.5 ORBITS OF MASSIVE PARTICLES 190 10.6 RADIAL CAPTURE OF A MASSIVE PARTICLE 193 Chapter 11· 11.1 Kinematical tests of General Relativity GRAVITATIONAL SHIFT OF SPECTRAL LINES 11.1.1 11.1.2 Redshift of
spectral lines in the weak-field limit Redshift of spectral lines in a strong gravitational field 197 197 201 201 11.2 THE DEFLECTION OF LIGHT 203 11.3 PERIASTRON PRECESSION 207
viii ■ Contents 11.4 THE SHAPIRO TIME DELAY 11.5 THE SHADOW OF A BLACK HOLE 11.5.1 212 The accretion disk of a black hole 214 218 Chapter 12· Gravitational waves_ 221 12.1 12.2 PERTURBATIVE APPROACH 221 GRAVITATIONAL WAVES AS PERTURBATIONS OF FLAT SPACETIME 223 12.3 HOW TO CHOOSE THE HARMONIC GAUGE 228 12.4 PLANE GRAVITATIONAL WAVES 229 12.5 THE TT GAUGE 230 12.6 12.7 12.8 HOW DOES A GRAVITATIONAL WAVE AFFECT THE MOTION OF A SINGLE PARTICLE GEODESIC DEVIATION INDUCED BY A GRAVITATIONAL WAVE GRAVITATIONAL WAVES AND MICHELSON INTERFEROMETERS Chapter 13· Gravitational waves 13.1 13.2 232 239 in the quadrupede approximation_ 243 THE WEAK-FIELD, SLOW-MOTION APPROXIMATION THE QUADRUPOLE FORMULA 13.2.1 13.2.2 13.2.3 232 The tensor-viriai theorem The quadrupole moment tensor Absence of monopolar and dipolar gravitational waves 243 245 245 247 247 13.3 HOW TO TRANSFORM TO THE TT GAUGE 248 13.4 GRAVITATIONAL WAVES EMITTED BY A HARMONIC OSCILLATOR 250 13.5 GRAVITATIONAL WAVE EMITTED BY A BINARY SYSTEM IN CIRCULAR ORBIT 252 13.6 ENERGY CARRIED BY A GRAVITATIONAL WAVE 13.6.1 13.6.2 The stress-energy pseudo-tensor of the gravitational field Energy flux of a gravitational wave Chapter 14· Gravitational wave sources_ 257 257 261 267 14.1 EVOLUTION OF A COMPACT BINARY SYSTEM 14.2 GRAVITATIONAL WAVES FROM INSPIRALLING COMPACT OBJECTS 271 14.2.1 14.2.2 14.2.3 14.2.4 14.2.5 September 14th, 2015: the detection of gravitational waves The chirp mass and the luminosity distance A lower bound for the total mass of the system
The final stages of the inspirai Merger and ringdown: identifying the nature of coalescing compact objects 14.2.6 More signals from coalescences 14.3 GRAVITATIONAL WAVES FROM ROTATING COMPACT STARS 14.3.1 14.3.2 Stars rigidly rotating around a principal axis Wobbling stars 267 273 274 277 278 279 280 284 284 289
Contents ■ ix 14.4 COSMOLOGICAL PARAMETERS 14.4.1 14.4.2 14.4.3 14.4.4 The cosmological redshift The Hubble constant Luminosity distance Standard sirens: coalescing binaries as standard candles Chapter 15· Gravitational 292 293 294 295 297 waves from oscillating black holes_ 299 15.1 A TOY MODEL: SCALAR PERTURBATIONS 299 15.2 PERTURBATIONS OF THE SCHWARZSCHILDSPACETIME 304 15.2.1 15.2.2 15.2.3 15.2.4 15.2.5 15.2.6 15.3 MASTER EQUATIONS FOR AXIAL AND POLARPERTURBATIONS 15.3.1 15.3.2 15.4 Linearized Einstein’s equations in vacuum Harmonic decomposition The Regge-Wheeler gauge Perturbations with l = 0 and 1 = 1 Linearized Einstein’s equations in vacuum Separation of the angular dependence The Regge-Wheeler equation for the axial perturbations The Zerilli equation for the polar perturbations 304 305 308 310 310 311 312 313 313 THE QUASI-NORMAL MODES OF A SCHWARZSCHILD BLACK HOLE 315 Chapter 16· Compact stars 321 16.1 STELLAR EVOLUTION IN A NUTSHELL 321 16.2 WHITE DWARFS 324 16.2.1 16.2.2 16.2.3 16.2.4 16.2.5 16.2.6 16.3 The discovery of white dwarfs Degenerate gas in quantum mechanics A criterion for degeneracy The equation of state of a fully degenerate gas of fermions The structure of a white dwarf The Chandrasekhar limit NEUTRON STARS 16.3.1 16.3.2 16.3.3 16.3.4 16.3.5 16.3.6 16.3.7 16.3.8 16.3.9 The discovery of neutron stars The internal structure of a neutron star Thermodynamics of perfect fluids in General Relativity The stress-energy tensor of a perfect fluid The equations of stellar structure in General Relativity The Schwarzschild solution for a homogeneous
star Relativistic polytropes Buchdahl’s theorem Stability of a compact star 324 325 327 330 334 338 343 343 343 346 352 353 358 360 365 367
x ■ Contents Chapter 17 ■ The far-field 17.1 limit of an isolated, stationary object_ 371 THE WEAK-FIELD CASE 17.1.1 17.1.2 The multipolar expansion The metric of the far-field limit in polar coordinates 372 373 377 17.2 THE STRONG-FIELD CASE 378 17.3 MASS AND ANGULAR MOMENTUM OF AN ISOLATED OBJECT 382 17.4 PRECESSION OF A GYROSCOPE IN A GRAVITATIONAL FIELD 386 17.4.1 Gyroscopes in the gravitational field of a rotating body: the Lense-Thirring precession 387 17.4.2 Moving gyroscopes: geodesic precession 388 17.4.3 Measurement of geodesic and Lense-Thirring frequencies: Gravity Probe В 392 Chapter 18· The Kerr solution_ 395 18.1 OBSERVATIONAL EVIDENCE FOR ROTATING BLACK HOLES 395 18.2 THE KERR METRIC IN THE BOYER-LINDQUIST COORDINATES 396 18.3 SYMMETRIES 399 18.4 BLACK HOLE HORIZONS 399 18.4.1 18.4.2 Horizon structure How to remove the singularity at the horizons 400 401 18.5 FRAME DRAGGING 405 18.6 THE ERGOSPHERE 406 18.6.1 18.7 Static and stationary observers 408 THE SINGULARITY OF THE KERR SPACETIME 409 18.7.1 18.7.2 18.8 THE INTERIOR OF AN ETERNAL KERR BLACK HOLE 18.8.1 18.8.2 18.9 The Kerr-Schild coordinates The metric in Kerr-Schild’s coordinates Extensions of the Kerr metric Causality violations GENERAL BLACK HOLE SOLUTIONS Chapter 19· Geodesic motion in Kerr's spacetime_ 409 411 413 414 416 417 421 19.1 THE EQUATIONS FOR t AND φ 422 19.2 THE EQUATIONS FOR f AND Θ 423 19.3 EQUATORIAL GEODESICS 426 19.3.1 19.3.2 19.3.3 19.3.4 19.3.5 19.3.6 Potentials for equatorial geodesics Null geodesics Energy of a
particle in Kerr’s spacetime Timelike geodesics Emission of light from the ergosphere Generalization of Kepler’s third law in Kerr’s spacetime 429 430 434 436 439 441
Contents ■ xi 19.4 ENERGY EXTRACTION FROM BLACK HOLES 19.4.1 19.4.2 The Penrose process Superradiant scattering 442 442 444 Chapter 20 ■ Black hole thermodynamics_ 447 20.1 IRREDUCIBLE MASS AND BLACK HOLE AREA THEOREM 447 20.2 THE LAWS OF BLACK HOLE THERMODYNAMICS 452 20.3 THE GENERALIZED SECOND LAW OF THERMODYNAMICS 456 20.4 THE HAWKING RADIATION 459 Bibliography 461 Index 469
General Relativity and its Applications Black Holes, Compact Stars and Gravitational Waves Containing the latest, groundbreaking discoveries in the field, this text outlines the basics of Einstein’s theory of gravity with a focus on its most important astrophysical consequences, including stellar structure, black holes and gravitational waves. Blending advanced topics - usually not found in introductory textbooks - with examples, pedagogical boxes, mathematical tools and practical applications of the theory, this textbook maximises learning opportunities and is ideal for master and graduate students in Physics and Astronomy. Key features: • Provides a self-contained and consistent treatment of the subject that does not require previous knowledge of the fie!d. • Explores the subject with a new focus on gravitational waves and astrophysical relativity, unlike current introductory textbooks. • Fully up-to-date, containing the latest developments and discoveries in the field. Valeria Ferrari is Professor of Theoretical Physics at Sapienza, University of Rome (Italy). She performed research in various fields of General Relativity, with special focus on black hole and stellar perturbations, and on gravitational wave sources. She has published over 125 papers in international journals, given over 100 invited talks, in addition to coordinating international research projects, the Virgo-Ego Scientific forum, and serving as part of several journal Editorial Boards. She co-authored textbooks in Physics for undergraduate students. She is a founding member of the Amaldi Research Center
for Gravitational Physics at Sapienza and co-Chair of the European Research Network COST Action GWverse. Leonardo Gualtieri is Associate Professor at Sapienza, University of Rome (Italy). His main research interests are gravitational theory and gravitational waves, with special focus on gravitational wave sources as probes of fundamental physics. He has published over 100 papers in international journals and given over 30 invited talks, in addition to coordinating TEONGRAV, the INFN network of Italian groups studying gravitational wave sources. Paolo Pani is Associate Professor of Theoretical Physics at Sapienza, University of Rome (Italy) and Junior Fellow at Sapienza’s School for Advanced Studies. He coordinates the DarkGRA project “Unveiling the dark universe with gravitational waves” funded by the European Research Council (ERC). He received the SIGRAV Prize and the Outstanding Referee award from the American Physics Society. He is co-author of the book “Superradiance” and of over 100 scientific publications on black-hole physics and gravitational-wave phenomenology, and their connections to fundamental physics. Physics CRC Press Taylor և Francis Group an informa business WWW. routledge.com CRC Press titles are available as eBook editions in a range of digital formats 9780367625320 |
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| spelling | Ferrari, Valeria 1952- Verfasser (DE-588)1227009879 aut General relativity and its applications black holes, compact stars and gravitational waves Valeria Ferrari, Leonardo Gualtieri, Paolo Pani First edition Boca Raton ; London ; New York CRC Press, Taylor & Francis Group 2021 xviii, 426 Seiten Illustrationen, Diagramme Breite 178 mm, Hoehe 254 mm txt rdacontent n rdamedia nc rdacarrier Literaturverzeichnis Seite 461-467 Relativitätstheorie (DE-588)4049363-5 gnd rswk-swf Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd rswk-swf Gravitationswelle (DE-588)4158119-2 gnd rswk-swf Schwarzes Loch (DE-588)4053793-6 gnd rswk-swf Relativitätstheorie (DE-588)4049363-5 s Allgemeine Relativitätstheorie (DE-588)4112491-1 s Gravitationswelle (DE-588)4158119-2 s Schwarzes Loch (DE-588)4053793-6 s DE-604 Gualtieri, Leonardo 1971- Verfasser (DE-588)1227010494 aut Pani, Paolo 1984- Verfasser (DE-588)112427975X aut Erscheint auch als Online-Ausgabe 978-0429-49140-5 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=032378666&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 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=032378666&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Ferrari, Valeria 1952- Gualtieri, Leonardo 1971- Pani, Paolo 1984- General relativity and its applications black holes, compact stars and gravitational waves Relativitätstheorie (DE-588)4049363-5 gnd Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd Gravitationswelle (DE-588)4158119-2 gnd Schwarzes Loch (DE-588)4053793-6 gnd |
| subject_GND | (DE-588)4049363-5 (DE-588)4112491-1 (DE-588)4158119-2 (DE-588)4053793-6 |
| title | General relativity and its applications black holes, compact stars and gravitational waves |
| title_auth | General relativity and its applications black holes, compact stars and gravitational waves |
| title_exact_search | General relativity and its applications black holes, compact stars and gravitational waves |
| title_exact_search_txtP | General relativity and its applications black holes, compact stars and gravitational waves |
| title_full | General relativity and its applications black holes, compact stars and gravitational waves Valeria Ferrari, Leonardo Gualtieri, Paolo Pani |
| title_fullStr | General relativity and its applications black holes, compact stars and gravitational waves Valeria Ferrari, Leonardo Gualtieri, Paolo Pani |
| title_full_unstemmed | General relativity and its applications black holes, compact stars and gravitational waves Valeria Ferrari, Leonardo Gualtieri, Paolo Pani |
| title_short | General relativity and its applications |
| title_sort | general relativity and its applications black holes compact stars and gravitational waves |
| title_sub | black holes, compact stars and gravitational waves |
| topic | Relativitätstheorie (DE-588)4049363-5 gnd Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd Gravitationswelle (DE-588)4158119-2 gnd Schwarzes Loch (DE-588)4053793-6 gnd |
| topic_facet | Relativitätstheorie Allgemeine Relativitätstheorie Gravitationswelle Schwarzes Loch |
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