A First Course In General Relativity, 2nd Edition

Clarity, readability and rigor combine in the second edition of this widely-used textbook to provide the first step into general relativity for undergraduate students with a minimal background in mathematics. Topics within relativity that fascinate astrophysical researchers and students alike are covered with Schutz's characteristic ease and authority - from black holes to gravitational lenses, from pulsars to the study of the Universe as a whole. This edition now contains discoveries by astronomers that require general relativity for their explanation; a revised chapter on relativistic stars, including new information on pulsars; an entirely rewritten chapter on cosmology; and an extended, comprehensive treatment of modern detectors and expected sources. Over 300 exercises, many new to this edition, give students the confidence to work with general relativity and the necessary mathematics, whilst the informal writing style makes the subject matter easily accessible. Password protected solutions for instructors are available at www.cambridge.org/9780521887052. Cover Half-title Title Copyright Contents Dedication Preface to the second edition Preface to the first edition 1 Special relativity 1.1 Fundamental principles of special relativity (SR) theory 1.2 Definition of an inertial observer in SR 1.3 New units 1.4 Spacetime diagrams 1.5 Construction of the coordinates used by another observer 1.6 Invariance of the interval 1.7 Invariant hyperbolae 1.8 Particularly important results Time dilation Lorentz contraction Conventions Failure of relativity? 1.9 The Lorentz transformation 1.10 The velocity-composition law 1.11 Paradoxes and physical intuition 1.12 Further reading 1.13 Appendix: The twin ‘paradox’ dissected The problem Brief solution Fuller discussion 1.14 Exercises 2 Vector analysis in special relativity 2.1 Definition of a vector 2.2 Vector algebra Basis vectors Transformation of basis vectors Inverse transformations 2.3 The four-velocity 2.4 The four-momentum Conservation of four-momentum Center of momentum (CM) frame 2.5 Scalar product Magnitude of a vector Scalar product of two vectors 2.6 Applications Four-velocity and acceleration as derivatives Energy and momentum 2.7 Photons No four-velocity Four-momentum Zero rest-mass particles 2.8 Further reading 2.9 Exercises 3 Tensor analysis in special relativity 3.1 The metric tensor 3.2 Definition of tensors Aside on the usage of the term ‘function’ Components of a tensor 3.3 The (0 1) tensors: one-forms General properties Basis one-forms Picture of a one-form Gradient of a function is a one-form Notation for derivatives Normal one-forms 3.4 The (0 2) tensors Components Symmetries 3.5 Metric as a mapping of vectors into one-forms The inverse: going from... Why distinguish one-forms from vectors? Magnitudes and scalar products of one-forms Normal vectors and unit normal one-forms 3.6 Finally: (M N) tensors Vector as a function of one-forms (M 0) tensors (M N) tensors Circular reasoning? 3.7 Index ‘raising’ and ‘lowering’ Mixed components of metric Metric and nonmetric vector algebras 3.8 Differentiation of tensors 3.9 Further reading 3.10 Exercises 4 Perfect fluids in special relativity 4.1 Fluids 4.2 Dust: the number–flux vector... The number density n The flux across a surface The number–flux four-vector... 4.3 One-forms and surfaces Number density as a timelike flux A one-form defines a surface The flux across the surface Representation of a frame by a one-form 4.4 Dust again: the stress–energy tensor Energy density Stress–energy tensor 4.5 General fluids Definition of macroscopic quantities First law of thermodynamics The general stress–energy tensor The spatial components of T, T ij Symmetry of Talphabeta in MCRF Conservation of energy–momentum Conservation of particles 4.6 Perfect fluids No heat conduction No viscosity Form of T Aside on the meaning of pressure The conservation laws 4.7 Importance for general relativity 4.8 Gauss’ law 4.9 Further reading 4.10 Exercises 5 Preface to curvature 5.1 On the relation of gravitation to curvature The gravitational redshift experiment Nonexistence of a Lorentz frame at rest on Earth The principle of equivalence The redshift experiment again Local inertial frames Tidal forces The role of curvature 5.2 Tensor algebra in polar coordinates Vectors and one-forms Curves and vectors Polar coordinate basis one-forms and vectors Metric tensor 5.3 Tensor calculus in polar coordinates Derivatives of basis vectors Derivatives of general vectors The Christoffel symbols The covariant derivative Divergence and Laplacian Derivatives of one-forms and tensors of higher types 5.4 Christoffel symbols and the metric Calculating the Christoffel symbols from the metric The tensorial nature of... 5.5 Noncoordinate bases Polar coordinate basis Polar unit basis General remarks on noncoordinate bases Noncoordinate bases in this book 5.6 Looking ahead 5.7 Further reading 5.8 Exercises 6 Curved manifolds 6.1 Differentiable manifolds and tensors Manifolds Differential structure 6.2 Riemannian manifolds The metric and local flatness Lengths and volumes Proof of the local-flatness theorem 6.3 Covariant differentiation Divergence formula 6.4 Parallel-transport, geodesics, and curvature Parallel-transport Geodesics 6.5 The curvature tensor Geodesic deviation 6.6 Bianchi identities: Ricci and Einstein tensors The Ricci tensor The Einstein tensor 6.7 Curvature in perspective 6.8 Further reading 6.9 Exercises 7 Physics in a curved spacetime 7.1 The transition from differential geometry to gravity 7.2 Physics in slightly curved spacetimes 7.3 Curved intuition 7.4 Conserved quantities 7.5 Further reading 7.6 Exercises 8 The Einstein field equations 8.1 Purpose and justification of the field equations Geometrized units 8.2 Einstein’s equations 8.3 Einstein’s equations for weak gravitational fields Nearly Lorentz coordinate systems Background Lorentz transformations Gauge transformations Riemann tensor Weak-field Einstein equations 8.4 Newtonian gravitational fields Newtonian limit The far field of stationary relativistic sources Definition of the mass of a relativistic body 8.5 Further reading 8.6 Exercises 9 Gravitational radiation 9.1 The propagation of gravitational waves The transverse-traceless gauge The effect of waves on free particles Tidal accelerations: gravitational wave forces Measuring the stretching of space Polarization of gravitational waves An exact plane wave Geometrical optics: waves in a curved spacetime 9.2 The detection of gravitational waves General considerations A resonant detector Bar detectors in operation Measuring distances with light Beam detectors Interferometer observations 9.3 The generation of gravitational waves Simple estimates Slow motion wave generation Order-of-magnitude estimates Exact solution of the wave equation 9.4 The energy carried away by gravitational waves Preview The energy flux of a gravitational wave Energy lost by a radiating system An example. The Hulse–Taylor binary pulsar 9.5 Astrophysical sources of gravitational waves Overview Binary systems Spinning neutron stars Gravitational collapse Gravitational waves from the Big Bang 9.6 Further reading 9.7 Exercises 10 Spherical solutions for stars 10.1 Coordinates for spherically symmetric spacetimes Flat space in spherical coordinates Two-spheres in a curved spacetime Meshing the two-spheres into a three-space for t = const Spherically symmetric spacetime 10.2 Static spherically symmetric spacetimes The metric Physical interpretation of metric terms The Einstein tensor 10.3 Static perfect fluid Einstein equations Stress–energy tensor Equation of state Equations of motion Einstein equations 10.4 The exterior geometry Schwarzschild metric Generality of the metric 10.5 The interior structure of the star General rules for integrating the equations The structure of Newtonian stars 10.6 Exact interior solutions The Schwarzschild constant-density interior solution Buchdahl’s interior solution 10.7 Realistic stars and gravitational collapse Buchdahl’s theorem Formation of stellar-mass black holes Quantum mechanical pressure White dwarfs Neutron stars 10.8 Further reading 10.9 Exercises 11 Schwarzschild geometry and black holes 11.1 Trajectories in the Schwarzschild spacetime Black holes in Newtonian gravity Conserved quantities Types of orbits Perihelion shift Binary pulsars Post-Newtonian gravity Gravitational deflection of light Gravitational lensing 11.2 Nature of the surface r = 2M Coordinate singularities Infalling particles Inside r = 2M Coordinate systems Kruskal–Szekeres coordinates 11.3 General black holes Formation of black holes in general General properties of black holes Kerr black hole Dragging of inertial frames Ergoregion The Kerr horizon Equatorial photon motion in the Kerr metric The Penrose process 11.4 Real black holes in astronomy Black holes of stellar mass Supermassive black holes Intermediate-mass black holes Dynamical black holes 11.5 Quantum mechanical emission of radiation by black holes: the Hawking process 11.6 Further reading 11.7 Exercises 12 Cosmology 12.1 What is cosmology? The universe in the large The cosmological arena 12.2 Cosmological kinematics: observing the expanding universe Homogeneity and isotropy of the universe Models of the universe: the cosmological principle Cosmological metrics Three types of universe Cosmological redshift as a distance measure Cosmography: measures of distance in the universe The universe is accelerating! 12.3 Cosmological dynamics: understanding the expanding universe Dynamics of Robertson–Walker universes: Big Bang and dark energy Critical density and the parameters of our universe 12.4 Physical cosmology: the evolution of the universe we observe Decoupling: forming the cosmic microwave background radiation Dark matter and galaxy formation: the universe after decoupling The early universe: fundamental physics meets cosmology Beyond general relativity 12.5 Further reading 12.6 Exercises Appendix A Summary of linear algebra Vector space Matrices References Index Clarity, Readability And Rigor Combine In The Second Edition Of This Widely-used Textbook To Provide The First Step Into General Relativity For Undergraduate Students With A Minimal Background In Mathematics. Topics Within Relativity That Fascinate Astrophysical Researchers And Students Alike Are Covered. Special Relativity -- Vector Analysis In Special Relativity -- Tensor Analysis In Special Relativity -- Perfect Fluids In Special Relativity -- Preface To Curvature -- Curved Manifolds -- Physics In A Curved Spacetime -- The Einstein Field Equations -- Gravitational Radiation -- Spherical Solutions For Stars -- Schwarzschild Geometry And Black Holes -- Cosmology. By Bernard Schutz. Previous Ed.: 1985. Includes Bibliographical References And Index.