Introducing General Relativity

ISBN-13: 9781118600719 / Angielski / Miękka / 2022 / 288 str.

Mark Hindmarsh; Andrew Liddle

Spis treści

Preface ixConstants and Symbols x1 Introducing General Relativity 12 A Special Relativity Reminder 32.1 The need for Special Relativity 42.2 The Lorentz transformation 62.3 Time dilation 82.4 Lorentz-Fitzgerald contraction 92.5 Addition of velocities 112.6 Simultaneity, colocality, and causality 122.7 Space-time diagrams 133 Tensors in Special Relativity 173.1 Coordinates 183.2 4-vectors 203.3 4-velocity, 4-momentum, and 4-acceleration 243.4 4-divergence and the wave operator 263.5 Tensors 283.6 Tensors in action: the Lorentz force 304 Towards General Relativity 374.1 Newtonian gravity 374.2 Special Relativity and gravity 394.3 Motivations for a General Theory of Relativity 414.3.1 Mach's Principle 424.3.2 Einstein's Equivalence Principle 424.4 Implications of the Equivalence Principle 444.4.1 Gravitational redshift 454.4.2 Gravitational time dilation 464.5 Principles of the General Theory of Relativity 474.6 Towards curved space-time 494.7 Curved space in two dimensions 505 Tensors and Curved Space-Time 575.1 General coordinate transformations 575.2 Tensor equations and the laws of physics 595.3 Partial differentiation of tensors 595.4 The covariant derivative and parallel transport 605.5 Christoffel symbols of a two-sphere 655.6 Parallel transport on a two-sphere 665.7 Curvature and the Riemann tensor 685.8 Riemann curvature of the two-sphere 715.9 More tensors describing curvature 725.10 Local inertial frames and local flatness 736 Describing Matter 796.1 The Correspondence Principle 796.2 The energy-momentum tensor 806.2.1 General properties 806.2.2 Conservation laws and 4-vector flux 816.2.3 Energy and momentum belong in a rank-2 tensor 836.2.4 Symmetry of the energy-momentum tensor 846.2.5 Energy-momentum of perfect fluids 846.2.6 The energy-momentum tensor in curved space-time 877 The Einstein Equation 917.1 The form of the Einstein equation 917.2 Properties of the Einstein equation 937.3 The Newtonian limit 937.4 The cosmological constant 957.5 The vacuum Einstein equation 968 The Schwarzschild Space-time 998.1 Christoffel symbols 1008.2 Riemann tensor 1018.3 Ricci tensor 1028.4 The Schwarzschild solution 1038.5 The Jebsen-Birkhoff theorem 1049 Geodesics and Orbits 1099.1 Geodesics 1099.2 Non-relativistic limit of geodesic motion 1129.3 Geodesic deviation 1139.4 Newtonian theory of orbits 1159.5 Orbits in the Schwarzschild space-time 1179.5.1 Massive particles 1179.5.2 Photon orbits 12010 Tests of General Relativity 12310.1 Precession of Mercury's perihelion 12310.2 Gravitational light bending 12510.3 Radar echo delays 12710.4 Gravitational redshift 12910.5 Binary pulsar PSR 1913+16 13110.6 Direct detection of gravitational waves 13511 Black Holes 13911.1 The Schwarzschild radius 13911.2 Singularities 14011.3 Radial rays in the Schwarzschild space-time 14111.4 Schwarzschild coordinate systems 14311.5 The black hole space-time 14511.6 Special orbits around black holes 14711.7 Black holes in physics and in astrophysics 14812 Cosmology 15512.1 Constant-curvature spaces 15612.2 The metric of the Universe 15812.3 The matter content of the Universe 15812.4 The Einstein equations 15913 Cosmological Models 16513.1 Simple solutions: matter and radiation 16513.2 Light travel, distances, and horizons 16913.2.1 Light travel in the cosmological metric 16913.2.2 Cosmological redshift 17013.2.3 The expansion rate 17113.2.4 The age of the Universe 17213.2.5 The distance-redshift relation and Hubble's law 17213.2.6 Cosmic horizons 17313.2.7 The luminosity and angular-diameter distances 17413.3 Ingredients for a realistic cosmological model 17513.4 Accelerating cosmologies 18014 General Relativity: The Next 100 Years 18314.1 Developing General Relativity 18314.2 Beyond General Relativity 18414.3 Into the future 187Advanced Topic A1 Geodesics in the Schwarzschild Space-Time 191A1.1 Geodesics and conservation laws 191A1.2 Schwarzschild geodesics for massive particles 192A1.3 Schwarzschild geodesics for massless particles 194Advanced Topic A2 The Solar System Tests in Detail 197A2.1 Newtonian orbits in detail 197A2.2 Perihelion shift in General Relativity 201A2.3 Light deflection 204A2.4 Time delay 205Advanced Topic A3 Weak Gravitational Fields and Gravitational Waves 209A3.1 Nearly-flat space-times 209A3.2 Gravitational waves 211A3.3 Sources of gravitational waves 214Advanced Topic A4 Gravitational Wave Sources and Detection 219A4.1 Gravitational waves from compact binaries 220A4.2 The energy in gravitational waves 223A4.3 Binary inspiral 224A4.4 Detecting gravitational waves 227A4.4.1 Laser interferometers 227A4.4.2 Pulsar timing 230A4.4.3 Interferometers in space 231Bibliography 233Answers to Selected Problems 237Index 263

O autorze

Mark Hindmarsh is Professor of Theoretical Physics with joint appointments at the University of Sussex, UK and the University of Helsinki, Finland. His research is focused on the physics of the Big Bang, and he is a member of the LISA consortium with particular expertise in the cosmological production of gravitational waves. He has taught at all levels of the undergraduate and postgraduate curriculum.Andrew Liddle is a Principal Researcher at the University of Lisbon in Portugal, with joint affiliations at the University of Edinburgh, UK, and the Perimeter Institute for Theoretical Physics, Waterloo, Canada. He researches the properties of our Universe and how these relate to fundamental physical laws, especially through understanding astronomical observations. He is involved in several international projects, including the Planck Satellite and the Dark Energy Survey.