com Contents RELATIVITY, GRAVITATION AND COSMOLOGY Introduction 9 Chapter 1 Special relativity and spacetime 11 Introduction 11 1.1 Basic concepts of special relativity 12 1.1 Events, frames of reference and observers 12 1.2 The postulates of special relativity 14 1.1 The Galilean transformations 16 1.2 The Lorentz transformations 18 1.3 A derivation of the Lorentz transformations 21 1.4 Intervals and their transformation rules 23 1.3 Consequences of the Lorentz transformations 24 1.3 The relativity of simultaneity 27 1.4 The Doppler effect 28 1.5 The velocity transformation 29 1.1 Spacetime diagrams, lightcones and causality 31 1.2 Spacetime separation and the Minkowski metric 35 1.3 The twin effect 38 Chapter 2 Special relativity and physical laws 45 Introduction 45 2.1 Invariants and physical laws 46 2.1 The invariance of physical quantities 46 2.2 The invariance of physical laws 47 2.2 The laws of mechanics 49 2.2 Relativistic kinetic energy 52 2.3 Total relativistic energy and mass energy 54 2.5 The energy–momentum relation 58 2.6 The conservation of energy and momentum 60 2.3 The laws of electromagnetism 67 5 www.1 The conservation of charge 67 2.2 The Lorentz force law 68 2.3 The transformation of electric and magnetic fields 73 2.4 The Maxwell equations 74 2.5 Four-tensors 75 Chapter 3 Geometry and curved spacetime 80 Introduction 80 3.1 Line elements and differential geometry 82 3.1 Line elements in a plane 82 3.2 Metrics and connections 90 3.1 Metrics and Riemannian geometry 90 3.2 Connections and parallel transport 92 3.1 Most direct route between two points 97 3.2 Shortest distance between two points 98 3.1 Curvature of a curve in a plane 101 3.2 Gaussian curvature of a two-dimensional surface 102 3.3 Curvature in spaces of higher dimensions 104 3.4 Curvature of spacetime 106 Chapter 4 General relativity and gravitation 110 Introduction 110 4.1 The founding principles of general relativity 111 4.1 The principle of equivalence 112 4.2 The principle of general covariance 116 4.3 The principle of consistency 124 4.2 The basic ingredients of general relativity 126 4.1 The energy–momentum tensor 126 4.2 The Einstein tensor 132 4.3 Einstein’s field equations and geodesic motion 133 4.1 The Einstein field equations 134 4.3 The Newtonian limit of Einstein’s field equations 138 4.4 The cosmological constant 139 Chapter 5 Schwarzschild spacetime 144 Introduction 144 5.1 The metric of Schwarzschild spacetime 145 6 www.1 The Schwarzschild metric 145 5.2 Derivation of the Schwarzschild metric 146 5.2 Properties of Schwarzschild spacetime 151 5.3 Coordinates and measurements in Schwarzschild spacetime 154 5.1 Frames and observers 155 5.2 Proper time and gravitational time dilation 156 5.4 Geodesic motion in Schwarzschild spacetime 160 5.1 The geodesic equations 161 5.2 Constants of the motion in Schwarzschild spacetime 162 5.3 Orbital motion in Schwarzschild spacetime 166 Chapter 6 Black holes 171 Introduction 171 6.1 Introducing black holes 171 6.1 A black hole and its event horizon 171 6.2 A brief history of black holes 172 6.3 The classification of black holes 175 6.2 Non-rotating black holes 176 6.1 Falling into a non-rotating black hole 177 6.2 Observing a fall from far away 179 6.3 Tidal effects near a non-rotating black hole 183 6.4 The deflection of light near a non-rotating black hole 186 6.5 The event horizon and beyond 187 6.3 Rotating black holes 192 6.1 The Kerr solution and rotating black holes 192 6.2 Motion near a rotating black hole 194 6.4 Quantum physics and black holes 198 6.2 Singularities and quantum physics 200 Chapter 7 Testing general relativity 204 Introduction 204 7.1 The classic tests of general relativity 204 7.1 Precession of the perihelion of Mercury 204 7 www.2 Deflection of light by the Sun 205 7.3 Gravitational redshift and gravitational time dilation 206 7.4 Time delay of signals passing the Sun 211 7.2 Satellite-based tests 213 7.1 Geodesic gyroscope precession 213 7.3 The LAGEOS satellites 215 7.1 Gravitational waves and the Einstein field equations 226 7.2 Methods of detecting gravitational waves 229 7.3 Likely sources of gravitational waves 231 Chapter 8 Relativistic cosmology 234 Introduction 234 8.1 Basic principles and supporting observations 235 8.1 The applicability of general relativity 235 8.2 The cosmological principle 236 8.2 Robertson–Walker spacetime 242 8.1 The Robertson–Walker metric 243 8.2 Proper distances and velocities in cosmic spacetime 245 8.3 The cosmic geometry of space and spacetime 247 8.3 The Friedmann equations and cosmic evolution 251 8.1 The energy–momentum tensor of the cosmos 251 8.2 The Friedmann equations 254 8.3 Three cosmological models with k = 0 256 8.4 Friedmann–Robertson–Walker models in general 259 8.4 Friedmann–Robertson–Walker models and observations 263 8.1 Cosmological redshift and cosmic expansion 263 8.2 Density parameters and the age of the Universe 269 8.3 Horizons and limits 270 Appendix 277 Solutions 279 Acknowledgements 307 Index 308 8 www.com Introduction On the cosmic scale, gravitation dominates the universe. Nuclear and electromagnetic forces account for the detailed processes that allow stars to shine and astronomers to see them. But it is gravitation that shapes the universe, determining the geometry of space and time and thus the large-scale distribution of galaxies. Providing insight into gravitation – its effects, its nature and its causes – is therefore rightly seen as one of the most important goals of physics and astronomy.
Through more than a thousand years of human history the common explanation of gravitation was based on the Aristotelian belief that objects had a natural place in an Earth-centred universe that they would seek out if free to do so. For about two and a half centuries the Newtonian idea of gravity as a force held sway. Then, in the twentieth century, came Einstein’s conception of gravity as a manifestation of spacetime curvature. It is this latter view that is the main concern of this book.
The story of Einsteinian gravitation begins with a failure. Einstein’s theory of special relativity, published in 1905 while he was working as a clerk in the Swiss Figure 1 Albert Einstein Patent Office in Bern, marked an enormous step forward in theoretical physics (1879–1955) depicted during the and soon brought him academic recognition and personal fame. However, it also time that he worked at the Patent showed that the Newtonian idea of a gravitational force was inconsistent with the Office in Bern. While there, he relativistic approach and that a new theory of gravitation was required.
Ten years published a series of papers later, Einstein’s general theory of relativity met that need, highlighting the relating to special relativity, important role of geometry in accounting for gravitational phenomena and leading quantum physics and statistical on to concepts such as black holes and gravitational waves. Within a year and a mechanics. He was awarded the half of its completion, the new theory was providing the basis for a novel approach Nobel Prize for Physics in 1921, to cosmology – the science of the universe – that would soon have to take account mainly for his work on the of the astronomy of galaxies and the physics of cosmic expansion. The change in photoelectric effect.
thinking demanded by relativity was radical and profound. Its mastery is one of the great challenges and greatest delights of any serious study of physical science. This book begins with two chapters devoted to special relativity. These are followed by a mainly mathematical chapter that provides the background in geometry that is needed to appreciate Einstein’s subsequent development of the theory.
Chapter 4 examines the basic principles and assumptions of general relativity – Einstein’s theory of gravity – while Chapters 5 and 6 apply the theory to an isolated spherical body and then extend that analysis to non-rotating and rotating black holes. Chapter 7 concerns the testing of general relativity, including the use of astronomical observations and gravitational waves. Finally, Chapter 8 examines modern relativistic cosmology, setting the scene for further and ongoing studies of observational cosmology. The text before you is the result of a collaborative effort involving a team of authors and editors working as part of the broader effort to produce the Open University course S383 The Relativistic Universe.
Details of the team’s membership and responsibilities are listed elsewhere but it is appropriate to acknowledge here the particular contributions of Jim Hague regarding Chapters 1 and 2, Derek Capper concerning Chapters 3, 4 and 7, and Aiden Droogan in relation to Chapters 5, 6 and 8. Robert Lambourne was responsible for planning and producing the final unified text which benefited greatly from the input of the S383 Course Team Chair, Andrew Norton, and the attention of production editor 9 www.com Introduction Peter Twomey. The whole team drew heavily on the work and wisdom of an earlier Open University Course Team that was responsible for the production of the course S357 Space, Time and Cosmology. A major aim for this book is to allow upper-level undergraduate students to develop the skills and confidence needed to pursue the independent study of the many more comprehensive texts that are now available to students of relativity, gravitation and cosmology.
To facilitate this the current text has largely adopted the notation used in the outstanding book by Hobson et al. General Relativity : An Introduction for Physicists, M. Lasenby, Cambridge University Press, 2006. Other books that provide valuable further reading are (roughly in order of increasing mathematical demand): An Introduction to Modern Cosmology, A.
Relativity, Gravitation and Cosmology : A Basic Introduction, T-P. Cheng, Oxford University Press: 2005. Introducing Einstein’s Relativity, R. d’Inverno, Oxford University Press, 1992.
Relativity : Special, General and Cosmological, W. Rindler, Oxford University Press, 2001. Weinberg, Cambridge University Press, 2008. Two useful sources of reprints of original papers of historical significance are: The Principle of Relativity, A.
Einstein et al., Dover, New York, 1952. Cosmological Constants, edited by J. Feinberg, Columbia University Press, 1986. Those wishing to undertake background reading in astronomy, physics and mathematics to support their study of this book or of any of the others listed above might find the following particularly helpful: An Introduction to Galaxies and Cosmology, edited by M.
Lambourne, Cambridge University Press, 2003. The seven volumes in the series The Physical World, edited by R. Norton et al., Institute of Physics Publishing, 2000. (Go to www.org for further details.) The paired volumes Basic Mathematics for the Physical Sciences, edited by R.
Further Mathematics for the Physical Sciences, edited by M.com Chapter 1 Special relativity and spacetime Introduction In two seminal papers in 1861 and 1864, and in his treatise of 1873, James Clerk Maxwell (Figure 1.1), Scottish physicist and genius, wrote down his revolutionary unified theory of electricity and magnetism, a theory that is now summarized in the equations that bear his name. One of the deep results of the theory introduced by Maxwell was the prediction that wave-like excitations of combined electric and magnetic fields would travel through a vacuum with the same speed as light. It was soon widely accepted that light itself was an electromagnetic disturbance propagating through space, thus unifying electricity and magnetism with optics. The fundamental work of Maxwell opened the way for an understanding of the universe at a much deeper level.
Maxwell himself, in common with many scientists of the nineteenth century, believed in an all-pervading medium called the ether, through which electromagnetic disturbances travelled, just as ocean waves travelled through water. Maxwell’s theory predicted that light travels with the same speed in all directions, so it was generally assumed that the theory Figure 1.1 James Clerk predicted the results of measurements made using equipment that was at rest with Maxwell (1831–1879) respect to the ether. Since the Earth was expected to move through the ether as it developed a theory of orbited the Sun, measurements made in terrestrial laboratories were expected to electromagnetism that was show that light actually travelled with different speeds in different directions, already compatible with special allowing the speed of the Earth’s movement through the ether to be determined. relativity theory several decades However, the failure to detect any variations in the measured speed of light, most before Einstein and others notably by A.