A First Course in General Relativity
Schutz (Bernard)
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Back Cover Blurb

  1. General relativity has become one of the central pillars of theoretical physics, with important applications in both astrophysics and high-energy particle physics, and no modern theoretical physicist's education should be regarded as complete without some study of the subject. This textbook, based on the author's own undergraduate teaching, develops general relativity and its associated mathematics from a minimum of prerequisites, leading to a physical understanding of the theory in some depth. It reinforces this understanding by making a detailed study of the theory's most important applications – neutron stars, black holes, gravitational waves, and cosmology – using the most up-to-date astronomical developments.
  2. The book begins by introducing four-vectors and spacetime diagrams for special relativity, and then studies tensor algebra and fluid dynamics within special relativity. The next chapters develop differential geometry, beginning with curved coordinates in the two-dimensional plane. There follows a discussion of the equivalence principle, of the curvature of spacetime, and of physics in curved spacetimes. Einstein's equations are introduced, followed by an extraction of their Newtonian and weak-field limits. The book's longest chapter is a thorough study of gravitational waves; their physical effects, their production by astrophysical sources, the associated radiation-reaction effects they produce, and methods for their detection in the laboratory. Strong gravitational fields are discussed in a chapter devoted to compact stars and a longer one on black holes, both spherical and rotating, which finishes with a discussion of the Hawking effect, the quantum mechanical emission of radiation from black holes. The book concludes with a chapter on cosmological models and the physical history of the universe. Each chapter is followed by numerous problems that range from easy to very difficult. A few require programmable calculators or computers. Many solutions are provided.
  3. The book is suitable for a one-year course for beginning graduate students or for undergraduates in physics who have studied special relativity, vector calculus, and electrostatics. Graduate students should be able to use the book selectively for half-year courses.

  1. Special relativity
      1.1 Fundamental principles of special relativity theory (SR)
      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
      1.9 The Lorentz transformation
      1.10 The velocity-composition law
      1.11 Paradoxes and physical intuition
      1.12 Bibliography
      1.13 Appendix
      1.14 Exercises
  2. Vector analysis in special relativity
      2.1 Definition of a vector
      2.2 Vector algebra
      2.3 The four-velocity
      2.4 The four-momentum
      2.5 Scalar product
      2.6 Applications
      2.7 Photons
      2.8 Bibliography
      2.9 Exercises
  3. Tensor analysis in special relativity
      3.1 The metric tensor
      3.2 Definition of tensors
      3.3 The (0 1) tensors: one-forms
      3.4 The (0 2) tensors
      3.5 Metric as a mapping of vectors into one-forms
      3.6 Finally: (M N) tensors
      3.7 Index 'raising' and 'lowering'
      3.8 Differentiation of tensors
      3.9 Bibliography
      3.10 Exercises
  4. Perfect fluids in special relativity
      4.1 Fluids
      4.2 Dust: The number-flux vector N
      4.3 One-forms and surfaces
      4.4 Dust again: The stress-energy tensor
      4.5 General fluids
      4.6 Perfect fluids
      4.7 Importance for general relativity
      4.8 Gauss' law
      4.9 Bibliography
      4 10 Exercises
  5. Preface to curvature
      5.1 On the relation of gravitation to curvature
      5.2 Tensor algebra in polar coordinates
      5.3 Tensor calculus in polar coordinates
      5.4 Christoffel symbols and the metric
      5.5 The tensorial nature of […]
      5.6 Noncoordinate bases
      5.7 Looking ahead
      5.8 Bibliography
      5.9 Exercises
  6. Curved manifolds
      6.1 Differentiable manifolds and tensors
      6.2 Riemannian manifolds
      6.3 Covariant differentiation
      6.4 Parallel-transport, geodesies and curvature
      6.5 The curvature tensor
      6.6 Bianchi identities; Ricci and Einstein tensors
      6.7 Curvature in perspective
      6.8 Bibliography
      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 Bibliography
      7.6 Exercises
  8. The Einstein field equations
      8.1 Purpose and justification of the field equations
      8.2 Einstein's equations
      8.3 Einstein's equations for weak gravitational fields
      8.4 Newtonian gravitational fields
      8.5 Bibliography
      8.6 Exercises
  9. Gravitational radiation
      9.1 The propagation of gravitational waves
      9.2 The detection of gravitational waves
      9.3 The generation of gravitational waves
      9.4 The energy carried away by gravitational waves
      9.5 Bibliography
      9.6 Exercises
  10. Spherical solutions for stars
      10.1 Coordinates for spherically symmetric spacetimes
      10.2 Static spherically symmetric spacetimes
      10.3 Static perfect fluid Einstein equations
      10.4 The exterior geometry
      10.5 The interior structure of the star
      10.6 Exact interior solutions
      10.7 Realistic stars and gravitational collapse
      10.8 Bibliography
      10.9 Exercises
  11. Schwarzschild geometry and black holes
      11.1 Trajectories in the Schwarzschild spacetime
      11.2 Nature of the surface r = 2M
      11.3 More-general black holes
      11.4 Quantum mechanical emission of radiation by black holes: The Hawking process
      11.5 Bibliography
      11.6 Exercises
  12. Cosmology
      12.1 What is cosmology?
      12.2 General-relativistic cosmological models
      12.3 Cosmological observations
      12.4 Physical cosmology
      12.5 Bibliography
      12.6 Exercises

    Appendix A: Summary of linear algebra
    Appendix B: Hints and solutions to selected exercises

Back Cover Blurb
  1. This book has evolved from lecture notes for a full-year undergraduate course in general relativity which I taught from 1975 to 1980, an experience which firmly convinced me that general relativity is not significantly more difficult for undergraduates to learn than the standard undergraduate-level treatments of electromagnetism and quantum mechanics1. The explosion of research interest in general relativity in the past 20 years, largely stimulated by astronomy, has not only led to a deeper and more complete understanding of the theory; it has also taught us simpler, more physical ways of understanding it. Relativity is now in the mainstream of physics and astronomy, so that no theoretical physicist can be regarded as broadly educated without some training in the subject. The formidable reputation relativity acquired in its early years (Interviewer: 'Professor Eddington, is it true that only three people in the world understand Einstein's theory?' Eddington: 'Who is the third?') is today perhaps the chief obstacle that prevents it being more widely taught to theoretical physicists. The aim of this textbook is to present general relativity at a level appropriate for undergraduates, so that the student will understand the basic physical concepts and their experimental implications, will be able to solve elementary problems, and will be well prepared for the more advanced texts on the subject.
  2. In pursuing this aim, I have tried to satisfy two competing criteria: first, to assume a minimum of prerequisites: and second, to avoid watering down the subject matter. Unlike most introductory texts, this one does not assume that the student has already studied electromagnetism in its manifestly relativistic formulation, the theory of electromagnetic waves, or fluid dynamics. The necessary fluid dynamics is developed in the relevant chapters. The main consequence of not assuming a familiarity with electromagnetic waves is that gravitational waves have to be introduced slowly: the wave equation is studied from scratch. A full list of prerequisites appears below.
  3. The second guiding principle, that of not watering down the treatment, is very subjective and rather more difficult to describe. I have tried to introduce differential geometry fully, not being content to rely only on analogies with curved surfaces, but I have left out subjects that are not essential to general relativity at this level, such as nonmetric manifold theory. Lie derivatives, and fiber bundles. I have introduced the full nonlinear field equations, not just those of linearized theory, but I solve them only in the plane and spherical cases, quoting and examining, in addition, the Kerr solution. I study gravitational waves mainly in the linear approximation, but go slightly beyond it to derive the energy in the waves and the reaction effects in the wave emitter. I have tried in each topic to supply enough foundation for the student to be able to go to more advanced treatments without having to start over again at the beginning.
  4. The first part of the book, up to Ch. 8, introduces the theory in a sequence which is typical of many treatments: a review of special relativity, development of tensor analysis and continuum physics in special relativity, study of tensor calculus in curvilinear coordinates in Euclidean and Minkowski spaces, geometry of curved manifolds, physics in a curved spacetime, and finally the field equations. The remaining four chapters study a few topics which I have chosen because of their importance in modern astrophysics. The chapter on gravitational radiation is more detailed than usual at this level because the observation of gravitational waves may be one of the most significant developments in astronomy in the next decade. The chapter on spherical stars includes, besides the usual material, a useful family of exact compressible solutions due to Buchdahl. A long chapter on black holes studies in some detail the physical nature of the horizon, going as far as the Kruskal coordinates, then exploring the rotating (Kerr) black hole, and concluding with a simple discussion of the Hawking effect, the quantum mechanical emission of radiation by black holes. The concluding chapter on cosmology derives the homogeneous and isotropic metrics and briefly studies the physics of cosmological observation and evolution. There is an appendix summarizing the linear algebra needed in the text, and another appendix containing hints and solutions for selected exercises. One subject I have decided not to give as much prominence to as other texts traditionally have is experimental tests of general relativity and of alternative theories of gravity. Points of contact with experiment are treated as they arise, but systematic discussions of tests now require whole books (Will 1981). Physicists today have far more confidence in the validity of general relativity than they had a decade or two ago, and I believe that an extensive discussion of alternative theories is therefore almost as out of place in a modern elementary text on gravity as it would be in one on electromagnetism.
  5. The student is assumed already to have studied: special relativity, including the Lorentz transformation and relativistic mechanics; Euclidean vector calculus; ordinary and simple partial differential equations; thermodynamics and hydrostatics; Newtonian gravity (simple stellar structure would be useful but not essential); and enough elementary quantum mechanics2 to know what a photon is.
  6. The notation and conventions are essentially the same as in "Misner (Charles W.), Thorne (Kip S.) & Wheeler (John Archibald) - Gravitation" (W. H. Freeman 1973), which may be regarded as one possible follow-on text after this one. The physical point of view and development of the subject are also inevitably influenced by that book, partly because Thorne was my teacher and partly because Gravitation has become such an influential text. But because I have tried to make the subject accessible to a much wider audience, the style and pedagogical method of the present book are very different.
  7. Regarding the use of the book, it is designed to be studied sequentially as a whole, in a one-year course, but it can be shortened to accommodate a half-year course. Half-year courses probably should aim at restricted goals. For example, it would be reasonable to aim to teach gravitational waves and black holes in half a year to students who have already studied electromagnetic waves, by carefully skipping some of Chs. 1-3 and most of Chs. 4, 7, and 10. Students with preparation in special relativity and fluid dynamics could learn stellar structure and cosmology in half a year, provided they could go quickly through the first four chapters and then skip Chs. 9 and 11. A graduate-level course can, of course, go much more quickly, and it should be possible to cover the whole text in half a year.
  8. Each chapter is followed by a set of exercises, which range from trivial ones (filling in missing steps in the body of the text, manipulating newly introduced mathematics) to advanced problems that considerably extend the discussion in the text. Some problems require programmable calculators or computers. I cannot overstress the importance of doing a selection of problems. The easy and medium-hard ones in the early chapters give essential practice, without which the later chapters will be much less comprehensible. The medium-hard and hard problems of the later chapters are a test of the student's understanding. It is all too common in relativity for students to find the conceptual framework so interesting that they relegate problem solving to second place. Such a separation is false and dangerous: a student who can't solve problems of reasonable difficulty doesn't really understand the concepts of the theory either. There are generally more problems than one would expect a student to solve; several chapters have more than 30. The teacher will have to select them judiciously. Another rich source of problems is the Problem Book in Relativity and Gravitation, Lightman et al. (Princeton University Press 1975).
  9. I am indebted to …


Useful first few chapters on Special Relativity

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  1. Blue: Text by me; © Theo Todman, 2020
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