A First Course in General Relativity

Citation

Schutz, B. F. (2009). A First Course in General Relativity (2nd ed.). Cambridge University Press.

Chapter Summary

The book has twelve chapters and an appendix summarising linear algebra, and each chapter ends with further reading and exercises. Schutz describes the first eight chapters as a sequence typical of many treatments, from special relativity through tensors and curved manifolds to the field equations, and the last four as applications chosen for their importance in modern astrophysics.

Chapter 1: Special Relativity

A review of special relativity from its geometrical point of view. It covers the two fundamental postulates, inertial observers, spacetime diagrams, constructing another observer's coordinates, the invariance of the interval, invariant hyperbolae, the Lorentz transformation, velocity composition and paradoxes, with an appendix dissecting the twin paradox. The opening makes clear that the reader is assumed competent in the elementary, algebraic version of the subject, and sets out to teach the geometrical one.

Chapters 2–3: Vector Analysis in Special Relativity; Tensor Analysis in Special Relativity

Chapter 2 defines vectors and vector algebra in spacetime and introduces the four-velocity, four-momentum, scalar product and photons. Chapter 3 builds the tensor machinery: the metric tensor, one-forms, tensors of higher rank, the metric as a map between vectors and one-forms, raising and lowering indices and differentiation of tensors.

Chapter 4: Perfect Fluids in Special Relativity

This chapter develops the description of matter needed later. It moves from dust and the number–flux vector to the stress–energy tensor, general fluids and perfect fluids, and ends with a section on the importance of all this for general relativity and one on Gauss' law.

Chapters 5–6: Preface to Curvature; Curved Manifolds

Chapter 5 introduces the relation between gravitation and curvature and practises tensor algebra and calculus in polar coordinates, including Christoffel symbols and noncoordinate bases. Chapter 6 develops differential geometry proper: differentiable and Riemannian manifolds, covariant differentiation, parallel transport, geodesics, the curvature tensor, the Bianchi identities and the Ricci and Einstein tensors.

Chapters 7–8: Physics in a Curved Spacetime; The Einstein Field Equations

Chapter 7 moves from differential geometry to gravity, with physics in slightly curved spacetimes and conserved quantities. Chapter 8 states the purpose and justification of the field equations, writes Einstein's equations, and examines weak fields and the Newtonian limit.

Chapter 9: Gravitational Radiation

The first of the applications chapters covers the propagation, detection and generation of gravitational waves, the energy they carry away and astrophysical sources. In this edition, the author expanded the treatment of detection by interferometers such as LIGO and the planned space-based LISA.

Chapters 10–11: Spherical Solutions for Stars; Schwarzschild Geometry and Black Holes

Chapter 10 derives static spherically symmetric spacetimes, the exterior and interior structure of a star and exact interior solutions, and ends with realistic stars and gravitational collapse. Chapter 11 follows trajectories in the Schwarzschild spacetime, examines the surface r = 2M, treats general black holes, observed black holes in astronomy and the Hawking process.

Chapter 12: Cosmology

Completely rewritten for the second edition and organized around an expanding universe that includes the cosmological constant. It covers cosmological kinematics (observing the expansion), cosmological dynamics (understanding it) and physical cosmology (the evolution of the observed universe). Schutz describes it as a brief taste of the subject.

Key Concepts

Chapters 1–4: Special Relativity in Geometric Form

• Invariance of the interval: the spacetime separation that all inertial observers agree on, the basis for the geometrical view.
• Four-vectors: four-velocity and four-momentum as geometric objects.
• Tensors and one-forms: the objects that carry physical laws in a form independent of coordinates.
• Stress–energy tensor: the description of matter and energy, such as perfect fluids, that sources gravity.

Chapters 5–8: Curved Spacetime

• Manifolds, geodesics and parallel transport: the geometric structure in which free particles move.
• Curvature tensor and Einstein tensor: measures of curvature built from the metric.
• Einstein field equations: the relation between spacetime curvature and its sources, with the weak-field Newtonian limit.

Chapters 9–12: Applications

• Gravitational waves: propagation, detection, generation and energy.
• Schwarzschild geometry and black holes: the exterior of a spherical mass and the horizon at r = 2M.
• Cosmological models: expanding universe models including the cosmological constant.

Critical Analysis

Strengths

• A stated aim and level. The preface says the book presents general relativity at the undergraduate level, assuming a minimum of prerequisites without watering down the subject, and lists the required background in full.
• Full treatment of the geometry. The author introduces differential geometry fully rather than relying only on curved surfaces, then uses the full nonlinear field equations, solving them in the plane and spherical cases.
• Modern applications. The second edition updates the last four chapters: gravitational-wave detection, neutron stars and pulsars, astrophysical evidence for black holes, and a rewritten cosmology chapter with the cosmological constant and accelerating expansion.
• Plenty of exercises. The publisher counts over 300 exercises, many new to this edition, and the first-edition preface says that doing a selection is essential, with several chapters containing more than 30.

Limitations

• Classical only. The author says the subject remains classical general relativity: apart from a brief discussion of Hawking radiation, there is no quantization.
• Selective in mathematics. Nonmetric manifold theory, Lie derivatives and fibre bundles are left out, and gravitational waves are studied mainly in the linear approximation.
• Little on experimental tests. The author gives systematic discussion of experimental tests and alternative theories less prominence than traditional texts, treating them as they arise.
• Cosmology is brief. The author says that Chapter 12 can give only a brief taste of what astronomers have learned about inflation, dark matter and dark energy.
• Solutions not in the book. The exercise solutions were removed from the second edition and are provided on the book's website; the publisher notes that solutions for instructors are password protected, so self-study readers may not have them.

Where It Fits in the Course

Schutz is an undergraduate-level first course in general relativity and sits after special relativity in this course list. Taylor and Wheeler's Spacetime Physics is a short introduction to special relativity built on the spacetime interval, and gives the geometric thinking that Schutz's Chapter 1 reviews. Krane's Modern Physics and Thornton, Rex and Hood's Modern Physics for Scientists and Engineers treat special relativity and quantum physics at the sophomore level (Thornton–Rex–Hood closes with chapters on general relativity and cosmology), and OpenStax University Physics Volume 3 introduces special relativity in a free calculus-based text. Sean Carroll's Spacetime and Geometry is the graduate-level alternative that develops manifolds and curvature carefully before moving to gravitation and goes on to quantum field theory in curved spacetime; Schutz is the gentler route to the same ground and a good preparation for it.

How to Use This Book

The preface lists the assumed background: special relativity including the Lorentz transformation and relativistic mechanics; Euclidean vector calculus; ordinary and simple partial differential equations; thermodynamics and hydrostatics; Newtonian gravity; and enough quantum mechanics to know what a photon is. The author designed the book to be read in sequence in a one-year course, and suggests it can be shortened.

A practical plan:

• Core: Chapters 1–8 in order, since each builds on the last; do exercises throughout, because the author states that the early easy and medium ones give practice without which later chapters are much less comprehensible.
• Applications to choose from: Chapters 9 (gravitational waves), 10–11 (stars and black holes) and 12 (cosmology). The preface suggests that a half-year course could aim for gravitational waves and black holes by skipping parts of Chapters 1–3 and most of 4, 7 and 10, or for stellar structure and cosmology by moving quickly through the first four chapters and skipping Chapters 9 and 11.
• Alongside: use the further reading at the end of each chapter. Because the second edition prints no exercise solutions, self-study readers may need to check answers with others or through the book's website.