# Spacetime and Geometry

### An Introduction to General Relativity

*Spacetime and Geometry* is a graduate-level textbook on general relativity, published by Addison-Wesley.

**Buy it:**

Amazon.com ● Addison Wesley

“For if each Star is little more a mathematical Point,

located upon the Hemisphere of Heaven by Right Ascension and Declination,

then all the Stars, taken together, tho’ innumerable,

must like any other set of points,

in turn represent some single gigantick Equation,

to the mind of God as straightforward as, say, the Equation of a Sphere,—

to us unreadable, incalculable.

A lonely, uncompensated, perhaps even impossible Task,—

yet some of us must ever be seeking, I suppose.”

— Thomas Pynchon, *Mason & Dixon*

## About the Book

In 1996 I taught a one-semester graduate course graduate course in general relativity at MIT. Along the way I typed up a detailed set of lecture notes. The book *Spacetime and Geometry* is a significantly revised and expanded version of these notes; about half of the finished book is completely new. The lecture notes will continue to be available for free online.

The philosophy of the book is to provide an accessible, useful, and pedagogical introduction to general relativity. In particular, no effort has been made to write a comprehensive reference book. More on the approach taken can be found in the preface.

You can order the book online from Amazon.com or from Addison Wesley.

### Contents

**Preface****Special Relativity and Flat Spacetime**- Prelude
- Space and time, separately and together
- Lorentz transformations
- Vectors
- Dual vectors (one-forms)
- Tensors
- Manipulating tensors
- Maxwell’s equations
- Energy and momentum
- Classical field theory
- Exercises

**Manifolds**- Gravity as geometry
- What is a manifold?
- Vectors again
- Tensors again
- The metric
- An expanding universe
- Causality
- Tensor densities
- Differential forms
- Integration
- Exercises

**Curvature**- Overview
- Covariant derivatives
- Parallel transport and geodesics
- Properties of geodesics
- The expanding universe revisited
- The Riemann curvature tensor
- Properties of the Riemann tensor
- Symmetries and Killing vectors
- Maximally symmetric spaces
- Geodesic deviation
- Exercises

**Gravitation**- Physics in curved spacetime
- Einstein’s equation
- Lagrangian formulation
- Properties of Einstein’s equation
- The cosmological constant
- Energy conditions
- The Equivalence Principle revisited
- Alternative theories
- Exercises

**The Schwarzschild Solution**- The Schwarzschild metric
- Birkhoff’s theorem
- Singularities
- Geodesics of Schwarzschild
- Experimental tests
- Schwarzschild black holes
- The maximally extended Schwarzschild solution
- Stars and black holes
- Exercises

**More General Black Holes**- The black hole zoo
- Event Horizons
- Killing Horizons
- Mass, charge, and spin
- Charged (Reissner-Nordström) black holes
- Rotating (Kerr) black holes
- The Penrose process and black-hole thermodynamics
- Exercises

**Perturbation Theory and Gravitational Radiation**- Linearized theory and gauge transformations
- Degrees of freedom
- Newtonian fields and photon trajectories
- Gravitational wave solutions
- Production of gravitational waves
- Energy loss due to gravitational radiation
- Detection of gravitational waves
- Exercises

**Cosmology**- Maximally symmetric universes
- Robertson-Walker metrics
- The Friedmann equation
- Evolution of the scale factor
- Redshifts and distances
- Gravitational lensing
- Our universe
- Inflation
- Exercises

**Quantum Field Theory in Curved Spacetime**- Introduction
- Quantum mechanics
- Quantum field theory in flat spacetime
- Quantum field theory in curved spacetime
- The Unruh effect
- The Hawking effect and black hole evaporation

**Appendices**- Maps between manifolds
- Diffeomorphisms and Lie derivatives
- Submanifolds
- Hypersurfaces
- Stokes’s theorem
- Geodesic congruences
- Conformal transformations
- Conformal diagrams
- The parallel propagator
- Non-coordinate bases

## Preface

General relativity is the most beautiful physical theory ever invented. It describes one of the most pervasive features of the world we experience — gravitation — in terms of an elegant mathematical structure — the differential geometry of curved spacetime — leading to unambigous predictions that have received spectacular experimental confirmation. Consequences of general relativity, from the big bang to black holes, are often what get young people first interested in physics, and it is an unalloyed joy to finally reach the point in one’s studies where these phenomena may be understood at a rigorous quantitative level. If you are contemplating reading this book, that point is here.

In recent decades, general relativity (GR) has become an integral and indispensable part of modern physics. For a long time after it was proposed by Einstein in 1916, GR was counted as a shining achievement that lay somewhat outside the mainstream of interesting research. Increasingly, however, contemporary students in a variety of specialties are finding it necessary to study Einstein’s theory. In addition to being an active research area in its own right, GR is part of the standard syllabus for anyone interested in astrophysics, cosmology, string theory, and even particle physics. This is not to slight the more pragmatic uses of GR, including the workings of the Global Positioning System (GPS) satellite network.

There is no shortage of books on GR, and many of them are excellent. Indeed, approximately thirty years ago witnessed the appearance of no fewer than three books in the subject, each of which has become a classic in its own right: those by Weinberg (1972), Misner, Thorne, and Wheeler (1973), and Hawking and Ellis (1975). Each of these books is suffused with a strongly-held point of view advocated by the authors. This has led to a love-hate relationship between these works and their readers; in each case, it takes little effort to find students who will declare them to be the best textbook ever written, or other students who find them completely unpalatable. For the individuals in question, these judgments may very well be correct; there are many different ways to approach this subject.

The present book has a single purpose: to provide a clear introduction to general relativity, suitable for graduate students or advanced undergraduates. I have attempted to include enough material so that almost any one-semester introductory course on GR can find the appropriate subjects covered in the text, but not too much more than that. In particular, I have tried to resist the temptation to write a comprehensive reference book. The only goal of this book is to teach you GR.

An intentional effort has been made to prefer the conventional over the ideosyncratic. If I can be accused of any particular ideological bias, it would be a tendency think of general relativity as a field theory, a point of view which helps one to appreciate the connections between GR, particle physics, and string theory. At the same time, there are a number of exciting astrophysical applications of GR (black holes, gravitational lensing, the production and detection of gravitational waves, the early universe, the late universe, the cosmological constant), and I have endeavored to include at least enough background discussion of these issues to prepare students to tackle the current literature.

The primary question facing any introductory treatment of general relativity is the level of mathematical rigor at which to operate. There is no uniquely proper solution, as different students will respond with different levels of understanding and enthusiasm to different approaches. Recognizing this, I have tried to provide something for everyone. I have not shied away from detailed formalism, but have also attempted to include concrete examples and informal discussion of the concepts under consideration. Much of the most mathematical material has been relegated to appendices. Some of the material in the appendices is actually an integral part of the course (for example, the discussion of conformal diagrams), but an individual reader or instructor can decide just when it is appropriate to delve into them; signposts are included in the body of the text.

Surprisingly, there are very few formal prerequisites for learning general relativity; most of the material is developed as you go along. Certainly no prior exposure to Riemannian geometry is assumed, nor would it necessarily be helpful. It would be nice to have already studied some special relativity; although a discussion is included in Chapter One, its purpose is more to review the basics and and introduce some notation, rather than to provide a self-contained introduction. Beyond that, some exposure to electromagnetism, Lagrangian mechanics, and linear algebra might be useful, but the essentials are included here.

The structure of the book should be clear. The first chapter is a review of special relativity and basic tensor algebra, including a brief discussion of classical field theory. The next two chapters introduce manifolds and curvature in some detail; some motivational physics is included, but building a mathematical framework is the primary goal. General relativity proper is introduced in Chapter Four, along with some discussion of alternative theories. The next four chapters discuss the three major applications of GR: black holes (two chapters), perturbation theory and gravitational waves, and cosmology. Each of these subjects has witnessed an explosion of research in recent years, so the discussions here will be necessarily introductory, but I have tried to emphasize issues of relevance to current work. These three applications can be covered in any order, although there are interdependencies highlighted in the text. Discussions of experimental tests are sprinkled through these chapters. Chapter Nine is a brief introduction to quantum field theory in curved spacetime; this is not a necessary part of a first look at GR, but has become increasingly important to work in quantum gravity and cosmology, and therefore deserves some mention. On the other hand, a few topics are scandalously neglected; the initial value problem and cosmological perturbation theory come to mind, but there are others. Fortunately there is no shortage of other resources. The appendices serve various purposes: there are discussions of technical points which were avoided in the body of the book, crucial concepts which could have been put in various different places, and extra topics which are useful but outside the main development.

Since the goal of the book is pedagogy rather than originality, I have often leaned heavily on other books (listed in the bibliography) when their expositions seemed perfectly sensible to me. When this leaning was especially heavy, I have indicated it in the text itself. It will be clear that a primary resource was the book by Wald (1984), which has become a standard reference in the field; readers of this book will hopefully be well-prepared to jump into the more advanced sections of Wald’s book.

This book grew out of a set of lecture notes that were prepared when I taught a course on GR at MIT. These notes are available on the web for free, and will continue to be so; they will be linked to the website listed below. Perhaps a little over half of the material here is contained in the notes, although the advantages of owning the book (several copies, even) should go without saying.

Countless people have contributed greatly both to my own understanding of general relativity and to this book in particular — too many to acknowledge with any hope of completeness. Some people, however, deserve special mention. Ted Pyne learned the subject along with me, taught me a great deal, and collaborated with me the first time we taught a GR course, as a seminar in the astronomy department at Harvard; parts of this book are based on our mutual notes. Nick Warner taught the course at MIT from which I first learned GR, and his lectures were certainly a very heavy influence on what appears here. Neil Cornish was kind enough to provide a wealth of exercises, many of which have been included at the end of each chapter. And among the many people who have read parts of the manuscript and offered suggestions, Sanaz Arkani-Hamed was kind enough to go through the entire thing in great detail.

I would also like to thank everyone who either commented in person or by email on different parts of the book; these include Tigran Aivazian, Teodora Beloreshka, Ed Bertschinger, Patrick Brady, Peter Brown, Jennifer Chen, Michele Ferraz Figueiro, Eanna Flanagan, Jacques Fric, Ygor Geurts, Marco Godina, Monica Guica, Jim Hartle, Tamas Hauer, Mark Hoffman, Daniel Holz, Ted Jacobson, Akash Kansagra, Chuck Keeton, Arthur Kosowsky, Eugene Lim, Jorma Louko, Robert A. McNees, Hayri Mutluay, Simon Ross, Itai Seggev, Robert Wald, and Barton Zwiebach. Apologies are due to anyone I may have neglected to mention. And along the way I was fortunate to be the recipient of wisdom and perspective from numerous people, including Shadi Bartsch, George Field, Deryn Fogg, Ilana Harrus, Gretchen Helfrich, Mari Ruti, Maria Spiropulu, Mark Trodden, and of course my family. (This wisdom often came in the form, “What were you thinking?”) Finally, I would like to thank the students in my GR classes, on whom the strategies deployed here were first tested, and express my gratitude to my students and collaborators, for excusing my book-related absences when I should have been doing research.

My friends who have written textbooks themselves tell me that the first printing of a book will sometimes contain mistakes. In the unlikely event that this happens here, there will be a list of errata kept at the website for the book: http://spacetimeandgeometry.net/. The website will also contain other relevant links of interest to readers.

During the time I was working on this book, I was supported by the National Science Foundation, the Department of Energy, the Alfred P. Sloan Foundation, and the David and Lucile Packard Foundation.

## Lecture Notes

##### This set of lecture notes on general relativity has been expanded into a textbook, * Spacetime and Geometry: An Introduction to General Relativity*, available for purchase online or at finer bookstores everywhere.

##### About 50% of the book is completely new; I’ve also polished and improved many of the explanations, and made the organization more flexible and user-friendly. The notes as they are will always be here for free.

These lecture notes are a lightly edited version of the ones I handed out while teaching Physics 8.962, the graduate course in General Relativity at MIT, during Spring 1996. Each of the chapters is available here as PDF. Constructive comments and general flattery may be sent to me via the address below. The notes as a whole are available as gr-qc/9712019, and in html from “Level 5” at Caltech. What is even more amazing, the notes have been translated into French by Jacques Fric. Je ne parle pas francais, mais cette traduction devrait etre bonne.

Dates refer to the last nontrivial modification of the corresponding file (fixing typos doesn’t count). Note that, unlike the book, no real effort has been made to fix errata in these notes, so be sure to check your equations.

In a hurry? Can’t be bothered to slog through lovingly detailed descriptions of subtle features of curved spacetime? Try the No-Nonsense Introduction to General Relativity, a 24-page condensation of the full-blown lecture notes (PDF).

While you are here check out the *Spacetime and Geometry* page — including the annotated bibilography of technical and popular books, many available for purchase online.

### Lecture Notes

1. Special Relativity and Flat Spacetime

(22 Nov 1997; 37 pages)

the spacetime interval — the metric — Lorentz transformations — spacetime diagrams — vectors — the tangent space — dual vectors — tensors — tensor products — the Levi-Civita tensor — index manipulation — electromagnetism — differential forms — Hodge duality — worldlines — proper time — energy-momentum vector — energy-momentum tensor — perfect fluids — energy-momentum conservation

2. Manifolds

(22 Nov 1997; 24 pages)

examples — non-examples — maps — continuity — the chain rule — open sets — charts and atlases — manifolds — examples of charts — differentiation — vectors as derivatives — coordinate bases — the tensor transformation law — partial derivatives are not tensors — the metric again — canonical form of the metric — Riemann normal coordinates — tensor densities — volume forms and integration

3. Curvature

(23 Nov 1997; 42 pages)

covariant derivatives and connections — connection coefficients — transformation properties — the Christoffel connection — structures on manifolds — parallel transport — the parallel propagator — geodesics — affine parameters — the exponential map — the Riemann curvature tensor — symmetries of the Riemann tensor — the Bianchi identity — Ricci and Einstein tensors — Weyl tensor — simple examples — geodesic deviation — tetrads and non-coordinate bases — the spin connection — Maurer-Cartan structure equations — fiber bundles and gauge transformations

4. Gravitation

(25 Nov 1997; 32 pages)

the Principle of Equivalence — gravitational redshift — gravitation as spacetime curvature — the Newtonian limit — physics in curved spacetime — Einstein’s equations — the Hilbert action — the energy-momentum tensor again — the Weak Energy Condition — alternative theories — the initial value problem — gauge invariance and harmonic gauge — domains of dependence — causality

5. More Geometry

(26 Nov 1997; 13 pages)

pullbacks and pushforwards — diffeomorphisms — integral curves — Lie derivatives — the energy-momentum tensor one more time — isometries and Killing vectors

6. Weak Fields and Gravitational Radiation

(26 Nov 1997; 22 pages)

the weak-field limit defined — gauge transformations — linearized Einstein equations — gravitational plane waves — transverse traceless gauge — polarizations — gravitational radiation by sources — energy loss

7. The Schwarzschild Solution and Black Holes

(29 Nov 1997; 53 pages)

spherical symmetry — the Schwarzschild metric — Birkhoff’s theorem — geodesics of Schwarzschild — Newtonian vs. relativistic orbits — perihelion precession — the event horizon — black holes — Kruskal coordinates — formation of black holes — Penrose diagrams — conformal infinity — no hair — charged black holes — cosmic censorship — extremal black holes — rotating black holes — Killing tensors — the Penrose process — irreducible mass — black hole thermodynamics

8. Cosmology

(1 Dec 1997; 15 pages)

homogeneity and isotropy — the Robertson-Walker metric — forms of energy-momentum — Friedmann equations — cosmological parameters — evolution of the scale factor — redshift — Hubble’s law

### Related sets of notes or tutorials

- General Relativity Tutorial, John Baez, UC Riverside

- Black Hole Lecture Notes, Paul Townsend, Cambridge

- Introduction to General Relativity (pdf), Gerard ‘t Hooft, Utrecht University

- Introduction to Differential Geometry and General Relativity, Stefan Waner, Hofstra University

### Links to GR resources

- Relativity Bookmarks, Syracuse University

- Relativity and Black Hole Links, Andrew Hamilton, University of Colorado

- Hyperspace, University of British Columbia

## Commentary/Reviews

**Customer reviews**are available at amazon.com and Barnes and Noble.

- The January 2005 issue of Physics Today has a joint review of my book and Jim Hartle’s book, by
**Jennie Traschen**. It’s only available online to subscribers, unfortunately. An excerpt:- Carroll’s text adds a new option to the list of solid general relativity texts appropriate for graduate students planning to do research in gravity, high-energy theory, or both. … Of course, every teacher has a different opinion about the amount of foundational mathematics to include in a general relativity class that has to fit into one semester. Carroll has successfully steered a middle course, and I believe his text will be very useful to students. … Both books are fine contributions that extend in useful and different ways the range of pedagogical choices available to instructors.

- A review by Eric Poisson appeared in
*Classical and Quantum Gravity*. Excerpt:- My favourite part of the book is the first three chapters on differential geometry. The presentation of the mathematical formalism is crystal clear and very enjoyable, and it comes with a large number of helpful (and attractive) diagrams… The ninth and final chapter of Carroll’s book is devoted to a topic that has never been covered in an introductory text: quantum field theory in curved spacetime. To include this was a truly inspired thought, and Carroll is to be congratulated for this initiative… A complete education in general relativity cannot exclude this important subject, and now we have a textbook that presents it in a clear, accessible way.

### Blurbs

- Sean Carroll won the Graduate Student Council Teaching Award for the general relativity course that he taught at MIT, and the course has remained legendary among our graduate students. This book can be expected to spread the legend worldwide. Starting at the very beginning, Carroll gives a clear, well-motivated, and intellectually lively development of general relativity, extending into cosmology and even the basics of quantum field theory in curved spacetime. The book is a gem. –Alan Guth,
*Massachusetts Institute of Technology*

- Sean Carroll brings a relaxed authority to the teaching of general relativity at the graduate level. The text is beautifully written and succeeds in linking modern mathematical formalism to physical applications.
*Spacetime and Geometry*continues the tradition of excellent relativity books and is likely to establish a common culture for the next generation of relativists. –Roger Blandford,*Kavli Institute for Particle Astrophysics and Cosmology*

- This is an excellent, up to date, graduate text in general relativity, written in an informal yet sophisticated style. It provides an efficient introduction to the gravitational physics essential for today’s understanding of black holes, gravitational waves, and the universe. –James Hartle,
*University of California, Santa Barbara*

- This is the best book on contemporary gravitational physics for everyone from beginning graduate students to established researchers. Anyone interested in modern high-energy physics needs to understand cosmology and black holes, and this is the book they should read. It will be the leading textbook on gravity for years to come. –Renata Kallosh,
*Stanford University*

- “Spacetime and Geometry” is well-presented, very accessible and right up-to-date. It provides a comprehensive treatment that invites rather than discourages new students of the subject. –Bernard Schutz,
*Max Planck Institute for Gravitational Physics*

- A seamless combination of mathematical sophistication and intuitive physical reasoning. –Arthur Kosowsky,
*Rutgers University*

- A very clear, crisp, and self-contained introduction to general relativity in the modern language of differential geometry that’s a pleasure to read. Most remarkably, the author presents the most intricate subjects with characteristic clarity, but without compromising their integrity. If you have any doubts read and compare Carroll against any other text in the market. –Ben Grinstein,
*University of California, San Diego*

## Bibliography

##### My Book

- S.M. Carroll,
*Spacetime and Geometry: An Introduction to General Relativity*(Addison-Wesley, 2003).

Not actually in the bibliography, but why not give you every chance to order it?

##### Special Relativity

- E.F. Taylor and J.A. Wheeler,
*Spacetime Physics*(Freeman, 1992).

A very nice introduction to special relativity, making a great effort to explain away the “paradoxes” this subject seems to engender. - A.P. French,
*Special Relativity*(W.W. Norton, 1968).

Somewhat less colorful than Taylor and Wheeler, but a straightforward introduction to special relativity.

##### Undergraduate General Relativity

- J.B. Hartle,
*Gravity: An Introduction to Einstein’s General Relativity*(Addison-Wesley, 2002).

Eases the exploration of GR by concentrating on examples of curved spacetimes and the behavior of particles in them, putting physics before formalism whenever possible. - B.F. Schutz,
*A First Course in General Relativity*(Cambridge, 1985).

This is a very nice introductory text, making a real effort to bridge the transition from common topics in undergraduate physics to the language and results of GR. - E.F. Taylor and J.A. Wheeler,
*Exploring Black Holes*(Benjamin/Cummings, 2000).

Uses black holes as a way to introduce physical principles of GR.

##### Graduate General Relativity

- Wald,
*General Relativity*(Chicago, 1984).

Thorough discussions of a number of advanced topics, including black holes, global structure, and spinors. An invaluable reference, this is the book to turn to if you need the right answer to a well-posed GR question. - C. Misner, K. Thorne and J. Wheeler,
*Gravitation*(Freeman, 1973).

The book that educated at least two generations of researchers in gravitational physics. Comprehensive and encyclopedic, the book is written in an often-ideosyncratic style that you will either like or not. - S. Weinberg,
*Gravitation and Cosmology*(Wiley, 1972).

A great book at what it does, especially strong on astrophysics, cosmology, and experimental tests. However, it takes an unusual non-geometric approach to the material, and doesn’t discuss black holes. Weinberg is much better than most of us at cranking through impressive calculations. - R. D’Inverno,
*Introducing Einstein’s Relativity*(Oxford, 1992).

A sensible and lucid introduction to general relativity, with solid coverage of the major topics necessary in a modern GR course. - A.P. Lightman, W.H. Press, R.H. Price, and S.A. Teukolsky,
*Problem Book in Relativity and Gravitation*(Princeton, 1975).

A sizeable collection of problems in all areas of GR, with fully worked solutions, making it all the more difficult for instructors to invent problems the students can’t easily find the answers to.

##### Advanced General Relativity

- S.W. Hawking and G.F.R. Ellis,
*The Large-Scale Structure of Space-Time*(Cambridge, 1973).

An advanced book which emphasizes global techniques, differential topology, and singularity theorems; a classic. - F. de Felice and C. Clarke,
*Relativity on Curved Manifolds*(Cambridge, 1990).

A mathematical approach, but with an excellent emphasis on physically measurable quantities. - R. Sachs and H. Wu,
*General Relativity for Mathematicians*(Springer-Verlag, 1977).

Just what the title says, although the typically dry mathematics prose style is here enlivened by frequent opinionated asides about both physics and mathematics (and the state of the world). - J. Stewart,
*Advanced General Relativity*(Cambridge, 2003).

A short but sweet introduction to some advanced topics, especially spinors, asymptotic structure, and the characteristic initial-value problem.

##### Mathematical Background

- B. Schutz,
*Geometrical Methods of Mathematical Physics*(Cambridge, 1980).

Another good book by Schutz, this one covering some mathematical points that are left out of the GR book (but at a very accessible level). Included are discussions of Lie derivatives, differential forms, and applications to physics other than GR. - T. Frankel,
*The Geometry of Physics: An Introduction*(Cambridge, 2001).

A rich, readable book on topics in geometry that are of real use to physics, including manifolds, bundles, curvature, Lie groups, and algebraic topology. - M. Nakahara,
*Geometry, Topology and Physics*(Institute of Physics, 2003).

An accessible introduction to differential geometry and topology, with an emphasis on topics of interest to physicists. - F.W. Warner,
*Foundations of Differentiable Manifolds and Lie Groups*(Springer-Verlag, 1983).

The standard text in the field, includes basic topics such as manifolds and tensor fields as well as more advanced subjects.

##### Specialized Topics

- J.D. Jackson,
*Classical Electrodynamics*(Wiley, 1999).

The classic reference for graduate-level electromagnetism. The problems have left indelible marks on generations of graduate students. - H. Goldstein et al.,
*Classical Mechanics*(Prentice-Hall, 2002).

The classic reference for graduate-level mechanics. An updated edition adds more discussion of nonlinear dynamics. - V.I. Arnold,
*Mathematical Methods of Classical Mechanics*(Springer-Verlag, 1989).

A scary book for some physicists, but an inspiring treatment of classical mechanics from a mathematically sophisticated point of view. A lot of good differential geometry here. - E.W. Kolb and M.S. Turner,
*The Early Universe*(Perseus, 1994).

Has become a standard reference for early-universe cosmology, including dark matter, phase transitions, and inflation. - A.R. Liddle and D. Lyth,
*Cosmological Inflation and Large-Scale Structure*(Cambridge, 2000).

Focusing on inflation and its implications for large-scale structure, gives a careful treatment of cosmological perturbation theory. - B.S. Ryden,
*An Introduction to Cosmology*(Addison-Wesley, 2002).

A very modern and physical introduction to topics in contemporary cosmology, aimed at advanced undergraduates or beginning graduate students. - S. Dodelson,
*Modern Cosmology*(Academic Press, 2003).

A graduate-level introduction to cosmology, emphasizing cosmological perturbations, large-scale structure, and the cosmic microwave background. - C.M. Will,
*Theory and Experiment in Gravitational Physics*(Cambridge, 1993).

A useful compendium of alternatives to GR and the experimental constraints on them, including a discussion of the parameterized post-Newtonian formalism. - S.L. Shapiro and S.A. Teukolsky,
*Black Holes, White Dwarfs, and Neutron Stars : The Physics of Compact Objects*(Wiley, 1983).

A self-contained introduction to the physics and astrophysics of compact stars and black holes. - M.E. Peskin and D.V. Schroeder,
*An Introduction to Quantum Field Theory*(Westview Press, 1995).

Has quickly become the standard textbook in quantum field theory. - J.G. Polchinski,
*String Theory*, Vol. 1 and Vol. 2 (Cambridge, 1998).

The standard two-volume introduction to modern string theory, including discussions of D-branes and string duality. - C.V. Johnson,
*D-Branes*(Cambridge, 1998).

A detailed introduction to the extended objects called D-branes which have become an indispensable part of string theory; prior knowledge of string theory itself not required. - E.E. Falco, P. Schneider, and J. Ehlers,
*Gravitational Lenses*(Springer-Verlag, 1999).

A thorough introduction to the theory and applications of gravitational lensing. - N.D. Birrell and P.C. Davies,
*Quantum Fields in Curved Spacetime*(Cambridge, 1994).

The standard book for people who want a practical introduction to quantum field theory in curved spacetime, including the Hawking effect. - R.M. Wald,
*Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics*(Chicago, 1994).

A careful and mathematically rigorous exposition of quantum fields in curved spacetimes; if you really want to know what a vacuum state is, look here.

##### Popular Books

- K.S. Thorne,
*Black Holes and Time Warps : Einstein’s Outrageous Legacy*(W.W. Norton, 1995).

Thorne is one of the world’s leading researchers in gravitational physics of all kinds, and he offers both a history of work in GR and an introduction to very up-to-date research topics. - R. Geroch,
*General Relativity from A to B*(Chicago, 1991).

A truly beautiful exposition of the workings of spacetime. - B. Greene,
*The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory*(W.W. Norton, 1999).

A timely and personal introduction to the physics of string theory. Not afraid to discuss quite advanced concepts, but aiming at a general audience all along; very well written. - A.H. Guth,
*The Inflationary Universe: The Quest for a New Theory of Cosmic Origins*(Addison-Wesley, 1998).

A thorough and lucid introduction to all of modern cosmology, focusing on inflation. - L. Smolin,
*Three Roads to Quantum Gravity*(Perseus, 1998).

The “three roads” are string theory, loop quantum gravity, and something more profound; Smolin is a partisan for loop quantum gravity, but the discussion should be interesting for everyone. - G. Kane,
*Supersymmetry: Unveiling the Ultimate Laws of Nature*(Perseus, 2001).

A nice introduction to supersymmetry, a hypothetical symmetry between bosons and fermions that may be within the reach of particle accelerators soon. - A. Einstein, H.A. Lorentz, H. Weyl, and H. Minkowski,
*The Principle of Relativity*(Dover, 1924).

Actually not a “popular” book at all; rather, a collection of the original research articles on special and general relativity, translated into English. - A. Pais,
*Subtle is the Lord: The Science and the Life of Albert Einstein*(Oxford, 1983).

A scientific biography of Einstein, complete with equations.

## Additional Resources

##### General-Interest Sites

- Einstein — Image and Impact, AIP

An exhibit on the life and times of Einstein. See also Albert Einstein Online, and the Einstein Archives Online. - Einstein speaks

Listen to Einstein explain his most famous formula. - LIGO, VIRGO, GEO, TAMA, ACIGA, AIGO, LISA

Gravitational-wave observatories. - Falling Into a Black Hole, Andrew Hamilton
- History of Special Relativity and of General Relativity
- Spacetime Wrinkles, NCSA
- GR Tensor

A symbolic tensor manipulation program. Originally developed for Maple, a Mathematica version is now available. There is even a java version you can run remotely. - Gravitational Lensing, Peter Newbury
- Beyond Einstein

NASA program to explore the boundaries of relativity. - Superstringtheory.com

The official string theory web site. (Look, I just link to them, I don’t come up with the names.) - sci.physics.research, sci.physics.relativity, sci.physics

Usenet discussion groups.

##### Online Notes and Tutorials

See also the Bibliography tab above for actual books.

- A No-Nonsense Introduction to General Relativity, Sean Carroll (postscript, also available in pdf)

A very short (24 pages) but serious overview of the workings of GR. - Lecture Notes on General Relativity, Sean Carroll (postscript, also in html)

The notes on which the book was based. - General Relativity Tutorial, John Baez
- Relativity FAQ
- MIT Physics 8.962, Ed Bertschinger, MIT
- Black Hole Lecture Notes, Paul Townsend
- Advanced Course in General Relativity, Eric Poisson (pdf)
- Tensors and Relativity, Peter Dunsby
- Introduction to General Relativity, Gerard ‘t Hooft (postscript)
- Introduction to Differential Geometry and General Relativity, Stefan Waner
- Lecture Notes on General Relativity, Matthias Blau (gzipped postscript).
- Applications of Classical Physics, Roger Blandford and Kip Thorne

An illuminating overview of topics in classical physics by two masters. (Advanced undergraduate level; GR is covered briefly at the end.) - Gravitational Waves: A Web-Based Course, Kip Thorne

A detailed introduction to gravitational wave production and detection. Includes notes, videos of lectures, exercises, and supplemental material. - Fields, Warren Siegel

An introduction to quantum field theory, including discussions of supersymmetry and string theory. - Cosmology Tutorial and Relativity Tutorial, Ned Wright
- CMB Tutorial, Wayne Hu

A multi-level introduction to cosmological perturbation theory and microwave background anisotropies. - Review Articles, Sean Carroll

A collection of reviews (mostly by other people) on field theory, relativity, and cosmology. - Relativity: The Special And General Theory, Albert Einstein

From 1920, a semi-popular exposition by Einstein himself, complete online.

##### Research and News Sources

- gr-qc at arxiv.org

The indispensable preprint server, containing essentially all respectable contemporary research papers. “gr-qc” stands for “general relativity and quantum cosmology”; look for related servers in astrophysics (astro-ph), high-energy theory and string theory (hep-th), and mathematics (math). - SPIRES High-Energy Physics Literature Database

A service which not only indexes papers, but keeps track of references and citations. Nominally devoted to high-energy physics, but gradually expanding to keep track of gravitation and astrophysics. - Living Reviews in Relativity

Online review articles on all aspects of relativity. - General Relativity News from Malcolm MacCallum at Queen Mary
- Matters of Gravity

A newsletter for the APS Topical Group on Gravitation, edited by Jorge Pullin.

##### Web Guides

- Relativity on the World Wide Web, UC Riverside

A carefully reviewed and annotated list of links, originally prepared by Chris Hillman and maintained by John Baez. - Relativity Bookmarks, Syracuse University

Includes a long list of research groups in GR. - Relativity and Black Hole Links, University of Colorado

## Errata

This page collects any mistakes that people have been able to find in the book. Those listed as being noticed before 10/14/03 have been corrected in the second printing. Dates refer to when the addition was made to this page, not necessarily when it was sent to me.

If you find a mistake that isn’t listed here, please contact me.

**Chapter One**

- p. 1: Most embarassing mistake yet. Equations (1.1) and (1.4) should both have minus signs; apparently my grasp of Newtonian gravity is a little shaky. Note that (4.1) is okay. [Thanks to T. Damour, 11/12/03.]
- p. 2: In the last full paragraph, “the one sitting a table” should be “the one sitting on a table.” [Thanks to D. Starr, 10/26/03.]
- p. 22: In the first line of the last full paragraph, “tangent vectors” should be “dual vectors.” [Thanks to D. Taylor, 11/10/03.]
- p. 28: 7th line from the bottom, “give us tensor” should be “give us tensors.” [Thanks to D. Taylor, 12/6/03.]
- p. 32: (1.106) should have a minus sign. [11/28/03.]
- p. 42: In the sentence containing (1.160), “stregth” should be “strength.” [Thanks to D. Hartmann, D. Taylor, and J. Collins, 11/10/03.]
- p. 43: In the sentence after (1.163), “tricker” should be “trickier.” [Thanks to J. Collins, 11/15/03.]
- p. 45: In Exercise 1, “mirror” should be “wall.” [Thanks to D. Taylor, 11/10/03.]

**Chapter Two**

- p. 49: In the middle of the page, “accelerating at a constant velocity” should be “accelerating at a constant rate,” of course. [Thanks to D. Hartmann, 11/28/03.]
- p. 52: In the paragraph beginning “Consider”, each appearance of “box” should be “rocket”, to agree with Figure 2.2. [Thanks to D. Taylor, 4/3/04.]
- p. 57: I goofed on the cone, when I said it was a non-smooth manifold. The
*metric*on the cone, inherited from its embedding in Euclidean space, is not smooth at the vertex, but it is perfectly possible to give the cone a smooth atlas (just project it down to the plane, and use the conventional atlas there). The irony is that I knew all of this perfectly well, but had somehow convinced myself that there was a more natural atlas for the cone, one that was not smooth. Upon reflection, I’m not sure where that conviction came from. [Thanks to P. Shepard and N. Jeevanje, communicated by R. Bousso, 1/29/04.] - p. 74: Near the beginning of the first full paragraph, “to see a sketch a proof” should just be “to sketch a proof.” [Thanks to J. Collins, 11/21/03.]
- p. 79: Near the end of the first paragraph, “a curve of zero length is chronal but not causal” is exactly backwards, and should be “a curve of zero length is causal but not chronal.” In the same sentence, the word “in” should be inserted between “not necessarily” and “its own.” [Thanks to D. Taylor, 12/6/03.]
- p. 80: In figure 2.24, the line from D^+(Sigma) should extend into the interior of the past light cone. [Thanks to D. Taylor, 12/11/03.]
- p. 84: In line 5 of the first paragraph, “the the” should just be “the”. [Thanks to D. Taylor, 12/11/03.]
- pp. 86-87: Maxwell’s equations are referenced as (1.88) and (1.89), while they are actually (1.96) and (1.97). [Thanks to D. Hartmann and J. Olson, 10/14/03.]
- p. 87: After (2.86), “phi” should be capitalized for consistency with Chapter One. [Thanks to D. Taylor, 1/10/04.]
- p. 87: A phrase was mistakenly dropped from the sentence containing (2.87), which should read “The other one of Maxwell’s equations, (1.96), can be expressed as an equation between three-forms: …” [10/31/03.]
- p. 92: Problem 9 doesn’t make sense in “Euclidean three-space”, it should be in ordinary four-dimensional Minkowski space. Only the ball in part (d) is in Euclidean three-space. [Thanks to D. Starr, 2/10/04.]

**Chapter Three**

- p. 96: After (3.11), it’s not precisely correct to say that the transformation properties of the tilded connection coefficients are the same as those of the original Gammas; in the analogue of (3.10), there would be a plus sign instead of a minus sign. [Thanks to D. Starr, 4/3/04.]
- p. 99: In (3.20), the index on
*V*in the second and third terms should be a nu, not a lambda. [Thanks to G. Chinnici, 4/3/04.] - p. 102: In (3.36), the second and third terms should be multiplied by a factor of 2, given the definition of the exterior derivative. [Thanks to D. Taylor, 1/10/04.]
- p. 112: In the fifth-to-last line of the (only) full paragraph, the “mu” in “the sought-after coordinates x^mu” should have a hat. [Thanks to D. Taylor, 1/10/04.]
- p. 108: The sentence containing (3.56) was mangled. After the semicolon should read “in fact, looking back to the expression (1.106) for the Lorentz force in special relativity, it is natural to guess that”, and then the equation. [Thanks to D. Hartmann, 11/28/03.]
- p. 111: In the first sentence of the first full paragraph, “convenient a way” should just be “convenient way”. [Thanks to D. Hartmann, 11/28/03.]
- p. 122: In the second line after (3.111), “the last term is simply the torsion tensor” would be better rendered as “the antisymmetrized connection coefficients in the last term are simply one-half times the torsion tensor”. [Thanks to D. Taylor, 2/11/04.]
- p. 126: To be perfectly clear, at the end of the first paragraph of section 3.7 we should say “reduce the number of independent components” rather than just “reduce the independent components”. [Thanks to R. Riklund, 1/7/04.]
- p. 130: After (3.148), instead of “the Appendices” we should have “Appendix G”. [Thanks to D. Taylor, 2/11/04.]
- p. 136: The long sentence at the bottom of the page was apparently mangled in the typesetting process; delete the word “using” at the beginning of the sentence, and change the comma after “generalizes to additional indices” to a colon. Not my most elegant sentence in any event. [Thanks to D. Taylor, 2/11/04.]
- p. 141: In (3.191), the nu in the last term should be an ordinary subscript, not a sub-subscript. [Thanks to D. Starr, 4/3/04.]
- p. 145: At the end of the paragraph containing (3.204), the last sentence should include the condition “(when
*t*is the proper time)”. [10/14/03.]

**Chapter Four**

- p. 179: In the sixth line, “out” should be “our”. [Thanks to D. Taylor, 4/3/04.]
- p. 187: After (4.144), “minus the square root of” should be “the square root of minus”. [Thanks to D. Taylor, 4/3/04.]
- p. 190: In the very first line, “will suppressed” should be “will be suppressed”. [Thanks to T. Keech, 10/20/03.]
- p. 190: In Exercise 1, just to be clear, let’s assume that the A_mu J^mu term does not contribute to the energy-momentum tensor. It won’t, for example, for charged point particles, since J^mu contains an implicit factor of 1/sqrt{-g}, as in the delta-function defined in (4.157). [Thanks to H. Haber, 4/5/04.]
- p. 191: In Exercise 5, “the Appendices” could be more usefully rendered as “Appendices D and F”. [Thanks to D. Taylor, 4/3/04.]

**Chapter Five**

- p. 202: After (5.39), delete “following”. [Thanks to D. Taylor, 4/3/04.]
- pp. 221, 223: In Figures 5.10 and 5.11, one side of the middle light cones should be perfectly vertical, to line up with the event horizon at
*r*=2*GM*. [Thanks to D. Taylor, 4/3/04.] - p. 225: “Kruskal-Szekres” should be “Kruskal-Szekeres”. [Thanks to R. Riklund, 1/7/04, and apologies to Prof. George Szekeres.]

**Chapter Six**

- pp 255-273: Late in the game I decided to use rationalized (Heaviside-Lorentz) units for electromagnetism. In these units there are no 4pi’s in Maxwell’s equations (see p. 29) but there are in the expression for the electric field of a point charge (p. 457). The one place the change didn’t happen is in the discussion of charged (RN) black holes in Chapter Six. I believe that every appearance of the charges
*Q*or*P*in that chapter should be divided by 4pi to make it okay. [Thanks to E. Abers, 4/3/04.]

**Chapter Seven**

- p. 304: There should not be a minus sign in (7.134). [Thanks to E. Abers, 4/3/04.]

**Chapter Eight**

- p. 343: Three lines below (8.94), “the the” should just be “the”. [Thanks to R. Riklund, 4/3/04.]
- p. 375: Problem 2 was completely fumbled. First off, it should be “
*e*^{2*Ht*}”, not “*e*^{*Ht*}” (if we want*H*to really be the Hubble parameter). Second and worse, comoving observers are precisely the wrong ones to ask this question about; for them,*t*actually is an affine parameter. The question should have asked about**non**-comoving observers. [Thanks to D. Garfinkle, 2/2/04.]

**Chapter Nine**

- p. 379: In (9.9), the exponent should be “i(omega t + alpha_0)”, with parentheses, rather than simply “i omega t + alpha_0”. [Thanks to T. Keech, 10/31/03.]
- p. 384: In (9.34), that should be a time-ordered exponential, similar in spirit to the path-ordering discussed in Appendix I. [Thanks to R. Riklund, 4/3/04.]
- p. 388: The left-hand-side of (9.57) really shouldn’t use mu as a dummy index twice. [Thanks to R. Riklund, 4/3/04.]
- p. 390: There should not be a comma in the square root in (9.70). [Thanks to R. Riklund, 4/3/04.]
- p. 403: The last term in (9.130) should be 1/alpha^2, rather than alpha^2. [Thanks to J. Fric, 11/26/03.]

**Appendices**

- p. 425: The left arrow in Figure A.2 should be labeled “V(p)”. [Thanks to D. Starr, 4/3/04.]
- p. 426: There should be parentheses around the last
*p*in the caption to Figure A.3. [Thanks to D. Starr, 4/3/04.] - p. 432: In the caption of Figure B.3, at the end of the second line “
*T*a point” should be “*T*at a point” . [Thanks to D. Starr, 4/3/04.] - p. 432: The pullbacks in (B.9) and (B.10) should have upper asterisks, not lower ones. [Thanks to G. Chinnici, 4/3/04.]
- p. 436: In (B.24) and (B.25), the vector
*V*should be thought of as infinitesimal; alternatively, think of*V*as finite and multiply each appearance by an infinitesimal parameter epsilon. [Thanks to D. Starr, 4/3/04.] - p. 436: Below (B.25), the definition of the energy-momentum tensor is actually (4.75), not (4.73). [Thanks to D. Starr, 4/3/04.]
- pp. 437, 469: The Exercises in the Appendices are numbered incorrectly; the correct version should be obvious. [10/14/03.]
- p. 439: In the definition of a submanifold, the condition on the inverse map should be that it is
*C*^infinity, not that it is one-to-one (which it always will be). Also, it should have been made clear that this definition, following Wald’s GR book, differs from one common in the math literature; Wald was careful enough to point this out, while I was not. In much of the math literature, what I called an “immersed submanifold” would simply be an “immersion”; this makes sense, since if the image is self-intersecting it’s not really a manifold. What I called an “embedded submanifold” would only be an “immersed submanifold”, and the definition of “embedded” would include the additional condition that the image not come arbitrarily close to intersecting itself (technically, that the map phi defines a homeomorphism between*S*and its image). [Thanks to D. Starr, 4/3/04.] - p. 439: In third line of second paragraph, “the the” should be “the”. [Thanks to D. Snead, 12/23/03.]
- p. 439: Each appearance of “phi[M]” in the second paragraph should be “phi[S]”. [Thanks to D. Snead and G. Chinnici, 12/23/03.]
- p. 440: In the third line of the first full paragraph, “
*m*function” should be “*m*functions”. [Thanks to D. Starr, 4/3/04.] - p. 441: In the first complete sentence after (C.4), “fit together to define a submanifold” should be “fit together to define a set of submanifolds”. [Thanks to D. Starr, 4/3/04.]
- p. 443: In (D.2), the index on the derivative should be “nu”. [Thanks to D. Snead and I. Sahabandu, 12/21/03.]
- p. 444: In the line between (D.10) and (D.11), the equation for zeta should be an equation for xi. [10/14/03.]
- p. 451: Each appearance of “Codacci” should be “Codazzi.” [Thanks to M. Godina, 10/29/03.]