Spacetime and Geometry: An Introduction to General Relativity by Sean M. Carroll

• Relativity
• bcrowell
In summary, Sean M. Carroll's Spacetime and Geometry: An Introduction to General Relativity is an excellent graduate level book that provides a concise and comprehensive overview of General Relativity.

For those who have used this book

• Strongly don't Recommend

• Total voters
10
bcrowell
Staff Emeritus
Gold Member

Code:
Contents:

1. Special Relativity and Flat Spacetime
1.1 Prelude
1.2 Space and Time, Separately and Together
1.3 Lorentz Transformations
1.4 Vectors
1.5 Dual Vectors (One-Forms)
1.6 Tensors
1.7 Manipulating Tensors
1.8 Maxwell's Equations
1.9 Energy and Momentum
1.10 Classical Field Theory
1.11 Exercises

2. Manifolds
2.1 Gravity as Geometry
2.2 What Is a Manifold?
2.3 Vectors Again
2.4 Tensors Again
2.5 The Metric
2.6 An Expanding Universe
2.7 Causality
2.8 Tensor Densities
2.9 Differential Forms
2.10 Integration
2.11 Exercises

3. Curvature
3.1 Overview
3.2 Covariant Derivatives
3.3 Parallel Transport and Geodesics
3.4 Properties of Geodesics
3.5 The Expanding Universe Revisited
3.6 The Riemann Curvature Tensor
3.7 Properties of the Riemann Tensor
3.8 Symmetries and Killing Vectors
3.9 Maximally Symmetric Spaces
3.10 Geodesic Deviation
3.11 Exercises

4. Gravitation
4.1 Physics in Curved Spacetime
4.2 Einstein's Equation
4.3 Lagrangian Formulation
4.4 Properties of Einstein's Equation
4.5 The Cosmological Constant
4.6 Energy Conditions
4.7 The Equivalence Principle Revisited
4.8 Alternative Theories
4.9 Exercises

5. The Schwarzschild Solution
5.1 The Schwarzschild Metric
5.2 Birkhoff's Theorem
5.3 Singularities
5.4 Geodesics of Schwarzschild
5.5 Experimental Tests
5.6 Schwarzschild Black Holes
5.7 The Maximally Extended Schwarzschild Solution
5.8 Stars and Black Holes
5.9 Exercises

6. More General Black Holes
6.1 The Black Hole Zoo
6.2 Event Horizons
6.3 Killing Horizons
6.4 Mass, Charge, and Spin
6.5 Charged (Reissner-Nordstrom) Black Holes
6.6 Rotating (Kerr) Black Holes
6.7 The Penrose Process and Black-Hole Thermodynamics
6.8 Exercises

7. Perturbation Theory and Gravitational Radiation
7.1 Linearized Gravity and Gauge Transformations
7.2 Degrees of Freedom
7.3 Newtonian Fields and Photon Trajectories
7.4 Gravitational Wave Solutions
7.5 Production of Gravitational Waves
7.6 Energy Loss Due to Gravitational Radiation
7.7 Detection of Gravitational Waves
7.8 Exercises

8. Cosmology
8.1 Maximally Symmetric Universes
8.2 Robertson-Walker Metrics
8.3 The Friedmann Equation
8.4 Evolution of the Scale Factor
8.5 Redshifts and Distances
8.6 Gravitational Lensing
8.7 Our Universe
8.8 Inflation
8.9 Exercises

9. Quantum Field Theory in Curved Spacetime
9.1 Introduction
9.2 Quantum Mechanics
9.3 Quantum Field Theory in Flat Spacetime
9.4 Quantum Field Theory in Curved Spacetime
9.5 The Unruh Effect
9.6 The Hawking Effect and Black Hole EvaporationAPPENDIXES

A. Maps Between Manifolds
B. Diffeomorphisms and Lie Derivatives
C. Submanifolds
D. Hypersurfaces
E. Stokes's Theorem
F. Geodesic Congruences
G. Conformal Transformations
H. Conformal Diagrams
I. The Parallel Propagator
J. Noncoordinate Bases

Bibliography
Index

Last edited by a moderator:

For a long time there's been a need for an up to date graduate text on GR. The classics are Wald and MTW, but at the interface with experiment, those both predate LIGO, Gravity Probe B, modern studies of CMB anisotropy, and the discoveries of supermassive black holes and the nonzero cosmological constant. Carroll's book is a little less austere and scary than Wald, more concise than MTW. At this point it's the book that I would point a first-year grad student to. It's wonderful that the book is available online for free (see the URL in the listing).

There are three things IMO about this book that stand out and make it great: even in the maths chapters he includes really cool physical examples, the appendices are awesome and should be required reading - especially the one on geodesic congruence, and the chapter on gravitational waves is IMO better than those in other standard introductory texts. Too bad the full text is insanely expensive and I am alas but a poor stable boy.

Joker93
This is by far my favorite GR text, and is usually my go-to when I have a question (if not then Wald).

Ben, it should be noted that the lecture notes you linked to above are not the same as his book. His book is more detailed, covers more topics (for example a particularly good discussion on classical field theory and an entire section about QFT in curved spacetime), has wonderful appendices, and has exercises for the reader. The lecture notes give you the general idea of what his book covers, though.

I particularly love the way he goes about describing the Equivalence Principle and Diffeomorphism invariance. Something, which I didn't find with such detailed explanation in Wald.

But then again, the way Wald (or Hawking & Ellis) describe the covariant derivative and other concepts of Differential Geometry is much more complete (though a bit terse).

But as bcrowell rightly said, best book for a first year grad student!

The most rewarding part of my undergraduate years was spent in the library studying this book for an independent study. I literally learned more math and physics from 6 months with this book than I did in the previous three and a half. Wish he would have included more worked examples, though!

binbagsss

1. What is the main concept of spacetime and geometry in general relativity?

The main concept of spacetime and geometry in general relativity is that the fabric of the universe is not flat, but rather it is curved due to the presence of massive objects. This curvature of spacetime is what causes objects to follow curved paths and explains the force of gravity.

2. How does general relativity differ from Newton's theory of gravity?

General relativity differs from Newton's theory of gravity in that it takes into account the curvature of spacetime caused by massive objects, whereas Newton's theory only describes gravity as a force between masses. General relativity also predicts that the path of a moving object is affected by the presence of massive objects, while Newton's theory predicts that the path of a moving object is a straight line.

3. What is the significance of the speed of light in general relativity?

The speed of light is a fundamental constant in general relativity and is used to define the geometry of spacetime. It is also the maximum speed at which any object can travel in the universe and plays a crucial role in the theory's predictions, such as time dilation and length contraction.

4. Can you explain the concept of spacetime curvature in general relativity?

In general relativity, spacetime curvature refers to the bending of the fabric of the universe due to the presence of mass and energy. This curvature is described by the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy. The greater the mass and energy in a given region of spacetime, the greater the curvature.

5. How has general relativity been tested and confirmed?

General relativity has been extensively tested and confirmed through various experiments and observations. One of the most famous examples is the deflection of light by the sun, which was predicted by general relativity and observed during a solar eclipse in 1919. Other tests include the precession of the orbit of Mercury, gravitational time dilation, and the bending of light by massive objects in space.

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