First published in 1973, *Gravitation* is a landmark graduate-level textbook that presents Einstein’s general theory of relativity and offers a rigorous, full-year course on the physics of gravitation. Upon publication, *Science *called it “a pedagogic masterpiece,” and it has since become a classic, considered essential reading for every serious student and researcher in the field of relativity. This authoritative text has shaped the research of generations of physicists and astronomers, and the book continues to influence the way experts think about the subject.

With an emphasis on geometric interpretation, this masterful and comprehensive book introduces the theory of relativity; describes physical applications, from stars to black holes and gravitational waves; and portrays the field’s frontiers. The book also offers a unique, alternating, two-track pathway through the subject. Material focusing on basic physical ideas is designated as Track 1 and formulates an appropriate one-semester graduate-level course. The remaining Track 2 material provides a wealth of advanced topics instructors can draw on for a two-semester course, with Track 1 sections serving as prerequisites.

This must-have reference for students and scholars of relativity includes a new preface by David Kaiser, reflecting on the history of the book’s publication and reception, and a new introduction by Charles Misner and Kip Thorne, discussing exciting developments in the field since the book’s original publication.

- The book teaches students to:
- Grasp the laws of physics in flat and curved spacetime
- Predict orders of magnitude
- Calculate using the principal tools of modern geometry
- Understand Einstein’s geometric framework for physics
- Explore applications, including neutron stars, Schwarzschild and Kerr black holes, gravitational collapse, gravitational waves, cosmology, and so much more

#### Awards and Recognition

- Kip S. Thorne, Co-Winner of the 2017 Nobel Prize in Physics

**Charles W. Misner** is professor emeritus of physics at the University of Maryland. **Kip S. Thorne**, co-winner of the 2017 Nobel Prize in physics, is the Feynman Professor Emeritus of Theoretical Physics at the California Institute of Technology. His books include *Modern Classical Physics* (Princeton), *Black Holes and Time Warps*, and *The Science of Interstellar*. John Archibald Wheeler (1911–2008) was professor of physics at Princeton University and later at the University of Texas, Austin. His books include *Spacetime Physics* and *Geons, Black Holes, and Quantum Foam*

- LIST OF BOXES
- LIST OF FIGURES
- FOREWORD TO THE 2017 EDITION
- PREFACE TO THE 2017 EDITION
- PREFACE
- ACKNOWLEDGMENTS
- Part I SPACETIME PHYSICS
- 1. Geometrodynamics in Brief
- 1. The Parable of the Apple
- 2. Spacetime With and Without Coordinates
- 3. Weightlessness
- 4. Local Lorentz Geometry, With and Without Coordinates
- 5. Time
- 6. Curvature
- 7. Effect of Matter on Geometry

- Part II PHYSICS IN FLAT SPACETIME
- 2. Foundations of Special Relativity
- 1. Overview
- 2. Geometric Objects
- 3. Vectors
- 4. The Metric Tensor
- 5. Differential Forms
- 6. Gradients and Directional Derivatives
- 7. Coordinate Representation of Geometric Objects
- 8. The Centrifuge and the Photon
- 9. Lorentz Transformations
- 10. Collisions

- 3. The Electromagnetic Field
- 1. The Lorentz Force and the Electromagnetic Field Tensor
- 2. Tensors in All Generality
- 3. Three-Plus-One View Versus Geometric View
- 4. Maxwell’s Equations
- 5. Working with Tensors

- 4. Electromagnetism and Differential Forms
- 1. Exterior Calculus
- 2. Electromagnetic 2-Form and Lorentz Force
- 3. Forms Illuminate Electromagnetism and Electromagnetism Illuminates Forms
- 4. Radiation Fields
- 5. Maxwell’s Equations
- 6. Exterior Derivative and Closed Forms
- 7. Distant Action from Local Law

- 5. Stress-Energy Tensor and Conservation Laws
- 1. Track-1 Overview
- 2. Three-Dimensional Volumes and Definition of the Stress-Energy Tensor
- 3. Components of Stress-Energy Tensor
- 4. Stress-Energy Tensor for a Swarm of Particles
- 5. Stress-Energy Tensor for a Perfect Fluid
- 6. Electromagnetic Stress-Energy
- 7. Symmetry of the Stress-Energy Tensor
- 8. Conservation of 4-Momentum: Integral Formulation
- 9. Conservation of 4-Momentum: Differential Formulation
- 10. Sample Application of ▼ · T = 0
- 11. Angular Momentum

- 6. Accelerated Observers
- 1. Accelerated Observers Can Be Analyzed Using Special Relativity
- 2. Hyperbolic Motion
- 3. Constraints on Size of an Accelerated Frame
- 4. The Tetrad Carried by a Uniformly Accelerated Observer
- 5. The Tetrad Fermi-Walker Transported by an Observer with Arbitrary Acceleration
- 6. The Local Coordinate System of an Accelerated Observer

- 7. Incompatibility of Gravity and Special Relativity
- 1. Attempts to Incorporate Gravity into Special Relativity
- 2. Gravitational Redshift Derived from Energy Conservation
- 3. Gravitational Redshift Implies Spacetime Is Curved
- 4. Gravitational Redshift as Evidence for the Principle of Equivalence
- 5. Local Flatness, Global Curvature

- Part III THE MATHEMATICS OF CURVED SPACETIME
- 8. Differential Geometry: An Overview
- 1. An Overview of Part III
- 2. Track 1 Versus Track 2: Difference in Outlook and Power
- 3. Three Aspects of Geometry: Pictorial, Abstract, Component
- 4. Tensor Algebra in Curved Spacetime
- 5. Parallel Transport, Covariant Derivative, Connection Coefficients, Geodesics
- 6. Local Lorentz Frames: Mathematical Discussion
- 7. Geodesic Deviation and the Riemann Curvature Tensor

- 9. Differential Topology
- 1. Geometric Objects in Metric-Free, Geodesic-Free Spacetime
- 2. “Vector” and “Directional Derivative” Refined into Tangent Vector
- 3. Bases, Components, and Transformation Laws for Vectors
- 4. 1-Forms
- 5. Tensors
- 6. Commutators and Pictorial Techniques
- 7. Manifolds and Differential Topology

- 10. Affine Geometry: Geodesics, Parallel Transport and Covariant Derivative
- 1. Geodesics and the Equivalence Principle
- 2. Parallel Transport and Covariant Derivative: Pictorial Approach
- 3. Parallel Transport and Covariant Derivative: Abstract Approach
- 4. Parallel Transport and Covariant Derivative: Component Approach
- 5. Geodesic Equation

- 11. Geodesic Deviation and Spacetime Curvature
- 1. Curvature, At Last!
- 2. The Relative Acceleration of Neighboring Geodesics
- 3. Tidal Gravitational Forces and Riemann Curvature Tensor
- 4. Parallel Transport Around a Closed Curve
- 5. Flatness is Equivalent to Zero Riemann Curvature
- 6. Riemann Normal Coordinates

- 12. Newtonian Gravity in the Language of Curved Spacetime
- 1. Newtonian Gravity in Brief
- 2. Stratification of Newtonian Spacetime
- 3. Galilean Coordinate Systems
- 4. Geometric, Coordinate-Free Formulation of Newtonian Gravity
- 5. The Geometric View of Physics: A Critique

- 13. Riemannian Geometry: Metric as Foundation of All
- 1. New Features Imposed on Geometry by Local Validity of Special Relativity
- 2. Metric
- 3. Concord Between Geodesics of Curved Spacetime Geometry and Straight Lines of Local Lorentz Geometry
- 4. Geodesics as World Lines of Extremal Proper Time
- 5. Metric-Induced Properties of Riemann
- 6. The Proper Reference Frame of an Accelerated Observer

- 14. Calculation of Curvature
- 1. Curvature as a Tool for Understanding Physics
- 2. Forming the Einstein Tensor
- 3. More Efficient Computation
- 4. The Geodesic Lagrangian Method
- 5. Curvature 2-Forms
- 6. Computation of Curvature Using Exterior Differential Forms

- 15. Bianchi Identities and the Boundary of a Boundary
- 1. Bianchi Identities in Brief
- 2. Bianchi Identity dR = 0 as a Manifestation of “Boundary of Boundary = 0”
- 3. Moment of Rotation: Key to Contracted Bianchi Identity
- 4. Calculation of the Moment of Rotation
- 5. Conservation of Moment of Rotation Seen from “Boundary of a Boundary is Zero”
- 6. Conservation of Moment of Rotation Expressed in Differential Form
- 7. From Conservation of Moment of Rotation to Einstein’s Geometrodynamics: A Preview

- Part IV EINSTEIN’S GEOMETRIC THEORY OF GRAVITY
- 16. Equivalence Principle and Measurement of the “Gravitational Field”
- 1. Overview
- 2. The Laws of Physics in Curved Spacetime
- 3. Factor-Ordering Problems in the Equivalence Principle
- 4. The Rods and Clocks Used to Measure Space and Time Intervals
- 5. The Measurement of the Gravitational Field

- 17. How Mass-Energy Generates Curvature
- 1. Automatic Conservation of the Source as the Central Idea in the Formulation of the Field Equation
- 2. Automatic Conservation of the Source: A Dynamic Necessity
- 3. Cosmological Constant
- 4. The Newtonian Limit
- 5. Axiomatize Einstein’s Theory?
- 6. “No Prior Geometry”: A Feature Distinguishing Einstein’s Theory from Other Theories of Gravity
- 7. A Taste of the History of Einstein’s Equation

- 18. Weak Gravitational Fields
- 1. The Linearized Theory of Gravity
- 2. Gravitational Waves
- 3. Effect of Gravity on Matter
- 4. Nearly Newtonian Gravitational Fields

- 19. Mass and Angular Momentum of a Gravitating System
- 1. External Field of a Weakly Gravitating Source
- 2. Measurement of the Mass and Angular Momentum
- 3. Mass and Angular Momentum of Fully Relativistic Sources
- 4. Mass and Angular Momentum of a Closed Universe

- 20. Conservation Laws for 4-Momentum and Angular Momentum
- 1. Overview
- 2. Gaussian Flux Integrals for 4-Momentum and Angular Momentum
- 3. Volume Integrals for 4-Momentum and Angular Momentum
- 4. Why the Energy of the Gravitational Field Cannot be Localized
- 5. Conservation Laws for Total 4-Momentum and Angular Momentum
- 6. Equation of Motion Derived from the Field Equation

- 21. Variational Principle and Initial-Value Data
- 1. Dynamics Requires Initial-Value Data
- 2. The Hilbert Action Principle and the Palatini Method of Variation
- 3. Matter Lagrangian and Stress-Energy Tensor
- 4. Splitting Spacetime into Space and Time
- 5. Intrinsic and Extrinsic Curvature
- 6. The Hilbert Action Principle and the Arnowitt-Deser-Misner Modification Thereof in the Space-plus-Time Split
- 7. The Arnowitt-Deser-Misner Formulation of the Dynamics of Geometry
- 8. Integrating Forward in Time
- 9. The Initial-Value Problem in the Thin-Sandwich Formulation
- 10. The Time-Symmetric and Time-Antisymmetric Initial-Value Problem
- 11. York’s “Handles” to Specify a 4-Geometry
- 12. Mach’s Principle and the Origin of Inertia
- 13. Junction Conditions

- 22. Thermodynamics, Hydrodynamics, Electrodynamics, Geometric Optics, and Kinetic Theory
- 1. The Why of this Chapter
- 2. Thermodynamics in Curved Spacetime
- 3. Hydrodynamics in Curved Spacetime
- 4. Electrodynamics in Curved Spacetime
- 5. Geometric Optics in Curved Spacetime
- 6. Kinetic Theory in Curved Spacetime

- Part V RELATIVISTIC STARS
- 23. Spherical Stars
- 1. Prolog
- 2. Coordinates and Metric for a Static, Spherical System
- 3. Physical Interpretation of Schwarzschild coordinates
- 4. Description of the Matter Inside a Star
- 5. Equations of Structure
- 6. External Gravitational Field
- 7. How to Construct a Stellar Model
- 8. The Spacetime Geometry for a Static Star

- 24. Pulsars and Neutron Stars; Quasars and Supermassive Stars
- 1. Overview
- 2. The Endpoint of Stellar Evolution
- 3. Pulsars
- 4. Supermassive Stars and Stellar Instabilities
- 5. Quasars and Explosions In Galactic Nuclei
- 6. Relativistic Star Clusters

- 25. The “Pit in the Potential” as the Central New Feature of Motion in Schwarzschild Geometry
- 1. From Kepler’s Laws to the Effective Potential for Motion in Schwarzschild Geometry
- 2. Symmetries and Conservation Laws
- 3. Conserved Quantities for Motion in Schwarzschild Geometry
- 4. Gravitational Redshift
- 5. Orbits of Particles
- 6. Orbit of a Photon, Neutrino, or Graviton in Schwarzschild Geometry
- 7. Spherical Star Clusters

- 26. Stellar Pulsations
- 1. Motivation
- 2. Setting Up the Problem
- 3. Eulerian versus Lagrangian Perturbations
- 4. Initial-Value Equations
- 5. Dynamic Equation and Boundary Conditions
- 6. Summary of Results

- Part VI THE UNIVERSE
- 27. Idealized Cosmologies
- 1. The Homogeneity and Isotropy of the Universe
- 2. Stress-Energy Content of the Universe—the Fluid Idealization
- 3. Geometric Implications of Homogeneity and Isotropy
- 4. Comoving, Synchronous Coordinate Systems for the Universe
- 5. The Expansion Factor
- 6. Possible 3-Geometries for a Hypersurface of Homogeneity
- 7. Equations of Motion for the Fluid
- 8. The Einstein Field Equations
- 9. Time Parameters and the Hubble Constant
- 10. The Elementary Friedmann Cosmology of a Closed Universe
- 11. Homogeneous Isotropic Model Universes that Violate Einstein’s Conception of Cosmology

- 28. Evolution of the Universe into Its Present State
- 1. The “Standard Model” of the Universe
- 2. Standard Model Modified for Primordial Chaos
- 3. What “Preceded” the Initial Singularity?
- 4. Other Cosmological Theories

- 29. Present State and Future Evolution of the Universe
- 1. Parameters that Determine the Fate of the Universe
- 2. Cosmological Redshift
- 3. The Distance-Redshift Relation: Measurement of the Hubble Constant
- 4. The Magnitude-Redshift Relation: Measurement of the Deceleration Parameter
- 5. Search for “Lens Effect” of the Universe
- 6. Density of the Universe Today
- 7. Summary of Present Knowledge About Cosmological Parameters

- 30. Anisotropic and Inhomogeneous Cosmologies
- 1. Why Is the Universe So Homogeneous and Isotropic?
- 2. The Kasner Model for an Anisotropic Universe
- 3. Adiabatic Cooling of Anisotropy
- 4. Viscous Dissipation of Anisotropy
- 5. Particle Creation in an Anisotropic Universe
- 6. Inhomogeneous Cosmologies
- 7. The Mixmaster Universe
- 8. Horizons and the Isotropy of the Microwave Background

- Part VII GRAVITATIONAL COLLAPSE AND BLACK HOLES
- 31. Schwarzschild Geometry
- 1. Inevitability of Collapse for Massive Stars
- 2. The Nonsingularity of the Gravitational Radius
- 3. Behavior of Schwarzschild Coordinates at r = 2M
- 4. Several Well-Behaved Coordinate Systems
- 5. Relationship Between Kruskal-Szekeres Coordinates and Schwarzschild Coordinates
- 6. Dynamics of the Schwarzschild Geometry

- 32. Gravitational Collapse
- 1. Relevance of Schwarzschild Geometry
- 2. Birkhoff’s Theorem
- 3. Exterior Geometry of a Collapsing Star
- 4. Collapse of a Star with Uniform Density and Zero Pressure
- 5. Spherically Symmetric Collapse with Internal Pressure Forces
- 6. The Fate of a Man Who Falls into the Singularity at r = 0
- 7. Realistic Gravitational Collapse—An Overview

- 33. Black Holes
- 1. Why “Black Hole”?
- 2. The Gravitational and Electromagnetic Fields of a Black Hole
- 3. Mass, Angular Momentum, Charge, and Magnetic Moment
- 4. Symmetries and Frame Dragging
- 5. Equations of Motion for Test Particles
- 6. Principal Null Congruences
- 7. Storage and Removal of Energy from Black Holes
- 8. Reversible and Irreversible Transformations

- 34. Global Techniques, Horizons, and Singularity Theorems
- 1. Global Techniques Versus Local Techniques
- 2. “Infinity” in Asymptotically Flat Spacetime
- 3. Causality and Horizons
- 4. Global Structure of Horizons
- 5. Proof of Second Law of Black-Hole Dynamics
- 6. Singularity Theorems and the “Issue of the Final State”

- Part VIII GRAVITATIONAL WAVES
- 35. Propagation of Gravitational Waves
- 1. Viewpoints
- 2. Review of “Linearized Theory” in Vacuum
- 3. Plane-Wave Solutions in Linearized Theory
- 4. The Transverse Traceless (TT) Gauge
- 5. Geodesic Deviation in a Linearized Gravitational Wave
- 6. Polarization of a Plane Wave
- 7. The Stress-Energy Carried by a Gravitational Wave
- 8. Gravitational Waves in the Full Theory of General Relativity
- 9. An Exact Plane-Wave Solution
- 10. Physical Properties of the Exact Plane Wave
- 11. Comparison of an Exact Electromagnetic Plane Wave with the Gravitational Plane Wave
- 12. A New Viewpoint on the Exact Plane Wave
- 13. The Shortwave Approximation
- 14. Effect of Background Curvature on Wave Propagation
- 15. Stress-Energy Tensor for Gravitational Waves

- 36. Generation of Gravitational Waves
- 1. The Quadrupole Nature of Gravitational Waves
- 2. Power Radiated in Terms of Internal Power Flow
- 3. Laboratory Generators of Gravitational Waves
- 4. Astrophysical Sources of Gravitational Waves: General Discussion
- 5. Gravitational Collapse, Black Holes, Supernovae, and Pulsars as Sources
- 6. Binary Stars as Sources
- 7. Formulas for Radiation from Nearly Newtonian Slow-Motion Sources
- 8. Radiation Reaction in Slow-Motion Sources
- 9. Foundations for Derivation of Radiation Formulas
- 10. Evaluation of the Radiation Field in the Slow-Motion Approximation
- 11. Derivation of the Radiation-Reaction Potential

- 37. Detection of Gravitational Waves
- 1. Coordinate Systems and Impinging Waves
- 2. Accelerations in Mechanical Detectors
- 3. Types of Mechanical Detectors
- 4. Vibrating, Mechanical Detectors: Introductory Remarks
- 5. Idealized Wave-Dominated Detector, Excited by Steady Flux of Monochromatic Waves
- 6. Idealized, Wave-Dominated Detector, Excited by Arbitrary Flux of Radiation
- 7. General Wave-Dominated Detector, Excited by Arbitrary Flux of Radiation
- 8. Noisy Detectors
- 9. Nonmechanical Detectors
- 10. Looking Toward the Future

- Part IX. EXPERIMENTAL TESTS OF GENERAL RELATIVITY
- 38. Testing the Foundations of Relativity
- 1. Testing is Easier in the Solar System than in Remote Space
- 2. Theoretical Frameworks for Analyzing Tests of General Relativity
- 3. Tests of the Principle of the Uniqueness of Free Fall: Eotvos-Dicke Experiment
- 4. Tests for the Existence of a Metric Governing Length and Time Measurements
- 5. Tests of Geodesic Motion: Gravitational Redshift Experiments
- 6. Tests of the Equivalence Principle
- 7. Tests for the Existence of Unknown Long-Range Fields

- 39. Other Theories of Gravity and the Post-Newtonian Approximation
- 1. Other Theories
- 2. Metric Theories of Gravity
- 3. Post-Newtonian Limit and PPN Formalism
- 4. PPN Coordinate System
- 5. Description of the Matter in the Solar System
- 6. Nature of the Post-Newtonian Expansion
- 7. Newtonian Approximation
- 8. PPN Metric Coefficients
- 9. Velocity of PPN Coordinates Relative to “Universal Rest Frame”
- 10. PPN Stress-Energy Tensor
- 11. PPN Equations of Motion
- 12. Relation of PPN Coordinates to Surrounding Universe
- 13. Summary of PPN Formalism

- 40. Solar-System Experiments
- 1. Many Experiments Open to Distinguish General Relativity from Proposed Metric Theories of Gravity
- 2. The Use of Light Rays and Radio Waves to Test Gravity
- 3. “Light” Deflection
- 4. Time-Delay in Radar Propagation
- 5. Perihelion Shift and Periodic Perturbations in Geodesic Orbits
- 6. Three-Body Effects in the Lunar Orbit
- 7. The Dragging of Inertial Frames
- 8. Is the Gravitational Constant Constant?
- 9. Do Planets and the Sun Move on Geodesics?
- 10. Summary of Experimental Tests of General Relativity

- Part X. FRONTIERS
- 41. Spinors
- 1. Reflections, Rotations, and the Combination of Rotations
- 2. Infinitesimal Rotations
- 3. Lorentz Transformation via Spinor Algebra
- 4. Thomas Precession via Spinor Algebra
- 5. Spinors
- 6. Correspondence Between Vectors and Spinors
- 7. Spinor Algebra
- 8. Spin Space and Its Basis Spinors
- 9. Spinor Viewed as Flagpole Plus Flag Plus Orientation-Entanglement Relation
- 10. Appearance of the Night Sky: An Application of Spinors
- 11. Spinors as a Powerful Tool in Gravitation Theory

- 42. Regge Calculus
- 1. Why the Regge Calculus?
- 2. Regge Calculus in Brief
- 3. Simplexes and Deficit Angles
- 4. Skeleton Form of Filed Equations
- 5. The Choice of Lattice Structure
- 6. The Choice of Edge Lengths
- 7. Past Applications of Regge Calculus
- 8. The Future of Regge Calculus

- 43. Superspace: Arena for the Dynamics of Geometry
- 1. Space, Superpace, and Spacetime Distinguished
- 2. The Dynamics of Geometry Described in the Language of the Superspace of the (3)y’s
- 3. The Einstein-Hamilton-Jacobi Equation
- 4. Fluctuations in Geometry

- 44. Beyond the End of Time
- 1. Gravitational Collapse as the Greatest Crisis in Physics of All Time
- 2. Assessment of the Theory that Predicts Collapse
- 3. Vacuum Fluctuations: Their Prevalence and Final Dominance
- 4. Not Geometry, but Pregeometry, as the Magic Building Material
- 5. Pregeometry as the Calculus of Propositions
- 6. The Black Box: The Reprocessing of the Universe

- Bibliography and Index of Names
- Subject Index

"*Gravitation* is a classic."—David Spergel, Princeton University

"*Gravitation* is the encyclopedic, definitive exposition of general relativity as of 1970. Any serious researcher of general relativity will want to have this book at hand."—Andrew Hamilton, University of Colorado, Boulder

"*Gravitation* has been a very influential book in the field, and it gives a remarkably thorough treatment of what's relevant in general relativity today."—Frans Pretorius, Princeton University

"This book is considered the Bible for everyone in the field of gravitation."—Alberto Vecchio, University of Birmingham